Applications of space-time adaptive processing

frequency F = —0.156, which is close to the clutter main beam for the spectral folding due to the low PRP value used in this analysis. Moreover, the threshold is set so as to yield a probability of false alarm Pfa = 10~3 in the absence of interfering targets. As a worst case, the interfering target is assumed to share the same DoA and Doppler frequency as that of the desired target. Its presence in the secondary data affects the estimate of the covariance matrix; therefore, the resulting adaptive detection scheme tends to equalise the space-time spectrum, thus putting a notch in the DoA and Doppler frequency of the target. As the INR increases, the notch becomes deeper and the desired target is quickly nulled. Obviously, using less secondary data reduces the probability that the secondary data are affected by interfering targets, therefore we consider both the cases of Q = 32 and Q = 16 secondary data. Notice that, when an interfering target is present in the 16 cells closest to the CUT, the JDL plots with Q = 16 start decreasing by about 5 dB before the corresponding plots with Q = 32 secondary data. This is because the undesired contribution in the sample covariance matrix is averaged with a smaller number of uncontaminated data. It must also be noticed that the JDL plots with Q = 16 secondary data show a lower probability of detection than the corresponding plots with Q = 32 secondary data even for very low INR values. This can be verified from Figure 2.1b, representing the P& as a function of the SNR under the same conditions in the absence of interfering targets. As apparent, the JDL with Q = 16 secondary cells yields about 3 dB of loss with respect to the case of Q = 32. Moreover further reductions of Q would show a drastic fall in the detection performance. Therefore, with the JDL, there is an intrinsic limit in the possibility of reducing the probability of interfering targets by reducing the size of secondary data, unless accepting the presence of significant detection losses even in the absence of interferences. Finally, it is interesting to observe from Figures 2.1a and 2. Ib that, with both 2 = 1 6 and Q = 32 secondary cells, the plots of MSMI and GLRT are very close to one another. Thus, in the following, only the JDL MSMI is considered. The detection performance against a non-homogeneous environment is demonstrated in this chapter with reference to a data set provided by the Naval Research Laboratory (USA), which consists of land, sea and land-sea interface scenarios of various terrain types and sea states. The data have already been processed by using systolic-based adaptive algorithms [6] and various STAP filters [7,8]. These data were recorded by a modified AN/APS-125 radar at UHF on a P-3A aircraft and include flight tests taken in 1979 and 1980 over the Appalachian mountain regions, Maryland, Virginia and midwestern areas of the United States. The receiving system consists of eight identical channels giving / and Q signals at 10 bits within a baseband of 5 MHz. For the purpose of our analysis, a non-adaptive digital beamforming is applied to the spatial channels, which combines, with appropriate phases, K = 3 overlapped uniform subarrays (OUS) of four adjacent channels. The processed file (DL075) refers to littoral clutter; the transition between ground and sea echoes is well apparent in Figure 2.2a, which shows the power received at the first channel and first PRI versus range. The data refer to 18 PRIs, each one containing 896 range cells for each channel. The first 450 samples refer to sea clutter and have a power level of about 1OdB;

power, dB output power, dB

range bin

test statistic, 101ogl0 (T)

range bin

c

Figure 2.2

range bin

Results of JDL against recorded data by AAFTE a Power level versus range for first antenna element and first pulse b JDL MSMI output power versus range for 1 target injected and Q = 32 c JDL MSMI test statistic versus range for 1 target injected and Q = 32

from sample 550 to 896 ground clutter echoes are received, which have a level of about 30 dB; samples in between are characterised by a high non-homogeneity, since they are reflected from the sea coast region which is primarily sandy beaches with small hills and cliffs near the water. To show the clutter cancellation performance of the JDL, a fictitious point-like target is injected at range cell 480 at

of the signal at the output of the normalised filter, Z(x), of the JDL detector (see equation (2.2)), with Mp = Kp = 3 and Q = 32, is portrayed in Figure 2.2b. It is noted that about 25 dB of clutter cancellation has been obtained on the sea area and about 35 dB on the land area and the target appears well above the clutter level. The behaviour in terms of detection is easily verified from the test statistics, T(x) (see equation (2.1)), in Figure 2.2c, which shows that using the JDL an appropriate threshold can easily be selected to detect the target without having false alarms. To show the performance of the JDL in the presence of interfering targets, we proceed as follows. Nine sets of two or three fictitious point-like targets have been added to the data set at the same target DoA 90° and Doppler —213 Hz, at range cells and power levels described in Table 2.1. Sets (A, C, E, F, I) named Group I, have two targets with the same power level; sets (B, D, G, H), named Group II, have targets with different power levels. The proposed STAP schemes have been applied to these data to assess their comparative performance. Notice that almost all targets are well below the clutter level (compare Figures 2.3a and 2.2a). Figures 2.3b,c show the CFAR statistic T(\) of the JDL MSMI versus range, with Q = 32 and Q = 20 secondary data, respectively. A few considerations apply: (i)

The JDL with Q = 32 correctly detects the targets in Group I (italic labels) but looses the smallest targets in Group II (bold labels). When the smallest target of a configuration from Group II is in the CUT, one or two stronger targets affect the estimation of the covariance matrix and the threshold setting. Following the above discussion - and recalling that all targets have been considered at the same DoA and Doppler frequency - this implies that a notch is set in the target space-time location so that the small target is not detected. For targets of the same power level, the same notch operates on the target in the CUT. However, due to its original power level, this still exceeds the threshold despite the notch in its space-time location. (ii) Strong false alarms appear at range cells 507, 577, 603; namely, when setting a threshold to detect most of the targets, a detection is declared also in these range cells, however no targets were injected there. This is independent of the presence of interfering targets. The high values of the test statistic, which are not Table 2.1 Synthetic targets test configuration Set Group Range bin Power (dB) Set Group Range bin Power (dB)

A I 90,93 35,35 F I 540,543 15, 15

B II 180,183,186 0,15,15 G II 630,633,636 15,0,25

C D E I II I 270,273 360,363 450,453 0,0 0,15 15,15 H / II I 720,723 810,813 25,0 0,0

input power, dB test statistic, 101og10 (T)

range bin

I i

test statistic, 10 log 10 (T)

range bin

c

range bin

Figure 2.3

Results of JDL against recorded data by AAFTE for the target test configuration of Table 2.1 a Synthetic targets test configuration b JDL MSMI test statistic versus range for Q = 32 c JDL MSMI test statistic versus range for Q = 20

related to injected targets, come from the non-homogeneous area of transition between sea and land. Such values were also present in Figure 2.2c; however, in the case of sequences of closely spaced targets (with various power levels and partial cancellation of each one due to the presence of the neighbours in the sample estimate of the covariance matrix) they directly yield false alarms. In fact, to detect all targets, we need to set a lower threshold than that in Figure 2.2c which is also exceeded by all these values.

(iii) The JDL with Q = 20 shows similar behaviour to the JDL with Q = 32, but there are significantly more false alarms. This shows that reducing the number of secondary data increases the fluctuation of the test output. The undesired effect from adjacent targets is removed only if the target distance is larger than Q/2 = 1 0 cells; however, it is not removed in the case of sequences of targets more closely spaced (as considered in our test configuration). A possible approach to reduce the effect of the interfering targets in the secondary data consists in devising adaptive filtering techniques that intrinsically need a very limited amount of secondary data, Q, so that the probability of incurring in the performance degradation due to interfering targets is largely reduced. In particular, only targets with distance less or equal to Q/2 show an interfering effect. Obviously, the JDL scheme with Mp = 3, Kp = 3 does not allow us to work with a number of secondary data Q lower than 18, unless accepting large adaptivity losses. Therefore, alternative approaches must be considered to reduce Q further. A two-dimensional FIR filtering scheme is presented in the next section that is able to operate with a very small number of secondary data, without significant adaptivity losses.

2.3

AR-based FIR filters

To derive the AR-based FIR (finite impulse response) filter, the space-time disturbance echoes (clutter-plus-noise) are represented with a two-dimensional autoregressive model: AR(L — 1, A, R) of order L — 1 and matrix parameters A and R: (2.3) d (p) being the ( ^ x I ) echoes vector at the /7th PRI, w(/?) is a sample from a white vectorial complex Gaussian process with zero mean value and (K x K) covariance matrix R, and A = [A^ - 1 , A £ _ 2 , . . . , A*]* the (K(L — 1) x K) matrix of the two-dimensional AR parameters. By assuming that the AR poles are not close to the unit circle, and extending to the two-dimensional domain the approximation in Reference 9, the joint PDF of the data vector in the CUT can be approximated under the hypothesis of target absence (HQ):

(2.4)

To simplify the notation of equation (2.4), we define the (K x KL) matrix H = [—A* 1^], where I^ is a ^-dimensional identity matrix. By rearranging the data into the (KL x (M - L + I)) matrix X:

(2.5) which implicitly defines the vectors x ^ , ra = l , . . . , M — L + l,it yields: (2.6) Similarly, the approximate PDF, /?i(x), under hypothesis H\ (namely assuming also the presence of a target with complex amplitude A and space-time steering vector s), is obtained by replacing in equation (2.6), X with (X — AS), where S = [S^1JL S ^ • • • s^jT L + 1 ) ] is the target matrix obtained by applying the same reordering operation as for X. By assuming known disturbance characteristics, namely known A and R matrices, the generalised likelihood ratio test is easily obtained by maximising the ratio p\(x)/po(x) with respect to the target complex amplitude, A. The maximisation yields the maximum likelihood (ML) value for the target complex amplitude A = fr{X*H*R-1HS}/^r{S*H*R-1HS}, and finally the logarithmic GLRT: (2.7) where XAR is the detection threshold. The denominator of equation (2.7), Tr{S*H*R~1HS} = E{jl}, is the expected value of the average clutter power jx at the output of the filter, estimated over the Q secondary data, thus the filter is intrinsically CFAR. As apparent, the denominator is a constant independent of the data matrix X in the CUT, so that the numerator defines the filter shape. By exploding the trace into the sum of its components, and using the identity s ^ = e^2nF^n~V)sA)F it is rewritten as:

(2.8)

where the adaptive weight vector w = H*R 1 Hs^] 7 is implicitly defined. Equation (2.8) shows that the GLR test is given by the application of the twodimensional adaptive filter w to the data with L ^-dimensional support, followed by a bank of Doppler filters (each one centred on a different target speed). It is important to observe that only a very limited number of coefficients (L taps for each of the K spatial channels) is required to implement this filter. Further - unlike the JDL - only the target steering vector depends on the Doppler frequency, therefore the adaptive portion (H*R~ 1 H) of the filter weight vector is independent of the considered Doppler channel and can be evaluated and applied to the data only once. This is a interesting advantage of the AR-based FIR filter, that can be classified as a pre-Doppler technique, where the optimisation is independent of the subsequent Doppler filter bank. These characteristics yield a low computational cost. Moreover, the space-time FIR filter is especially appealing since it was shown to be coincident with the optimum adaptive processor (OAP) in References 10 and 11, under specific assumptions on the clutter characteristics. In the more general case under analysis, the improvement factor (IF) obtained with this AR-based two-dimensional FIR filter is compared in Figure 2.4 with both the OAP and the JDL detector, for the case of a radar operating with K = 3 spatial channels, M = 64 PRIs, a Gaussian-shaped temporal clutter correlation, CNR = 16 dB and beam pointing at
Figure 2.4

Comparison of IF curves for K = 3, M = 64,

for AR filters with L = 3 and L = 6 taps while the JDL operates with Kp — 3 and Mp = 3. As apparent, the considered clutter model does not follow exactly the two-dimensional AR model; however, the two-dimensional FIR filter shows only a limited performance loss close to the notch with respect to both OAP and JDL, despite the large reduction of the computational load. Moreover, as the number of taps, L, increases from L = 3 to L = 6, the performance of the AR-based FIR filter close to the clutter notch also gets closer to the JDL, which shows a trade off between computational cost and performance. Further increases of the number of taps L would not yield better performance close to the clutter notch. A different behaviour is present far from the clutter notch, where filters with more taps show larger losses. This performance loss far away from the clutter notch is due to the fact that a FIR filter with L taps uses a shortened coherent CPI of (M — L + 1) samples, because the filter does not slide over the edges of the data record. It is to be noticed that the use of FIR filters deriving from a two-dimensional AR model of the clutter returns was first introduced by Klemm and Ender in References 12 and 13. In contrast to our GLRT approach, based on the approximate PDF in equations (2.4)-(2.6), they applied the minimum mean square error criterion at the filter output, obtaining slightly different expressions for the filter parameters. The performance of the two filters is largely comparable, thus similar considerations as above apply. For the practical application of the AR detection scheme, the clutter parameters A and R cannot be considered known, and must be estimated from the data themselves. To this purpose, we use the independent echo vectors received from Q adjacent range cells that are considered homogeneous and target free. As for the primary data, the KM echo vector y^ in each secondary range cell is rearranged into the matrix Y^, for k = 1 , . . . , <2; then these matrices are collected into the global (KL x (M — L + 1) Q) matrix of the secondary data Y = [Yi Y2 • • • Yg]-As shown in the Appendix (Section 2.8), the ML estimates of the matrix parameters A and R are obtained from the sample covariance matrix My = YY*. For convenience, we decompose the My in blocks of size K(L — 1) and L, respectively: (2.9) Therefore, we have the ML estimates:

(2.10)

where [M F 1]L,L is the last KxK block in the main diagonal of the inverse of My. These values are used inside the practical detection scheme to replace the known values. Further consideration is in order for the practical application of the AR-based two-dimensional FIR filter. The denominator in equation (2.7) is coincident with the normalising CFAR term only if the data follow exactly the two-dimensional AR model. For the general case, where real or simulated data with unknown spectral

shape are fed into the detector, the denominator must be replaced by the estimated clutter power /i(Y):

(2.11) The performance of the AR-based FIR filter with L = 3 taps is compared with the JDL with Kp — 3 and Mp = 3 in Figure 2.1b (solid lines). A few comments are in order: (i) the FIR filter with Q = 32 cells has the best performance (ii) the FIR filter with 2 = 8 cells shows a loss of 2 dB with respect to the FIR filter with Q = 32 and less than 1 dB with respect to the JDL with Q = 32, but it largely outperforms the JDL with 2 = 16 (iii) the FIR filter with Q = A cells is comparable to the JDL with 2 = 16. Comparatively, the JDL scheme suffers high adaptivity losses that largely compensate the little advantage shown in the IF. The case of L = 6 could achieve better performance, but it also requires slightly more secondary data (larger Q) corresponding to the larger number of filter parameters to estimate. An important point to be noticed is that the AR-based FIR filters take great advantage from a much more accurate estimate of the clutter covariance matrix. This is obtained by using the sample matrix My, which contains the average made inside the CPI (other than on the range cells), and follows from the exploitation of the AR properties. The possibility of operating with 2 = 8, instead of Q = 32, secondary data largely reduces the probability of having interfering targets in the secondary data. Moreover, clutter edges have very little influence on the detector performance. On the other hand, when interference is present, its effect is more devastating than with Q = 32, since it is not largely averaged with uncontaminated data. This is apparent from Figure 2.1a, where we consider the unfavourable case of an interfering target present in the secondary data of the AR with Q = 32, 2 — 8 a n d 2 = 4. As expected, the reduction of Q implies a faster decrease in the detection capability. This is the price to be paid to reduce the probability of having interfering targets in the secondary data. Figure 2.5 shows the application of the adaptive FIR filters to the AAFTE data under the same condition as Figure 2.2. Figure 2.5a shows the output power versus range, when operating with L = 3, 2 = 8 for the same case as for Figure 2.2b; a comparable (slightly better) clutter cancellation is obtained with the AR-based FIR filter. Figure 2.5b depicts the corresponding test statistic that clearly shows the effective detection of the target. Figures 2.6a-c show the detection statistic for the target test configuration of Table 2.1, respectively with Q = 8, Q = 20 and Q = 32 secondary data.

output power, dB test statistic, 10 log , 0 (T)

range bin

b

Figure 2.5

range bin

Results of the AR-based two-dimensional FIR filter against recorded data by AAFTE for 1 target injected and Q = 8 a Output power versus range b Test statistic versus range

The following considerations are in order: (i)

The AR-based FIR filter with Q = 8 is slightly worse than the JDL with Q = 20 (Figure 2.3c). (ii) The AR-based FIR filter with Q = 20 detects all targets of Group I except for set A, while it fails to detect the smaller targets in Group II, but has less false alarms than does the JDL with Q = 20. (iii) The AR-based FIR filter with Q =32 is comparable to the JDL with Q =32. Notice that it also loses set A, which has higher power than the clutter and could be detected before cancellation. However, the number of false alarms is largely reduced with respect to the JDL (namely, setting a threshold to detect the same targets, it yields a lower number of false alarms). Therefore, the use of two-dimensional FIR filters with short temporal support and limited adaptivity losses does not solve all of the problems. A number of techniques have been investigated in the literature to cope with the problems deriving from the presence of interfering targets, by properly selecting or discarding the range cells of secondary data (see for example References 14 and 15). However, these techniques appear to have a very high computational cost. Alternative techniques are investigated below, based on a non-linear combination of filters.

test statistic, 101og10(T) test statistic, 101og10 (T)

range bin

test statistic, 101og10 (T)

range bin

c

range bin

Figure 2.6

2.4

Test statistic versus range for the AR-based two-dimensional FIR filter against recorded data by AAFTE for the targets test configuration of Table 2 J a Q=S b Q = 20 c Q = 32

Non-linear combination of non-adaptive filters

When considering the expected characteristics of the space-time cancellation filter, the following considerations are in order. Whichever the radar and clutter characteristics are, the clutter spectrum mainly appears as a ridge with known shape in the Doppler frequency-angle plane that is modulated by the transmitting antenna

pattern. The slope of the F-coscp clutter trajectory (namely the footprint in the F-cos

a bank of P fixed filters is designed, with notches of different widths, and normalised so as to give the same output when fed with the expected target (ii) the average power at the filters' output is evaluated on Q range cells (iii) the filter with minimum output power is selected; notice that, due to the normalisation, at point (i), the filter with the widest notch does not necessarily yield the minimum output power (iv) the output of the selected filter corresponding to the CUT is scaled by the corresponding normalising CFAR term and compared with the detection threshold to test for the target presence. The scheme of this filter is depicted in Figure 2.7 for a bank of P = 3 filters: filter A, filter B and filter C. This non-linear combination of non-adaptive filters has apparently a very limited level of flexibility, consisting in the automatic selection of the filter with minimum output power for each range cell [17]. However, if the filters are carefully designed, this detection scheme can yield very convincing results. Its computational cost is P times higher than for a fixed weights linear filter, but much less than for the OAP filter, which requires the real-time inversion of the KM x KM estimated clutter covariance matrix.

2.4.1 Filter bank design To design the filter bank, it is useful to recall that the clutter covariance matrix Q is a Toeplitz block matrix with blocks related to the spatial channels. We assume a spatially homogeneous scene and consider a ratio a = d/(2vT) between spatial and temporal sampling intervals, being d the antenna spacing and v the platform velocity [1,18]; a is also referred to as the space-time slope. In general, for side-looking configurations, Q is obtained from the usual space-time correlation coefficient expression [19]: (2.12)

where pt(At) is the temporal correlation coefficient, which depends on the ICM and is a function of the temporal displacement At = mT of the two clutter echoes. ps[At + As/(2v)] is the spatial correlation coefficient, which is independent of the ICM and encodes the decorrelation due to the combined effect of the radar platform motion (and thus of the different view angle over the scene) and the antenna pattern. Since this term depends on the spatial position of the receiver, it is a function of both the temporal displacement At (which implies a different position of the platform) and the spatial displacement of the antennas in the array As. Let us assume that: (i) the nominal clutter returns describe a diagonal of the Doppler-angle plane with slope a, (ii) the target DoA is cp, and (iii) the expected normalised target Doppler frequency is F; then a suitable two-dimensional filter has the main beam at (
clutter filter null COS(T)

Figure 2.7

Sketch of the non-linear space-time processor a Filter A b Filter B c Filter C d Detector scheme

being 8(x) the Kronecker delta function, and appropriate pt(At) correlation function. In particular, a Gaussian filter bank is obtained by considering a Gaussian shaped Pt(At) with one-lag correlation coefficient pt. To this purpose, we select P = 3 values: pt\ = 0.9, ptB = 0.99 and ptQ = 0.999, and we build, respectively, the covariance matrix QA, of filter A, QB of filter B and Qc of filter C, corresponding to clutter echoes with different intensity of internal motion. The optimum processor is obtained by a linear combination of the KM echoes with weights w = Q - 1 S. Therefore, a Gaussian filter bank is easily obtained by considering P = 3 optimum filters corresponding to the three covariance matrices above. To properly select the best filter in the bank, each filter is normalised so that it produces unity output when fed with the target steering vector, s. As mentioned above, this normalisation guarantees that the non-linear scheme does not always select the filter with the widest notch, thus yielding a performance degradation of the very slow targets. The instantaneous power for the CUT at the output of filter A is:

(2.13) The CA-CFAR normalising term for this filter, evaluated over Q adjacent range bins, is an estimate of the clutter power at its output and is given by: (2.14) where the explicit dependence on x refers to the selection of the samples y&, adjacent to the CUT with echo vector x. Similar expressions are valid for filters B and C. The non-linear detector selects the filter with the minimum estimated clutter output power on the Q adjacent range cells:

(2.15) Finally, the non-linear scheme compares it to the CA-CFAR detection threshold corresponding to the selected filter. In analogy to the case of the GLRT, it can be written in terms of test statistic:

(2.16)

which is to be compared to the threshold G, to test for detection. The set of three twodimensional filters is repeated for the different values of the expected target Doppler frequency.

2.4.2 Detection threshold and performance A closed-form expression of the false alarm probability (as well as of the P& for Swerling I targets) has been obtained for P = 2 filters [7]. Define the clutter power at the output of the two filters as o\ = E{Z2A},G^ = £{Z|}, the output cross correlation coefficient, p = \E{ZAZ^}\/(GA<JB), and the target power, erf. When selecting the filter based on the Q adjacent range cells, Pd = PA + PB, where: (2.17) with GA 0

=

GB

V W B + 1 + (1 -

=

G, aA 2

P )MA1

2

= 2

G2JaI - [1 + (1 - P 2 )MAL £A

=

2

- 4 p a i / ^ , and /xA = G A /(1 + crf/a A). PB is

obtained from PA by exchanging suffix A with B. The Pfa is easily obtained by setting at = 0. Even in the presence of P > 2 filters, assuming that the clutter background is approximately homogeneous, the non-linear filter has to choose essentially between the two filters which are closest to the ideal optimum, while the others give a negligible contribution to the detection performance. When evaluating the performance, the theoretical values of the filters are known, so that we are able to choose the two correct filters and apply the theoretical formula. Also, the detection threshold can be easily set using equation (2.17). Figure 2.8a shows the power at the output of the non-linear scheme, Z(x) (see equations (2.13) and (2.15)), when operating with Q = 20 against the AAFTE data file with an injected target as in Figure 2.2b. It is now apparent that the output power has a low fluctuation level and performs at least as well as the JDL with Q = 32, even if it only operates selecting the most appropriate filter out of a set of three. Moreover, the target is now clearly visible. Obviously better cancellations could be achieved by using full adaptivity and larger regions of secondary data, at the expense of higher computational cost. However, when dealing with small regions of data we show that the behaviour of the non-linear non-adaptive detection scheme is comparable to the best results obtained by the reduced DoF techniques. The behaviour in terms of detection is easily verified from the test statistics, 7\x) (see equation (2.16)), in Figure 2.8b, which shows that the non-linear processor properly detects the target. The filter selection made by the non-linear non-adaptive detector is portrayed in Figure 2.8c. The narrowest filter (C) is selected only for a fraction of returns along range and mainly inside the land area. In contrast, in the sea area, there are large regions where the widest filters (A) and (B) are selected (bins 50 to 180 and 290 to 390). Moreover, in the highly non-homogeneous areas, the selection largely oscillates among all three filters; this certainly applies to the sea to ground transition (bins about 450 to 480) and to the land region in cells 810 to 890, containing probably a river bank.

output power, dB test statistic, 101og10(T)

range bin

filter selection

range bin

c

Figure 2.8

range bin

Results of the non-linear non-adaptive Gaussian-based filter versus range against recorded data by AAFTE for 1 target injected and Q = 20 a Output power b Test statistic c Non-linear filter selection

2.4.3 AR-based non-linear detector Although the use of filters based on Gaussian-shaped autocorrelation functions is a simple and effective way to obtain a set of similar filters with notches of different width, two considerations are in order: (i) the Gaussian filters are generally not an optimal choice, since the Gaussian model is arbitrary, and (ii) the filters have the

full implementation cost of MK complex products, since they have MK non-zero coefficients. A significant reduction of the implementation cost can be achieved by replacing them with FIR filters with a small temporal support. A non-adaptive version of the two-dimensional AR-based FIR filters presented in Section 2.4 can be used to this purpose. To derive the filter coefficients for this case, we proceed as follows. From the theory of the stationary two-dimensional-AR processes, it is known that the relationship between AR parameters, A and R, and the covariance matrix Q can be written as H • QAF = [0K • • • 0K R], where QAF = £ { x ^ x ^ T } for any value of m and is coincident with the generic rath (KL x KL) block on the main diagonal of Q. Therefore, H + R" 1 = Q ^ [ O * • • • O^ I*]* so that H* R" 1 is the last (KL x K) block column of Q ^ , and R" 1 is the last (K x K) block in the main diagonal of Q^f. An AR-based FIR filter bank can be obtained by using the relationships above to approximate the two-dimensional Gaussian filters with twodimensional AR-based FIR filters with L taps. Figures 2.9a,b show, respectively, the output power and the test statistic for the same case as Figure 2.8a,b. As apparent, with L = 3 taps the clutter cancellation is very similar to the case of the Gaussian filters and the target is easily extracted by the filter. The corresponding filter selection is reported in Figure 2.9c. In this case, the selection shows fewer fluctuations than in the case of the Gaussian filters and the wider filter A is selected most of the time. The filter selection is still largely fluctuating in most non-stationary clutter areas. The performance of the non-linear non-adaptive detection scheme based on the bank of AR FIR filters against the target test configuration of Table 2.1 is portrayed in Figure 2.10, to investigate the robustness of this detection scheme to the presence of interfering targets in the secondary data. Notice that the secondary data are only used here to select the best filter in the bank and to provide the CFAR normalisation. Figure 2.10a shows the test statistic as a function of the range cell. As apparent, the targets of Group I are properly detected, but the smallest targets of Group II are not detected by the filter. Notice also that set I is not clearly extracted; this is probably due to its closeness to a highly homogeneous area of ground clutter (probably containing the river banks). The filter selection is expected not to be affected by the presence of the interfering targets in the secondary data, since all filters in the bank are normalised so as to yield unity gain for the desired target. Therefore, the degraded performance of the technique must be mainly attributed to the interfering targets affecting the CA-CFAR normalisation. To counteract this effect the following scheme is proposed: (i) (ii) (iii)

(iv)

select the best filter as before (the filter yielding the minimum output power, as estimated on all the secondary data) split the secondary data into P blocks of Q cells evaluate the P normalised filter outputs, obtained by dividing the filter output from the CUT by the mean filter output on each block of Q range cells (this is the CA-CFAR output obtained by considering only the secondary cells of the individual block) take the median of the P normalised filter outputs.

t a

I o

range bin

P O

i O

range bin

I 1 c

Figure 2.9

range bin

Results of the non-linear non-adaptive AR-based filter versus range against recorded data by AAFTE for 1 target injected and Q = 20 a Output power b Test statistic c Non-linear filter selection

The median operator is selected in order to reject the anomalous outputs that are corrupted by the presence of the interfering target. This is obviously based on the assumption that only a small number of blocks is expected to be affected by the interfering targets. The resulting test statistic is shown in Figure 2.10b for P = 5 blocks with Q = 4 cells, for a total of 20 secondary data. As apparent, almost all targets are clearly extracted and a threshold could be selected to detect all of them without causing false alarms. This is not quite true for target set I, due to the above mentioned

test statistic, 101og10(T) test statistic, 10 log 10 (T)

range bm

block selection

range bin

c

Figure 2.10

range bin

Results of the non-linear non-adaptive AR-based filter versus range against recorded data by AAFTE for the targets test configuration of Table 2.1 and Q = 20 a Test statistic b Test statistic with block selection c Block selection around target set D

non-homogeneity. Notice also that the strong effect of the target in the CFAR scaling factor, yielding very low returns around the target positions in Figure 2.10a, is not present in Figure 2.10b. To demonstrate the effectiveness of this median selection for the CFAR, Figure 2.10c shows the selected CFAR block versus range cells, around the targets of set D. The small circles represent the blocks affected by interfering targets,

and the solid line represents the block selected by the median operator. As apparent, the blocks affected by interfering targets are never selected. As a final step in the non-adaptive non-linear detector, we notice that the selection of the best filter of the bank can be done without using the secondary data. The non-linear selection strategy can operate directly by choosing the filter yielding the minimum output power on the CUT, after normalisation as in equation (2.13), namely: Z(x) = min{ZA(x), Z B (x), Z c (x)}

(2.18)

Test statistic, 10 log.0 (T)

The same CA-CFAR scheme above is then applied, consisting of the normalisation by the median of the average output powers on the P blocks of Q range cells. The resulting test statistic is reported in Figure 2.11a, which is largely comparable with Figure 2.10b. Similarly Figure 2.11b, shows the correct selection of a block of secondary data not affected by interfering targets. As apparent, this filter yields a further reduction in computational cost, since only one of the three filters must be applied to all range cells. Moreover, it yields a very effective detection scheme that is robust to the presence of interfering targets. However, target set I is still not detected which is due to the clutter non-homogeneity.

block selection

range bin

b

range bin

Figure 2.11 Results of the non-linear non-adaptive AR-basedfilter with block selection and test on the CUT versus range against recorded data by AAFTE for the targets test configuration of Table 2.1 and Q — 20 a Test statistic b Block selection around target set D

2.5

Non-linear combination of adaptive AR-based two-dimensional FIR filters

From the analysis of the previous sections, we observe that both AR-based adaptive FIR filters and non-linear combinations of non-adaptive filters are able to work with a limited amount of secondary data, moreover: (i) The non-linear bank of non-adaptive filters performs very well against the nonhomogeneous background and in the presence of interfering targets, but it largely depends on the appropriate selection of the shape of the filters in the bank. The filters should be changed in accordance to the platform flight conditions (pitch, bank, roll, crab, wind speed, ground speed), and the radar parameters (antenna pointing, PRF). Moreover, it might happen that none of the used filters is really appropriate to clutter cancellation, with the consequent performance degradation. (ii) The AR-based adaptive FIR filters are able to adapt to any kind of mismatch, such as the variations in platform flight conditions and radar parameters. However, their performance is largely degraded by the presence of interfering targets in the secondary data. To obtain a robust detection scheme that at the same time copes with the changing radar and platform conditions and is not largely affected by the presence of interfering targets in the adjacent range cells, non-linear adaptive schemes (in contrast to the nonlinear non-adaptive schemes of Section 2.4) are presented. In these adaptive schemes, a set of P AR-based FIR filters is obtained by estimating the covariance matrix on P different blocks of Q range cells around the CUT and estimating the corresponding matrices A and R to be used inside equation (2.8). Two non-linear adaptive schemes are obtained by applying a non-linear selection among the outputs of P linear adaptive filters, according to two possible criteria [20]: Minimum residual power (MRP) Each filter is normalised so as to give the same output to the received target, as for the non-linear combination of non-adaptive filter of Section 2.4. Thus, for filter A:

(2.19) where HA and RA are estimated from the first block of Q secondary range cells; similarly ZB(X), ZC(X), . . . are obtained from the second, third, . . . block of Q secondary data. Then, each of the P filters is applied to the CUT and the non-linear scheme selects the filter that yields the minimum output power. Finally, the selected

filter is normalised by the CFAR scaling factor computed on the range cells of the corresponding block; namely, for filter A:

(2.20)

before comparing to the detection threshold. In the presence of interfering targets in one or more blocks of secondary data, the filters obtained from the estimates on the contaminated blocks yield a worse clutter cancellation than the filters based on the non-contaminated blocks. Therefore, the latter filters are selected, which makes the non-linear adaptive detector robust to the presence of interfering targets. Median test output (MTO) The output of the CFAR test statistic 7} = Z///x r , i = A, B, . . . is evaluated for each of the P filters (see Figure 2.12 for P = 5 blocks yielding filters A to E). The median value of the P outputs on the CUT is selected and compared to the detection threshold. Assuming that interferences don't affect more than (P — l)/2 blocks of secondary data, namely (P — l)/2 filter outputs, the median output should correspond to a non-contaminated AR filter. The performance of these non-linear adaptive detectors that use the AR-based FIR filter are reported in Figures 2.13a and 2.13b, respectively, for Q = 4 and Q = 8 cells in each of the P blocks. In both Figures the P& of the linear AR-based FIR filter with global number of cells 2 = 4, 8,32 is reported for comparison. We notice that the MRP detector operating with blocks of Q cells always provides a P$

median

CUT

Figure 2.12

Scheme of the MTO detector with P = 5 blocks of Q secondary cells

ARQ = 32 ARQ = 8 ARQ = 4 MRPP = 2Q = 4 MRPP = 4Q = 4 MTOP = 3Q = 4 MT0P = 5O = 4

SNR, dB ARQ = 32 ARQ = 8 ARQ = 4 MRPP = 2Q = 8 MRPP = 4Q = 8 MTOP = 3Q = 8 MTOP = 5Q = 8

b

Figure 2.13

SNR, dB

Comparison of non-linear adaptive schemes to the linear AR-based filter with Q = 32,8,4. Pd versus SNR for K = 3, M = 64, target at (p = 45° and F = -0.156 a MRP and MTO with blocks of Q = 4 b MRP and MTO with blocks of Q = 8

comparable with the linear FIR filter with the same total number of cells Q; this is almost independent of the value of P. In contrast, the MTO detector operating with P blocks of Q cells, is approximately comparable with the linear AR-based FIR filter operating with P • Q cells. Figures 2.14a and 2.14b show, respectively, Pfa and P& for the case of one interfering target in the secondary data, as a function of the INR, using blocks with Q = 4

JDL MSMI Q = 32 ARQ = 32 ARQ = 4 MRPP=2Q=4 MRPP=4Q=4 MTOP = 3 Q = 4 MTOP = 5 Q = 4

MRPP=2Q=4 MRPP = 4 Q = 4 MTOP=3Q = 4 MTOP = 5 Q = 4

INR, dB

INR, dB

MRP P=2 Q=4 MRP P=4 Q=4 MTO P=3 Q=4 MTO P=5 Q=4

MRP P=2 Q=4 MRP P=4 Q=4 MTO P=3 Q=4 MTO P=5 Q=4

probability of correct selection

probability of correct selection

JDL MSMI Q=3; AR Q=32 ARQ=4 MRP P=2 Q=4 MRP P=4 Q=4 MTO P=3 Q=4 MTO P=5 Q=4

MRP P=2 Q=4 MRP P=4 Q=4 MTO P=3 Q=4 MTO P=5 Q=4

INR, dB

Figure 2.14

Comparative performance of linear and non-linear adaptive schemes using AR-based FIR in the presence of interfering targets a Pfa versus JNR for one interference b Pd versus JNR for one interference c Probability of correct filter selection for one interference d Pfa versus JNR for two interferences e Fd versus JNR for two interferences / Probability of correct filter selection for two interferences

data. Notice that the Pfa of the standard JDL, as well as of the linear AR-based FIR filter, shows a fast decrease, which also implies a corresponding loss of the detection performance (as observed from Figure 2.1a). As apparent, all non-linear adaptive techniques show a Pfa fairly constant and independent of the interference power.

The Pd of the MTO detector both with P = 3 and P = 5 clearly outperforms the MRP detector with both P = 2 and P = 4. Moreover, the MTO detector is almost unaffected by the presence of the interference; MRP shows performance degradation when the INR is not large enough to allow a clear identification of the corrupted block of secondary data. As expected, a larger P yields better performance for both schemes, which is paid in terms of an increase of computational cost. Figure 2.14c shows the probability of correct filter selection, namely the probability of selecting a filter with coefficients computed from a block not affected by the interfering target. As apparent, the MTO non-linear adaptive detector always selects uncorrupted blocks, whereas the MRP detector is unable to perform a correct selection before the ESFR is high enough; this causes its performance degradation. Figures 14d-f show similar plots for the case of two interfering targets, displaced so as to affect two different blocks of secondary data. As apparent, only the filters with P > 4 are able to maintain a false alarm rate approximately constant (Figure 2.14d); moreover, only the MTO detector with P = 5 yields a high detection performance (Figure 2.14e). This is also confirmed by the probability of correct filter selection in Figure 2.14f. Notice that with this new non-linear adaptive filter, globally P • Q secondary data are used, so that the probability of including interfering targets is P times larger than when using a single block of Q secondary data. However, in the presence of closely spaced target sequences these interferences cannot be avoided and the presented non-linear adaptive filters are effective despite their presence. This can be easily verified from Figures 2.15 and 2.16 which show the results of the nonlinear adaptive AR schemes against the real data set, with the test configuration of Table 2.1. The non-linear adaptive MRP detector with P = 2 blocks of Q = 16 cells and the non-linear adaptive MRP detector with P-A blocks of Q = 8 are shown in Figures 2.15a and 2.15b, respectively. They yield a very similar detection capability except for set G, where all targets are extracted when operating with P = 4, and the central target is lost when operating with P = 2. In fact, in the latter case both adjacent blocks of secondary data are affected by the strong interfering targets, while the central target of set G is small. Figure 2.16a shows the test statistic of the MTO detector with P = 3 and Q = 10. As apparent, all targets are extracted; when setting a threshold to detect all of them, only a single false detection cannot be avoided. Figure 2.16b shows the result obtained using the MTO detector with P = 5 blocks of Q = 4 cells. As apparent, the targets are extracted better from the clutter and they can be very easily detected by setting a threshold, without suffering any false alarm. Finally, Figure 2.16c shows that increasing the size of the individual blocks does not yield any further performance improvement. Figures 2.17a and 17b show the filter selection around target set D, corresponding to the results of Figures 2.15b and 2.16b, respectively. Notice that in this case there are some errors in the filter selection; however, these errors never appear in a cell containing a real target to be detected (cells 360 and 363). Moreover, the test statistic always has a low value in correspondence of such errors, implying that the filter operates correctly and no targets are detected in those range cells. This justifies the good performance of the non-linear adaptive detectors. It is also interesting to observe that both non-linear adaptive MRP and MTO detectors are

test statistic, 10 log 10 (T) test statistic, 101Og10 (T)

range bin

range bin

Figure 2.15

MRP test statistic versus range against recorded data by AAFTE for the targets test configuration of Table 2.1 a P = 2 and Q = 16 b P = 4 and 0 = 8

also able to correctly extract target set I, which was lost by the non-linear non-adaptive schemes. This can easily be explained recalling that this target set is close to a largely non-homogeneous area, containing a river and its banks. Under such conditions, the bank of adaptive filters (MTO with P = 5 blocks of Q = 4 cells) clearly outperforms the bank of non-adaptive filters (AR-based non-linear with median CFAR on P = 5 blocks of Q = 4 cells).

2.6

Conclusions

In the practical application of adaptive STAP filters against real data, one of the main problems is represented by non-homogeneity of the clutter background and the possible presence of interfering targets among the secondary data used to derive the filter parameters. The latter problem was recognised to be critical for the possibility of properly extracting sequences of closely spaced target echoes. Since the probability that interfering targets affect the secondary data is directly proportional to the number of secondary range cells, a first way to mitigate the problem can be to use STAP techniques that are effective even when operating with a very limited amount of secondary data. A two-dimensional FIR filtering scheme, based on the AR model for the disturbance, was considered to this purpose. The filter was shown to perform

test statistic, 101og10(T) test statistic, 101og10(T)

range bin

test statistic, 10 log 10 (T)

range bin

c

Figure 2.16

range bin

MTO test statistic versus range against recorded data by AAFTE for the targets test configuration of Table 2.1 a P = 3 and Q = 10 b P = 5 and Q = 4 c P = 5 and Q = 6

effectively against the real environment. Specifically for our case even with Q = S secondary data such a filter was shown to yield a detection performance very close to the JDL with Q = 32 secondary data. Moreover, the AR-based FIR filter is a pre-Doppler technique, where the optimisation is independent of the subsequent Doppler filter bank. This also requires a largely reduced computational cost, which is especially important for real-time applications. Unfortunately, when using only few secondary data, the possible presence of an interfering target among them produces a much more devastating effect. This is because the undesired contribution in the sample covariance matrix is averaged with

filter selection filter selection

range bin

b

Figure 2.17

range bin

Filter selection versus range against recorded data by AAFTE for targets test configuration of Table 2.1 (zoom around target set D) a MRPP = 4 a n d £ = 8 b MTO for P = 5 and Q = 4

a smaller number of uncontaminated data. Therefore, the use of two-dimensional FIR filters with short temporal support and limited adaptivity losses does not solve all of the problems, and alternative techniques were considered. In particular, a nonlinear non-adaptive scheme was considered, which is based on the selection of the most appropriate non-adaptive filter out of a bank of filters with different cancellation characteristics. The scheme was shown to yield a good clutter cancellation against the real data and to select the most appropriate filter even in the presence of interfering targets in the secondary data. However, the subsequent CA-CFAR normalisation is still largely affected by the presence of the interference. To solve the problem, a new non-linear non-adaptive filter was introduced that first selects the most appropriate non-adaptive two-dimensional cancellation filter and then selects a block of secondary data that is likely to be uncontaminated. This is obtained by splitting the secondary data in P blocks of range cells and applying the median operator to their mean power, based on the assumption that only a few range cells are expected to be contaminated by other targets. Moreover, to reduce the computational cost of the non-linear filter, the Gaussian filters are replaced with AR-based non-adaptive FIR filters and the filter selection is performed using only the output power corresponding to the CUT. This scheme is shown to be effective against the real data even when a few sets of closely spaced targets are injected. The only weakness of this filter consists in the limited capability of such a scheme to adapt to different and changing clutter conditions as well as to mismatches between assumed and real platform motion parameters and to potential calibration errors in the antenna array.

Finally, two new non-linear adaptive STAP schemes have been presented for the target detection against a non-homogeneous environment. These filters are obtained by selecting the most appropriate filter among a set of two-dimensional AR-based adaptive FIR filters, obtained by estimating the AR parameters on P different blocks of secondary data. Their effectiveness against the presence of interfering targets has been shown both by means of simulated analysis and by application to a real data set. The adaptive nature of the individual filters ensures that the non-linear adaptive scheme is able to cope with any mismatch in system and environment conditions. Moreover, the low computational cost of the proposed schemes, together with their robustness to non-homogeneity and interference, makes them appealing for the practical real-time application. 2.7

Acknowledgments

The authors gratefully thank Dr Alfonso Farina (AMS) for many fruitful discussions on these topics and for his encouragement in this work. They also acknowledge the possibility to use the AAFTE data set collected during the work of Dr Fred Staudaher and Dr Fred Lee and their group at the Radar Division of NLR. Finally, they acknowledge the collaboration with FIAR Spa that originated some of the theoretical and simulated results presented in Sections 2.3 and 2.5. 2.8

Appendix: ML estimation of two-dimensional AR parameters

Using the matrix Y, the joint PDF of the secondary data is simply written as:

(2.21) By defining the matrix L so that R it yields:

x

= LL* and M^y (2.22)

where B =

(2.23)

For convenience, we decompose the matrix MLY in blocks of size K(L — 1) and K, respectively:

(2.24)

Using equation (2.24), the value of B that maximises the exponent of equation (2.22), under the constraint of having the last ^-dimensional block equal to the identity matrix, is:

(2.25)

Therefore, we have the ML estimate A = M^00MyOi • Using this value, we have for the trace: tr{K*KMLY} = ?r{-M* y o l M-^ 0 0 M L y 0 i + M L n i } = f r f - L L ^ M ^ 1 ] ^ } (2.26) where [My ]^,L is the last KxK block in the main diagonal of the inverse of My. Therefore, the maximum of the PDF with respect to A can be rewritten as: max{/?0(Y)} =

(TT-K\R\)^M-L+1)

exp[-;r{R~ 1 PV^ 1JZ i l l

A

By maximising equation (2.27) with respect to R, 1/(2(M-L +I))[M"1]^.

(2.27)

'

it yields R

=

References 1 KLEMM, R.: 'Principles of space-time adaptive processing-principles and applications' (IEE, London, 2002, 2nd edn.) 2 KELLY, E. J.: 'An adaptive detection algorithm', IEEE Trans. Aerosp. Electron. Syst., 1986, AES-22, (1), pp. 115-127 3 BRENNAN, L. E., MALLET, J. D., and REED, I. S.: 'Adaptive arrays in airborne MTI radar' IEEE Trans. Antennas and Propagation, 1976, AP-24, (5), pp. 607-615 4 WANG, H. and CAI, L.: 'On adaptive spatial-temporal processing for airborne surveillance radar systems', IEEE Trans. Aerosp. Electron. Syst., 1994, AES-30, (3), pp. 660-670 5 LOMBARDO, P., GRECO, M. V., GINI, R, FARINA, A., and BILLINGSLEY, J. B.: 'Impact clutter spectra on radar performance prediction', IEEE Trans. Aerosp. Electron. Syst., 2001, AES-37, (3), pp. 1022-1038 6 FARINA, A., GRAZIANO, R., LEE, F., and TIMMONERI, L.: 'Adaptive space-time processing with systolic algorithm: experimental results using recorded live data'. Proceedings of international conference on Radar, Radar 95, Washington D.C., 8-11 May 1995, pp. 595-602 7 FARINA, A., LOMBARDO, R, and PIRRI, M.: 'Non-linear space-time processing for airborne early warning radar', IEEProc, Radar Sonar Navig., 1998,145, (1), pp. 9-18 8 FARINA, A., LOMBARDO, P., and PIRRI, M.: 'Non-linear STAP processing', Electron. Commun. Eng. J., 1999, 11, (1), pp. 41-48

9 KAY, S. M. and NAGESHA, V.: 'Maximum likelihood estimation of signals in autoregressive noise', IEEE Trans. Signal Process., 1994, SP-42, (1), pp. 88-101 10 LOMBARDO, P.: 4DPCA processing for SAR moving target detection in the presence of internal clutter motion and velocity mismatch'. Microwave sensing and synthetic aperture radar, EUROPTO, 2958, September 1996, Taormina, Italy, pp. 50-61 11 LOMBARDO, P.: 'Optimum multichannel SAR detection of slowly moving targets in the presence of internal clutter motion'. CIE-ICR'96, international Radar conference, Beijing, China, 8-11 October 1996, pp. 321-325 12 KLEMM, R. and ENDER, J.: 'Multidimensional digital filters for moving sensor arrays'. Proceedings IASTED on Signal processing and digital filtering, June 1990, Lugano, Switzerland, pp. 9-12 13 KLEMM, R. and ENDER, J.: 'Two-dimensional filters for radar and sonar applications'. Signal Processing V EUSIPCO, September 1990, Barcelona, Elsevier Science Publisher, B.V., pp. 2023-2026 14 WICKS, M. C , MELVIN, W. L., and CHEN, P.: 'An efficient architecture for nonhomogeneity detection in space-time adaptive processing airborne early warning radar'. IEE international Radar conference, Radar'97, October 1997, Edinburgh, UK, pp. 295-299 15 ADVE, R. S., HALE, T. B., and WICKS, M. C : 'Practical joint domain localised adaptive processing in homogeneous and nonhomogeneous environments. Part 2: Nonhomogeneous environments', IEEProc, Radar SonarNavig., 2000,147, (2), pp. 66-74 16 FARINA, A.: 'Linear and non-linear filters for clutter cancellation in radar system', Signal Process., 1997, (59), pp.101-112 17 FARINA, A., LOMBARDO, P., and CARAMANICA, F.: 'Non-linear nonadaptive clutter cancellation for airborne early warning radar'. IEE international Radar conference, Radar'97, Edinburgh, October 1997, pp. 420^24 18 KLEMM, R.: 'Adaptive clutter suppression for airborne phased array radar', IEE Proc. F and H, 1983,130, (1), pp. 125-132 19 LOMBARDO, P.: 'Echoes covariance modelling for SAR along-track interferometry'. IEEE international symposium IGARSS '96, Lincoln, Nebraska, USA, May 1996, pp. 347-349 20 LOMBARDO, P. and COLONE, F.: 'Non-linear STAP filters based on adaptive 2D-FIR filters'. IEEE Radar conference, Alabama, USA, May 2003, pp. 51-58

Chapter 3

Space-time techniques for SAR Alfonso Farina and Pierfrancesco Lombardo

3.1

Summary

This contribution describes the application of STAP (space time adaptive processing) to synthetic aperture radar (SAR) systems. SAR is a microwave sensor that allows us to have a high resolution mapping of electromagnetic (EM) backscatter from an observed scene. A two-dimensional image is provided in the radar polar coordinates, i.e. slant range and azimuth. High resolution in slant range is obtained by transmitting a coded waveform, with a large value of the time-bandwidth product, and coherently processing the echoes in a filter matched to the waveform. High resolution along the transversal direction is achieved by forming a synthetic aperture. This requires us to: (i) put the radar on board of a moving platform, e.g. an aircraft or a satellite (ii) record the EM signals from each scatterer which is illuminated by the moving antenna beam in successive instants of time (iii) coherently combine the signals - via a suitable azimuthal matched filter - thus focusing the sliding antenna pattern in a narrower synthetic beam [I]. The advantage of combining SAR and STAP is evident: the detected moving target is shown on top of the SAR image of the sensed scene. This Chapter describes in detail the multichannel SAR (MSAR). 3.2

Description of the problem and state of the art

In many applications (e.g. surveillance) of SAR, it is desirable to detect and possibly produce focused images of moving objects. A moving low RCS object is not easily detectable against strong echoes scattered from an extended fixed scene. When detected, its resulting image is smeared and ill positioned with respect to the stationary background. These shortcomings are a direct consequence of the SAR image

formation process. The cross-range high resolution in an SAR is obtained by taking advantage of the relative motion, supposed known, between the sensor and the scene. If, however, there is an object moving in an unpredictable manner, the image formation process does not function properly. Basically, the main degradations due to the target motion are: (i)

(ii)

The range migration through adjacent resolution cells (due to the radial velocity of target with respect to radar) causes a reduction of the signal-to-clutter power ratio, which can seriously impair the detection capabilities. Furthermore, range migration causes a decrease in the integration time and a consequent loss of resolution. Even in the absence of range migration, or after its correction, the phase shift induced by the motion causes: an ill-positioning (along track) of the target image with respect to ground, mainly owing to the range component of the relative radar-target velocity; a smearing of the image due to the uncompensated cross-range velocity and/or range acceleration.

An initial possibility that has been studied for discriminating the moving target signals from the fixed scene returns is on the basis of their different Doppler frequency spectra; see, for instance, Reference 2 and Reference 37. In fact, the target spectrum has a Doppler centroid approximately linearly proportional to the along-range velocity of target and a spectrum width depending on the azimuthal velocity and the radial acceleration components of target. Assuming that the radar pulse repetition frequency (PRF) is high enough to make available a region in the Doppler frequency domain not occupied by the stationary scene, the method works as follows (see Section 3.4.2 for details). First, transform a sequence of radar target returns to the frequency domain. Second, locate spectral bands, outside the narrowband frequency around the origin corresponding to stationary scene, and determine the centre frequencies of such bands. Third, translate each outlying spectral band to the origin, convert the resulting signal back to the time domain and correlate with the reference function of the conventional SAR. The correlator output will show the peaks in the correct locations of the targets. A refinement of this basic technique aims at the image formation of each target: it is obtained by also matching the width, not only the mean value, of the spectral band outside the stationary scene Doppler spectrum. The method suffers, however, from three shortcomings: (i)

It requires the use of a high PRF which causes a corresponding reduction of the SAR swath width and an increase of the data throughput. (ii) It does not correctly focus the image of a target having a quite arbitrary path. The spectrum of the target echoes alone is not sufficient; we need to know the instantaneous phase law to form the synthetic aperture with respect to the moving target. (iii) It does not succeed with a target whose motion has a small range velocity component, so that its spectrum is superimposed on the clutter (i.e. on the stationary scene) spectrum.

A distinct advantage of this system relates to the possibility of using it with conventional, non-multichannel phased array radar antennas. More powerful methods have been conceived to overcome these drawbacks; they are based on the use of more than one antenna, on board the moving platform, to cancel the clutter and detect the slowly moving targets. The radar system uses an array of antennas, mounted on the platform along the flight direction, and corresponding receiving channels. This makes available a certain number of space samples (echoes received from different antenna elements) and time samples (echoes collected at different time instants). These echoes are coherently combined, with proper weights, in a space-time processor to cancel the echo backscattered from the ground and enhance the target echo. Space-time effectively reduces the lower bound on the minimum detectable target velocity that would be established by using only frequency filtering. It measures the relative phase between two or more coherent signals, received from different antennas, rather than the Doppler frequency shift within a single receiving channel. Instead of using mono-dimensional filtering, clutter cancellation is the result of a powerful two-dimensional (in the temporal frequency, i.e. Doppler, and in the spatial frequency, i.e. azimuth angle) filtering. Furthermore, this method does not require necessarily to work with high PRF values. This is the STAP approach: see References 3, 4 and 5 for details. However, the conventional STAP techniques have to be extended to be made compatible with the MSAR case, which is intrinsically characterised by a long integration time, during which both target and clutter Doppler and direction of arrival (DoA) change. These characteristics are reviewed in Section 3.3, and Section 3.4.1 presents a taxonomy of a number of MSAR processing techniques proposed in the literature to jointly exploit the effectiveness of SAR and STAP techniques. The way to fully-fledged STAP wasn't immediate: after leaving the single-channel approach (Section 3.4.2) it passed through ATI applied to SAR (described in Section 3.4.4) and DPCA (see Section 3.4.3); both techniques being essentially based on the use of the echoes captured by two antennas. A refinement of ATI-SAR was the VSAR (velocity SAR) method [6]. In a way similar to the progression from a two-pulse canceller to a bank of Doppler filters to reject clutter and detect moving targets in a conventional search radar, the technique can be generalised to a linear array of identical antennas. For each channel a complex SAR image is focused. A Fourier transform along the physical aperture (i.e. the channel number) is applied to each pixel, and this corresponds to a multiple beam former. For each Fourier cell the related image shows the scene for a certain range of radial velocities (velocity SAR image [6]). This method wasn't, however, originally designed to suppress clutter, since no attempt is made to subtract signals. The requirement to cancel the echoes from stationary scatterers leads directly to adaptive space-time filtering. Two classes of advanced STAP techniques are considered in particular for the MSAR case: (i) time-domain reduced DoF (degrees of freedom) adaptive two-dimensional finite impulse response (FIR) filters (Sections 3.4.5.2, [7-11] and Section 3.4.6) (ii) frequency-domain reduced DoF adaptive processing [12].

Finally, Section 3.4.6 considers the optimum MSAR approach in a special case for the clutter characteristics, to gain insight into the properties of the optimum filter and its effectiveness. This can be used for comparison of the different filters and yields the upper bound for achievable performance. Once the presence of a moving target has been detected, we have to estimate its phase modulation law to be able to form a high resolution image of it. A method for providing the estimate is by means of time-frequency analysis of the received signal (Section 3.4.5.1). This, combined with STAP, brings us to the joint spacetime-frequency processing presented in detail in Section 3.4.5.2. The time-frequency representation is obtained by evaluating the Wigner-Ville distribution (WVD) of the signal. This distribution is a signal representation consisting in the mapping of the signal onto a plane whose coordinates are time and frequency. The WVD, in particular, produces a mapping such that the signal energy is concentrated along the curve of the instantaneous frequency. This frequency is obtained as the centre of gravity of the WVD; the instantaneous phase is derived by integration of the instantaneous frequency. The clutter echoes are cancelled by the adaptive space-time processor, where the space-time covariance matrix of clutter is estimated online and used to evaluate the optimal weights of the two-dimensional filter. The time-frequency analysis provides an estimation of the instantaneous frequency of the possibly present moving target, and - by integration - the original instantaneous phase. The phase is used to compensate for the shift due to the relative target-radar motion.

33

Model of MSAR echoes

3.3.1 Aberrations due to target motion There are some interesting effects that occur with moving targets. A moving target with a radial component of velocity vr results in a Doppler shift on each echo of: /o = ^

(3.1) C

where /o is the radar carrier frequency and c is the velocity of propagation. Thus the Doppler history of the sequence of echoes is shifted in frequency (solid line of Figure 3.1), and is matched filtered with a reference chirp (dashed line of Figure 3.1). This produces a shift in the azimuth position, which is given by the product of the target Doppler shift and the slope of the Doppler history

yb-£ = ^

0.2)

where v is the platform speed, r is the platform-target range and d is the along-track dimension of the antenna (the real aperture). It is well known that an image from an aircraftborne SAR of a moving train having a component of velocity in the range direction appears shifted, so it appears to be travelling not along the railway track, but displaced to the side! As another example, a ship in a satellite SAR image with a radial velocity of 10 m/s at a range of 1200 km would experience an azimuth shift of

5 CN

Il ^S

JO

I I ex

a O

1 synthetic aperture length = rl/d

Figure 3.1 A target with a radial velocity (solid line) is matched filtered with the SAR azimuthal reference chirp (dashed line) 1.7 km. On the other hand, the ship wake (which is stationary) appears in the correct position. Thus the ship appears not at the tip of the wake, but displaced in azimuth. This effect is visible in a number of spaceborne SAR images. From knowledge of the geometry, and of the azimuth shift, it is possible to estimate the target velocity.

3.3.2 Space-time-frequency representation The MSAR allows Doppler and angular information to be decoupled by means of the along-track interferometric processing. The derivation of optimum clutter cancellation schemes for moving target detection requires the definition of a model for the covariance of the MSAR space-time clutter echoes to take full advantage of this potentiality. Moreover, a proper representation space for the resulting MSAR space-time model is required to adequately interpret the results. Both models and representations are available separately for single-channel SAR, where the long observation time and the consequent non-linear phase history of the echoes are the main issues, and for multichannel systems [4, Chapters 2 and 3], which assume, on the contrary, a short integration time and focus on the angular characteristics. In this section, a simplified closed-form model for the MSAR echoes is described, which takes into account both non-linear phase modulation and angular position at the same time and defines a proper representation space for it. Even though simple to handle, the model encodes all the main characteristics of the scattering from the observed scene [13]. 3.3.2.1 Echoes from homogeneous correlated clutter A simplified condition is assumed in the following, for which the radiation patterns of transmitting and receiving antennas have the same width. This allows us to reduce again the analytical complexity of the analysis, while considering the effect of parameters. The simplified model is used in the present section to include the effect

TX/RX array antenna

Figure 3.2

The multichannel SAR observation geometry (PRT:pulse repetition time)

of the terrain reflectivity correlation. With reference to the slant plane geometry in Figure 3.2, assume that the radar transmitting antenna moves, with constant velocity v, along a straight trajectory at a fixed distance Ro from the g-axis (crossing the r-axis at time t = 0) and has a fixed pointing, orthogonal to the flight path. The sequence of echoes relative to the TV pulses, transmitted with pulse repetition time PRT, is received by a uniform linear array of K antennas, parallel to the radar trajectory. The K receiving antennas have separate receiving channels and phase centres at distance Sk = (k — (K — l)/2) A, k = 0 , . . . , K — 1 from the transmitter. A pointlike scatterer T is assumed to be for t = 0 on the g-axis at the position xq and to move with velocity components vq and vr along the two axes q and r, named along-track and cross-track. It is further assumed that: (i) the range cell migration is negligible or has been corrected for (ii) the two-way antenna pattern, for the channel k, has the Gaussian shape gk [y] = \/{y/nXq) exp[— (y — sk/2)2/X^\, Xqy being the along-track position of the scatterer relative to the phase centre of the transmitter (iii) quadratic terms in the receiving antennas displacements can be neglected and the Fresnel approximation is valid for the distance of the scatterer from the transmitter (narrow antenna pattern). Thus the echo received by the &th receiving antenna at time tn = (n — (N — l)/2) PRT, n = 0 , . . . , Af — 1 can be modelled as:

(3.3)

where Ao and O are constant amplitude and phase terms and fy = 2vr/X, the Doppler frequency due to the cross-track velocity for the wavelength X. Equation (3.3) takes into account the central Doppler frequency and DoA of the scatterer, which are the usual model for space-time processing techniques based on a short integration time. Moreover, the quadratic term encodes the time varying Doppler and DoA, observed in the long SAR integration time. The clutter echo Ck (t), received at time t and receiving element Jc, is obtained by integrating the echoes from the infinitesimal clutter patches of dimension dxq, modelled as equation (3.3) with instantaneous reflectivity Ao = IJi(xq, t)dxq. The correlation of the clutter echoes received at times tn x = t,tn2 = t+r and receivers in Sk1, Sk2 = Sk1 + 2rj is:

(3.4) the symbol (•) standing for the statistical expectation. The model for homogeneous correlated clutter is obtained in the hypothesis of factorisation of the spatial and temporal correlations, px(x) and pt(t), of the clutter reflectivity: (3.5) Using a Gaussian spatial correlation, with variance o2,px(x) = (\f2nGc)~x 2 exp[—x /(2 a*)], the double integration in equation (3.4) becomes a Gaussian function of the space-time displacement, z = (v — vq)r + rj9 of the two-way phase centre and is thus stationary in both space and time:

(3.6) where E 2 = 1/[1/X2 + (2nXq/(XRo))2] is the square of the maximum SAR resolution. By taking the bidimensional Fourier transform of equation (3.6) with respect to time: t —> f, and space: r\ —> sin(0)/X ^ 6/X, we obtain the clutter power spectral density (PSD) as a function of frequency / and angle 0: (3.7) where Ft [/], is the Fourier transform of pt(r). In the absence of temporal decorrelation (pt (r) = 1), P(f, 0) reduces to a sheet for / = fd + (v — vq)0/X, whose Gaussian shaped amplitude is controlled by the variance £ 2 + o2. This reduces to the square of the SAR resolution for uncorrelated reflectivity and grows with the spatial correlation a2 otherwise. This can be especially important for the new generation of VHF SAR systems, to which the reflectivity of a natural scene can appear more correlated than

to microwave sensors. It applies also in a number of remote sensing applications over smooth surfaces, such as sea or ice covered regions and in some planet exploration missions, where the correlation of the surface scattering is not negligible and also a coherent contribution can be present in the echo. This is evident from the behaviour of the PSD that for a\ —> oo behaves as a Dirac delta for 0 = 0. As expected, the mean cross-track velocity of the observed surface (as, for instance, the sea surface) produces a constant frequency shift, whereas a mean along-track velocity changes the slope of the clutter ridge. The presence of temporal decorrelation spreads the PSD around the straight line above, according to the function FVf/], and Xq controls its amplitude, affecting the angle over which the transmitted power is spread.

3.3.2.2 Representation of MSAR echoes Adequate representation strategies are available separately for single-channel SAR, where the long coherent integration interval (CPI) and the consequent non-linear phase history of the echoes are the main issues, and for multichannel MTI systems, which assume, on the contrary, a short CPI and focus on the angular characteristics. In the present section it is shown that the representation planes used for the systems above are inadequate to fully represent the MSAR echoes. The need for a proper representation space for the MSAR echoes is thus illustrated, which takes into account both non-linear phase modulation and angular position at the same time. The angle-frequency plane (O9 f) in Figure 3.3 is the preferred representation space for the clutter covariance in multichannel systems [4, Chapters 2 and 3]. A short observation time is assumed, so that Doppler frequency and DoA of the single scatterer are constant and the array of receivers is used to exploit the scatterers' angle. The PSD in equation (3.7) is directly mapped into this plane, which seems thus appropriate for homogeneous clutter. However, when the observation time is long, both DoA and Doppler frequency of the clutter echo from the single surface scatterer depend on time (equation (3.3)). This can't be represented in the (O, f) plane, where the contribution from each patch of an homogeneous clutter moves in time along a line with constant slope (see arrows in Figure 3.3). Since the scatterers that quit the antenna beam are replaced by new ones entering, on average, the same PSD is observed. Moreover, since the time-varying characteristic of the echo's DoA and Doppler spreads its power over a line in the (O9 / ) plane, the gain of the SAR coherent integration over the noise can't be related directly to Doppler or angular filtering in this plane. This applies also for the detection of targets, with a motion different from that of the clutter patches. The time-frequency plane (t, / ) in Figure 3.4 is the preferred representation space for single channel SAR, where a long observation time is considered. It allows the correct representation of the time-varying characteristic of the Doppler frequency but doesn't contain angular information. Thus the Doppler frequency / of the generic scatterer, shown in this plane, relates to a combination of cross-track velocity and angular position: this ambiguity cannot be resolved. This is shown in Figure 3.4, where solid and dashed lines represent the echo from the same homogeneous clutter

Figure 3.3 Frequency angle plane: typical of STAP (PRT: pulse repetition time)

Figure 3.4

Frequency-time plane: typical of SAR (PRT: pulse repetition time)

patch, received by different channels. The clutter edge is localised in time in the (t9 / ) plane and the SAR coherent integration gain is now clear. From the analysis above it can be seen that, to interpret the MSAR echoes properly, a higher dimensional space is required, which allows both Doppler frequency

Figure 3.5

Space-frequency-time volume: typical of MSAR (PRT: pulse repetition time)

and DoA to be represented as functions of time [13]. The space-time-frequency space (0, t, / ) , in Figure 3.5, is thus introduced. Its integration over the temporal axis collapses it back into the (O, f) plane, and integration over the angular axis collapses it into the (t, / ) plane. In this space the motion of the scatterer echoes in the Doppler angle limits of the antenna beam are correctly represented for both homogeneous and non-homogeneous scene: ( ). Moreover, the generic target echo: (_._), with velocity components different from the clutter background can be effectively represented in the volume of Figure 3.5 and discriminated from the clutter echoes. This representation can be used to get inside the characteristics of the different processing schemes for the processing of MSAR data and fully understand the effect of their application.

3.4

Processing schemes

In the following several processing schemes for the detection and imaging of moving targets are described.

3.4.1

Taxonomy of processing schemes for MSAR

A wide range of processing architectures has been derived in the last few years to detect and focus moving targets using SAR systems with K antennas, based on the digitised echoes received during the long sequence of N PRI from the Q range cells under analysis. The above mentioned architectures are characterised by different approaches to deal with the three main parameters of the received echo signals: alongtrack velocity (Doppler rate), cross-track velocity (Doppler), and DoA. A brief review of the different processing architectures is reported below.

3.4.1.1

1st processing architecture: optimum filter with very long integration time (due to the extended synthetic aperture required to achieve high spatial resolution) With respect to the conventional optimum STAP filter [4, Chapter 4], this optimum filter (see Figure 3.6) is characterised by a target steering vector with Doppler frequency and DoA that cannot be taken constant during CPI. As apparent from Section 3.3.2, with MSAR these quantities might change considerably during long CPI. This filter has the obvious advantage of yielding an optimum approach, however, it has the full computational complexity due to the three-dimensional filter bank (Doppler rate, Doppler, DoA). Only in special cases can this optimal solution be easily derived and interpreted [10, H]. Moreover, the adaptivity of the filters (ideally the filter weight vector is w = s^R" 1 where H stands for complex conjugate transpose) requires very many taps for each filter of the bank and a very large training area with homogeneous clutter, to estimate the clutter covariance matrix R.

3.4.1.2

2nd processing architecture: STAP filtering + coherent integration for SAR processing [9] This suboptimum scheme (see Figure 3.7) is obtained by using a STAP cancellation filter which performs the clutter cancellation and collapses the K channels into a single ideally clutter-free channel. A two-dimensional filter bank (Doppler rate, Doppler) follows to detect the target. This is based on the consideration that cross-track and speed both contribute to modify the linear phase term (Section 3.3.1). Moreover, the filter effect of STAP is related independently to cross-track speed and DoA. The effect of the STAP filter on the second stage is also analysed in Reference 14. This architecture is less computationally demanding and yields a reasonable suboptimum processing scheme.

K channels

Figure 3.6

target detection & imaging

Optimum processing scheme of MSAR

K channels

Figure 3.7

STAP: 3D filter bank: •DoA •Doppler •Doppler rate

STAP Cancellation filter

1 channel

2D filter bank: •Doppler •Doppler rate via Wigner-Ville

Space-time-frequency processing

target detection & imaging

3.4.1.3

3rd processing architecture: formation of SAR image & detection of moving targets This processing scheme requires the application of a conventional MTI clutter canceller on each channel separately. Then the formation of K SAR images is performed (see Figure 3.8), with the corresponding phase compensation (as it happens for Interfermetic SAR (InSAR)). This is followed by a one-dimensional (Doppler) filter bank for the final target detection. The reduced computational cost of this processing architecture yields largely degraded performance. It is especially difficult to compensate for the along-track speed of the target different from the clutter. This technique does not exploit the spatial processing dimension of the multichannel antenna for clutter cancellation. It is therefore (except for the signal to noise power ratio (SNR)) equivalent to a single-channel MTI SAR. 3.4.1.4

4th processing architecture: joint domain localised (JDL) for MSAR [15] The scheme is depicted in Figure 3.9. First a time-varying transformation is applied to the space-time data to compensate for the time-varying Doppler frequency of the clutter echoes. Therefore, a large reduction of degrees of freedom is performed in the Doppler-DoA plane, after two-dimensional FFT. This is followed by an adaptive optimum generalised likelihood ratio test applied in the reduced domain [16]. This

MTI

SAR

MTI

SAR

MTI

SAR

ID filter bank: •Doppler

K channels

Figure 3.8

Channelised MTI&SAR processing time-varying transformation

NK samples

Figure 3.9

2DFFT

discard data

Yes target GLRT

No target

to compensate for cluster of 2D beams adaptivity on generalised Doppler frequency in Doppler & spatial a reduced number likelihood and DoA variation frequencies domain OfDoF ratio test during CPI

Joint domain localised (JDL) on MSAR

RX-K

SAR processor 1

SAR processor 2

TX

SAR processor K

velocity processor

Figure 3.10

Velocity SAR

scheme has low computational complexity, close to optimum detection performance, and reduced size of required homogeneous clutter area. However, it requires a proper time-varying transformation and is less intuitive for targets with non-negligible alongtrack velocity component. 3.4.1.5 5th processing architecture: velocity SAR (VSAR) [6] As sketched in Figure 3.10, the set of complex video signals is processed by SAR processors to produce complex images: Yk(x, y), k = 0 , . . . , K — 1 with x, y azimuth, range coordinates. Phase of formed images is preserved. The post-processing of complex images brings out the velocity dimension. This step involves processing of each pixel for all images. Consider the whole set of images as a single threedimensional image: Yk(x9 y, k); the velocity processing applies along the rc-axis and provides the resulting three-dimensional image: Z(;t, y, v) in azimuth, range and velocity of the observed scene. Velocity processing in its simple form involves Fourier transformation of K points along the A>axis. Finally, each velocity plane is shifted by an amount proportional to its velocity to correct the motion-induced azimuth displacement. All the planes are summed to produce the final undistorted VSAR. The method is not designed to suppress clutter, since no attempt is made to subtract signals. The requirement to cancel the echoes from stationary scatterers leads directly to adaptive space-time filtering. 3.4.1.6 6th processing architecture: processing in the frequency domain This method finds its mathematical foundation in the stochastic stationary characteristic of the clutter processes that have natural description in the Fourier domain via the spectral density matrix. The Fourier transforms (FT) of the process for fixed

FFT

FFT

FFT receiving channels

spatial null for each frequency bin target match in space

target match in frequency Threshold

Figure 3.11 Frequency domain processing for MSAR [12] frequencies are asymptotically independent when the time base tends to infinity. Thus cross-correlations between frequency channels are negligible. Therefore, frequency processing is valid asymptotically, since the spectral representation provides a time signal extending from — to + infinity. For SAR we have in the limit an infinite sequence of equispaced samples along the pulse-to-pulse time (slow time). This is in contrast to the finite pulse train of airborne early warning (AEW) search radar. For azimuth and range compressions the FT is performed in any case. Moderate time base (= synthetic aperture length) is sufficient to make negligible the cross-terms between the frequency channels. In Doppler domain processing, the clutter energy is collected in one-dimensional subspace. This is the key to using projection methods (see Figure 3.11) with the advantage that the clutter is cancelled perfectly without sensitivity to its amplitude variations. A good estimate of the covariance matrix is achievable since there is a large number of range bins. Even though clutter to noise power ratio (CNR) varies with range, the structure of underlying clutter subspace remains unchanged. The computational load is affordable if only a few spatial channels (three is the minimum) are used for long sample sequences. SMI (sample matrix inverse) and eigendecomposition of 3 by 3 matrices are performed very fast. For details see Reference 12. 3.4.1.7

7th processing architecture: displaced phase centre antenna (DPCA) for SAR The concept of DPCA requires an antenna divided in two subarrays and the transmission of two pulses. Due to the aircraft movement, the transmitting and receiving

antennas form two bistatic configurations for the two pulses. If the subarrays' displacement, the platform speed and the radar PRF are properly selected, the global two-way phase centres of the bistatic antenna configurations are coincident. The stationary clutter is seen at two different time instants from the same radar location and it can be cancelled by subtracting the corresponding echoes. The DPCA corresponds to a kind of spatial MTI: returns from stationary objects cancel, while moving target echoes are maintained. The extreme simplicity of this approach is paid for by very strong constraints on the antenna, PRF and platform speed, which are often unacceptable. Moreover, the technique has no ability to compensate for any mismatch from this idealised condition by means of adaptivity. 3.4.1.8

8th processing architecture: along-track interferometty SAR (ATI-SAR) This operates similarly to DPCA using two displaced antennas connected to two receiving channels. An SAR image is generated for each channel; thereafter, the time delay between the azimuth signals is compensated for using two different azimuth reference signals. By multiplying the first image by the complex conjugate of the second, the remaining phase is zero for stationary objects and non-zero otherwise. Basically this technique is closely related to DPCA and STAR It may suffer from constraints imposed on the processing according to SAR requirements (SAR requires low PRF values to have a wide along-range swath width, and STAP needs large PRF to detect also fast moving targets).

3.4.1.9 9th processing architecture: pulse Doppler and MTI [2] This technique is derived from the well known moving target detector (MTD) processing widely used in conventional ground-based radar. After range compression, an azimuth processor is developed to detect the presence of a moving point-like target in each range cell and to measure its velocity components. The method is straightforward and simple to implement under the conditions that the target velocity is bounded within a minimum detectable value and the unambiguous value, and that range migration is negligible. In the following we will describe in some detail the following processing schemes: MTI + PD (Section 3.4.2), DPCA (Section 3.4.3), ATI-SAR(Section 3.4.4), joint space-time-frequency (Section 3.4.5), optimum filter for a special case (Section 3.4.6). We prefer to follow the historical derivation rather than the optimum approach from which all the suboptimum schemes can be derived.

3.4.2 MTI+ PD The material of this section is derived from Reference 2 which describes the work done by colleagues of the authors. The processing technique, referred to as moving target detection and imaging (MTDI), has been derived by the well known MTD processing widely used in conventional ground-based radars. A range-Doppler SAR processing is adopted; the raw data are first processed along the range direction and then along the azimuth. An azimuth processor is developed to detect the presence of

beam footprint

Figure 3.12

Geometry of SAR and moving target

a moving point-like target in each range cell and to measure its velocity components. The method is valid under the following hypotheses, namely: • •

the target velocity is bounded within minimum detectable and maximum unambiguous values (i.e. PRF/2) the SAR system and the moving target cause a negligible range migration.

It should be noted that with a one-channel antenna no detection of endoclutter targets (i.e. targets with Doppler frequency within the clutter spectrum) is possible. Increasing the PRF may not be desirable because of an increase in range ambiguities, and likewise for SAR applications. The mathematical model of the echo received by the radar antenna during the synthetic aperture time interval is reported in this section. Figure 3.12 sketches the geometry of an SAR system and a moving object of interest. The antenna, on board an aircraft, moves along the azimuth direction x. The antenna beampattern is directed orthogonal to the flight path; 0 (the off-nadir angle) is the angle formed by the normal to the ground and the line from the radar to the central point of the scene; Vx and vy are the velocity components of the target along the reference coordinates. Along the azimuth, the radar transmits pulse trains with repetition frequency PRF, each pulse having a linear frequency modulation (chirp). To account for some unpredictable changes in the environment wherein the transmitter operates, an unknown initial phase 0o» modelled as a random variate uniformly distributed in (0,2n), is assumed in the transmitted signal. At the nth azimuth position of the antenna, the transmitted pulse is expressed as: Tn(t) = A cos(2jtf0t + at2 + 0O),

~


X

-

(3.8)

where A and r are the pulse amplitude and the pulse width, respectively, /o is the carrier frequency, a = 2JTB/T is the chirp rate, and B is the chirp bandwidth. Consider now a point-like scatterer on a completely absorbing background; the echo received by the antenna at the nth azimuth position, after down frequency conversion, can be modelled as: Sn(t) = AlGOg0(O, v sin 0 Fdc = - 1 Y -

F =

( 3 - 10 >

2(V - vx)2

* —5i*T~

(3 11}

-

where V is the platform speed, Vx and vy are the target velocity components respectively along the x- and y-axes, and Ro is the range between the platform and the scene centre. The above approximation leads to a useful expression for the bandwidth of the azimuth chirp, i.e.: Bd =

2{V Vx)

~

= FjrTi

(3.12)

LJ

with L being the along-track antenna length and 7} the integration time, i.e. the time during which the target is illuminated by the antenna main lobe. The signal Sn (t), sampled at the proper frequency Fc, is stored as a column in the holographic matrix S. The dimensions, Nsr and Nsa, of the matrix S represent the number of range and the azimuth samples, respectively. The entry Smn of S can be reparameterised to elicit the dependence on the Doppler centroid Fdc and the Doppler rate F^ r , namely: Smn = Alcrogo($,
(3.13)

where tm = m/Fc is the range time (fast time) and tn is the azimuth time (slow time). A conventional SAR processor, designed to provide the image of a stationary scene, works separately along the range and the azimuth directions. Specifically, at the first stage each column of the holographic matrix is convolved with the impulse response function (IRF) hr of the range filter yielding the range-compressed data. At the second

stage, each row of the resulting matrix is convolved with the IRF ha of the azimuth filter, thus providing the final image. Useful expressions for hr and ha, derived from equation (3.13), are: M*m) = expOa£),

m = 1,2,...,JV,

(3.14)

ha(tn)

n = l,2,...,Na

(3.15)

= exp(j27tFdr0tl),

where Fjro is the Doppler rate (equation (3.11)) for Vx = 0. The conventional processing fails when the convolutions are implemented by means of equations (3.14) and (3.15) on the signals coming from moving objects for which F& ^ F^rO a n d F^c 7^ 0. In MTD technique the Doppler frequency shift is used as a means of detecting moving targets embedded in a strong ground backscatter (clutter). A delay-time canceller behaves as a filter to reject the low Doppler components associated with the clutter and to preserve the high Doppler components of the moving targets. The canceller is cascaded with a bank of narrowband filters (channels), uniformly spread in the PRF interval. When matched filtering is performed, only that Doppler channel containing the target echo will supply a significantly non-zero output. These concepts are amenable to extension to SAR application: in fact, for sensing and imaging purposes, the stationary scene resembles clutter and the point-like moving object represents the target. It appears convenient to refer to this approach as MTDI, since it is the natural extension of MTD to SAR. However, some novel problems, arise when we deal with SAR. First, the representation of the signal returned by a scatterer results in an intrinsically two-dimensional problem. In some airborne cases, however, the dwell time is usually short and the range cell migration is avoided. Whenever the SAR system parameters do not induce range migration by themselves, a moving target could still be displaced over more range lines if its radial velocity is high enough to encompass more range lines during the integration time, i.e. if: (3.16) where Afrcm is the number of range migration cells and 8r is the range resolution of the SAR system. Assuming an airborne SAR with 7} = 0.2 s, 6 = 45°, 8r = 4 m, and substituting the numerical values in equation (3.16), a maximum target speed of 30m/s along the range direction is allowed to avoid range migration (Nrcm < 1). Under these assumptions the pattern is not truly bidimensional and the processing can be performed by a separate filtering of the rows and the columns of the holographic matrix. Another problem which arises when dealing with moving targets is the aliasing effect due to the broadening and shifting of the signal spectrum. Assume, in fact, that PRF = Bd'. in this case a radial velocity component of the target causes a Doppler shift according to equation (3.10) and a significant aliasing cannot be avoided. Furthermore, if the target velocity Vx is opposite to that of the platform, an increase of the nominal bandwidth occurs according to equation (3.12). As a consequence of these concurrent phenomena, the azimuth compression cannot be perfectly achieved and the target is not correctly imaged. If instead the

sampling frequency PRF is ^> Bd, then the discrete and the analogue representations can be considered equivalent. More precisely, PRF must be constrained by the inequality: (3.17) where F j c m a x and F^ r m a x are available from equations (3.10) and (3.11) on substituting the quantities Vx and vy with their maximum values. The basic considerations leading to the MTDI algorithm as a means of detecting moving targets are discussed below. When the system PRF complies with equation (3.17), the ground echo and the echoes of the moving targets are spectrally separated: the former is concentrated in [-Bd/2, BdIT) and the latter are allocated in (-PRF/2, -Bd/2) U (Bd/2, PRF/2). The scheme of the MTDI processor is shown in Figure 3.13. The raw data are stored in the holographic matrix and fed into the range processor. Then, the spectrum of the azimuth line is computed via a fast Fourier transform (FFT) and split throughout the Ndc channels, each one encompassing a bandwidth as large as Bd. The rejection of the ground echo is performed leaving out the channel centred on the zero Doppler frequency, after phase compensation of the platform speed V. The processing that follows performs azimuth compression on each stream of data. The procedure consists in evaluating the product between the data coming from a single channel and a suitable reference function, making an inverse Fourier transform and comparing the maximum value attained by the signal with a threshold. Finally, if a moving point is detected, a fine estimate of the chirp parameters - as discussed in

reference chirp FFT M/N pt.

to Doppler estimator channel 1 holographic matrix

range processing

reference chirp

range-compressed transformed data

IFFT

threshold

IFFT

threshold

Fin Fie, estimation image formation estimation

channel 2 Mpt. FFT

channel N IFFT

Figure 3.13 Processing scheme of MTDI [2]

threshold

Fdr,

F^

estimation

Reference 2 - is provided to improve the performance of the image-forming algorithm. The range of detectable velocities is limited between a minimum detectable and a maximum unambiguous values. The minimum and maximum velocities correspond to a Doppler displacement equal to the first and the last available Doppler channel centre frequencies, respectively. Conservative values (with respect to the off-nadir angle) are: (3.18) (3.19) For airborne SARs operating at millimetric wavelengths, values of 1 m/s for vm[n and 20 m/s for vmax are found using a PRF of 10 kHz. In Reference 2 the problem of probing the presence of a chirp signal embedded in a disturbance environment (backscattering by fixed scene and thermal noise) is modelled as a binary decision test; also the detection performance is found. Details are in the quoted Reference. A high quality image of moving targets calls for suitable procedures for estimating the chirp parameters. Here we introduce two algorithms for Doppler centroid and Doppler rate estimation. The algorithms are suited for airborne SARs and pointlike targets. The chirp parameters are related to the moving point velocities through equations (3.10) and (3.11), the knowledge of the former allows one to recover the latter. 3.4.2.1 Doppler centroid estimate The target radial velocity causes a Doppler frequency shift given by equation (3.10). The algorithm discussed here involves a correlation between the power spectrum of the azimuth signal and that of a reference signal, namely a chirp signal with no Doppler shift and nominal Doppler rate. The correlation peak is centred on the searched Doppler shift. More formally, the algorithm consists of the following steps: (a)

Discrete Fourier transformation (DFT) of the azimuth signal: (3.20)

where Nt,a is the azimuth FFT block length, and computation of the power spectrum: Sx(m) = \X(m)\2 (b)

(3.21)

Extraction of a subset of samples S(m) from Sx(m) according to the following rule: once the channel where detection occurs is selected, pick up all samples encompassing a bandwidth as large as 2 Bj centred around the channel reference frequency. So we are ensured that only a small amount of the signal energy is left.

(c)

Evaluation of the cross-correlation Rs(m) between Sim) and reference spectrum Sr(m)\ Rs{m) = S(m) ® Sr(-m)

(d)

(3.22)

where
where Q\ indicates the set of samples belonging to the lower frequencies (m = 1, . . . 5 ni, say) and ^2 indicates the corresponding set belonging to the upper frequencies {m = n\ + 1 , . . . , np, say) (np is the total number of points of Rs(m)). We select n\ so that the difference \e\ —£21 is minimum. The knowledge of n\ can be used for the evaluation of the Doppler centroid. 3.4.2.2 Airborne SAR auto foe us The Doppler rate is the other parameter required to perform a sharp focusing of the target echo. A widely used algorithm for Doppler rate estimation is based on the multilook technique; unfortunately, this technique does not work well if applied to airborne SARs. Consider, in fact, the Doppler rate resolution achievable by this algorithm [2]: (3.24)

where: Ni is the number of looks, A / is the centre-looks distance. Table 3.1 reports the values assumed by the above-specified parameters, together with the Doppler resolution, for typical spaceborne and airborne SAR systems. It is easy to observe that the resolution is good for Seasat SAR but poor for the airborne SAR. In Reference 2 a new algorithm is introduced which performs an accurate estimation of the Doppler rate for both spaceborne and airborne SARs provided that the Doppler centroid has been previously corrected. Consider the modulus of the mutual energy € between the received signal x(t) and the reference signal xr(t): (3.25) where /S is the Doppler rate to be estimated. The received chirp x(t, fi), after Doppler centroid correction, can be simply expressed as: X(t,Fdr)=exp{jFdrt2}

(3.26)

Table 3.1

fdr0 A/ Bd NL 8fdr

Some typical parameters for spaceborne and airborne SARs SEASAT SAR

Airborne SAR

520Hz/s 967Hz 1389Hz 4 0.8Hz/s

3112Hz/s 300Hz 400Hz 4 292Hz/s

If we choose: xr(t,l3) = exp{jl3t2}

(3.27)

and collect all inessential terms in a complex constant K, we obtain: (3.28) which can be approximated as: (3.29) Expression (3.29) can be regarded as the integration of Na pulses having the same modulus, but time-varying phase. The integration process leads to the maximisation of the cross-energy modulus if and only if (Fdr — /3) is equal to zero or, in other words, if the estimated Doppler rate is equal to the true one. Thus, the estimation rule amounts to evaluating the maximum of expression (3.29), with respect to j6, namely: (3.30)

where Qp is a suited domain for /3. After the estimation of Fdc and Fdr, the azimuth signal can be correctly compressed by means of a classical SAR azimuth processing to obtain the position and the reflectivity of targets. Moreover, vx and vy can be evaluated by applying equations (3.10) and (3.11). The algorithms above have been simulated to check their ability to generate correctly focused images of moving and stationary point-like targets over a fixed background; the successful results are reported in Reference 2.

3.4.3

DPCA

The concept of DPCA, displaced phase centre antenna, was conceived by F. R. Dickey, F. M. Staudaher and M. Labitt; in 1991 they received the IEEE-AESS Pioneer award for this invention [17]. One major application of DPCA is in the Joint-STARS (joint

two-element DPCA array position when the 1st pulse is transmitted subarray used for receiving echoes of 1st Tx pulse subarray used for receiving echoes of 2nd Tx pulse

'platform y ' platform

array position when the 2nd pulse is transmitted position of Tx/Rx phase centres for the two pulses

Figure 3.14

Working principle of DPCA

strategic target attack radar system) [18]. Figure 3.14 shows the working principle of the system; it depicts the radar antenna moving along track when two pulses are transmitted and corresponding echoes received. The antenna is divided into two subarrays; the whole antenna transmits two successive pulses, the corresponding echo of the first pulse is received by the fore subarray, the echo of the second pulse is received by the aft subarray. Due to the aircraft movement, the transmitting and receiving antennas form two bistatic configurations for the two pulses. However, if the antenna dimension, the platform speed and the radar PRF are properly selected, the phase centre of the bistatic antenna configuration for the first transmitted-received pulse coincides with the phase centre of the second transmitted-received pulse. Thus, the clutter is seen at two different time instants from the same radar location: the platform motion has been compensated, consequently the Doppler spectrum of clutter is not spread due to platform motion. The clutter is cancelled by subtracting the echoes of the first pulse received by the fore subarray from the echo of the second pulse received by the aft subarray; DPCA corresponds to a kind of spatial MTI. Stationary targets should cancel, but echoes from moving targets will give a non-zero result from the subtraction, so they should remain. Coe and White [19] have carried out a theoretical and practical evaluation of this technique. They have found that cancellation of the order of 25 dB is possible. The major DPCA sources of limitations are: loss of receiving aperture, PRF-velocity constraint, performance limited by subarray mismatch of radiation patterns, two degrees of freedom (DoFs) only, the system is not adaptive. STAP is the natural generalisation of DPCA, in fact it has the following features: • • • • •

generalisation of adaptive array of antennas for jammer nulling generalisation of DPCA for clutter cancellation freedom in shaping the null fewer constraints on the spacing of the antenna elements compensation of platform motion along (as DPCA) and orthogonal to the array thus avoiding clutter spectral spreading.

3.4.4 Along-track interferometry (ATI)-SAR In a similar manner to DPCA (see Section 3.4.3), along-track interferometry (ATI) uses two displaced antennas connected to two receiving channels. For each channel

an SAR image can be generated. The time delay between the azimuth signals can be compensated for during the azimuth compression using two different reference signals, incorporating the azimuth chirps generated in the two channels by a common point scatterer. If the first image is multiplied by the complex conjugate of the second, the remaining phase is zero for stationary objects and non-zero otherwise. If the two receivers are spatially separated by the distance J, the interferometric phase is approximated by tp = —(27t/X)d(vr/va), where vr is the radial velocity of the target. An example of parameters for ATI-SAR is: d = 1 m, va = lOOm/s, A = 3 cm, vr = 1 m/s, cp = 120°. ATI-SAR was originally developed to measure the speed of ocean surfaces; in Reference 12 the capability of SAR to detect slowly moving targets is shown on live data. In Reference 20 an in-depth analysis of the capabilities of ATI-SAR to detect moving targets is presented. In this section some analytical and simulation results concerning detection of a moving target are presented. ATI-SAR has the following limitation: to have a high phase sensitivity, the two antennas have to be widely separated, but this leads to a comb of blind velocities Vbhnd = kvah/d, where k is an integer: this limits the useful target velocity interval; moreover, the above mentioned distortions in the SAR image of moving target (Section 3.3.1) remain. The probability density function (PDF) of the interferogram phase (p for just stationary clutter and thermal noise has been found to be [21]: (3.31) where

Clutter and noise are independent random processes with zero mean and Gaussian PDF. Radar echoes i\, /2 received by the two antennas at the same time have a correlation coefficient y which depends on the endogenous correlation coefficient of clutter Yc and the clutter-to-noise power ratio CNR = Pc/Pn: thus the presence of noise reduces the clutter coherence. In practice, the interferometric phase is estimated by averaging L looks as follows: (3.32)

The corresponding PDF becomes:

(3.33)

where 2^i( # ) denotes the Gaussian hypergeometric function. For L = I equation (3.33) reduces to equation (3.31). Recently, a mathematical expression of the PDF has been presented in Reference 20 also for the case of target presence. In the following we illustrate some simulation results concerning the PDF of an interferogram in the presence of target, clutter and noise. The mathematical expression (3.33) plays an important role in finding the detection threshold for a prescribed value of false alarm probability (P/ fl ). By numerical integration the following numerical values for the threshold Th have been found: y = 0.9 -* Th = 1.0505; y = 0.98 -> Th = 0.3802; y = 0.99 -> Th = 0.2301; the prescribed false alarm probability, the clutter-to-noise power value and the number of averaged looks are: P/a = 10~4, CNR = 2OdB, L = 4. The variation of the detection threshold with the number of averaged looks for CNR = 2OdB, y = 0.98 and pfa = 10- 4 is as follows: L = 1 -> Th = 2.8414; L = 2 -> Th = 0.9505; L = 4^ Th = 0.3802;« = L -> Th = 0.3002; L = 10 -> Th = 0.2201. The target has been modelled as Swerling I. Figure 3.15 depicts the histogram of a phase interferogram for the following parameters: two antennas, L = 4, clutter Doppler phase = 0, CNR = 2OdB, y = 0.98, target Doppler phase: target = 1.3 rad. It is noted that when the signal-to-clutter power ratio (SCR) is very small in dB, the histogram coincides with the one predicted by equation (3.33); with the increase of the SCR the peak of the interferogram migrates on the target Doppler phase. Figure 3.16

Figure 3.15

Histogram of phase interferogram

SCR, dB

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cp, rad

Figure 3.16

Contour curves with constant Pd versus target Doppler phase and signal to clutter power ratio

illustrates the contour curves with constant detection probability versus target Doppler phase and SCR. The relevant parameters are: two antennas, CNR = 20 dB, number of averaged looks L = 4, correlation coefficient: y = 0.98, Pfa = 10~4 (detection threshold = 0.3802 rad). The following comments are in order: the contour curves are aliased in the interval cp = [—Tt9 Jt]; detection is practically zero for low values of the target Doppler phase because of the numerical value of the detection threshold; the detection increases with the increase of the SCR value. The next areas of research are in the field of removal of aliasing by using different values of carrier frequency [22]. In References 23 and 24 the possibility to do ATISAR with a one-bit coded SAR signal is demonstrated for moving target detection: this is another lively area of research; it has a bonus in terms of simplicity of processing implementation.

3.4.5 Processor in the space-time-frequency domain In this section a method for detecting and imaging objects moving on the ground and observed by an SAR is described. The method is based on the combination of two processing steps: space-time processing which exploits the motion of an antenna array for cancelling the echo from background, and time-frequency processing which exploits the difference in time allocation of the instantaneous spectrum corresponding to echoes from the ground or from moving objects, for an adaptive time-varying filtering and for the estimation of target echo instantaneous frequency, necessary for producing a focused image of it. The design and performance of the space-time

SCR, dB

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cp, rad

Figure 3.16

Contour curves with constant Pd versus target Doppler phase and signal to clutter power ratio

illustrates the contour curves with constant detection probability versus target Doppler phase and SCR. The relevant parameters are: two antennas, CNR = 20 dB, number of averaged looks L = 4, correlation coefficient: y = 0.98, Pfa = 10~4 (detection threshold = 0.3802 rad). The following comments are in order: the contour curves are aliased in the interval cp = [—Tt9 Jt]; detection is practically zero for low values of the target Doppler phase because of the numerical value of the detection threshold; the detection increases with the increase of the SCR value. The next areas of research are in the field of removal of aliasing by using different values of carrier frequency [22]. In References 23 and 24 the possibility to do ATISAR with a one-bit coded SAR signal is demonstrated for moving target detection: this is another lively area of research; it has a bonus in terms of simplicity of processing implementation.

3.4.5 Processor in the space-time-frequency domain In this section a method for detecting and imaging objects moving on the ground and observed by an SAR is described. The method is based on the combination of two processing steps: space-time processing which exploits the motion of an antenna array for cancelling the echo from background, and time-frequency processing which exploits the difference in time allocation of the instantaneous spectrum corresponding to echoes from the ground or from moving objects, for an adaptive time-varying filtering and for the estimation of target echo instantaneous frequency, necessary for producing a focused image of it. The design and performance of the space-time

filter is described in References 25-27 and 9. Some of the theoretical aspects of STAP of interest for this chapter are discussed also in a number of publications; Reference 11 discusses the optimum processing scheme for a special case and an approximation of it realised with a two-dimensional FIR (finite impulse response) filter. Other relevant References are 28-30. Here we concentrate on the time-frequency step (Section 3.4.5.1) and on the joint space-time-frequency (Section 3.4.5.2). 3.4.5.1 Joint time-frequency analysis by Wigner-Ville distribution (WVD) The aim of this section is to show how time-frequency representation by WVD of the echoes received by an SAR provides a useful tool for detection of moving objects and estimation of instantaneous phase shift induced by relative radar-object motion [27]. The phase history is then used to compensate for the received signal and to form a synthetic aperture with respect to the moving object, necessary to produce a high resolution image. A method for extracting the instantaneous phase is based on the time-frequency (TF) distribution of the received signal. The WVD has been chosen here because it presents some important features concerning detection and estimation issues. There are simple methods for analysing signals in the TF domain, such as the short-time Fourier transform (STFT), but they do not exhibit the same resolution capabilities in the TF domain as does the WVD. In particular, since the STFT is based on an FT applied to a time-windowed version of the signal, with the window central instant varying with time, the frequency resolution is inversely proportional to the window duration. The narrower the window, the better is the time resolution, but the worse is the frequency resolution and vice versa. Conversely, the WVD does not suffer from this shortcoming. The WVD provides a higher concentration of signal energy in the TF plane, around the curve of signal instantaneous frequency (IF). This allows a better estimation of IF in the presence of noise and this information is exploited for the synthesis of the long aperture with respect to a moving object. On the other hand, the WVD poses other problems since it is not a linear transformation. This causes the appearance of undesired cross-products when more than one signal is present. Mapping of the received signal in the TF plane provides a tool for the synthesis of the optimal receiver filter without a priori knowledge of the useful signal, provided that the signal-to-noise ratio is sufficiently large. The TF representation provides a unique tool for exploiting one of the most relevant differences between useful signals and disturbances in the imaging of small moving objects, namely the instantaneous frequency and the bandwidth. In fact, it can be shown that, although the bandwidth occupied by a target echo during the observation interval necessary to form the synthetic aperture mainly depends on radar-object motion, the instantaneous bandwidth is proportional to object size. Therefore, the echo corresponding to a small target can occupy a large band during the overall observation time, but its instantaneous bandwidth is considerably narrower (i.e. the echo backscattered by a point-like target has zero instantaneous bandwidth but it may exhibit a large overall bandwidth). Conversely, the echo from the background and the receiver noise have a large instantaneous bandwidth. Therefore, even if the useful signal and the disturbance may have a large total band, the possibility of tracking the instantaneous bandwidth,

made available by the TF representation, allows a discrimination of the useful signal from the disturbance not possible by conventional processing. Another important and unique advantage related to use of the WVD is that it allows the recovery of the echo phase history even in the case of undersampling, as shown in Reference 31. This is particularly important in SAR applications since it allows us to work with a PRF lower than the limits imposed by the signal bandwidth occupied during the observation interval. Owing to the target motion, this bandwidth may be considerably larger than the bandwidth occupied by the background echo. According to conventional processing, we should then use a correspondingly higher PRF. Conversely, if the useful signal has a large total bandwidth, but a narrow instantaneous bandwidth, the TF representation prevents superposition of spectrum replicas created by undersampling because, even if the replicas occupy the same bandwidth, they occur at different times. This property allows us to recover the desired information even from undersampled signals. Since the PRF value imposes a limit on the size of the monitored area, due to time - and then range - ambiguities, the possibility of using a low PRF prevents the reduction of the region to be imaged, as well as an increase of the data rate. The WVD of a signal is defined as: (3.34)

where s(t) represents the analytic signal. The estimation of the instantaneous frequency of the signal is done as follows. Express the signal in terms of its envelope and phase: s(t)=a(t)exp{j
(3.35)

It can be shown that the local or mean conditional frequency of the WVD distribution, defined as: (3.36) is equal to the signal IF: (3.37) The estimate of the mean conditional frequency of the WVD then provides the information about the signal IF. This is exactly the information we need for rephasing the received signal in order to produce a focused image. We will explain now the rationale for using the WVD to detect a moving target and estimate its parameters with SAR data. The received signal is the sum of echo from the moving object, whose phase modulation is unknown, plus clutter (the echo from fixed background), plus noise. The echoes from the fixed scene represent, in our case, a disturbance. The aim is to detect the presence of moving objects and to estimate their motion parameters. The detection and parameter estimation performance depend on the signal-to-clutter power ratio (SCR). If this ratio is small, we have to first process the received signal in order to improve it as much as possible.

This operation can be carried out by matched filtering. However, the matched filter can be defined only if the shape of the useful signal is known, and this is not the case. It turns out that the two operations, detection and parameter estimation, cannot be separated: estimation of the useful signal parameters can be carried out only after having detected the presence of a useful signal; a reliable detection, on the other hand, requires the knowledge of the signal parameters, in order to carry out a proper matched filtering, before the detection itself. It is then necessary to carry on these two kinds of operation contemporaneously. The time-frequency analysis of the received signal, in particular the WVD, provides a powerful tool for achieving the aforementioned requirements and extracting the desired information, namely the energy and the phase history. These two pieces of information are what we need for our purposes: the detection of the presence of a moving target is made by comparing the energy with a suitable threshold; the instantaneous phase is used to phase-compensate the received signal for a correct coherent integration, necessary to imaging purposes. As regards the effect of disturbances in the received signal, it is useful to recall that matched filtering can always be performed by cascading: a clutter cancellation followed by a coherent integration. The first operation does not require knowledge of the useful signal shape, whereas the second operation does. In particular, a correct coherent integration requires knowledge of the signal phase history. The time-frequency analysis aims to facilitate extraction of the signal phase history. Therefore, the sequence of operations to be performed on the received signals is the following: a clutter cancellation is performed first and the output of the cancellation filter is analysed in the TF domain. If the clutter has been reduced to a power level sufficiently smaller than the useful signal, the parameters estimated by the WVD are correct, within an error depending on the achieved SCR. These parameters provide the information necessary to set up a correct coherent integration. This rather intuitive reasoning for using the WVD in detection and estimation problems is formalised into the framework of the matched filter theory in Reference 27; here we give a summary of the main results concerning the application to SAR problems. The signal received by an SAR is given by the sum of the echoes from the ground, the echoes from a possible moving object and receiver thermal noise. The echo from the ground can be modelled as a correlated random process, whose power spectral density is proportional to the antenna power radiation pattern. The echo from a moving object is characterised by an unknown modulation, induced by the relative motion between the radar and the object. In SAR imaging, we are interested in the phase modulation induced by the relative radar-target motion. If the transmitted signal is: p(t)=a(t)Qxp(j27tf0t)

(3.38)

where a(t) is the analytic signal and /o is the carrier frequency, the echo received by a point-like scatterer at a distance r(t) from the radar is proportional to: (3.39) During the observation interval, the amplitude of each backscattering coefficient can be assumed constant since the aspect angle does not vary by an amount such as

to justify a change in the reflectivity characteristics (this assumption underlies all SAR signal processing). The only modulation of interest then is phase modulation. In the formation of the synthetic aperture, we are interested in the last term of the equation (3.39). The conventional techniques for autofocusing extended scenes compensate only linear and quadratic phase shifts. This means that only slow fluctuations (with respect to the integration time) are compensated. Conversely, the TF approach allows estimation and compensation of any kind of phase history. The distance r(t) can be modelled as the sum of slow and rapid fluctuations, with respect to the observation interval. Slow variations can be expressed as a low-order polynomial, whereas fast variations follow a sinusoidal behaviour, whose period is shorter than the time observation interval. The change in the distance causes a phase modulation, which must be compensated for to obtain a correct image. The variation of distance can cause migration of the received echo over different range cells, depending on the ratio between the amount of variation and the range resolution. The range migration problem can be faced according to the double resolution strategy outlined in the Section 3.4.5.2. According to the strategy, all the phase estimations are carried out on data whose range resolution is such as to consider the migration negligible. According to the formulation of matched filtering in terms of the WVD, the processing scheme for detecting and estimating the parameters of the echoes from moving objects is sketched in Figure 3.17. The received signal is first range compressed. Then it is processed by an MTI filter to reduce the clutter contribution; in Section 3.4.5.2 we explain how to use STAR The analysis of the signal in the TF domain allows estimation of

MTI

estimate instantaneous frequency

WVD

range compressed data

integration

generate reference signal

FFT

modulator

display

detection threshold

Figure 3.17

Processing scheme which uses the Wigner—Ville distribution

the signal instantaneous frequency. Some smoothing can be applied on the WVD to improve the estimation accuracy. The instantaneous frequency is then integrated to obtain the phase modulation to be used for phase compensation of the received signal. At this point an FFT is sufficient to provide the high cross-range resolution image. A threshold is then applied to the envelope signal to check for detection. An example of an image relative to a moving point-like target superimposed on a fixed extended scene is reported in Reference 27; the simulation results convey a positive feeling on the performance of the processing scheme of Figure 3.17. 3.4.5.2 Joint space-time-frequency analysis Space-time and time-frequency processing can be combined to form a processor which allows both the cancellation of the clutter echoes and the compensation of the target motion, necessary for the formation of a high cross-range resolution image. The overall space-time-frequency processing is shown in Figure 3.18. The echo from the ground is cancelled by means of space-time processing. Having cancelled the background echo, we proceed to the estimation of the target instantaneous frequency which is done by resorting to the WVD. Before proceeding to the estimation, however, some care must be devoted to the range migration problem. In fact, the relative radartarget motion causes not only a phase shift, but also a range migration which cannot be neglected if it overcomes the range resolution. The radar echoes are sampled and arranged in a matrix, whose columns are relative to successive transmitted pulses and the entries of each column contain the echoes coming from different distances. Given a certain point, if its distance (during the time on target) from the radar does not vary by an amount bigger than the size of a range resolution cell, the echoes from that point are all stored on one column. If all the points composing the observed scene satisfy this condition, no range migration occurs and the samples can be first processed in range,

adaptive space-time processor on range compressed data

corner turning

compute Wigner-Ville distribution estimate instantaneous frequency

FFT

generate reference signal

integration

Figure 3.18

Processing scheme of joint space-time-frequency

display

column by column, and then in cross range, row by row. If, however, the echoes from some point occupy more than one row, cross-range processing cannot be performed directly on the collected data, row by row. Some algorithm for compensating for range migration must be applied in order to realign the data, before cross-range processing. Range migration compensation techniques have already been examined in the literature in the imaging of stationary scenes. The problem is harder in the imaging of moving targets, however, since the radar-target distance is not known. Two main problems arise when dealing with moving targets, in presence of range migration: the migration causes a spreading of the target energy through many range cells, therefore the signal-to-noise ratio, for each range cell, decreases and this causes a loss of detectability, and due to the lack of knowledge of the target range migration law, it is not known a priori which samples of the matrix of the collected data are to be taken for carrying out the estimation of the phase history. Here we follow a two-step approach for compensating for the migration problem: a coarse range resolution, and a fine range resolution analysis (see Figure 3.19). The rationale of the approach is based on the fact that the migration problem can be neglected if the amount of migration does not exceed the range resolution. Therefore, we can initially work in a low-resolution mode, where the range resolution has been degraded by an amount such as to make the migration negligible. A first detection is performed on these data: if a detection occurs, the processing chain for the estimation of the target echo phase history is enabled. The estimation is carried out in the time-frequency domain, according to the criterion described previously. Having once estimated the phase 0(0, we can evaluate the law of variation of the distance by exploiting the relationship between phase shift and distance:

(3.40)

high-resolution channel

fine range alignment

compute distance

low-resolution range compression

full resolution compression

generate compensation signal

estimate instantaneous phase

Low-resolution channel

Figure 3.19

Two-step approach to account for range migration problem

By using d(t), we can compensate for the range migration by properly rearranging the data in the matrix containing the received samples. These data can now be phase shifted to compensate for the phase shift induced by the target motion. The data then undergo a full range and cross-range compression for obtaining a high-resolution image. The price paid by adopting the two-step resolution approach is the SNR loss inevitably related to the degradation of the range resolution in the coarse resolution mode. It is important to point out that, even if the cause of both phase shift and migration problems is the same, namely the variation of the radar-target distance, the phase shift is much more sensitive to range variation than to range migration. In fact, for example, in an X-band radar (X = 3 cm), having a range resolution of 1 m, an error on the distance of 7.5 mm causes a phase shift of n radians, while the corresponding range migration is absolutely irrelevant. This means that, having once estimated the phase history by the described procedure, the resulting value for the distance variation d(t) is estimated with a good accuracy. The performance of the space-time-frequency processing scheme has been evaluated by a simulation program which generates the echoes from an extended surface and from a point-like target and then applies the proposed algorithm [9]. The target has been supposed moving on the terrain (shadowing effects have been neglected) at a constant velocity, in a direction oblique with respect to the radar motion. The velocity parameters have been chosen to make evident the presence of range migration and of cross-range smearing of the target image. The ground reflectivity has been assumed equal to the target reflectivity (this is a quite pessimistic assumption, because in many cases of practical interest the target reflectivity is higher). The thermal noise, 4OdB below the target return, has also been assumed for the received signal. The ground echo is first cancelled, by using a STAP with a two-element antenna and two time samples. The two antennas are spaced by d = vT. An SAR image is then formed by conventional techniques. The result is shown in Figure 3.20 (from Figure 14 of

* * * *

range

Figure 3.20

Image of moving target, after clutter cancellation (reproduced with permission from Figure 14 of Reference 9)

range

Figure 3.21 Image of moving target, after range migration andphase compensation (reproduced with permission from Figure 15 of Reference 9) Reference 9). The smearing of the moving point-like target is evident. Given the motion parameters, the target has migrated over six range cells. This is the result of the broadening of the target image even in range, as well as in cross range. Anyway, the target echo causes a detection and initialises the motion estimation channel. The high resolution data are smoothed in range to decrease the range resolution. Then the processor looks for the range cell with the maximum energy content and computes the WVD of that cell only. The frequency history, and then the phase history, are evaluated, according to the procedure outlined previously. The phase history is then used for compensating for the range migration and the phase shift on the high-resolution range data. The final image is shown in Figure 3.21 (from Figure 15 of Reference 9). The sharpening of the target image is quite evident. The method above corrects only the (translation) motion of a point-like target. If a focused image of a three-dimensional object is required, for recognition purposes, the motion parameters must be estimated to compensate correctly for the relative signals [32, 33]. Since the motion of any rigid body can be decomposed as the sum of the translation of a point plus the rotation of the other points around that point, in some cases, depending on the target dimension and kind of motion, compensation for only the translation motion is not sufficient and we still have to compensate for the rotational motion. This second compensation can be efficiently performed in the frequency domain, by applying, for example, the method proposed in Reference 32. Algorithms have been conceived to produce fine resolution images of moving targets having any translation or rotation motions. They require the presence of multiple prominent points in the target image. The echo from a first point is initially analysed. Its phase is computed and subtracted pulse-to-pulse from the phase of the incoming signal. This operation removes the effect of the target translation motion and makes this first point effectively the new centre of the scene. At this point, if the rotational motion is negligible, conventional SAR processing yields the focused image. If,

conversely, the rotation cannot be neglected, two other prominent points are required to estimate the rotation parameters. These parameters are used to compensate for the rotational motion in the frequency domain and to help in applying the SAR processing correctly. The algorithm requires that the prominent points be separable and that their phases be estimated without interfering with each other. In some cases, however, this assumption might not be met: at high resolution, separability is more likely to occur, but the range migration could complicate the phase estimation problem; at low resolution, the range migration could be negligible, but it is more likely that some prominent points might occupy the same range resolution cell. Further analyses are necessary in the imaging of extended targets when more dominant scatterers occupy the same resolution cell. In this case, in fact, the bilinearity of the WVD creates undesired cross-product terms which can seriously impair the estimation of the instantaneous frequency. A reduction of these undesired terms could be achieved by resorting to other time-frequency representations, such as the Choi-Williams distribution [34]. Another approach to reduce the appearance of undesired crossproduct terms is described in Reference 35. It is shown that it is still possible to estimate the signal parameters, even for multicomponent chirp signals embedded in noise, by combining WVD with the Hough transform.

3.4.6

Optimum processing for MSAR

This section focuses on the optimum MSAR detection of slowly moving targets (SMT), whose Doppler frequency is well inside the clutter spectrum, by means of space-time processing [4, Chapters 4 and H]. Clutter cancellation is easily performed by space-time processing assuming the idealised condition of DPCA condition (PRF matched to the ratio between antenna displacement and platform speed) and absence of the intrinsic temporal decorrelation of the clutter echoes, known as internal clutter motion (ICM). However, it is well known that the cancellation can be highly affected by the presence of ICM which has the strongest impact on the detection of the SMT, where the MSAR provides a large performance improvement over conventional multichannel systems. Moreover, the exact matching of DPCA condition is difficult to achieve in practice. Therefore, to obtain a significant analysis of its effectiveness we consider separately two cases of mismatch from the idealised conditions: (i) the presence of ICM and (ii) the velocity mismatch. A complete analysis is derived for a special case [10,13]. A space-time exponential model is considered for the clutter covariance matrix, related to a Cauchy antenna pattern (i.e. antenna gain G(O) = Go • (O2 + 0J))"1, 0 being the azimuth angle, #o a parameter that controls the beamwidth and Go a suitable coefficient to encode the antenna gain) which yields a double exponential Doppler spectrum for the fixed clutter and a single pole description of the internal motion process. The simplified model allows closed form expressions to be identified, both for optimum detector and detection performance. In the following, we first consider the different possibilities for ordering the long sequence of MSAR echoes, then we discuss the special structure of the clutter covariance matrix in the two above mentioned mismatch conditions. The special structure

allows us to obtain a formal expression for the space-time cancellation filter and its properties, and finally to discuss its performance. 3.4.6.1 Echoes ordering strategy Initially, we assume, as in Reference 36, that the DPCA condition is verified. With reference to the slant plane geometry in Figure 3.22, and assuming that: (i) the effect of the antenna pattern modulation on the target echoes in the observation time is negligible, and (ii) the far field approximation can be used for the antenna element displacements and the Fresnel approximation for the distance of the target from the transmitter, the target echo received by the A:th (k = 0 , . . . , K — I5) element at Xn = (n-(N - l)/2) PRT, n = 0 , . . . , N - 1 can be rewritten as:

(3.41) where Ao is the amplitude of the echo, O is a constant phase term and A the carrier wavelength. xq = xq/(vPRT) is the normalised target along-track position, fy = 2vrPRT/X is the normalised Doppler frequency due to the cross-track velocity component and sq = vq/v is the relative along-track velocity. Also, d — A/(vPRT) is the ratio between spatial and temporal sampling, being A the antenna element separation, and /x = 2(vPRT)2/(XRo) is the rate of the linear frequency sweep for

time

along-track spatial axis

Figure 3.22

Bidimensional space-time data collection plane (K = 4, N = 10)

a fixed target. The terms in the first line of equation (3.41) take into account the central target Doppler frequency and DoA. They are the usual model for space-time processing techniques based on a short integration time. The terms in the second and third lines are characteristic of the SAR and encode the time-varying Doppler and DoA observed in the long SAR integration time. Assuming that the clutter echoes can be modelled as a zero mean Gaussian random process, derived from a spatially independent Gaussian distributed scene reflectivity (see Section 3.3.2.1) with temporal correlation (ICM) pt(At), the correlation of the echoes received at times ^i, t2 by elements ^ i , Ski is: (3.42)

where Pc is the total clutter power. Since the NxK echoes arise from the space-time sampling, they are by their nature bidimensional, and have not any natural unidimensional order. This is illustrated in Figure 3.22, where the space-time location of the phase centres relative to the combination of transmitting and receiving antenna for each echo is sketched. As in Reference 36, the DPCA condition is assumed, namely PRT and platform speed are matched so that d = 2. It is useful to arrange these echoes in a vector, which is an ordered sequence of blocks s = [sj, sj9... ]T. Three strategies can be devised to build the blocks [11], keeping fixed a parameter, which correspond to the three directions in which the points in Figure 3.22 can be aligned: (A)

fixed-time blocks (Figure 3.23a):

(B)

fixed element blocks (Figure 3.23b):

(C) fixed spatial position blocks (Figure 3.23c):

The rotation matrices T ^ , Tca and Tcb are defined, to convert different ordering strategies: SB = T ^ S A , SC = TCflSA, Sc = T ^ S B - It is worth recalling that all the elements of the matrices above are zero except for one element equal to one in each row and column and have the properties T ^ T ^ = T ^ T ^ = I, T ^ T ^ = T^T Cfl = I, TcfcT^ = TlbTcb = I and Tcb = TcaTla, being I the NK x NK identity matrix (see also Reference 30).

Figure 3.23

Strategies for collecting and processing the SAR echoes

3.4.6.2 Covariance matrix properties: a special case for ICM An interesting condition, which allows us to study the detection performance and the interpretation of the optimum detection structure, is obtained under the hypothesis that the transmitting antenna pattern has a Cauchy shape [H]. Antenna sidelobes are not taken into account but this is not essential here, since targets competing with the mainlobe clutter are considered. Thus an exponential spatial correlation is obtained: ps(i) = p[l\ where ps is the correlation coefficient at a spatial distance equal to half the inter-element spacing. The worst case arises when the temporal correlation, which encodes the ICM, also has an exponential shape, pt(i) = p\ , being pt the one-lag correlation coefficient, since it corresponds to the maximum entropy spectrum. With reference to ordering strategy (A), the normalised covariance matrix is symmetric, due to the broadside antenna pointing and has a Toeplitz-block-Toeplitz structure:

(3.43)

where the K x K block P ; is given by:

(3.44)

For the interpretation of the optimum detector, this case is especially appealing, since the covariance matrix in equation (3.44) has the properties below. We define the KxK matrices H and H / :

(3.45)

which are related via a rotation around their centre, and are apparently not full rank. Under the hypotheses above, the blocks P1- can be written as the product of the ith power of one of the matrices times the first block Po as P; = FTPo = Po(H7*) . Thus RA = TA diagyyjPo,..., Po), where:

(3.46)

and diag^fPo,..., Po) is a block diagonal matrix, with TV blocks Po- It can be shown that FA can be factored as

(3.47)

where the blocks Q; are square matrices of order /, with exponentially decaying elements with one-lag correlation coefficient pt and U is a N x N upper triangular matrix, obtained from an exponential square matrix, with one-lag correlation coefficient ps, pt, nulling its lower diagonal part. From equations (3.46) and (3.47) and from the symmetry of RA we have R^ 1 = diag^fP^ 1 ,..., P^ 1 }T~^1

(3.48)

Moreover, using HH'H = H and H H H ; = H', it can be shown that the inverse matrix F^ 1 is block tridiagonal. Thus, R^ 1 is block tridiagonal and symmetric:

(3.49)

Moreover each of its blocks is itself a tridiagonal band matrix: Y = (1 — P^)P 0 1 + P^P-'CHH' + H'H); Z = P Q 1 H ; Y 0 = P^ 1 + p^1HH; % = Y~x + p , 2 P ^ H H ' . Moreover, the decomposition R^ = LeLg applies, where:

(3.50) and blocks

(3.51)

Interpretations of the optimum detector Assume that the data vector x (NK x 1) is ordered according to strategy (A). As well known, the optimum detector for the target vector s, known aside for amplitude and initial phase, against a Gaussian clutter with covariance matrix RA, is obtained comparing the modulus of s ^ R ^ x to a threshold. Since we focus on the detection of SMT, whose power spectrum competes with the main clutter return, thermal noise is neglected, assuming the ICM to be the dominant disturbance. Using the decomposition of the matrix RA, the following interpretations of the optimum filter s^R^ x

data in space only filter a rotation unidim. space-time filter b rotation time only filter c rotation match to signal

Figure 3.24

Decomposition of the optimum detector in a sequence of three uni-dimensional filters

yield [H]: (i)

The bidimensional space-time cancellation filter in the optimum detector, corresponding to R^ l , can be thought of as the cascade of three mono-dimensional filters along the three data ordering directions. This follows from the decomposition in equation (3.48) for R^ 1 , which can be implemented as depicted in Figure 3.24: first the matrix diag^fP^" 1 ,... ,P^"1} is applied to the data, ordered with strategy (A). It operates separately on its blocks, performing a spatial filtering only. The data are then rearranged by T ^ into ordering (B), where the filter diag^{U~ lr , I , . . . , I, U" 1 } along the diagonal of the space-time plane is performed. Thereafter, the filtered data are rotated to ordering (C), where the pure temporal filtering B d i a g A ^ + ^ _ i ( Q ^ 5 Q - 1 , . . . , Q " 1 , . . . , Q " 1 , . . . ,Q^ 1 ,Q^ 1 J is applied. Finally, the filtered data are rotated back to the initial ordering; this can be avoided by storing s with strategy (C). Thus, the three ordering strategies combine in the detector. It is instructive to observe that the filtering along the diagonal of the space-time plane affects only the data in the first and last antenna elements, corresponding to a border effect only. It is required to compensate for the absence of the adjacent element samples. This is interesting, since the selected model is a separable random Markov field, for which in theory it should be possible to split the filter into the cascade of two monodimensional filters. Unfortunately, the data are collected on an area of this plane, which is not aligned with the main correlation axes. Thus the physical sampling process induces in the optimum filter a coupling of the two directions even in this ideal case of decoupled exponential correlations, (ii) The filtering operated in R^ x is a shift invariant space-time FIR filter with nine taps, which uses the nine echoes collected at the nine most adjacent space and time positions, except for the border effects. This is a consequence of the block-tridiagonal structure of the R^ 1 , with blocks being three-element band matrices, as outlined in equation (3.49). Thus the inverse covariance matrix operates combining the data received at a given

time and element with only the data received one element and one PRT apart, as illustrated in Figure 3.23d (a). It is to be noted that the shift invariance is not available for the first and last blocks, where border effects must be compensated for. Moreover, the filtering is non-causal, since both preceding and following samples are used even though this is not an issue for the FIR characteristic, (iii) S77R^1X is obtained filtering the data with a shift invariant causal space-time filter with four taps, which uses only the data already received at the closest space-time sampling positions, matching them to an equally filtered reference signal. This is a similar interpretation to (ii), and derives from the structure of the decomposition in equations (3.50) and (3.51): s ^ R ^ x = (LgS)77LgX. This is illustrated in Figure 3.23d (fi). This one-lag space-time memory is obviously a consequence of the double exponential model, which can be regarded as being a worst case analysis. In fact, the exponential autocorrelation is the one characterised by the maximum entropy, once the one-lag decorrelation is constrained. Specifically, using only adjacent space-time samples prevents the whole length of the receiving array from being used to obtain a narrow null along the clutter ridge. The width of the null is instead controlled by the interelement spacing. The three interpretations are of high interest for the long sequence of SAR echoes, since the batch processing is replaced with a space-time FIR filter, with a major reduction in the computations. In the general correlation case the smallest rectangular region around each space-time sample is not a sufficient support for the optimum filter. 3.4.6.3 Covariance matrix properties: a special case for velocity mismatch Using the exponential spatial autocorrelation function, another case of departure from the ideal conditions can also be analysed in closed form: the presence of a mismatch between the radar platform velocity and the PRF, so that the DPCA condition is not any more exactly matched. We assume, for the sake of simplicity, the absence of internal clutter motion and a small mismatch. Under these conditions, the ordering strategies above can still be used with obvious extensions from the case of perfect DPCA. Assuming that the velocity mismatch is a fraction y of the theoretical velocity, we have for the actual one-lag correlation coefficient pse = ps • pe, where pe = pYs. Under the hypothesis that \K • y\ < 1, namely that the mismatch is smaller than the antenna spacing, using ordering strategy (C), the covariance matrix Rc takes the form:

(3.52)

time

along-track spatial axis

Figure 3.25

Bidimensional space-time data collection plane (K = 4, N = 10). SAR echoes selection strategy in the presence of velocity mismatch, v ^> 0

where the KxK

block M / is given by:

(3.53)

for y > 0 (see Figure 3.25) and by its transpose for y < 0 (see Figure 3.26). It is worth observing that the matrix M/ is not just obtained from the matrix Qh by replacing pt with pe. In particular, the elements of M/ can have modules larger than one, since the velocity error causes either reduction or increase to the ideal spatial correlation, depending on the positions of the elements. This was not the case in the presence of internal clutter motion, which was always decreasing the correlation between samples. The structure of the matrices in equations (3.52) and (3.53) is evident from the sketch of observation geometry and correlations in Figures 3.25 and 3.26. In the following we proceed with reference to the case of y > 0, which can be extended to the case y < 0. To get a decomposition of the covariance matrix similar to the one obtained for the internal clutter motion [10], we define the K x K

along-track spatial axis

D

Figure 3.26

Bidimensional space-time data collection plane (K = 4, N = 1O)SAR echoes selection strategy in the presence of velocity mismatch, y < 0

matrices He and Hfe:

(3.54)

which are related via a rotation around their centre; namely it yields the property H'e — &KHe&K, where $ „ is the (« x n)-dimensional rotation matrix:

(3.55)

Thus the ith block can be rewritten as M1- = pleHeVe = p^VeH^ = pe(UfeVe)T, being Ve a K x K exponential matrix with one-lag correlation coefficient pe. Thus the covariance matrix can be decomposed as Rc = Fc • diag^{P^,..., Ve}, and:

(3.56)

The special structure of the matrix Fc yields a closed-form inverse as a block tridiagonal matrix. By setting be = pse/{\ — p2sepj ), the inverse covariance matrix R^ 1 is also block tridiagonal:

(3.57)

where Ye0 = V;1+bepsep-K+l?JlUe*K;Y'eo = Vjl+bepsepJK+lVjl UfeK = 1 l K+l 1 1 *KYeo*K; Ze = PJ H*; \ e = V~ + bePsepJ (VJ H6 + VJ H.'e)*K are themselves tridiagonal matrices. Moreover, a triangular block decomposition applies, similar to equations (3.50) and (3.51). These properties of the covariance matrix Rc with ordering strategy (C) - obtained in the case of DPC A velocity mismatch, but in the absence of ICM - yield an optimum detector with very similar interpretations to the case of covariance matrix RA with ordering strategy (A) - obtained in the case of matched DPCA speed and presence of ICM: (i)

The bidimensional space-time cancellation filter in the optimum detector, corresponding to R^ 1 , can be thought of as the cascade of three mono-dimensional filters along the three data ordering directions. (ii) The filtering operated in R^ x is a shift invariant space-time FIR filter with nine taps, which uses the nine echoes collected at the nine most adjacent space and time positions, except for the border effects. (iii) s^R^ x is obtained filtering the data with a shift invariant causal space-time filter with four taps, which uses only the data already received at the closest space-time sampling positions, and matching them to an equally filtered reference signal. Unfortunately, it is to be noticed also that the structure of the clutter covariance matrix does not maintain a similar pattern when both internal clutter motion and velocity mismatch are present. In particular, its inverse is no longer a block-tridiagonal matrix. Therefore these exact closed-form interpretations do not strictly apply. However, a

suboptimum detection scheme can be obtained by extending such a support region, so as to maintain the FIR characteristic, while achieving detection performance close to the optimum. It is interesting to compare the scheme above with those derived in References 7 and 8, which have been the first to the author's best knowledge to introduce FIR filters for SAR detection. Those filters are similar to the one in (iii). The main difference is that in References 7 and 8 the filter LB is derived by using the criterion of the minimum mean square error for clutter cancellation, whereas here it is derived, for a special case, from a maximisation of the detection probability. This study of a special case allows the interpretation of the optimum detector in closed form. This shows that the scheme of References 7 and 8 has the same structure as that of the optimum filter. It is also important to recall that for long integration times (large covariance matrix), the DFT matrix gets very close to the matrix of the eigenvectors, of the clutter covariance matrix, so that the clutter sample at the different frequencies tends to have a low correlation. In this case, the Doppler processing is also very close to the optimum filter, so that both classes of STAP techniques for the MSAR case: (i) time domain reduced DoF (degrees of freedom) adaptive two-dimensional FIR filters (ii) frequency domain reduced DoF adaptive processing, can be interpreted as low-cost approximations of the optimum STAP filter for MSAR. 3.4.6.4 Performance analysis Thanks to the simple form of the inverse covariance matrix the signal-to-clutter ratio of the optimum filter SCR = S^R -1 S can be evaluated in closed form. The analysis is shown with reference to the case of the presence of ICM [H]. For simplicity, only the condition with xq = sq = O (see Figure 3.2) is considered, this is a broadside target with no along-track velocity component:

(3.58) where /3= sin(n\iN)/(N sin(nfi)) and it has been considered that sm(nfjiK)/ (K sin(niJi)) & 1, assuming N ^$> K. The SCR is shown as a function of the normalised Doppler frequency fy in Figure 3.27 for different integration times corresponding to N = 128, 256 and 512 samples, K = A and setting AQ/PC = 1. Three temporal correlation coefficients pt = 0.9,0.99 and 0.999 and a fixed ps = 0.5 are considered. The figure shows that pt has a strong influence on the SCR, which grows with it at almost all Doppler frequencies. However, assuming equal total clutter power, an opposite behaviour appears for the stationary targets, since some clutter power spreads away from the diagonal of the space-time plane and a fixed target

SCR, dB

Figure 3.27

Optimum SCR as a function of fN: (*)p, = 0.999, (+)p r = 0.99, (o) pt = 0.9, ( )N = 512, ( - - -)N = 256, {....)N = 128

very close to this diagonal competes with a reduced clutter power. It also appears, from both equation (3.58) and Figure 3.27, that the SCR grows with the integration time N. In fact, only one adjacent temporal and spatial sample is necessary to whiten the clutter spectrum, as required by the optimum detector. The whitening implies also a partial target cancellation, which strongly affects the performance especially in the SMT region (see the sinusoidal term of equation (3.58)). The longer sample sequence of the SAR allows the coherent integration to gain in SCR over the whitened clutter. This extra gain can be essential in the region of the SMT, where the SCR is low, to permit detection. As a final consideration, it is interesting to observe that both the scheme of References 7 and 8 and the scheme of Reference 12 share the same structure for the optimum filter in this special case, and have in general also very similar detection performance and are therefore attractive for the practical implementation.

3.5

Conclusions

In this chapter processing techniques have been described to combine the SAR and STAP functions; the goal is to obtain a well focused target image in the right place on the image of the stationary scene. A taxonomy of different processing schemes has been presented, with different characteristics in terms of computational cost,

detection performance and operational limitations. Among the presented techniques, some are recognised to be better suited to providing a proof of concept (DPCA, ATI-SAR), while others can be only exploited for interpretation purposes (optimum filter). However, some approaches are recognised as being appropriate for practical use. Two classes belong to this group of techniques: (i) the cascade of a space-time FIR filter and a two-dimensional filter bank detector (space-time-frequency approach and approximation of optimum filter), and (ii) the processing in a confined Doppler plane (Doppler processing and MSAR JDL). As considered in detail in Section 3.4.6, these two schemes share the same structure as the optimum filter in special cases, and have in general also very similar detection performance, resulting therefore in being attractive for the practical implementation. When imaging is a main issue the use of time-frequency analysis combined with the STAP techniques appears to be applicable to the case of a very general target motion trajectory and to give good detection and imaging performance.

3.6

Acknowledgments

Moving target detection and imaging with SAR has been the subject of R&D work for a number of years in Selenia/Alenia (now AMS). The authors acknowledge with thanks the cooperation of a number of colleagues, notably Professor S. Barbarossa (today with the University of Rome 'La Sapienza') and Dr. E. D'Addio. More recently Dr. A. Gabrielli (AMS) has cooperated in the development of ATI-SAR work.

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8 KLEMM, R. and ENDER, J.: 'Two-dimensional filters for radar and sonar applications'. Signal Processing V EUSIPCO, September 1990, Barcelona, Elsevier Science Publisher, pp. 2023-2026 9 BARBAROSSA, S. and FARINA, A.: 'Space-time-frequency processing of synthetic aperture radar signals', IEEE Trans. Aerosp. Electron. Syst, April 1994, AES-30, (2), pp. 341-358 10 LOMBARDO, P.: 'DPCA processing for SAR moving target detection in the presence of internal clutter motion and velocity mismatch'. Microwave sensing and synthetic aperture radar, EUROPTO series, 2958, September 1996, Taormina, Italy, pp. 50-61 11 LOMBARDO, P.: 'Optimum multichannel SAR detection of slowly moving targets in the presence of internal clutter motion'. CIE-ICR '96, international Radar conference, Beijing, China, 8-11 October, 1996, pp. 321-325 12 ENDER, J. H. G.: 'Space-time processing for multichannel synthetic aperture radar', Electron. Commun. Eng. J. (Special issue on STAP), February 1999 11, (1), pp. 29-40 13 LOMBARDO, P.: 'Echoes covariance modelling for SAR along-track interferometry', IEEE international symposium IGARSS '96, Lincoln, Nebraska, USA, May 1996, pp. 347-349 14 LOMBARDO, P.: 'Estimation of target motion parameters from dual-channel SAR echoes via Time-Frequency analysis'. IEEE national Radar conference, NatRad'97, Syracuse, NY, May 1997, pp. 13-18 15 LOMBARDO, P.: 'A joint domain localized processor for SAR target detection' . Presented at European conference on Synthetic aperture radar, EUSAR'98, Friedrichshafen, Germany, 25-27 May 1998, pp. 263-266 16 WANG, H. and CAI, L.: 'On adaptive spatial-temporal processing for airborne surveillance radar systems', IEEE Trans. Aerosp. Electron. Syst, July 1994, 30, (3), pp. 660-670 17 'IEEE AESS 1991 award to F. R. DICKEY, F. M. STAUDAHER, and M. LABITT', IEEE Aerosp. Electron. Syst. Mag., 32, May 1991, p. 32 18 SHNITKIN, H.: 'Joint STARS phased-array radar antenna', IEEE Aerosp. Electron. Syst. Mag., October 1994, pp. 3 4 ^ 1 19 COE, D. J. and WHITE, R. G.: 'Experimental moving target detection results from a three-beam airborne SAR'. Eusar 96, Konigswinter, Germany, 1996, pp. 419-422 20 GIERULL, C : 'Moving target detection with along-track SAR interferometry'. DREO Technical Report 2002-000, 09 January 2002 21 LEE, J. S., HOPPEL, K. W., MANGO, S. A., and MILLER, A. R.: 'Intensity and phase statistics of multilook polarimatric and interferometric SAR imagery', IEEE Trans. Geosci. Remote Sens., 1994, 5, pp. 1017-1028 22 PASCAZIO, V. and SCHIRINZI, G.: 'Estimation of terrain elevation by multifrequency interferometric wide band SAR data', IEEE Signal Process. Lett., 2001, 8, pp. 7-9 23 PASCAZIO, V., SCHIRINZI, G., and FARINA, A.: 'Along track interferometry by one bit coded SAR signals', in POSA, F. and GVERRIERS, L. (eds): SAR Image Analysis, Modeling and Techniques III, SPIE, 2000, 4173, pp. 259-266

24 PASCAZIO, V., SCHIRINZI, G., and FARINA, A.: 'Moving target detection by along track interferometry'. Proceedings of IGARSS 2001, 1 Sidney, Australia, July 2001, pp. 3024-3026 25 BARBAROSSA, S. and FARINA, A.: 'A novel procedure for detection and focusing moving objects with SAR based on the Wigner-Ville distribution'. Proceedings IEEE international Radar conference, Arlington, VA, May 7-10, 1990, pp. 44-50 26 BARBAROSSA, S. and FARINA, A.: 'Detection and imaging of moving objects with SAR by a joint space-time-frequency processing'. Proceedings of the Chinese international conference on Radar, Beijing, China, October 22-24,1991, pp. 307-311 27 BARBAROSSA, S. and FARINA, A.: 'Detection and imaging of moving objects with synthetic aperture radar. Part 2: Joint time-frequency analysis by Wigner-Ville distribution', IEE Proc. F, Radar Signal Process., 1992, 139, (1), pp. 89-97 28 FARINA, A. and TIMMONERI, L. 'Space-time processing for AEW radar'. Proceedings of IEE international conference on Radar 92, Brighton, UK, 12-13 October 1992, pp. 312-315 29 LOMBARDO, P. and FARINA, A.: 'Dual antenna baseline optimisation for SAR detection of moving targets'. ICSP - international conference on Signal processing, Beijing, China, 14-18 October 1996, pp. 431^34 30 LOMBARDO, P.: 'Data selection strategies for radar space time adaptive processing'. Presented at IEEE Radar conference, Radar'98, Dallas, Texas, USA, May 12-13, 1998, pp. 201-206 31 BARBAROSSA, S.: 'Parameter estimation of undersampled signals by WignerVille analysis'. IEEE international conference on Acoustics, speech and signal Processing, ICASSP '91, Toronto, May, 1991, pp. 3253-3256 32 WERNESS, S., CARRARA, W., JOYCE, L., and FRANCZAK, D.: 'Moving target imaging algorithm for SAR data', IEEE Trans. Aerosp. Electron. SySt., 1990, AES-26, (1), pp. 57-67 33 FIENUP, J. R.: 'Detection of moving targets in SAR imagery by focusing', IEEE Trans. Aerosp. Electron. SySt9 July 2001 AES-37, (3), pp. 794-809 34 COHEN, L.: 'Time-frequency distributions - a review', Proc. IEEE, 1989 77, (7), pp. 941-981 35 BARBAROSSA, S. and ZANALDA, A.: 'A combined Wigner-Ville and Hough transform for cross terms suppression and optimal detection and parameter estimation'. Proceedings of the IEEE international conference on Acoustics, speech and signal processing, ICASSP '92, March 1992, San Francisco, pp. 173-176 36 BARILE, E. C , FANTE, R. L., and TORRES, J. A.: 'Some limitations on the effectiveness of airborne adaptive radar', IEEE Trans. Aerosp. Electron. Syst, October 1992, AES-28, (4), pp. 1025-1032

Chapter 4

EA-STAP: an efficient, affordable approach for clutter suppression Hong Wang, Richard A. Schneible, Russell D. Brown and Yuhong Zhang

We now focus on adaptive clutter suppression with sum (E) beam and difference (A) beams. It is assumed for this chapter that spatial adaptive presuppression of jammers has been applied as necessary prior to STAR The most well known applications of A-beams are monopulse tracking [1] and platform motion compensation [2], with the latter being also called the EAimplementation of displaced phase centre antennas (DPCA). Griffiths [3] uses E A-beams for wideband adaptive beamforming of non-pulsed systems. Brown et al. [4, 5] propose STAP with E A-beams for clutter suppression of airborne surveillance systems, which is called EA-STAP. The most important feature of E A-STAP stems from the fact that antenna engineers have excelled in the design of low-sidelobe E A-beams, whether it is for a phased array or reflector-feed antenna. In fact, analogue beamforming techniques for E A-beams are so well developed that the more expensive digital beamforming method for E A-beams does not seem necessary. In other words, STAP with E A-beams may well be an affordable approach to high-performance airborne surveillance radars.

4.1

Definition of the difference (A) beams

Consider a planar aperture of either a phased array or a reflector-feed antenna which has, in addition to the sum-beam channel, at least one more receiver channel. Let G(0a, 0e) be the response pattern of the E-beam with 0a and 0e being the azimuth and elevation angles, respectively. Let F(0a, 0e) be the response pattern of the second channel. For every given steering angle (Oao,Oeo) of the E-beam, we assume: G(OaO9Oe0) = maxG(0a,0e)

(4.1)

If correspondingly we have for any 6e: F(9a0,ee) = 0

(4.2)

the second channel is said to produce an azimuth difference beam to be denoted as Aa. Similarly, an elevation difference beam, Ae, is defined by: F(O09OeO) = O

(4.3)

and an azimuth-elevation difference beam, Aae, by: F(OaO9Se) = O and

F(O09Oe0) = 0

(4.4)

The above definition is much idealised and simplified only to preserve the most basic aspect of the A-beams for signal processing oriented readers. More general descriptions can be found in antenna books such as Reference 6. Figure 4.1 shows an example of Aa-9 Ae- and Afle-beams along with the E-beam, where the steering (E-beam pointing) angle is assumed at (Oao9 6eo) = (O90). Usually, all three A-beams can be obtained simultaneously from a single aperture, resulting in a system with four receiver channels.

beam norm, gain, dB

norm, gain, dB

beam

I Figure 4.1

A ae -beam norm, gain, dB

Ae-beam PQ

An example of E- and A-beams of a planar array with tapering: H-azimuth 35 dB Taylor, Y-elevation of 25 dB Taylor, A-azimuth of 30 dB Bayliss, and A-elevation of 25 dB Bayliss

Z-beam A-beam Taylor(35)/Bayliss(30) normalised antenna pattern, dB

Nc=l6

sin 6

Figure 4.2

An example of null-peak aligned H A-beams with tapering: 35 dB Taylor for E and 3OdB Baylissfor A

Figures 4.2 and 4.3 are two examples of different A^-beams with their respective E-beams, where the idealised planar array is assumed to have fixed column combiners and thus does not have Ae- or Aae-beams. The purpose of these two Figures is to show the null alignment of the E- and A-beams of different tapering without any array error, which affects the performance of EA-STAP.

4.2

XA-STAP algorithms

Figure 4.4 illustrates the block diagram of general EA-STAP where the dimension of the processor's spatial DoF, Nps, is determined by the number of A-beams. Under the assumption of no redundant A-beams, Nps is equal to the number of A-beams. Much of this block diagram follows the general STAP configuration. Let Xs, Nt x 1, be the E-channel data of a range cell before temporal DoF reduction, where Nt is the number of the pulses in the CPI. Let XA, NpsNt x 1, be the stacked A-channel data of the same range cell before temporal DoF reduction. We can express the data after DoF reduction as: (4.5) (4.6)

normalised antenna pattern, dB

Z-beam A-beam Hanning/full-cycle sine Nc=16

Figure 4.3 An example of null-aligned T^A-beams with tapering: Hanningfor E and full-cycle sine for A where T, Nt x (Npt +1), is the temporal DoF reduction matrix, and Npt represents the processor's temporal DoF whose specification is determined by the clutter suppression need and the available sample support, and I(^W is the Nps by Nps identity matrix. The estimated correlation matrix can be written as: (4.7) where (4.8)

(4.9)

(4.10)

(4.11) with y-£,k, yAk, k = 1,2,3,..

.,K being the selected/conditioned samples.

From phased array beamformer or reflector antenna with X-A feed system Z (sum) channel

A (difference) channel(s)

receiver and A/D

receiver and A/D

temporal DoF reduction

temporal DoF reduction

joint space-time domain adaptive processing with Nps=no. of A-channels and Npt = the required and supportable temporal DoF

clutter suppressed outputs at all Doppler bins and all range cells for further processing

Figure 4.4

Block diagram of general E A-STAP where Nps (Npt) is the processor s spatial (temporal) DoF

Let S/, Nt x 1, be the temporal steering vector of a chosen Doppler bin, and denote: (4.12) (4.13) and (4.14) For EA-STAP, the test statistic is computed using modified sample matrix inversion (MSMI) or generalised likelihood ratio (GLR). The MSMI and GLR tests are, respectively: (4.15)

and (4.16) where the filtering weight vector, (Nps + l)(Npt + 1) x 1, is given by: (4.17) We note that equations (4.15) and (4.16) differ from the general MSMI and GLR expressions only because of the deep central nulls of the A-beams. As an example we now consider a typical S A-STAP algorithm of Figure 4.5 where a single A-beam is employed and the temporal DoF reduction is the Doppler-domain localised processing approach (DDL). Under the assumption of no zero padding and a uniform pulse train, the matrix T in equations (4.5) and (4.6) is just the standard DFT From phased array beamformer or reflector antenna with Ir-A feed system E (sum) channel

A (difference) channel

receiver and A/D

receiver and A/D

1-D DFT of min. Nt points

1-D DFT of min. Nt points

modified SMI or GLR algorithm of 6th order (as shown) for theyth Doppler bin with Nps = 1 and Npt = 2

clutter suppressed outputs at they'th Doppler bin and all range cells for further processing

Figure 4.5

Block diagram of a typical HA-STAP with a single A-channel and DDL(2)

matrix for Npt + 1 frequency bins, and st of equation (4.12) becomes a vector of Npt zeros centred around the only non-zero entry of 1.0. Obviously, further computation reduction can be achieved with such st in equation (4.15) or (4.16). This algorithm with Npt = 2 has been applied to measured airborne radar data [4] and the results are presented in section 4.4.

4.3 4 J.I

Analytical performance formulas of EA-STAP SINR potential

The SES[R potential is defined as the output SINR of the filtering portion of the STAP when the correlation matrix is exactly known. The filtering weight vector for the known correlation matrix Q is:

(4.18)

(4.19)

(4.20)

(4.21) (4-22) (4.23) (4.24)

(4.25)

with (4.26) (4.27) (4.28) (4.29) The SINR potential can be found by: (4.30) (4.31) where Ps is the power of the input signal.

4.3.2 Probabilities of detection andfalse alarm 4.3.2.1 MSMI The test of equation (4.15) leads to the probability of false alarm [7]: (4.32) where (4.33) (4.34) (4.35) The probability of detection is given by: (4.36) where, for the case of non-fluctuation target:

(4.37) with (4.38) being the output SINR potential, and a is the amplitude of the input signal.

For the case of the Swerling I target fluctuation model where the target amplitude is assumed to be Gaussian with zero mean and variance o^: (4.39) with (4.40) 4.3.2.2

GLR

The probability of false alarm [7] is: (4.41) The probability of detection is: (4.42) where fp is defined in equation (4.33), and for the case of non-fluctuation target:

(4.43)

with fi of equation (4.38). For the case of the Swerling I target, we have: (4.44) where /3 is given in equation (4.40).

4.4

A real-data demonstration of DA-STAP

To illustrate its simplicity and yet good performance, we will use airborne radar data for a performance demonstration of I] A-STAP against a popular STAP approach, and the conventional non-adaptive approach involving the E-channel only. The airborne radar referenced here was developed by the Air Force Research Laboratory, Rome NY site, and was called the MCARM (multichannel airborne radar measurement) system. The L-band MCARM system has sixteen half-wavelength spaced columns each with a receiver channel, i.e. Nc — Ns = 16. Separately, an analogue beamformer delivers a E-beam of an over —50 dB sidelobe level (Figure 4.6) and a A^-beam with a

—25 dB central null (Figure 4.7). One should note that the above analogue beamforming performance with such a small array is rarely achieved by digital beamforming with current hardware. The data set is part of the Flight 2 data collected over a rural area in the eastern shore region south of Baltimore, MD in early 1995.

magnitude, dB sat

14.4 (is, RxBeam 7, Pos 64, 1270 MHz, RG 70 cells, IPP #1.4

Points @ 0.2 deg. Beam W on Edge, **51.3 dB RMSide**, & 42.1 deg N width 16 az by 8 el Elements, File; 25053011, Time: 5:16:25; 7/25/94

angle, deg

Figure 4.6

Measured E receive pattern at boresight of MCARM system

magnitude, dB sat

14.4 |is, RxBeam 7, Pos 64, 1270 MHz, RG 70 cells, IPP #1.4

PonjtS^O.2 deg. Beam W on Edge, **43.3 dB RMSide**, & 55.9 deg N width 16 az by 8 el Elements, File; 25053091, Time: 5:16:47; 7/25/94

angle, deg

Figure 4.7

Measured A receive pattern at boresight of MCARM system

The three processing approaches compared are: 1

EAa-STAP with DDL(2) and MSMI(6): (see Figure 4.5) Doppler domain localised adaptive processor and modified sample matrix inversion 2 FA-STAP with MSMI( 16): factored approach STAP (FA-STAP) [8] with modified sample matrix inversion and Blackman temporal tapering 3 Conventional PD with CA(18): the conventional pulse-Doppler processor (twopulse canceller) followed by a Doppler filter bank with Blackman temporal tapering and a cell average CFAR using eighteen adjacent cells (CA-CFAR[18]).

test statistic, rjMSm

For all the three approaches given above, a known target signal of —40 dB input SINR is injected at range bin 290, and the thresholds for the same false alarm probability of 10~6 are calculated and marked in the respective output range-Doppler plots of Figures 4.8-4.10. For each range bin, adjacent secondary range bins provide samples for its correlation matrix and/or CFAR threshold estimation. Reliable detection of the same injected target under the given conditions can be achieved only by the EA^-STAP among the three, even though the FA-STAP uses more receiver channels, more samples for correlation matrix estimation and more computation time than does the EA^-STAP. One should note that the above real data demonstration does not involve any effort to optimise the performance of each approach through data conditioning or sample selection, other than excluding the cell under test from the secondary data. Even without such optimisation, the SINR potential of EA^-STAP compares favourably to factored approach STAP and low-sidelobe aperture pulse-Doppler processing. The results shown in Figure 4.11 show the advantage of the E Aa-STAP over the competing processors in the mainlobe clutter region, under conditions similar to the MCARM experiment.

injected target:

CTXTD —

range bin index

Figure 4.8

normalised Doppler frequency

MCARM range-Doppler plot: Y>ka-STAP with DDL(2) and MSMI(6)

test statistic, rjMsm

injected target:

SINR =

range bin index

Figure 4.9

normalised Doppler frequency

MCARM range-Doppler plot: FA-STAP with MSMI(16)

test statistic

injected target:

SINR = PD(CA-CFAR: 18 cells)

range bin index

Figure 4.10

normalised Doppler frequency

MCARM range-Dopplerplot: conventional PD with CA-CFAR(18)

One may also notice that the MCARM system has a very good, low sidelobe, SVbeam to begin with; thus it is interesting to see whether EA^-STAP can still offer a significant improvement over the conventional PD approach if the implementation of the E-beam is not as good. Figures 4.12-^1.14 serve to answer this question, where we have omitted the effects of the temporal DoF reduction in the SINR potential calculation to concentrate on the effects of the S A^-beams. For the three pairs

SINR potentials, dB

input: INR = 50 dB SNR = 0 dB

Bound Hanning full-cycle sine FA (Blackman) PD (Blackman)

normalised Doppler frequency

Figure 4.11 SINR potentials: E A-STAP versus FA-STAP and PD of EA a -beams of —30, —40, —50dB peak E-beam sidelobe levels, respectively, the E A^ can provide significant detection performance in the mainlobe clutter region, outperforming conventional techniques. As anticipated, the SES[R improvement in the sidelobe clutter region is significant, even if the sidelobe clutter level using the chosen transmitter and E-beam is relatively high to begin with. EA^-STAP places nulls in the adapted antenna pattern without significantly changing the existing sidelobe level of the E^-beam.

4.5

Desired A-beam characteristics

Based on fundamental principles of interference cancellation, we may hypothesise the following two desired characteristics of A-beams for EA-STAP: 1

Except the central null, all other nulls of the A-beam should be located at the same angles as the E-beam nulls, at which the interference components in the E-channel have already been significantly attenuated. Null alignment is required since the A-beam cannot be used to improve performance at angles where nulls occur. 2 At each non-null angle of the E-beam, the magnitude of the A-beampattern should be sufficiently larger than that of the E-beampattern. This characteristic reduces errors in adaptive weight estimation, and ensures that the resulting weight for the A-channel is reasonably small so as to minimise degradation of output SINR.

normalised antenna pattern, dB

E-beam A-beam

Nc=l6

SINR potentials, dB

sin 6

SINR bound XA-STAP MTI-DFT with 7OdB cheb.

normalised Doppler frequency

Figure 4.12 a E A-beams b comparison of SINR potentials of EA-STAP (Npt = Nt - 1) and non-adaptive £-only processor: peak sidelobe level (PSL) of E-beamat -3OdB

normalised antenna pattern, dB

X-beam A-beam

Nc=l6

SINR potentials, dB

sin 6

SINR bound IA-STAP MTI-DFT with 7OdB cheb.

Nc=\6 N = 32 L-.PSL =-4OdB (T : 0.22 m/s

normalised Doppler frequency

Figure 4.13 a E A-beams b comparison of SINR potentials of E A-STAP (Npt = Nt - 1) and non-adaptive S-only processor: PSL of E-beam at —40 dB

normalised antenna pattern, dB

Z-beam A-beam

SINR potentials, dB

Nc=\6

SINR bound SA-STAP MTI-DFT with 70 dB cheb.

N = 16 N = 32 Z : PSL =-5OdB <Jv: 0.22 m/s

normalised Doppler frequency

Figure 4.14 a E A -beams b comparison of SINRpotentials of EA-STAP (Npt = Nt — l) and non-adaptive E-only processor: PSL of E-beam at —50 dB

radar platform

ground clutter patch

Figure 4.15

Radar observation geometry and coordinates

The analysis of this section is provided to confirm the above conditions. To simplify the analysis as much as possible, we will use the output SINR potential with the omission of the temporal DoF reduction effects. We will examine the SINR potential as a function of the angle of arrival of a single scatterer, and of the H A^-beampatterns which are formed by a phased array. Consider a ground scatterer located at the angle ^r relative to the platform velocity, as shown in Figure 4.15. Let u = cos V^, and ft(u) = 2Vpu/(XFpr). The correlation matrix due to the single scatterer and receiver noise is: (4.45) where (4.46) with the assumption (4.47) and (4.48) Equation (4.45) can be rewritten as: (4.49)

Denote: (4.50) (4.51) and %c(u)=a2c(u)/cr2n

(4.52)

To avoid lengthy expressions, we temporarily drop the argument u in the following derivation. Denoting:

and

we have: (4.53) (4.54) (4.55) We note that x/> and Q^ differ from x and Q by the temporal DoF reduction. Using the Sherman-Morrison formula [9], we arrive at: (4.56) Further assume that Wj is even symmetric and WA odd symmetric so that: wgw A = 0

(4.57)

Under the above condition, we obtain: (4.58) (4.59) and (4.60)

Therefore we have from equations (4.26) and (4.31): (4.61) and (4.62) where as = W^s5. This is the output SINR potential for a single scatterer with angle of arrival \j/ under the stated assumptions. If F^(u) ^ 0, it is a monotonically increasing function of |F A (w)| 2 , and its minimum is at F&(u) = 0 with F^(u) ^ 0. First of all, one should notice that F^ (u) and FA (M) of equations (4.50) and (4.51) are equivalent to the E- and A-beampatterns, respectively. If F^ (u) = 0 for some u = cos yjr, equation (4.62) becomes: (4.63) which does not depend on F&(u). In other words, the output SINR potential of equation (4.62) will not decrease due to F&(u) = 0 only if F^(u) = 0 at the same u = cos \fs. This confirms our first hypothesis stated at the beginning of this section. Figure 4.16 illustrates the null-alignment effect for a linear array.

SINR potentials, dB

SINR potentials of ZA^-STAP

SINR bound DPCA patn-taylor (30 dB) Hanning/full-cycle sine Taylor (40 dB)/Bayliss (30 dB) Tx antenna: uniform

Nc=\6 Nt=\6 Npt=Nr\ av=0mJs

normalised Doppler frequency

Figure 4.16

Effect o/SA -patterns on S A -STAP: null-aligned patterns versus null peak aligned patterns

The second observation follows from equation (4.62). Moreover, one should note that equation (4.63) is also the highest SINR potential. We can thus define the loss of the SINR potential as the ratio of equation (4.62) over equation (4.63), denoted by L. If we require L > Lj, the specified tolerable SINR potential loss, then we must have: (4.64) For large Nt, strong clutter, or large \F^(u)\ (i.e., in the mainlobe region), we would like to have: (4.65) For example, Lj = 0.5(—3 dB) is corresponding to |FA(W)| 2 > |Fs(w)| 2 and Lj = 0.8(-ldB)tol01og(|F A (w)| 2 /|F E (w)| 2 ) > 6 dB. Equation (4.64) or equation (4.65) can serve as a convenient guideline for EA-beam design.

4.5.1 Mathematical equivalence ofsubarray and T1A-STAP We now consider the case of two subarrays of Figure 4.17 whose beampatterns, E\ (0) and E2(0), are not necessarily equal. If one applies STAP directly to the subarray outputs x, its SINR potential would be: SINR= PssHQ~ls

(4.66)

where the correlation matrix is Qx = E(xxH), and the steering vector s is: (4.67)

array

Now consider applying STAP to: x = Tf x

Figure 4.17

Subarray ing and I] A-STAP

(4.68)

with (4.69) where I ^ is the Nt by Nt identity matrix. Then: (4.70) (4.71) with the new steering vector: s = Tf s

(4.72)

Therefore, the SINR potential remains the same, i.e.:

(4.73)

We note that this equivalence still holds even if the A'-beam is not a A-beam. Although the above is a special case of the general SINR-invariance under the orthogonal transformation, HA-STAP provides the following advantages: 1 2

4.6

Using a real A-beam eliminates the need to calibrate and store the steering vector information as the subarray STAP does with equation (4.69). Even if both subarrays are designed to have low sidelobes, combining the two does not ensure a E-beam of the same low sidelobe level, which may even result in a need to have a separately designed transmit beam and thus increase the overall system cost. The sidelobe level (and gain) of the transmit beam has a direct impact on the overall system performance.

Summary

We summarise this chapter in terms of the advantages of the EA-STAP approach and its limitations.

4.6.1 Advantages of the S A -STAP approach With only two channels, we have shown that the EA-STAP approach can lead to a clutter suppression performance potential as high as that of other STAP approaches that require many more channels. The following are its advantages over others: 1 Affordability - affordability has long been an issue with STAP-based systems. Although progress has been made in reducing the expense of front-end analogue electronics for phased array transmit-receive (TR) modules, the associated digital electronics for A/D conversion, matched filtering, and channel equalisation

remain part of the onboard processing system. Analogue beamforming and minimisation of the number of digitised receiver channels provides a substantial payoff in total system cost. Reliability is increased and maintenance costs are reduced by simplifying system interconnects. In these areas, EA-STAP greatly reduces system cost. 2 Data efficiency - correlation matrix estimation for EA-STAP can be performed with training sets of fewer than 20 secondary data vectors. This feature provides good performance in severely non-homogeneous environments where other STAP approaches break down, regardless of how high their performance potentials are with known a priori clutter statistics. With an integrated non-homogeneity detector, the performance of EA-STAP can be further improved by selection of the secondary data vectors to be used among those in the neighbourhood of the test cell. 3 Channel calibration - channel calibration is a problem issue for many STAP approaches. In order to minimise performance degradation, the channels with other STAP approaches must be matched across the signal band, and steering vectors must be known to match the array. Considering the fact that channels generally differ in both elevation and azimuth patterns (magnitude as well as phase) even at a fixed frequency, the calibration difficulty has long been underestimated, as revealed by recent STAP experiments. The so-called element-space approaches, with an adaptive weight for each error-carrying element which hopefully can be modelled by a complex scalar, offer one possible solution to the calibration problem. This comes at a significantly increased system implementation cost, since each element then needs a digitised receiver channel. Unfortunately, such a solution can rarely be realised for an AEW system with a practical aperture size operated in non-homogeneous environments. With spatial DoF reduction required to bring down the number of adaptive weights to a sample-supportable level, the element errors are no longer directly accessible by the adaptive weights, and thus the embedded robustness of the element-space STAP approaches is lost. In contrast, EA-STAP uses two well structured channels to begin with, and its corresponding signal (steering) vector remains well calibrated for target detection as long as the central null of the A-beam is correctly placed (a reasonable property of good antenna design). Therefore, EA-STAP greatly simplifies the calibration issue in practice. 4 Response pattern - STAP has long been known to exhibit adapted spatial response patterns that are undesirable in some applications, e.g. very high sidelobe levels in some interference-free regions, loss of the mainlobe gain and significantly shifted mainlobe peak. These undesirable beam characteristics can be largely attributed to excessive spatial DoF and associated unconstrained estimation errors. With its spatial DoF equal to one and with the critical null location of the A-beam, EA-STAP offers much more desirable and predictable response patterns than do other STAP approaches with excessive DoF. This property can be maintained even when spatial presuppression of jammers is performed ahead of the EA processor.

5

Computation load - although the trend is toward more affordable computing hardware, STAP processing still imposes a considerable processing burden which increases sharply with the order of the adaptive processor and radar bandwidth. In this respect, EA-STAP reduces computational requirements in matrix order N^ adaptive problems. Moreover, the signal vector characteristic (sparse) can be exploited to further reduce test statistic numerical computations. 6 Applicability to existing systems - with other approaches, the application of STAP almost always requires completely new system hardware, from an expensive phased array to multichannel receivers which may well lead to a demand for a new platform. A significant advantage of SA-STAP, in contrast to other STAP approaches, is its applicability to existing radar systems, both phased array and continuous aperture. Adaptive clutter rejection in the joint angle-Doppler domain can be incorporated into existing radar systems by digitising the monopulse difference channel, or making relatively minor antenna modifications to add such a channel. Such a relatively low cost add-on can significantly improve the clutter suppression performance of an existing airborne radar system, whether its original design is based on low sidelobe beamforming or EA-DPCA. In fact, it is not difficult to see from the discussions of this chapter that the EA-STAP is a very natural combination of the traditional antenna design-based approaches and modern signal-processing-based approaches, making the best use of the strength of each and avoiding the weakness of each. For example, the traditional low sidelobe design and EA-DPCA have been operating at their hardware limits for the very difficult tasks of further lowering the sidelobe level or matching the DPCA patterns. With EA-STAP, one can still take whatever the traditional approaches can offer and at the same time use adaptive processing to improve performance towards the theoretical limit, as well as lifting the PRF-V^ constraint of the EA-DPCA. In this regard, EA-STAP does not attempt to do the whole job alone, and utilises the inherent desirable properties of the traditional non-adaptive aperture. It avoids many of the pitfalls of other adaptive techniques such as the dependence of quality sample support and uncontrolled response patterns with excessive DoF. EA-STAP also offers a new look at the so-called beam-space class of STAP approaches which usually involve identically shaped beams only. By starting with a set of well developed, relatively low-cost beams, a new class of practical STAP system concepts can emerge and may serve to fulfil different system application needs.

4.6.2 Limitations ofllA-STAP We will concentrate on the EA^-STAP for discussion herein. The results can be generalised to EA^- and EA^-STAP. 1 Spatial ambiguities - any two-channel system will lack sufficient spatial DoF for clutter suppression whenever spatial ambiguities occur due to the use of a very low PRF and/or high clutter Doppler induced by a very fast moving platform. The E Aa-STAP is not exceptional. Although the PRF selection for STAP-based

systems requires design attention, the spatial ambiguity problem may limit the application of EA43-STAP without the use of additional adaptive beams. 2 Beampattern mismatch - another limitation comes from the design of the E A^-beams. A deep null of the E A^-beam in the sidelobe region of the E-beam can cause poor clutter suppression performance, unless it is aligned with a null of the E-beam or the transmit beam has a sufficiently low sidelobe level. Before one applies the EA^-STAP, therefore, the S A^-beampatterns should be carefully measured and optimised if possible. 3 Jammer cancellation -jammer suppression is not a feature of EA^-STAP. However, possible application of spatial presuppression of jammers with additional auxiliary channels can make EA^-STAP suitable for such applications. 4 Limited monopulse capability - E A-STAP, as formulated here, does not provide a clutter-free difference channel for monopulse tracking. This capability still exists, however, in regions where clutter can be separated from the target by conventional Doppler processing.

4.6.3 Potential applications of E A-STAP As a large inventory of various airborne radars already exists worldwide, the potential market of a E A-STAP-based add-on upgrade can be huge. It is important to demonstrate the performance gain of such an upgrade, in order for system users to make informed and reasonable decisions. In many cases, a new multichannel system may have a performance gain that is smaller than that which can be achieved by a much lower cost EA-STAP upgrade of an existing system.

References 1 SHERMAN, S. M.: 'Monopulse principles and techniques' (Artech House, 1980) 2 STAUDAHER, F. M.: 'Airborne MTF, in M. I. Skolnik (Ed.): 'Radar handbook' (McGraw-Hill Inc., 1990) chapter 16 3 GRIFFITHS, L. J.: 'Adaptive monopulse beamforming'. Proceedings of IEEE, August 1976, pp. 1260-1261 4 BROWN, R. D., WICKS, M. C , ZHANG, Y, ZHANG, Q., and WANG, H.: 'A space-time adaptive processing approach for improved performance and affordability'. Proceedings of IEEE national Radar conference, Ann Arbor, MI, May 13-16, 1996, pp. 321-326 5 BROWN, R. D., SCHNEIBLE, R. A., WANG, H., WICKS, M. C , and ZHANG, Y.: 'STAP for clutter suppression with sum and difference beams', IEEE Trans. Aerosp. Electron. Syst., April 2000, 36(2), pp. 634-646 6 ELLIOTT, R. S.: 'Antenna theory and design' (John Wiley & Sons, Inc., Hoboken, New Jersey, 2003) p. 142 7 CAI, L. and WANG, H.: 'Performance comparisons of modified SMI and GLR algorithms', IEEE Trans. Aerosp. Electron. Syst, May 1991, AES-27(3), pp. 487^91

8 DiPIETRO, R. C. 'Extended factored space-time processing for airborne radar systems'. Proceedings of 26th Asilomar conference on Signals, systems, and computers, Pacific Grove, CA, November 1992, pp. 425-430 9 GOLUB, G. H. and VAN LOAN, C. F. 'Matrix computations' (John Hopkins University Press, London, 1996) p. 50

Chapter 5

STAP with omnidirectional antenna arrays Richard Klemm

5.1

Introduction

Airborne radar is an essential tool for military surveillance and reconnaissance. There are two stringent requirements related to surveillance and reconnaissance radar: •

•

GMTI (ground moving target indication) capability for detection of moving targets on the ground. The detection of low Doppler targets, i.e. targets whose Doppler frequency appears within the clutter Doppler band requires space-time processing techniques. 360° azimuthal coverage is an important operational requirement to obtain a complete picture of the area of interest. Existing systems such as AWACS use a mechanically rotating antenna. Joint-STARS has GMTI capability, but is based on side-looking antennas which permit only the view in a limited sector.

In this chapter we discuss possibilities of replacing the mechanically steered antenna by a phased array. The phased array antenna offers various advantages such as look agility through electronic beam steering, adaptive beamforming including jammer cancellation and superresolution, and slow target detection by STAP-based GMTI techniques. The major part of this chapter has been reported at a RTO IST/SET symposium [2]. In the following we discuss the possibility of detecting low Doppler targets with an array antenna with 360° azimuthal coverage. Is this contradictory?

5.1.1 Preliminaries on STAP antennas Basically, one can conduct STAP with any kind of antenna array configuration if the array is fully digitised, i.e. all the individual array elements have their own receive channels including an analogue-to-digital converter. The associated optimum STAP

processor exploits the echo data received by all the array elements. Obviously, for realistic array apertures having several hundreds to thousands of elements, this may cause unsolvable problems in various respects: • • • •

processing power computing time, particularly in view of real-time processing requirements availability of a sufficient amount of training data for the adaptive process numerical accuracy.

Suboptimum techniques with the potential of real-time processing for adaptive beamforming and STAP for fully digitised arrays have been proposed by Mather etal. [15]. The linear equispaced array has unique properties among all possible array configurations. With a linear equispaced array suboptimum processor architectures can be designed which achieve near-optimum clutter rejection performance while using a very limited number of weighting coefficients. This can be explained as follows. A linear equispaced array can be subdivided in displaced subarrays of uniform shape, see the top of Figure 5.1. Overlapping or disjoint subarrays are possible. The degree of overlap determines the locations of the subarray phase centres or, in other words, the spatial sampling. It can be shown that best STAP performance can be achieved by choosing the PRF so that spatial and temporal sampling are equal. All subarray elements are combined by subarray beamformers which altogether point in the same look direction. If the shapes of the subarrays are uniform they have the same directivity pattern, which in turn receives the same clutter Doppler spectrum. There is only a spatial phase between the individual subarray output signals due to the spatial displacement of the subarrays. Therefore, clutter cancellation can be done just by subtracting two of these spectra from each other after weighting one of them with the spatial phase factor. In other words, only one weighting coefficient is required. For more details on subarray forming for STAP the reader is referred to Klemm [8, 9, chapter 6] and the literature quoted there. If the idea of uniform subarrays is combined with some technique to reduce the temporal dimension [9, chapters 7, 9] almost optimum clutter rejection can be achieved by using a very small number of spatial and temporal weighting coefficients. Reduction of the number of degrees of freedom requires, of course, that the tolerances and errors in the receive channels are small.1 The disadvantage of linear (or planar) arrays is that their field of view is limited to about —60° . . . 60° in azimuth because of the directivity patterns of the individual array elements. In the following we discuss a few alternative array configurations. These concepts are based on an omnidirectional radiating element as has been used in the crow's nest antenna (Wilden and Ender [16]), combined with the principle of forming uniform subarrays.

If you want to reach the optimum at low expense you have to pay for it in terms of accuracy

beamformers

inverse of beam time covariance matrix

subarray combination (secondary beamformer)

Doppler filter bank

test function

Figure 5.1 STAP processor with overlapping subarrays

5. L 2

The circular ring array concept

The UESA radar (UHF electronically steered array) is an experimental array radar with 360° azimuthal coverage and electronic beam steering. Figure 5.2 shows the geometry of the UESA array. The elements are placed on a circle which lies in a horizontal plane. One third of the elements are active whereas the others are idle. This limitation is caused by the horizontal directivity of the radiating elements. Several authors (Bell et al. [3], Fuhrmann and Rieken [6], Li et al. [14], Zatman [18]) have discussed issues arising in STAP processing for such an array configuration. Some properties of ring arrays with a moving active sector should be mentioned: •

Power budget - the SNR (signal-to-noise ratio) achieved by an active array is the product of the transmitted power times transmit antenna gain times receive

excited elements idle elements

Figure 5.2

Circular ring array

gain. Each of these quantities is proportional to the number of array elements N, Therefore: SNR (X N3

•

•

5.2

(5.1)

This means that the SNR obtained by an array that uses only N/3 elements is 14.3 dB lower than that achieved by an array with N elements. Since the minimum detectable velocity (width of the clutter notch) depends strongly on the SNR this results in a degradation in slow target detection. Array curvature - the curvature of the active sector does not permit the design of identical subarrays looking in the same direction. It is, therefore, not easily possible to design subarray beams with uniform beampatterns in order to reduce the number of antenna channels which means a reduction of the signal vector space in the spatial dimension. This aspect has not been discussed in the literature quoted above. Directivity pattern - the array curvature will affect the directivity pattern. Some broadening of the main beam and some rise of the sidelobes can be expected.

Array configurations for 360° coverage

In this chapter three array configurations are discussed which offer both the aptitude for STAP and 360° azimuthal coverage. Other than the UESA array these concepts are based on a magnetic ring dipole as shown schematically in Figure 5.3. This dipole has been developed for the crow's nest antenna (Wilden and Ender [16]) which is

Figure 5.3

Circular dipole

a spherical array with 360° coverage. The two-way gain of this dipole is slightly above 1 dB whereas a directive dipole on the front of a metal plate as used in linear or vertical planar arrays as well as in the UESA ring array has about 6 dB two-way gain. We assume in the subsequent discussion that the directive dipole in front of a metal plate has 6 dB gain over the omnidirectional ring dipole. The three concepts are: • • •

displaced ring arrays randomly thinned circular planar array octagonal array.

The three concepts will be analysed in terms of the achievable SNR and the directivity pattern. Moreover, these concepts are compared with the concept of four linear (or rectangular planar) arrays. If not denoted otherwise the radar parameters listed in Table 5.1 have been used in this investigation.

5.2.1 Four linear arrays We start the discussion with the concept of four linear arrays arranged in a horizontal plane as depicted in Figure 5.4. One of the arrays is active while the other three

Table 5.1 Radar parameters Platform velocity Range Number of elements Element spacing Number of processed echoes Clutter-to-noise ratio, single element, single pulse Signal-to-noise ratio, single element, single pulse Wavelength ^^Nyquist

vp = 240 m/s 10 km N = 960 X/2 (Nyquist) 24 20 dB — 10 dB A = 0.03 m 32 000Hz

idle elements

active elements

Figure 5.4

Four linear (or rectangular) arrays

are idle. In practice rectangular or trapezoidal planar arrays are used so as to focus the transmitted energy and the receive gain in the vertical dimension. There are operational radar systems existing which follow this design concept. Compared with an array that uses all transmit and receive elements simultaneously, the achievable SNIR is reduced according to equation (5.1) by 4 3 (about 18 dB). Figures 5.5 and 5.6 show the SCNR plotted versus the target velocity for a sidelooking (array aligned with the flight path) and a forward-looking (array perpendicular to the flight path) linear uniformly spaced array. The number of transmit and receive elements was assumed to be 240 (one fourth of the total number of array elements according to Table 5.1). Both arrays look in broadside direction, that means, the sidelooking array is steered in the cross-flight direction whereas the forward-looking array looks in the flight direction. In both examples four curves have been plotted

SNlR, dB

Linear array, side-looking. Nc: 2, 3, 4, 6 (lowest to highest curve)

SNIR, dB

Figure 5.5

Figure 5.6

Linear array, forward-looking. Nc: 2, 3, 4, 6 (lowest to highest curve)

for different numbers of array channels Nc = 2,3,4,6 which have been generated by forming overlapping uniform subarrays. The subarray displacement has been chosen so that the spatial (subarray displacement) and temporal sampling (PRF) are both at Nyquist frequency. That means, the element displacement is chosen to be half the wavelength, and the PRF is chosen so that the phase advance between any two

azimuth, deg

Figure 5.7

Directivity pattern of the linear array versus azimuth (deg)

adjacent elements is equal to twice the distance the array moves during one pulse repetition interval (PRI). This is also called the DPCA condition. A very narrow clutter notch can be noticed in both examples. It is remarkable that for the given set of parameters the SNIR curves are independent of the number of subarrays. This follows from the earlier discussion on subdividing a linear array in uniform subarrays. For the side-looking array the clutter notch appears at zero velocity (tangential motion of the clutter relative to the radar). For the forward-looking array the clutter velocity is determined by the platform velocity times the cosine of the depression angle. Figure 5.7 shows the associated directivity pattern. Notice that the array uses 'directive elements; therefore, no backlobes show up. The picture shows the far sidelobes. The near sidelobes can be seen in Figure 5.21a.

5.2.2

Displaced circular rings

The concept of displaced circular rings has been evolved from a discussion of the properties of the ring array with active sector, see Section 5.3.2. We replace the directive sensors in the ring array by omnidirectional ring dipoles according to Figure 5.3 and use all of them simultaneously. As mentioned above the crow's nest antenna [16] has been designed in this way. However, a circular array has obviously no aptitude for forming subarrays with uniform shape according to Figure 5.1. How can we use STAP with a circular ring array, avoiding the unrealistic option of using a fully adaptive processor. The solution might be an array configuration as shown in Figure 5.8. A number of identical circular ring arrays are arranged in such a way that a group of identical subarrays displaced by

quadrapacks

circular subarray

Figure 5.8

Displaced circular ring arrays

a certain distance is obtained. In practice, such a configuration can be generated by designing groups of sensors (for instance, quadrapacks as in the example Figure 5.8). All elements belonging to a certain ring form a subarray and have to be combined by a subarray (or primary) beamformer. All subarray beams have the same look direction. Now we have a sequence of uniform subarrays displaced by a constant distance in the flight direction. In essence we designed a linear side-looking array with circular ring-shaped radiating elements. In Figure 5.9 the SNIR for different numbers of channels are plotted. We notice that, in contrast to the linear array of Figures 5.5 and 5.6, the width of the clutter notch depends strongly on the number of subarrays. The more subarrays that are formed the narrower the clutter notch is. Notice that the gain outside the clutter notch is 12 dB higher than that obtained by the linear array. Recall from Section 5.2.1 that the reduction of the number of elements by a factor of four results in a loss of 18 dB. Taking into account the gain of the individual element antenna (we assume that the gain of the ring dipole according to Figure 5.3 and Section 5.3.2 is 6 dB less than that for a directive dipole in front of a metal plate) we come up with a difference in SNIR of 12 dB between the displaced ring arrays and four linear arrays concept. Directivity patterns of the displaced ring antenna are shown in Figure 5.10 for different numbers of antenna channels. The maximum of the far sidelobes is about —20 dB. The near sidelobes are depicted in Figure 5.21b. In comparison with Figure 5.21a we notice that the displaced ring antenna produces a broader mainlobe and higher sidelobes than the linear array using 960/4 = 240 array elements.

5.2.3

Circular planar array with randomly distributed elements

Randomly distributed elements have been used for radar antenna design in the ELRA system (Groger et al. [7], Wirth [17, pp. 390]) and in the crow's nest antenna

SNIR, dB

Figure 5.9

Figure 5.10

Displaced ring arrays. Nc: 2, 3, 4, 6 (lowest to highest curve)

Directivity pattern of the displaced ring array versus azimuth (deg) a Nc = 2 b Nc = 3 c Nc = 4 d Nc = 6

(Wilden and Ender [16], Wirth [17, pp. 77]). By distributing the array elements in a random fashion thinned arrays can be generated which offer a narrower beam than an array with half-wavelength spacing, however at the expense of raised sidelobes. We consider now a horizontal planar circular array with random distribution of omnidirectional elements according to Figure 5.3. It is obvious that such an array cannot be subdivided into a number of uniform subarrays. We apply therefore again the technique used already for the displaced ring arrays. At the position of each element a doublet, triplet or quadruplet etc. of sensors is placed. Of course, as in the previous section, the sensors within such a group have to be aligned in the flight direction and displaced by a constant amount so as to design a linear side-looking array with uniform circular planar subarrays as elements. Examples of such nested arrays are shown in Figures 5.11 to 5.14. Notice that the number of elements has been kept constant. As before, uniform subarrays are obtained by combining, for example, all first, second etc. elements of all doublets, triplets etc. of the array. Comparing the four figures it is obvious that the shape of the total array is influenced by the number of subarrays. This was also the case in the displaced ring array concept. The SNIR curves in Figure 5.15 are very similar to those obtained with displaced ring array concept in Section 5.2.2. Again we notice that, in contrast to the linear array, the width of the clutter notch depends on the number of array channels. As can be seen from Figures 5.16 and 5.21c one gets a very narrow beam. The maximum sidelobe is at about —20 dB (lower than for the 240 elements linear array), the average sidelobe level is about —30 dB.

Figure 5.11 Randomly thinned array using sensor doublets (axes in m)

Figure 5.12

Randomly thinned array using sensor triplets (axes in m)

Figure 5.13

Randomly thinned array using sensor quadruplets (axes in m)

5.2.4

Octagonal planar array

The octagonal array has been proposed already in Klemm [9, p. 193]. In the example in Figure 5.17 it is shown how the octagonal array is subdivided into four overlapping uniform subarrays. In this example the subarray displacement is equal to the element

Randomly thinned array using sensor sextuplets (axes in m)

SNIR, dB

Figure 5.14

Figure 5.15

Randomly thinned array. Nc: 2, 3, 4, 6 (lowest to highest curve)

spacing. If the element displacement is half the wavelength then this kind of subarray formation is in agreement with the PRF chosen to be Nyquist of the clutter Doppler bandwidth (in our examples: PRF = 32 000 Hz), i.e. the spatial frequency is equal to the Doppler frequency.

Figure 5.16

Directivity pattern of the randomly thinned array versus azimuth (deg) a Nc = 2 b Nc = 3 c Nc = 4 d Nc = 6

Figure 5.17

Formation of overlapping subarrays in an octagonal array

Octagonal array with 976 elements (axes: element numbers)

SNIR, dB

Figure 5.18

Figure 5.19

Octagonal array. Nc: 2, 3, 4, 6 (lowest to highest curve)

In Figure 5.18 we find a realistic design example with 976 elements which is about the same size as the arrays discussed before. In Figure 5.19 we find the SNIR curves for the four chosen numbers of subarrays (Afc = 2,3,4,6). As can be seen, these SNIR curves are again much more similar to

azimuth, deg

Figure 5.20

Directivity pattern of the octagonal array versus azimuth (deg)

those obtained with linear arrays (Figures 5.5 and 5.6). The clutter notch is narrow compared with those of the displaced ring and random planar arrays. The directivity pattern (Figure 5.20) exhibits a relatively broad beam which is a consequence of the small aperture. Notice that for this array the elements are displaced by half the wavelength. The diameter of this array is about 50 cm whereas the randomly thinned array has a diameter of about 4 m. Of course, if the budget permits, much larger octagonal arrays can be designed, leading to narrower beamwidth, lower sidelobes and even narrower clutter notches. In addition, because this is a regularly half-wavelength spaced array, it can be tapered so as to reduce the sidelobes even further. For irregular or thinned arrays tapering does not give any advantage.

5.3

Discussion

5.3.1 Directivity patterns In Figure 5.21 we compare the directivity patterns of the four array architectures discussed above in the near neighbourhood of the mainbeam. The apertures of the four antennas are approximately: a = 3.6m; b = 2m; c = 4m; d = 0.55m. The different sizes of the apertures result from the requirement that for a given number of elements (960) no spatial ambiguities in the directivity patterns (grating lobes) must occur. The randomly distributed array appears to achieve the best compromise between beamwidth and the level of the near sidelobes.

Figure 5.21

Comparison of main lobe patterns a 4 linear arrays b displaced circular rings c randomly thinned array d octagonal array

5.3.2 Range-ambiguous clutter If the PRF is chosen so that the radar is range ambiguous the received clutter is a superposition of the clutter in the actual range bin and clutter components coming from ambiguous range bins. In the case of a linear side-looking array the Doppler frequency of clutter returns is range independent. Therefore, the ambiguous clutter returns coming from different range bins exhibit altogether the same Doppler frequency. This means, all ambiguous clutter arrivals contribute to the same clutter notch, the clutter notch is practically the same as in the case of range-unambiguous operation. For all other array configurations ambiguous clutter returns exhibit Doppler frequencies different from the Doppler in the actual range bin. Therefore, the ambiguous clutter returns cause a broadening of the clutter notch or even additional clutter notches, which results in degraded moving target detection performance. Even the performance of a side-looking linear array can suffer from aircraft crabbing caused by cross wind [H]. Let us compare the behaviour of one of our planar arrays, for example the randomly thinned array, with the four linear arrays concept described in Section 5.2.1. If there is no aircraft crabbing, ambiguous clutter returns do not alter the clutter notch

SNIR, dB

Figure 5.22

Effect of range ambiguities. Comparison of linear (thick) and randomly spaced planar (thin) array. Nc = 4

as long as those arrays parallel to the flight path are used. However, if one of the arrays arranged in the cross-flight direction is in operation the ambiguous clutter returns have a strong influence on the SNIR curve. For illustration, consider Figure 5.22. The thick curve has been calculated for a linear forward-looking array (orientation in the cross-flight direction) whereas the thin line belongs to a randomly thinned array as treated under Section 5.2.3. Both curves were calculated for PRF = 32 000 Hz. For the assumed geometry 40 ambiguous range bins occur. The influence of the ambiguities on the forward-looking array is dramatic (compare with Figure 5.6 where the ambiguous clutter returns were ignored). It is, furthermore, remarkable that the SNIR curve of the randomly distributed circular array according to Figure 5.13 is not affected by the range ambiguities (compare with the third lowest curve in Figure 5.15). The reason for this beneficial behaviour is the fact that by composing the array of doublets, triplets, etc. arranged in the flight direction, linear arrays with Nc highly directive elements (subarrays with beamformers) are generated whose axes are in the flight direction. This is the architecture of a side-looking array which, as is well known, is not affected by ambiguous clutter returns because the clutter Doppler is range independent. If the linear forward-looking arrays in Figure 5.4 are replaced by vertical planar arrays with vertical degrees of freedom (subarrays in the vertical array dimension) ambiguous clutter returns can be cancelled by placing vertical nulls in the directions of the ambiguous clutter returns. Of course, when considering a constant total number of elements, this means reducing the horizontal aperture by a certain factor, with the natural consequences, such as a broadened mainbeam and a broadened clutter notch.

5.4

Effect of array tilt

So far we have assumed that the various array configurations, i.e. the displacements of the subarrays, are aligned with the flight path. In the following we briefly discuss the impact of a horizontal tilt angle on the STAP performance of the above described array configurations. Horizontally tilted geometries may be generated intentionally, for example, by mechanical steering of space-based radar systems [12], or may be caused by wind drift (aircraft crabbing [H]). We assume a range-ambiguous mode such as medium PRF (MPRF) or high PRF (HPRF) which are both ambiguous in range, Doppler and, hence, produce ambiguous clutter returns.

5.4.1

Side-looking linear and rectangular arrays

In Figure 5.23 the impact of a horizontal tilt angle on the improvement factor achieved by the ASEP ([9], Chapter 9) is illustrated for a side-looking linear array in a rangeambiguous radar mode. The upper curve shows the performance without tilt angle, the lower curve has been calculated for a tilt angle of 5°. For the parameters chosen the number of ambiguous clutter returns between the radar and the line of sight amount to 40. The effect of the tilt angle on the chosen look direction (pi = 0° has been corrected by beam squint. As has been shown in Reference 9, chapter 3, for a side-looking linear array without tilt the clutter Doppler is range independent. Therefore, ambiguous returns assume the same Doppler frequency so that all the ambiguous clutter arrivals fall into the same clutter notch of the STAP filter.

Figure 5.23

Side-looking linear array, horizontal tilt angle 0°, 5°

Figure 5.24

Side-looking planar array, horizontal tilt angle 20°. Rectangular planar array; number of columns c = 24\ number of rows r = +7; o 2; * 4; x 6

Comparing the two curves in Figure 5.23 one can notice that even for a small tilt angle (5°) the clutter notch is broadened considerably by the ambiguous clutter returns which leads to losses in the detection of slow targets. For larger tilt angles the losses would increase. As mentioned above, crab angles up to 17° due to cross wind have been observed during a flight campaign with the AERII SAR instrument [5]. In Figure 5.24 a vertical rectangular array of radiating elements was assumed. The horizontal tilt angle is 20°. The curves have been plotted for different numbers of rows of radiating elements, i.e. different numbers of vertical degrees of freedom of the STAP processor. As can be seen, the ambiguous clutter returns are perfectly cancelled if the number of rows in the array is equal to four or larger. Obviously the vertical degrees of freedom of the planar array serve for vertical nulling of ambiguous clutter arrivals which appear at different depression angles.

5.4.2

Omnidirectional arrays

Let us now consider the effect of range-ambiguous clutter returns on the clutter rejection performance of the above described array configurations with 360° azimuthal coverage. In Figure 5.25 the effect of array tilt in the presence of range-ambiguous clutter on the improvement factor achieved by a displaced ring configuration (see Figure 5.8) is shown. The upper curve has been calculated for zero tilt angle and serves as a reference. It can be noticed that some degradation occurs on the left-hand side of the clutter notch (negative Doppler) similar to the effects observed for side-looking arrays (see Figure 5.23). However, much stronger losses can be observed at positive

Figure 5.25

Effect of antenna tilt on STAP performance (displaced horizontal circles), tilt angle: tilt angle = o 0°, * 5°, x 10°, + 20°

Doppler frequencies. Even for small tilt angles (e.g. 5°) dramatic losses show up. These losses come through the array backlobe due to the fact that we used omnidirectional radiating elements. In contrast, for the linear array (Figure 5.23) cos2-shaped directivity patterns were assumed. For the other two omnidirectional array concepts (displaced randomly thinned array, octagonal array) similar results are obtained as can be seen in Figures 5.26 and 5.27. Following the result given in Figure 5.24 it appears that the only remedy against the degrading effect of an array tilt is to use a horizontal array with two-dimensional degrees of freedom, i.e. two-dimensional subarrays. Such an array is shown schematically in Figure 5.28. Originally this kind of array was designed for the nose radar of a fighter aircraft, such as the AMSAR project [I]. In Figure 5.29 improvement factor curves are shown for the circular array shown in Figure 5.28. In contrast to the other array configurations which are in essence linear arrays with complex subarray structures, this array has a two-dimensional subarray structure and has, therefore, degrees of freedom in the azimuth and elevation dimensions. This array is obviously capable of nulling the ambiguous clutter arrivals in the vertical dimension. Therefore, the improvement factor curves for 0° and 20° coincide.

5.5

Conclusions

Several array architectures with 360° azimuthal coverage and STAP aptitude have been compared. Most of the concepts are based on circular dipoles. An important feature of such array antennas is that all radiating elements are always active.

Figure 5.26

Randomly thinned circular horizontal array, tilt angle = o 0°, * 5°, x 10°, + 20°

Figure 5.27

Octagonal horizontal array tilt angle = o 0°, * 5°, x 10°, + 20°

The results can be summarised as follows: 1

Four horizontal linear arrays • Only one fourth of the elements is busy (in our example 240 out of 960 elements). About 12 dB loss in SNIR has to be taken into account.

Figure 5.28

Circular planar array with checkerboard subarrays

Figure 5.29

Checkerboard circular array, tilt angle = o 0°, * 20°

• •

Very narrow clutter notches (corresponding to the MDV) are produced by the STAP filter. The width of the clutter notch does not depend much on the number of array channels.

•

The directivity pattern exhibits a relatively narrow beam and the usual sidelobe pattern. Since half-wavelength spacing is used, tapering for reducing the sidelobe is possible. • The performance of the forward-looking arrays is affected by ambiguous clutter returns which cause additional clutter notches. • Compensation for range-ambiguous clutter returns can be accomplished by replacing linear arrays by vertical planar arrays with vertical degrees of freedom. The adaptive STAP processor will cancel the ambiguous returns by spatial nulling. 2 Displaced ring arrays concept • Clutter notches are broader than for a linear array. • This disadvantage is offset by the fact that the achievable SNIR is 12 dB higher than for the 240 elements linear array. • The width of the clutter notch increases with the number of subarrays (array channels). • Compared with the linear 240 elements array the beamwidth is enlarged and the sidelobes are raised. • The displaced rings form a side-looking linear array with directive elements generated by the ring-shaped subarrays. Therefore, range-ambiguous clutter returns fall altogether on the same Doppler frequency and thus do not cause any distortion of the clutter notch. • Reduction of the sidelobes by tapering is probably not possible. 3 Randomly thinned circular planar arrays • Clutter notches are broader than for a linear array (similar to the displaced rings concept). • As before, this disadvantage is offset by the fact that the achievable SNIR is 12 dB higher than for the 240 elements linear array. • The width of clutter notch increases with the number of subarrays (array channels). • The achievable beamwidth is comparable to linear 240 elements array, and the near sidelobes are lower. • The randomly thinned circular planar array form a side-looking linear array with directive elements generated by the shifted uniform subarrays. Rangeambiguous clutter returns fall altogether on the same Doppler frequency and thus do not cause any distortion of the clutter notch. • Reduction of the sidelobes by tapering is not possible. 4 Octagonal array. All the subsequent aspects are also valid for a quadratic or rectangular array. It is expected, however, that the octagonal shape approximates better a circular array than does a quadratic array. It is expected that the beam shape does not vary with azimuth angle as much as for a quadratic array. • Clutter notches are narrow. • The width of the clutter notch is almost independent of the number of subarrays. • The achievable SNIR is 12 dB higher than for the linear 240 elements array. Excellent clutter rejection performance is obtained with two subarrays only.

•

The achievable beamwidth is much larger than for the linear array because all elements are distributed in a plane instead of a line. A narrow beam can be obtained only by increasing the number of elements. • The randomly thinned circular planar array form a side-looking linear array with directive elements generated by the shifted uniform subarrays. The subarray configuration has the properties of a side-looking array and, hence, receives clutter returns with Doppler frequencies constant with range. The clutter notch is not affected by range-ambiguous clutter returns. • Effects of aircraft crabbing on the detectability of slow moving targets can be mitigated by subdividing the array in the cross flight dimension. • Reduction of the sidelobes by tapering is possible because we deal with a half wavelength spaced array. 5 Effect of array tilt • Linear arrays as well as omnidirectional arrays based on a linear subarray structure are very sensitive to horizontal tilt if the radar is operated in a rangeambiguous mode. • Among those omnidirectional arrays considered the octagonal array appears to be the preferable configuration. For small tilt angles (about 5°) the losses are tolerable. • Due to the two-dimensional nature of the clutter returns (azimuth, elevation) the array needs a two-dimensional subarray structure in order to spatially cancel with range-ambiguous clutter returns. Perfect cancellation of the range-ambiguous clutter returns can be achieved with an array with a two-dimensional subarray structure. • The price for the robustness against array tilt by using an array with twodimensional subarray structure is an increase in required degrees of freedom in two array dimensions (array channels). The minimum amount of degrees of freedom required is a topic for further studies. • The geometry of range-ambiguous clutter arrivals is range dependent. Therefore, the STAP processing has to be carried out in a range-dependent fashion (as has been proposed in Reference 15) which involves high computational workload. • Doppler compensation techniques as used in References 13 and 4 may be a future way to avoid range-dependent STAP processing. Currently no technique for Doppler compensation of range-ambiguous clutter data is available.

References 1 ALBAREL, G., TANNER, J. S., and UHLMANN, M.: 'The trinational AMSAR programme: CAR active antenna architecture'. Radar '97, 14-16 October 1997, Edinburgh, Scotland, pp. 344-347 2 KLEMM, R.: 'Adaptive antennas for ground surveillance radar'. Proceedings of RTOIST/SET symposium, 7-8 April 2003, Chester, UK

3 BELL, K. L., VAN TREES, H. L., and GRIFFITHS, L. J.: 'Adaptive beampattern control using quadratic constraints for circular arrays'. ASAP 2000 workshop, 13-14 March 2000, MIT Lincoln Laboratory, pp. 43-48 4 BICKERT, B., ENDER, J., and KLEMM, R.: 'Verification of adaptive signal enhancement algorithms based on STAP techniques using four channel AER-II radar data in a forward looking air-to-ground GMTI mode'. Proceedings of TIWRS 2003, Elba, Italy, 15-18 September 2003 5 ENDER, J.: 'Experimental results achieved with the airborne multi-channel SAR system AER-II'. Proceedings of EUSAR'98, 25-27 May 1998, Friedrichshafen, Germany, pp. 315-318 6 FUHRMANN, D. R. and RIEKEN, D. W.: 'Array calibration for circulararray STAP using clutter scattering and projection matrix fitting'. ASAP 2000 workshop, 13-14 March 2000, MIT Lincoln Laboratory, pp. 79-84 7 GROGER, L, SANDER, W., and WIRTH, W. D.: 'Experimental phased array radar ELRA with extended flexibility'. Radar '90, Arlington, VA, 1990, pp. 286-290 8 KLEMM, R.: 'Antenna design for airborne MTI'. Proceedings of Radar 92, October 1992, Brighton, UK, pp. 296-299 9 KLEMM, R.: 'Principles of space-time adaptive processing' (IEE Publishing, London, UK, 2002) 10 KLEMM, R.: 'A planar antenna for GMTI radar with 360° coverage'. Proceedings of IEE Radar, 2002, Edinburgh, Scotland 11 KLEMM, R.: 'Effect of aircraft crabbing on sidelooking STAP radar'. Proceedings of EUSAR, 2002, Cologne, Germany, pp. 203-208 12 KOGON, S. M. and ZATMAN, M.: 'Techniques for range-ambiguous clutter mitigation in space-based radar systems', in: KLEMM, R. (Ed.): 'Applications of space-time adaptive processing' (IEE Publishing, London, UK, this volume) 13 KREYENKAMP, O. and KLEMM, R.: 'Doppler compensation in forward looking STAP radar', IEEProc, Radar Sonar Navig., 2001,148, (5), pp. 253-258 14 LI, T., SIDIROPOULOS, N. D., and GIANNAKIS, G. B.: 'PARAFAC-STAP for the UESA radar'. ASAP 2000 workshop, 13-14 March 2000, MIT Lincoln Laboratory, pp. 49-54 15 MATHER, J. L., REES, H. D., and SKIDMORE, I. D.: 'Adaptive clutter and jammer cancellation for element-digitised airborne radar'. Proceedings of 33rd Asilomar conference, Pacific Grove, CA, 24-27 October 1999, pp. 92-97 16 WILDEN, H. and ENDER, J.: 'The crow's nest antenna - experimental results'. IEEE international Radar conference, Arlington, VA, 1990, pp. 280-285 17 WIRTH, W. D.: 'Radar techniques using array antennas' (IEE Publishing, Stevenage, Herts., UK, 2001) 18 ZATMAN,M.: 'Circular array STA?\ IEEE Trans. Aerosp. Electron. Syst., April 2000, 36, (2), pp. 518-527

Part II

Space-slow time processing for space-based MTI radar

Chapter 6

SAR-GMTI concept for RADARSAT-2 Christoph H. Gierull and Chuck Livingstone

6.1

Introduction

6.1.1 Background The transportation infrastructure of today's world has been readily observable from space as road systems, seaports and airports since the early days of earth observation satellites. As the spatial resolution of spaceborne sensors becomes finer, increased infrastructure detail can be observed. The construction, maintenance, management and evolution of transportation routes and facilities are major financial investments for all levels of government in all nations. The data used for infrastructure planning, design and operation has been based on observations of traffic flow and local spot tests. Up to the present time, there has been no economically feasible way to provide velocity and density measurements of traffic from remote sensing systems. The fundamental technology needed to map moving vehicles is, in fact, well known. Due to the urgent information requirements and high risks associated with military activities, extensive research into motion measurement radars (for example Reference 1) has resulted in the development of operational airborne facilities for detecting and mapping the movement of vehicles on the earth's surface (ground moving target indication or GMTI radars). The Joint-STARS radar ground surveillance aircraft [2,3], developed in the United States in the late 1980s, is a system of this type. Although Joint-STARS has been shown to be effective in its military role, the effective coverage area is relatively small and the cost of operation is high (even by military standards). Manned, airborne GMTI systems will not provide economically viable data sources for civilian applications. Many studies [4-7] have shown that space-based GMTI radar systems can solve the area coverage problem at spatial scales suitable for both civilian and military requirements. To date, the technology needed to implement the GMTI space-based radars (SBRs) has not been sufficiently mature and the estimated capital costs were

prohibitive. Recent advances in efficient, compact, satellite design and in active array antennas have changed the economics of SBR GMTI to a point where the development of a custom designed SAR GMTI demonstrator satellite is being planned. The payoff of proposed applications of low-cost satellite technologies that include active array antennas to commercial SAR satellites has made investment more appealing. At present (2004), no civilian spaceborne radar system has a GMTI capability; however, several spaceborne SAR systems have been, are being and will be flown in the near future. Today's civilian spaceborne SARs were an outgrowth of applications developed for civilian airborne SAR data. The airborne SAR systems evolved from military SAR reconnaissance radars developed in the 1960s and 1970s. The demand for spaceborne SAR data has now grown to the point where significant commercial funding of spaceborne radar system development is feasible. Due to the large capital costs associated with the development and operation of airborne GMTI sensors, these sensors have not been used for transportation system monitoring or other uses. All development drivers exist only in the public sector and are defined by national (primarily military) needs. The air to space path followed by SAR systems is unlikely to occur for GMTI systems. The data to develop volume markets for GMTI measurements must come from space. A fully functional GMTI radar system searches large areas in a sector scan mode to detect targets of interest, dwells on selected measurement tiles several times during the tile access time to measure velocity vectors and compiles target tracks. These radars are much more complex than SARs and must acquire and process or acquire and downlink much more data than an SAR system in the same observation time. SAR modes can be incorporated as a subset of functions into a full GMTI radar. Design studies for spaceborne GMTI emphasise the need for large, real-time, onboard processors to reduce downlinked data volume. Although suitable processing capability can now be developed for ground installations at reasonable cost and weight, space qualified (radiation tolerant) versions are still in the future. Although all of the processes needed for GMTI have been developed for airborne systems, differences in platform velocity, range to target and accessible depression angles between airborne and spaceborne radars result in several unknown parameters for the optimisation of spaceborne GMTI sensors. Cost, complexity, available technology and design risk have all combined to preclude the construction and launch of a spaceborne GMTI.

6.1.2 Addition of MTI modes to spaceborne SAR Instead of deploying a fully functional spaceborne GMTI a polarimetric SAR system that has two parallel receiver channels and associated data recorders and downlinks can be easily converted into a two-channel displaced phase centre antenna (DPCA) radar if a full corporate feed is used for each of the polarisation channels. The essential conversion is a switch that replaces the first splitter link in the feed chain and allows the two halves of the antenna to be routed to the two receiver channels. If an active antenna (distributed transmit/receive (TR) modules) design is used, additional beamforming controls can be imposed to minimise azimuth sidelobes and to match the two subbeams. Provided that the radar is designed to operate as an interleaved

pulse (phase coherent) polarimeter, the SAR PRF will be in a range suitable for DPCA or DPCA-like STAP processing. When the GMTI operation is constrained to the SAR beams defined by the rest of the radar control, GMTI and SAR functions can be simultaneously performed in parallel streams in ground processors. Limited azimuthal beam steering may or may not be available depending on the SAR antenna design drivers. If, in addition to the multi-aperture antenna capability, the radar control is designed to generate the SAR operating parameters from look-up tables of actuator settings on board the spacecraft, the addition of a limited set of GMTI modes requires only additional memory space. Since the SAR is being built regardless of GMTI, this is the lowest cost approach to developing an experimental spaceborne GMTI radar. Although the subset of possible GMTI operating modes available from a radar of this type is small, it can be used to validate GMTI parameters and algorithms needed for more sophisticated radars (experimental functions). The radar can also be used to begin the investigation of possible high-volume applications of GMTI data.

6.1.3 RADARSA T-2 moving object detection experiment RADARSAT-2 is the second commercial Canadian SAR satellite and, like its predecessor, RADARSAT-I, represents a major advance in space-based SAR capability. Figure 6.1 shows an artistic conceptional view of the RADARSAT-2 system deployed in space, which will be owned and operated by MacDonald

Figure 6.1 Artist s conceptional view ofRADARSAT-2 (© Canadian Space Agency)

Dettwiler & Associates (MDA). The RADARSAT-2 system is being designed to provide extended access to the RADARSAT-I modes (with an added cross polarisation channel, HV). These are augmented by a set of new modes that include: an expanded RADARSAT-I mode set (VV and VH polarisations), polarimetric modes, high-resolution modes and an experimental GMTI mode. RADARSAT-2 increases terrain access opportunities by allowing the satellite to roll so that all modes can be used either to the right or the left of the satellite track. As the satellite traverses its orbit it will be mechanically steered in azimuth to coarsely compensate for earth rotation (yaw steering). RADARSAT-2 achieves its expanded capability by taking advantage of the inherent flexibility of an active array antenna design in which transmitter, receiver and control functions are integrated with, and distributed over, the antenna structure. Other functional features that arise from technical advances include increased attitude knowledge and control precision (star tracker attitude measurement), improved orbit knowledge (GPS receivers), programming flexibility (computer architecture advances and distributed control), increased recorder service life and input data rates (solid state recorders) and increased output data rates (dual data downlinks). Further details can be found in Reference 8. RADARSAT-2 MODEX will provide the first opportunity to routinely measure and monitor vehicles moving on the earth's surface from space. In historical terms this is a GMTI mode. The RADARSAT-2 SAR antenna design allows the antenna to be partitioned into two halves along the direction of flight and thus permits two closely spaced observations to be made of the same scene to observe temporal changes [9]. As the radar system is fundamentally a strip mapping SAR, the total observation time for any point on the earth's surface is limited to the time that the radar beam illuminates that point as it sweeps by. The dwell time for velocity measurements cannot exceed the real aperture time of the radar. Object motion will be measured with the following SAR-GMTI techniques: along-track interferometry (SAR-ATI), SAR-DPCA and space-time adaptive processing (SAR-STAP). The RADARSAT-2 MODEX capability is being built into the satellite system by the developer MDA, with collaboration and sponsorship from the Canadian Space Agency (CSA) and the Canadian Department of National Defence (DND). One of the key objectives of the DND GMTI project is to assess the strengths and weaknesses of space-based SAR GMTI systems for measuring cultural activities on the earth's surface. The development of the ground processing/analysis segment for RADARSAT-2 MODEX will be based upon models derived from theoretical understanding of the measurement process. These models are validated by simulations [10] and airborne SAR-GMTI experiments. The experimental airborne GMTI measurements use the two aperture ATI mode of the CV-580 SAR system operated by Environment Canada to provide experimental data for RADARSAT-2 resolution and incidence angles [H].

6.2

Analysis of SAR-GMTI modes for RADARSAT-2

The aim of this section is to evaluate RADARSAT-2's potential detection performance for the different SAR-GMTI techniques based on a statistical description of the

detection problem. In the subsequent analysis, we limit the model complexity (degrees of freedom (DoF)) to the number of antenna elements (phase centres) rather than the complete dimension of space-time samples, commonly used in STAR In fact, the spatial snapshot vectors are considered as mutually stochastically independent along the time (azimuth) or frequency (Doppler) direction, respectively, therefore reducing the model complexity to the space dimension only. The underlying reason for this is the practical execution of such techniques for combined SAR-GMTI purposes. On one hand, there are the classical methods such as DPCA and ATI which are applied on the processed SAR images. Adjacent SAR image pixels are independent as sampling meets the Nyquist criterion. On the other hand, there are raw data-based techniques like raw DPCA [12] or STAP which are efficiently implemented in the Fourier or Doppler domain, where the frequency bins are asymptotically independent. The statistical dependence between the samples is negligible when the time base for the Fourier transform becomes sufficiently large, which is easily satisfiable in case of SAR with a large number of transmit pulses [13].

6.2.1

Background

6.2.1.1 Classical GMTI radar The classical airborne or spaceborne GMTI radar is not an imaging system. The radar outputs are the position coordinates and radial velocity vector components of moving targets. When the radar is designed to observe a scene containing moving targets from different azimuth angles (angle measured from the direction of travel of the radar) the velocity vector of the targets can be inferred. For both airborne and spaceborne radar systems, the motion of the radar and the angular width of the radar beam are combined to embed a component of the radar velocity into the radar returns from stationary scene components. From the radar's viewpoint, the moving scene elements are embedded in a moving scene. The Doppler spectral lines associated with the moving targets are embedded in the scene Doppler spectrum. One effective technique for separating the spectra of moving targets from that of the static scene is to partition the radar antenna into two or more spatial subapertures distributed along the radar's direction of motion. Each spatial subaperture is coupled to its own receiver channel and defines a spatial DoF for adaptive processing of the channel output set. Proper synchronisation of the radar sample frequency (PRF) and the along-track position of the phase centres allows the use of a DPCA background (clutter) cancellation technique for enhancing the moving target signal. However, the alignment of the receiver channels with the flight direction is not imperative because STAP, for instance, works also for forward-looking arrays. 6.2.1.2 Combining SAR and GMTI Because it is not an imaging sensor, a classical GMTI radar only needs to detect and extract measurements from a relatively small number of points (target and potential target locations) within the illuminated scene. Appropriate choices of beam scanning and sampling parameters (PRF) allow the radar to accommodate range (or azimuth)

ambiguous measurements and to search large areas for moving targets by rapidly stepping the beam position to dwell on scene tiles for predefined dwell times. When the radars operate in these modes the acquired data does not meet SAR imaging criteria and the GMTI and SAR modes of radar operation are incompatible. When the radar beam orientation is fixed for long periods of time and when the radar PRF, range gate delay and swath are selected to minimise range and azimuth ambiguities, the same data stream can be processed, through separate processing paths, to produce SAR images, temporal SAR interferometer velocity images and GMTI outputs. In this case, the opportunity exists to apply knowledge gained through one process to the results of another with a net increase in information output. The measurement of object motion using SAR requires two operations. These operations are: the detection of the movers in the SAR data, and the estimation of their velocity vectors and azimuthal locations in the ground-range plane. Target detection and parameter estimation can either be performed incoherently, with a single SAR aperture, or coherently (with much higher fidelity), with two or more apertures [13-20]. 6.2.1.3 SAR imaging of moving targets A conventional range/azimuth coordinate system is assumed in which the azimuth direction on the imaged surface is taken to be parallel to the motion of the radar. Assuming range compression, the imaging geometry model for any target point (x, v, z) on the imaged surface can be expressed in terms of the systematic phase history. If the radar position is given by Rp(O = [xp, yp, zp]T, and the target position is given by Rr(O = [XT, yr, ZTY\ a n d t is defined to be zero when the radar is broadside to the target, then the relative systematic phase history is given by:


te {-|, | J

(6.1)

where range R(O = Rp(O - Rr(OFor an SAR operation, the duration, T9 is the maximum usable synthetic aperture time determined by the two-way antenna pattern.
assumptions. They will generally be displaced in azimuth from their proper image position and will be superimposed on correctly positioned radar returns from unrelated terrain. Signals from badly mismatched objects may not be discernible at all easily in the SAR image. The image-domain representation of mismatched moving targets will be dominated by four effects: range walk, azimuth impulse response broadening, azimuth smearing and azimuth displacement. Depending on the characteristics of target motion, any combination of the above four effects may be apparent [22-24]. 6.2.1.4 SAR measurement of target motion Once a mover is located, the range walk, azimuth displacement, azimuth smear and defocusing effects all provide non-coherent clues to target motion in SAR data. Single aperture techniques can be used to provide a coarse estimation of target velocity for sufficiently strong, sufficiently fast targets. Successful velocity estimation will be contingent upon reliable separation of target size effects and motion spreading effects. If the SAR signal spectrum is filtered into overlapping subbands in the azimuth direction, each subband will then correspond to a subdivision of the radar's azimuth illumination beam into overlapping subbeams. When each subbeam data set is focused to an SAR image, each of the images will have observed the same scene elements at different times. Features whose range location has shifted from one subbeam scene to the next are moving target candidates. The shift divided by the difference in observation time provides a first, rough estimate of radial velocity. Moving target candidates that are not visible in an SAR image processed with a SWMF can, for instance, be found by processing the scene data set for several hypothetical radial velocities that are selected to be in a reasonable range. Image features that become more prominent at non-zero hypothetical speeds are identified for further investigation. A combination of subbeam centroid shift and target enhancement as a function of matched filter velocity hypothesis provides a refined radial velocity estimate. The sensitivity of this non-coherent approach and the resulting accuracy of the inferred target velocities are dependent on the fundamental resolution of the radar mode used to generate the images. If the radar antenna is partitioned into two or more apertures that are distributed along the direction of motion of the radar, and if the design allows each aperture to be assigned to a separate, physical receiver channel, then coherent processing of simultaneously received signals can be used to detect moving targets and estimate their velocities and positions. The fundamental principle of this process is that each aperture observes the scene from the same point in space at a different time. In consideration of sampling ambiguity effects, the azimuth sampling of the radar (PRF) must be increased to adequately sample each aperture. Partition of the antenna into two apertures results in each aperture having a larger azimuth beamwidth than the combined antenna. The synthetic aperture time T will increase and thus the observation time available for the incoherent (impulse response-based) estimation processes discussed previously will increase. For an aperture phase centre separation of distance d the time lapse between terrain observations from the same point in space is: r = d/1|\p ||, where T « J1 and \p is the radar platform velocity. Typically r

has values between 0.5 ms and a few ms. When relatively calibrated SAR data from each aperture are focused to a complex image (using the same matched filter for each image) and then spatially registered to each other, the image content will be common except for scene changes over time r. For land (as opposed to ocean) scenes, 'fixed' elements are stationary over time r and their image representations will be very nearly identical. Moving objects, however, will be displaced between the two scenes and these displacements can be measured to fractional wavelength accuracies as phase shifts. When a so-called three-beam DPCA process is used, monopulse processing techniques can be employed between beam pairs to refine estimates of the azimuth position of observed targets. Existing, operational GMTI radars use three physical phase centres to provide robust DPCA clutter suppression and monopulse target position measurement operation. So-called adaptive DPCA clutter suppression algorithms use both spatial and temporal degrees of freedom to detect and measure moving targets. These algorithms are special cases of space-time adaptive processing (STAP) [25,26]. Its successful application to multichannel SAR data has, for instance, been described in References 13 and 14.

6.2.2 Statistical models of measured signals As long as the SWMF is strictly a linear operator on the received data, the shown statistical properties are valid for both the raw data and the processed SAR images. Even though in reality one has to deal with a variety of clutter properties, particularly heterogeneous terrain or water surfaces, the following performance analysis concentrates for clarity on homogeneous clutter. Extensions to the heterogeneous case can be found elsewhere, e.g. Reference 27. 6.2.2.1 Stationary clutter The quadrature components of a radar channel for the stationary world (clutter) are often modelled as zero-mean Gaussian processes which implies Rayleigh distributed magnitude. The model is warranted for rough (on the scale of the wavelength) but homogeneous backscatter magnitude terrain because the sum of independent (rough), identically (homogenous) distributed scatterers, will be complex Gaussian distributed by the central limit theorem. In a typical SAR image, the model results in granularity commonly denoted as speckle. The model has been validated over homogeneous agricultural and natural areas, but breaks down over more heterogenous or smooth surfaces such as urban or sea surfaces, see e.g. Reference 28. For a multichannel SAR system, the one-channel analysis has to be extended to several channels. As a starting point consider Af channels and define the column vector Q = [ Q i , . . . , QNV with elements Qt = y ^ J Z / , where Z^ is a standard complex normal distributed variable, i.e. Z/ ~ M (0,1). The mean power level of the /th channel is denoted by Prj for i = \,...,N and T is the transpose operator. The vector Q is modelled as a multivariate complex Gaussian random vector with density /q(q) Z=Z ^-N det(Q)""1 exp(—q*Q -1 q), where Q denotes the clutter (plus noise)

covariance matrix, the superscript * means complex conjugate transpose and det(Q) is the determinant of Q [29]. The following matter considers RADARSAT-2 where only two channels are involved, i.e. the dimension of Q is chosen to be N — 2. The covariance matrix can be written to:

(6.2) where Prt = Pcj -f Pn, p = y/Pc,\ Pc,i/(Pr,\ Pr,2), I is the identity matrix and p exp (jO) is the complex correlation coefficient between the two channel outputs. The magnitude of the complex correlation coefficient p will be referred to as coherence for simplification of notation. Cross-channel correlation is a potential source of information in different areas of remote sensing. It is a composite measure of all effects which lead to decorrelation of the signals, for example, temporal decorrelation due to surface backscatter variation and the unavoidable receiver or sensor noise. Typical values for solid surfaces in single-pass across-track interferometry as well as for along-track interferometry are larger than 0.95 but can be significantly lower for water surfaces p < 0.8. In order to reduce the speckle, the data are multilook processed which requires averaging several independent one-look interferograms. The n-look sample covariance matrix is given as: (6.3) where n is the so-called number of looks and q^ the &th one-look or single-look sample. Even though the following theoretical derivation of the PDFs is based on the assumption of statistically independent samples, it has been shown that this PDF can still be applied if certain dependence between samples occurs. In such a case, compensation with an effective number of looks (which is smaller than the averaged sample size) can account for the statistical dependence [30,31]. The random matrix A = nQ is well known to be complex Wishart-distributed, A ~ Wp(n,Q) [29], with probability density:

/A (A)

= ,r ( n)rt ( -\Vdet( Q r exp(-tr{Q"'A})

(6 4)

-

For across-track interferometry the interferometric phase 6 in equation (6.2) depends on the illumination geometry and the terrain elevation, for along-track interferometry this value is often assumed to be zero (0 = 0), i.e., the clutter is assumed to be stationary. In cases of internal clutter motion, such as a sea surface current, 0 is a function of the current and wave surface velocities.

6.2.2.2 Moving targets Deterministic target model - As a first intuitive model, a deterministic moving target signal s(#) = \/T^[l,e~J^]T can be considered as superimposed upon the stationary clutter returns in each channel. Without loss of generality, the Doppler phase of targets in the fore channel can be set to zero. The real amplitude *J~FS determines the signal-to-clutter ratio (SCR). Since n single-look cells in equation (6.3) are summed, the resulting statistics depend on both the number of moving targets (included in the entire multilook cell) and their dimension compared to the multilook cell size. In the most general case, each single-look cell contains a moving point scatterer with different RCS and different radial velocities. This scenario might happen when observing either urban areas with a potentially large number of movers or extended targets where different scattering centres may have different radial velocity components, such as large ships rolling and pitching in the sea. In such cases, the /th channel output is:

Xi(Jc) = s(k,») + qiik) = y[pMe>m

+ qt(k\

k = 1,2, . • • ,ft

(6.5)

The composite random vectors x(k) = [x\(k),X2(k)]T are mutually independent identical complex normal distributed with expectation s(fc, #) = ^ Ps(k){\,e^k)Y fork = 1 , . . . , n and covariance matrix Q, i.e. X(A:) ~ Af?r(s(k), Q). In contrast to equation (6.3), the random matrix A = YHc=\ X(A;)X(Zc)* is now non-central complex Wishart distributed, A ~ Vl^p (ft, Q, Q)9 with non-centrality matrix Q = Q - 1 MM*, where: (6.6) The density function of the positive definite A is [32]:

As this density function contains a hypergeometric function of matrix arguments, analytical analysis is exceedingly complicated. However, for two practical cases statistical evaluation becomes possible. First, regarding SAR systems with resolutions of a few metres to submetre range, the assumption of a single moving target within a multilook cell for a reasonable number of looks (MO) seems adequate, i.e. s(k, 0) = S(1O-). This mover can either be a single-point scatterer or an extended target consisting of several scattering centres with approximately the same amplitudes and radial velocities, such as a large truck etc. Under the assumption that the target covers only / < n single-look cells, the matrix of expectation vectors in equation (6.6) is given as:

where the vector e/ has / components equal to one and the rest zeros. In this case the matrix product MM* (and therewith also the non-centrality parameter

Q = Q 1 MM*) has rank one. The non-centrality matrix can be written as £2 = lQ~ls($)s(&)*. The density function of A now involves hypergeometric functions of scalar arguments:

(6.7) The equivalent density function for real variables (when the non-centrality matrix has rank one) was originally derived by Anderson and Girshick [33] where they applied the identity 0F\(a;X) = V(a)/(VX)a~lIa~\(2VX) for the confluent hypergeometric function. / v ( ) denotes the modified Bessel function of order v. Second, when the number of looks becomes sufficiently large, the second moment matrix A can be simplified because the cross terms (of independent random variables with zero mean) tend to zero for increasing n. Hence it can be rewritten as a superposition of the target signal matrix on the clutter covariance matrix: (6.8) Gaussian target model - In the previous moving target model it was assumed that the single mover amplitude was constant from one-look cell to the next one-look cell. In practise this will not be the case because varying propagation conditions, system instabilities and, primarily, variation of the target RCS caused by changing aspect angles are reasons for amplitude fluctuation. If a target consists of many single reflectors, then the backscattered signal is composed of many single contributions with quasi random relative phases. Accordingly, it can be approximated as complex normal distributed. Over short periods of time, e.g. corresponding to the fast time or range direction in SAR, the aspect angle changes only a little and the amplitude can be assumed to be constant. This model is commonly named the Swerling I case. In the flight direction of the SAR, i.e. slow-time or azimuth direction, the pulses may be far enough apart that the amplitudes of the single returns can be considered as stochastically independent. This so-called Swerling II case is investigated in the subsequent sections. Depending on the geometric resolution compared with the target dimensions, two Swerling II applications have to be distinguished. If the mover dimension is of the order of the multilook cell size the resulting covariance matrix for signal plus interference (clutter and noise) can be modelled analogously to equation (6.5), but where the deterministic target signals have to be replaced by the random variables 5/ (k) for / = 1,2. As mentioned above, the composite random vectors S(Jc) = [S\(k)9S2(k)]T are assumed to be mutually independent complex normal distributed S ~ TV^(O, Qs) with expectation zero and covariance matrix:

where ps denotes the coherence of the moving target signals. It is important to note that in the absence of temporal decorrelation, the coherence between the two channel outputs will be exactly identical for the target and the clutter. With temporal decorrelation, the target coherence can either be larger or smaller than the surrounding clutter coherence. A perfectly steady vehicle, for instance, smoothly moving over a terrain of vegetation might have a larger coherence than the vegetation. The contrary, where the vehicle is rolling and bouncing over terrain which is otherwise perfectly stationary, leads to a lower coherence for the target compared with the surrounding terrain. However, this motion-induced decorrelation will not change the principle form of the optimum filter (equation (6.12)). Therefore, by setting ps = 1, one can concentrate on the deterministic signal vector s(#) = ^fP~s[l,exp(—j$)]T. Since the clutter and the target are mutually stochastically independent, the sum of them will still be complex normal distributed, i.e. X ~ A/^ (O9 Qs + Q)- The resulting sample covariance matrix: (6.9)

with \(k) = q(ifc) + s(ifc) is therefore complex Wishart distributed A ~ W*p(n,R) with n degrees of freedom. The composite covariance matrix is given as R = Qs + Q which has the same structure as in equation (6.2). In cases where the moving target dimension is smaller than the multilook cell size, the model in equation (6.9) is no longer adequate. Instead, we get for the sample covariance matrix: (6.10)

when it is assumed that the mover is only contained in the first / G {2,..., n] onelook cells. Therefore, the random matrix A consists of a sum of two independent complex Wishart distributed matrices B ~ W*p(Z,R) and C ~ W ^ (n - /,Q). To the best knowledge of the authors the corresponding density function has not yet been derived in closed form.

6.2.3 SCNR optimum processing 6.2.3.1 Known covariance matrix Let the signal output, i.e. the test statistics on which the actual detection is based, be the magnitude of y = b*X, where b is the filter or beamformer vector and X = [Xi, X2]T is the sensor outputs. Due to one of the most basic facts in detection theory, the probability of detection of a signal in noise depends mainly on the SNR. Therefore, one criterion for adjusting the beamformer vector b could be maximisation of the signal-to-clutter-plus-noise ratio (SCNR) K of the signal output when the processor

is matched to Doppler phase #: (6.11) where s(#) = y^[l,exp(—jtf)] 7 ' is the given target signal vector at this Doppler phase. It is easy to verify that the optimum filter is: bopt(#) = K T 1 / 2 s ( # ) =

YQ~1S(#)

(6.12)

The optimum filter can be interpreted as a two-step process, the decorrelation (whitening) of the clutter with Q" 1 / 2 and a matched filtering with the adapted signal s = Q~1//2s. Applying the optimum filter (e.g. for y = 1) to the measured data vector x, yields for the signal output: y= sWQ-1!

(6.13)

where the data vector x contains only clutter plus thermal noise, i.e. X = Q - A/2 (0, Q), the beamformer output (equation (6.13)) will be a linear combination of normal variables which is also complex normal distributed with zero mean and variance: a 2 = E|v| 2 = S(^) + Q" 1 EXX*Q~1s(#) = s(^)*Q~ 1 sW

(6.14)

Therefore, the magnitude \y\/cry is Rayleigh distributed and the detection threshold can be calculated via the cumulative distribution function of the Rayleigh distribution for any given false alarm rate a. If, for instance, a deterministic target signal s(#) is included in x, such that the corresponding random vector X = s(#) + Q, the magnitude \y\/cry will be Rice distributed with parameter s(^)*Q - 1 s(#)/<7y, from which the probability of detection Pd for any arbitrary Doppler phase # can be directly calculated. If the target signal S itself can be considered as a Gaussian random variable, i.e. X — A/2 (0, s(#)s(#)* + Q) see subsection 6.2.2, the output y is zero-mean complex normal distributed with variance: a] = JWQ- 1 S(O)(I + 8(O)+Q-1S(O))

(6.15)

and the magnitude \y\/cry follows a Rayleigh distribution. Figure 6.2 illustrates the histogram of \y\ for the deterministic target and Figure 6.3 for the Gaussian target with # = 0.5 rad over varying SCR based on simulated data. The superimposed solid curves are the corresponding Rayleigh and Rice density functions; a perfect match can be recognised. Since the random target model performance is generally worse in terms of lower probability of detection, it will serve as the worst case scenario for the remainder of this chapter. Depending on the spatial resolution of the SAR system, one possibility for increasing Pd is to average several independent time samples (snapshots) xm for m = 1 , . . . , n in order to reduce the noise component. However, applying the average on the complex data, i.e. \y\ = 1/M| YTm=\ s (#)*Q~ lx ml does not yield the desired effect, because the expectation and variance of the Rayleigh and Rice distribution are reduced

Figure 6.2

Histograms and theoretical PDFs of beamformer output for SCNR-optimum processing over varying SCR with deterministic target model

in the same way. In other words, this average only leads to a rescaling of the abscissa in Figures 6.2 and 6.3 and hence Pd remains unchanged. Instead, it is advantageous to average the magnitudes, i.e. \y\ = 1 / M ^ = 1 |s(^)*Q~ 1 x m |, therefore reducing the variance by a factor of \/n while keeping the expectation constant. However, assuming that the moving vehicle does not cover the entire multilook cell (which is likely for the rather moderate resolution of spaceborne SAR systems), the effective SCR will decrease by roughly a factor of l/n if / denotes the number of single-look cells covered by the target. As a consequence, there is a trade off in terms of reduction of noise versus reduction of SCR. Figure 6.4 shows the resulting probability of detection Pd for the Gaussian target model versus radial velocity for varying SCR for a number of temporal samples n — 9, when only / = 3 samples contain the target. Since the PDF in this case is unknown (see equation (6.10)), a numerical study has been performed instead. The Doppler phase, which is approximately given as:

(6.16)

Figure 6.3 Histograms and theoretical PDFs of beamformer output for SCNR-optimum processing over varying SCR with Gaussian random target model

was determined in accordance to RADARSAT-2 system parameters: wavelength X = 0.0556 m, sensor spacing d = 7.5 m and platform velocity vp = 7500 m/s. The false alarm rate was Pf — 10~4. One can see, for instance, that a target moving with a radial velocity of 18 km/h (5 m/s) and an SCR of 10 dB can be detected with a 97 percent probability, which corresponds to an improvement of about 17 percent compared with the single-look detector (due to limited space, the corresponding single-look result is not shown here). 6.2.3.2 SMI, LSMI and EVP In practice the covariance matrix is not known and is usually estimated from a set of sample vectors with the same clutter statistics, but which have to be free of moving target signals. Using the sample covariance matrix of equation (6.3) instead of the asymptotic (Q in equation (6.13)) is commonly called the sample matrix inversion (SMI) technique. Its loss compared with the optimum solution was first studied in Reference 34. If only a very few samples n are available for computing Q, a loading of the diagonal of Q prior to inversion can reduce the SCNR loss significantly. This method called loaded SMI (LSMI) has been studied, e.g. in Reference 35.

SCR = O SCR = 5 SCR=IO SCR= 15 SCR = 20 SCR=25

Figure 6.4

Probability of detection of the SNCR optimum processing for the Gaussian target model versus radial velocity for varying SCR (dB) after 9-look averaging. Only three single-look cells contain the target, Pf = 10~4

However, for side-looking SAR-GMTI the entire range dimension can support the covariance matrix estimation, assuming a sufficiently broad antenna pattern in elevation. Therefore the estimation loss can be neglected in most practical cases. Due to range dependence or heterogeneity of clutter returns, this is not the case, for instance, for squinted or forward-looking geometries [36]. Another interesting class of techniques, the so-called subspace or projection methods are a special case of the inverse of the covariance matrix when the CNR tends to infinity. Rewrite equation (6.2) as: (6.17) Use the matrix inversion lemma to get: (6.18)

Equation (6.18) tends for infinite CNR, P«/||p|| 2 = Pn/(Pc,i + Ptf) -* 0, towards a multiple of the projection PjJ- = (I — pp*/||p|| 2 ) onto the complement of the clutter subspace spanned by the vector p. The importance of such techniques for practical applications stems from the fact that projections suppress the clutter theoretically down to zero (independently of the actual clutter power) whereas the amount of clutter suppression achieved by SMI and LSMI is inversely proportional to the average clutter power level within the training data [26,37]. The arguably best known approach to estimate the clutter subspace basis vector p for finite sample size is based on the eigenvector decomposition of the sample covariance matrix. This eigenvector projection method (EVP) has been intensively investigated in the past by many authors for different radar applications such as superresolution (here it is traditionally called MUSIC) or interference suppression in large phased arrays, e.g. Reference 38. However, the EVP features an extensive computation oftentimes preventing its use in practical applications. Faster and computationally much less intensive techniques possessing similar fidelity have, for instance, been proposed [37] which will also be considered for RADARSAT-2-GMTI.

6.2.4 SAR displaced phase centre antenna The classical DPCA two phase centre clutter suppression can be regarded as a special case, or more precisely as an approximation, of the SCNR optimum processing. The inverse of the 2 x 2 covariance matrix of equation (6.2):

(6.19)

If one now assumes that the power in the two channels is identical, Pr?i = Pr\ = Pr and that 0 = 0, i.e. perfect calibration (removal of any channel imbalances), equation (6.19) yields: (6.20) Instead of scanning for a variety of values of ft in the matched filter, it has been found to be sufficient to use a fixed beamformer vector s(#o) = V^* P» exp(—y^o)]7, i.e. a single mismatched steering vector. Choosing, for instance, fto = ^ 5 which corresponds in the terminology of conventional MTI radar to the optimum speed, the signal output will be y = const.* (JCI — #2). In other words, for #0 = n, the optimum filter simplifies to the difference between the channel outputs regardless of the coherence [16,18,19]. Recalling that X ~ A^(O 5 Q), the magnitude of the difference \Y\/ay = \X\ X2I is analogous to the SCNR optimum processor which is Rayleigh distributed for the clutter-only case and Rice distributed, with parameter 2y/7^ sin(ft/2)/ay, for the deterministic target case. The variance is given as ai — Pr,i + Pr,2 — 2p/Pr~JP~^ COS(O).

In case of the Gaussian target model, i.e. X ~ №2 (Q9 R) with:

(6.21) the resulting beamformer output \Y\/ay is again Rayleigh distributed with &2y = 2(Pr + P5)(I - pcosO), where:

(6.22)

The receiver operating characteristics calculated on the basis of this PDF look similar to Figure 6.4, except that a loss of 2-3 dB compared with the optimum solution, i.e. when the optimum filter is optimally matched to the signal s(#), can be found. In particular, the probability to detect the aforementioned target moving with 18 km/h and a SCR of 1OdB is reduced to 93 percent from 97 percent for the optimum processing.

6.2.5

SAR along-track interferometry

In contrast to the previously introduced techniques, SAR along-track interferometry (ATI) exploits only the interferometric phase, i.e. the phase difference between the channels and ignores the magnitude information. After coregistering the channels and under ideal conditions the two channel signals are identical for stationary terrain (clutter). Hence, it can be cancelled out by computing the phase difference (i.e. the interferogram), leaving only the moving targets in the differential data. In practice, the unavoidable phase noise limits the degree of cancellation [17,32]. The argument of the off-diagonal elements A12 = A^1 in equation (6.3) describes the phase of the multilook interferogram for the clutter: (6.23)

The integration of the PDF of equation (6.4) with respect to the diagonal elements An and A22 then leads to the joint density function of the multilook interferogram's phase and magnitude [39,40], and further integration with respect to the magnitude

r\ leads to the marginal phase density:

(6.24) for — TV < r/r < it and where 2F\{a^b\c\y) is Gauss's hypergeometric function. It should be noted that, although n is an integer in the derivation of the complex Wishart density function, the function in equation (6.24) is still a density function when n is any real number n e IR greater than one [31]. The statistics of the interferometric phase when a moving target signal is superimposed upon the clutter have first been investigated in Reference 32. For a deterministic target signal s(#) and a sufficiently large number of looks n, i.e. when equation (6.8) holds, the joint density function of the interferometric phase \jr and normalised magnitude rj was determined to be:

(6.25) where <5 denotes the SCR 8 = Ps/^Pr,\Pr,2> Unfortunately, integrating equation (6.25) with respect to rj to get the marginal phase PDF seems to be intractable. Alternatively, numerical integration can be used. For the Gaussian target model, the PDF of x/r has the same form as equation (6.24) except that the correlation coefficient p has to be replaced by p and the phase 0 by 6 given in equation (6.22). For increasing SCR values, the phase density becomes wider than the original clutter PDF because the factor p is always less than one if # ^ 0, assuming that ps = p. At the same time the peak of the PDF migrates towards the target phase value #. When the SCR tends to infinity (Ps -> oo), p tends to p and 0 towards ^, i.e. the phase density function in that case is identical to the original clutter PDF except for the Doppler shift towards the target frequency. This behaviour of the PDF is illustrated in Figure 6.5, where the perfect agreement between theory and simulation can be seen. The target Doppler phase was chosen to 1.3 rad. Figure 6.6 shows the corresponding probability of detection as a function of target radial velocity for n = 9 and / = 3. It is found that the probability of detection is always less than that of DPCA. Particularly, the example target moving with 18 km/h and possessing a SCR of 10 dB can only be detected with 60 per cent chance compared with 93 per cent for multilook DPCA. However, even though the performance of DPCA seems to be significantly better than that for ATI, one has to bear in mind that this is only true for homogeneous

ATI phase, rad

Figure 6.5

Histograms and theoretical PDFs ofinterferometricphase over varying SCR with Gaussian random target model True phase \jr = 1.3 rad, n = 1

terrain. It was shown in numerous publications (e.g. References 27 and 40) that the interferometric phase is invariant against inhomogeneity of the clutter, hence preserving its detection capability in fairly heterogeneous composite terrain. In contrast, the PDF for DPCA will show larger tails in such cases, i.e. a higher probability of larger amplitudes. In order to keep the false alarm rate constant, the detection threshold must inevitably be increased which in turn decreases the probability of detection. A quantitative comparison of the performance loss caused by extremely heterogeneous terrain, such as urban areas, is currently under way, e.g. Reference 41.

6.3

SAR-STAP scheme for RADARSAT-2

6.3.1 Detection This section gives a brief summary of the theoretical background and the resulting algorithms for adaptive space-time (or more precisely space-frequency) processing in conjunction with SAR for RADARSAT-2. An elaboration of this topic and its successful application to experimental airborne SAR data can be found in References 13 and 14 and also in Chapter 3 in this book.

SCR = 0 [dB] SCR = 5 [dB] S C R - 1 0 [dB] SCR= 15 [dB] SCR = 20 [dB] SCR = 25 [dB]

v rad , m/s

Figure 6.6

Probability of detection of ATI for the Gaussian target model versus radial velocity for varying SCR (dB) after 9-look averaging, Pf = 10~4

Although for the statistical performance analysis in Section 6.2.3 it was sufficient to look at the problem in a static way, such as using two processed SAR images, STAP working on the raw data requires consideration of the time history of the received clutter and moving target echoes. Let x(/,S) be the received echo consisting of a moving target signal s(t,%) superimposed upon the stationary clutter qO) and the unavoidable thermal receiver noise n(t). The parameters describing the moving target, such as velocity in the along-track direction va, velocity in across-track or range direction vr and its location xo at time t = 0 are combined in the vector S = [va,vr,xo]T. In the following, the Fourier transform of the signal will be denoted as X(&>,§) = .^{x^S)}. Even though the correlation time r c of the clutter q(t) in the time domain is rc = l/BC9 where Bc = 2up@3dBA is the clutter bandwidth constrained by the antenna beamwidth @3dB> the frequency bins in the Doppler domain are asymptotically mutually independent. For a sufficiently long time base of the Fourier transform, the receiver output vector can be considered as being stochastically independent and complex normal distributed, i.e. X(O)9 S) - Np(S(CQ9 S), Q((o)), where S(ty, S) is the Fourier transform of s(t, S) and Q(co) the so-called spectral power density matrix of the clutter plus noise. Under this premise, the SCNR optimum filter for a particular frequency is given, analogously

to equation (6.14), y(oo) = S(co, ^TQ'1 (co)X((o) with the optimum SCNR: /CoPt(Co) = S(co, §)*Q~1 (a>)S(G>, §)

(6.26)

Figure 6.7 shows an experimental data example of this SCNR optimum processing in the Doppler range domain for an airborne two-channel side-looking SAR system [11], with S (co, £) = S chosen as fixed to S = [ 1, — 1 ] T . The left-hand image shows y before any clutter cancellation, i.e. y(co) = S*X(co) (the clutter energy spread over the entire clutter bandwidth can be clearly recognised) and the right-hand one after suppression with SMI, y(co) = S*Q~1X(ct>). The clutter covariance matrix was estimated over all the range bins and hence possesses a very low variance. In the right-hand image, one can clearly recognise the enhanced moving target signals which were partly or fully covered by the clutter. The fast moving cars on the highway (upper part) are at least partly outside of the clutter bandwidth, whereas the slow moving vehicle in the bottom part was completely covered. Since the target energy in S(&>, £) spreads over a certain bandwidth, depending on the parameter vector £, the optimum filter is modified as an integration over the target bandwidth:

range bins

range bins

(6.27)

Doppler frequency, Hz

Figure 6.7

Doppler frequency, Hz

Doppler range image of two-channel SAR data before (left) and after STAP processing (right)

To analyse the potential performance of SAR-STAP it is advantageous to have a closer look at the model for the received target signal: Si(t,$)

= Di(ute))e*PU2PRite))

(6-28)

where R(t) denotes the slant range distance and ui(t) the direction history (directional cosine) from the /th antenna to the moving target on the ground. D1-(M) describes the two-way antenna pattern of the /th channel. In the far-field assumption, u\ (t) = u \ (t) and the distances can be written as R((t) = 2R(t) + u\(t)d, where d is the spacing between the receivers and R(t) the slant range distance from any reference channel to the scatterer. The two-dimensional signal vector becomes: (6.29) In array processing terminology the vector a(w) is called the steering or direction of arrival (DoA) vector. Using a special property of chirp signal possessing a large time bandwidth product, the Fourier transform of equation (6.29) can be written in analytical form to: S(CD,!;) = y(coMu(co,i;))

(6.30)

[13], i.e. as a multiple of the DoA vector in direction:

(6.31)

For non-moving objects this dependency tends to a straight line u(co) — —co/(2fivp) where the slope is determined by the platform velocity vp. For spaceborne systems with vp ^$> va, equation (6.31) can further be simplified to u(co, £) = —co/(2fivp) + vr/Vp for any ground moving vehicle. Inserting the normalised vector of equation (6.30) into the optimum SCNR of equation (6.26) leads to the so-called space-time characteristics (STC) or transfer function: (6.32) which is an excellent tool for the examination of moving array systems for GMTI [13]. Figure 6.8 shows the anticipated STC of the two-channel RADARSAT-2 antenna when the signal is transmitted from one half of the antenna and received on both halves, resulting in a phase centre separation of about 3.75 m. For simplicity, the single-element pattern was chosen of exponential form with a beamwidth 0 3 dB = 0.42° (horizontal dashed lines) corresponding to the 7.5 m subaperture.

directional azimuth, deg

Doppler frequency, Hz

Figure 6.8

Space-time characteristics of the two-channel RADARSAT-2 antenna, d= 7.5m

The vertical dashed lines represent RADARSAT-2's maximum PRF of 3.8 kHz and combined with the horizontal lines indicate the GMTI-relevant Doppler direction plane. Bright white indicates perfect clutter suppression, i.e. a detection capability as it would be without any clutter present. Obviously, a fully white plane would be desirable. The adaptive filter (even though it is only of dimension two) forms a sharp notch along the clutter trajectory u(co) = —co/(2pvp) (white solid line). Target motion will create a deviation from this straight line, see equation (6.31), i.e. will more or less fall in the white area and hence become detectable. Most significantly, two ambiguity zones (notches) caused by the spatial undersampling of the array at either side of the plane are visible. Targets with a Doppler direction trajectory getting close to these notches will be partly suppressed as well and will be less detectable. However, the first ambiguous target velocity Vy can be seen to be at the corresponding direction u(v^) = 0.42°, resulting in vay = 0.007vp = 200km/h, which might, at least for ground moving vehicles, only create problems in rare circumstances such as fast traffic on a German autobahn. If, in contrast, the signals are transmitted from the entire aperture of 15 m and received on both halves, the effective two-way antenna beamwidth decreases to 03 ^B = 0.27° (the phase centre separation remains the same)

and the ambiguity notches are (although still existing) less pronounced. However, the clutter-free Doppler zone significantly increases (target detection is straightforward in this area) because of the narrower clutter bandwidth caused by the smaller antenna beamwidth.

6.3.2 Parameter estimation The entire slow-time range of the scene under consideration can be divided into segments of equal duration which are individually transformed in the Doppler domain. The detection and parameter estimation is then done in the Doppler range domain separately for each segment, for instance, by comparing the clutter-suppressed pixels with a predefined CFAR threshold. The duration (length) of the time segments can be chosen as the maximum time stationary targets stay in one Doppler cell. Under the assumption of statistical independency between the Doppler bins, the clutter suppression can be done individually for each frequency bin. For a side-looking SAR, the sample covariance matrix can be averaged over the entire range dimension, usually providing a very low variance of the estimate. Applying the inverse (or eigenvector projection) to the data in this Doppler bin means clutter suppression (with a performance specified by the STC of the moving array), leaving only the moving targets in the data. Having the Doppler frequency ft and the slant-range position Rt determined at this stage, the only remaining unknown parameter is the azimuth location xt or equivalently the target direction ut = xt/Rt. For a multichannel SAR system with N > 2, Ender [42] has proposed a very elegant way to estimate that direction. He used the measured array manifold (estimated via the first eigenvector of the sample covariance matrix) to yield high resolution azimuth spectra for DoA estimation. Unfortunately, this approach cannot be applied to a two-channel system like RADARSAT-2 which has only one spatial degree of freedom (DoF). This DoF is spent suppressing the clutter and hence there is none left to retrieve the direction information. One possible way to overcome this dilemma is the use of several space-time (or frequency) samples to increase the dimensionality. For instance, taking M subsequent time segments, i.e. time segments of same length but staggered by 1 , . . . , M PRIs, transformed into the Fourier domain will all cover exactly the same Doppler frequencies.1 For each Doppler bin the M, Af-element vectors are concatenated to form the NM-dimensional space-time vector X(&>) = [Xi (&>),... ,XM(CO)]T. The estimation of the corresponding space-time covariance matrix is done analogously along the entire range direction. As a main difference, the clutter will not be compressed into one large eigenvalue but spread over N + /3(M — 1) eigenvalues, where /3 denotes the number of half interelement spacing traversed by the platform during one PRI [25,26,37]. For example, if we chose M = 3, and recalling N = 2 and p = 0.52 (PRF = 3.8 kHz) for RADARSAT-2, the clutter rank will be roughly three, i.e. another three DoF would theoretically be available to estimate 1

In classical terminology for airborne MTI this technique is known as PRI-staggered STAP

the target direction. One disadvantage is that many slower targets (which are close to the clutter subspace) will have significant power distributions in the second and third eigenvalues. Hence, adapting and suppressing two or three eigenvalues would also suppress target energy. In other words, there is a trade off between improved DoA estimation and reduced detectability. However, as a compromise the detection could preferably be done without staggering (keeping also the computational load smaller) and the enlarged dimensionality only be used to estimate the location of the target. A drawback of this approach is the increased computational complexity due to the order of the space-time covariance matrix. The inversion or decomposition of such matrices is numerically very intensive. However, many fast methods which avoid these intensive operations but possess almost optimum fidelity are known from classical radar array applications such as jammer suppression or superresolution [26,37,43].

6.4

Conclusions

RADARSAT-2 is a two-aperture SAR interferometer. When used for GMTI measurements, RADARSAT-2 will use beams in the 40° to 50° incidence angle range to maximise the radial velocity component of vehicle motion. The airborne experimental SAR reported here was designed to replicate the RADARSAT-2 GMTI mode resolution and observation geometry as closely as possible and to test data processing algorithms that will be migrated to the RADARSAT-2 GMTI processor. The greatest difference between the airborne and space-based SAR/GMTI capabilities arises from the relationship between the platform velocity and the along-track velocities of moving targets. In the airborne case, the target speeds are a significant fraction of the radar speed and reasonably accurate azimuthal target speed estimates can be made. This is not true for space-based radars. The RADARSAT-2-GMTI processor, which is currently under development, will likely be composed of all presented techniques, i.e. ATI, DPCA and STAR Each technique has strengths and weaknesses. For instance, ATI and DPCA are computationally simple and robust but, because they are based on the processed SAR images, are strongly dependent on the stationary world processing fidelity. On the other hand STAP, working in the raw data domain, circumvents the processing problem, but is computationally much more demanding, particularly for SAR where often several thousands of pulses are used to form the image. Another essential criterion is the robustness against varying degrees of heterogeneity of the underlying terrain in order to provide similar GMTI performance even in challenging environments such as sea surface or urban areas. Feeding the received echoes into parallel GMTI processing chains will result in an estimated target parameter pool, where the redundant information can be used to reject doubtful hits and enhance stable detections and estimates. Last but not least, RADARSAT-2 Modex is designed as an experimental rather than operational/commercial mode with the goal of identifying the potential and weaknesses of single-pass spaceborne SAR-GMTI.

6.5

List of symbols

a b ft d Di K k n N PCi Pn Pri Ps \/r q Q Q Qs pe-i9 R(t) R s S ft u va Vp vr frad jco x X £ y

direction of arrival (DoA) vector beamformer weights wave number sensor spacing horizontal sensor antenna pattern signal-to-clutter-plus-noise ratio wavelength number of independent samples (looks) number of sensors (N = 2 for RAD ARS AT-2) clutter power at /th sensor noise power received echo power at ith sensor signal power interferometric phase clutter-plus-noise vector, clutter-plus-noise covariance matrix estimated clutter-plus-noise covariance matrix signal covariance matrix complex correlation coefficient between sensors range signal-plus-clutter-plus-noise covariance matrix signal vector signal spectrum vector Doppler phase directional cosine along-track component of target velocity platform velocity across-track (range) component of target velocity radial target velocity along-track target location at t = 0 received echo vector spectrum of received echo vector signal parameter vector beamformer output, output signal

References 1 STONE, M. L. and INCE, W. J.: 'Air-to-ground MTI radar using a displaced phase center phased array'. Proceedings of IEEE international Radar conference, 1980, pp. 225-230 2 BROADBENT, S.: 'Joint-stars: force multiplier for Europe', Janes Defense Weekly, April 1987, (19), pp. 729-731

3 COVAULT, C.: 'Joint-stars patrols Bosnia', Aviation Week and Space Technology, February 1996, (9), pp. 4 4 ^ 9 4 TSANDOULAS, G.: 'Space based radar', ScL, 1987, (237), pp. 257-262 5 BIRD, J. and BRIDGEWATER, A.: 'Performance of space-based radar in the presence of earth clutter', IEE Proc. F, Commun. Radar Signal Process., 1984, 131, (5), pp. 491-501 6 CANTAFIO, L. J.: 'Space-based radar handbook', (Artech House, Dedham MA, 1989) 7 CURRY, G. R.: 'A low-cost space-based radar system concept', IEEE Aerosp. Electron. Syst. Mag, September 1996, pp. 21-24 8 Canadian Space Agency and MacDonald Detwiler & Associates: 'RADARSAT-2 programme', http://www.space.gc.ca/radarsat-2, http://radarsat.mda.ca, 2002 9 THOMPSON, A. A. and LIVINGSTONE, C. E.: 'Moving target performance for RADARSAT-2'. Proceedings of IGARSS, July 2000 10 CHIU, S. and LIVINGSTONE, C : 'A simulation study of RADARSAT-2 GMTI performance'. Proceedings of IGARSS, Sydney, Australia, 2001 11 LIVINGSTONE, C , SIKANETA, L, GIERULL, C. H., CHIU, S., BEAUDOIN, A., CAMPBELL, J., BEAUDOIN, J., GONG, S., and KNIGHT, T.A.: 'An airborne SAR experiment to support RADARSAT-2 GMTI', Can. J. Remote Sens., December 2002, 28, (6), pp. 1-20 12 GIERULL, C. H. and SIKANETA, I. C : 'Raw data based two-aperture SAR ground moving target indication'. Proceedings of IGARSS, Toulouse, France, 2003 13 ENDER, J. H. G.:' The airborne experimental multi-channel SAR system AER-II'. Proceedings of EUSAR '96 conference, Konigswinter, Germany, 1996, pp. 49-52 14 ENDER, J. H. G.: 'Space-time processing for multichannel synthetic aperture radar', Electron. Commun. Eng. J., February 1999, pp. 29-38 15 BARBAROSSA, S.: 'Detection and estimation of moving objects with synthetic aperture radar; part 1: optimal detection and parameter estimation theory', IEE Proc. Ff Radar Signal Process., 1992,1, (139), pp. 79-88 16 SUN, H., SU, W., GU, H., LIU, G., and NI, J.: 'Performance analysis of several clutter cancellation techniques by multi-channel SAR'. Proceedings of EUSAR conference, Munich, Germany, 2000, pp. 549-552 17 YADIN, E.: 'Evaluation of noise and clutter induced relocation errors in SAR-MTI'. Proceedings of IEEE international Radar conference, 1995, pp. 650-655 18 YADIN, E.: 'A performance evaluation model for a two port interferometer SAR-MTI'. Proceedings of IEEE national Radar conference, 1996, pp. 261-266 19 STOCKBURGER, E. F. and HELD, D. N.: 'Interferometric moving ground target imaging'. Proceedings of IEEE international Radar conference, 1995, pp. 438^43 20 SIKANETA, I. C , GIERULL, C. H., and CHOUINARD, J. -Y: 'Metrics for SAR-GMTI based on eigen-decomposition of the sample covariance matrix'. Proceedings of international Radar 2003, Adelaide, South Australia, 2003

21 FRANCESCHETTI, G. and LANARI, R.: 'Synthetic aperture radar processing' (CRC Press, 1999) 22 RANEY, R. K.: 'Synthetic aperture imaging radar and moving targets'. IEEE Trans. Aerosp. Electron. Syst., 1971, 7, (3), pp. 499-505 23 FREEMAN, A.: 'Simple MTI using synthetic aperture radar'. Proceedings of IGARSS, 1984, pp. SP-215 24 ENDER, J. H. G.: 'MTI-SARprocessing'. Carl-Cranz-Gesellschaft, course notes SE 2.06 on SAR-principles and applications, 1997 25 WARD, J.:'Space-time adaptive processing for airborne radar'. Technical report TR-1015, MIT Lincoln Laboratory, December 1994 26 KLEMM, R.:' Space-time adaptive processing' (IEE Publishing, Stevenage, UK, 1998) 27 GIERULL, C H . : ' Statistical analysis of multilook SAR interferograms for CFAR detection of ground moving targets: IEEE Trans. Geosci. Remote Sens., April 2004,42, (4), pp. 691-701 28 CONTE, E., LONGO, M., and LOPS, M.: 'Modelling and simulation of nonRayleigh radar clutter'. IEE Proc. F, Radar Sonar Navig., April 1991, 138, (2), pp. 121-130 29 GOODMAN, N. R.: 'Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)', Ann. Math. Stat, 1963 (152), pp. 152-180 30 JOUGHIN, I. R. and WINEBRENNER, D. P.: 'Effective number of looks for a multilook interferometric phase distribution'. Proceedings of IGARSS9 Pasadena, CA, 1994, pp. 2276-2278 31 GIERULL, C. H. and SIKANETA, L: 'Estimating the effective number of looks in interferometric SAR data', IEEE Trans. Geosci. Remote Sens., August 2002, 40, (8), pp. 1733-1742 32 GIERULL, C. H.: 'Moving target detection with along-track SAR interferometry - a theoretical analysis'. Technical report TR 2002-084, Defence R&D Canada, Ottawa, Canada, 2002 33 ANDERSON, T. W. and GIRSHICK, M. A.: 'Some extensions of the Wishart distribution', Annals of Mathematical Statistics, December 1944, 15, (4), pp. 345-357 34 REED, L. S., MALLETT, J. D., and BRENNAN, L. E.: 'Rapid convergence rate in adaptive arrays', IEEE Trans. Aerosp. Electron. Syst, 1974, AES-10, (6), pp. 853-863 35 CHEREMISIN, O. P.: 'Efficiency of adaptive algorithms with regulized sample covariance matrix (in Russian)', Radiotechnology and Electronics (Russia), 1982, 2, (10), pp. 1933-1941 36 KREYENKAMP, O. and KLEMM, R.: 'Doppler compensation in forwardlooking STAP radar', IEE Proc. F, Radar Sonar Navig., 2001, 148, (5), pp. 253-258 37 GIERULL, C. H. and BALAJI, B.: 'Minimal sample support space-time adaptive processing with fast subspace techniques', IEE Proc. F, Radar Sonar Navig., October 2002,149, (5), pp. 209-220

38 NICKEL, U.: 'On the application of subspace methods for small sample size', AEUInt. J. Electron. Commun., 1997, 51, (6), pp. 279-289 39 LEE, J.-S., HOPPEL, K. W., MANGO, S. A., and MILLER, A. R.: 'Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery', IEEE Trans. Geosci. Remote Sens., September 1994, 32, (5), pp. 1017-1028 40 JOUGHIN, I. R., WINEBRENNER, D. R, and PERCIVAL, D. B.: 'Probability density functions for multilook polarimetric signatures', IEEE Trans. Geosci. Remote Sens., May 1994, 32, (3), pp. 562-574 41 SIKANETA,I. C. andGIERULL, C. H.: 'Parameterestimationforphase statistics in interferometric SAR'. Proceedings of IGARSS, Toronto, ON, Canada, 2002 42 ENDER, J. H. G.:' Signal processing for multi-channel SAR applied to the experimental SAR system AER'. Proceedings of international Radar conference, Paris, 1994 43 WIRTH, W.: 'Radar techniques using array antennas' (IEE Publishing, Stevenage,UK,2001)

Chapter 7

STAP simulation and processing for spaceborne radar Tim J. Nohara and Peter Weber

7.1

Introduction

Spaceborne radar (SBR) has been proposed for various military and civilian applications [I]. Military applications include wide-area surveillance (WAS), theatre defence and disarmament functions. Civilian ones include remote sensing, air-traffic control, space exploration and law enforcement. The 1970s saw the first spaceborne synthetic aperture radars (SAR) with GEOS-C launched in 1975 and SEASAT in 1978 [2]. These provided remote sensing of the earth, including data such as high-resolution topography and ocean dynamics. Spaceborne SAR has matured and today includes RADARSAT-2, which is due to launch in 2005. SBR for WAS and theatre defence, however, is still in the experimental domain. In the 1980s and early 1990s, research focused on WAS and airborne moving target indication (AMTI) designs [ 1,3-6]. More recently, focus has shifted to theatre defence applications where ground moving target indication (GMTI), and combined SARGMTI modes are of interest [7]. MTI (i.e. GMTI and AMTI) radars exploit space-time adaptive processing (STAP) techniques to detect targets that would otherwise be buried in clutter. In Canada, an experimental GMTI mode is being developed for RADARSAT-2. This programme, along with earlier programmes such as the United States' Discoverer-II programme, are intended to provide much needed design and performance data, which are necessary to take spaceborne MTI and SAR-MTI radars from experimental to operational systems in the next ten years. To help mitigate the exorbitant cost of fielding and testing spaceborne radar prototypes, engineers have come to appreciate the key role that simulation technologies can play in the development process. Recently, many texts have been published that are dedicated to simulating electronic systems such as radars [8-10], and there are now organisations dedicated to promoting the use of simulation technologies through

the development of standards and other support [H]. In addition, there are technical conferences focused entirely on modelling and simulation [12]. Today, good mathematical models are available to represent the entire spaceborne radar, along with sufficient, general-purpose computing power to implement and exercise these models in reasonable times. As a result, spaceborne MTI radar designs can be accurately modelled, their performance evaluated and trade-offs can be carried out over a wide range of design parameters and operating conditions, all before a prototype is built and launched for experimental validation. The principal objective of this chapter is to discuss the design of computer simulation tools suitable for modelling and evaluating the performance of spaceborne MTI radars employing STAP techniques. This objective is met by first reviewing spaceborne MTI radar applications and radar design. This is followed by a review of the STAP techniques typically considered for spaceborne MTI radar. With this background, the design of spaceborne radar (SBR) simulation tools is examined in detail.

7.2

Spaceborne radar applications and design

Reviews of both spaceborne radar MTI applications and typical MTI radar designs are presented. These allow the extraction of modelling and processing requirements for computer simulation tools that can be used for design trade-offs and performance assessments of an SBR.

7.2.1

Spaceborne MTI radar applications

Two key advantages of SBR are a greatly increased field of view (FoV), and global coverage without the political, strategic or geographic issues associated with surfacebased and airborne radars. The portion of the earth visible to a spaceborne platform is referred to as the FoV. As much as one third of the earth's surface can be within the FoV at one time. As a result, large search rates can be achieved and a constellation of SBRs can be designed to meet coverage requirements. WAS is used to protect nations with large geographical areas from threats such as ICBMs or enemy aircraft. Such nations would create search fences surrounding their territories wherein threats are detected. The fence acts as a tripwire when crossed. SBR would detect the intrusion, and report the location to friendly forces to respond. WAS requires radars with airborne moving target indication (AMTI) capability. Theatre defence is used when conflict occurs in some region, and timely intelligence concerning ground-troop movement is needed. During the Persian Gulf War (1990-91), the Joint-STARS and AWACS airborne radars were able to provide long-range air-to-ground and air-to-air surveillance. Such intelligence is only available while assets are deployed in the region. An SBR constellation can provide continuous surveillance in several regions, and can also be quickly deployed to new trouble spots. Theatre defence requires radars with ground moving target indication (GMTI) capability. Spaceborne MTI radar applications are illustrated in Figure 7.1. A radar waveform is transmitted from a transmit antenna whose beam is steered towards the desired

transmit waveform characteristics

receiver front-end SBR motion

external noise antenna characteristics: • main • auxiliaries

ionospheric effects Faraday rotation scintillation attenuation

tropospheric effects: volumetric clutter attenuation refraction

target characteristics

jammers

ianu

sea discrete earth's rotation

Figure 7.1 Spaceborne MTI radar applications footprint on the ground. The signal interacts with the ionosphere and is affected to some degree by Faraday rotation, scintillation and attenuation. The signal then interacts with the troposphere and is affected by volumetric clutter (e.g. rain), further attenuation and refraction. In an AMTI application, the signal of interest reflects off airborne targets, and in a GMTI application, the signal of interest reflects off ground targets, before propagating back towards the receive antennas. Only a single receive antenna is shown, but in practice at least two are needed for MTI operation using STAP techniques. Auxiliary antennas may be available for other functions such as electronic counter counter measures (ECCM), to deal with sidelobe jammers, for example. The receive antennas have moved with respect to the transmit location due to the high speed of the orbiting satellite (typically several km/s) and the long distance to the ground. The clutter signals reflected off the earth's surface and the signals transmitted by jammers towards the SBR are interference signals, which must be suppressed by the radar signal processor, so that the desired targets can be reliably detected. The large FoV is evident in the Figure.

7.2.2

Spaceborne MTI radar design

Several radar and system elements are designed and traded-off in order to meet particular mission requirements. The SBR orbit altitude and inclination are two key design considerations. Circular orbits are preferred for SBR because of the uniform coverage

iiilllillilii iiliiiliilili

||||li||p;||;| iiiiiiiiii

nw

Figure 7.2

|||||JjJ|||i||

SH||:iipi|j||||ii

iiwilliiililiil l^^llllHI ;||||||||||||||||||

Electronic steering geometries for SBR

provided and because an elliptic orbit has large variations in power-aperture requirements. Circular orbits also minimise the constellation size for global coverage. For weight and cost reasons, altitude is constrained by the Van-Allen belts. Higher radiation levels need more (heavier) shielding to protect onboard electronics and thus have higher launch costs. Altitudes of interest are thus limited to approximately 2800 km. Spacecraft at higher altitudes provide a larger field of view, and require fewer satellites for efficient global coverage. Below 800 km, the number of satellites needed to provide a given level of global coverage increases rapidly. However, coverage of limited regions can be accommodated at lower altitudes. Lower altitude SBRs are cheaper because of reduced power-aperture and shielding requirements. For a given altitude, the orbit inclination determines the geographical coverage provided. An equatorial orbit will only cover regions within a band on either side of the equator; a highly inclined orbit will cover all of the earth's surface. Antenna design is one of the key elements that influences radar performance. Whereas rotating antennas have been proposed for some MTI radars, electronicallysteerable antennas are preferred (see Figure 7.2 for illustrations of the features described below). Conventional SAR systems (e.g. RADARSAT-I) usually only provide elevation beam steering. With this type of antenna, stripmap SAR imaging can be performed. The antenna beams can be electronically scanned in elevation to move the beam inward or outward from the satellite track. However, azimuth scanning, which would allow the beams to move (almost) instantly forward and backward, is not supported. Combined azimuth and elevation scanning is needed for MTI SBRs and represents a significant engineering challenge. In surveillance applications, scanning a specified volume, where beams are steered and tiled on the ground to interrogate the specified region, is often required (rather than a stripmap). Certain look directions may be revisited for confirmation or tracking dwells. Information from other sensors may be used to cue the radar to look at a given area at a certain time. If the dwell time

for a given look is long enough, the antenna may require spotlighting, which involves steering the antenna during the dwell so that its boresight remains fixed on the area of interest. All of these scanning features require electronic steering to work well. Active phased-array antennas provide the ability to quickly look to arbitrary directions, and to reconfigure the aperture for STAP operation (i.e. forming multiple simultaneous subapertures on receive). Usually the array is aligned with its face either down (nadir-pointing) or inclined to the side (side-looking) and with its longer dimension parallel to the flight direction. In this case, phase shifts that vary across the length of the array cause the beam to steer forward and backwards (i.e. azimuth steering), while phase shifts varying across the width of the array cause the beam to shift up and down (i.e. elevation steering). In practice, the FoV available for surveillance is between the grazing angles 3° and 50°. The lower limit is due to atmospheric effects such as attenuation, whereas the higher limit is due to stronger clutter returns and lower target Doppler. Furthermore, the small antenna footprint at higher grazing angles makes surveillance inefficient. The circular region bounded by 50° grazing angle is referred to as the nadir hole (in surveillance coverage). WAS applications often favour the lower grazing angles due to the longer footprint and higher search rates achievable, whereas theatre defence applications usually work with the higher grazing angles (lower incidence angles). For SBR, the choice of RF is one of the most complex and important design decisions. Low-frequency systems are generally simpler and cheaper, but some hardware is bulkier and heavier. Higher frequencies offer better angular resolution and thus lower minimum detectable velocity (MDV) (which is important for MTI) for a given aperture size. Low frequencies suffer less atmospheric and rain attenuation, but have more problems with the ionosphere. Clutter CTQ tends to increase with RF. Proposed SBRs have varied from UHF to X-band. A typical WAS AMTI system provides long-range surveillance, where high power is needed, which is cheaper at lower frequencies. Since AMTIMDV requirements are not too stringent, L-band provides a reasonable overall compromise. Theatre defence GMTI systems, on the other hand, require more stringent MDV, which is more easily attainable at X-band. Rain is less problematic because of the higher grazing angles. STAP techniques are needed to provide suitable MDV with SBRs. Consider the mainbeam clutter bandwidth, which is nominally 2vp/L Hz, where vp is the spacecraft velocity in m/s and L is the antenna length in m. Given a typical low-earth orbit satellite speed of 7350 m/s and an antenna length of say 10 m, the mainbeam clutter will occupy a Doppler bandwidth of 1470Hz. This translates to mainbeam clutter velocities spreading ± 110 m/s at L-band and ± 11 m/s at X-band. Sidelobe clutter can easily fill the remaining spectrum as illustrated in Figure 7.3. As a result, subclutter visibility is needed for all small targets, and for large targets that are not fast enough (in radial velocity) to shift in Doppler out of the mainbeam clutter. STAP techniques filter or cancel the clutter in two dimensions (space and time) in order to provide the required MDV for targets of interest. In Reference 6, a simple expression for the MDV achievable is derived, which is given by MDV = 0.23 * Vp * X/L. This expression says that an MDV of 5 m/s (for a GMTI application) can be achieved with a 10 m antenna at X-band using STAR STAP

jammer mainbeam clutter

jammer interference sidelobe clutter SBR antenna pattern

small target large slow target

large fast target

SBR direction of motion Doppler

Figure 7.3

Typical SBR clutter spectrum

requires a radar system with a minimum of two receive antennas, suitably spaced, each with dedicated and well matched receivers.

7.3 7.3.1

STAP processing for SBR Typical GMTI signal processing

A review of signal processing techniques typically considered for spaceborne MTI radars is presented next. In this section, both adaptive and non-adaptive MTI signal processing algorithms will be briefly examined. Although this examination is not exhaustive, it illustrates the kinds of signal processing typically considered. Later in this chapter, signal processing requirements for computer simulation tools will be derived, based on this discussion. Space-time adaptive processing techniques are effective for cancelling clutter viewed from fast-moving platforms. STAP techniques operate adaptively on time samples collected from several spatially distinct receive antennas, in order to suppress unwanted clutter. An MTI filter that operates in only one dimension (time) does reduce the mainbeam clutter power. Unfortunately, with moving platforms, it also annihilates targets. STAP filters that operate in both dimensions (time and space) attenuate mainbeam clutter while not overly harming moving targets. Possible STAP domains are time and aperture (space), time and angle, Doppler and aperture, and Doppler and angle. In all cases, STAP is applied quasi-independently over the range dimension. In the time and aperture domain, adaptive clutter cancellation weights are computed from and applied to the pulsed (slow-time) signals from each aperture. MASR (multiple antenna surveillance radar) DPCA [16] and subCPI STAP

are algorithms in this domain. In the Doppler and aperture domain, pulsed-Doppler processing is performed before STAP, which works on the Doppler signals from each aperture. ASAR (arrested synthetic aperture radar) DPCA [5], PRI-staggered STAP and factored STAP [17] are examples of Doppler aperture domain algorithms. A variety of two-dimensional STAP processors that operate simultaneously in time and space are described in Reference 18. Non-adaptive space-time processing can also be applied successfully, using the displaced phase centre antenna (DPCA) technique. The DPCA technique is a form of STAP that works well with only two receive apertures. The whole antenna aperture is used on transmit. On receive, the aperture is divided into halves, each of which feeds its own receiver. The DPCA condition assumes that the phase-centre separation between the two is twice the distance that the platform moves in one PRI. If the condition holds, then on any pulse, the ranges to clutter scatterers for the leading aperture are the same as those for the trailing aperture one pulse later. Therefore, clutter can be cancelled by a conventional (non-adaptive) MTI filter operating on the pulse streams from the two subapertures (shifted by one pulse with respect to each other). Since a target moves over the PRI, its range changes, and it is not cancelled by the filter. The DPCA technique works for clutter from any look direction, and can be generalised to three or more receive apertures. Adaptive cancellation can also be performed effectively with DPCA. A candidate signal processing baseline suitable for GMTI operation is shown in Figure 7.4. If DPCA operation is desired, then the delays shown must correspond to one pulse repetition interval (T) or a multiple thereof. If an algorithm such as PRI-staggered STAP is employed, then a few delays (not shown) would have to be provided for each antenna channel. If a factored STAP algorithm is employed, then the delays are not needed. A minimum of two antenna channels is needed for MTI operation; three channels are shown in Figure 7.4, which allows monopulse estimation to be carried out concurrently with MTI operation. Pulse Doppler (PD) and pulse compression (PC) operations are performed on each channel; these provide the required coherent integration against noise. (Often, PC precedes PD; however, the ordering shown in Figure 7.4 supports the use of non-Doppler tolerant waveforms.) Each STAP filter combines its set of input channels into a single clutter-cancelled signal. The signals from multiple bursts are integrated noncoherently by the NCI function. The detection function thresholds the data and assembles a detection list, which includes position and velocity information, to complete the processing. Adaptive nulling (ECCM) is also shown in Figure 7.4. This function suppresses sidelobe and mainbeam jammers. Sidelobe nulling uses signals from the auxiliary antennas to cancel the jammer signals in the main antenna channels. To illustrate typical processing steps associated with the STAP filtering, the PRIstaggered STAP technique is described in some detail. For each antenna channel, PRI-staggered STAP uses the signals from a number of temporal taps. All temporal taps consist of the same number of pulses, but they differ in their starting pulse. For example, tap 1 's signal consists of pulses 1 to ND, while tap 2 has pulses 2 to No -f-1. A particular tap from a particular antenna is referred to as a space-time channel,

Bogpfef

STAP

auxiliaries

Figure 7.4

Typical MTIprocessor architecture

and corresponds to an adaptive degree of freedom in the STAP algorithm. With PRIstaggered STAP, the number of degrees of freedom is KN, where K is the number of taps and N is the number of channels. The total number of pulses range-compressed is M = No + K — 1, in order to handle all of the taps. Each space-time channel is Doppler processed after range compression. The input to the STAP processing is ^Vrange-Doppler arrays, each with NR range gates and ND Doppler bins, or NRNO resolution cells. For each range-Doppler cell, which has a length KN vector of samples spanning the space-time channels, STAP processing is applied. This involves collecting the vector of samples from the given cell (x) and a size KN by Ns matrix of snapshot vectors from neighbouring cells (X). The snapshot vectors are from the same Doppler bin as the current cell, at ranges surrounding it. The snapshot region specifications (Ns, guard region) are algorithm design parameters. Outer products of the snapshot vectors are averaged to form a covariance matrix:

which is an estimate of the clutter covariance across the space-time channels. R is then inverted. Two options are available for forming the weight vector used to cancel clutter: STAP without steering vectors and STAP with steering vectors. STAP without steering vectors forms the product xHR~ 1X, which cancels the clutter while taking the power of the target plus residual noise. The product, arranged as a range-Doppler array, is the STAP output signal, and is the input to the CFAR detection processing. Optimal detection matches the received vector to all possible steering vectors, and then picks the best one. A steering vector is the response across the space-time channels to an ideal target with a specified position and velocity. To generate the STAP output signal in each range-Doppler cell, the statistic xHR~ls is computed for all steering angles, and the one with maximum amplitude is kept. This is maximum likelihood (ML) processing. The operation, like STAP without steering vectors, uses R~l to cancel the clutter components in x. The advantage with steering is in the suppression of the noise components in x by matching with s. The steering vectors may be scaled by (sHR~ls)~1/2 as described in Reference 13; this is required for proper angle estimation.

7.3.2 Extension to other modes It is unlikely that an SBR would only provide a single mode of operation such as GMTI. Other modes, such as AMTI, pulse Doppler (PD), SAR and SAR-MTI, would also be considered. Trade-offs concerning the performance of any given mode against the other modes is necessary. Therefore, simulation tools used to assist in these trade-offs would need to support a variety of modes. AMTI operation, for example, requires similar processing to that shown in Figure 7.4, but would typically use a waveform optimised for faster targets and greater search rates (i.e. lower resolution). Additional processing considerations, such as range ambiguity removal, would be required. The pulse Doppler mode is suitable for fast targets that are clear of clutter, as well as for the detection of very large targets such as ships. Only a single antenna channel is needed. PC and PD operations are performed, followed by NCI and detection. SAR operation is similar to pulse Doppler in that a single antenna channel is required. PC and PD operations are replaced by an SAR algorithm, which includes range processing similar to PC, range cell migration correction to account for range walk due to the longer dwells and azimuth processing to form the synthetic apertures. NCI is replaced with multilook processing to reduce image speckle, and automated detection, if present, is usually image based. SAR-MTI operation combines features of MTI and SAR to allow moving targets to be detected and overlaid onto an SAR image. At least two channels are needed to support the MTI operation. In a conceptually simple case, SAR processing is performed first on each channel, followed by MTI processing and automated detection. Delay taps can be used as illustrated in Figure 7.4 to implement a DPCA condition, or to support MTI algorithms such as PRI-staggered STAR

7.3.3 Other issues There are a few system and environment issues that impact STAP performance and hence require special modelling considerations. The following issues are discussed below before leaving this section: (i) (ii) (iii) (iv)

effect effect effect effect

of internal motion of random sidelobes of the earth's rotation of jammers and rain.

Clutter internal motion is an inherent characteristic of clutter that fundamentally limits the ability of STAP algorithms to cancel clutter. Sea scatterers are moved about by sea swell, waves and wind, and sea clutter can have a spectral spread of the order of 1 m/s. Land clutter (e.g. vegetation and trees) vibrates due to the wind to a lesser degree, and can have a spectral width of the order of 0.1 m/s. This random motion introduces random phase shifts that limit the amount of cancellation otherwise achievable. Other system elements also impact cancellation (e.g. how well matched and calibrated the antennas and receivers are). Therefore, clutter internal motion effects should be considered when evaluating STAP performance. As alluded to above, the level of matching associated with the receive antennas impacts STAP performance. Consider an active phased array that is an ideal candidate for an SBR antenna. The beam pattern is formed by summing the element responses, suitably weighted and phase shifted. Due to imperfections in manufacturing, there are random variations in the spacing and gains associated with the elements. These variations have two effects. First, they result in the presence of random sidelobes in the beam pattern of a given antenna. Second, the patterns (main beam and sidelobes) of different antennas will be different. As a result, perfect clutter cancellation will not be possible due to these imperfections. In order to mitigate STAP performance degradations that would otherwise result due to random sidelobes, STAP algorithms adaptively compute separate weights for each Doppler bin, which has the effect of making piecewise gain corrections to the antenna patterns. Random sidelobe effects should be considered when evaluating SBR designs or trading-off STAP performance. Since an SBR orbits around it, the earth's rotation imparts different radial velocities on clutter scatterers, depending on their look directions to the radar. As a result, earth's rotation causes clutter spectral spreading in addition to that caused by the motion of the radar platform. To mitigate the effects of earth's rotation, the radar antenna can be mechanically slewed so that the receive antennas align with a certain vector: the sum of the satellite orbital motion vector and the earth's rotational vector (which varies with latitude). Alternatively, if a programmable active phased array is used, receive antennas can be properly aligned electrically, by controlling their aperture shading functions. Earth's rotation and slewing (mechanically or electrically) should be taken into account when modelling and evaluating the performance of STAP radars. The presence ofjammers or rain has the effect of requiring more adaptive degrees of freedom to maintain the same STAP performance. In the case of rain, this is because

the total clutter spectrum becomes more complex in the Doppler dimension. Rather than a simple notch, a more elaborate filter response is needed. Thus extra temporal degrees of freedom (e.g. more taps from each subaperture) may be needed to mitigate rain clutter. Dealing with jammers requires spatial diversity in the radar antennas, so that angular nulls (regions of low gain) can be steered toward the jammer. By varying multiplicative weights on the antenna signals before summing their signals, interferometric nulls can be moved to arbitrary directions. Differing strategies are needed for dealing with either mainbeam or sidelobe jammers. Because the gain towards a sidelobe jammer is low, the interferometer can be set up with a low gain auxiliary antenna and the main aperture. Main beam jammers are more problematic. Nulling them requires similar procedures as for clutter cancellation, namely dividing the main aperture into a small number of large subapertures. Then the interferometer is set up with apertures having similar gains towards the jammer. Combining clutter cancellation with main beam jammer nulling requires more spatio-temporal degrees of freedom than for either individually. It also requires a STAP algorithm designed to handle both forms of interferes

7.4

Simulation and processing for SBR

SBR simulators are used to estimate the performance of space-based radar systems and their signal processing algorithms. They are also used to design space-based radar systems by trading-off and optimising design parameters to achieve a specified performance. In the previous sections, it is seen that GMTI radar detection concepts rely on the cancellation of strong clutter signals, in order to allow the detection of weak target returns. The clutter signals that are combined for cancellation by the STAP filter originate from different apertures at different times. To draw meaningful conclusions, it is imperative that proper correlation (in space and time) of both clutter and target signals be modelled. It is also important to model the effects of real hardware because, as discussed earlier, it impacts radar performance. A simulator must properly model platform motion, range and Doppler ambiguities, clutter internal motion, antenna patterns (differing between apertures) and earth's rotation. It should support a myriad of design choices, with selectable radar, antenna, platform and signal processing parameters so that different radar designs can be traded-off or optimised. It should also be able to simulate arbitrary scenarios, with selectable targets, clutter, and jamming so that particular applications or scenarios of interest can be evaluated. All of these requirements point to a baseband signal simulation, where the radar return signal is generated as samples in range for each pulse. The simulator models the full, expanded, transmit pulse, and then implements pulse compression, rather than modelling an effective compressed pulse. This approach results in greater fidelity and also provides the mechanism for efficiently implementing receiver response mismatch. Also, by starting off with an expanded pulse, imperfections in pulse compression operation can be embedded naturally. The full signal path must be modelled, including the environment, the antennas, the radar analogue and digital parts (for both

transmit and receive) and the signal processor. All of the effects in Figure 7.1 (motion, atmosphere, large FoV etc.) must be dealt with. As return signals are convolutions of the transmitted pulse with each scatterer's response function, range equation terms are needed for each scatterer (gains, range, Doppler, RCS) so that its sampled return signal can be generated. The difficulty with baseband signal simulation is in generating the returns from the many scatterers contributing to the return signal. Although the mathematics for computing their returns is (relatively) straightforward, the sheer volume of computations can easily render a simulator unable to generate the signals in an acceptable time frame. Clutter generation is the single most computationally intensive operation, and care must be taken to optimise its speed and fidelity. Efficiency in generation is a key requirement in radar simulator design. A well designed graphical user interface (GUI) is important to the user, who is the person using the tool to carry out design studies or performance evaluations. It eases the entry of parameters that describe the radar system and scenario, and the subsequent running of simulation experiments. The GUI also permits the quick derivation and cataloguing of results from the experiments, including the plotting of images and curves. The user should be able to specify the design of a radar system and scenario of interest, generate the corresponding complex baseband signals, and then process them with algorithms and parameters that he/she selects. In this section, the aforementioned high-level simulator requirements are broken down into a set of design elements, which describe a candidate baseband signal simulator of space-based radars. Quantities that are described as being modelled, computed, converted, transformed etc. are operations internal to the simulator code. The presentation below organises the design elements into functional areas, beginning with the GUI and followed by discussion of the models needed for the radar, the environment, the baseband signal generation, the GMTI signal processing, and tools needed for evaluation of the results. Radarsim™ SBR [15] is a space-based radar simulator codeveloped by the authors that satisfies the requirements described herein. Selected screen-captures taken from this tool will be used to illustrate design concepts, where appropriate.

7.4.1

User interface

For convenient use of a simulator, a properly designed graphical user interface (GUI) is important, since radar systems and engagement scenarios have so many variables. GUI designs that intuitively manage related groups of parameters make the user's job easier. Below is sampling of logical parameter groupings needed to specify a typical space-based radar, followed by a description of how the environment might be conveniently defined via a GUI. 7.4.1.1 Simulation parameters Reference to the spaceborne radar section yields several logical groupings of radar system parameters that need to be specified by the user: orbit, waveform, receiver,

Table 7.1 Orbit parameters Parameter

Typical value

Altitude Inclination Subsatellite point

800 km 80° latitude less than inclination; any longitude north or south

Direction

Table 7.2

Waveform parameters

Parameter

Typical value

Peak power Carrier frequency Expanded pulse width Pulse bandwidth Burst length Nominal PRF Pulse modulation Fill pulse duration PRI compensation? ZRT compensation? Spotlighting?

5 kW 10 GHz 50 |xs 200 MHz 100 ms 2000Hz linear FM 5 ms yes yes yes

antenna, generation and environment parameters. These are summarised in the following series of tables. Circular orbits are preferred for GMTI radars. A parameter set suitable for specifying a circular orbit is shown in Table 7.1. The subsatellite point and direction relate the SBR to the environment scenario. Pulse Doppler waveforms are appropriate for space-based GMTI (and other modes). Such waveforms can be modelled using the parameters shown in Table 7.2. The pulse compression ratio is defined by the product of its expanded width and its bandwidth. Other forms of modulation include phase coding, non-linear FM and none. The fill pulse duration should be long enough to fill the footprint with ambiguous pulses. The effects of non-ideal receivers are important because of their impact on clutter cancellation. The receivers can be modelled according to the noise added, their frequency responses (including differences between channels) and their non-linearities. The IF filter parameters in Table 7.3 allow the nominal response for each channel

Table 7.3 Receiver parameters Parameter

Typical value

Noise temperature Radar system loss IF bandwidth IF filter type and order IF centre frequency Channel to channel mismatch Number of mismatch ripples A/D sampling rate A/D quantisation level Number of A/D bits Phase noise spectrum

10000K 3 dB 200MHz Chebyshev, 8 1500 MHz —40 dB 8 200 MHz — 120 dB m 8 levels (dB c) at a discrete set of frequencies

Table 7.4 Antenna parameters Parameter

Typical value

Receive aperture locations

+0.5 m azimuth +0 m elevation 4 m azimuth by 1 m elevation Taylor —45 dB azimuth; —30dB elevation to place footprint at desired location 30° elevation tilt; auto-yaw for minimum clutter spread — 50 dB 1.5 cm

Aperture sizes Aperture shadings Electronic steering Mechanical positioning Random sidelobe levels Element spacing

to be computed; the mismatch parameters are used to compute each error response. Additional parameters are used to model the A/D and phase noise characteristics. A phased array antenna system, with multiple subapertures on receive, is a part of most high-performance GMTI designs. Parameters to describe such a system are listed in Table 7.4. Aperture sizes and shadings are specified for the transmit and all receive apertures. Locations are of the given receive aperture's centre, relative to the transmit aperture centre. Random sidelobes are due to aperture errors, whose level can be derived from the entered sidelobe level. An auto-yaw capability is provided, which sets the positioning to minimise the effects of earth's rotation on clutter.

Table 7.5 Generation parameters Parameter

Typical value

Noise seed Clutter seed Aperture error seed Jammer seed

large large large large

integer integer integer integer

Table 7.6 Clutter patch parameters Parameter

Typical value

Position Size Scatterer spacing Backscatter statistics a0 Spectral width Mean radial velocity Height Rain rate

within range swath and main beam 5 km by 5 km 1 m range; 10 m cross-range log-normal, with 10 dB spread -20dBm 2 /m 2 0.1 m/s 2 m/s 3 km 5 mm/hr

Random number seeds (such as those in Table 7.5) for all statistical processes modelled should be user-set parameters, so that simulations can be repeated, and so Monte-Carlo experiments can be performed. Modelling realistic clutter returns is required in order to properly assess the target detection capability of SBR. Parameters to describe clutter patches and rain cells are listed in Table 7.6. (Land clutter should have zero mean radial velocity.) Height and rain rate are only appropriate for rain cells. Rain cells are modelled as sources of clutter as well as in terms of the attenuation they impart on propagating signals. Typical target and jammer parameters of interest are shown in Tables 7.7 and 7.8. These parameters are most easily obtained from the user using a dialogue window. Figure 7.5 shows one such window for some of the receive aperture parameters. 7.4.1.2 Environment GUI Although dialogue windows are good for obtaining parameters such as those discussed in Tables 7.1 to 7.8, they are not well suited to placement of targets and clutter.

Table 7.7 Target parameters Parameter

Typical value

Position Velocity Mean RCS Spectral width

within range swath and main beam 10 m/s; heading towards radar 10 m 2 0.1 m/s

Table 7.8 Jammer parameters Parameter

Typical value

Position Velocity ERP Centre frequency Bandwidth Type Modulation Start and end times

anywhere in FoV stationary 500W 9.5 GHz 1 GHz barrage not pulsed extends over duration of radar waveform

Editing parameters for receive subaperture 1 Select subaperture: | Rx aperture 1 Azimuth (X) taper: | taylor Az. sidelobe level (dB): | X width (m): [ X position (m): [

jjj

jjjt| Elevation (Y) taper: j taylor El. sidelobe level (dB): |

jf|

-40

j

T

|

Y width (m): [

-30 T

^3

]

Y position (m): ["""

0

Figure 7.5 Parameter window A graphical display tool for specifying the locations of clutter patches, rain cells, targets and jammers is a better design. It also allows the placement of the antenna beam and the range swath. The display can provide a map of the earth with latitude/longitude grid lines. Zoom and pan capability can also be provided. The display can also have km rulers as a guide to the distortions introduced by the projections and zooming. Objects can be placed in the environment using controls on the side of the display. After selecting the object to place, the mouse can be used to click and place the object

(or to click and drag out the region occupied by a patch). Once placed on the display, objects can be edited by first selecting them, and then raising a window for their parameters. The subsatellite point can be indicated on the display to assist the user in locating objects. Other useful indicators that can be shown on the display include the orbit ground track, a world map and the azimuth, ground range and slant range to any selected object. In addition, isodop and isorange contours (including the horizon) can be drawn. Surface scatterers on a given isorange line (circle) have the same range to the radar. Stationary scatterers on a given isodop line (hyperbola) have the same radial velocity with respect to the radar. Having these contours helps the user to design good test experiments by allowing him/her to easily place objects so that they appear at desired locations within the radar images. Land and sea patches can be indicated in the display by rectangles of a specified size at the indicated centre location. Targets can be indicated by an appropriate symbol placed at the location of the target, with an attached arrow denoting the direction of motion at that location. Figure 7.6 shows a sample environment display (zoomed in) for an example scenario. Targets are denoted by T symbols, and the solid square is a land patch. The range swath is between the dotted lines (the O symbol is the userdefined swath centre). Solid lines are isodops. The dashed curves are the antenna beam contours (the + symbol is the electronic scanned boresight).

environment editing area (axes in degrees) to add objects select object and press define

!attitude, degrees

Land Patch

axes dimensions (km): x-top: 16.3021 x-middle: 16.3954 x-bottom: 16.4886 y: 40.3316 instructions:

longitude, degrees object geometry from SSP (deg/km): azimuth, slant range, ground range zoom: state: 88.196 817.564 415.123

Figure 7.6

Environment display

zoom:

pan:

7.4.2

Model the radar

In the next two subsections (model radar and environment), suitable models for the platform, antenna, radar and environment are described which form the basis of a simulator for SBR applications. Model-related operations computed as initialisations before core generation operations are also discussed. 7.4.2.1 Platform geometry Standard circular orbits are modelled, and Kepler's laws are used to determine the orbital speed, given the altitude. The satellite orbit is computed in its plane given its inclination, altitude and subsatellite point at time zero. Orbiting platforms are first computed in an ECI (earth-centred inertial) frame, within which the earth rotates. Orbit positions are then transformed into the ECR (earth-centred rotating) frame, where the target and clutter scatterers are represented, by applying earth's rotation. The positions of the transmit and receive antennas are computed at time instants spanning the duration of the waveform. The antenna orientation is computed at the same instants; it consists of the three-dimensional rotation matrices required to convert scatterer ECR coordinates to antenna coordinates (which are azimuth (w), elevation (t>) and boresight (w)). Coordinate transformations are a fundamental aspect of the simulation of radar return signals. Since these transformations are executed many times, they are designed and implemented as efficiently as possible. The positions of the radar and the environment are both described in the ECR coordinate system, in order to difference them for range and bearing computations. They are then converted to antenna coordinates for gain computations. The conversion between the coordinate systems uses time-varying rotation matrices. The user enters the antenna boresight at the beginning of the waveform. It is entered either as its intersection with the earth's surface or as angles in the satellite FRD (forward-right-down) frame. The antenna is assumed to have the same FRD orientation throughout the waveform. The boresight orientation and the aperture positions are combined to determine the rotation matrices. 7.4.2.2 Antenna patterns The antennas are modelled according to the voltage distribution across their apertures. Aperture shadings are computed on two-dimensional x-y grids. Antenna errors (deviations from ideal in amplitude and phase for each element) lead to increased antenna sidelobes. The shadings include aperture errors that, for active phased arrays, are correlated for overlapping subapertures. (STAP processing needs the formation of multiple simultaneous subapertures on receive.) The errors cause the patterns to differ between otherwise identical subapertures. These pattern differences lead to reduced clutter cancellation (relative to ideally matched patterns) and therefore must be modelled. Antenna patterns (in azimuth and elevation) are computed as Fourier transforms of the shading functions. Electronic steering is modelled by applying phase shifts to the element shadings.

7.4.2.3 Receiver responses The receiver frequency responses are modelled as standard (Chebyshev, Butterworth) filter responses. Real receivers have imperfections in their responses that differ between channels. Channel-to-channel mismatch is introduced by multiplying each channel's nominal response with a different error response. The user specifies the level of error and mismatch. The channel responses are later applied to return signals (as filtering operations). 7.4.2.4 Waveform Pulse Doppler waveforms (identical pulses transmitted at an approximately constant repetition interval) are used for SBR. The pulse parameters (duration, bandwidth, RF, PRF etc.) are user-selectable. If selected, motion compensation (ZRT) attempts to keep a desired spot or track at the middle of the range swath. This is done by first computing the range to the spot/track for each pulse, and then delaying the start of sampling (i.e. zero range) appropriately. PRI compensation (if selected) has the pulses transmitted at constant azimuth separation, rather than at a constant time interval. The radar only samples and processes the signal returned from within a userdefined range swath. The swath width must be less than the PRI. The swath's position on the earth varies with time as the platform moves. The swath has range ambiguities, where returns from ranges within the swath plus or minus an integer number of PRIs are received. Range-ambiguous clutter returns can be more troublesome to cancel, thus modelling them is needed for a true assessment of performance. Fill pulses can be transmitted, to ensure that ambiguities of the swath have the same number of pulses returned to the radar. The transmit and receive times for each pulse are computed. Note that these times must be separated by the two-way delay to the swath. 7.4.2.5 Errors All forms of amplitude and phase mismatch between channels reduce the achievable cancellation of strong clutter signals. These include mismatches between antenna gains as a function of angle, those between receiver channels as a function of frequency and those between the I and Q channels of digitisers. All can be helped (but not eliminated) by various calibration and equalisation schemes. Short-term transmitter stability is also important for cancellation, which operates over different pulses. Each of these forms of error is modelled by applying the appropriate distortions to the received signals.

7.4.3

Model the environment

The environment includes models for targets, clutter, jammers and noise. 7.4.3.1 Positions of scatterers In order to properly model their returns across range and Doppler, distributed clutter patches are described as two-dimensional arrays of scatterers, located at the centres

of nominal resolution cells. A position is computed for each scatterer within a patch. Scatterer positions are laid out on a range/cross-range grid. This allows the scatterer spacing to account for the resolution available in each dimension. Target positions are computed as they move over the duration of the waveform, and are not represented on any grid. The atmospheric and rain attenuation are computed at each scatterer location. 7.4.3.2 Reflectivity of scatterers Land and sea clutter amplitude distributions (over area) are in general non-Rayleigh, especially with high range resolution. The distributions one should model include log-normal, K, Rayleigh and fixed. Consider a log-normally distributed patch as an example. Log-normally distributed variates are generated, and these become the mean powers of the scatterers within the patch. However, each scatterer is temporally Rayleigh amplitude distributed. That is, as time evolves, its amplitude will vary according to a Rayleigh distribution, with a user-specified spectral width. This is a realistic model, since a given ground scatterer is unlikely to be log-normal temporally. The target models can be chosen from the same distribution families, and also include a spectral width for their scintillation. The temporal variation models clutter internal motion; doing so properly is critical to deriving performance in realistic clutter. The scattering time sequences, which are their radar cross sections (RCS) described as a (complex) voltage for each pulse, are computed for each scatterer. This can be done in either of two ways. The first is to generate a random spectral sequence shaped by the amplitude spectrum, and then inverse Fourier transform. The second is to generate a white time sequence and then FIR filter with a response shaped by the correlation. The choice depends on the scatterer bandwidth relative to the PRF. These techniques are described in Mitchell [14].

7.4.3.3 Noise For each channel, a white noise signal is first computed, and then the receiver response is applied. This approach works for receiver channels originating from independent apertures. To implement proper receiver noise modelling, its channelto-channel correlation across overlapping subapertures of phased arrays must be included. This is important because adaptive algorithms can be impeded by this correlation. A selectable noise temperature completes the model. 7.4.3.4 Jammers Jammer generation is performed in a manner similar to noise generation. Extra steps include applying antenna gains, propagation loss (radar equation terms) and jammer modulation. When aperture separations and receiver bandwidths are both large, it is important to model jammer decorrelation between channels, since it leads to degradations in achievable cancellation.

7.4.4

Generate the signals

This part of the simulation generates the return signals from the scatterers. The received signal is the sum of scatterer returns, noise and jamming. It is of dimensions (number of receivers) by (number of pulses) by (number of samples in range swath), typically 2 by 128 by 4 K. The objective of STAP processing is to cancel the strong clutter return that originates from a large number of scatterers. It is thus important that the amplitudes and phases for all contributing range equation terms for each scatterer at each time instant be consistent. Any discontinuities in the models can lead to spurious clutter residue that really does not exist. In particular, the following terms must be simulated with high fidelity: • • • •

range and bearing (as functions of time) from each aperture to each scatterer cross section (as function of time) for each scatterer gain (as function of angle) for each aperture towards each scatterer distortions (as functions of time or frequency) for each channel.

7.4.4.1 Range equation terms On each pulse, the range, azimuth (w), elevation (i>), Doppler, transmit and receive gains, carrier phase and cross section are computed for each scatterer. These are obtained by differencing scatterer and satellite positions, rotating to antenna coordinates, differencing ranges at the two time instants, looking up and interpolating the antenna pattern grids, respectively. Ranges to each aperture are computed using their respective positions at the receive time. Accurate range computations are important to get the correct carrier phase relation between apertures, which is needed to accurately model the achieved cancellation. Doppler is needed for interpolations in the scatterer summation/convolution. Bistatic range computations are necessary for high fidelity modelling of spacebased radars. These account for motion between the transmission and reception of the radar pulses. The algorithm interpolates backwards from receive times to find reflection and then transmit times for each scatterer, and then interpolating the satellite position to the transmit time. For moving targets, their range computations also include interpolation of the target position to the reflection time. The range, transmit power, system loss, RCS, antenna gains etc. are combined together as range equation terms into a (complex) return signal voltage for each aperture. The return from each scatterer now conceptually consists of a delta function at its range delay, weighted with this voltage. The sum of delta functions is next convolved with the transmit pulse. The result of the convolution is the received reflected signal. Depending on whether there was a transmit pulse for a given scatterer's range ambiguity (i.e. if there were enough fill pulses), its returns may be suppressed for the first or last few pulses. 7.4.4.2 Scatterer summation The purpose of the scatterer summation is to generate the complex voltage signal that is the superposition of the pulse returns from all scatterers. For all of the scatterers, the

range delays t;(0), received voltages Vix and Doppler frequencies Di(O) have been computed (/ denotes scatterer and t denotes continuous time starting at the zero-range return for the given pulse):

T[ is the range to the /th scatterer and fc is the carrier frequency. The total voltage is the sum of the individual scatterer voltages convolved with the expanded pulse response p(t): VcumiO = J2

Vix

P(f ~ T 'W) exp(-j27r/ c r/(0)

i

Each scatterer pulse return requires a complex exponential (representing the expanded pulse) to be evaluated at every receiver sampling time, which is very expensive if done by brute force. The process of optimising this computation is most challenging in terms of maintaining both high fidelity and high execution rates. 7.4.4.3 Receiver responses Receiver responses and models for the non-linearities (including the A/D), phase noise and eclipsing are applied to received signals. Receiver responses are implemented as filtering operations, while the other effects are single-point functions applied to each received sample.

7.4.5

Model the processing

Simulating the processing is more straightforward, since most algorithms can be simulated exactly as they would be implemented. The challenge is in simulating most of the many potentially viable algorithmic alternatives. The user first selects one out of a number of candidate baselines, and then chooses a set of appropriate parameters. The baselines of interest for SBR in general consist of coherent integration followed by STAP and then by detection. 7.4.5.1 Coherent integration Range and Doppler processing form a bank of matched filters to the waveform over the span of resolvable target ranges and velocities. The user specifies which channels and how many pulses are required as part of the processing parameters, as well as window parameters to control range and Doppler sidelobes. Each pulse from each spatial channel is range compressed. Range compression produces NR range gates for each pulse. Doppler processing immediately follows range compression, and produces No Doppler bins from Np pulses, and is performed on each space-time channel.

7.4.5.2 Clutter cancellation STAP processing then filters across the space-time channels. Covariance matrix estimates are formed for each range-Doppler cell, and these are used to compute a weight vector. The reference cell parameters should be user entered. The weight vector is used for summing the signals over the channels, resulting in a single cluttercancelled matrix of ND by NR resolution cells. With this part of the processing, the numerical precision of the SBR target computer should be modelled, since it has important effects on adaptive algorithms that are attempting to cancel interference that is considerably stronger than target returns. 7.4.5.3 Non-coherent processing After clutter has been cancelled, detection, estimation and imaging operations are typically performed. CFAR processing typically includes either the cell-averaging (CA) or the ordered statistic (OS) algorithm in order to produce a list of target detections. The user selects the threshold and the CFAR reference cell region (including guard bins) spanning both the range and Doppler dimensions.

7.4.6

Evaluate the results

Important performance measures for a radar system relate to the detectability of targets: how small? how slow? at what range? in what clutter? A simulator needs tools to measure the detection (and estimation) performance of the radar system being modelled. It also needs the ability to analyse the signals after various processing steps, in order to determine their effectiveness. 7.4.6.1 Intermediate products During processing, intermediate products (the signals coming out of each processing stage) should be stored in data structures so that the user can analyse them. The twodimensional signal after each stage can be displayed as an image depicting amplitude as a function of range and Doppler (or pulse). Intensity/colour denotes amplitude and each signal dimension takes a Cartesian axis. The following images can be displayed, depending on the processing baseline and options: 1 2 3 4 5

input signal: amplitude versus fast-time and pulse range-compressed signal; amplitude versus range and pulse post PC/PD signal: amplitude versus range and Doppler clutter-cancelled signal: amplitude versus range and Doppler CFAR normalised signal: amplitude versus range and Doppler

For each image, the user selects the two-dimensional region to display and the dynamic range. For images 1 to 3, the channel number is also selected. Target positions can be overlaid with symbols at their true (known a priori) locations; this helps greatly in determining the effectiveness of various processing steps. Figures 7.7 and 7.8 show images of the post-PC/PD and of the clutter-cancelled signals for an example scenario.

Doppler, Hz

ASAR post PC-PD signal, channel 1

range, m

Figure 7.7 Signal before STAP processing

Doppler, Hz

ASAR post-STAP signal

range, m

Figure 7.8 Signal after STAP processing

In the second image, notice that the clutter has been significantly cancelled, and that targets now appear as bright regions surrounding their true locations. 7.4.6.2 Analysis products Target statistics can be extracted from (small) range-Doppler regions surrounding their true positions. Each target's level, SES[R and improvement factor is computed as the maximum over its respective region. Clutter statistics (level, cancellation ratio)

can be extracted from a larger range-Doppler region. All target regions are excluded from the clutter statistics. Statistics are extracted from the signals both before and after STAP processing. This allows improvement factors to be computed. Target statistics can be displayed versus target number, Doppler, range, RCS or ground velocity to provide additional information. The detection process also creates a detection list (Doppler bin, range bin and power). Clutter statistics can be displayed as histograms, or as scatter plots versus Doppler or range.

7.5

Discussion and conclusions

Computer simulation tools for modelling space-based radars employing STAP have been presented. Using these tools, the performance of SBR designs can be evaluated, and design parameters can be optimised. Spaceborne radar applications and design were reviewed. SBR has a large FoV and is capable of global coverage and large search rates. WAS is used to protect large geographical areas from airborne threats and requires AMTI. Theatre defence is used for monitoring a localised region and requires GMTI. STAP techniques for space-based radar were reviewed. Clutter signals must be suppressed, so that targets can be detected. With fast-moving platforms, STAP filters the clutter in space and time in order to cancel it. This provides the required MDV for targets. STAP requires a minimum of two displaced receive antennas with well matched receivers. Pulse compression and pulse Doppler provide the coherent integration. The STAP algorithm combines the channels into a single clutter-cancelled signal, where targets can be detected. Active phased array antennas with electronic steering are preferred for SBR, because they provide beam agility, spotlighting and the ability to form multiple subapertures on receive. Clutter internal motion, ambiguities, pattern and receiver mismatches, earth's rotation, rain clutter and jammers all limit the ability of STAP to cancel clutter with SBR. To mitigate, STAP algorithms adaptively compute weights, have extra degrees of freedom and have their antennas slewed. SBR simulators are used to estimate performance and to help design systems. STAP cancels large clutter signals in order to detect small target returns. To ensure accuracy, proper correlation of signals must be modelled, as must the above limiting factors. Thus baseband signal simulation is recommended, where the radar return signal is generated as samples in range for each pulse. The implementation of a baseband simulator is difficult because of the large number of scatterer returns that must be included. The simulator should also support many design options, parameters and scenarios. Because of the many parameters needed to describe SBR, a well designed GUI is important. It eases parameter entry and allows the running of experiments. To generate the return signals, there are many details to be modelled: positions and orientation of antennas, ranges and bearings of scatterers, aperture shadings with errors, antenna patterns, pulse Doppler waveforms, the range swath with its ambiguities, clutter patches, scatterer distributions over area, and scatterer temporal variation. These are

all used to derive range equation terms that are convolved with the transmit pulse and combined to form the return signal voltage. It is important that the amplitudes and phases of modelled functions be accurate and consistent. It is also important to optimise the signal generation routines, in order to run in reasonable time. Optimising the convolution with the expanded pulse is typically the most difficult of the speed-fidelity trade-offs. A simulator needs tools to measure performance. It also needs the ability to analyse the signals after various processing steps, in order to determine their effectiveness. The two-dimensional signals after each stage can be displayed as images. Target and clutter statistics can be extracted both before and after STAP processing, and be displayed as histograms, curves or scatter plots. The statistics from a series of experiments can be extrapolated to determine whether the SBR design can meet mission requirements. SBR simulators should become increasingly more useful to radar designers in the future, as long as the models within them are accurate and flexible, with appropriate care taken so that both fidelity and speed are achieved. Their implementation on general-purpose office computers, making them widely available, should result in future SBRs being designed and launched with significantly reduced development costs. Finally, it is worth noting that the simulation requirements for SBR are common to other applications such as airborne MTI radars; hence, simulators for them can be designed using similar strategies to those described herein.

References 1 CANTAFIO, L. J. (Ed.): 'Space-based radar handbook' (Artech House 1989) 2 SKOLNIK, M.: 'Radar handbook' (McGraw-Hill, 1990, 2nd edn.) 3 BIRD, J. S. and BRIDGEWATER, A. W. 'Performance of space-based radar in the presence of earth clutter' IEE Proc. F, Radar Sonar Navig., August 1984, 131, (5), pp 491-500 4 BROOKNER, E. and MAHONEY, T. F.: 'Derivation of a satellite radar architecture for air surveillance', Microw. J., February 1986, pp 173-191 5 NOHARA, T. J.: 'Design of a space-based radar signal processor', IEEE Trans. Aerosp. Electron. Syst, April 1998, 34, (2), pp 366-377 6 NOHARA, T. J., WEBER, P., and PREMJI, A.: 'Space-based radar signal processing baselines for air, land, and sea applications', Electron. Commun. Eng. J., October 2000, pp 229-239 7 NOHARA, T. J., WEBER, P., PREMJI, A., and LIVINGSTONE, C : 'SARGMTI processing with Canada's Radarsat 2 satellite'. Proceedings of the IEEE symposium 2000 on Adaptive systems for signal processing, communications, and control, Lake Louise, Alberta, Canada, October 1^4, 2000 8 YAKOV D. SHIRMAN: 'Computer simulation of aerial target radar scattering, recognition, detection and tracking' (Kharkov Military University, Editor, Artech House Publishers, 2002)

9 SERGEY A. LEONOV: 'Handbook of computer simulation in radio engineering, communications and radar' (Artech House Publishers, 2001) 10 BASSEM R. MAHAFZA: 'Radar systems analysis and design using MATLAB' (Chapman & HALL/CRC, 2000) 11 U.S. Defense Modeling and Simulation Office (DMSO), http://www.dmso.mil 12 Modeling and simulation conference, Association of Old Crows, U.S.A., Orlando, FL, July 17-18, 2002 13 KELLY, E. J.: 'An Adaptive Detection Algorithm', IEEE Trans. Aerosp. Electron. Syst, AES-22, (1), March 1986, pp. 115-127 14 MITCHELL, R. L.: 'Radar signal simulation' (Artech House, Dedham, MA, 1976) 15 Radarsim®SBR Space-based radar design and performance evaluation tool, Sicom Systems Ltd., www.sicomsystems.com 16 KELLY, E. J., and TSANDOULAS, G. N.: 'A displaced phase center antenna concept for space-based radar applications'. IEEE Eascon, Washington, 1983, pp.141-148 17 TIM J. NOHARA: 'Comparison of DPCA and STAP for space-based radar.' Proceedings of the 1995 IEEE international Radar conference, Arlington, VA, USA, May 8-11, 1995, pp. 113-119 18 KLEMM, R. and ENDER, J.: 'New aspects of airborne MTF. Proceedings of the 1990 IEEE international Radar conference, Arlington VA, USA, May 7-10, 1990, pp. 335-340

Chapter 8

Techniques for range-ambiguous clutter mitigation in space-based radar systems1 Stephen M. Kogon and Michael Zatman

8.1

Introduction

Space-based radar (SBR) systems provide several important capabilities for the detection of moving targets that are not possible from an airborne platform. These advantages include continuous access to important tactical areas as well as rapid surveillance of large regions on the ground [I]. Although SBR systems have been studied for some time now, it is only recently with the rapid development of computational and antenna array technology that these systems have come closer to realisation. As a result, a large research investment has been made into the development of future SBR systems [2]. SBR systems show great promise for the detection of ground moving targets, a mode commonly referred to as ground moving target indication (GMTI) which is enabled by ground clutter cancellation using space-time adaptive processing (STAP). A GMTI mode typically utilises a pulse-Doppler waveform, consisting of a series of pulses transmitted at a constant rate known as the pulse repetition frequency (PRF). The advantage of these waveforms is the approximate decoupling of range and Doppler that allows for efficient processing with fast Fourier transforms (FFTs). However, these pulse-Doppler waveforms are plagued by ambiguities in both Doppler and range [3, Chapter 17]. In SBR geometries, these ambiguities intercept clutter and cause problems for GMTI detection [I]. Doppler ambiguities arise due to the high platform velocity in an SBR but can be effectively managed through sidelobe control and STAP algorithm design. Range ambiguities, on the other hand, are a problem

1

This work was supported by the U.S. Air Force under Air Force Contract # F19628-00-C-0002. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Government

radar pulses

Figure 8.1 Range-ambiguity problem for an SBR at low grazing angles or long ranges for an SBR at low grazing angles or equivalently long ranges when the area on the ground illuminated by the radar exceeds the radar receive window of each pulse. An illustration of this problem is shown in Figure 8.1. These ambiguities are much more difficult to handle. Mechanical steering of the SBR array to maintain boresight on an area of interest produces a variation in clutter angle-Doppler characteristics with range. This non-stationarity leads to multiple Doppler frequencies with SINR loss holes from range-ambiguous clutter and can severely compromise detection performance. The use of STAP in the presence of range ambiguities and strategies for coping with these ambiguities are the topics of this chapter. For this chapter, we focus on a notional SBR platform in low-earth orbit (LEO) at 1000 km altitude with an antenna 20 m in length (azimuth) and 2 m in height (elevation). First, notation and metrics are established for moving target detection from an SBR with STAP followed by a discussion of the unique characteristics of ground clutter returns in SBR. In particular, the problem of range-ambiguous clutter is described. Next, we demonstrate the impact of range-ambiguous clutter on STAP performance and describe two means of overcoming the range-ambiguous clutter problem for pulse-Doppler waveforms: PRF diversity and an increase in elevation aperture. Last, the use of an alternative new waveform, namely a long single pulse waveform, that eliminates ambiguities both in range and Doppler, is considered for an SBR system.

8.2

Moving target detection with SBR

An SBR is a very powerful asset for surveillance applications since it can observe a large area on the ground below due to its high platform altitude. This large observable area gives an SBR a great deal of flexibility in where it looks and offers the capability of rapid search rates. The difficulty that arises from a large observable area is the management of all the returns. A large azimuthal aperture or array length focuses the SBR to a particular azimuth or cross-range area. In addition, aperture in the azimuth dimension limits mainbeam clutter Doppler spread which in turn determines performance against slow moving targets. The other dimension of range has an extent starting at the subsatellite point out to the horizon. This large range extent can present problems for a pulse-Doppler waveform. Unlike airborne applications which have

platform velocity

aimpoint

mechanical steering angle

Figure 8.2

Mechanical steering of SBR with angles with respect to array

much smaller range extent, an SBR utilising a pulse-Doppler waveform typically cannot remain range unambiguous.2 Therefore, an SBR must control range by using elevation aperture in a two-dimensional antenna array to focus to a particular range relying on elevation sidelobe control to reject range-ambiguous clutter. Since the deployment of two-dimensional arrays in space can be very costly, a limit must be placed on the total area of an array aperture. Therefore, elevation and azimuthal aperture inevitably must be traded off. Increasing azimuthal aperture improves crossrange accuracy and GMTI performance against slow moving targets, while elevation aperture can be used to control range and in turn range-ambiguous clutter. Another unique aspect of an SBR is that in an effort to optimise accuracy, the array is mechanically steered to keep the array face pointing at the aimpoint. Maintaining array broadside to the aimpoint is commonly referred to as boresighting the antenna. Like rotating arrays in airborne applications, mechanical steering leads to a mismatch in the platform velocity heading and the orientation of the array. A depiction of a mechanically steered array is shown in Figure 8.2 where (/> is used as the cone angle with respect to the array and ^ is the cone angle with respect to the velocity vector. The difference in these two angles: #mech=0-^

(8-1)

For example, at an altitude of 1000 km the PRF to remain range unambiguous to the horizon would be 55.5 Hz

is the mechanical steering angle. The angle between the array axis and the velocity heading is often referred to as the crab angle in certain airborne systems [4, 5]. This mismatch of the array axis with velocity heading leads to a range dependence of clutter Doppler and has certain consequences on STAP both in terms of training adaptive weights [6, 7] and range-ambiguous clutter. Although electronic steering can also be employed,3 for the analysis in this chapter we will restrict ourselves to a mechanically steered array without any electronic steering, i.e. array broadside 0 = 90°.

8.2.1 STAP for SBR systems Space-time adaptive processing (STAP) has become a well established research area, in large part due to a heavy research effort over the past 15 years, most of which has been well documented [4, 5]. Although the cancellation of ground clutter in radar systems using STAP had been introduced back in the 1970s [8], STAP has matured to the point that a recent textbook has been devoted to the topic [5]. Although most STAP work to date has focused on airborne platforms, space-based platforms have been considered more recently [9, 10]. STAP is the two-dimensional adaptive filtering for ground clutter cancellation used in radar applications with a moving platform, airborne or space-based, for the detection of moving targets. The principle that STAP exploits is that clutter returns constitute interference from the non-moving ground. These returns have a unique spatial and Doppler structure determined by the platform velocity and the orientation of the array. Moving targets, on the other hand, have a different Doppler than ground clutter due to the additional Doppler shift from their own radial velocity i>t relative to the radar. The Doppler frequency of a moving target is given by: ft = -r1 + - ^ c o s V t A

(8.2)

A

where up is the radar platform velocity, A is the radar wavelength and i/rt is the cone angle between the platform velocity vector and the direction to the target.4 The Doppler frequency of clutter coming from the same angle as the target is: 2 Un

/ c = - ^ cos Vt

(8.3)

A

Hence, the use of the two dimensions of space and time makes it possible to separate moving targets from clutter that is not possible in either angle or Doppler alone. Although most STAP analyses assume an TV element uniform linear array with A/2 spacing, SBRs typically use a two-dimensional array that is so large that forming digital channels on each element is not practical. Thus, we consider an SBR twodimensional array with subarrays formed in two-dimensional panels to create an array of TV spatial channels, i.e. subarrays, with a uniform spacing of D metres. Note that the spatial channels are formed in one dimension along the azimuthal axis as the 3

Electronic steering is used to alleviate mechanical steering requirements or to simultaneously cover multiple areas on the ground 4 V = 0° is forward, \Jr = 180° is aft and f = 90° is broadside

entire elevation dimension has been beamformed to the aimpoint for each subarray.5 Consider the problem of detecting a moving target at a Doppler frequency / t and an angle with respect to the array 0t- The pulsed waveform uses M coherent pulses transmitted at the pulse repetition frequency (PRF) /PR. The time between pulses, known as the pulse repetition interval (PRI), is simply the reciprocal of the PRF. The returns from these M pulses make up a coherent processing interval (CPI). The space and time response vectors of this target are given by: (8.4) (8.5) respectively. The combined space-time response vector of the target is then simply the Kronecker product: v(0t,/t) = b ( / t ) ® a ( 0 t )

(8.6)

Let us consider a space-time snapshot containing a target signal and given by: x ( / i ) = a t v ( 0 t , / t ) + Xi+n(n)

(8.7)

where oft, 4>u and / t are the target amplitude, angle with respect to the array, and Doppler frequency, respectively. Xi+n is the interference-plus-noise signal and n is the snapshot index. Here, we only consider interference consisting of ground clutter returns from the transmitted radar signal. The optimum space-time weight vector6 steered to 0o a t Doppler frequency /o is given by [8]: (8.8) where Qi+n = E{xi +n (n)xj +n (n) // } is the interference-plus-noise covariance matrix due to clutter coming from all angles. This version of the optimum STAP weights has been normalised for unit gain in the look direction. The performance of the optimum space-time processor from equation (8.8) is measured via the output signalto-interference-plus-noise ratio (SINR). As the name implies, SINR is simply the ratio of output target signal and interference-plus-noise powers: (8.9) where orf is the target signal power in a subarray channel. Many times, we want to compare the SINR to the maximum SINR that could possibly be achieved. This upper limit is determined by the ideal matched filter for the interference-free case, i.e. thermal noise only. Normalising the SINR by the SNR of the ideal (interference-free) The use of two-dimensional array channels is not considered in this chapter; adaptation for threedimensional STAP with two-dimensional degrees of freedom requires more complicated algorithms and training due to the range-dependent nature of clutter in the elevation dimension ^ STAP weights are optimum when (pQ = (pt and /o = ft

matched filter SNRo yields: (8.10) which is known as SINR loss [4]. Many times SINR loss is computed across angles and/or Doppler frequencies. An SINR loss of unity (0 dB) indicates perfect interference cancellation. Although the SINR loss metric indicates the losses associated with the presence of clutter, a more meaningful metric for a GMTI radar is the slowest velocity it is able to reliably detect, known as the minimum detectable velocity (MDV). A common measure of MDV is the minimum velocity for an acceptable SINR loss, e.g. LSINR = - 5 dB.

8.3

Clutter characteristics of pulse-Doppler waveforms in SBR

In this section, we examine the characteristics of clutter in an SBR system using pulse-Doppler waveforms that lead to some unique issues for STAP ground clutter cancellation. These characteristics arise from the range-Doppler ambiguity function for a pulse-Doppler waveform shown in Figure 8.3. Doppler ambiguities occur at regular spacings of the PRF /PR, and range ambiguities occur at integer multiples of the PRI, i.e. inversely proportional to the PRF. Control of these two ambiguity types is clearly at odds with one another. As we will see, requirements on a minimum PRF for Doppler considerations can lead to range ambiguities and degraded STAP performance. In addition, the Doppler resolution is determined by the coherent integration time (CPI length) and the range resolution by the radar bandwidth as shown in Figure 8.3. The implications of this pulse-Doppler ambiguity surface for an SBR arise from some unique aspects of the SBR platform. These aspects include:

azimuth beamwidth

Doppler

1 large platform velocity 2 high altitude 3 full mechanical steering (360°).

elevation beamwidth 2*bandwidth

integration time

range

Figure 8.3 Range-Doppler ambiguity surface for a pulse-Doppler waveform

The high platform velocity of an SBR leads to large Doppler frequencies for clutter and a large Doppler spread of mainbeam clutter. As a result, the radar has a minimum PRF that must be maintained for effective clutter mitigation and MDV performance. On the other hand, a high platform altitude leads to a large range to the ground and therefore a large illuminated area on the ground. To remain range unambiguous over the illuminated area, the radar must operate at a maximum PRF corresponding to the range extent of this illuminated area. Finally, range-ambiguous clutter is complicated by the mechanical steering of the array which leads to misalignment with the platform motion. The clutter has range-dependent angle-Doppler characteristics resulting in non-stationary behaviour in range complicating the clutter mitigation problem. We detail these characteristics and their implications on STAP in the following section.

8.3.1

Clutter Doppler ambiguities

SBR systems, unlike airborne radars, have such a high platform velocity that it is not possible to operate at a PRF that does not result in ambiguities in Doppler. Recall that the Doppler frequency of clutter from any point on the ground is given by: /c = ^COSi/rc A

(8.11)

where \j/c is the angle to the clutter with respect to platform velocity. The total Doppler extent of clutter is then the difference between forward ^c = 0° and ^c = 180° and is given by: 4v A/c = - 1

(8.12)

A

Since the PRF is essentially the Doppler sampling frequency, it must be greater than the total clutter Doppler extent in equation (8.12) to avoid any aliasing, i.e. to be unambiguous. For example, for a low-earth orbit SBR with an altitude of 1000 km and a platform velocity of 7000 m/s operating at / = 10 GHz, the total Doppler extent is 933.3 kHz. Choosing a PRF to be Doppler unambiguous in this case is clearly not practical.7 The fact that can be exploited in order to manage Doppler ambiguous clutter is that clutter has a unique angle-Doppler correspondence. As a result, Doppler ambiguous clutter can be rejected spatially either with low two-way transmit/receive (Tx/Rx) azimuth sidelobes or with STAP spatial degrees of freedom. For low grazing angles, i.e. shorter range, received clutter power is stronger and spatial DoFs can be used to handle clutter that leaks through azimuth sidelobes. At smaller grazing angles (long ranges), clutter power is weaker and azimuth sidelobes should be used to suppress Doppler ambiguities. Since we can rely on sidelobes and STAP to handle Doppler ambiguities, the key requirement is that we do not allow any Doppler ambiguities 7

A PRP this high would yield an extremely low range extent and would be highly range-ambiguous. In addition, this PRF would be severely oversampled and therefore redundant in terms of potential target velocities placing an unnecessary processing burden on the SBR

to exist within the receive azimuth mainbeam. Satisfying this condition allows for effective nulling of the clutter Doppler ambiguity and maintains acceptable SINR loss and MDV performance.8 For an array with an azimuthal aperture L az , the PRF must be chosen to be larger than the mainbeam clutter extent to avoid mainbeam Doppler ambiguities. Clutter within the mainbeam already requires STAP to perform mainbeam nulling to detect targets within the mainbeam clutter spread. For an array of length L32, the nulls for an untapered beam fall at COs^11 ^ = A/L az and the null-to-null beamwidth is: Acos

(mb)^|i

(8.13)

Therefore, the mainbeam clutter spread measured from null to null of the beamwidth is found by substituting equation (8.13) into equation (8.11) and is given by:

A/ffi = ^

(8-14)

L az

Note that this Doppler spread is independent of frequency or wavelength. As a result, the minimum PRF for effective STAP clutter suppression and to maintain good MDV performance is:

/PR > ~ L

(8.15)

az

This PRF is commonly referred to as the displaced phase centre array (DPCA) PRF which is the PRF that has a pulse-to-pulse delay T equal to the time for equivalent monostatic phase centres from the front and back halves of the array to spatially align from consecutive pulses [4]. Using a time delay between pulses that spatially aligns the front and back half apertures results in matching clutter characteristics for the two spatial channels. In theory, this condition allows for the perfect cancellation of clutter by subtracting consecutive pulses from the two half-aperture phase centres.

8.3.2

Clutter range ambiguities

The amount of unambiguous time associated with a pulse in a pulse-Doppler waveform is the pulse repetition interval (PRI) given by: ^P = y -

( 8 - 16 )

/PR

and the amount of time associated with a range extent AR on the ground is: At=™*

(8.17)

C

Provided the coherent integration time is long enough to give sufficient Doppler resolution and to ensure STAP performance is aperture limited, e.g. r c o h > ^r 2 -

Therefore, the amount of unambiguous range associated with a PRI, considering two-way propagation from the radar to the ground, is:

A*=-f-

(8.18)

2/PR

We will strictly concern ourselves with elevation mainbeam clutter since remaining range unambiguous out to the radar horizon is not practical for SBR. Instead, we will rely on the elevation Tx/Rx sidelobes and proper radar management of clutter power to ensure that ambiguities in the elevation sidelobes can be ignored. The amount of range illuminated by an SBR antenna array is determined by the elevation aperture Lei and the range to the ground. Although, for a flat earth assumption the illuminated range can be approximated by R\\ ~ /?A0ei, SBR applications must account for earth curvature and such a simple expression for illuminated range is generally not possible. However, the illuminated range is easily computed by computing the angles of elevation beamwidth nulls and computing the grazing angle, and therefore range, associated with these elevation angles. From the illuminated range, we can compute the range-unambiguous PRF by substituting for AR in equation (8.18) and solving for /PR. Figure 8.4 shows the range-unambiguous PRF, i.e. the maximum PRF for which the radar does not have any range ambiguities in the elevation mainbeam, as a function of grazing angle for our SBR example at / = 10 GHz with a platform

range-unambiguous PRF, Hz

5 metre 2 metre 1 metre

grazing angle, deg

Figure 8.4

Range-unambiguous PRF versus grazing angle for SBR with platform altitude h = 1000 km, f = 10 GHz and elevation apertures of1 m (solid line), 2 m (dotted line) and 5 m (dashed line)

altitude of 1000 km for elevation apertures of 1, 2, and 5 m. Recall the PRF for effective clutter cancellation from equation (8.15) is a function of the azimuth aperture. Usually, azimuth aperture will be L32 > 10 m for MDV considerations and 1500 > /PR > 3000 Hz. This requirement for range-unambiguous PRF is in clear violation of our minimum PRF requirement to maintain MDV performance from Section 8.3.1, especially for low grazing angles (long range). For these grazing angles, an SBR must simply accept the fact that range-ambiguous clutter exists within the elevation mainbeam. As we also see from Figure 8.4, more elevation aperture allows an SBR to operate range-unambiguously out to lower grazing angles. We explore this option as a tool to combat range-ambiguous clutter in Section 8.5.2.

8.4

Impact of range-ambiguous clutter on STAP performance

In the previous section, we discussed ambiguities in range and Doppler. Doppler ambiguities of clutter can be effectively handled with proper sidelobe control and STAP spatial degrees of freedom. Range-ambiguous clutter, however, is much more difficult to alleviate since STAP does not have a dimension to discriminate ambiguities from one another. Here, we give examples of the impact of range-ambiguous clutter and show the two aspects that affect the impact of range-ambiguous clutter: grazing angle or range to the radar and mechanical steering of the array. As discussed in Section 8.3.2, range ambiguities arise at long ranges due to the range extent of the elevation mainbeam. Mechanical steering of the array, however, is the mechanism which makes range-ambiguous clutter a big problem for GMTI performance. Consider the depiction of range-ambiguous clutter for the cases with and without mechanical steering shown in Figure 8.5. For a mechanically steered array, the misalignment between the velocity vector and the array axis produces isoDoppler and isocone angles that no longer overlay. As a result, clutter will have angle-Doppler characteristics that vary with range. This problem arises for STAP training across range, but its impact is much more severe in the case of range-ambiguous clutter. The result is multiple clutter ridges for all ambiguities which result in multiple Doppler frequencies that have clutter for any given angle, as shown in Figure 8.5. As we will see, the result is multiple SINR loss notches that create several blind velocity zones. To illustrate the impact of range-ambiguous clutter on STAP clutter cancellation performance, we show SINR loss versus target velocity for a few SBR scenarios.9 Throughout this chapter, SINR loss is computed from true covariance matrices and is intended to provide an upper bound on STAP performance that could be achieved in an actual SBR for the given geometry. The results do not reflect any losses arising from estimated covariances or from training STAP weights with range-varying clutter snapshots. In both cases, we look at the performance of a 20 x 2 m array operating 9 SINR loss is plotted versus target velocity and not Doppler frequency. Clearly, for mechanical steering angles other than broadside # mec h = 0°, the Doppler shift at array broadside is /dOpp ¥= 0- These SINR loss curves versus target velocity reflect the Doppler contribution of the target only and can be thought of as having compensated for SBR platform-induced Doppler on the target

isocone isoDoppler ambiguity aimpoint ambiguity

Doppler

no mechanical steering (azimuth = 90°)

angle mechanical steering

Doppler

range ambiguities

angle

Figure 8.5

Non-stationarity ofclutter ridges in angle-Dopplerfor range-ambiguous clutter

at a 1000 km altitude, / = 10 GHz and a PRF of / P R = 2500 Hz at the two ranges of 1300 km and 2700 km (47° and 11° grazing angles). The number of subarray channels is N = 8 and the CPI length is M = 64 pulses. The clutter-to-noise ratio in this case is 25 dB. SINR loss is shown for target velocities between ±100 km/hr (Doppler/velocity unambiguous out to ±67 km/hr for /PR = 2500 Hz), a typical range of velocities to expect for ground moving vehicles.10 First, we examine the performance for the array mechanically steered to broadside of the radar <9mech = 0° shown in Figure 8.6a. We see that the only difference in the two ranges is that the SINR loss notch at i>t = 0 km/hr is deeper for the long range case with the range ambiguities. The deeper notch is due to the fact that the clutter range ambiguities aligned in Doppler and angle and therefore just have an additive effect on overall CNR. Since the elevation mainbeam contains three ambiguous ranges at 11° grazing angle, the SINR loss is approximately —4.5 dB deeper. Next we look at SINR loss for the case when the array is mechanically steered forward to #mech = 60° shown in Figure 8.6b. Now clutter range ambiguities no longer align in Doppler and there are multiple SINR loss notches. The effect is that in addition to low velocity targets several other target velocities are undetectable, commonly referred to as blind velocities. *l

10 Note that Doppler frequency of a moving target corresponds to the radial velocity of the target with respect to the radar platform 11 Blind velocities are typically associated with Doppler frequencies separated from the Doppler at array broadside by multiples of the PRF but in this case are due to range ambiguous clutter

target velocity, km/hr

Cl

Figure 8.7

b

target velocity, km/hr

U

SINR loss, dB

SINR loss, dB velocity, km/hr

mechanical steering, deg

SINR loss without range-ambiguous clutter (1300 km range) and with range-ambiguous clutter (2700 km range) for mechanical steering of (p = 0° (broadside to platform) and 0 = 60° (forward scanned) a mechanical steering = 0° b mechanical steering = 60°

mechanical steering, deg

Figure 8.6

SINR loss, dB

SINR loss, dB a

velocity, km/hr

SINR loss versus mechanical steering angle without range-ambiguous clutter (1300 km range) and with range-ambiguous clutter (2700 km range) a Range = 1300 km (grazing angle = 47°) b Range = 2700 km (grazing angle =11°)

These complete losses in coverage are clearly undesirable. To get a better feel for the full effect on the coverage of an SBR, we show the SINR loss performance at both ranges for all mechanical steering angles from platform broadside (#mech = 0°) to forward (#mech = 90°) in Figure 8.7. Here, the effect on SBR coverage is quite dramatic. The region 0° < #rnech < 40° is almost completely lost as the range ambiguities have separated in Doppler and are all within the azimuth mainbeam. STAP cannot place multiple mainbeam nulls and as a result performance is clutter

limited rather than noise limited. As the array is steered forward (#mech > 40°), it is able to resolve the clutter range ambiguities and at least restore performance between the resulting blind velocities from the range-ambiguous clutter. In contrast, for the shorter range without clutter range ambiguities, coverage is complete for all mechanical steering angles.

8.5

Range-ambiguous clutter mitigation techniques with pulse-Doppler waveforms

We have now outlined the problem of range-ambiguous clutter for pulse-Doppler waveforms and shown their effect on GMTI performance with STAR Examining the pulse-Doppler ambiguity function from Figure 8.3, we clearly have two means of controlling clutter range ambiguities: lowering the PRF to increase the unambiguous range extent for a PRI from equation (8.16) or reducing the elevation beamwidth of the SBR by increasing the elevation aperture. Since PRF can only be reduced to the limit imposed by mainbeam clutter spread, we instead will consider a scheme of using multiple range-ambiguous PRFs to cover all potential target velocities, referred to as PRF diversity. For increasing the SBR elevation aperture, we will consider the case of maintaining the same azimuth aperture for MDV purposes. The resulting twodimensional array, therefore, has a larger array total area and is more costly in terms of deployment. This analysis may be used to gain insights into the amount of improvement additional elevation aperture provides to determine if the performance gains justify the additional cost. Note that we do not consider a true two-dimensional array with spatial channels in both the azimuthal and elevation dimensions. We assume that the elevation dimension is increased to reduce the elevation mainbeam and do not consider the use of elevation degrees of freedom for clutter cancellation [5, Chapter 10].

8.5.1 PRF diversity As has been shown, range ambiguities have the effect of creating additional blind velocities other than v\ = Okm/hr as illustrated in Figure 8.6b. The location of these blind velocities is determined by the range-ambiguous PRF falling at the Doppler frequencies of the clutter range aliases. By changing the alias ranges, accomplished by changing the PRF, we can alter the Doppler frequencies of the aliases and therefore move the blind velocities. In this section, we explore the use of multiple PRFs, or PRF diversity to cover all target velocities of interest. Note that this technique results in a loss in terms of surveillance rate since now multiple CPIs with different PRFs must be devoted to each area on the ground. We will not attempt to quantify this loss here since its specifics are highly dependent on the radar system parameters. Wereturntothecaselookedatearlierwitha20x2marrayat/z = 1000 km altitude mechanically steered to #mech = 60°. The range of the SBR to the aimpoint is 2700 km (Bp = 11°). Recall the SINR loss for a PRF / P R = 2500 Hz in Figure 8.6b. This plot is repeated in Figure 8.8 as the dash-dot line. Here, we also consider the two other

SINR loss, dB

PRF =1500 Hz PRF = 2000 Hz PRF = 2500 Hz target velocity, km/hr

Figure 8.8

SINR loss for multiple PRFs at 2 700 km range and mechanical steering ofcp = 60°. Solid line is / P R = 1500 Hz, dotted line is / P R = 2000Hz and dash-dot line is /PR = 2500 Hz

PRFs of/PR = 2000 Hz and 1500 Hz shown in the dotted and solid lines, respectively. Note that both of these PRFs are above the minimum PRF for this array which from equation (8.15) is / ^ m ) = 1400Hz. Using all three PRFs, we see that there is no additional blind velocity zone (LSINR < —5 dB) other than the ft = 0 km/hr zone. Clearly, three PRFs are sufficient to accomplish coverage for ±100 km/hr, although with the resulting losses in surveillance rate associated with dwelling on the same area with three different PRFs. Next, we consider full mechanical steering for PRFs of 1800 Hz and 2500 Hz shown in Figure 8.9. The lower PRF has reduced the number of total range ambiguities and therefore has fewer blind velocities, especially noticeable as the array looks forward. Important to note is that the region of mechanical steering angles of 0° < #mech < 40° has very poor performance for both PRFs, lacking detectability for all velocities. The reason for the complete loss of performance is that range ambiguities are now separated in Doppler due to the mechanical steering, yet both ambiguities fall within the azimuth mainbeam. STAP cannot cope with multiple mainbeam nulls and performance therefore becomes clutter limited since clutter cannot be effectively nulled. Once the mechanical steering is beyond #mech > 40°, the clutter range ambiguities fall outside the azimuth mainbeam and only affect the velocity corresponding to their associated Doppler frequency. Again, all the STAP performance results shown here are with true covariance matrices and do not reflect various real-world effects, e.g. estimation/training or nonstationary clutter. The training of the STAP adaptive weights with range-ambiguous clutter is complicated by the non-stationary nature of clutter for any mechanical steering. With multiple clutter points that all vary differently with range, techniques such as

Figure 8.9

SINR loss, dB

mechanical steering, deg

SINR loss, dB

mechanical steering, deg

velocity, km/hr

velocity, km/hr

SINR loss versus target velocity and mechanical steering angle for 20 x 2m array at range of 2700km a PRF = 2500Hz b P R F = 1800Hz

Doppler warping [6, 7] will not be very effective. The best alternative is to have the weights vary with range and update rapidly in range or to use some technique that accounts for non-stationarity.

8.5.2 Aperture trade offs The complete solution to the range-ambiguous clutter problem for SBR is to increase the elevation aperture to the point where range ambiguities do not exist. Of course, in most cases this increase in aperture comes at a very large cost. We will not attempt to get into this aspect of larger SBR antenna arrays but instead will only attempt to show how the use of larger apertures can enable full coverage for an SBR. We again consider the case of an SBR at an altitude of h = 1000 km operating at / = 10 GHz and a range to the aimpoint of 2700 km (0& = 11°). We look at the STAP performance for the same PRFs (/ PR = 1800 and 2500 Hz) as in Figure 8.9 for the 20 x 2 m aperture. However, in this case, we consider a 20 x 5 m aperture. The larger elevation aperture reduces the range extent illuminated on the ground and therefore the radar can still be range unambiguous at higher PRFs. The performance of the 20 x 5 m aperture is shown in Figure 8.10 for mechanical steering angles from #mech = 0° to #mech = 90°. This aperture can operate range-unambiguously for /PR = 1800 Hz while still having two range ambiguities for /PR = 2500 Hz. Clearly, the /PR = 1800 Hz could operate at this range without any problems. Even the higher PRF /PR = 2500 Hz has much better performance, even for 0° < #mech < 50° when the ambiguities cause multiple mainbeam nulls. Overall, the increased aperture is the most attractive solution for the rangeambiguous clutter problem since it completely eliminates the ambiguities. It should be noted that we demonstrated performance for a 20 x 5m array at #gr = 11° or 2700 km range. Further ranges still have range ambiguity problems and the amount

Figure 8.10

SINR loss, dB

mechanical steering, deg

SINR loss, dB

mechanical steering, deg

velocity, km/hr

velocity, km/hr

SINR loss versus target velocity and mechanical steering angle for 20 x 5 m array at range of 2700km a PRF = 2500Hz b PRF = 1800Hz

of elevation aperture needed to remain range unambiguous for all ranges depends on the maximum range for the SBR application.

8.6

Long single pulse phase-encoded waveforms

It has already been established that more aperture and multiple coherent processing intervals with different PRFs can be used to overcome the ambiguous nature of the returns from pulse-Doppler waveforms. Another way of handling this problem is to use waveforms which are effectively unambiguous in both range and Doppler. In this section, the use of unambiguous long single pulse waveforms is discussed.12 Examples of appropriate modulation include random binary phase encoding [3, Chapter 10.6] and chaotic modulation [H]. In this section, we consider the binary phase-encoded waveforms for illustration purposes. The Doppler resolution for single pulse waveforms is simply the inverse of the pulse length, i.e. the total coherent integration time. For conventional monostatic radars, which are unable to transmit and receive simultaneously, the pulse length is limited to less than half the propagation time to the target, since there must be enough uneclipsed range samples to fill the pulse-compression filter. We use the following well known relationships of the propagation time Jpuise> Doppler resolution A/dOpP

The objective could be viewed as designing a waveform whose ambiguities, if they exist, do not coincide with clutter returns, i.e. ambiguities are moved in such a way that all returns from the earth are unambiguous. We focus merely on the concept and do not investigate waveform design

resolution, km/hr

range, km

Figure 8.11 Doppler (velocity) resolution versus pulse length for a 10 GHz radar

of the pulse, and Doppler frequency /dopp(8.19) where R is the range to the target, c is the speed of light (propagation velocity), v is velocity and k is the wavelength. From these relations, we can derive the Doppler (velocity) resolution: Aw=^

(8.20)

Figure 8.11 shows the limit on Doppler resolution as a function of range for a radar operating at 10 GHz. The Doppler resolution of a GMTI radar should be significantly better than the specified target MDV for the radar. Airborne GMTI radars typically look for targets at ranges of under 300 km, equating to Doppler resolutions which are worse than 25 km/hr and clearly inadequate for typical GMTI performance requirements. However, SBR systems have considerably longer ranges to the target. For the SBR example explored in this chapter with a 1000 km orbit and a minimum range of 1200 km, the Doppler resolution without simultaneous transmit and receive is better than 7 km/hr, which is adequate for most GMTI applications. Figure 8.12 pictorially summarises the concept of the long single pulse waveform for SBR.

amplitude

phase encoded waveform

time

Concept of long single pulse waveform for SBR

response, dB

velocity, km/hr

Figure 8.12

range (km)

Figure 8.13 Ambiguity surface for a 10 ms 5 MHz chip rate phase-encoded pulse at 10 GHz 8.6.1

Properties of long single pulse phase-encoded

waveform (LSP W)

Except for a few special cases [12-14], phase-encoded waveforms have mean range and Doppler sidelobes levels that are inversely proportional to the time-bandwidth product given by: SLL = — — (8.21) TxB For example, a time-bandwidth product of one million is needed to achieve mean range-Doppler sidelobes of -6OdB. This mean sidelobe level is combined with a point-like ambiguity function as shown in Figure 8.13. These waveforms are

unambiguous in range, and unambiguous in Doppler up to the sample or chip rate of the waveform. As we will discuss below, this mean sidelobe level is a limitation of the phase-encoded LSPW, and care must be used in designing the radar to cope with this limitation. In conventional pulse-Doppler radars that utilise chirp waveforms or short phase-encoded pulses, the large amount of range-Doppler coupling makes the pulsecompression and Doppler filtering operations approximately separable. Since the range-Doppler coupling of the phase-encoded LSPW described here is minimal, the pulse-compression and Doppler filtering operations of the radar are not separable, and a Doppler optimised pulse-compression filter is required for each Doppler bin. There are both pros and cons to having separate pulse-compression filters for each Doppler bin. The pro is that for constant velocity targets there are no range walk losses due to target motion over the length of the single pulse. The Doppler-dependent pulse-compression filters are already tuned to the motion. The con is the amount of computation required. For example, an X-band SBR system with a 3000 km (20 msec) pulse has a Doppler resolution of 2.7km/hr from equation (8.20). To cover target velocities of ±110km/hr, eighty separate pulse-compression filters are needed. For waveforms with large time-bandwidth products, the computational complexity of the pulse-compression and Doppler filtering becomes phenomenal with conventional FIR implementations of the pulse compressor. However, the use of overlap-add FFTs helps ameliorate the computational burden. One example of an LSPW STAP processing chain is shown in Figure 8.14. In this case, the Doppler-dependent pulse-compression filters naturally feed the adjacent bin post-Doppler STAP algorithm [15]. However, this is not the only STAP algorithm which may be applied to the LSPW. Application of an inverse-discrete-Fourier transform to the output of all the pulse-compression filters on a per-range-gate basis (Figure 8.15) converts the output into a pre-Doppler pulse-like space with an apparent

pulse comp. Doppler bin 1 N channels pulse comp. Doppler bin 2 digital receiver pulse comp. Doppler bin...

pulse comp. Doppler bin M

Figure 8.14

Adjacent-bin post-Doppler STAP for a long single-pulse waveform

Af channels pulse comp. Doppler bin 2 digital receiver

pulse comp. Doppler bin...

inverse Fourier transform

pulse comp. Doppler bin 1

M 'Pulses' (nyquist sampled for highest Doppler)

pulse comp. Doppler bin M

Figure 8.15

Pulse-Doppler matched filtering for LPSW radar with an inverse DFT across Doppler frequency allows LPSW radar to emulate a conventional pulse-Doppler radar

PRF of: (8.22) where / ^ a x and /^ 11 I are the Doppler frequencies of the highest and lowest Doppler bins used. With this transformation any STAP algorithm that can be applied to pulseDoppler waveforms may also be used with an LSPW. The LSPW advantages of no Doppler or range ambiguities are still retained.

8.6.2 Integrated sidelobe clutter levels One of the key issues with LSPWs is the integrated sidelobe clutter level. If the integral of all the clutter outside the range-gate and Doppler bin of interest is significant compared with the noise floor, then the radar will be desensitised. Therefore, a fundamental requirement on an LSPW radar is an average range-Doppler sidelobe level that is better than the integrated clutter level. The integrated clutter level is computed by integrating the radar range equation over the illuminated clutter [16]: (8.23) where Pt is the transmit power, G(4>az, R) is the antenna gain as a function of azimuth angle 0 ^ and range R, o~(R) is the clutter reflectivity as a function of range, t the pulse length, X the wavelength, k Boltzman's constant, TQ = 290 K, L the radar system losses and F the noise figure. In determining the integrated clutter level, the radar designer only has control over the first three terms in the equation, transmit power and transmit and receive gain.

By approximating the azimuth distribution of the antenna pattern by the Rayleigh beamwidth b and modelling the clutter reflectivity with the constant Gamma clutter model [17] where #gr represents the grazing angle, equation (8.23) becomes:

(8.24)

CNR, dB

The middle term of equation (8.24) represents the clutter reflectivity multiplied by the area illuminated between the ranges R and R + Rg (Rg is the range resolution of the radar). It is interesting to note that since the area illuminated by the radar is approximately inversely proportional to the antenna's gain, the integrated clutter level is approximately proportional to the power aperture of the radar. Since the received target power is proportional to the power aperture squared of the radar, for the same target SNR, radars with larger apertures (and hence less transmit power) will exhibit smaller integrated clutter levels. For the X-band SBR described earlier, with a peak transmit power of 30 dBw, a 20 x 2m aperture (gain of 57.5 dB), noise figure of 4dB, 1OdB of losses, range resolution of 1 m, pulse length equal to the propagation time and a 50 dB receive taper in elevation, Figure 8.16 depicts the clutter-to-noise ratio as a function of range for a beam illuminating the ground at a range of 2000 km with a clutter y of —12 dB.

range, km

Figure 8.16

Clutter-to-noise ratio versus rangefor an LSP W with the example radar system

level, dB

constant power CNR mean sidelobe level variable power CNR range, km

Figure 8.17

Clutter-to-noise ratios for constant and variable transmit power and mean sidelobe level versus range

The CNR is positive for a range extent of about 150 km, or 150000 range gates. Integrating over range gives an integrated clutter-to-noise ratio of 65.2 dB. Figure 8.17 shows both the integrated clutter level and mean range-Doppler sidelobe level (assuming the LSPW length is the same as the propagation time and a bandwith of 150 MHz) as a function of range. If the transmit power is kept constant irrespective of the range which is being illuminated, then the integrated CNR is larger than the mean sidelobe level at ranges less than 2565 km. This would make the radar's performance clutter limited rather than noise limited. However, it should be noted that the target SNR grows faster with decreasing range than does the unsuppressed clutter, so target detectability still improves. However, if the transmit power is varied so as to keep constant SNR on target, then the ratio of the integrated CNR to the mean sidelobe level drops with decreasing range. The effective transmit power to the target can be adjusted in two ways: either by reducing the gain of the transmit amplifiers, or by broadening the transmit beam and forming multiple high gain receive beams. The latter method has the advantage of maximising the radar's search area rate. If the radar is designed to meet the SNR on target and the integrated CNR requirements at maximum range, then performance will be better at all closer ranges. As an example, consider the design of a radar which needs to attain a target SNR of 7.5 dB per pulse at 3000 km. Figure 8.18 shows how the integrated CNR changes as a function of array area and transmit power at 3000 km for a bandwidth of 150 MHz. The integrated CNR is approximately proportional to the radar's power aperture. Contours

integrated CNR, dB

area, m2

7.5 dB SNR on target 63 dB ICNR contour 66 dB ICNR contour peak power, dBw

Figure 8.18

Integrated CNR as a function of area and peak power

of constant integrated CNR and a constant target SNR of 7.5 dB are also shown in the Figure. To design a 150 MHz bandwidth radar which would achieve 7.5 dB SNR per pulse and have the integrated CNR equal to the mean range-Doppler sidelobes requires an array area of 56 m2 and a peak power of 1.25 kW. A radar with the integrated CNR 3 dB below the mean range-Doppler sidelobes at 3000 km requires an area of 111 m2 and a peak power of 330 W.

8.6.3 STAP simulations Here, we look at STAP simulations using the 40 m2 aperture SBR described earlier, and an LSPW with -7OdB range-Doppler sidelobes. The target range is 2500 km and the mechanical steering angle is 30° from the velocity vector, i.e. #mech = 60°. Two hundred snapshots are selected from between 2490 km and 2510 km to estimate the clutter-plus-noise covariance matrix. There are eight spatial degrees of freedom (subarrays) and the pulse is 16 msec long. Three different processing schemes are considered: 1

Pulse-Doppler with a PRF of 2 kHz, Doppler warping [6] and PRI-staggered STAP [5, Chapters 10, 18 and 19] 2 LSPW with adjacent-bin post-Doppler STAP from Figure 8.14 3 LSPW with transformation back to pulse-space from Figure 8.15, Doppler warping and PRI-staggered STAR In case 1, the /PR = 2000 Hz PRF means that range-ambiguous clutter at ±75 km will also be above the noise floor. Doppler warping is applied in cases 1 and 3 to compensate for the change in the Doppler of the mainbeam clutter as a function of

SINR loss, dB

velocity, km/hr

Figure 8.19

STAP performance for processing case 1

range due to the Doppler. However, in case 1 the Doppler warping is only precisely correct for the clutter around 2500 km, not the range-ambiguous clutter, leading to further degradation of the STAP performance. Since the Doppler warping is applied across the pulses on a range gate by range gate basis prior to the Doppler filtering, it is incompatible with the processing in case 2. The PRI staggered processing is chosen for cases 1 and 3 since this algorithm is known to exhibit better performance than the adjacent bin algorithm. The results are plotted in Figures 8.19 to 8.21. In all cases optimum refers to full dimension STAP with exact knowledge of clutter covariance (impossible in practice). Figure 8.19 shows the SINR performance of processing case 1. The nulls at about —32 km/hr and 27 km/hr are due to the range-ambiguous clutter from 2425 km and 2575 km. In this case, the range-ambiguous clutter nulls are only slightly wider than the correct null at 0 km/hr, since the Doppler warping is close to being correct for these ambiguities. With higher CNR the broadening of these nulls would be more severe. Figure 8.20 shows the STAP performance of processing case 2. In this case there is only a single null, but it is broad. Since processing chain 2 is incompatible with Doppler warping, the change in the Doppler of the mainbeam clutter as a function of range cannot be compensated for in the training data. Figure 8.21 depicts the STAP performance of processing case 3. There is only a single narrow null. The combination of the LSPW with a transformation back into the time domain from the Doppler domain, Doppler warping and PRI staggered STAP combine to give near-optimal performance. The only losses with respect to optimum

SINR loss, dB

optimal adjacent bin

velocity, km/hr

STAP performance of processing case 2

SINR loss, dB

Figure 8.20

optimal PRI stagg.

velocity, km/hr

Figure 8.21

STAP performance of processing case 3

STAP are due to the finite sample support and low sidelobe weighting on the Doppler filters in the PRI staggered algorithm.

8.7

Summary

In this chapter, we have discussed the problem of range-ambiguous clutter and its effect on the performance of an SBR system used for GMTI. Range-ambiguous clutter is a problem that arises at long ranges when the array is mechanically steered forward or aft. The misalignment of the array with the velocity vector creates a change in the angle-Doppler characteristics of range-ambiguous clutter and results in multiple blind velocity regions for GMTI. Pulse-Doppler waveforms can utilise multiple PRFs in order to shift the Doppler of these ambiguities and cover all velocities. The cost of this PRF diversity approach is the loss in search rate associated with using multiple PRFs to cover each spot on the ground. Also, PRF diversity is only effective at overcoming clutter range ambiguities that fall outside the mainbeam. For mechanical steering angles from broadside #mech = 0° to #mech = 40°, the 20 x 2m array considered in this chapter could not place multiple mainbeam nulls and, as a result, suffered large losses for all velocities. A more effective means of mitigating range-ambiguous clutter for pulse-Doppler waveforms was to increase the elevation aperture, completely eliminating or reducing the number of range ambiguities depending on the grazing angle. More aperture is clearly the preferred method of overcoming range-ambiguous clutter, the problem becomes a matter of the large cost of deploying a large aperture for an SBR. Finally, we discussed the new, novel approach to range-ambiguous clutter that uses an alternate waveform consisting of a single, long pulse utilising phase encoding to remain range and Doppler unambiguous. The length of the pulse is determined by the range of the SBR to the aimpoint so that at longer ranges, longer pulses are possible. Recall that the problem of range-ambiguous clutter arises at longer ranges. One of the main issues with a long phase-encoded pulse waveform is integrated sidelobe levels which are a problem due to the range-Doppler sidelobe levels of this waveform. Another issue is the amount of computation associated with the matched filtering in range and Doppler of the phase-encoded waveform. Two efficient architectures for STAP were given along with simulations demonstrating the viability and benefit that a range and Doppler unambiguous waveform could provide. This alternative waveform shows promise for SBR applications and bears further investigation and experimentation. References 1 CANTAFIO, L. J.:' Space-based radar handbook' (Artech House, Norwood, MA, USA, 1989) 2 DAVIS, M. E.: 'Technology challenges in affordable space-based radar'. Proceedings of IEEE international Radar conference. Washington, USA, 2000, pp.18-23

3 SKOLNIK, M.: 'Radar handbook' (McGraw-Hill, New York, USA, 1990) 4 WARD, J.: 'Space-time adaptive processing for airborne radar'. MIT Lincoln Laboratory TR 1015, ESC-TR-94-109, 1994 5 KLEMM, R.: 'Principles of space-time adaptive processing' (IEE, London, England, 2002) 6 BORSARI, G. K.: 'Mitigating effects on STAP processing caused by an inclined array'. Proceedings of IEEE Radar conference, 1998, pp. 135-140 7 KREYENKAMP, O. and KLEMM, R.: 'Doppler compensation in forwardlooking STAP radar', IEE Proc, Radar Sonar Navig., October 2001, 148, (5), pp. 253-258 8 BRENNAN, L. E. and REED, I. S.: 'Theory of adaptive radar', IEEE Trans. Aerosp. Electron. Syst., March 1973, 9, (2), pp. 237-252 9 NOHARA, T. J.: 'Design of a space-based radar signal processor', IEEE Trans. Aerosp. Electron. Syst., 1998, 34, (2), pp. 366-377 10 KOGON, S. M., RABIDEAU, D. J., and BARNES, R. M.: 'Clutter mitigation techniques for space-based radar'. Proceedings of IEEE international conference on Acoustics, speech, and signal processing, 1999, pp. 2323-2326 11 OPPENHEIM, A. V. and CUOMO, K. M.: 'Chaotic signals and signal processing', in MADISETTI, V. and WILLIAMS, D. (Eds.): 'Digital signal processing handbook' (CRC Press, Boca Raton, USA, 1997) pp. 71.1-71.13 12 BARKER, R.H.: 'Group synchronizing of binary digital systems', in 'Communication theory' (Academic Press, 1953) pp. 272-287 13 ZIERLER, N.: 'Linear recurring sequences', /. Industrial Applied Mathematics, 1959, 7, pp. 31-48 14 WELCH, L. R.: 'Lower bounds on the maximum cross correlation of signals', IEEE Trans. Inf. Theory, 1914, 20, pp. 397-399 15 DIPIETRO, R.: 'Extended factored space-time processing for airborne radar systems'. Proceedings of Asimolar conference on Signals and systems, 1992, pp. 425-430 16 VAN TREES, H.: 'Detection, estimation, and modulation theory, part III (John Wiley & Sons, New York, USA, 2001, 2nd edn.) 17 BILLINGSLEY, J. B.: 'Low angle radar land clutter measurements and empirical models' (SciTech Publishing, Norwich, NY, 2002) 18 BRENNAN, L. E. and STAUDAHER, R M.: 'Subclutter visibility demonstration'. Adaptive Sensors Inc. technical report, RL-TR-92-21, 1992 19 WARD, J. and STEINHARDT, A. O.: 'Multiwindow post-Doppler space-time adaptive processing'. Proceedings of 7th workshop on Statistical signal and array processing, 1994, pp. 461-464

Part III

Processing architectures

Chapter 9

Parallel processing architectures for STAP Alfonso Farina and Luca Timmoneri

9.1

Summary and introduction

This chapter describes methodologies for online processing of received radar data by a set of N antennas and M pulse repetition intervals (PRIs) for the calculation of space-time adaptive (STAP) filter output. The numerically robust and computationally efficient QR-decomposition (QRD) is used to derive the so-called MVDR (minimum variance distortionless response) and lattice algorithms; the novel inverse QRD (IQRD) is also applied to the MVDR problem. These algorithms are represented as systolic computational flow graphs. The MVDR is able to produce more than one adapted beams focused along different angular directions and Doppler frequencies in the radar surveillance volume. The lattice algorithm offers a computational saving; in fact, its computational burden is 0(N2M) in lieu of 0(N2M2). An analysis of the numerical robustness of the STAP computational schemes is presented when the CORDIC (coordinate rotation digital computer) algorithm is used to compute the QRD and the IQRD. Benchmarks on general purpose parallel computers and on a VLSI (very large scale integration) CORDIC board are also presented.

9.2

Baseline systolic algorithm

The detection of low flying aircraft and/or surface moving targets, and the standoff surveillance of areas of interest require a radar on an elevated platform like an aircraft. The AEW (airborne early warning) radars pose a number of interesting technical problems especially in the signal processing area. The issue is not new: detect target echoes in an environment crowded with natural (clutter), intentional (jammer) and other unintentional radiofrequency (especially in the low region of microwaves, e.g. VHF/UHF bands) interference. The challenge is related to the large dynamic range of the received signals, the non-homogeneous and non-stationary nature of

the interference, and the need to fulfil the surveillance and detection functions in real time. One technique proposed today to solve the problem is based on STAP [2-4,9,10,14-18]. Essentially, the radar is required to have an array (for instance, a linear array along the aircraft axis) of Af antennas each receiving M echoes from a transmitted train of M coherent pulses. Under the hypothesis of disturbance having a Gaussian probability density function and a Swerling target model, the optimum processor is provided by the linear combination of the NM echoes with weights w = M - 1 S*, envelope detection and comparison with threshold. M is the spacetime interference covariance matrix, i.e. M = E{z*zT} where z (dimension NM x 1) is the collection of the NM disturbance echoes in a range cell, s, the space-time steering vector, is the collection of the NM samples expected by the target and (*) stands for complex conjugate. A direct implementation (via sample matrix inversion, SMI) of the weight equation w = M - 1 S* is not recommended. One reason is related to the poor numerical stability in the inversion of the interference covariance matrix especially when a large dynamic range signal is expected during the operation; another one is the very high computational cost. There is a need of extremely high arithmetic precision during digital calculation. Note that double precision costs four times as much as single precision. The situation would be different if, instead of operating on the covariance matrix M, we would operate directly on the data snapshots z(k), k = 1,2,... ,ft where n is the number of snapshots (i.e. range cells) used to estimate the weights w. It can be shown that the required number of bits to calculate the weights, within a certain accuracy, by inversion of M is two times the number of bits to calculate the weights operating directly on the data snapshots z(k). This is so because the calculation of power values is avoided, and thus the required dynamic range is halved. The algorithms that operate directly on the data are referred to as data domain algorithms in contrast to the power domain algorithms requiring the estimation of M. Figure 9.1 depicts both approaches; in this chapter we will develop the algorithms based on the data domain approach. The QRD is a numerical technique for solving least squares problems, like the one in STAP, that avoids direct computation and inversion of interference covariance matrix [1,5]. Indicate with Z the n x (NM)-dimensional matrix which collects the n data snapshots: (9.1) The weight equation can be written as follows: (9.2) where (m)H stands for the complex conjugate transpose. Taking the data matrix Z and operating on it with unitary (i.e. covariance preserving) matrix Q (with dimension ft x /i) we are able to transform the matrix Z in an upper triangular matrix R (with dimension NM x NM): (9.3)

data cube CUT linear combination of weights and signals from the cell under test (CUT)

antenna elements

adapted output

range cells PRT

covariance matrix estimation

weight calculation

power domain approach

data matrix triangulation data domain approach

Figure 9.1

The power and data domain approaches for STAP

thus equation (9.2) can be rewritten as: R^Rw = s*

(9.4)

which is now easily solved by forward and back-substitution steps as follows. Indicating by a new vector, t, the product Rw, equation (9.4) becomes: R H t = s*

(9.5)

that can be solved in t. Subsequently, the additional equation: Rw = t

(9.6)

is solved in w. A noticeable improvement of the basic technique allows us to calculate the STAP output without extracting the weights, i.e. without performing the two substitutions above (see, for instance Reference 1 at page 147, see also Section 9.10, Appendix A). In summary, either the weight vector w or the output signal of the STAP are obtained without forming and inverting any covariance matrix. By using a large number of bits the data domain algorithm provides the same results as the power domain algorithm which estimates the covariance matrix ZH(n)Z(n) and derives the weight vector by the conventional Cholesky factorisation of that matrix in equation (9.2). The noticeable result is related to the far superior performance of the data domain algorithm when using a limited number of bits; in fact, the data domain algorithm needs half the number of the bits required by the power domain method to reach good interference cancellation and target coherent integration. The triangularisation of the data matrix, see equation (9.3), can be done with the following known methods: Givens rotations, Householder reflections (a generalisation of Givens rotations) and Gram Schmidt [I]. Another method to obtain a sparse (actually a diagonal in lieu of triangular) data matrix is singular value

decomposition (SVD); the Jacobi and Hestenes are recursive parallel algorithms to efficiently obtain the SVD. The Lanczos is another numerically efficient candidate to solve our real-time STAP problem [6]. The preferred approach in this chapter is the one based on Givens rotations (see Section 9.10, Appendix A). All these methods enj oy the possibility of being mapped onto a parallel processor such as a systolic array. This means that the algorithm is readily transformed in a computer architecture; this is not the case for the equation (9.2) where a single processor computer has the task of performing the indicated operations. Today it is possible to implement a systolic array with custom VLSI technology thus providing compact processors requiring limited prime power. An additional advantage is related to the large data throughput of the parallel processor representing a suitable means of reaching real-time operation. A remarkable implementation of a systolic algorithm on VLSI chips is called MUSE; it was developed by C. Rader and colleagues at MIT-Lincoln Laboratory (USA) (see third entry of the table in Section 9.14, Appendix E). The baseline architecture considered for the STAP problem is the trapezoidal one depicted in Figure 9.2 [5]. This constitutes the generalisation of a method, which was originally developed for MVDR beamforming, by QRD. The TVM-dimensional triangular array ABC receives the snapshots of data from a set of range cells and outputs from the right-hand side the matrix R produced as the data descend through the array. The matrix is passed to the right-hand column of cells DE which serves to steer the main beam in the desired angular direction and Doppler frequency. Multiple beams can be formed simply by adding right-hand columns as depicted in Figure 9.2; they are constraint post-processors. The bulk of the computation, i.e. the QRD, is common to all of the separate beamforming tasks, and only needs to be performed once. The MVDR processor in Figure 9.2 is designed to operate in the following manner [5]. The triangular processor, in normal adaptive mode (selected by setting an input binary flag / = 1), is fed with sufficient data snapshots to form a good statistical estimate of the environment. The triangular array is then frozen (by setting the input binary flag / = 0) while a look-direction constraint is input as though it were a data vector z(n) emerging from the multichannel tapped delay line. This serves to calculate the vector a = ( R ^ ) " 1 s* which is captured and stored in the right-hand column (also operating in mode / = 0); this vector is needed to determine the STAP output e(n) = zT(n)R~l (RH)~l s*. Once the vertical columns have been initialised, the adaptive mode of operation ( / = 1) is selected for both the main triangular array and the right-hand columns and more data snapshots are presented to the processor. The processor then updates its estimate of the environment (via the stored quantities R and a) and simultaneously outputs the STAP signals from the bottom of the columns DE. The number of processing elements in the triangular systolic arrays is 0.5(MN + X)-MN. The MVDR algorithm has a noticeable computational advantage with respect to the SMI which requires O(N3 M3) arithmetic operations per sample time. Two types of processing element are needed within the triangular array: one calculates the sine and cosine of an angle between two input data values, the other rotates the remaining data of the same angle. The calculation of the rotation and the application of the rotation is repeated for each row of the triangular array. A third cell type is used in

triangular array

Figure 9.2

detection

detection

no target target present

no target target present

Baseline QRD based MVDR flow graph [5]

in the look-direction constraint columns. Every processing cell of the triangular array should perform on average 24 floating-point operations per data snapshot. Let d be the desired data rate, i.e. the snapshots per second to process, the systolic machine should perform HdM2N2 flops. As an example, let d be 1 MHz and NM = 1 0 0 the corresponding processing power needed is 100 Gflops approximately. By down sampling (see also Section 9.4) the radar data by a factor often, the required processing power is 10 Gflops.

9.3

Lattice and vectorial lattice algorithms

An advanced processing architecture referred to as the MVDR lattice processor requires 0(N2M) arithmetic operations per sample time; it is described in Reference 5. It takes advantage of the time-shift invariance1 associated with STAR 1 This requirement is fulfilled only if the PRI is constant (no PRI staggering) and no platform motion perturbations (e.g. rapid acceleration) occur

trapezoidal array

1st stage

delay 2nd stage

MVDR column final stage

beamformer residual

Figure 9.3 MVDR lattice processor [5]

The data entering the triangular array change very little from one PRI to the next which means that a large part of the computation is being repeated on successive PRIs albeit in different parts of the array. This repetition is eliminated in the lattice algorithm where the big trapezoidal array is decomposed in a lattice of smaller (i.e. of dimension N) trapezoidal arrays; the lattice has M stages (see Figure 9.3). The lattice-based MVDR operates in a similar manner to the big trapezoidal array; details are found in Reference 5 (see also Section 9.11, Appendix B). If M = N = 10 and the update rate is one tenth of 1 MHz, the required computational power is 1 GfIop. The lattice

algorithm has also been designed and tested with simulated data for wideband STAP [12]; this architecture is particularly useful: (i) to deal with wideband radar, (ii) to compensate for amplitude and phase mismatching between the receiving channels, and (iii) to combat the hot clutter. The processing architecture, named the vectorial lattice, operates on an array of Af antennas, M PRIs and P samples taken within the radar range cell. The lattice has again M stages, each having trapezoidal arrays of dimension NP. The computational complexity of the scheme is 0(MN2P2). In the above mentioned processing architectures, namely the MVDR, the lattice and the vectorial lattice, the common processing module is the triangular systolic array. In Sections 9.5 and 9.6 we report the results concerning the mapping of the triangular systolic array onto parallel processors.

9.4

Inverse QRD-based algorithms

A further improvement of the triangular systolic array for STAP processing is called IQRD (inverse QR decomposition) and promises an additional decrease of the required computational power. The need to reduce the computational requirements of the triangular systolic array was discussed in Section 9.2. The possibility of down sampling of a factor ten, say, the update of the triangular array was mentioned. This tacitly requires the extraction of the adaptive weights of the STAP at the low update rate and the application of the weights to the radar echoes at the natural rate of the data. This approach has the following practical problem. The MVDR systolic array of Figure 9.2 could extract the adapted weights via back substitution; however, pipelining the two steps of triangular update and back-substitution seems impossible. There are two possibilities to overcome this problem. The first is to use a triangular array in addition to the main one, the second triangular array being reversed with respect to the array which updates the matrix R [14, p. 332]. This approach requires more hardware to be integrated on the chip. The second approach exploits a recursive equation which updates ( R ^ ) " 1 instead of R. The update of ( R ^ ) " 1 serves the purpose of extracting the weights. This algorithm, referred to as IQRD, can be implemented with just one triangular systolic array, which has a specular orientation of the basic triangular array to update R. Figure 9.4 illustrates the processing scheme for the sidelobe canceller (an adaptive system with a main antenna and a number of auxiliaries); by resorting to the concept of generalised sidelobe canceller, it is possible to show that a slightly modified version of the scheme is applicable to STAR The complex valued set of data x e CN are received by the N auxiliary channels; y is the data collected by the main antenna; x, y are processed to give the set of adapted weights w. The quantity e = y — x r w is the adapted residue; the parameter 8/8* occurs to update the weights as more data are received, during time, by the radar; see Section 3.1 of Reference 14 for details. A limitation of this approach is related to the difficult schedule of the various processing steps. A detailed comparative analysis of the IQRD and QRD-based MVDR algorithms is presented in Reference 14. Also, an implementation of the corresponding systolic architectures, with the use of the CORDIC algorithm as a basic building block, is discussed.

Figure 9.4

9.5

RLS-IQR array [14]

Experiments with general purpose parallel processors

This section summarises the findings described in detail in Reference 8; today this study seems out of date for the advancement in signal processing hardware, nevertheless it is still very instructive. We study the use of parallel processors of MIMD (multiple instruction streams multiple data streams) and SIMD (single instruction stream multiple data streams) types available on the market (early 1990s). This approach is meant to be preliminary to the VLSI solution. In fact, it provides guidelines for the design of the processing architecture to be implemented on silicon. The problems of synchronisation of the whole systolic array by a master clock and the data transfer between processors can also be investigated. Additionally, an estimate of the computational performance of several candidate processing architectures is also possible. With reference to the MIMD machine, a reconfigurable transputer-based architecture (the MEIKO computing surface, using up to 128 T800 INMOS transputers) has been adopted and three solutions have been proposed. The first uses a ring of transputers. Then an improvement of performance is reached by diminishing the amount of communication required; such a result has been achieved by using a linear array of processors. The mapping of the algorithm onto a triangular array of processors has also been studied. This solution allows the use of an arbitrary number B of processors provided that B = p(p + l)/2, p being an integer number. This mapping shows performance better than does linear mapping. The investigation on MIMD computers is concluded with a comparison of the results achieved by using the nCUBE2 with 64 processing elements. With reference to SIMD machine, tests on the Connection Machine CM-200 and the MasPar MP-I have been performed. CM-200 is equipped with 8192 single bit processors, whereas MP-I has 4096 four bit processors. The QRD has been mapped onto a mesh architecture for both machines.

Table 9.1 Pros and cons of COTS Pros

Cons

programmable and flexible

complex infrastructure including I/O control and protocols high speed data buses high speed memory and memory control

robust to technology obsolescence reuse of previously developed software essential in design trajectory of VLSI custom architecture (search for trade-off between flexibility and modularity, parallelisation options)

multi-DSP infrastructure requires extra-overhead which brings to a decline of ideal linear increment of computational power

Without going into the details, which are described in Reference 8, the main conclusions of the work are the following. The experimental work done suggests mapping the systolic array for the QRD algorithm onto an MIMD machine configured as a triangular array. An achievable data throughput is in the order of 1OkHz for a STAP with MN = 16 and 120 PEs using the MEIKO Computing Surface. A data throughput of the order of 100 kHz should be feasible either with advanced transputers or with devices like the Texas TMS320C40. These conclusions, which date back ten years, should be reconsidered in the light of the more powerful COTS (commercial off the shelf) machine available today; see Section 9.7. A preliminary evaluation of the pros and cons of the hardware implementation of STAP based on COTS is summarised in Table 9.1.

9.6

Experiments with VLSI-based CORDIC board

To explore the possibility of achieving better computational performance and using compacter systems - for installation in an operational radar - a QRD algorithm has been mapped onto an application specific prototyping platform which contains four VLSI CORDIC ASICs (application specific integrated circuits) and some FPGAs (field programmable gate arrays) [H]; this work was done in cooperation with the Technical University of Delft (The Netherlands). For details on the CORDIC algorithm and its use in adaptive beamforming see Sections 9.13 and 9.14, Appendices C and D, respectively. The test-bed platform mainly consists of a large (modular) memory buffer that is connected to a Sun workstation via a VME (versa module eurocard) bus. The memory buffer stores data that flow through the application board, back into the buffer. The application board consists of four CORDIC processors which are mesh-connected. These four processors perform complex rotations on two-dimensional complex vectors. The CORDIC processor is a pipeline processor operating in block floating

point. The physical connections between the CORDIC have been implemented via Xilinx chips. In the benchmark described in Reference 11, the triangular systolic array was mapped onto the 2 x 2 CORDIC application board of the tested platform. This four CORDIC mesh corresponds functionally to one of the processing nodes constituting the triangular systolic array. However, as the CORDIC processors are pipelined processors, many of these rotations can be performed at a very high throughput rate (the clock rate of the pipe), provided a schedule can be found such that the pipe can be kept filled. Such a schedule can indeed be found for the QRD algorithm. The results of the benchmark may be briefly summarised as follows. With a 100 per cent pipeline utilisation of the CORDIC, the throughput can be computed simply as: throughput =

clockfreqCORDIC —\ number of rotations

(9.7)

where clockfreqCORDIC is the clock frequency of the CORDIC processor (only 5 MHz in the experiments, just to show that no extremal values are needed), and number of rotations is the number of rotations (vectorisations included) for the case where we simulate a system of MN degrees of freedom. For MN = 1 0 the throughput is approximately 8OkHz, i.e. 80000 input vectors could be processed per second, which is better than the results reported in Section 9.5 where larger computers and higher clock frequencies were used. In a non-experimental implementation of the CORDIC system described in this section, clock frequencies up to 40 MHz are easily achievable; this would improve the throughput even further within a factor of 8. Table 9.2 summarises the pros and cons of the hardware implementation of STAP based on custom VLSI. Selection between COTS and VLSI is still an open question; the specialised technical literature reports descriptions of experimental systems using both the two technologies: a consensus has not been found yet on which technology to use, even though the trend seems today in favour of COTS. Further considerations on this problem are reported in Section 9.7. Also Section 9.14, Appendix E, lists several implementation examples of STAP taken by the open technical literature; these testify the wide spectrum of technologies used.

Table 9.2 Pros and cons of VLSI Pros

Cons

extremely high throughput (bulk processing) limited size and power consumption

low degree of flexibility expensive for limited number of pieces to produce

9.7

Modern signal processing technology overview and its impact on real-time STAP

In the chapter we have mentioned the role of the VLSI custom chip and the relevance of chips that may implement the CORDIC algorithm. In this section we give attention to more commercial technology and evaluate its possible use for real-time STAR The impact of modern signal processing technology on real-time STAP is provided in this section to complement the algorithmic aspects of STAP described up to now in the chapter. We summarise the state of the art of relevant devices for signal processing and the way to design complex signal processing schemes which are of interest for the real-time implementation of STAR Perhaps one of the most significant advances in radar in the past 30 years has been the application of digital technology to allow the radar designer to make practical what in the past were only academic curiosities. An impressive drawing of the advancement of digital technology is in Figure 9.5. It illustrates Moore's law, named for Intel cofounder Gordon Moore, which predicts that transistor density on microprocessors will double every 18 months. This prediction so far has proven amazingly accurate. In recent years the processing technology adopted for radar systems has evolved along the following steps: • •

design and implementation of proprietary circuits which, however, have the following cons: high life cycle cost and obsolescence proprietary circuits exploiting the digital signal processing (DSP) devices available from the market pentium processor 1993 intel 486 microprocessor

number of transistors 10 million

286 microprocessor 1982 1 million 8086 microprocessor 1978 - i n t e l 386 microprocessor 1985 microprocessor 8080 microprocessor 1974

Figure 9.5

Moore s Law, named for Intel co-founder Gordon Moore, predicts that transistor density on microprocessors will double every 18 months. This prediction, so far, has proven amazingly accurate. If it continues, Intel processors should contain between 50 and 100 million transistors by the turn of the century (From IEEE Aerosp. Electron. Syst. Mag., October 2000, 15, (10), Jubilee Issue, p. 13 (©2000 IEEE))

•

•

a tremendous interest in COTS technology which, however, has the following cons: need of engineering resources to track technology evolution and developing tools the more recent system COTS (SCOTS) also embedding communication, operative system and developing tools.

Today modern radar systems, including those with adaptive features, require a wide spectrum of technologies; e.g. ASIC, FPGA, DSP, fibre optic communication channel to use each one matched to a suitable purpose. Examples of the application of heterogeneous technology in adaptive radar, and also STAP, are: • • • •

ASIC for analogue to digital converter (ADC), filtering and channel equalisation DSP for matrix algebra calculation reduced instruction set computer (RISC) for data processing fibre optic for communication channels to distribute/collect data and command.

A taste of figures for recent technologies are (this is a not exhaustive list and by no means a commercial indication, but just a sample taken from the specialised literature): •

• • •

Analog Device's very recent processing board is characterised by the following features: eight ADSP-TSlOl Tiger Shark give 9000MFlops, the processor is running at 250 MHz, has 64 bits at 66 MHz on compact PCI (peripheral computer interconnect), with three banks of memory with 64 Mbytes; it is programmable in C, with library and software tools in Sharklab linked to Matlab status of the art for FPGA (year 2002): 10 Mgates, they are reprogrammable via software status of the art for ADC technology: 14 bits at 100 MHz sampling rate, 10 bits at 1.5GHz, 8 bits at 3 GHz status of the art for FFT: DoubleBW 1 K complex points FFT with windowing in 10 iis [21].

These technologies are heterogeneous but need to be approached, in the design phase of the whole radar system, in an homogeneous fashion; this has prompted the so-called concurrent design technology, or system codesign. It is a methodological approach based on: tools for cost estimate and analysis of requirements (e.g. Rational Rose), algorithmic analysis (with Matlab and Simulink), functional design (Ptolemy II, System C, Handel C), core library etc. In more detail, the rationale of codesign technology is the following. Systems are becoming more and more complex in terms of both functionality and hardware architecture. The need to include interaction with other design domains, such as data processing, control processing, input/output etc. is also increasing. Therefore there is a need for a true system level design capability which not only allows the design/simulation of the constituent parts of the system but also their interaction. To bring together the different design domains, system design languages are being developed which allow a model of the behaviour of the entire system to be generated, thus speeding up the design of the entire system by finding problems early in the design cycle rather than at the system integration stage.

Codesign techniques permit the functional specification to be explicitly mapped onto a model of a candidate architecture, in terms of both functional processing, memory access and communication interfaces between hardware and software elements. The resulting partitioned design can then be analysed for performance and the suitability of the candidate architectures investigated. Modifications of the architectural structure and mapping of the functional specification onto the modified architecture can then be made until the design meets the requirements. The level of abstraction of the architecture model can then be increased until the designer is satisfied that an implementation of the design will be 'right first time'. In addition to this simulation capability, codesign methodologies and supporting tools need to be able to export the design information to hardware-software coverification environments and implementation tools.

9.8

Processing of recorded live data

The data recorded by the Naval Research Laboratory (NRL-USA) airborne multichannel radar system have been processed by a systolic trapezoidal array which implements the STAP [9]. The performance of the algorithm has been evaluated against ground clutter, littoral clutter and jammer. Vehicular traffic has also been detected. The systolic array processing has been emulated with a MATLAB software tool. The airborne radar system used by NRL for its STAP flight test program is a modified AN/APS-125 surveillance radar; the operating frequency is 420-450 MHz. The side-looking linear array consists often hooked dipole antennas spaced approximately a half wave length apart, mounted in a 90° corner reflector to provide elevation pattern shaping. The two outer dipoles are terminated yielding eight channels with roughly equivalent element patterns and — 3 dB beam widths of 80° for both azimuth and elevation. The array was energised with a high power corporate feed which applied a taper on transmit such that the maximum azimuth sidelobe level is 25 dB down with respect to the main beam. The receiving system consists of eight identical channels with each channel having a UHF preamplifier, mixer, VHF amplifier bandpass filter and a synchronous demodulator. The synchronous demodulator consists of two demodulators, one referenced to the coherent oscillator (COHO) and the other referenced to the COHO shifted by 90°. This yields two bipolar video channels, one in phase (I), the other quadrature phase (Q). Each I and Q signal is converted to digital by a 10 bit, 5 MHz ADC. The radar pulse repetition frequency (PRF) is 300/750 pps. The output of the receiving system is 16 digital channels for a total digital word width of 160 bits with a clock of 200 ns. This yields a data bandwidth of 800 Mbps which is buffered in real time and stored on magnetic tape.

9.8.1 Systolic algorithm for live data processing As indicated in Figure 9.2, the radar has an array of N = 8 antennas and receiving channels. Each of these receives M echoes from a transmitted train of M (up to 18 in the actual radar) coherent pulses with a PRI (pulse repetition interval) of T = Kz s

where r is the Nyquist sampling period (i.e. typically, the range cell duration) and K is the range cell number in the PRI. The STAP provides a two-dimensional filter in the direction of arrival (DoA) Doppler frequency (/b) plane with a main beam focused towards the target and a wide null in the regions of the ' D O A - / D ' plane containing the interference. QRD constitutes the fundamental component of the voltage-domain algorithm. It operates recursively by using each snapshot of data to update the online estimation of the disturbing environment without forming the interference covariance matrix and only requires 0(N2M2) arithmetic operations to be performed every sample time. The scheme of Figure 9.2 has been applied to the data recorded by the NRL radar.

9.8.2 Data files used in the data reduction experiments This section describes the data files, recorded by NRL radar, used for space-time processing experiments. The files refer to ground clutter, land-sea clutter interface and jamming. The following information has been extracted by the data files, namely: (i) echo power in a radar receiving channel versus range, (ii) the probability density function (PDF) of the amplitude and phase of the radar echoes, (iii) the eigenvalues spectrum, and (iv) the two-dimensional power spectral density of the clutter versus / D and DoA. In this chapter, just a subset of this information is enclosed. 9.8.2.1 Ground clutter Two data files were examined, namely DL050 and DL087. For these files we have calculated the amplitude and phase histograms of the radar echoes. The histograms have been estimated using 896 echoes along range. The amplitude histograms show visually a good fit with the Rayleigh PDF. One more test to verify whether the histogram adequately matches the Rayleigh PDF is to calculate the mean to median ratio. The estimated value is 1.115, and the exact value is 1.442. The histogram of phase is approximately uniform. For file DL050 the spectrum of eigenvalues of the interference covariance matrix is reported in Figure 9.6. The number of antennas is 8, and the number of PRIs is the parameter of the curves ranging from 1 to 18. The covariance matrix has been estimated by averaging 896 independent samples along range, and the maximum eigenvalue has been normalised to O dB. The minimum eigenvalue, corresponding to the curve labelled '18', gives a good estimate of the noise floor in each receiving channel; before normalisation this value is about 10 dB. The clutter-plus-noise power value amounts to 45 dB in each receiving channel; this value has been determined by averaging along range the received signal on the first antenna. Thus the input average clutter-to-noise power ratio has been assumed to be equal to 35 dB for each receiving channels, i.e. for each antenna and for each range sample. 9.8.2.2 Land-sea clutter Figure 9.7 portrays the power versus range of the echoes collected by the first antenna for the data DR075. At the 480th range cell the transition from sea to land is clearly visible. The sea clutter power, estimated along the first 200 range cells,

amplitude of eigenvalues, dB

number of eigenvalues

Eigenvalue spectrum for data file DL050 (ground clutter) [9]. The parameter of the curves represents the number of PRIs

power (1st ant.)

Figure 9.6

range

Figure 9.7

Power versus range of the radar echoes collected by the first antenna of the array [9], At the 480th range cell the transition from sea to land is clearly visible

amplitude of eigenvalues, dB

number of eigenvalues

Figure 9.8

Spectrum of eigenvalues of jamming interference. Curve a: N = 8 antennas and M = I PRI; curve b: N = 8, M = 2 [9]

amounts to 12.8 dB. The land clutter power estimated from 600th to 800th range cells measures 30.2 dB. 9.8.2.3 Jamming The data file DWO15 refers to jamming overland. The jammer appears at the end fire, i.e. DoA = 90°. Figure 9.8 reports the eigenvalues (normalised to OdB) of the estimated covariance matrix (over 300 range cells) for N = 8 antennas and M = 1 PRI (curve a) and N = S and M = I (curve b). The presence of one principal eigenvalue in curve a indicates the presence of one jamming source. We also estimate (over 300 range cells) from the data file that the jammer plus noise power is equal to 36.5 dB. The thermal noise, evaluated by the minimum eigenvalue of the interference covariance matrix is 30 dB. Thus the jammer-to-noise power ratio is 6.5 dB.

9.8.3 Performance evaluation The detection performance of the systolic array of Figure 9.2 depends upon the array parameters, the interference environment and the target signal features. The parameters defining the trapezoidal array are: (i) the dimension NM of the data snapshot vector which equals the number of input lines to the triangular systolic canceller, (ii) the forgetting factor (which controls the adaptation speed of the canceller) of the QRD canceller, and (iii) the number L of linear columns for constraints. Synthetic targets as well as signals injected into the receiver are used to determine the integration of target echoes. Performance during steady state is measured in

terms of: (a) improvement factor (IF) defined as the ratio of the signal-to-total disturbance power ratios at the output and input of STAP, (b) visibility curve, i.e. IF versus target / D sweeping across the PRF, and (c) the two-dimensional response of the adaptive system versus DoA and /D9.8.3.1 Performance against ground clutter Consider the file DL087. Assume a trapezoidal array with one antenna (N = 1), eighteen pulses (M = 18) and L = 3 linear columns (processing cells DE of Figure 9.2). The constraints in the three columns are set to detect a target having the following Doppler frequencies: 0.5 PRF, 0.25 PRF and 0 PRF. A synthetic target having a Doppler frequency value of 0.5 PRF was added at the 264th range cell. Figures 9.9a, b and c show the power in dB of the residue signals at the output of the three columns. Note that the target echo appears only in Figure 9.9a as expected (being the constraint set at / b = 0.5 PRF); the estimated IF is 35 dB. 9.8.3.2 Performance against sea-land clutter The file DR075 contains a test target, injected in the receiver at the 3547th range cell. The Doppler frequency of the target is 0.5 PRF and the DoA is 0°. Figure 9.10 portrays the power in dB of the residue signal obtained by adaptively processing the echoes received by N = S and M = 18 PRIs. The trapezoidal systolic array has one vertical column (L = 1) with the constraints / D = 0.5 PRF and DoA = 0° which are fully matched to the target signal. The spike appears at the 3691st cell which differs from the original target range due to the space-time filter delay which is equal to the total number of degrees of freedom, i.e. 144. The visibility curve for a fictitious target having DoA = 0° and Doppler frequency sweeping across the radar PRF is reported in Figure 9.11; the visibility curve is approximately flat except around / D = 0 which is the mean Doppler frequency of clutter after compensation of the platform speed. The maximum value of the clutter cancellation equals the clutter-to-noise power ratio which has been estimated to be 23.9 dB while the maximum gain in the target direction of arrival and Doppler frequency is equal to 21.5 dB having used all the 144 degrees of freedom; this results in an IF of 45.5 dB. From the visibility curve the maximum IF value amounts to 44 dB, while the optimum IF would be 45.5 dB which is just few dBs higher than the values shown in visibility curve. 9.8.3.3 Performance against jammer The improvement factor of an array of N = 8 antennas, one PRI (M = 1) and one column constraint (the constraint is set along the expected target direction of arrival) is shown in Figure 9.12 as a function of the DoA of a simulated target scanning the angular interval [-90°, +90°]. The jammer is that described in Section 9.8.2.3. It is noted that the maximum IF is about 13 dB, while the optimum IF value would be 17 dB. The 4 dB loss is due to the adaptation of the systolic arrays [9, p. 600]. We note that adaptation loss is higher for jamming than for clutter; possible explanations are the following: (i) the number of spatial degrees of freedom is 8, which is lower than

residue power, dB

target

residue power, dB

range cells

residue power, dB

range cells

c

Figure 9.9

range cells

Processing of ground clutter live data [9]

target

power, dB

initialisation of space-time filter

range

Power of residue signal for data file DRO 75 [9]

IF, dB

Figure 9.10

F

Figure 9.11

Doppler/PRF

Visibility curve for data file DR075 [9]

the number of temporal degrees of freedom (=18), (ii) the jammer-to-thermal noise power has been estimated as being equal to 6.5 dB [9, p. 598], considerably lower than the clutter-to-thermal noise power. This will cause higher loss due to the need for precise estimation of jamming direction of arrival.

degree

Figure 9.12 IF versus DoA of a simulated target against jamming [9] 9.8.4

Detection of vehicular traffic

The detection of vehicular traffic has been attempted along US route 50 (see, for details, Reference 9). Four points on the route have been selected (bearing angle relative to the array normal, with positive values coming from the right-hand side of the array): 1st point: range = 39268 m, azimuth = —5.8° 2nd point: range = 39268 m, azimuth = —3.4° 3rd point: range = 39429 m, azimuth = -0.6° 4th point: range = 39429 m, azimuth =1.0°. The systolic array processes the snapshots along the range cells received by 8 antennas and 18 PRIs (i.e. it works with the maximum number of adaptive degrees of freedom). The adapted residue along the range cells has been further processed by a constant false alarm rate (CFAR) thresholding device based on the cell average (CA) technique. The CFAR-CA has two guard range cells on each side of the range cell under test and twenty range cells on each side to estimate the detection threshold. The CFAR-CA has been set to guarantee a PFA of 10~4. Figure 9.13 depicts the adapted residue versus range when the receiving antenna pattern is focused at —5.8°, which is the azimuth value corresponding to the first point on the US route 50. The analysed Doppler frequency is 0.225 PRF which corresponds to a radial speed of 23.2 m/s (i.e. 83.52 km/h) compatible with vehicular traffic. A detection appears at the 932th range cell that (it can be shown) comfortably compares with the expected location of the target in the first point. Similar results have been obtained for the other three points on the US route 50 [9].

range cells

Figure 9.13 Adapted residue power and detection threshold curves versus range cell [9]

9.9

Concluding remarks

The research work described in this chapter and the enclosed references is also relevant for other radar applications, sometimes simpler than the STAP, namely (i) ground-based or ship-borne radars for clutter cancellation and (ii) ground-based or ship-borne radars equipped with a multichannel phased array antenna for jamming cancellation. The STAP reverts to the first application by setting Af = 1, while it becomes the second application for M = 1. Thus, the adaptive processing architectures described in this chapter are applicable also to the systems in (i) and (ii). In general, the number of degrees of freedom involved is one order of magnitude less than for the STAP case; this makes less critical the implementation of a VLSI-based systolic array. A practical application of systolic processing for classical ground-based or ship-borne radar is described in Reference 19 where it is shown how to combine in one systolic scheme the two functions of adaptive interference cancellation and sidelobe blanking. The application of STAP to synthetic aperture radar for detecting and imaging of slowly moving targets is discussed in References 7, 15 and Chapter 3 of this book. In this respect the procedure to form the SAR image by one-bit processing plays a role; this procedure is also applied in along-track interferometry (ATI)-SAR to detect moving targets [20]. It can be shown that this approach offers a considerable computational advantage; FPGA technology has been successfully applied to implementing one-bit SAR processing. The enormous progress made in signal processing technology is under our eyes; this progress is also exploited and, at the same time, motivated by STAR Today the

key words are: heterogeneous processing (i.e. based on VLSI, ASIC, FPGA, RISC, MEMS, photonic technology etc.), virtual and rapid prototyping, modularity and flexibility of processing architectures, reuse and porting of the same, COTS approach to software and hardware, software language (e.g. System C; Handel C for FPGA), design tools like Ptolemy. All these techniques and technologies are conceived to counteract the obsolescence which is one of the most important problems to face today in signal processing.

9.10

Appendix A: Givens rotations and systolic implementation of sidelobe canceller

The QRD, which mainly performs orthogonal rotations, may be efficiently implemented by a recursive application of Givens rotations. A complex Givens rotation is an elementary transformation of the form: (9.8) where /3 is a scaling factor. The rotation coefficients, c and s, satisfy: (9.9) These relationships uniquely specify the rotation coefficients, c and s: (9.10) (9.11) The QRD by Givens rotations may efficiently be mapped onto a systolic array computer. A systolic computer is an array of processing cells. Each cell has a local memory and is connected with its neighbouring cells in the form of a regular grid. The more common configurations of the systolic array are linear and triangular. The operations performed by the systolic array are synchronised by a clock. On each clock cycle, every cell receives data from its neighbouring cells and performs operations. The resulting data are stored within the cell and passed to the neighbouring cells on the next clock cycle. The triangular systolic array is shown in Figure 9.14 for the simple case of an SLC (sidelobe canceller) system equipped with one auxiliary antenna. The triangular systolic array comprises three types of computational cell: the boundary cell (circular cell in Figure 9.14), the internal cell (rectangular cell in Figure 9.14) and the final cell which is a simple two-input multiplier. The function of each computational cell is specified in the Figure. Each cell performs the specified functions on its input data and delivers the appropriate output values to the neighbouring cells. The least-squares residue constitutes the noise-reduced output signal from the adaptive beamformer.

boundary cell

otherwise

internal cell

radio frequency (RF), intermediate frequency (IF) and baseband (BB) receiving channels

residual

Figure 9.14

Implementation of an SLC with one auxiliary antenna by means of a triangular systolic array (Adapted from WARD et al., IEEE Trans Antennas Propag., AP-34, (3), March 1986, pp. 338-346, (©1986 IEEE))

The processing scheme can be applied to the STAP case as shown in Figure 9.15. One limitation of this scheme derives from the need to reinitialise the systolic array every time that the look direction for searching the target is changed; in fact, the processor of Figure 9.15 focuses one quiescent beam to gather the echoes from a target possibly present in a specific region of the space-time plane. Because the direction of arrival and the Doppler frequency values of the target are not known a priori, it needs to monitor a number of lines of residue power for target detection purpose. The above mentioned limitation is overcome with the scheme of Figure 9.2.

triangular systolic array

processing elements

Figure 9.15

9.11

residue

Calculation of the adapted STAP residue via a sequence of Givens rotations implemented with a triangular systolic array. Comparison of residue to a detection threshold X to check for the two alternative hypotheses: H\ (target presence) versus HQ (no target)

Appendix B: lattice working principle

The lattice systolic array requires three working phases to correctly process the data samples. Indicate with i — PRT (pulse repetition time: PRT = PRI) the data matrix having dimensions TV (rows) and k (columns) containing the data pertinent to the /th pulse; N denotes the number of antennas and k the number of data snapshots to process in the adaptation phase. Indicate also with (/ — j) PRT the data matrix (of dimension N by k), pertinent to the data of the ith PRT after the application (by means of the squared array) of the rotation coefficients computed in the triangular array starting from the data captured during the jth PRT. For the study case described here the number (M) of PRI is three. The first working phase of the lattice processor is shown in Figure 9.16. The triangular and squared systolic processors, also shown in Figure 9.3, have dimension N by N; the number of operations of this phase is k by O(N2). Following the above mentioned notations, the second lattice working phase is reported in Figure 9.17; note that this phase costs 3k O(N2) operations. Finally, in Figure 9.18 is presented the third lattice work phase. Note also that this phase costs 3k O(N2) operations. The total number of operations is Ik O(N2) which is approximately 2 by M (i.e the number of pulses) by k O(N2); thus, the number of operations k 0(M2N2) required by the full triangular array of Figure 9.2 of the text has been reduced.

1 st phase data matrix of 2 PRT

data matrix of 1 PRT

squared array of processing elements data matrix of (1-2) PRT total number of operations = k O(N2)

Figure 9.16

1st working phase of lattice

2nd phase

(2-3) PRT total number of operations = k O(N2)

(3-2) PRT total number of operations = k Q(N2)

(l-2)-(3-2)PRT total number of operations = k O(N2)

Figure 9.17

9Al

2nd working phase of lattice

Appendix C: the CORDIC algorithm

Since QR-based algorithms mainly perform orthogonal rotations (see Section 9.10, Appendix A), the CORDIC algorithm may be selected for implementing the processing cells of the above described systolic arrays. A further motivation comes from the fact that some CORDIC-based VLSI processor arrays have already been developed for radar and more general signal processing applications.

3rd phase 3 PRT fictitious PRT = O total number of operations = k 0(N2)

(2-3) PRT total number of operations = k O(N2)

(l-2)-(3-2)PRT total number of operations = k O(N2)

Figure 9.18

3rd working phase of lattice

The basic idea underlying CORDIC [22] is to decompose a desired rotation angle O into a weighted sum of a given number n (e.g. n > 6) of predefined elementary rotation angles Qf(O, such that the overall rotation can be carried out via a sequence of n shift-and-add operations, called /JL-rotations. More specifically, given the vector Xin of components [x in ,y in ], the CORDIC algorithm transforms it, by means of a sequence of /x-rotations, into a new vector xout = [xout, youtlThe rotations are attained through the following operations: Xout = *in COSQf(Z) -

)>jit SnIQf(O

yout = xin sin Qf(Z) - yin cos a(i) ' IgVd) = Pd)I-1 pd) = ±1 = sign(xin)sign(yin)

^AZ)

The CORDIC algorithm can operate either in rotating or in vectoring modes. In rotating mode, the rotation angle 6 encoded by the sequence a(i) is applied to the vector [xinyin\T to give a new rotated vector [xoutyoutY'• Conversely, in vectoring mode, the CORDIC algorithm computes the coding sequence of the rotations which when applied to [xinyin\T yields \xout 0 ] r . The input/output description of the CORDIC cell considered for implementation of systolic algorithms is given in Figure 9.19 where m is a control bit selecting either the vectoring or rotating modes, and a represents the /[x-rotations sequence. The CORDIC algorithm operates on real valued vectors while, in adaptive beamforming, complex valued vectors must be handled. In particular, for the vectoring

Figure 9.19

Schematic of CORDIC processing function

mode

mode 0-CORDIC

-CORDIC

CORDIC

Re(r)

ImO) Re

Figure 9.20

twl

Im[X0UtI

CORDIC supercell for circular rotation on complex valued numbers. Adapted from Figure 4 of: CM. Rader, Wafer-Scale Integration of a Large Scale Systolic Array for Adaptive Nulling', The Lincoln Laboratory Journal, 1991, 4, (1), pp. 3-29

mode, a unitary transformation Q must be computed, which annihilates the second component of a given vector [rx]T, with r e R (i.e. r is a real valued number) and x e C (i.e. x is a complex valued number). This step occurs in QR and inverse QR algorithms [14, equation 37]. It is possible to see that such a transformation can be represented as [14]: Q

Tcos0 -[-sin0

sin0iri 0 I cos0][o e-J°\

(9 13)

'

with 0 — arc ^g (Im Qt)/Re (x)) and 0 = arcfg(|;t|/r). Subsequently, in the rotating mode the transformation Q can be applied to a generic two-dimensional complex vector. An example of a processor realising rotations on complex valued vectors, and using the same structure for vectoring and rotating mode, is shown in Figure 9.20. The processor is obtained by interconnecting three CORDIC cells, of the type of

Figure 9.19, named O and 0 CORDICs and two registers to store the real and imaginary parts of r (r e C when the processor is working in rotating mode). For the vectoring mode, this architecture allows us to annihilate the second component of a given vector [rx]T. In more detail, with reference to Figure 9.20 (r e R, thus Im(r) = 0), the imaginary part of x is annihilated in the 0 CORX)IC cell and, subsequently, \x\ is annihilated in the left 0 CORDIC cell. The right 0 CORDIC cell is not operating. The Q transformation of equation 9.13 is therefore coded by the sequence (fie, / ^ ) and they will be applied in the rotating mode. In the rotating mode the rotation angle 6 is applied to the incoming vector [Re(jc) Im(x)] r in the O CORDIC and, subsequently, the rotation angle 0 is applied by the left and right 0 CORDIC cells, respectively, to the real and imaginary parts of r (now r 6 C, as requested by QR and IQR algorithms) and the rotated vector x.

9.13

Appendix D: the SLC implementation via CORDIC algorithm

The sidelobe canceller (SLC) may be implemented by means of the CORDIC cells described in Section 9.12. Figure 9.21 refers to the case of a main antenna and one auxiliary antenna (the same application example as in Section 9.10, Appendix A, Figure 9.14). On the left-hand side of the Figure, the SLC algorithm is realised by means of a systolic triangular array computing the Givens rotations. On the right-hand side of Figure 9.21, the same algorithm is implemented by means of the CORDIC arithmetic. The architecture is basically composed of four computational cells having as input the data from the main and the auxiliary antennas, plus a computational cell auxiliary channel

main channel

auxiliary channel

main channel

<j> CORDIC vectoring

(f> CORDIC rotating

9 CORDIC vectoring

6 CORDIC rotating

residue

residue

Figure 9.21

The SLC implemented with CORDIC algorithm

6 CORDIC rotating

required for normalisation of the residue. The cells may work either in vectoring mode or in rotating mode: the following operations are performed in vectoring mode. Given the vector [x(0), y(0)] the rotation sequence is computed such that: (9.14) where ns is the required number of rotations. In rotating mode, given the sequence /x(7), the following operations are applied: y(ns)

= V(O) - QfX(O)

(9.15)

The cells (numbers 1 and 3 in the Figure) working in vectoring mode operate directly on the auxiliary data, computing the /x-rotations required to annihilate the second coordinate of the complex data xi = Qtir,Jti/), which changes in x'i = (x'lr,0). The rotating cells (numbers 2 and 4 in the Figure) operate on the main channel data applying the /x-rotations computed by the adjacent vectoring cells; the remaining computational cell (number 5 in the Figure) works as the previous ones and is required to normalise the cancellation residue.

9.14

Appendix E: an example of existing processors for STAP

This appendix summarises in a tabular form some of the most relevant processors for STAP as they are perceived by the authors in the open technical literature. The table is organised in three columns that report, respectively, the name of the processor and of the organisation, the features and some relevant references. The table entries are in accordance to the publication date of the references.

Name

Features

References

Rome Air Development Center RADC (USA)

Systolic array implemented with digital (chips from ESL Company) and optical technologies. QU factorisation based on Givens method; systolic weight computer and a digital weight applier; ESL systolic chip is a custom VLSI chip with 32-bits floating points. Acousto-optic adaptive processor: use of Bragg cells, a liquid spatial light modulator and an

LIS, S. etal.: 'Digital and optical systolic architectures for airborne adaptive radars'. Agard conference proceedings 381, Multifunction radar for airborne applications, pp. 18-1, 18-13, 1986

Name

Features

References

optical detector; GaAlAs semiconductor diode laser provides the illumination for the time integration correlator. Hazeltine, funded by RADC (USA)

Systolic array brass-board of 1.25 billion floating-point operations/s; solution of multiple simultaneous equations with twelve unknowns; weight update every 50 |xs. In 1988 brass-board was integrated into the flexible adaptive spatial signal processor test bed of RADC.

LACKEY, R. J., BAURLE, H. R, and BARILE, J.: 'Application specific supercomputer', SPIE, 977, Real Time Signal Processing XI, 1988, pp. 187-195

MUSE, MIT-Lincoln Laboratory (USA)

Matrix update systolic experiment (MUSE) for 64 degrees of freedom. 96 CORDIC processors update the 64 weights - to apply, say, to 64 receiving channels - on the basis of 300 new observations in only 6.7 ms; equivalent to 2.8 Ginstructions/s. MUSE has been realised on a single large wafer of 5 in by 5 in by using restructurable VLSI. Each CORDIC cell has 54 000 CMOS transistors. 50 dB of signal-to-interference-plus-noise power ratio is achievable. Mapping strategies and estimation of achieved computational throughput of QR decomposition by using: Meiko surface computer with 128 processing elements (PEs), nCube2 with 64 PEs, connection machine CM-200 with 8192 PEs, and MasPar MP-I with 1024 PEs. Application to an adaptive array of antennas and STAR

RADER, C. M.: 'Wafer scale integration of a large scale systolic array for adaptive nulling', Line. Lab. J., 1991, 4,(1), pp. 3-29 RADER, C. M.: 'VLSI systolic arrays for adaptive nulling', IEEE Signal Process. Mag., July 1996, pp. 29-49

Benchmarks on general purpose parallel computers (It)

D ' A C I E R N O , A.,

CECCARELLI, M., FARINA, A., PETROSINO, A., and TIMMONERI, L.: 'Mapping QR decomposition on parallel computers: a study case for radar applications', IEICE Trans. Commun., October 1994, E77-B, (10), pp. 1264-1271

References

Name

Features

Test bed (MIT-Lincoln Laboratory)

One of the first world-class systems MARTINEZ, D. R. and McPHEE, J. V.: ever developed to demonstrate an end-to-end real-time STAP 'Real-time test bed for processing capability designed STAP'. IEEE 1994 around commercial off-the-shelf Adaptive antenna integrated hardware components. systems symposium, The peak throughput amounts to Long Island, 7-8 22 Gops/s. The system is divided November 1994, into front end digital system with pp. 135-141 capability of 19 Gop/s and a back MARTINEZ, D. R., end programmable processor with MOELLER, T. J., and capability of 2.5 Gflops/s. This back TEITELBAUM, K.: end is based on DSP TI TMS320C30 'Application of microprocessor. The processor reconfigurable architecture is sufficiently flexible in computing to a high programming to implement different performance front-end classes of STAP algorithms. The radar signal processor', mapping of algorithm is on identical J. VLSI Signal Process., PEs operating under the same May/June 2001, 28, set of program instructions (1-2) but on different data input streams. Application of reconfigurable computing to high throughput front-end radar.

CORDIC test-board (It, Ne)

A board with four VLSI CORDIC chips used to implement the QR decomposition for adaptive array processing; the estimation of achieved computational throughput is also obtained.

KAPTEIJN, P., TIMMONERI, L., DEPRETTERE, E., and FARINA, A.: 'Implementation of the recursive QR algorithm on a 2*2 CORDIC test-board: a study case for radar applications'. 25th European Microwave conference, 4-7 September 1995, Bologna, Italy

Name

Features

References

Maui High Performance Computer Center (MHPCC), MIT-Lincoln Laboratory (USA)

A network configuration with the 400-way IBM supercomputer SP2 hardware and software, visualisation hardware, mass storage system, file servers, parallel tools, compilers, preprocessors, debuggers, parallel operating environment etc. Crest environments I and II for STAP related to data of Mountaintop program.

1996 Adaptive Sensor Array Processing (ASAP) workshop, Maui High Performance Computer Center (MHPCC) Training Session, 12 March 1996

Mountaintop is UHF radar with 18 channels (16 of which are active) operating with the concept of inverse displaced phase centre antenna (IDPCA). An auxiliary transmitting aperture is added to a stationary radar. The auxiliary array is linear with the number of elements equal to the number of pulses in a coherent processing interval; the element spacing is chosen to create the desired aircraft motion. Mercury Computer Systems, Inc. (USA)

Description of strategies to distribute three-dimensional data set for STAP over available computing elements in a parallel computer system.

SKALABRIN, M. F. and EINSTEIN, T. H.: 'STAP processing on a multi-computer: distribution of 3-D data sets and processor allocation for optimum interprocessor communication'. Proceedings of Adaptive sensor array processing (ASAP) workshop, 13-15 March 1996, pp. 429-447

Name

Features

References

STAP on GP parallel computer, Honeywell Inc. (USA)

Coarse grained data flow mapping for STAP; up to 234 nodes on Rome Lab. 321 node machine; sequential C code for four benchmarks for Doppler filtering, adaptive processing etc.

SAMSON, R., GRIMM, D., MORRILL, K., and ANDRESEN, T.: 'STAP performance on a Paragon ™Touchstone system'. IEEE National Radar Conference, Natrad 96, Ann Arbor, MI, 13-16 May 1996, pp. 315-320

Massive parallel processor (USA)

Experience in porting the MTI-Lincoln Laboratory STAP benchmark programs onto the IBM-SP2, Cray T3D and Intel Paragon. Benchmark performance results along with scalability analysis on machine and problem size.

HWANG, K. and XU, Z.: 'Scalable parallel computers for real-time signal processing', IEEE Signal Process. Mag., July 1996, pp. 50-66

MeshSP, MIT-Lincoln Laboratory (USA)

McMAHON, J. O.: Investigates the application of commercially available massively 'Space-time adaptive parallel processors for STAR These processing on the mesh processors are sufficiently flexible to synchronous processor', accommodate different STAP MIT Line. Lab. J., 1996, architectures and algorithms and are 9, (2), pp. 131-152 scalable over a wide parameter space to support the requirements of different radar systems. The mesh synchronous processor is an SIMD architecture with an array of processors connected via a two-dimensional or three-dimensional nearest-neighbour mesh. It incorporates the single monolithic processor element (PE) of the Analog Devices ADSP-21060 SHARC. Each PE permits 120 Mflops/s peak performance and 512 kB of internal memory,

Name

Features

References

six processor communication ports, each capable of 40 MB/s peak throughput and two I/O ports, each capable of 5 MB/s peak throughput. A MeshSP processing board (7 inc by 13 inc) dissipates 100 W, contains 64 SHARCs and is capable of 7.7 Gflops/s peak performance. The higher order post Doppler (HOPD), an embodiment of STAP algorithm, has been mapped onto the MeshSP. In the HOPD the method for computing the QR decomposition of the data matrix has been the Householder reflection algorithm. It was found that the real-time processing for a study case of 48 channels, 128 Doppler bins and 1250 range gates required 16 boards for a total of 123 GFlops. ONEST: on line experimental space time, FGAN-FFM (Ge)

Implements algorithms of moving target extraction in SAR by means of STAR Use of heterogeneous DSP network, VME bus boards; Sharp processors for range and azimuth compression FFT; bunch of 24 -i- 32 TMS320C40 for filtering of slow moving targets; 16 TMS320C40 for moving target detection, position finding and concurrent SAR image generation.

JANSEN, W. and KIRCHNER, C : 'ONEST: concept of a real time SAR/MTI processor'. EUSAR96, Konigswinter, Germany, 1996, pp. 349-352

High Performance Computer (HPC), Rome Lab. (USA)

Honeywell ruggedised Touchstone used in four flight experiments, 2 5 + 4 processing nodes; each node has three i860 processors; 300 Mflops per node, 7.5 Gflops overall; weights update with QRD, sustained overall throughput 3.15 Gflops, efficiency 48 per cent. Data are passed from IF digital down conversion to the 29 nodes via a

LINDERMAN, M. H. and LINDERMAN, R. W.: 'Real time STAP demonstration on an embedded high performance computer'. IEEE National Radar Conference, Natrad 97, 13-15 May 1997,

Name

Features

References

high performance parallel interconnect (HiPPI) channel having 100 Mbytes/s. Hardware housed in two racks (19 in).

Syracuse, NY, pp. 54-59

Real-time multi-channel airborne radar measurements (RT-MCARM) with onboard Intel Paragon computer with 25 compute nodes running Sunmos operating systems. Each node has three i860 processors accessing common memory of 64 Mbytes as shared resource; two HiPPI; two service nodes. Linear speed up was obtained for up to 236 compute nodes. The adaptive beamforming is done applying the QRD. Real-time multichannel airborne radar measurements (RT-MCARM), Rome Lab. (USA)

In May-June 1996 Rome Laboratory conducted experiments of real-time STAP on board the BAC-III. Two processing chains have been tested contemporaneously: a) conventional analogue beamforming without STAP on a Mercury computer, b) 16 simultaneous beams provide digital data to the ruggedised 28 nodes of Paragon using SUNMOS as operating system and a Pentium PC to control radar, processing chain and display. Four flights included urban and rural clutter, land-sea interface, target was a Sabreliner, a moving target simulator and various targets of opportunity plus a CW jammer operating during random periods. Paragon functions: digital beamforming (six receiving beams within a wide transmitter beam),

CHOUDARY, A., LIAO, W-., WEINER, D. et al.: 'Design implementation and evaluation of parallel pipelined STAP on parallel computers', IEEE Trans. Aerosp. Electron. SySt., April 2000, 36, (2), pp. 528-548

LITTLE, M. V. and BERRY, W. P.: 'Real time multichannel airborne radar measurements'. IEEE national Radar conference, Natrad 97, 13-15 May 1997, Syracuse, NY, pp. 138-142

Name

Lockheed Martin (USA)

Features

References

pulse compression, two alternative STAP algorithms, CFAR detection, data recording. The flights demonstrated the feasibility of using high performance computer to conduct STAP of radar data in real time: a unique capability within DoD USA. High performance scalable computer MANSUR, H. H.: 4 (HPSC) employs 716 Analog STAP architecture Devices Share processing elements implementations using (PEs); two identical chassis and only high performance two board types are needed. scalable computer Sustained processing throughput of (HPCS)MEEE national the order of 32 Gflops. The Share Radar conference, PEs are grouped in four forming a Natrad97, 13-15 May processing node. Myrinet switching 1997, Syracuse, NY, technology (129Mbytes/s) provides pp. 325-330 the connections between the nodes. The recursive modified Gram Schmidt QR algorithm is implemented. The HPCS is currently (1997) under development.

References 1 FARINA, A.: 'Antenna based signal processing techniques for radar systems' (Artech House, 1992) 2 WARD, J.: 'Space-time adaptive processing for airborne radar'. MIT-Lincoln Laboratory, TR 1015, 13 December 1994 3 KLEMM, R.: 'Principles of space-time adaptive processing' (IEE, UK, 2002) 4 FARINA, A. and TIMMONERI, L.: 'Space-time processing for AEW radar'. Proceedings of international Radar conference, Radar 92, Brighton, UK, 12-13 October 1992, pp. 312-315 5 TIMMONERI, L., PROUDLER, I. K., FARINA, A., and McWHIRTER, J. C : 'QRD-based MVDR algorithm for multipulse antenna array signal processing', IEEProc. Radar, Sonar Navig., April 1994,141, (2), pp. 93-102 6 FARINA, A., BARBAROSSA, S., CECCARELLI, M., PETROSINO, A., TIMMONERI, L., and VINELLI, F.: 'Application of the extreme eigenvalue

7

8

9

10

11

12

13

14

15

16

17

18 19

analysis to signal and image processing for radar'. Invited paper, Colloque International sur Ie Radar, Paris, 3-6 May 1994, pp. 207-213 FARINA, A. and BARBAROSSA, S.: 'Space-time-frequency processing of synthetic aperture radar signals', IEEE Trans. Aerosp. Electron. Syst, 1994, 30, (2), pp. 341-358 D'ACIERNO, A., CECCARELLI, M., FARINA, A., PETROSINO, A., and TIMMONERI, L.: 'Mapping QR decomposition on parallel computers: a study case for radar applications', IEICE Trans. Commun., October 1994, E77-B, (10), pp.1264-1271 FARINA, A., GRAZIANO, R., LEE, F., and TIMMONERI, L.: 'Adaptive spacetime processing with systolic algorithm: experimental results using recorded live data'. Proceedings of the international conference on Radar, Radar 95, Washington DC, 8-11 May 1995, pp. 595-602 FARINA, A. and TIMMONERI, L.: 'Antenna based signal processing techniques and space-time processing'. Tutorial, international conference, Radar 95, Washington DC, USA, 8-11 May 1995 KAPTEIJIN, P., DEPRETTERE, E., TIMMONERI, L., and FARINA, A.: 'Implementation of the recursive QR algorithm on a 2 x 2 CORDIC test-board: a case study for radar application'. Proceeding of the 25th European Microwave conference, Bologna, Italy, 1995, pp. 490-^95 FARINA, A., SAVERIONE, A., and TIMMONERI, L.: The MVDR vectorial lattice applied to space-time processing for AEW radar with large instantaneous bandwidth', IEEProc, Radar Sonar Navig., February 1996,143, (1), pp. 4 1 ^ 6 FARINA, A. and TIMMONERI, L.: 'Parallel algorithms and processing architectures for space-time adaptive processing', CIE international conference on Radar, ICR96, Beijing, P. R. of China, invited paper for the workshop on STAP, 1996, pp. 771-774 BOLLINI, P., CHISCI, L., FARINA, A., GIANNELLI, M., TIMMONERI, L., and ZAPPA, G.: 'QR versus IQR algorithms for adaptive signal processing: performance evaluation for radar applications', IEE Proc, Radar Sonar Navig., October 1996,143, (5), pp. 328-340 LOMBARDO, P. and FARINA, A.: 'Dual antenna baseline optimization for SAR detection of moving targets'. Proceedings of ICSP96, Beijing, RR. of China, 1996, pp. 431-433 FARINA, A. and LOMBARDO, P.: 'Space-time adaptive signal processing'. Tutorial, IEE international conference on Radar, Radar 97, Edinburgh, UK, 13 October 1997 FARINA, A. and TIMMONERI, L.: 'Real time STAP techniques'. Proceedings of IEE symposium on Space time adaptive processing, London, 6th April 1998, pp. 3/1-3/7 FARINA, A. and TIMMONERI, L.: 'Real time STAP techniques', Electron. Commun. Eng. J., Special Issue on STAP, February 1999,11, (1), pp. 13-22 FARINA, A. and TIMMONERI, L.: 'Systolic schemes for joint SLB, SLC and adaptive phased-array'. Proceedings of the IEEE international conference Radar 2000, Washington DC, 7-12 May 2000, pp. 602-607

20 PASCAZIO, V., SCHIRINZI, G., and FARINA, A.: 'Moving target detection by along track interferometry'. IGARSS 2001, Sidney, Australia, July 2001 21 BIERENS, L.: DoubleBW Systems B.V., May 2002, Private communication www.doublebw.com/brochures.htm 22 VOLDER, J. E.: 'The CORDIC trigonometric computing technique', IRE Trans. Electron. Computers, September 1959, pp. 330-334

Part IV

Clutter inhomogeneities

Chapter 10

STAP in heterogeneous clutter environments William L. Melvin

10.1

Introduction

Aerospace radar systems must detect a variety of target types in the presence of severe, dynamic clutter and jamming signals. Signal diversity - the exploitation of azimuthal, elevation, Doppler, range and polarisation measurement spaces - is a necessary component of advanced detection architectures. Space-time adaptive processing1 (STAP) improves the detection of slow moving and/or low radar cross section (RCS) targets competing with mainlobe and sidelobe ground clutter returns [I]. Additionally, in light of the equivalence between the maximum signal-to-interference-plus-noise ratio (SINR) filter and the minimum variance beamformer, we recognise STAP as a member of the class of superresolution algorithms [2]. For this reason, STAP is a key element of radar systems whose electrically small apertures, and hence relatively large beamwidths, would otherwise seriously affect clutter-limited detection performance. Adaptive filters adjust their response in accord with estimates of interference characteristics. A critical distinction exists between optimal and adaptive filters. Specifically, the optimum filter design requires clairvoyant knowledge of interference statistics (e.g. known covariance matrix), while the adaptive implementation relies on necessarily imperfect estimates of unknown interference parameters. A training stage generates estimates of these unknown parameters. Hence, STAP is a datadomain implementation of the optimum filter. The optimum filter defines the upper bound on STAP's detection performance potential. In the multivariate Gaussian case, maximising SINR equivalently maximises the probability of detection (PD) for a fixed probability of false alarm (PFA) [I]1 In the case of ground clutter suppression, we consider spatial and slow time (Doppler) degrees of freedom, thereby taking advantage of the clutter's angle-Doppler coupling. On the other hand, cancellation of jammer multipath requires spatial and fast time (range) degrees of freedom

Consider an N-channel array receiving M pulses. Given the space-time snapshot for the &th range, Xk e cMNxl, the optimal weight vector in the maximum SINR sense takes the well known form, Wk = /xQ^ 1 S 8 -^ where \i is an arbitrary scalar; Qk = £{Xk///0x£yH } e cNMxNM; Xk/H0 is the zero-mean, interference-only (nullhypothesis, Ho) space-time snapshot; and, ss_t € CNMxl is the target space-time signal vector [3]. In practice, both Qk and ss_t are unknown, and so the adaptive processor substitutes the estimate Qk for Qk and the surrogate steering vector vs_t for ss_t; the adaptive weight vector is then Wk = AQk vs-t> where it is common to set l//x = y v ^ Q ^ V s - t in an attempt to normalise the output noise to unit power. Using secondary (auxiliary, training) data taken from other range cells within the CPI, the processor computes Qk. Ideally, secondary data exhibit statistical behaviour identical to the null-hypothesis condition characterising the primary (test) range cell k. The bandwidth and PRF limit the maximum number of unambiguous range cells in the coherent dwell to Qtot = Ru/^R total vectors, where R11 is the unambiguous range extent and AR is range resolution.2 Just as the optimal processor bounds the performance of the STAP, finite sample support of <2tot — 1 independent and identically distributed (HD) secondary data vectors limits the capability of the adaptive processor. Moreover, suppose the receiver bandwidth B is 5 MHz, indicating AR = 30 m. If M = 32 and Af = 11 (typical values), then sample support of twice the processor's degrees of freedom (DoF) is 2NM = 704; in this case, training data come from a range interval of 21.1 km! Based on the variable nature of ground clutter, it is highly unlikely that data exhibit statistical similarity over this range swath. Even minimal STAP training intervals - arising, for example, from the application of reduced-dimension/reduced-rank methods [ 3 , 4 ] - require relatively large numbers of samples and may suffer from discrete outliers. Additionally, limitations on computing capability may restrict training to block selection, further increasing the likelihood of mismatch between primary and secondary data features. STAP dynamically adjusts its two-dimensional frequency response, H(fs, / D ) = w^v s _ t (/y, / D ) , where fs and / D are spatial and Doppler frequencies, to suppress ground clutter returns coupled in angle and Doppler. When the clutter angle-Doppler properties vary over range, filter mismatch occurs and the output SINR deviates from a value one predicts under the HD assumption. Two factors lead to range-varying angle-Doppler behaviour: changing cultural features, such as clutter reflectivity or shadowing, and non-ideal sensor geometry. In the latter instance, non-linear or canted arrays, combined with deviation from the perfectly side-looking (zero crab) case, result in non-proportionality between clutter Doppler and spatial frequencies [3,5,6]. This type of variation is deterministic when platform velocity vector and array manifold are known, and so the STAP radar system designer has some control over this non-stationary effect. In contrast, changing cultural behaviour is environmentally driven, leading to covariance matrix estimation errors (viz., E[Q]x] ^ Qk) For example, eclipsing, pulse compression, sensitivity time control and antenna pattern gain variation decrease the number of usable range cells

and consequent STAP performance loss. We refer to this latter effect as clutter heterogeneity. References 7-15 highlight the impact of heterogeneous clutter effects on adaptive radar performance. Nitzberg considers the impact of amplitude heterogeneous clutter on improvement factor (IF) when using adaptive Doppler processing, consequently finding small losses as long as the processor accurately estimates clutter centre Doppler frequency and spectral width [7]. Armstrong et al. extend Nitzberg's work by including simultaneous amplitude and spectral mismatch, thereafter reporting significant IF degradation of the adaptive Doppler processor and proposing some ameliorating techniques, including non-adaptive prefiltering [8]. Futernik and Haimovich consider STAP performance in the presence of edge transition and Weibull range-dependent clutter effects [9]; they compare the performance of sample matrix inversion, the generalised likelihood ratio test and the eigencanceller [16], reporting superior performance for the latter processing method. In References 10 and 11, Melvin et al. hypothesise a taxonomy of space-time clutter heterogeneity and compute STAP performance loss for several cases, including range-angle varying amplitude fluctuations, range-angle varying spectral mismatch, clutter-to-noise ratio (CNR) dependent variations over range leading to amplitude and spectral mismatch, edge effects, and target-like signals in the secondary data (TSD). Kalson considers a parameterised generalised likelihood ratio test (GLRT), where adjustment of the parameter minimises the impact of mismatched signals - such as sidelobe discretes and TSD on detection performance [12]; the TSD problem is described further in Reference 13. Cai and Wang describe the impact of signal contamination and clutter edge heterogeneity on the performance of the generalised likelihood ratio test (GLRT) STAP detection algorithm in Reference 14. McDonald and Blum present exact expressions describing STAP performance in the presence of steering vector and clutter mismatch in Reference 15, further performing sensitivity analysis on detector characteristics in light of this mismatch.

10.1.1 Adaptivity with finite sample support Several different approaches are available for computing the adaptive weight vector. Among the competing methods, the maximum likelihood estimate (MLE) is the most popular [17] and is given by:3

(10.1)

where Xs e CNMx@ is the collection of secondary data vectors. Substituting Q^ 1 for Q^ 1 leads to the sample matrix inverse (SMI) formulation. Reed, Mallett and 3

This formulation assumes the secondary data are zero mean, Gaussian and HD

Brennan further demonstrate in Reference 17 that: P(wk;vs_t = ss_t) =

' ^ T " ^

(10-2)

^YiVA I Wk

where w k and Wk correspond to STAP and optimal weight vectors, is bound between zero and unity and follows a beta distribution, in which case: (10.3) The famous Reed-Mallett-Brennan (RMB) rule follows from equation (10.3): to achieve an average SINR loss of 3 dB between optimal and adaptive implementations, set Q = 2NM — 3. The SMI approach is preferable to other weight vector calculation schemes since convergence of the weight vector to the optimal condition depends solely on the number of HD secondary data vectors used in equation (10.1). In Reference 18, Boroson derives the variance of equation (10.2) under the conditions of unknown signal direction or corruption of the training data by the desired signal. When the clutter and interference is of low numerical rank, alternate processing approaches can yield 3 dB average SINR loss with sample support relaxing to 2/c, where K = rank(Qk) [19]. Clutter is often of low numerical rank due to redundant spatio-temporal sampling of the environment [3,16]. Each narrowband noise jamming source contributes M independent components to the space-time rank due to the complete decorrelation of the jamming source in slow time [3]. An estimate of the interference rank for an environment composed of ground clutter and / noise jammers is then: (10.4) where |"-~| denotes rounding to the next highest integer. The first term on the right-hand side of equation (10.4) is known as Brennan's rule [3]. Due to the generally low numerical rank of clutter, alternate STAP techniques with reduced degrees of freedom provide acceptable performance while minimising sample support requirements. Additionally, weight computation strategies involving diagonal loading or aperture smoothing tend to reduce requisite sample support with potentially small penalties in performance. Section 10.9.2 describes these issues in more detail.

10.1.2 STAP performance metrics The radar detection problem involves binary hypothesis testing to determine either target absence (Ho) or target presence (H\). The two hypotheses are given as: # o : x k = c k + j k + nk Hl:xk = s + xk/H0

^'D)

c k , j k and n k represent clutter, interference and noise snapshots, while s is the target signal vector. In the non-fluctuating target case, s = ys s _t(/s,/b), where y

is a complex scalar with unknown peak amplitude and uniformly distributed phase (n.b., since it is understood that the space-time steering vector ss_t is a function of angle and Doppler, we often neglect the corresponding argument as a matter of notational convenience). By exploiting signal diversity, STAP greatly improves detection performance over a variety of competing methods: it enables endoclutter detection performance by overcoming diffraction-limited characteristics of the spacetime receive aperture, suppresses sidelobe clutter masking the detection of low radar cross section targets, and supports simultaneous ground clutter and noise jammer cancellation. This section highlights key STAP metrics. Consider an arbitrary weight vector, Wk, applied to the space-time snapshot and thus yielding the output: (10.6) An optimum detection statistic for equation (10.6), under the Gaussian assumption X k/Ho ~ CN(Q, Qk), follows from the likelihood ratio test and appears as [1,3]:

(10.7) The performance of equation (10.7) is given by:

(10.8)

where PFA is the probability of false alarm, Po is the probability of detection, fir is a normalised detection threshold, /o(-) is the modified zero-order Bessel function of the first kind and a equals the square-root of the peak output signal-to-interferenceplus-noise ratio (SINR).4 In light of equation (10.6), we find: (10.9)

4

Peak (signal amplitude) SINR is often used in detection analyses. Most radar and communication engineers prefer to use power, and so the corresponding SINR equals one-half the peak value. By using the term SINR in this chapter, we imply the use of power

CJ-2 — £[|y| 2 /2] is the single channel, single pulse signal power. Equation (10.8) is a monotonic function of a, and hence a2. Thus, maximising SINR likewise maximises PD for a fixed value of PFA- For this reason, SINR is a critical STAP metric. When comparing algorithms or determining the impact of training on performance, it is convenient to define SINR loss factors [3]. Two commonly used SINR loss factors are LSi\ (/?, / b ) and Ls^(fs, Jb) I each loss term is bound between zero and unity. LSfi(fS9fv) compares clutter/interference-limited performance to the noiselimited condition for arbitrary weight vector WR: (10.10) Q s = cr2ss-ts^_t is the signal correlation matrix and a2 represents the single channel, single pulse noise variance. Upon substituting Wk = wopt, where wopt = MQ1^* s s-t is the optimal weight vector, equation (10.10) specifies the upper bound on performance in the maximum SINR sense. Since computing w opt requires the known covariance matrix, LSi\ (fs, / D ) is sometimes called the clairvoyant SINR loss. On the other hand, Ls^(fs, / D ) determines the loss between an implementation requiring estimated statistics and the clairvoyant case (e.g. adaptive versus optimum): (10.11) In the absence of steering vector mismatch, we find Ls^(fs, Jb) = p(w k ;v s - t = s s _ t ). Radar system characteristics, the severity of the interference environment, and the particular signal processing approach all strongly influence LSi\ (fs, /b) . Accurate parameter estimation relies on both fidelity and quantity of secondary data; typical radar environments provide limited, heterogeneous secondary data, leading to smaller values ofLSi2(fs, / D ) . Since: (10.12) with SNR(fs) the angle-dependent signal-to-noise ratio, those effects reducing Ls,2(fs, / D ) will likewise reduce PD in accordance with equation (10.8). For nonadaptive signal processing methods, such as displaced phase centre antenna (DPCA) processing or digital beamforming cascaded with a weighted Doppler processor,

Lsa(fS9jb) = Wf59Jp. Those target velocities closest to the dominant clutter component, and exhibiting SINRloss above some acceptable value, viz. Ls^\ (/?, JD)-LS^(JS, JD) > £, determine the minimum discernible velocity (MDV). For example, suppose we calculate SNR to be 13 dB, thereby yielding PD = 0.87 for PfA — IE — 6 according to equation (10.8). If our minimum detection requirement is PD — 0.5 for this same false alarm rate, then SINR must be greater than or equal to 11.25 dB. This indicates a tolerable combined SINR loss of 1.75 dB, or e = 0.668.

The notion of SINR loss equally applies to fluctuating target models. For example, in the Swerling I case: (10.13) where o^ is the sum of (Gaussian) clutter, noise and interference variances and a^vg denotes the average SINR computed from equation (10.12) after replacing SNR with the value computed using the average target radar cross section. Improvement factor (IF) is given as: (10.14) with O^ representing the total clutter single pulse power received by a single element. In the noise-limited case, equation (10.14) defaults to the space-time integration gain (nominally, NM). IF closely relates to the preceding SINR loss definitions.

10.1.3

Covariance matrix errors

The adaptive filter employs training data to estimate the unknown covariance matrix, as equation (10.1) suggests. In an HD training environment, the finite sample support leads to differences between estimated and actual covariance matrices, Qk and Qk. These differences consequently result in mismatch between optimum and adaptive filter responses. Reed, Mallett and Brennan characterised the impact of covariance errors by deriving the distribution for the SINR loss term given in equation (10.2) under homogeneous, yet finite, training support conditions. Low sample support conditions tend to perturb the noise floor estimate, leading to a substantial degree of loss. A variety of STAP methods improve upon the convergence rate identified by the RMB rule; Section 10.9.2 suggests some minimal sample support methods. In contrast, heterogeneous clutter environments can lead to significant deviation between Qk and Qk. The impact of covariance matrix error depends not only on the error itself, but also on the angle-Doppler location of the target with respect to the mismatch. Suppose: Ak ^ Q k 1 - Q k 1

(10.15)

which exists when both inverses exist. The output SINR of the adaptive processor is (10.16) Letting vs_t = ss_t, the SINR loss due to the covariance mismatch alone is:

(10.17)

Under the matched condition, Qk -> Qk, and so Ak = 0 and Ls -> 1. Otherwise, equation (10.17) suggests an SINR loss dependent on both the magnitude of the co variance error - given by || Ak || ^, for instance - and the projection of the target steering vector onto the columns of the error matrix, s^ t Ak- For example, a large clutter discrete substantially displaced from the target look-direction may affect performance much less than a smaller discrete at an angle-Doppler location in proximity to the target. By specifying the specific nature of Ak, equation (10.17) can be used to characterise the impact of certain forms of clutter heterogeneity on STAP performance. We accomplish this goal implicitly in subsequent sections.

10.2

Classes of space-time clutter heterogeneity

A general model for the clutter space-time snapshot is: N0

N

P

Ck = ^^r,oc/k.m^n(cLt(k\m,n)

O st(./b/*;#w,/i))

m=\ n=\

( a s ( ^ , n ) 0 s s ( / f c n ) )

(10.18)

where we approximate a continuum of statistically independent clutter scatterers along an unambiguous isorange contour k by a summation accounting for Na rangefolded cells, each composed of Np discrete, independent clutter patches. Additionally, ^c/k-m n describes the average power for the mnth. clutter patch and fcth range, at(k; m, n) is a vector describing the normalised pulse-to-pulse voltages, St(fD/k;m,n) is the length M clutter patch temporal steering vector, as(k;m,n) accounts for spatial decorrelation, ss(fs/k;m,n) is the length TV clutter patch spatial steering vector, and O and ® denote Schur and Kronecker products. Also, At = E[cttct^] and As = E[asa^] are the corresponding temporal and spatial correlation matrices. In the HD case, the terms comprising equation (10.18) exhibit no dependence on k. The clutter-plus-noise space-time snapshot follows from equation (10.18): Xk = Ck -b nk, where nk ~ CAf(O, (J^INM), &n *s t n e n °ise variance, Xk ~ CN(O, Qc/k + O^INM), andQc/k = £[c k c^]. The clutter voltage decorrelates from pulse-to-pulse due to intrinsic clutter motion (ICM), system jitter, antenna flexing and rotation, and radar cross section scintillation. ICM is a function of the type of scatterer, e.g. moving vegetation or ocean waves lead to Doppler fluctuations, whereas 'hard' scatterers, such as buildings, yield narrower Doppler spectra. Commonly used ICM models include the Gaussian [20,8,10J and exponential [21] correlation functions. The frequency spectrum of the Gaussian model is: (10.19)

where |gol2 is an arbitrary power gain, Ao is the centre wavelength, / is frequency and av is the RMS velocity spread in metres per second [20]; Reference 21 describes a contrasting model. The correlation coefficient is the normalised inverse Fourier transform of equation (10.19); it is Gaussian too, taking the form: (10.20) /o = COO/2TZ is the centre frequency. The argument in equation (10.20) corresponds to the i — yth pulse pair. The pulse-to-pulse correlation matrix is then: A t = Toeplitz(p/(0), P1(I)9 P1(I)9...,

Pl(M

- I))

(10.21)

By summing the variances, equation (10.20) accommodates multiple, independent sources of temporal decorrelation, i.e. o\ = Y^m a\im' The channel-to-channel voltage decorrelates due to dispersive effects. Assuming a bandlimited receive waveform - appearing flat over the bandwidth &>o — TCB < a* < coo + xB, where co is the frequency in radians - yields the corresponding spatial correlation coefficient of the form: ps(m,n) = smc(7tB(rm - Xn))

(10.22)

where xp is the time delay between the pth channel and a suitable reference point due to direction of arrival (DoA). Bandwidth, the DoA embodied in the time delay, and the size of the array (influencing the maximum time difference between samples) affect ps (m, n). The channel-to-channel correlation matrix is then:

(10.23)

which is Toeplitz if the spatial sampling is uniform. (Non-dispersive errors yield non-Toeplitz structure; the model of equation (10.18) can incorporate such errors independent of equation (10.23).) As subsequently shown in Section 10.8.1, the model given by equation (10.18) is a good approximation to actual measured data. To summarise, the key components of the clutter space-time model include: the amplitude scaling term <7c/k;m,n used to set the clutter patch power; the particular spatial and temporal response of the patch, fs/k;m,n and fo/kim^nl the RMS spectral spread of the patch, ov/k;m,n\ the spatial decorrelation term of equation (10.22) characterising the process as(k;m,n). Range variation among these components leads to heterogeneous clutter conditions. Table 10.1 provides a taxonomy of predominant heterogeneous clutter effects and also lists the corresponding cause and effect relationship. This Table includes target-like signals - a significant nuisance effect, especially for ground moving target indication (GMTI) radar - as a related effect. Additionally, the clutter angleDoppler loci can vary over range due to non-linear array geometries, non-side-looking

Table 10.1

Taxonomy of clutter heterogeneity and related effects

Heterogeneity type

Causes

Impact on adaptive radar

Amplitude (range variation in

shadowing and obscuration, range-angle dependent change in clutter reflectivity, strong stationary discretes, sea spikes, urban centres, land-sea interfaces etc.

null depth depends on eigenvalue ratio - MLE 'averaging' leads to underestimated eigenvalue magnitude and, consequently, uncancelled clutter and increased false alarm rate null width set to mean spread too narrow for some range cells, too wide for others thereby leading to either increased clutter residue or signal cancellation; degrades MDV same impact as spectral mismatch

a

c/k;m,n)

Spectral (range variation in Gv/k;m,n)

CNR-dependent spectral mismatch (range variation in <*c/k\mji

leads t0

apparent spectral width variation) Moving scatterers

Some other effects (e.g. variation in the pair, (fs/k;m,rn fD/k\m,n))

intrinsic clutter motion due to soft scatterers (trees, windblown fields etc.), ocean waves, weather effects

modulation of principal components and other low power signal terms rise above the noise floor with increases in CNR ground traffic, weather, insects and birds, air vehicles chaff, hot clutter, multibounce/multipath, impact of platform geometry (e.g. non-side-looking or bistatic) on angle-Doppler behaviour over range

mainlobe nulling, false sidelobe target declarations, distorted beampatterns, exhausts DoF combination of above effects

arrays, or bistatic sensor configuration [3,5,6]. Section 10.7 briefly describes the cause of range variation in the angle-Doppler pair (fs/k;m,n, fD/t,m,n) for monostatic radar. Clutter heterogeneity can occur among the training data, within the cell under test, or simultaneously among both training and test cells. At issue here is the difference between average and instantaneous performance. The covariance estimate resulting from equation (10.1) captures the average nature of the training data set. For this reason, the estimated covariance matrix can appear significantly mismatched with respect to the interference in any given test cell. As well as designing for worse case circumstances, one should also consider performance achievable in any given test cell as an upper bound.

10.2.1

General simulation characteristics

In the ensuing sections we further consider the forms of clutter heterogeneity listed in Table 10.1. Nominal simulation results correspond to a 'typical' airborne radar scenario with the following salient parameters: L-band transmit frequency, 1.2 m by 1.2 m side-looking array (8 elements in azimuth by 8 elements in elevation), eight azimuthally oriented spatial channels (N = 8), 25 dB Taylor weighting on transmit, cosine-squared element pattern (180° null beamwidth), no weighting on receive, PRF set slightly higher than the DPCA condition, sixteen pulse dwell (M = 16), platform velocity of lOOm/s and height of 3000 m, and clutter reflectivity varied to yield a CNR of value specified within the text. It is also worthwhile at this stage to note the following: we use both asymptotic and finite sample support analysis in succeeding sections of this chapter. In the asymptotic case we let Qk -> £[QkL thereby allowing us to examine the expected impact of heterogeneity on detection performance. The finite sample support condition, on the other hand, employs a training interval to compute a covariance estimate, thus enabling STAP performance characterisation in the most natural adaptive implementation. We identify the corresponding analysis as belonging to either class asymptotic or finite sample support - as the various examples appear within the text, and include the number of training samples for the finite sample case. The simulation results provide a controlled environment since we are able to specify the known, ideal covariance matrix Qk.

10.3

Amplitude heterogeneity

Amplitude heterogeneity is perhaps the most obvious source of mismatch occurring in adaptive radar signal environments. In this section we examine three different views of amplitude variation affecting practical STAP implementation: (a) strong discrete clutter returns in an otherwise homogeneous texture; (b) range-angle variation of clutter reflectivity; (c) abrupt changes in the clutter environment due to transition or edge regions.

10.3.1

Clutter discretes

Strong clutter discretes can corrupt the cell under test (CUT), the training data, or both. For simplicity, assume a space-time processor operating in a homogeneous, distributed clutter background with either a strong discrete in the CUT or a strong discrete in the training data. In the former case, residual clutter degrades SINR and also increases PFA • The latter case results in overnulling and the potential for signal cancellation. Suppose a clutter discrete resides in the CUT. The corresponding null-hypothesis covariance matrix takes the form: Qk = Qc+n/k + crJsdS^

(10.24)

where Qc+n/k is the covariance matrix of the homogeneous, distributed clutter-plusnoise component5 and crj is the power of the discrete clutter return with space-time signal vector Sd located on the clutter ridge. Given a homogeneous training set, we let Wk -» Wk = £[wk] = MQc^ n /k v s-t t 0 evaluate asymptotic behaviour. The corresponding asymptotic SINR loss is:

(10.25) Letting vs_t = ss_t, equation (10.25) expands to: 6

(10.26) \/2

1/2

Defining p = Qc_|_n/kss-t and u = Qc+n/kSd> equation (10.26) can be written: (10.27) where cos 2 (p,u) = Ip^up/HpH^llull^ and sin 2 (p,u) = 1 — cos 2 (p,u). From equation (10.27) we make the following expected observations: 0 < Ls < l;whenaj = 0, we find Ls = 1; when Sd = vs_t (u = p) and a^ J=. 0, then Ls = 1; as aj —• oo, Ls -> 0; while in general, as a j ^9L5 | as long as Sd ^= v s _ t . We numerically verified the equivalence between equations (10.27) and (10.25). Discretes in the CUT will adversely affect the false alarm rate. Suppose the design threshold, vj, is set for the expected output noise power: (10.28) whereas the actual output noise power in the CUT is: (10.29)

5 In this case, Q c + n /k = Qc+n V k since we assume the distributed clutter environment is homogeneous ^ The SINR loss due to target mismatch is typically small in comparison with the loss associated with clutter heterogeneity. Since clutter heterogeneity is our focus, we'll assume perfect match between hypothesised and actual space-time steering vectors

The expected and actual false alarm rates, PfA,e and PFA,CI, are then: (10.30) from which we find (10.31) PFA,e serves as the design false alarm rate in this instance. Equation (10.31) indicates the following: (a) if Pa = Pe, then PFA,a = PFAy, (b) if Pe > Pa, then PFA,a < PFA*',

and, more importantly for the case of a discrete in the CUT, (c) if Pe < Pa, then PFA,a > PFA,e- Also, as a result of the exponential term, a small mismatch between the design and actual power leads to large changes in false alarm rate. For the case with the discrete in the training interval, we let w^ -* ^k — £[Wk] = ^(Qc+n/k+a^SdS^)" 1 v s _ t , where a j incorporates the 'averaging' process, to evaluate asymptotic performance degradation. The corresponding SINR loss is:

(10.32) Equation (10.32) can be rewritten:

(10.33)

Qc+n/ksd- Using the prior definitions for p and u, equation (10.33) becomes: (10.34) We observe the following from equation (10.33): 0 < Ls < 1; Ls = 1 when o2d = 0; Ls = 1 when Sd = vs_t (u = p) and
Rather than increase the false alarm rate, a discrete residing in the training data generally leads to an upward bias of the threshold. Consequently, target masking is a concern in this situation. Ideally, we want PFA,O, — PFA,e- Equation (10.30) then indicates a threshold bias of ^/Pe/ Pa from the optimal setting, viz. vj = V Pe I ^a VT, optimal •

SINR loss, dB

Consider the typical airborne radar configuration described in Section 10.2.1. The clutter reflectivity is set to yield approximately 64 dB integrated CNR. The nominal slant range of interest is 32 km. We generate a clutter discrete of specified power <7j and angular position and compute the impact on STAP performance for the two cases described above: clutter discrete in the cell under test and clutter discrete in the training data (post averaging). Figure 10.1 shows the consequent asymptotic loss in the former case for varying values of aj over azimuth angle measured from the antenna boresight. As expected, loss depends on both power and position of the discrete, as equation (10.27) suggests; the loss is seen to be substantial in some regions. In contrast, Figure 10.2 shows the asymptotic loss in the latter case, where the discrete resides in the training data segment. The loss in this latter instance tends to be quite small, but the reader should keep in mind the somewhat ideal nature of the otherwise homogeneous, distributed clutter environment. Subsequent sections describe other forms of heterogeneity that, in combination, further exacerbate STAP performance. Finally, Figure 10.3 suggests the very sensitive dependence of the probability of false alarm, P ^ , on the output noise residue; from this Figure it is seen that only 1-2 dB of residue leads to a tremendous increase in PFA-

target location: antenna boresight (270°) 200 Hz Doppler (endoclutter)

discrete's azimuthal location, deg

Figure 10.1

SINR loss due to discrete in CUT

SINR loss, dB

target location: antenna boresight (270°) 200Hz Doppler (endoclutter)

discrete's azimuthal location, deg

Figure 10.2

SINR loss due to discrete in training data

probability of false alarm

actual P F A design P F A

actual output noise power, dB

Figure 10.3

Impact of discrete on false alarm rate

10.3.2 Range-angle varying clutter RCS In Reference 7, Nitzberg analyses adaptive Doppler filter performance degradation due to clutter power variation in range by considering the range-dependent, temporal covariance matrix: Qk = PcJcQx+ o*lM

(10.35)

where pc,k is the clutter power at the £th range with Weibull or gamma probability density, Q x is a normalised clutter temporal covariance matrix and a% is the uncorrelated noise power. The Fourier transform of Q x describes the normalised clutter power spectrum. Expression (10.35) is a composite model: each temporal snapshot is multivariate complex Gaussian, but non-Gaussian fluctuation in clutter radar cross section leads to power fluctuation over range. The corresponding asymptotic MLE is: (10.36) Generally, the mean clutter power will not precisely match the instantaneous clutter power at any given range, i.e. pc ^ /?c?£, and so estimated and actual covariance matrices are non-convergent. The consequent mismatch between adaptive and optimal filters leads to detection performance degradation. As described in Reference 10, the model of equation (10.35) applies under the following circumstances: (a) the spectral width and location of correlated clutter returns in the frequency domain appear fixed over range; (b) variation in clutter reflectivity is constant with azimuth, thereby implying uniform scaling of all clutter subspaces. In practice, we expect spectral width to vary with clutter type and environmental conditions, and clutter power to exhibit dependency over both range and angle. Shadowing, obscuration and varying clutter types lead to apparent changes in clutter RCS with range and angle. Additionally, non-side-looking sensor configurations induce angle-Doppler contour dependency with range irrespective of any cultural variations (see Section 10.7 and References 3, 5 and 6). Returning to the space-time case, equation (10.18) accommodates range-angle variation of clutter power and spectral width through terms (Jt,m,n and at(k;m,n). Assume narrowband conditions and no other effects giving rise to spatial decorrelation so that ocs (k; m9n) = 1. Further, note that clutter reflectivity, yc, is constant over range and angle in the homogeneous case. The single channel, single pulse clutter-to-noise ratio for the mnth patch at range k is then approximated as:

(10.37) where Pt is peak transmit power, Gt{-) is transmit antenna gain, ((pm,n,0k;m) is the azimuth-elevation pair, r is slant range, i/fg/k is the grazing angle, Acc is the clutter

patch area, A. is wavelength, gr(-) is the receive channel antenna gain, Nin is the input noise power, Fn is the receiver noise figure and Lr/ represents radiofrequency (RF) system losses [20]. The constant gamma model, ao = yc sin^g/fc), evident in equation (10.37), was proposed by Barton based on extensive analysis of measured data [22]; this model is commonly used in radar system analysis. Sensitivity time control is used to equalise range-dependent space loss [20]. Antenna gain characteristics and the impact of grazing angle on clutter backscatter determine the range-angle variability of patch clutter power under the homogeneous assumption. To accommodate heterogeneous clutter reflectivity, let yc/k-m,n represent the reflectivity of the mnth clutter patch at range k. Heterogeneous clutter power follows from equation (10.37) as: 2

(Gt((/)m,n,0t,m)gr((f)m,n,0k;m)

K G °c/k;m,n c/k;m,n = = K\\ (

~4 ~A

sin(^rg/ik) \

— J) Yc/k;m,n Yc/k;m,n

(10.38)

where K\ is a constant. The fluctuations embodied in yk;m,n result in range-dependent amplitude variation across the clutter ridge. Suppose the range-angle variation in clutter reflectivity - and hence, clutter power - follows the gamma probability distribution: ,v p(ycc)) = p(y

kVY

11

(~Yc\ ^ - a-\ I (-Yc\ — vr exp —z— —z— ;

~ E[yc]22 £[/c] = aa = ;

ar £~ vvar[y l>c]c] 6= = B

,m-iON (10.39)

K J T(S)F \ P ) var[yc ]' H E[yc] where a and ft are shape parameters. The selection of the gamma distribution represents a likely choice for overland surveillance [7,8], Notice that as var[}/c] approaches zero while the mean is held constant, the environment tends to the homogeneous case. Figure 10.4 shows finite sample support SINR loss based on 150 Monte Carlo trials for varying levels of clutter amplitude heterogeneity for our nominal airborne radar simulation example. Figure 10.5 depicts the CNR sample mean, standard deviation (STD) and maximum values specified on a single channel, single pulse basis. The basic analysis procedure, described in Reference 10, involves the following:

(a)

compute 2MN + 1 space-time realisations for the nominal airborne radar scenario, where the clutter reflectivity for each clutter patch is randomly chosen to abide by the distribution of equation (10.39) with fixed mean and specified variance (b) compute and save the clairvoyant covariance matrix for the first space-time realisation (c) compute the covariance matrix estimate using the MLE and the remaining 256 realisations (RMB loss in an HD environment is —2.96 dB in this instance) (d) compute the adaptive weight vector for the transmit angle over all Doppler (e) calculate SINR loss (f) repeat in Monte Carlo fashion, averaging the various trials. Results given in Figure 10.4 are consistent with those described by Nitzberg in Reference 7, and seem to suggest the robustness of STAP to range-angle varying clutter reflectivity. Generally, the adaptive filter's null depth inadequately suppresses

SINR loss, dB

(l)mean/std=1000 (2)mean/std=100 (3)mean/std=10 (4)mean/std = 2/3 (5)mean/std=l/2 (6)mean/std=l/3 (7)mean/std=l/10 (8)mean/std=l/12 RMB Rule (-2.96 dB)

Doppler filter

Figure 10.4

SINR loss due to range-angle varying clutter reflectivity following a gamma distribution ([10J, © 2000 IEEE)

large clutter fluctuations and so residual clutter leads to performance degradation. Figure 10.4 indicates that the predominant loss occurs about the centre of the clutter spectrum, with only slight impact on MDV in this instance. However, the reader should keep in mind that the Figure does not capably capture instantaneous performance (i.e. the Monte Carlo procedure smoothes the results) or the impact on false alarm rate.

10.3.3

Clutter edges

Shadowing and edges lead to abrupt changes in the clutter power profile over range. Terrain obscuration leads to shadowing, while littoral or rural-urban interfaces are examples of edges. Both amplitude and spectral characteristics can vary across clutter edges or transition regions. For the moment we consider amplitude effects. The power variation describing a linear transition between distinct clutter regions A and B takes the form:

(10.40)

CNR sample mean CNR sample std.

case number

CNR sample maximum

case number

Figure 10.5

Mean, standard deviation, and maximum single channel, single pulse CNR for example in Figure 10.4 ([1O], © 2000 IEEE)

with AAB — (PC/B — Pc/A)/(?B — VA) representing the change in power over the transition region. The asymptotic clutter covariance matrix is: (10.41) rm is the rath discrete range sample and Q cx is a normalised clutter covariance matrix, where tr(Qcx) = 1. If an abrupt edge characterises the clutter environment, equation (10.41) takes the form:

(10.42)

Both formulations (10.41) and (10.42) assume uniform scaling of all clutter subspaces. In the latter instance, QrcA and QrcB characterise the clutter covariance matrices of clutter snapshots taken from either region A (r^) or region B (r#), respectively.

Consider the following cases: (i)

Scenario A: primary region tr(QrcA) = 65 dB,

a 2 = 0.0001 m 2 /s 2

secondary region tr(QrcB) = 53.5 dB,

a 2 = 0.01 m 2 /s 2 .

(ii) Scenario B: primary region tr(QrcA) = 76.5 dB,

o2v = 0m 2 /s 2

secondary region tr(QrcB) = 43.5 dB,

a2v = 0.25 m 2 /s 2 .

Scenario C: Same as Scenario A, but with primary and secondary region characteristics reversed, (iv) Scenario D: Same as Scenario B, but with primary and secondary region characteristics reversed.

SINR loss, dB

SINR loss, dB

(iii)

Scenario A Scenario C Scenario D

Scenario B

Figure 10.6 Asymptotic SINR loss for clutter edge scenarios ([W]t O 2000 IEEE)

Figure 10.6 shows the asymptotic loss due to the four edge scenarios listed above for the worst case Doppler, with the computation assuming that the test cell resides in the primary region. (Li = QA, L = Q). The abscissa indicates the per cent of training data taken from the primary region. Based on the characteristics listed above, we anticipate small losses for Scenarios A and C: the degree of mismatch between the two scenarios is relatively small. Undernulled clutter is the culprit leading to the larger losses seen for Scenario B. In summary, the modest degree of mismatch among the four scenarios, and the otherwise idealised nature of the clutter environment and platform configuration, tends to limit the severity of the observed losses.

10.4

Spectral heterogeneity

SINR loss, dB

Spectral heterogeneity results from the varying range-angle responses of different clutter classes to environmental conditions [7,8,10]. For instance, the intrinsic clutter spread of a windblown field differs from that of an urban region. Adaptive radar design for systems operating in littoral zones, over sea clutter, in the presence of weather effects, or under otherwise diverse conditions, should be particularly mindful of range-varying spectral properties. Since the adaptive processor tends to a response characterising the average behaviour of clutter in the training region, spectral

mean/std=1000 mean/std = 10 mean/std = 2 mean/std=l/2 mean/std= 1/5 RMB rule (-2.96 dB)

Doppler filter

Figure 10.7

SINR loss for varying degrees of spectral heterogeneity ([10], O 2000 IEEE)

heterogeneity leads to mismatch between adaptive and optimal filter notch widths. The inappropriately set adaptive notch width either increases clutter residue or partially cancels the target signal. In this section we examine the impact of spectral heterogeneity on STAP performance using both finite sample support Monte Carlo analysis and asymptotic evaluation. Taking an approach similar to that of Section 10.3.2, suppose the RJVIS clutter spread exhibits range-angle dependence given by the gamma distribution of equation (10.39) with av replacing y. We set E[av] = 0.1 m/s and alter the variance to cover cases of increasing spectral heterogeneity. The single channel, single pulse CNR is held constant at 25 dB (46 dB integrated CNR). Figure 10.7 shows the SINR loss based on 150 Monte Carlo trials. As in the case of range-angle varying clutter reflectivity considered in Section 10.3.2, we generated 257 space-time realisations per trial, using the latter 256 vectors to estimate the covariance matrix. As seen from the Figure, losses are small for the more homogeneous cases where the gamma distribution shape parameters abide by mean/STD >2. In accord with our expectations, the loss is negligible in the bin encompassing main beam clutter (Doppler filter 1), and the greatest losses occur in adjacent bins (Doppler filters 2 and 16). The roughly 3 dB loss in the Doppler filters adjacent to main beam clutter for the most heterogeneous case clearly indicates the deleterious nature of spectral heterogeneity on the radar MDV. In general, we expect the impact of spectral heterogeneity to further depend on CNR (by typical accounts, 46 dB CNR is modest). We examine this issue in a slightly different context in the next section.

eigenvalue number

Figure 10.8

Eigenspectrafor varying levels ofspectral spread ([1OJ, © 2 000 IEEE)

SINR loss, dB

Next Page

CNR1 = 58 dB

Doppler filter number

Figure 10.9 Asymptotic SINR loss for ([10], ©2000 IEEE)

varying levels of spectral spread

Next, suppose we train the adaptive filter in a region whose dominant spectral features differ from the test cell region (e.g. littoral zone or rural-urban interface). Figure 10.8 depicts the clutter-plus-noise eigenspectra for six distinct training regions. We assume the primary data exhibit an RMS clutter velocity spread of 0.8 m/s. Next, we suppose the training region encompasses one of the six regions identified in Figure 10.8. Figure 10.9 depicts the corresponding asymptotic loss over sixteen Doppler filters for the varying levels of spectral heterogeneity. The integrated CNR is set to 58 dB, a 12 dB increase over the preceding example. No loss occurs when training in the region with 0.8 m/s RMS spread, since this represents the matched condition. For values of RMS velocity spread of less than 0.8 m/s, the clutter notch width is inadequate, thereby leading to increased clutter residue and losses up to 3 dB in addition to those losses already associated with finite training. In contrast, the penalty for training in a region with greater spectral spread than the primary data is an apparently lesser degree of degradation. However, the result is dependent on the temporal resolution; we anticipate an increasingly severe, observable loss upon resolving the clutter spectrum via a longer dwell time. Additionally, degradation is CNR-dependent.

10.5

CNR-induced spectral mismatch

CNR-induced spectral mismatch refers to the range variation of clutter spectral width due to fluctuating CNR. ICM, clutter scintillation, dispersion, timing jitter,

SINR loss, dB

Previous Page

CNR1 = 58 dB

Doppler filter number

Figure 10.9 Asymptotic SINR loss for ([10], ©2000 IEEE)

varying levels of spectral spread

Next, suppose we train the adaptive filter in a region whose dominant spectral features differ from the test cell region (e.g. littoral zone or rural-urban interface). Figure 10.8 depicts the clutter-plus-noise eigenspectra for six distinct training regions. We assume the primary data exhibit an RMS clutter velocity spread of 0.8 m/s. Next, we suppose the training region encompasses one of the six regions identified in Figure 10.8. Figure 10.9 depicts the corresponding asymptotic loss over sixteen Doppler filters for the varying levels of spectral heterogeneity. The integrated CNR is set to 58 dB, a 12 dB increase over the preceding example. No loss occurs when training in the region with 0.8 m/s RMS spread, since this represents the matched condition. For values of RMS velocity spread of less than 0.8 m/s, the clutter notch width is inadequate, thereby leading to increased clutter residue and losses up to 3 dB in addition to those losses already associated with finite training. In contrast, the penalty for training in a region with greater spectral spread than the primary data is an apparently lesser degree of degradation. However, the result is dependent on the temporal resolution; we anticipate an increasingly severe, observable loss upon resolving the clutter spectrum via a longer dwell time. Additionally, degradation is CNR-dependent.

10.5

CNR-induced spectral mismatch

CNR-induced spectral mismatch refers to the range variation of clutter spectral width due to fluctuating CNR. ICM, clutter scintillation, dispersion, timing jitter,

clutter-only, yc = -20 dB clutter-plus-noise, yc = -20dB clutter-only, yc = -l0 dB clutter-plus-noise, yc = -10 dB

noise floor

ranked eigenvalue number

Figure 10.10

Illustration of eigenspectra variation for clutter spectral spread of 1.5 m/s and variable CNR

antenna motion and system instabilities, for instance, lead to subspace leakage [23]. Expressions (10.21) and (10.23) effectively model the temporal and spatial spectral spreading mechanisms; their effective application takes the form of a covariance matrix taper (CMT) [23,24]. As CNR increases, modulated components rise above the noise floor, thereby altering the width of the clutter spectrum. The adaptive processor must alter its response to mitigate these new coloured noise subspaces. Let us reconsider our 'typical' airborne radar example with the clutter spectral spread nominally set to 1.5 m/s and variable CNR controlled by the clutter reflectivity, yc. Figure 10.10 shows the corresponding clutter-only and clutter-plus-noise eigenspectra. The CNR-dependent nature of the dimension of the clutter subspace is evident in the Figure, viz. the apparent clutter rank increases with CNR. An increase in the rank suggests the presence of additional signal components - the modulated components resulting from signal decorrelation - and a consequent increase in clutter Doppler spectral spread. Guerci and Bergin describe this effect in detail in Reference 23. System effects, scintillation, multipath and other practical, non-ideal effects influence the nature of the spectral spread. Thus, all other effects held constant, if the CNR varies over range simply due to varying clutter RCS, a heterogeneous clutter condition arises distinct from simple amplitude mismatch. The corresponding spectral mismatch leads to either increased clutter residue or cancellation of potentially

SINR loss, dB

Doppler filter 1 Doppler filter 2 Doppler filter 3 Doppler filter 4 Doppler filter 5

asymptotic power, dB

Figure 10.11 Asymptotic SINR loss for CNR-induced spectral mismatch detectable target signals. As an example, Figure 10.11 depicts the asymptotic SINR loss for different Doppler filters as a function of CNR. The primary data has CNR set to 52 dB. Both training and primary data incorporate 0.45 m/s RMS spectral spread into the normalised covariance structure. The Figure suggests that training in a region with lower CNR markedly degrades MDV as a result of the joint amplitude and spectral mismatch; losses are substantial for only modest deviation in CNR. In contrast, training in a region with CNR greater than the primary data leads to a lesser degree of loss in this specific example. However, as the losses in Doppler filter 2 - the bin adjacent to main beam clutter - suggest, MDV is still significantly affected. Implementations employing overnulling strategies for adaptive training will suffer for this reason. An increase in the nominal RMS spectral spread (0.45 m/s is modest) will lead to an increase in the observable SINR degradation. To summarise, inappropriate filter notch width is the physical mechanism leading to performance degradation in the presence of CNR-induced spectral spread. Additionally, we note this scenario differs from that described in Section 10.3.2 as follows: in Section 10.3.2, the clutter RCS varied over angle and range in accordance with the gamma distribution in a manner representative of spiky clutter conditions; in contrast, the CNR-induced class of clutter heterogeneity indicates a persistent change in CNR over range that also leads to a consequent mismatch in spectral width, as described herein. Section 10.8 presents a measured data example corroborating CNR-dependent spectral spread and providing further clarification of this issue (see Figure R5 in the colour signature).

10.6

Targets in the secondary data

In this section we consider the impact of targets in the secondary data (TSD). From a practical perspective, the likelihood of target-like signals corrupting the secondary data set in a GMTI scenario is high. TSD affects STAP performance in several ways: from a filtering perspective, TSD leads to whitening of the desired signal component; TSD distorts the adapted pattern; TSD leads to inefficient use of the adaptive processor's DoFs; depending on the weight training strategy, TSD can also modulate the output power of the STAP, thereby biasing the constant false alarm rate (CFAR) threshold applied in subsequent processing; and, TSD impacts the variance of the target's angle of arrival (AoA) estimate. We now examine these points in further detail. To begin, we consider the following simple modification of equation (10.1):

(10.43)

where STSD is the space-time signal vector for a corruptive, target-like signal with power a\SD and Qk is the covariance matrix estimate of an assumed homogeneous clutter-plus-noise component. The corresponding weight vector is then:

(10.44)

Two main cases of interest arise: (a) perfect match to the target response, SJSD = v s _t; (b) mismatch between target and TSD responses, SJSD = vs_t + 5, where 8 represents the steering vector mismatch. In case (a) it is readily seen that W^/TSD is proportional to Qj^ vs_t, and so no penalty in SINR results from the TSD. Next, considering case (b), we express the weight vector in equation (10.44) as:

(10.45)

with p = Q^" ' s s _ t and z = Q^ ' S representing whitened target response and mismatched steering vectors, respectively, and where we otherwise assume v s _ t = s s _ t (i.e. no array manifold errors or mismatch to target angle-Doppler response). Several observations are in order: as expected, r\ - • 0 as a\SD -^ 0; the magnitude of T] depends on both TSD power, Oj
To evaluate the asymptotic SES[R loss due to TSD, let Qk -» Qk in equations (10.43) and (10.45). We may then express the additional loss with respect to noise-limited and optimal cases as [13]:

(10.46) Consequently, SINR(f59 JD) = SNR(J5) x LSy\ (/., Jh) x L5jSD(fs, Jh)9 which does not include further losses due to finite sample support. Expression (10.46) identifies the key parameters affecting performance: the TSD power and location of the corruptive target-like signal (measured by the projection of the generalised vectors, Q~ 1 / 2 s s _ t and Q^ 1/2<S), as well as the normalised SINR term ||p||^ = s^Qj^Sg-t. Observe that 0 < LsjsD(fs, Jh) < 1 a n d LsjsD(fs, Jh) = 1 when 6 = 0. The term T]Q^ 8 in equation (10.45) - or its asymptotic value - is responsible for SINR degradation via target whitening; this effect is most apparent when viewing the filter response. We now consider the aforementioned typical airborne radar interference scenario comprising ground clutter and receiver noise with 60 dB integrated CNR. Recall, eight elements make up the array radar in the azimuth dimension, with eight elements in elevation beamformed into a single spatial channel. A 25 dB Taylor weighting is applied on transmit, with no spatial weighting on receive. The receive half-beamwidth is roughly six degrees in azimuth. The PRF is set slightly higher than the displaced phase centre antenna (DPCA) requirement [3] and the array is side-looking with respect to the platform velocity vector. Figure P.I (see colour signature) plots LSJSD(JS* JD) for a single Doppler filter (bin 2, adjacent to main beam clutter) with the TSD component stepped over angle and power. In this case, we compared the calculation in equation (10.46) against: (10.47) with SINRopt(Js, Jh) denoting the optimal SINR and Wk/TSD = Qk/TSDvst> finding both expressions to be numerically equivalent. The embedded power of the TSD measures the corruptive power included in the covariance estimate and is essentially the TSD-to-noise ratio (noise floor at 0 dB) after the averaging process leading to equation (10.43). Moving scatterers spread over the range extent, such as vehicles on a highway, contribute to TSD. Additionally, tractor-trailers and multiple vehicles in a resolution cell lead to a strong echo. As expected, we observe zero loss when the target and TSD have the same direction of arrival and Doppler. However, slight misalignment between target and TSD directions leads to substantial losses within the main beam due to signal whitening. The null near seven degrees corresponds with a null in the quiescent pattern. Figure 10.12 depicts the spatial filter responses in Doppler bin 2 for cases of moderate and strong main beam TSD; for comparison, we also show the optimal

normalised gain, dB

target and TSD separation: 1°

optimal strong TSD (22 dB) moderate TSD (15 dB)

direction of arrival, deg

Figure 10.12

Filter response distortion due to target-like signal corrupting covariance matrix estimate ([13], © 2001 IEEE)

filter response. The Figure clarities the issue of signal cancellation due to TSD: observe the decreased gain in the target direction of 0° due to the migrating adaptive null when comparing with the optimal case. As will be shown in Section 10.8.1, we observe this same effect in actual measured airborne radar data taken in regions with ample roadways in the radar field of view. Upon detecting a target, a bearing estimate is then necessary to initiate target tracking. A method seamlessly integrating with STAP processing is desirable. Relevant approaches based on maximum likelihood estimation are discussed in References 25 and 26. In developing the bearing estimator, first consider the alternative hypothesis space-time snapshot: (10.48) where y is a complex constant, g = [fs / b ] , and ntot is the interference-plusnoise (total noise) vector. The basic problem is to estimate g with g. If ntot ~ CN(O, Q k ), then: (10.49)

Given this likelihood function, we first require an estimate for the complex constant. Equation (10.49) is maximal when: n(K,g) = (*k - KSs-ICg))^Qk1CXk - yss-t(g))

(10.50)

is minimal. Differentiating equation (10.50) with respect to y and setting the result to zero yields: (10.51) Next, substituting equation (10.51) into (10.50) leads to: (10.52) Differentiating equation (10.52) with respect to g and setting the result to zero yields the MLE for g, given as g. Consequently, the estimator is: (10.53) In practice, we replace Qk in equation (10.53) with QkFigure 10.13 shows the DoA cost surface of equation (10.53) for a 2OdB SNR target with a true AoA of—2 degrees. The target signal competes with 60 dB integrated CNR and receiver noise for the typical eight-channel, side-looking airborne array. For comparison, we also include the MLE in the presence of TSD by incorporating a target-like component with crjSD = 20 dB and a true AoA of 3 degrees. By searching for the peak of each of the two cost surfaces, we find unbiased estimates of the target AoA. However, flattening of the MLE cost surface in the presence of TSD translates into a substantial increase in the estimate's variance. Constant false alarm rate (CFAR) performance is highly desirable in radar detectors; the false alarm rate is a critical design factor influenced by computational and algorithmic processing capabilities. In some cases, the normalisation applied to the STAP weight vector influences CFAR performance. For example, Robey et al. describe the adaptive matched filter (AMF) as a CFAR detection statistic [27]. The AMF takes the form: (10.54) where §i is the threshold and vs_t = ss_t (no steering vector mismatch). The AMF follows from the STAP weight vector when \x = \/Jv^_tQ^ v s _t, and normalises the output noise power to unity; this characteristic is highly desirable, since in theory it enables a single threshold setting for all angle, Doppler and range to achieve specified false alarm behaviour. The output noise for range cell k under the AMF setting,

no TSD TSD

true target AoA: -2° TSD AoA: 3°

direction of arrival, deg

Figure 10.13

Target angle of arrival estimate in the presence of TSD

WAMF/k, is:

(10.55)

Equation (10.55) suggests an output noise level roughly set to unity, and for this reason the AMF is sometimes referred to as the unit-power constraint. Suppose we now incorporate the TSD model of equation (10.43) into equation (10.54). Using the matrix inversion lemma, assuming vs_t = ss_t, and coaligned TSD and target signal (8 = 0), we find: (10.56) Thus, while TSD does not degrade SINR in this instance, as previously shown, it does effectively bias the threshold upwards by 1+ ^j5Z>ss^tQk Ss ~t' thereby leading to target masking. This same effect is seen in traditional scalar CFAR algorithms. Since mismatch and clutter residue are always concerns, it is common to follow the AMF with a traditional scalar CFAR with adaptive threshold; depending on the STAP training strategy (e.g. sliding window), the AMF normalisation in the presence

of TSD suppresses the target response prior to the scalar CFAR, thereby effectively biasing the target response below the adaptive threshold. Interestingly, some non-CFAR-type normalisations do not suffer from the preceding effect. The unit norm constraint on the weight vector, w^Wk = 1 ? leads to v s-tQk vs-t> a n d the well known minimum variance distortionless response (MVDR) constraint, w^s s _ t = 1, leads to: AMVDR = l / ( v ^ t Q k l y s-t)- In the case where TSD and desired target steering vectors are coaligned, one can confirm the impervious nature of the decision statistic to corruptive TSD by employing equation (10.43) with STSD = v s _t. These latter normalisations are less desirable in theory than the AMF since the detection processor requires a different threshold setting for each angle-Doppler resolution cell, at least in a range homogeneous environment. More to the point, in a heterogeneous environment, these latter normalisations tend to modulate the output noise in the range dimension. Now we consider the impact of TSD on STAP performance when using finite sample support. To wit, we employ a Poisson distribution to seed targets in various range-angle sectors of the radar field of regard. Otherwise, we assume a homogeneous clutter environment so that we may specifically isolate the effects of TSD. Melvin and Guerci presented related analysis in Reference 13. A more complete treatment using site-specific clutter and road layout is given by Bergin et al. in Reference 28. Suppose dense traffic regions occur in specific range-azimuth sectors, with certain densities Acars and Atrucks defining the expected number of vehicles per square kilometer. A Poisson distribution is a natural selection for seeding targets within the space-time data cube. Figure 10.14 shows a single trial seeding for the parameters of Table 10.2. We generate a space-time signal for each vehicle location shown in the Figure; additionally, we assume each vehicle response follows a Swerling I target model, with a mean radar cross section of 10 dBsm for cars/light trucks and 22 dBsm for tractor trailer trucks. Next, we generate 279 radar clutter-plus-noise data realisations from a starting range of 32 km towards the end of the unambiguous range interval at 75 km. We use the same typical airborne radar parameters as in the preceding asymptotic analysis, with a waveform bandwidth set to 1 MHz (150m resolution). Using all 279 data vectors in equation (10.1) yields E[LS^] ~ —3dB in an otherwise homogeneous clutter environment. Beginning with the homogeneous clutter-plus-noise data cube of dimension TV = 8 by M = 16 by Q = 279, we then independently seed the Poisson-distributed TSD for varying trials using the parameters in Table 10.2. The TSD is uniformly distributed over the specified range-angle sector. We set the TSD Doppler response to correspond with ground moving targets - cars and truck on roadways in the radar field of view - nominally spread over velocities in the range of ±30 m/s. The Doppler is fixed by region (see Table 10.2), as one might expect for a roadway with a specific orientation, but uniformly spread over the corresponding Doppler filter width. Figure 10.15 compares SINR loss for the HD and homogeneous clutter-plus-TSD cases shown over a ±30 m/s target velocity interval. In this Figure we show the complete loss, LSt\ • L8^, characterising finite sample support, TSD, radar system

slant range, km

cars trucks

azimuth, deg

Figure 10.14

Target seeding scenario, one trial ([13], © 2001 IEEE) Table 10.2 Poisson target seeding scenario Region

1

2

3

4

A-carsCkm"2) ^trucks (km"2) Start range (km) Stop range (km) Azimuth (degrees)

0.5 0.1 32 39 - 5 to - 3

1 0.09 48 56 6 to 7

0.2 0.1 64 66 6 to 10

0.01 0.02 43 60 12 to 15

and homogeneous clutter effects. The upper bound on performance is also shown in Figure 10.15 and labelled as 'known covariance'. We obtained the TSD curve by averaging 200 different Poisson trials. As expected from our earlier discussion, TSD in a finite sample support scenario leads to significant performance loss when using the proposed Poisson model. Figure P.2 (see colour signature) shows SINR loss over ±50 m/s for each individual Poisson trial. The impact of TSD on the adaptive radar's MDV is evident in this plot, which we may further contrast against SES[R loss in the homogeneous (no TSD) case in Figure P.3 (see colour signature). TSD degrades the MDV in this instance by roughly a factor of three.

SINR loss, dB

known covariance estimated: with TSD (200 trials) estimated: HD secondary data

velocity, m/s

Figure 10.15

SINR loss comparison using finite training data and 200 Poisson trials to seed TSD ([13], © 2001 IEEE)

In Section 10.8.1 we show measured multichannel airborne radar measurements (MCARM) data illustrating that TSD is a practical concern when implementing STAP and corroborating the effects discussed in this section of the chapter using synthetic data. Section 10.9.4 suggests some strategies to mitigate TSD effects.

10.7

Joint angle-Doppler mismatch and clutter heterogeneity

STAP maximises SINR by filtering ground clutter in the angle-Doppler domain. Radar geometry determines the filter null location in this higher-dimensional space. When the null location varies over range, the adaptive filter produces an incorrect frequency response. Indeed, the filter tends to an average response with the potential for very poor instantaneous performance. We briefly highlight angle-Doppler properties for the monostatic radar case; relevant discussion applicable to bistatic geometry is given in References 3, 5 and 6. Consider a right-handed coordinate system with the x-axis pointing north, the y-axis pointing west and the z-axis pointing upwards. A unit vector pointing from the platform to a stationary point on the ground is: k(0,0) = cos 0 sin 0x + cos 0 cos 0y + sin Oi

(10.57)

where 0 is azimuth measured positive in the clockwise direction from the y-axis and 0 is elevation measured negative in the downward direction from the horizon. The direction vector to the mth subarray is: (10.58) while the platform velocity vector is: (10.59) The spatial phase at the mth channel is: (10.60) A denotes wavelength. Also, the normalised Doppler frequency of a stationary point is: (10.61) where T is the pulse repetition interval. If the channel spacing is d and we consider a side-looking array configuration (i.e. (Is9Hi = (m — l)d\) we can define the spatial frequency as: (10.62) where 0cone represents cone angle [3]. Normalised Doppler in the side-looking case can be written: (10.63) In other words, the angle-Doppler properties of a continuum of stationary points (ground clutter) fall on a line with the slope determined by platform velocity, pulse repetition interval and sensor spacing, and there is no dependence of the angle-Doppler contours with range (elevation angle). Hence, platform geometry does not induce non-stationary behaviour. The forward-looking geometry is a contrasting case. With the platform in level flight in the x-direction: (10.64) The angle-Doppler contours are ellipses that vary over range. As a rule-of-thumb, when the slant range divided by the platform height is less than five, range dependence is significant [3]; otherwise, the beam traces tend to align with the isodops at farther range, and so the angle-Doppler contour locations tend to stabilise. This example shows that geometry induces non-stationary angle-Doppler behaviour.

Range-varying angle-Doppler loci further exacerbates culturally induced heterogeneous clutter effects. For example, residue from clutter discretes further increases in the presence of adaptive filter null migration associated with the non-stationary clutter mechanism described in this section.

10.8

Site-specific examples of clutter heterogeneity

10.8.1 Measured multichannel airborne radar data This section briefly examines measured data taken from the multichannel airborne radar measurements (MCARM) programme [29]. Figure 10.16 shows the MCARM platform with a port-mounted, multichannel radar system housed within the radome. Table 10.3 provides some characteristics of the MCARM system. We consider data taken from Flight 5, Acquisition 575. Our objectives in considering this data are two fold: (a) we wish to corroborate the model of equation (10.18); (b) we identify a few instances of heterogeneous clutter behaviour described in the preceding sections.

Figure 10.16 Table 10.3

• • • • • • • • • •

MCARM radar system Some MCARM

characteristics

L-band transmit frequency 15 kW peak transmit power variable PRF (0.5 kHz, 2 kHz, 7 kHz) LFM or gated-RF 0.8 microsecond range resolution 0.8 MHz receiver bandwidth 7.5 degree Tx beam or blob (3 x) pattern for broad coverage 1.25 MHz IF centre frequency 5 MHz IF sampling rate (4 x oversample for digital I/Q) test manifold for channel balancing

• • •

• • • •

range-measured steering vectors 32 transmit subarrays (16 over 16 planar configuration) 24 receivers (sum, delta, and 11 over 11 additional channels oriented in a planar configuration) 128 radiating elements total (32 subarrays x 4 elements per subarray) 1 acquisition = 1 CPI nominally 128 pulses per CPI at the 2 kHz PRF collection region: DelMarVa Peninsula, USA

estimated SINR loss, dB

simulated

Doppler, Hz

Figure 10.17

Estimated SINR lossfor MCARM 575 data, different training intervals

As a means of corroborating the model of equation (10.18), Figure P.4 (see colour signature) compares the minimum variance distortionless response (MVDR) spectra for the actual data against simulation. The plots generally exhibit close correspondence. Clutter ridge slope and shape show good match; the additional azimuthal power spread in the actual data is a likely result of near-field scattering, an effect not included in the simulation model. Figure P.4 (see colour signature) confirms the suitability of equation (10.18) as a basic space-time model for ground clutter. Using test manifold data, we estimate the MCARM receiver noise floor. Next, we select training data in block regions from near range to far and estimate the clutter-plusnoise covariance matrix. Figure 10.17 shows estimated SINR loss for training data taken over different regions, identified in the legend, as well as the simulated curve using the model of equation (10.18). Good correspondence exists between actual and simulated clutter null locations. However, the impact of amplitude heterogeneity and TSD affects the shape of the measured data SINR loss curves. For example, scalloping in the loss curves is most likely due to TSD. To verify this conjecture, Figure 10.18 shows estimated SINR loss when training over a contiguous block, and after removing roadways intersecting the main beam (MB) and first sidelobe (SL) regions; we also apply some diagonal loading (denoted as DL) to stabilise the noise floor after removing training samples. We use map data (see Figure 10.20) and ownship navigation data to identify roadways in the radar field of view. As can be seen, the scalloping disappears after excising certain training data potentially containing target-like signals,

estimated SINR loss, dB

excise Hwy 15 MB excise Hwy 15 SL excise Hwy 15 SL, 0 dB DL simulated

Doppler, Hz

Figure 10.18

Estimated SINR loss after removing training data overlaying roadways

thereby yielding much closer correspondence between actual and simulated curves. Figure 10.19 shows the adaptive filter patterns within a single Doppler bin for the same training intervals (labelled by range bin number) as shown in Figure 10.17. The quiescent beam points to 1 degree off boresight. Migration of the null due to TSD is apparent in the upper two plots where the corresponding training data comes from regions with dense roadway networks. Varying clutter reflectivity affects the depth of the clutter null shown in Figure 10.17. Figure P.5 (see colour signature) shows range-Doppler and power versus range information for the subject acquisition. The power versus range curve clearly shows three distinct regions corresponding, from near to far range, to the predominantly farmland terrain of the DelMarVa Peninsula, USA, Delaware River and terrain in New Jersey, USA, opposite the Delaware River. A map of the collection region, along with the locations of several adjacent acquisitions, is shown in Figure 10.20; the aircraft flew a southerly route, with the array normal pointing almost due east. The three distinct terrain regions, along with many roadways in the radar field of view, are evident from viewing the map. Additionally, the CNR-dependent nature of the main beam clutter spread is seen in the range-Doppler map. From this single measured data acquisition we find examples of TSD, rangevarying clutter edges (three distinct regions) and CNR-dependent spectral spread, thereby corroborating several heterogeneous clutter models described in prior sections. Additionally, from Figure P.4 (see colour signature), we find that equation (10.18) adequately characterises the basic features of ground clutter.

Figure 10.19

10.8.2

azimuth angle, deg

azimuth angle, deg

azimuth angle, deg

azimuth angle, deg

Comparison of adaptive patterns for MCARM 575 data, different training regions ([13], © 2001 IEEE)

Site-specific

simulation

Realistic clutter environments, as the measured data from the prior section suggests, are site-specific. A variety of databases describing terrain cultural features are available to researchers investigating STAP performance in site-specific, heterogeneous clutter environments [28,30-34]. In addition, the signal processor can use these databases to enhance STAP implementation (e.g. by improving training data selection, predicting clutter edges, or prefiltering the data) [28,30,32]. For example, the United States Geological Survey (USGS) database includes land use and land cover (LULC) data and National Land Characterisation Data (NLCD) classifying predominant clutter types by geographic coordinate; digital line graph (DLG) data characterising discrete features, such as railway tracks, power transmission lines, etc.; digital elevation model (DEM) data providing terrain height information [30,32,33]. The US Census Bureau provides a variety of mapping products, including the Topologically Integrated Geographic Encoding and Referencing (TIGER/Line) road overlay data [28,30,34]. Combining cultural databases, such as those described in the preceding paragraph, with the clutter model of equation (10.18) enables site-specific clutter simulation. Different reflectivities are assigned to different clutter classes based on a query of the database [28,30]. Incorporating ground moving vehicles involves placing road

Figure 10.20

Mapping data of acquisition region showing roadways and Delaware River [34]

power, dB

MCARM acquisition 575 site-specific model

slant range, km

Figure 10.21 Measured and site-specific synthetic power versus range curves for MCARM 575 scenario ([30], © 2003 IEEE)

segments - given, for instance, by the TIGER/Line database - onto the earth's surface [28,30] and seeding targets using a particular probability distribution, such as the Poisson distribution employed in Section 10.6 or Reference 13. Database information can potentially provide very accurate prediction of ground clutter characteristics. Figure P.6 (see colour signature) shows a site-specific clutter RCS map of the data collection region for the MCARM Flight 5 data analysed in the prior section. The shape of the reflectivity map corresponds with our expectations based on the US Census Bureau map in Figure 10.20. Using the RCS map of Figure P.6 in the model of equation (10.18) leads to a remarkable match between the actual and site-specific, synthetic MCARM power versus range curves, as shown in Figure 10.21. Finally, Figures P.7 and P.8 (see colour signature) contrast the site-specific and homogeneous (bald earth) synthetic range-Doppler maps for the MCARM scenario. (Note: the ordinate has units of velocity, obtained simply by scaling Doppler frequency by half the wavelength.) Figure P.7 compares favourably against the measured MCARM range-Doppler map shown in Section 10.8.1.

10.9

STAP techniques in heterogeneous environments

In this section we highlight some STAP techniques applicable in heterogeneous clutter environments. This listing of methods is by no means exhaustive. A thorough examination and comparison of the various methods is beyond the scope of this chapter. We provide corresponding references for readers interested in delving further.

10.9.1 Data-dependent training techniques Data-dependent training techniques are valuable in heterogeneous clutter environments. The non-homogeneity detector (NHD) [35,36], power-selected training (PST) [37-39] and map-based training selection [28,30] are three examples of data-dependent training schemes. The non-homogeneity detector (NHD) assumes that gross changes in the underlying data structure lead to degraded performance [35,36]; the processor enhances adaptive capability by detecting and excising secondary data realisations significantly deviating from the surrounding realisations. The processor's goal is to select the most homogeneous set of secondary data based on a measure of covariance structure. The generalised inner product (GIP) is one viable metric and is given by: (10.65) Next, observe that the GIP can be written: (10.66) where Xk represents the whitened data vector, Ek is the matrix of eigenvectors resulting in the unitary similarity transform and Lk is a diagonal matrix containing the eigenvalues ofQk. Further notice that E[XkX1^] = Q^ QkQk , thereby implying

that when Qk = Qk(10.67) where am are the Karhunen-Loeve coefficients and e m are the eigenvectors of Qk. In this sense, the GIP measures the similarity in covariance structure between Qk and QkNotice that an ideal measure of difference in covariance structure is dk =
N

MSMI> dB

2 x DoF sliding window

slant range, miles

2 x DoF most homogeneous NMSMI* dB

injected target

slant range, miles

Figure 10.22

Comparison of sliding window (top) and NHD (bottom) training schemes applied to MCARMFlight 5, Acquisition 575, Doppler bin 10 (1 mile = 1.61 km) ([11], © 2000 IEEE)

to select a homogeneous training set, but masked by surrounding interference in the sliding window case. To implement the NHD, the GIP is applied as follows. First, the data are Doppler processed and then the processor passes a minimum size sliding window through the secondary data to compute a covariance matrix estimate. Next, the processor sorts and ranks the GIP, identifying the most homogeneous samples to compute the adaptive weight vector. Figure 10.23 depicts the GIP over range for the tenth Doppler bin. Using mapping data, the peaks of the GIP - corresponding to heterogeneous samples - are found to generally align with roadways in the radar field of view, thereby suggesting TSD as the culprit leading to the performance loss of the sliding window approach shown in Figure 10.22. Power-selected training (PST) uses an adaptive procedure to select secondary data for covariance matrix estimation [37,38]. The basis for PST is the presumption that clutter heterogeneity is a result of power variations over range, and that these power variations result in inadequate clutter null depth. The adaptive processor does not fully cancel clutter as a result of insufficient null depth, leading to an increase in

Rt.9 SL, Edge Del. River Intracoastal Waterway Back Edge of Del. River Acq.l,575 Broadside Doppler 10, 4xDoF Sliding Window RU5SL, Rts. 13,71 MB Rts. 9.13 SL Rt. 9 MB

SL = sidelobe, MB = mainbeam

range, miles

Figure 10.23

Generalised inner product versus range, MCARM Flight 5, Acquisition 575, Doppler bin 10 (1 mile = 1.61km) ([11], © 2000 IEEE)

the false alarm rate. PST involves choosing stronger clutter samples, thereby yielding deeper clutter nulls. Specifically, the optimal weight vector may be written: (10.68) where am = e^s s _t, e m is the mth eigenvector with eigenvalue Am, and"knoiseis the noise floor eigenvalue. The beamformer voltage pattern gain is:

(10.69) where Qv(g) is the quiescent response, Emv(g) is the eigenbeam response and Amv(g) is the projection of the eigenbeam onto the quiescent pattern. Notice that Amv (g) scales the eigenbeam equal to the quiescent response at g = [(/) O]. Therefore, the level of nulling below the quiescent pattern at g is: (10.70) or twice the eigenvalue spread. This suggests training on those snapshots exhibiting the strongest power, thereby increasing Xm and consequently driving the adaptive null to a greater depth. The procedure for implementing post-Doppler PST (see Table I in Reference 38) is as follows: (a) (b) (c)

a jammer-nulled beam is pointed in the direction of clutter for each Doppler bin the processor sorts the beamformed outputs by power (in each Doppler bin for the Doppler-factored STAP approach) the processor selects the data snapshots for covariance matrix estimation in one of two suggested ways, either the single training set or the multiple training sets approach. Using the single training set method, the processor chooses the strongest outputs over all range for covariance matrix estimation. In the multiple training sets approach, the processor selects those range samples whose estimated power exceeds the power in the primary cell.

Variations on the PST method are given in Reference 38. One alternate implementation, referred to as 'power-selected deemphasis (PSD)', involves including some proportion of the primary test cell and surrounding guard cells in the covariance matrix estimation procedure to potentially improve the representation of test cell interference. In this case, the covariance matrix estimate takes the form: (10.71) where & = 0 excludes the primary data from the MLE and # = 1 fully includes the primary data. Projection deemphasis (PDE), also proposed in Reference 38, is one

approach for avoiding the selection of #. The idea behind PDE is to remove target energy from the primary data. One potential choice for the projection operator, for a post-Doppler STAP approach, is: (10.72) where vc is the clutter spatial steering vector. PST reduces the false alarm rate in heterogeneous clutter environments. Principal concerns with this training approach include signal cancellation - particularly when close correlation exists between clutter and target, such as in GMTI applications and degraded MDV resulting from CNR-dependent spectral heterogeneity. An example of map-dependent training selection was given in Figure 10.18. Mapping information, such as the TIGER/Line data [34], along with ownship data, can be used to identify training regions likely to contain ground moving targets. The processor then excises suspect range cells to mitigate target whitening and other TSD-related degradation.

10.9.2 Minimal sample support STAP Distributed clutter may exhibit substantial similarity over localised regions. For example, if a series of clutter transitions (e.g. rural to urban to littoral) exists, it stands to reason that training independently on each clutter type - versus a more global training strategy - will enhance performance. Reduced-dimension (RD) and reduced-rank (RR) STAP methods minimise requisite training data, hence enabling localised training selection. Operating directly in the space-time domain, STAP serves as a data-domain implementation of the optimum filter; both optimum and adaptive filters involve weighting and combining complex baseband voltage outputs of Af spatial channels over M temporal samples, as equation (10.6) suggests. Additional space-time DoFs increase the requisite training range interval and computational burden. Decreasing either the spatial or temporal aperture is one approach to reducing DoF, but the corresponding information loss typically yields unacceptable detection performance since it is more difficult to separate the target signal from competing ground clutter returns. A preferred approach to DoF reduction involves linear filtering and/or subspace selection. Reduced-dimension (RD) STAP methods apply data-independent transformations to project space-time data to a lower-dimensional subspace of size P <^ NM [3]. In this case, the computational burden decreases from 0(N3M3) to O(P3), and nominal sample support tends to IP. Given the linear transformation matrix TRD € CNMxP, the reduced-dimension data vector is: (10.73) The optimal linear filter maximising SINR in the lower-dimensional subspace follows from prior results and has weight vector: (10.74)

Estimating the unknown PxP null-hypothesis covariance matrix proceeds in the same fashion as the full DoF space-time case: the processor uses the MLE of equation (10.1) applied to the transformed data. Although the clairvoyant performance of the filter given by equation (10.74) is inferior to the full DoF approach, the adaptive performance using finite training data can potentially yield better results. Reduced-rank (RR) STAP involves selecting a lower-dimensional subspace to represent clutter and jamming signals; each selected basis vector retains the original length NM. Subspace calculation commonly involves a computationally expensive singular value decomposition (SVD), but RR-STAP improves statistical convergence over the full DoF case from 2NM to 2/c, where K = rank (Qk) is ideally much less than NM [4,19]. In practice, rank determination is challenging. Unlike the RD-STAP case, the performance of the RR-STAP can identically match the full DoF processor when the interference subspace of the null-hypothesis covariance matrix is known a priori. Diagonal loading involves scaling an identity matrix in proportion to the nominal noise floor level and adding the result to the covariance estimate [40]. Diagonal loading tends to compress the noise eigenvalues, thereby stabilising the adaptive filter sidelobes and emphasising the covariance matrix principal components. Furthermore, diagonal loading enhances the covariance matrix condition number. The processor can reduce the requisite amount of training data without concern over ill conditioning. Forward-backward (F/B) averaging [41] and space-time aperture smoothing [42] further enable operation with reduced sample support. Both approaches exploit symmetry in the space-time snapshot to enhance statistical convergence. In the latter case, degradation of space-time aperture (resolution) is the corresponding cost for improved convergence. The processor can blend F/B averaging and the extent of smoothing to alter the range extent of the training region. F/B averaging and smoothing techniques are common in spectral estimation. The supposition that localised training enhances STAP performance is inherent when applying minimal sample support techniques to combat heterogeneity. Localised training is particularly valuable in those regions where different classes of distributed clutter dominate. Less clear, however, is the value of localised training when operating in the presence of clutter discretes and TSD. Furthermore, the variation of distributed clutter is unlikely to appear homogeneous over range and angle within the confines of a local range segment. One approach to circumvent the deleterious impact of heterogenous clutter is to completely forsake training in the range dimension. Sarkar et al. present a variant of space-time aperture smoothing in Reference 43 involving a least-squares formulation with training (parameter estimation) completely confined to the snapshot under test. Farina, Lombardo and Pirri describe a filter-bank approach in References 44 and 45 involving the design of a deterministic space-time filter bank, with a cell-averaging constant false alarm rate processor following each filter. A non-linear detector selects the filter exhibiting minimal estimated output clutter power, thereafter comparing the corresponding output signal in the range dimension to a CFAR threshold. The deterministic space-time clutter filter design does not require training, although the CFAR stage requires a small training interval in comparison with typical STAP approaches.

Farina et al. successfully compare the performance of this non-linear, non-adaptive processing scheme against fully adaptive STAP methods using measured radar data in Reference 44.

10.9.3

Clutter discretes

In Reference 46, Richmond analyses the performance of an adaptive sidelobe blanker configuration to reduce the false alarm rate of the adaptive matched filter in the presence of undernulled sidelobe clutter and discretes. The adaptive sidelobe blanker implementation involves two stages: the adaptive matched filter, resulting from the magnitude-squared output of the STAP with normalisation jl = \/J v^_ t Q^ v s _t; followed by the adaptive coherence estimator (ACE). The ACE determines whether a main beam or sidelobe origination best characterises the likely target response, operating in a manner similar to a traditional blanker. The power-selected training approach of References 37 and 38, and described in some detail in Section 10.9.1, attempts to minimise the impact of clutter discretes by training on the most powerful samples, thereby driving the adaptive null to greater depth. This approach will likely find greater application in airborne moving target indication. As Klemm suggests in Reference 3, orthogonal projection methods - such as the principle components inverse approach of Reference 47 - can mitigate clutter bleed through associated with amplitude heterogeneity and clutter discretes by offering (theoretically) infinite adaptive null depth.

10.9.4

Targets in training data

By minimising main beam distortion, diagonal loading can potentially mitigate TSD effects [13]. Additionally, some researchers have suggested foregoing adaptivity to avoid challenges associated with TSD; for example, the displaced phase centre antenna processing approach is a non-adaptive solution for suppressing ground clutter returns [3]. Figures 10.24 and 10.25 compare SINR loss for the Poisson target seeding example described in Section 10.6 when applying diagonal loading and DPCA methods. In Figure 10.24, the DPCA condition (see Reference 3) is violated, and so DPCA performance is quite poor. The 1OdB diagonally loaded STAP yields a 3—4 dB SINR improvement. Figure 10.25 shows performance under the DPCA condition; we find that the diagonally loaded STAP performance approaches the DPCA result, and that traditional STAP exhibits substantial losses due to TSD. Incorrect PRF selection, non-ideal platform attitude and various system errors exacerbate practical DPCA implementation; adaptive processing is commonly used in lieu of DPCA to accommodate these practical issues. Constrained adaptive processing is another approach to combating TSD effects. For example, derivative constraints mitigate main beam distortion associated with TSD [48]. The NHD [33,34] and map-based training approaches discussed in Section 10.9.1 are also potentially useful approaches when operating in environments contaminated by TSD. Finally, as another example, the adaptive multistage

SINR loss, dB

STAP with TSD (PRF - 2 kHz) DPCA (PRF = 2 kHz) diagonal loading (PRF = 2 kHz)

velocity, m/s

Figure 10.24

Comparison of conventional STAP, DPCA and STAP with 1OdB diagonal loading (DL) in a TSD environment ([13], © 2001 IEEE)

median cascaded canceller - proposed by Picciolo, Gerlach and Goldstein - is also an approach for coping with moving discretes in the training interval [49]. The method incorporates a median sample selection function to mitigate the impact of outliers on detection performance.

10.9.5

Covariance matrix tapers

Covariance matrix tapers (CMTs), described in References 23 and 24, provide robustness to spectral mismatch and modulation of the dominant subspace due to varying CNR conditions. The CMT concept involves applying a complex taper, t, to the space-time data to tailor the adaptive notch width. Specifically, let: xk = t O x k

(10.75)

The covariance matrix of x k is then: (10.76) Generally, ACMT will possess full rank. As previously mentioned, the application of the correlation matrices At and As described in Section 10.2 is tantamount to a CMT operation. With the particular heterogeneous clutter environment in mind, the STAP designer chooses the correlation matrix ACMT, or the corresponding taper t, to create a desirable adaptive filter response.

SINR loss, dB

STAP with TSD (PRF =1.3 kHz) DPCA (PRF =1.3 kHz) diagonal loading (PRF = 1.3 kHz)

DPCA condition

velocity, m/s

Figure 10.25

DPCA mitigates TSD losses - since it is non-adaptive ~ but requires rigid control over PRF, velocity vector and system errors ([13], © 2001 IEEE)

The CMT operation subsumes diagonal loading: ACMT tf£)L№zg(Qk))~ 1 INMJ where o2DL is the diagonal loading level. 10.9.6

=

1+

Knowledge-aided space-time processing

Oftentimes, a preponderance of a priori information is available to partially characterise aspects of the clutter environment. References 30 to 32 suggest a variety of knowledge sources, including cultural databases, inertial navigation unit (INU) and GPS data, and information from other sensors. The adaptive processor can incorporate this additional information to accomplish the following: • • • •

enhance training data selection [30-32,50,51] preadaptively null discretes [30,50,51] minimise the number of unknowns requiring estimation [52,53] condition the space-time data through pre-filtering [30,52,53].

The non-linear, non-adaptive filtering method proposed by Farina et al. in References 44 and 45, for instance, employs INU/GPS to estimate the clutter ridge slope; by default, this approach is knowledge-aided. Hiemstra proposes a coloured loading scheme in Reference 53; coloured loading replaces adaptive degrees of freedom with deterministic filtering objectives. Interestingly, the coloured loading approach provides a mechanism for incorporating a priori knowledge in a convenient manner.

Since clutter environments are generally site specific, knowledge-assistance will likely prove invaluable in further enhancing STAP capability for airborne and spaceborne radar systems.

10.10

Summary

STAP-based radar systems must operate in heterogeneous clutter environments. This chapter has introduced the reader to aspects of space-time clutter heterogeneity, further describing the impact of clutter heterogeneity on STAP performance and briefly highlighting some techniques to enhance STAP implementation when operating in heterogeneous clutter environments. We introduced five basic classes of clutter heterogeneity and related effects: amplitude heterogeneity, spectral mismatch, CNR-induced spectral mismatch, moving (Doppler-shifted) discretes, and related effects such as mismatched angle-Doppler loci. Using a basic model to describe space-time clutter characteristics, we investigated the potential impact of each class of heterogeneity on detection performance. Using measured airborne radar data, we also corroborated a fundamental space-time clutter model and several heterogeneous clutter models. It is important to note that performance curves provided in this chapter are specific to the particular simulation scenario and are generally intended to isolate singular effects, such as the impact of either amplitude or spectral mismatch. Practical environments are much more complicated than the simplified models described in this chapter might suggest. Specifically, real-world environments are site specific and involve a simultaneously occurring conglomeration of various heterogeneous clutter effects. Since actual clutter scenarios generally appear heterogeneous, an understanding of clutter heterogeneity is essential to the development of mitigation techniques. Certainly, much work remains.

10.11

Acknowledgments

The author gratefully acknowledges his collaboration on the development of STAP implementations with Dr. Joseph R. Guerci of DARPA's Special Projects Office, and thanks Dr. Guerci for his encouragement and support. The author further acknowledges the support of his Georgia Tech colleagues; special thanks go to Dr. Daniel Leatherwood, Dr. Greg Showman and Dr. Daren Zywicki, members of the technical staff at the Georgia Tech Research Institute. Additionally, the author gratefully acknowledges Dr. Zywicki for generating the site-specific clutter simulation results discussed in Section 10.8.2, and Dr. Showman for proofreading the manuscript and providing valuable suggestions for improvement. The author further thanks his colleagues at the Air Force Research Laboratory for introducing the author to STAP, for many stimulating interactions over the years, and gratefully acknowledges the support of Dr. Mark Davis, Mr. Bill Baldygo, Mr. Mike Callahan, Dr. Mike Wicks and Mr. Jerry Genello.

References 1 BRENNAN, L. E. and REED, I. S.: 'Theory of adaptive radar', IEEE Trans. Aerosp. Electron. Syst9 March 1973, 9, (2), pp. 237-252 2 HAYKIN, S.: 'Adaptive filter theory' (Prentice-Hall, Upper Saddle River, NJ, 1996, 3rd edn.) 3 KLEMM, R.: 'Principles of space-time adaptive processing' (IEE Publishing, UK, 2002, 2nd edn.) 4 GUERCI, J. R., GOLDSTEIN, J. S., and REED, I. S.: 'Optimal and adaptive reduced-rank STAP', IEEE Trans. Aerosp. Electron. Syst, April 2000, 36, (2), pp. 647-661 5 MELVIN, W. L., CALLAHAN, M. J., and WICKS, M. C : 'Adaptive clutter cancellation in bistatic radar'. Proceedings of 34th Asilomar conference, Pacific Grove, CA, 29-31 October 2000, pp. 1125-1130 6 KLEMM, R.: 'Comparison between monostatic and bistatic antenna configurations for STAP', IEEE Trans. Aerosp. Electron. Syst, April 2000, 36, (2), pp. 596-608 7 NITZBERG, R.: 'An effect of range-heterogeneous clutter on adaptive Doppler filters', IEEE Trans. Aerosp. Electron. Syst9 May 1990, 26, (3), pp. 475-^80 8 ARMSTRONG, B. C , GRIFFITHS, H. D., BAKER, CJ., and WHITE, R. G.: 'Performance of adaptive optimal Doppler processors in heterogeneous clutter', IEEProc, Radar Sonar Navig., August 1995,142, (4), pp. 179-190 9 FUTERNIK, A. and HAIMOVICH, A. M.: 'Performance of adaptive radar in range-heterogeneous clutter', J. Franklin Inst., 1998, 335B, pp. 71-87 10 MELVTN, W. L.: 'Space-time adaptive radar performance in heterogeneous clutter', IEEE Trans. Aerosp. Electron. Syst, April 2000, 36, (2), pp. 621-633 11 MELVIN, W. L., GUERCI, J. R., CALLAHAN, M. J., and WICKS, M. C : 'Design of adaptive detection algorithms for surveillance radar'. Proceedings of IEEE 2000 international Radar conference, Alexandria, VA, 7-12 May 2000, pp. 608-613 12 KALSON, S.: 'An adaptive array detector with mismatched signal rejection', IEEE Trans. Aerosp. Electron. Syst, January 1992, 28, (1), pp. 195-207 13 MELVIN, W. L. and GUERCI, J. R.: 'Adaptive detection in dense target environments'. Proceedings of 2001 IEEE Radar conference, Atlanta, GA, 1-3 May 2001, pp. 187-192 14 CAI, L. and WANG, H.: 'Further results on adaptive filtering with embedded CFAR', IEEE Trans. Aerosp. Electron. Syst, October 1994, 30, (4), pp. 1009-1020 15 McDONALD, K. P. and BLUM, R. S.: 'Exact performance of STAP algorithms with mismatched steering and clutter statistics', IEEE Trans. Signal Process., October 2000, 48, (10), pp. 2750-2763 16 HAIMOVICH, A.: 'Theeigencanceler: adaptive radar by eigenanalysis methods', IEEE Trans. Aerosp. Electron. Syst, April 1996, 32, (2), pp. 532-542

17 REED, I. S., MALLETT, J. D., and BRENNAN, L. E.: 'Rapid convergence rate in adaptive arrays', IEEE Trans. Aerosp. Electron. Syst, November 1974, 10, (6), pp. 853-863 18 BOROSON, D. M.: 'Sample size considerations for adaptive arrays', IEEE Trans. Aerosp. Electron. Syst, July 1980, AES-16, (4), pp. 446-451 19 GIERULL, C. H.: 'Statistical analysis of the eigenvector projection method for adaptive spatial filtering of interference', IEE Proc, Radar Sonar Navig., April 1997,144, (2), pp. 57-63 20 SKOLNIK, M. L: 'Introduction to radar systems' (McGraw Hill, New York, NY, 1980, 2nd edn.) 21 BILLINGSLEY, J. B., FARINA, A., GINI, R, GRECO, M. V., and VERRAZZANI, L.: 'Statistical analysis of measured radar ground clutter data', IEEE Trans. Aerosp. Electron. Syst., April 1999, 35, (2), pp. 579-593 22 BARTON, D.: 'Land clutter models for radar design and analysis', Proc. IEEE, February 1985, 73, (2), pp. 198-204 23 GUERCI, J. R. and BERGIN, J. S.: 'Principal components, covariance matrix tapers, and the subspace leakage problem', IEEE Trans. Aerosp. Electron. Syst, January 2002, 38, (1), pp. 152-162 24 GUERCI, J. R.: 'Theory and application of covariance matrix tapers for robust adaptive beamforming', IEEE Trans. Signal Process., April 1999, 47, (4), pp. 977-985 25 DAVIS, R. C , BRENNAN, L. E., and REED, I. S.: 'Angle estimation with adaptive arrays in external noise fields', IEEE Trans. Aerosp. Electron. Syst., March 1976, AES-12, (2), pp. 179-186 26 MANOLAKIS, D. G., INGLE, V. K., and KOGON, S. M.: 'Statistical and adaptive signal processing: spectral estimation, signal modeling, adaptive filtering and array processing' (McGraw Hill, Boston, MA, 2000) 27 ROBEY, F. C , FUHRMAN, D. R., KELLY, E. J., and NITZBERG, R.: 'A CFAR adaptive matched filter detector', IEEE Trans. Aerosp. Electron. Syst., January 1992, 28, (1), pp. 208-216 28 BERGIN, J., TECHAU, P., MELVIN, W. L., and GUERCI, J. R.: 'GMTI STAP in target-rich environments: site-specific analysis'. Proceedings of 2002 IEEE Radar conference, Long Beach, CA, 22-25 April 2002, ISBN 0-7803-7358-8 29 FENNER, D. K. and HOOVER, W. F.: 'Test results of a space-time adaptive processing system for airborne early warning radar'. Proceedings of 1996 IEEE national Radar conference, Ann Arbor, MI, 13-16 May 1996, pp. 88-93 30 ZYWICKI, D. J., MELVIN, W. L., SHOWMAN, G. A., and GUERCI, J. R.: 'STAP performance in site-specific clutter environments'. Proceedings of 2003 IEEE Aerospace conference, Big Sky, Montana, 8-15 March 2003, ISBN 0-7803-7651-X, paper no. 1145 31 MAHER, J., CALLAHAN, M., and LYNCH, D.: 'Effects of clutter modeling in evaluating STAP processing for space-based radars'. Proceedings of 2000 IEEE international Radar conference, Alexandria, VA, 7-12 May 2000, pp. 565-570

32 WEINER, D. D., CAPRARO, G. T., CAPRARO, C. T., BERDAN, G. B., and WICKS, M. C : 'An approach for utilizing known terrain and land feature data in estimation of the clutter covariance matrix'. Proceedings of 1998 IEEE national Radar conference, Dallas, TX, 12-13 May 1998, pp. 381-386 33 National Land Characterization Project website, http://landcover.usgs.gov/ classes.html 34 TIGER/Line website, http://tiger.census.gov/cgi-bin/mapbrowse-tbl 35 MELVIN, W. L. and WICKS, M. C : 'Improving practical space-time adaptive radar'. Proceedings of 1997 IEEE national Radar conference, Syracuse, New York, May 13-15, 1997, pp. 48-53 36 CHEN, P., MELVIN, W. L., and WICKS, M. C : 'Screening among multivariate normal data', J. Multivariate Anal, March 1999, 69, pp. 10-29 37 RABIDEAU, D. J. and STEINHARDT, A. 0.: 'Improving the performance of adaptive arrays in non-stationary environments through data-adaptive training'. Proceedings of 30th Asilomar conference, Pacific Grove, CA, 3-6 November 1996, pp. 75-79 38 RABIDEAU, D. J. and STEINHARDT, A. 0.: 'Improved adaptive clutter cancellation through data-adaptive training', IEEE Trans. Aerosp. Electron. SySt., July 1999, 35, (3), pp. 879-891 39 MARSHALL, D.: 'Evaluation of STAP training strategies with mountaintop data'. MIT Lincoln Lab, TR MTP-5, Bedford, MA, February 1996 40 CARLSON, B. D.: 'Covariance matrix estimation errors and diagonal loading in adaptive arrays', IEEE Trans. Aerosp. Electron. Syst., July 1988, 24, (4), pp. 397-401 41 PILLAI, S. U., KIM, Y. L., and GUERCI, J. R.: 'Generalized forward/backward subaperture smoothing techniques for sample starved STAP', IEEE Trans. Signal Process., December 2000, 48, (12), pp. 3569-3574 42 FANTE, R. L., BARILE, E. C , and GUELLA, T. P.: 'Clutter covariance smoothing by subaperture averaging', IEEE Trans. Aerosp. Electron. SySt., July 1994, 30, (3), pp. 941-945 43 SARKAR, T. K., WANG, H., PARK, S., et al.: 'A deterministic least-squares approach to space-time adaptive processing (STAP)', IEEE Trans. Antennas Propag., January 2001, 49, (1), pp. 91-103 44 FARINA, A., LOMBARDO, P., and PIRRI, M.: 'Nonlinear nonadaptive spacetime processing for airborne early warning radar', IEEProc, Radar Sonar Navig., February 1998,145, (1), pp. 9-18 45 FARINA, A., LOMBARDO, P., and PIRRI, M.: 'Nonlinear STAP processing', Electron. Commun. Eng. J., February 1999, pp. 41-48 46 RICHMOND, C. D.: 'Statistical performance analysis of the adaptive sidelobe blanker detection algorithm'. Proceedings of 31st Annual Asilomar conference, Signal, systems & computers, Pacific Grove, CA, 2-5 November 1997, pp. 872-876 47 TUFTS, D. W., KIRSTEINS, L, and KUMARESAN, R.: 'Data-adaptive detection of a weak signal', IEEE Trans. Aerosp. Electron. Syst., March 1983, AES-19, (2), pp. 313-316

48 JOHNSON, D. H. and DUDGEON, D. E.: 'Array signal processing: concepts and techniques' (Prentice-Hall, Englewood Cliffs, NJ, 1993) 49 PICCIOLO, M. L., GERLACH, K., and GOLDSTEIN, J. S.: 4An adaptive multistage median cascaded canceller'. Proceedings of 2002 IEEE Radar conference, Long Beach, CA, 22-25 April 2002, ISBN 0-7803-7358-8 50 ANTONIK, P., SCHUMAN, H. K., MELVIN, W. L., and WICKS, M. C : 'Implementation of knowledge-based control for space-time adaptive processing'. Proceedings of 1997 IEE international Radar conference, Edinburgh, Scotland, 14-16 October 1997, pp. 478-482 51 MELVIN, W., WICKS, M., ANTONIK, P., SALAMA, Y, LI, P., and SCHUMAN, H.: 'Knowledge-based space-time adaptive processing for AEW radar', IEEE Aerosp. Electron. Syst. Mag., April 1998, pp. 3 7 ^ 2 52 TECHAU, P. M. and BERGIN, J. S.: personal communication, March 2001 53 HIEMSTRA, J. D.: 'Colored diagonal loading'. Proceedings of 2002 IEEE Radar conference, Long Beach, CA, 22-25 April 2002, ISBN 0-7803-7358-8

Chapter 11

Adaptive weight training for post-Doppler STAP algorithms in non-homogeneous clutter1 Stephen M. Kogon

11.1

Introduction

The mission of a surveillance radar is the detection of moving targets, either airborne or ground-based, known as moving target indication (MTI). Traditionally, MTI radars have been implemented with a pulsed waveform using antenna reflectors or phased arrays with a limited number of spatial channels. Future systems, however, will most likely be implemented with antenna arrays with a large number of spatial channels to greatly enhance detection performance, especially in the presence of interference, through adaptive processing. The spatial channels formed from the antenna array, either beams or subarrays, provide angle measurements, and pulses are used to extract velocity information on potential targets. For each pulse, the radar returns are digitally sampled in time where each sample corresponds to a range or distance from the radar. The entire collection of spatial, pulse and range samples makes up a three-dimensional datacube. The key challenge to an MTI system is the suppression of interference, either due to other transmitted signals, such as jamming, or from the ground reflections of the transmitted radar signal, known as clutter. The fundamental problem that ground clutter presents is that, despite the fact that it is not physically moving, it has a nonzero Doppler frequency due to the platform motion of the MTI radar. As a result, the detection of moving targets that share the same Doppler with ground clutter is severely impaired without adaptive clutter mitigation. The physical characteristic that can be exploited, however, is the fact that ground clutter with the same Doppler as that

1

This work was supported by the DARPA under Air Force Contract # F19628-00-C-0002. Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by the United States Government

of a moving target has a different angle of incidence. Thus, by adaptively combining elements and pulses, clutter can be cancelled by placing a null in angle and Doppler. Two-dimensional adaptive processing of spatial channels and pulses is commonly referred to as space-time adaptive processing (STAP) [1—3]. STAP computes an adaptive weight vector for every beam (angle) and Doppler frequency bin for which a target detection hypothesis is to be formed. Since the exact clutter characteristics, i.e. array response and power level, are not known a priori, the STAP weights must be estimated from the data and the processing is thus data adaptive. Since the adaptation is performed in angle and Doppler, training is done using range samples assuming that angle/Doppler characteristics of clutter remain unchanged over the training samples.2 Although, training methods are often specific to a particular application, generally STAP weights are computed for an entire range interval or for all ranges. The motivation is not only to minimise computations but to also make the ensuing angle estimation problem tractable.3 Primary performance considerations for selecting a training strategy are clutter mitigation and detection of slow moving targets, also referred to as minimum detectable velocity (MDV). Since performance of a training method is highly dependent on actual clutter characteristics, it is crucial to apply any proposed method to experimental data to assess performance. STAP simulations, although intuitive, are less reliable since clutter characteristics are often very difficult to simulate accurately. One difficulty associated with training STAP weights is the non-homogeneous nature of clutter power4 [7]. Many times sharp peaks are present in the level of clutter returns that are a great deal stronger than the average clutter power, often referred to as clutter discretes. The result of training over all clutter is that these clutter discretes may be under nulled by STAP since the weights are computed for the average clutter level in the training cells. The resulting STAP weights are diluted by the averaging process and the null on clutter is not deep enough to effectively mitigate the clutter discretes. A training method aimed at combating the non-homogeneity of clutter and specifically for the mitigation of false alarms due to clutter discretes is power-selected training [8, 9]. Here, the strongest clutter returns are used for training to set the depth of the null based on these strongest returns instead of on the average clutter level. Although other weaker clutter is over-nulled, the consequence in terms of detection performance is usually negligible.5 Another problem can arise when basing the inclusion of the snapshots in STAP training on power level alone. If targets are present that may be either strong or at a Doppler frequency with weak clutter levels, these targets may

2 Certain radar scenarios can produce non-stationary angle/Doppler characteristics of clutter for which compensation methods are available that can approximately remove this non-stationarity [4-6] ^ When STAP is employed, angle estimation algorithms should account for the non-deterministic adaptive response of the STAP weights, particularly for slow moving targets near or in mainbeam clutter Non-homogeneous clutter refers to the power of the clutter returns. It does not indicate a nonstationarity of clutter angle/Doppler characteristics in range as occurs for cases when the platform motion vector does not align with the array axis, e.g. forward-looking radars Slow moving targets that compete with mainbeam clutter may suffer more loss due to over-nulling clutter which can slightly degrade minimum detectable velocity

be included in a training strategy if power is the only selection criterion. The effects of including targets in the STAP training set are potentially significant target self-nulling and an overall degradation in clutter cancellation performance. Both of these effects result in compromised detection capability. In this chapter, we introduce a new technique which attempts to solve the problem of targets included in the STAP adaptive weight training for power-selected training. Since any moving target produces its own non-zero Doppler contribution that adds to the radar-platform-induced Doppler, a target with the same Doppler as clutter must come from a different angle. Since for a given Doppler frequency, clutter comes from a common angle for all range cells over which STAP weights are trained, any significant deviation in angle indicates that the cell contains significant non-clutter returns. To avoid the deleterious effects of including targets in STAP training, a snapshot with significant target contributions should be excluded from the adaptive weight training. A method of excluding targets is based on angle or phase. One means of estimating the angle of a return is to measure the phase difference between antenna phase centres across the array. Since angle directly corresponds to phase on the array, this measurement is essentially an interferometric angle measurement whose quality improves with the strength of the incoming signal, i.e. those snapshots selected based on power. If the measured phase differs significantly from the expected phase (angle) of clutter for the given Doppler bin, it should not be included in STAP training. Using such an additional phase criteria, strong target signals are excised from the STAP adaptive weight training. The proposed technique is successfully demonstrated on experimental data collected with an airborne radar sensor.

11.2

Training of STAP algorithms

Space-time adaptive processing (STAP) has become a major focus of research activity in radar applications over the past decade [1, 3]. For airborne radars, the detection of moving targets, either airborne or ground-based, is impeded by the returns from ground clutter. Using an antenna array and a series of pulses, an airborne radar has the ability to spatially discriminate with the sensor array and distinguish based on Doppler frequency using the set of pulses. For a moving target with a given Doppler, the radar must be able to separate this target from clutter with the same Doppler frequency. The velocity of targets is determined by measuring both the spatial location (angle) and Doppler frequency of detected targets. STAP is the two-dimensional adaptive processing in angle and Doppler employed for the purposes of clutter mitigation to enable the detection of moving targets. For the STAP processing of radar signals, we want to emphasise signals that arrive at the radar from a certain angle and Doppler frequency, while rejecting all other significant energy. For this chapter, we assume a linear array of uniformly spaced elements. We begin by defining the space-time steering vector for a signal arriving from an angle 4> with a Doppler frequency / v(0,/) = b(/)®a(0)

(11.1)

where b ( / ) = [1 eJ2nf/f™ . . • GJ27u(L-l)f/fPRf

(1L2)

a(0) = [1 e i 2 ^ s i n 0 A ) _

(11.3)

% Qj2n(M-l)(dsm(f)/k)^T

are the temporal and spatial steering vectors, respectively. The radar wavelength is X9 the physical spacing between elements is d, L is the number of pulses in a coherent processing interval (CPI) and M is the number of spatial channels, /PR is the rate at which the radar system transmits and receives pulses, known as the pulse repetition frequency (PRF). Likewise, the time between pulses is the pulse repetition interval (PRI). Note that in many applications the antenna array must have a large aperture to achieve the necessary angular resolution. In such cases, only a limited number of spatial channels are digitised for adaptive processing. A common method of reducing the number of spatial channels is to form subarrays of the full aperture by combining the A/2-spaced elements from a section of the array prior to forming the digital channels. Radar returns from spatial channels and pulses are often represented by a spacetime data vector x(n) for a given range sample n. A space-time snapshot containing a target signal is given by: x(n) = a t v(0t, ft) + x i+n (n)

(11.4)

where Q^, 0t> and /t are the target amplitude, angle, and Doppler frequency, respectively. Xi+n is the interference-plus-noise signal and n is the snapshot index. Here, we only consider interference consisting of ground clutter returns from the transmitted radar signal. The optimum space-time processor is given by References 1 and 3 (p. 118): (11.5) where Q i + n = E{xi +n (n)xi +n (n) // } is the interference-plus-noise covariance matrix. The space-time steering vector v((/>o, /o) determines the angle and Doppler frequency of interest. Note that the Doppler frequency of a clutter patch is given by: /c = ^ s i n 0 c

(11.6)

A

where t>p is the radar platform velocity and c is cone angle of the clutter patch with respect to both the velocity axis of the radar platform and the array assuming a side-looking radar with the array axis aligned with the platform velocity vector [1,3, Chapter 3]. The performance of the optimum space-time processor from equation (11.5) is measured via the output signal-to-interference-plus-noise ratio (SINR). As the name implies, this is simply the output signal power divided by the interference-plus-noise

power: (11.7) where a^ is the signal power at the element level. Many times, we want to compare the SINR to the maximum SINR that could possibly be achieved. This upper limit is determined by the ideal matched filter for the interference-free case, i.e. thermal noise only. Normalising the SINR by the SNR of the ideal (interference-free) matched filter SNR0 yields: (11.8) which is known as SINR loss. Many times SINR loss is computed across angles and/or Doppler frequencies. An SINR loss of unity (0 dB) indicates perfect interference cancellation or no loss due to the presence of clutter. A processor is evaluated by how closely it comes to achieving this goal. Suboptimum partially adaptive processors are judged on their performance with respect to the optimum processor. Another common STAP metric is to determine the minimum velocity with an acceptable SINR loss, e.g. LSINR = —5 dB, often referred to as minimum detectable velocity (MDV). Since the optimum STAP weights are not known, they must be estimated from data samples or snapshots. Thus, STAP weights are computed using a sample covariance matrix given by: (11.9) /V—1

where Xtr = [x, r (l)x, r (2). -.xtr(K)]

(11.10)

is the data matrix of STAP training data samples. The training samples xtr(k) are selected from the overall set of radar returns x(n) for n = 1 to N. When Rx is substituted for Q i + n into the STAP weight equation from equation (11.5), we then have STAP weights given by: (11.11) This technique is commonly referred to as sample matrix inversion (SMI) [12]. The assumption made with this substitution of the sample covariance matrix is that the training samples in equation (11.10) are clutter-plus-noise snapshots and do not contain target signals and have clutter with the same angle/Doppler characteristics. Two major considerations go into the choice of data samples for STAP training: how many samples are necessary to get a good estimate of the covariance matrix and which data samples share the same statistics with the data sample to which the STAP weights are to be applied. Many times when the statistics of the clutter vary rapidly, these

two requirements are at odds with one another. We want to use as many samples as possible in order to obtain a good estimate of the covariance matrix that minimises the estimation loss [12]. On the other hand, the use of snapshots over a limited range near the data sample to be processed is desirable from the point of view that these samples most accurately reflect the local interference-plus-noise statistics.

11.3

Post-Doppler STAP algorithms

In practice, STAP usually must be implemented using a reduced degrees of freedom architecture that uses a deterministic preprocessor prior to adaptive processing to project into a reduced dimension subspace. The resulting algorithms are known as partially adaptive STAP [1,3, Chapter 5]. In the spatial dimension, the full array data can be reduced to either a set of subarray channels or a set of non-adaptive beams spanning the angular sector of interest. In the time domain, Doppler filtering is performed on the pulses prior to STAR This resulting class of algorithms is known as postDoppler STAR Since clutter comes from potentially all angles, performing Doppler filtering can reduce the STAP problem to cancelling clutter from the same Doppler frequency arriving from a different angle than the target, as shown in Figure 11.1. The viability of such a post-Doppler STAP approach is predicated on the assumption that the Doppler sidelobe levels are low enough to reduce clutter from other Doppler frequencies to below the thermal noise floor. Note that although interfering clutter may only be from a certain angle for a given Doppler frequency, limitations on Doppler resolution result in a clutter spread in angle over the Doppler frequency bin

Doppler

angle/Doppler resolution cell of STAP weights

STAP nulls

angle

Figure 11.1 Post-Doppler STAP approach with Doppler sidelobe suppression of mainbeam clutter

repeat for every Doppler pulses

spatial channels

DFT

degrees of freedom

Figure 11.2 PRI-staggered post-Doppler STAP algorithm block diagram and still necessitate spatial and temporal DoFs to cancel for two-dimensional adaptive processing, i.e. STAR Post-Doppler STAP algorithms are typically implemented with a DFT filter bank for the Doppler filtering component. Among these algorithms, the PRI-staggered post-Doppler STAP algorithm has been shown to be very effective, achieving near optimal performance [11]. A block diagram of the PRI-staggered STAP algorithm is shown in Figure 11.2. This algorithm uses two highly overlapped windows over pulses to produce staggered or delayed bins at the same Doppler frequency. The algorithm can be viewed as a frequency-domain (Doppler) adaptive filter using a time tap at a delay equal to an integer multiple of the PRI. The implementation requires two independent DFTs over the two highly overlapped windows over pulses. The delay between the two pulse windows, i.e. the pulse spacing, is known as the stagger and is ideally tuned to the PRF as in displaced phase centre array processing. This delay between bins is a temporal degree of freedom that STAP uses to suppress clutter. This algorithm was used for the experimental STAP results obtained in Section 11.5 of this chapter.

11.4 Phase and power-selected training for STAP The goal of any STAP training method is to determine the space-time snapshots x(A:) to use in the STAP data training matrix in equation (11.10). Many times, the assumption is made that ground clutter is homogeneous. In practice, clutter power levels can vary a great deal over range. The result of these non-homogeneities is that although clutter averages out to a certain power level, the resulting STAP weights computed with data samples over a large range extent will only have a clutter null deep enough to cancel clutter with the average power level. When the clutter contains large values, known as clutter discretes, the result will be under-nulled clutter which

in turn produces false alarm detections in the STAP output. Power-selected training is a data-adaptive STAP training method that attempts to combat this problem [8,9]. By using the strongest returns for each Doppler bin for training, STAP weights are based on the strongest clutter and, therefore, set the null depth according to the strongest clutter. Range cells with weaker clutter returns, in turn, have clutter interference over-nulled, although with little impact on SINR loss. One major problem with power-selected training arises when actual target signals are strong. Strong targets can arise from targets being at close ranges with respect to the radar or from large radar cross sections. Including targets in the STAP training can result in significant target self-nulling due to mismatches of the target space-time response and the assumed STAP steering vector [13]. In addition, since STAP is typically using reduced degrees of freedom, clutter cancellation performance suffers as a result of devoting a degree of freedom to the target in the STAP training data. Using power-selected training only compounds this problem since strong targets are almost guaranteed to be included in the STAP training. This problem is magnified when these targets are in Doppler bins with low clutter levels. Therefore, a desirable STAP training methodology would be to include strong clutter cells for maximum clutter nulling while recognising that any cells that contain targets need to be excluded from the training. This philosophy is what is behind phase and power-selected STAP training. For a given Doppler cell, clutter is expected to come from a certain angle given by: sin0c =

^_/

(11.12)

where / is the Doppler frequency, k is the wavelength and t>p is the radar platform velocity. Since angle translates to phase across the array, the expected relative phase of clutter is: Ax XJf0 = In—sin0c

(11.13)

where Ax is the physical distance between spatial phase centres. In contrast to ground clutter, moving targets have their own velocity which induces a Doppler frequency shift in addition to that Doppler produced by the motion of the radar platform. The angle of a moving target at the Doppler frequency / is therefore: sint= X' f~2vt 2vp

(11.14)

where vt is the radial velocity of the target with respect to the radar. Therefore, an effective means of discriminating between clutter and targets is to examine the phase between elements or phase centres to determine if the return has an angle of incidence consistent with the expected angle of clutter. If not, then this cell should be excluded from the STAP training.

For each potential STAP training cell within each Doppler bin, the phase can be estimated between two phase centres as: \Ir = arctan

(11.15)

where x\ (n) and x2(n) are data samples for the two phase centres on the array in the STAP Doppler bin for time or range sample H. Note that this STAP training method based on both phase and power is fairly robust to target amplitude. Since the accuracy of such an interferometric angle measurement is tied directly to signal-to-noise ratio, strong targets yield a significantly different phase than clutter. Range cells with only a weak target, on the other hand, are rejected for training on the basis of power via the power selection criterion. In addition, this method is computationally inexpensive since it only requires measuring a phase difference across the array for potential STAP training cells. Extensions of this method to other techniques that use angle to identify valid clutter training cells, especially for arrays with a large number of phase centres, is possible. The final STAP weight vector for a Doppler bin is then shared over a large range extent or even by all range cells. Note that target excision from STAP training data based on this phase criterion falls into a larger class of non-homogeneity detectors [7]. The combination of the power and phase selection criteria, however, has a different goal than these other non-homogeneity detectors [7]. These methods, which include the generalised inner product and an initial SMI test statistic metric, remove snapshots from STAP training that differ from the sample co variance matrix formed from the remainder of potential training cells. A relative distance between a potential training cell and the sample co variance is measured using the specified metric. In all of these metrics, however, no distinction is made based on the type of non-homogeneity, i.e. power or phase. Moving targets are removed from the training set based on phase-induced non-homogeneity from the target Doppler. Likewise, a snapshot containing the return from very strong clutter will fail these up-front tests based on the power non-homogeneity. The removal of strong clutter returns from STAP training is in contrast to the goal of power-selected training which seeks to include them. As a result, comparison of these STAP training methods with the phase and power-selection method outlined here is not appropriate since the methods are designed with different objectives.

11.5 Experimental results In this section, the proposed phase and power-based STAP training technique is demonstrated on experimental data collected with an airborne sensor. The aircraft had an altitude of approximately 5500 m and a velocity of 130 m/s. The radar was at X-band ( / = 10 GHz) with a PRF of/ P R = 1338Hz. The CPI length was 131 pulses or approximately 100 ms. The array consisted of three spatial channels formed with uniformly-spaced subarrays or phase centres each with a 0.6 m aperture. The subarrays have an overlap of 50 per cent. The experiment consisted of three ground moving vehicles driving on a road network with the range to the radar of approximately

20 km. The vehicles were a jeep, a pick-up truck and a long tractor-trailer. In addition to the road network, the experiment site also contained several large buildings that provided plenty of opportunity for large clutter discretes. Before looking at actual STAP results, we demonstrate the procedure for using the phase and power selection criterion using experimental data.

11.5.1 Example of phase/power selection We demonstrate the use of power and phase selection criteria for STAP training data with the experimental collection. Figures 11.3 and 11.4 show power and relative phase versus range for Doppler bin 42 (/dopp = —230 Hz) where a 128-point DFT was used for the Doppler filtering stage. The relative phase is measured between the first and third phase centres (subarrays). For this Doppler bin, the expected phase of clutter is approximately 150 degrees. Instead of relying on a predicted phase, however, we can use the median to estimate the clutter phase for this Doppler bin. Since clutter is present at all range cells, targets will produce phase outliers and the median is a robust estimator of mean in the presence of these outliers. An estimated phase from the data is actually more robust than relying on a predicted phase due to uncertainties that might arise in the data, e.g. unknown array orientation such as with aircraft crab. Next, a threshold is set around this median phase. Any samples with a phase outside of the acceptable range of deviations from the median phase are rejected or excluded from STAP training. These deviations in phase are due to either the presence of targets or clutter non-homogeneities. Also, many range gates may have a random phase due to noise when clutter power is weak. Using a tolerance of ±5 degrees, a total of 423

target # 2

power, dB

target # 1

range gate

Figure 11.3 Power versus range gate (Doppler =

-230Hz)

target # 2

phase, deg

target # 1

range gate

Figure 11.4 Relative phase versus range gate (Doppler = -23OHz) out of the 640 available range gates satisfy the phase criterion for inclusion in STAP training for this Doppler bin. Two target returns are actually present in this Doppler bin as illustrated in both Figures 11.3 and 11.4. Since these targets are effectively removed from the training based on phase, they cannot impact STAP performance. Once the phase criterion has been satisfied, power-selected training can be employed to combat strong clutter discretes, using the strongest cells that also satisfied the phase criterion to train STAP weights.

11.5.2 STAP results The STAP algorithm used was the PRI-staggered post-Doppler algorithm applied to the three-channel or phase centre array. A total of 131 pulses per CPI was used for the Doppler filter bank, with a — 65 dB Taylor taper and two staggers of 0 and 3 pulses (128 pulses per stagger). This staggering corresponds to two 128 pulse windows on the 131 pulse CPI similar to Figure 11.2 except that the stagger or delay between the two windows is now three pulses or roughly 2.2 ms. This delay was chosen to correspond to the time delay needed for one of the phase centres to fly into the other with the given PRF and platform velocity, commonly referred to as the displaced phase centre antenna (DPCA) condition. For PRI-staggered STAP, this stagger optimises STAP performance [H]. The array was approximately aligned with the velocity vector of the aircraft, i.e. no aircraft crab. With three phase centres and two staggers, the total degrees of freedom for this post-Doppler STAP algorithm is then 6. A total of 640 range cells was available for STAP weight training.

range gate

output power-to-noise, dB

Using the PRI-staggered post-Doppler STAP algorithm, we examine the performance of various STAP training methods. The performance of the proposed power and phase-based STAP training is compared with simple fixed target-free training and power-selected training. For all three methods, the number of training cells is fixed at 30 out of the total 640 which is five times the six STAP degrees of freedom (approximately a - 1 dB sample matrix inversion loss [12]). The STAP output is in the form of a range-Doppler power map for the beam steered to 0 = 0° (broadside) and is plotted in decibels. Doppler frequency has been converted to a target radial velocity hypothesis (at beam centre) in km/hr. The STAP normalisation is unit gain in the look direction (distortionless response), known as minimum variance distortionless response (MVDR), so that the zero velocity Doppler bin contains clutter at the expected clutter power level. Note that the data has been normalised up front so that the noise power level is set to 0 dB. Target-free training refers to using the last 30 range gates of each Doppler bin to compute STAP weights. The STAP training set is target free since it is known a priori that no targets are in these range gates. In practice, such an assumption cannot be made but these results are included to illustrate performance of STAP training without any considerations of clutter power levels. The outputs for target-free training (last 30 range gates), power-selected training and combined power and phase-selected training are shown in Figures 11.5, 11.6 and 11.7, respectively. Examining the STAP output from the target-free training, we see a large number of clutter discretes making it through STAP, especially at the Doppler bins corresponding to velocities between —30 and —20 km/hr. These range and Doppler cells correspond to the locations of buildings in the experiment producing large clutter discretes. Among these false alarms is also the actual target at —22 km/hr in range gate 280. Other false alarms are

velocity, km/hr

Figure 11.5 Range-Doppler STAP output for training on last 30 target-free range gates (broadside beam)

output power-to-noise, dB

range gate

velocity, km/hr

range gate

output power-to-noise, dB

Figure 11.6 Range-Doppler STAP output with power-only-based training selection (broadside beam)

velocity, km/hr

Figure 11.7 Range-Doppler STAP output with phase and power-based training selection (broadside beam) produced at approximately range gate 270 and velocity 18 km/hr and throughout the 6-7 km/hr Doppler bins. The remaining two targets are at range gate 300 and Doppler bin - 1 0 km/hr and range gates 205-225 and Doppler bins - 5 to - 1 0 km/hr. This last target is the tractor-trailer (extended in range) that is in a turn during the CPI. The different parts of this extended target produce different radial velocities and therefore spread over multiple Doppler bins in STAR Clearly, this high number of false alarms

is undesirable with more false alarms than actual detections. A high number of false alarms creates problems for subsequent tracking of target detections. For the power-selected training, the false alarms are cleaned up almost completely. However, since strong targets have been included in the STAP training, these targets have experienced a significant amount of self-nulling and clutter cancellation performance in these Doppler frequency bins/velocity hypotheses has been severely degraded. In fact, the target at —22 km/hr in range gate 280 has been almost completely removed. Also, for each of the Doppler bins that included targets in the STAP training, performance is degraded across all range due to the target self-nulling. The entire bin is corrupted from the high white noise gain of the STAP weights [14].6 By contrast, using both phase and power for STAP training selection criteria has successfully exposed all three targets by excluding them from the STAP training. Performance against false alarms due to clutter discretes is similar to power-selected training. Again, the three targets are at approximately —22 km/hr and range gate 280, — 10 km/hr and range gate 300, and between —5 and —10 km/hr and between range gates 205 and 225. Combining phase-based target excision from STAP training with power-selected training ensures adequate nulling on strong clutter returns helping STAP to realise the improved suppression of clutter discretes from power-selected training while avoiding the performance degradations associated with including the targets in the STAP training.

11.5.3 Experimental versus theoretical STAP performance Lastly, we compare the performance of STAP achieved on experimental data with the proposed phase and power-based STAP training to that predicted from theoretical STAP for a clutter-to-noise ratio of 25 dB (estimated from the data). Both the theoretical (dashed line) and experimental (solid line) SES[R losses, shown in Figure 11.8, reflect that the loss incurred by the Doppler taper (—65 dB Taylor) is approximately —3 dB. SINR loss is estimated from the experimental data using 200 target-free range gates. The result shows good agreement between experimental and theoretical SINR loss. A slight widening of the SINR loss notch at 0 km/hr is seen in the experimental data. Possible causes are array and channel errors as well as slight over nulling at low velocities. Note that internal clutter motion due to wind-blown clutter is not considered to be a likely culprit since the experiment site lacked any significant vegetation.

11.6

Summary

A new method for the training of STAP adaptive weights has been introduced and demonstrated on experimental data collected with an airborne sensor. The method 6

Note that, in general, white noise gain control methods are not usually considered for STAP since low MDV usually implies mainbeam nulling. White noise gain limits would hinder performance against low velocity targets

HP 'ssoi ^NLIS

Theoretical Experimental radial velocity, km/hr

Figure 11.8 Comparison of experimental (solid line) and theoretical (dashed line) SINR loss versus target velocity hypotheses uses a phase-based criterion, as well as power, to choose samples for STAP training and is considered as an improvement to power-selected training for post-Doppler STAP algorithms. The use of phase effectively removes targets whose inclusion in the training results in target self-nulling and a degradation in clutter cancellation performance. The new technique was compared with power-selected training on experimental data collected with an airborne radar using the PRI-staggered STAP algorithm. The performance improvement achieved over the power-selected training is significant as the target signals were properly excluded from the training set, yet the method still achieves effective mitigation of large clutter discretes. In addition, SINR loss estimated from the STAP applied to the experimental data was compared with a theoretically predicted SINR loss, showing good agreement over all Doppler frequencies.

References 1 WARD, J.: 'Space-time adaptive processing for airborne radar'. MIT Lincoln Laboratory TR 1015, ESC-TR-94-109, 1994 2 WARD, J.: 'Space-time adaptive processing for airborne radar'. Proceedings of IEEE international conference on A coustics, speech, and signal processing, 1995, pp. 2809-2812 3 KLEMM, R.: 'Principles of space-time adaptive processing' (IEE, London, England, 2002, 2nd edn.)

4 BORSARI, G. K.: 'Mitigating effects on STAP processing caused by an inclined array'. Proceedings of IEEE Radar conference, 1998, pp. 135-140 5 KREYENKAMP, O. and KLEMM, R.: 'Doppler compensation in forwardlooking STAP radar', IEE Proc, Radar Sonar Navig., October 2001, 148, (5), pp.253-258 6 HIMED, B., ZHANG, Y., and HAJJARI, A.: 'STAP with angle-Doppler compensation for bistatic airborne radar'. Proceedings of IEEE Radar conference, Long Beach, CA, 2002, pp. 311, 317 7 MELVIN, W. L. and WICKS, M. C : 'Improving practical space-time adaptive radar'. Proceedings of IEEE Radar conference, 1997, pp. 48-53 8 BORSARI, G. K. and STEINHARDT, A. O.: 'Cost-efficient training strategies for space-time adaptive processing algorithms'. Proceedings of Asilomar conference on Signals, systems, and computers, Pacific Grove, CA, 1995, pp. 650-654 9 RABIDEAU, D. J. and STEINHARDT, A. O.: 'Improved adaptive clutter cancellation through data-adaptive training', IEEE Trans. Aerosp. Electron. SySt., 1999, 35, (3), pp. 879-891 10 BRENNAN, L. E., MALLETT, J. D., and REED, I. S.: 'Adaptive arrays in airborne MTI radar', IEEE Trans. Antennas Propag., September 1976, AP-24, pp. 607-615 11 WARD, J. and STEINHARDT, A. O.: 'Multiwindow post-Doppler space-time adaptive processing'. Proceedings of 7th IEEE workshop on Statistical signal and array processing, Quebec City, 1994, pp. 461^-64 12 REED, I. S., MALLETT, J. D., and BRENNAN, L. E.: 'Rapid convergence in adaptive arrays', IEEE Trans. Aerosp. Electron. Syst., 1974, AES-IO, (6), pp. 853-863 13 COX, H.: 'Resolving power and sensitivity to mismatch for optimum array processors', J. Acoust. Soc. Am., 1973, 54, pp. 771-785 14 COX, H., ZESKIND, R. M., and OWEN, M. M.: 'Robust adaptive beamforming', IEEE Trans. Acoust. Speech Signal Process., 1987, 35, (10), pp. 1365-1376

Chapter 12

Application of deterministic techniques to STAP Jeffrey T. Carlo, Tapan K. Sarkar, Michael C. Wicks and Magdalena Salazar-Palma

12.1

Introduction

This chapter presents a deterministic least-squares adaptive signal processing technique for nulling interferers and estimating signals of interest (SoI) in the presence of clutter and transient electromagnetic interference. Unlike stochastic methods which rely upon auxiliary training data to estimate the statistics of the interference, in order to place nulls, this approach operates on a snapshot-by-snapshot basis to determine adaptive weights. Due to the reduced training data set (one range ring only - the range ring under test) and because of the particular least-squares techniques employed, this approach can readily be implemented in real time in fleld-deployable sensor signal processing systems. From a scientific standpoint, a deterministic method provides an estimate which comes closer to the Cramer-Rao bound than do the results provided by a stochastic methodology [I]. Another advantage of this method over conventional stochastic methods is that it operates effectively in highly non-stationary environments and is effective against fluctuating interferers as well1. In fact, this may be its greatest strength. The deterministic least-squares adaptive signal processing technique introduced here has application in many fields, including: communications, radar, sonar, medical imaging etc. The application addressed in this chapter is airborne radar, of which a description can be found in the literature [2,3]. In airborne radar surveillance systems, the detection of air and ground targets is complicated by many factors, as illustrated in Figure 12.1 [3]. In this Figure, the target signal competes with sidelobe clutter at the same Doppler frequency, in addition to the mainlobe clutter at the same angle, and jammers at potentially many different discrete angles, but spread over a wide band of frequencies. Changes in the velocity of the

Interferers fluctuating yet coherent with the radar signal

Figure 12.1 An airborne radar signal environment [3] airborne radar platform also impact the system performance. The Doppler spread of clutter is a function of platform velocity and the cosine of the angle of arrival (AoA) of the clutter energy with respect to the velocity vector of the platform. Therefore, an increase in the velocity of the platform reduces the clutter-free spectrum of the system and will degrade the minimum discernible velocity (MDV), which complicates the detection of slower moving ground targets. Compounding this are near-field airframe effects, in addition to size, weight and power constraints imposed by the platform, which limit the power aperture product of the radar (radar range equation effects). In addition to uncertainties introduced by the platform, the radar system must also contend with a severely non-stationary environment (consisting of jammers, multipath, clutter and discrete blinking interferences) while attempting to detect targets such as a low observable (LO) aircraft or objects employing concealment, camouflage and deception (CCD). One simple, albeit expensive, way to design a system that will detect these targets of interest is to have a very large aperture with very low sidelobes. This would provide a narrow search beam on transmit/receive, and low sidelobes to help mitigate interference. Beyond size constraints, one problem with this solution is that a low-cost, lightweight, air qualified array would be challenging to design and build and would be very expensive. Another solution, and the one discussed here, is to use adaptive space-time processing to suppress the interferences and enable the system to detect potentially weak target returns. In this analysis, we assume that the system consists of a linear array of A^ equally spaced antenna elements as shown in Figure 12.2 (with a digital receiver behind each element). Methods for handling directive antenna elements in a conformal array, including the effects of mutual coupling and near-field effects, are available elsewhere [1] and will not be discussed in this presentation. Here, d is the spacing between the antenna elements and O is the angle of incidence of the various signals impinging on the array with respect to the normal (the aperture is collinear with the platform velocity vector). Even though we are dealing with isotropic point radiators in this

Mh element

Figure 12.2 A uniform linear antenna array

pulses

snapshot

range bins

Figure 12.3 A radar data cube scenario, use of directive antenna elements can also be employed with a deterministic least-squares methodology, and is discussed elsewhere [1,4]. This configuration is comparable to that of a planar array, where elements in a column are combined using an analogue beamformer and, in turn, the output of the analogue beamformer is down converted and digitised. This analysis can easily be extended to the case where there is a digital receiver behind every element in the planar array. We also assume that the system processes a number of pulses (M) within a coherent processing interval (CPI). Each pulse repetition interval (PRI) consists of the transmission of a pulsed waveform of finite bandwidth and the reception of the reflected energy by the radar aperture and receiver(s), with a bandwidth matched to that of the radar pulse. In the receive chain, measured signals are down converted, matched filtered and stored, either digitally (IF sampling) or in analogue (baseband digitisation). In this manner, complex samples (/ and Q) are generated at R range bins for M pulses and at N elements (channels). To facilitate working with the array output the baseband samples can be arranged into a three-dimensional matrix commonly referred to as a data cube, as illustrated by Figure 12.3. The three axes of the data cube correspond to the spatial channel, pulse and range dimensions. At a particular range bin, n, the sheet or slice of the data cube is referred to as a space-time snapshot or simply the snapshot, as seen in Figure 12.3. We assume that the signals entering the array are narrowband, with a planar phase front, and consist of an SoI along with interference plus noise. The noise

(thermal noise) originates in the receiver and is understood to be independent across elements and pulses. The interference is external to the receiver and consists of clutter (reflection of the transmitted electromagnetic energy from terrain and sea, the atmosphere of the earth, including weather phenomena, etc.), jammers, signal multipath (due to the SoI, clutter, and/or jamming) and mutual coupling effects. We assume that for each jammer, the energy impinging on the array is confined to a particular direction of arrival (DoA) and is spread in frequency. The jammers may be blinking. The received complex envelope of the SoI at the nth element corresponding to the rath pulse is given by: a s e x p M 2 7 r n - s i n ( 0 ) + 27rm— M

(12.1)

where /d is the Doppler frequency of the signal, / r is the pulse repetition frequency (PRF), ofs is the complex amplitude and A is the wavelength at the centre frequency of the radar. Having stored the complex space-time-range samples, the adaptive signal processing algorithms can now be initiated. Traditionally, a range-dependent statistical approach to adaptive array processing has been taken [2,3,5-9]. Earlier approaches adjusted the spatial weights iteratively in an attempt to either maximise the signalto-noise ratio (SNR) [5] or minimise a mean square error [6]. Brennan and Reed showed [7] that in the case of Gaussian interference the space-time filter weights that maximise the probability of detection for a fixed false alarm rate, are given by: w

= k'№~1 -v*

(12.2)

where A: is a complex constant, 3fc is the covariance matrix of the interference, v is a steering vector and * represents the complex conjugate. This joint-domain (elementpulse) optimum processor requires knowledge of the statistics of the interference. Since these statistics are not known a priori in a fielded radar system, especially one in which the interference is rapidly changing, an adaptive approach is needed. One adaptive method, the sample matrix inversion method (SMI) [5], involves estimating the interference covariance matrix from measured radar data. This is achieved using the element-pulse returns from adjacent range rings which are assumed to be independent and identically distributed (HD) to the interference in the cell under test. This method of estimation uses the sample covariance matrix [8], which is written as: 1

n

»= -Ex^

( 12 - 3 >

where r\ is the number of range samples and x r is the received vector for the rth range ring. Here it is assumed that the training data does not contain any signal energy. It has been estimated [5] that approximately r] = 2£ HD samples are required to accurately estimate the covariance matrix, where £ is the order of the covariance matrix. It has been pointed out [3] that the number of required samples can be as high as three to five times £, depending on the environment. In an airborne surveillance radar, the clutter environment can be highly non-stationary and the aperture quite

large (in order to resolve long-range targets in angle). As such, the requirements on the training data sample size and HD conditions can severely limit the applications of such a methodology to radar. For this reason, researchers have investigated alternative approaches, which require less training data. One class of alternative processor to joint domain SMI is the factored or cascaded signal processing approach. In these algorithms, the raw data is Doppler processed first, and then adaptive beamforming is applied, or alternatively beamforming is followed by adaptive Doppler processing. These approaches perform one-dimensional processing sequentially (with adaptive processing in only the second dimension). This results in nulls being placed across the other (second) dimension. A partial taxonomy of various space-time processing methods is identified in the literature [3]. There is debate over which factored approach is superior. For the three different cases evaluated by Wang and Cai [9] the factored approach performed significantly poorer, or at best, equally to the joint domain approach, but under certain restrictive conditions. This poor performance by the factored approach, which requires a significant amount of secondary data, motivated Wang and Cai to develop an algorithm which operates in the joint transform (angle-Doppler) domain across a smaller dimensional space. This algorithm, the joint domain localised (JDL) processor, was thefirstpractical algorithm to show significant improvement over the factored approach. This JDL processor still requires training data, although significantly less than for other approaches. Another shortcoming of the factored approach includes poor performance in non-stationary environments. While many researchers were investigating suboptimum space-time adaptive approaches which relies upon estimating the underlying statistics using auxiliary training data, other researchers investigated deterministic ways to suppress interference. Luthra [10] proposed an algorithm which could null K-I coherent jammers, using data from N = 2 K — 1 elements (spatial information) while maintaining a fixed gain along the desired look direction. This approach was improved upon by Sarkar and Sangruji [11] using a new and simpler data matrix, and solved for the adaptive weights using the conjugate gradient method (CGM). Further improvements were made by Schneible and Park [12,13], where they implemented multiple constraints to preserve the SoI which might arrive slightly off the look direction (steering vector). Then Sarkar and coauthors [14-16] presented two more variations to this approach, the backward processor (BK) and the forward-backward (FB) processor. These algorithms were extended to the space-time domain, where Park and Sarkar [13,21] presented a generalised eigenvalue processor and Sarkar et al. [14-16] implemented the direct data domain least-squares (D3LS) processor.

12.2

Direct data domain least-squares (D3LS) approach, one dimension

The one-dimensional D3LS forward algorithm, developed in Reference 11, uses element space data and a look direction constraint to null out interference and estimate

the amplitude of the SoI. This algorithm is capable of nulling both coherent and noncoherent jammers/interferers. The number of nulls that can be placed is (N — l)/2, where N is the number of the receive antenna elements. A brief review of this method is provided below. Given the Af-element uniform linear array of Figure 12.2, assume that a desired sinusoidal signal arrives at an angle Os degrees from broadside (measured from the array normal) with amplitude of as. In addition, K — 1 far-field coherent jammer signals enter the array with angles of arrival #; and amplitude(s) <2/, i — 1,2,..., K-I, and clutter and thermal noise signals are also present. In this algorithm, array weights are computed based upon the samples taken across the array for a given pulse and range ring. Let Xn represent the baseband received signal (SOI, jammers, clutter and noise) arriving at the nth antenna element and is given by: K-I

{

d

\

Xn = a, eJWnd/k) sin(es) + J^ at exp f j2n — Sm(O1) J + cn

(12.4)

where Cn is the contribution due to clutter and thermal noise. The strength of the SoI, a s , is the desired unknown parameter which will be estimated using the snapshot of data, x, collected at time t. The vector of complex voltages measured at time t at the N elements of the array can then be represented as: x T = [Xl

X2

...

xN]

(12.5)

where T denotes the transpose of a matrix. Let X be a matrix, which contains data from the N elements of the vector x defined in equation (12.5). Let us define another matrix S whose elements comprise the complex voltages received at the antenna elements due to unity amplitude coming from the direction #s. The actual complex amplitude of the desired signal is a, which is to be determined. Then we form the matrix pencil using these two matrices, which are defined by: X-a-S

(12.6)

where

(12.7)

(12.8)

and the components of S, are defined by: (12.9)

Equation (12.6) represents the contribution of all the undesired signals, due to clutter, thermal noise, coherent multipaths and other interferences (i.e. all the undesired components except the signal). In the following method K = (N + l)/2 degrees of freedom {when Af is assumed to be odd} can be used to null the interference. One could form the undesired noise power from equation (12.6) and estimate a value of a by using a set of weights w, which will minimise the noise power. This results in [14-16]: (X-Qf-S)-W = [O]

(12.10)

Alternately, one can view the left-hand side of equation (12.10) as the total noise signal at the output of the adaptive processor due to interferences and thermal noise: N out - Q - W = { X - o f - S j - w

(12.11)

Hence, the total undesired power is given by: Undesired = WW{X - a • S]" • {X - « • S}w

(12.12)

where the superscript H denotes the conjugate transpose of a matrix. The objective is to set the undesired power to a minimum by selecting w for a fixed value of the signal strength a. This yields the generalised eigenvalue equation given by equation (12.10). Therefore: X - w = Qf-S-w

(12.13)

where of is given by the generalised eigenvalue and the weights w are given by the generalised eigenvector. Even though equation (12.8) represents a K x K matrix, the matrix S is only of rank one. Hence, equation (12.13) has only one eigenvalue and that generalised eigenvalue is the solution for the SoI. For real-time applications, it may be computationally burdensome to solve the reduced rank generalised eigenvalue problem efficiently, particularly if the number of weights and the dimension K are large. For this reason we convert the solution of a non-linear eigenvalue problem in equation (12.13) to a linear matrix equation which can be solved more efficiently. We observe that the first and the second elements of the matrix Q in equation (12.11) are given by: Qx9X

= Xi - o f -sx

(12.14«)

Ig 12 = X 2 -Qf-S 2

(12.146)

where xi and X2 are the voltages received at antenna elements 1 and 2 due to the signal, interferences and thermal noise, and si and S2 are the desired signals at these same elements due to an incident signal of unity strength. Defining: (12.15)

where Os is the angle of arrival corresponding to the desired signal. Then the following difference equation: Qi?i— Z" 1 • Q12 = undesired signals (= interference + noise)

(12.16)

removes the contribution due to the SoI, since: (12.17a) (12.176) Therefore one can form a reduced rank matrix [V](K-i)xK generated from Q. This matrix can then be set equal to a null vector to minimise the interference and noise:

(12.18) This matrix consists of N — K cancellation rows, for N = 2 • M — 1. In order to make the matrix full rank, we fix the gain of the subarray by forming a weighted sum of the voltages Ylf=\ w*x/ along the direction of arrival of the SoI. The gain of the subarray is C in the direction of Os. This provides an additional equation resulting in:

(12.19) or, equivalently: F w= y

(12.20)

Once the weights are solved using equation (12.20), the signal component as may be estimated from: (12.21) The proof of equations (12.20) and (12.21) is provided in Reference 11. As noted in Reference 14, equation (12.20) can be solved very efficiently by applying the fast Fourier transform (FFT) and the conjugate gradient method (CGM), which may be implemented to operate in real time utilising, for example, a DSP32C signal processing chip [17,18].

To solve equation (12.20), the conjugate gradient method starts with an initial guess wo for the solution and generates [18]: P 0 = -b-i • ¥H • r 0 = -b-i¥H

• {F • W0 - y}

(12.22a)

At the nth iteration the conjugate gradient method develops:

(1222b) (12.22c)

(I222d) (\222e) (12.22/) The norm is defined by: HF-PK II2 = p n " . F " . F . p n

(12.23)

The above iterative procedure continues until a prespecified error criterion is satisfied. In our case, the error defining the stopping criterion for the iteration is given by: (12.24) where £ denotes the effective number of bits of data associated with the measured voltages. Hence, the iteration stops when the normalised residuals are of the same order as the error in the data. The computational bottleneck in the application of the conjugate gradient method involves the various matrix vector products. That is where the FFT plays an important role as the equations involved exhibit a Hankel structure and therefore use of the FFT reduces the computational complexity by an order of magnitude without sacrificing accuracy [18]. The matrix vector products are done between a vector and the system matrix F which has a Hankel structure. The Hankel nature of a matrix always arises when one is performing a convolution. The conventional matrix-vector product usually requires an order of magnitude greater computation time than when performing the matrix-vector product using the FFT when the system matrix has a Hankel structure. The details are available in Reference 1. One advantage obtained using the conjugate gradient method arises out of the fact that the iterative procedure will converge even if the matrix F is singular. As such, it is suitable for real-time applications. Second, the number of iterations taken by the conjugate gradient method to converge to the solution is dictated by the number of independent eigenvalues of the matrix F. This often translates into the number of dominant signal components in the data. Thus, the conjugate gradient method has the advantage of a direct method as it is guaranteed to converge after a finite number of steps, as well as that of an iterative method, since the round off and truncation error in the computation are limited only to the last stage of computation (iteration).

Next we illustrate how to increase the degrees of freedom (DoF) associated with equation (12.20). It is well known in the spectral estimation literature that a sampled sequence can be estimated by observing it either in the forward direction (FW) or in the reverse direction [19,20]. This latter approach we term as the backward (BK) model as opposed to the forward model outlined above. If we now conjugate the data and form the reverse sequence, then we get an equation similar to equation (12.20) for the solution of weights w^ as:

(12.25)

where * denotes the complex conjugate, and C is the corresponding look direction constraint. Equivalently: B • wb = y b

(12.26)

The signal strength a can again be determined by: (12.27) Note that in both equations (12.20) and (12.26), dealing with the matrices F and B, we have K = (N + l)/2, where Af is the total number of antenna elements. We can increase the number of weights significantly by combining these approaches to produce the forward-backward model. In this way we exactly double the amount of data by considering the data not only in the forward direction but also conjugating it and reversing the increment direction of the independent variable as well. However, the number of weights increases only by a half. This approach can be applied so long as the series to be estimated can be approximated by an exponential function with a purely imaginary argument. This is always true in the adaptive array case. There is an additional benefit as well. For both the forward and the backward method, the maximum number of weights we can consider is (N + l)/2. Hence, even though all the antenna elements are being utilised, the number of DoF available is essentially half the number of antenna elements. For the forward-backward method (FB), the number of degrees of freedom can be significantly increased without increasing the number of antenna elements. For this case, the number of degrees of freedom [1] A'7 can reach (N + 0.5)/1.5. It is important to note, that for any method, be it deterministic or stochastic, this is the largest number of coherent interferers one can cancel using any method. For the stochastic method one needs a block of data to form the covariance

matrix where the same goal is achieved using a single snapshot. Thus one can have approximately 50 per cent more DoF for the FB case than for either of the two FW or BK cases. The equation in the combined forward-backward model is obtained using equation (12.20) and (12.26) to be:

(12.28)

or, equivalently: FB w ft = y ft

(12.29)

Once the increased degrees of freedom are used to compute the weights, the complex amplitude for the signal of interest is determined from equation (12.21) or (12.27), where in the summation K is replaced by the new degrees of freedom K''. Also, as before, the matrix FB now has a block Hankel structure. Thus we have illustrated how the direct data domain least-squares approach can be implemented in real-time using a single snapshot of data. For a stochastic method, when multiple steering vectors are available, one does not have to invert the covariance matrix as a simple multiplication is all that is necessary to get the desired weights from equation (12.2). In this case, one can carry out a similar procedure but the computations are involved, as the steering vector is part of the system matrix. Therefore, one can partition the system matrix into blocks and invert the dominant block of the system matrix which is still block Hankel and can do similar cost saving computations.

12.3

D3LS approach with main beam constraints

The D3LS approach defined in the last section is very effective in nulling interferers and estimating the SoI when there are (N + l)/2 interferers or less and the SoI arrives at exactly 6S, the look direction. In a radar system the main beam of the antenna is pointed in various discrete directions in search of targets. The target may not be at that exact angle. Also, due to atmospheric refraction, diffraction or vibration in the array, the assumed direction of arrival for the SoI may be in error. When the SoI

arrives slightly off the look direction, 0S, then the performance of the processor will be significantly degraded. One way to improve this is to add multiple constraints within the main beam [12], as a direct data domain method is a superresolution method. The manner in which this is done differs between the generalised eigenvalue processor and the least-squares process, but the effect is the same. In both cases, the additional constraints prevent the algorithm from treating the SoI as an interferer when the SoI arrives off the look direction, but within the a priori constraints of the adapted receive beampattern. Multiple constraints can be added to the generalised eigenvalue processor by modifying the S matrix [13]. The single constraint for this processor is given by equation (12.9), where the look direction is# s . Let#i = 0S, then additional constraints, #2? #3 etc., can be included by altering Sn such that: (12.30)

where L is the number of constraints. It is assumed that the weighting amplitude of each constraint is unity. Using equation (12.30) we can now form the generalised eigenvalue equation with multiple constraints as: (X-QfS 7 J-W = [O]

(12.31)

where the KxK matrix X is constructed using equation (12.7) and the KxK matrix S7 is constructed using equations (12.8) and (12.30). Multiple constraints can also be implemented in the D3LS approach [12,15]. Recall that the single look direction constraint is implemented in the first row of matrix F in equation (12.19). Additional constraints can be incorporated by adding multiple constraint rows to F and replacing the corresponding zeros in the excitation vector y equation (12.19) with non-zero entries. The format of these constraint rows is given by:

[l

Z1

Z\

Z3

...

Zf~l]

(12.32)

where I = 1 , . . . , L is the index number of the constraint. The format of the cancellation rows of the matrix F remains the same as in equation (12.19). For each constraint row added, a cancellation row must be removed so that the matrix F remains square. As defined earlier, the number of cancellation equations is given by N — K, where N is the number of elements and K is the number of weights. By adding L constraint equations the relationship between L, K and N becomes: K = L + N-K

=* N = 2K-L

(12.33)

Given a system with K weights and L constraints, the number of jammers that can be nulled is (K — L)/2. The price the processor pays for the additional constraints is that fewer jammers may be nulled for a given number of antenna elements N.

12.4

A D3LS approach with main beam constraints for space-time adaptive processing

The adaptive algorithms that have been presented up to this point have used data from a single space snapshot, which consists of samples from the antenna elements across the array at a particular instant in time. In this section we present four different algorithms that operate across pulses and elements, increasing the degrees of freedom over those of the element domain techniques. The first processor that will be described implements a generalised eigenvalue equation [1,16], and the last three processors implement a least-squares solution to a linear matrix equation [1,16]. Using the linear array defined in Figure 12.2, we now take advantage of the temporal diversity provided by the coherent pulses within a CPI, as well as the spatial diversity provided by the elements of the array. Using this data from a given range bin r as illustrated in Figure 12.3, the adaptive weights can be obtained deterministically, on a snapshot-by-snapshot basis, and implemented in the space-time architecture shown in Figure 12.4. Assuming a narrowband signal, the complex envelope of the received SoI with unity amplitude, for the mth pulse at the nth antenna element can be described by: (12.34)

antenna elements

receiver A/D

coherent pulses

receiver A/D

Figure 12.4 A two-dimensional space-time filter

for m = 1,2, . . . , M and n = 1,2, . . . , N. In this equation
(12.35)

where as is the complex amplitude of the SoI entering the array and S(m,n) is the received space-time snapshot due to only the SoI but of unity amplitude only. For each pulse-element cell (given range bin r) the difference equation: X(m9n) -asS(m9n)

(12.36)

removes the SoI from the nth antenna element and mth pulse data, leaving noise plus interference. Based on equation (12.36) a two-dimensional matrix pencil equation can be created whose solution will result in a weight vector that nulls interferers and extracts the SoI. The elements of this matrix pencil can be constructed by sliding a window (box) over the space-time snapshot as shown in Figure 12.5. By creating a vector using the elements in the window, each window position generates a row in the S and X matrices resulting in:

(12.37)

(12.38)

The window size along the element dimension is Na, and along the pulse dimension, Nt. Selection of Afa determines the number of spatial degrees of freedom, and Nt determines the temporal degrees of freedom. 7Va and Nt must satisfy the following

pulses

window #

elements

Figure 12.5 A space-time data snapshot equations for the single constraint case:

Wa < ^p-

(12.39)

N1 < Z±±

(12.40)

The product of the spatial and temporal degrees of freedom, Q, is: Q = NaxNt

(12.41)

Given the physical constraints (size, weight, power etc.) of the platform, most airborne radar systems contain more temporal degrees of freedom than spatial.

12.4.1 Space-time D^LS eigenvalue processor Using equations (12.37) and (12.38), the two-dimensional generalised eigenvalue system can then be constructed as: {X - as • S}w = [0]

(12.42)

Once again, the generalised eigenvalue provides the amplitude of the SoI, while the generalised eigenvector provides the weights necessary to suppress the interferers. Additional look-direction constraints can be added to this processor by modifying the signal matrix, S, in a manner similar to that for equation (12.30), except now constraints can be offset from the look direction in angle or Doppler, or both. The

constraint rows can be written as: (12.43) for n = 1, . . . , N and m = 1, . . . , M. Here Oi and ft determine the space-time look direction of the €th constraint. The generalised eigenvalue equation with multiple constraints can then be defined as: {X - a • S"} • w = [0]

(12.44)

where the Q x Q matrix X is constructed from equation (12.38) and the matrix S" is constructed by replacing Sm?n in equation (12.37) with S^ n in equation (12.43).

12.4.2 Space-time Z)3 LS forward processor The formulation of the D3LS space-time algorithm [17] can be obtained through an extension of the one-dimensional case. As before, we start by developing the forward case, and then present the backward and the forward-backward algorithms. For all the cases, the matrix equation to be solved is defined by: F w= y

(12.45)

where F is the system matrix and w is the vector of space-time weights which will null the interferers. The system matrix, F, contains the angle-Doppler information of the SoI, as well as the cancellation rows which contain angle-Doppler information describing the interference. This interference information is obtained through a difference equation similar to equation (12.16), where the contribution of the SoI is removed leaving only the undesired signals. For the two-dimensional case these difference equations are performed with elements offset in space, in time and jointly in space and time. Let us define the information off-set of the SoI in space and time, respectively, as: (12.46) (12.47) Again, the SoI has an angle of arrival of Os and a Doppler frequency of /d. The three types of difference equation for the FW, BK and the FB methods are then given by: (12.48) (12.49) (12.50) The cancellation rows of the matrix F can now be formed using equations (12.48) to (12.50) and using the windowing procedure depicted in Figure 12.5. In this case the dots in Figure 12.5 represent the induced voltages, X(m, n) in equation (12.35), for a

example: computing a cancellation row for Na = 3 and Np = 4 ^window 1

^ window 2 ^ 1

»

converting to a row vector

Figure 12.6

Creating a cancellation row in matrix [T]

given element-pulse location. Just as was done for the generalised eigenvalue algorithm, a space-time window is passed over the data. For each location of the window function, three rows in matrix F are formed by implementing equations (12.48) to (12.50), which removes the SoI. The rows are created by performing an element-byelement subtraction using the values contained within the offset windows and then arranging the resulting data into a row vector as illustrated in Figure 12.6. The window is then displaced one space to the right and three more rows are generated, and this procedure is continued along the space and time axis. After this window has reached the second column from the far right and three rows are generated, the window is reduced (by a row) and shifted to the left-hand side of the data array. The generation of rows continues, and is repeated until (Q-I) cancellation rows have been formed, where Q = Na x Nt. As in the one-dimensional case, the first row of the system matrix is used to set the gain of the system along the look direction. For the two-dimensional case the look direction is specified by the angle of arrival, 6S, and the Doppler frequency, /d, of the SoI. The elements of this row can be obtained by placing window #1 in Figure 12.5 over the matrix S of equation (12.37). Using the same method as above (Figure 12.6) the elements of the row constraint can then be properly arranged. This results in the following row vector:

(12.51)

In a similar manner, the elements of the weight vector, w, in equation (12.45) are arranged in the same fashion as the elements in the row constraints, except that the vector is a column vector. Setting the product of F and w equal to a column vector y then completes the matrix equation. The first element of y consists of the complex gain constant C, and the remaining Q-I elements are set to zero in order to complete the cancellation equations. The resulting matrix equation is:

(12.52)

In solving this equation one obtains the weight vector w which places space-time nulls along the direction of the interferers while maintaining a fixed gain in the direction of the SoI. The amplitude of the SoI can now be estimated from: (12.53) The above analysis is done for the single constraint case. As in the one-dimensional case, the SoI for the two-dimensional case could arrive at the antenna slightly offset in angle, in Doppler, or both. In order to prevent the processor from nulling the SoI, multiple constraints can be implemented. The added constraints would reduce the number of degrees of freedom, but given the antenna beam width and the Doppler filter width, the constraints could help to maintain system gain over this finite look direction. In a manner similar to that of the single constraint case, additional control can be implemented using L row constraints where the look direction of the €th row is determined by Og and ft. For the €th constraint we have: (12.54) (12.55) and the £th row of F, denoted as F (€,:), becomes:

(12.56)

12.4.3 Space-time D^LS backward processor A second direct data domain least-squares space-time processor can be implemented by conjugating the element-pulse data and processing this data in reverse. The basis for this algorithm, the backward two-dimensional processor, is given in References 16, 19 and 20. The form of this linear matrix equation is similar to that of the forward

algorithm and is given by: B-wb=yb

(12.57)

where B is Q x Q, and w and y are both Q x 1. The constraint rows in B are implemented in the same manner as the constraint rows in F. The difference between B and F is in the cancellation equations. For the backward method, these equations are formed by first conjugating the space-time snapshot illustrated in Figure 12.5. Then using a windowing procedure similar to that for the forward method three cancellation rows are generated for each position of the window, except that now the window starts in the lower right-hand corner of the space-time snapshot of Figure 12.5. This window is then moved to the right and up the snapshot. The three difference equations used to cancel the SoI are given by: (12.58) (12.59) (12.60) where X* is the complex conjugate of X. Using equations (12.58) to (12.60) the SoI is removed from the windowed data. This data is then converted from a matrix to a vector, as shown in Figure 12.6. Then the elements of the vector are reversed in order, to form the rows of B. A vector similar to y used in the forward method is also used in the backward method. For the single constraint case the only non-zero entry in yb will be the first element, which again is set to an arbitrary constant C The weight vector w b can now be solved by substituting B and yb into equation (12.57) and applying the conjugant gradient method. The estimate of the amplitude of the SoI is then obtained from: (12.61) For systems where the AoA and Doppler frequency of the SoI are not exactly known, but approximately known (e.g. within the mainbeam of the antenna and within a given Doppler filter), then multiple constraints can be implemented to preserve the SoI. The procedure for doing this is identical to that for the forward method. For each additional constraint a new equation replaces a cancellation equation in B and the corresponding amplitude is placed in y b . The constraint equations are generated using equation (12.56).

12.4.4 Space-time D 3 LS forward-backward processor The forward and the backward methods described in the previous sections both provided independent solutions for the problem through equation (12.52) and (12.57), using Q = Na x Nt degrees of freedom. The third D3LS processor, the forwardbackward processor, provides another independent solution to the space-time adaptive problem. By applying the data in both forward and backward manners,

one can increase the degrees of freedom by as much as 50 per cent over that of the forward or backward methods alone. Once again, one can use the samples from N antenna elements and M time pulses to formulate the following matrix equation: FB-Wfb^yfb

(12.62)

The system matrix FB consists of at least one constraint row and numerous cancellation rows. The constraint rows preserve the SoI during the adaptive process by defining multiple 0s — /d look direction constraints in which to maintain the SoI, as defined in equation (12.56). The remaining rows in FB consist of the cancellation equations that are formed in the forward and backward directions. The forward direction cancellation equations are defined by equations (12.48) to (12.50), and those for the reverse direction are given by equations (12.58) to (12.60). The important thing to remember is that for a given number of antenna elements TV and pulses M the selection of Na and N't for the FB method can be significantly larger for the forward-backward algorithm than Na and Nt used in either the forward or backward algorithm. Following the steps similar to the forward and backward methods, a non-zero element is placed in the vector yft, for each constraint row. Using the conjugant gradient method, the matrix FB and vector yfb, the weight vector Wft, in equation (12.62) can be determined. Having obtained Wfb, an estimate of the amplitude of the SoI can be computed using equatoin (12.53), with Na and Nt replaced by Na and N[9 respectively, and w by Wft>.

12.5

Determining the degrees of freedom

The spatial and temporal degrees of freedom (DoF) for the three direct data domain least-squares algorithms are constrained by the number of antenna elements and pulses, TV and M, respectively. Recall that for each given box position in Figure 12.5, three cancellation equations can be generated. Also, the number of possible box positions can be determined from the number of shifts in each dimension. Let Sa be the number of shifts in the spatial dimension and St the number of shifts in the temporal dimension, such that: N-Na

= Sa

(12.63)

M-Nx

= St

(12.64)

This results in a total of G possible box positions where: G = Sa x Sx

(12.65)

For both the forward and the backward methods the maximum number of cancellation equations possible is given by Nc: Nc = 3xG

(12.66)

Meanwhile, the number of elements for each cancellation row, and hence the number of elements in the weight vector, is Q which is given by: Q = NaxNt

(12.67)

In order to set up a system of equations with at least one solution, and possibly many from which a minimum norm solution can be chosen, Na and Nx must be selected such that: Na x Nx < Nc - L

(12.68)

where L is the number of system constraints, L > 1. Therefore, the number of spatial and temporal DoF for the forward or backward methods should be chosen such that: Na x Nx < 3 x (N - ATa) x (M - Nx) - L

(12.69)

The choice of N& and Nx for the forward-backward method is slightly different. For a system with N antenna elements and M coherent pulses the forward-backward method provides up to a 50 per cent larger product of degrees of freedom (PDoF) than either the forward or the backward method alone. In this case the choice of the spatial DoF, A^, and the temporal DoF, N[, can be chosen in the following manner. The number of elements in the weight vector is similar to that of equation (12.67), and is given by: Q' = N'a x N[

(12.70)

Likewise, the number of shifts in each dimension, Sa in the spatial and S[ in the temporal, is given by: Sa = K-N'a S[ = M-

(12.71) N[

(12.72)

Therefore the total number of possible box positions G' is given by: G' = Sa x St'

(12.73)

Since the data will be applied in both the forward and the reverse directions, the maximum number of cancellation equations possible for the forward-backward method is N'c, and is given by: N'c = 2x3xGf

= 6x(N-N'a)x(M-

N't)

(12.74)

Once again, in order to set up a system of equations with at least one solution, and possibly many solutions from which a minimum norm solution can be chosen, we must select N'a and N{ so that: K *N[
(12.75)

where L is the number of system constraints, L > 1. Therefore, the number of spatial and temporal DoF for the forward-backward method should be chosen such that: Nax

N{ <6x

(N - Na) x (M - N{) - L

(12.76)

12.6

An airborne radar example

12.6.1 Simulation setup The following example illustrates the utility of the D3LS space-time processing algorithms that have been presented in the previous sections for a real world scenario. We do not consider the stochastic algorithms in these cases because as it has been shown for the measured MCARM data sets [1,16] the performance of the direct data domain methods has superceded those of a stochastic-based methodology at a fraction of the computational cost. This scenario consists of a ground moving target indicator (GMTI) radar on board a medium sized unmanned air vehicle (UAV), such as the Predator shown in Figure 12.7 [22]. From a target detection point of view this type of platform is well suited for the GMTI mission due to its low velocity. The slow platform speed reduces clutter spectral spread, enhancing the system's ability to separate slow moving targets from the background clutter. On the other hand, the size and lift capability of the vehicle limits the radar system size and weight, providing a challenge to radar designers to not only improve performance versus weight of the hardware, but also that of the signal processing algorithms being implemented. Using the FAS website on the world wide web [22], the performance and dimensions of the airborne platform were obtained and from this an estimate was made of an aperture and radar system that might fit on such a UAV. Table 12.1 summarises the radar, platform, signal sources and geometric parameters for our sidelooking radar simulation. To simplify the implementation, the PRF was selected to keep clutter returns unambiguous. The clutter radar cross section, G0, in Table 12.1 was obtained from Figure 13.8 in Reference 23 for a side-looking radar, and applies to cultivated terrain at grazing angles at approximately 10 degrees for a horizontally polarised X-band radar. The size of the clutter area Ac can be estimated using: Ac = R • 0 3 d B • AR - secant (grz)

Figure 12.7 An example of the Predator UAV

(12.77)

Table 12.1

Parameters related to the simulation of a side-looking radar

Radar Frequency Wavelength Bandwidth Comp. pulse width Orientation Pulse rep. freq. Antenna dimensions Rx beam width, #3 ^B Doppler filter Range resolution

9.5GHz -0.0316 m 10 MHz 100 |xs side-looking 4 kHz 4' x Y ^ 0 . 1 rad ^ 6 deg ^444 Hz, 222 Hz 15m

Platform Altitude Radar velocity Target velocity Target RCS

17.4kft^5.3km 60 knts ^ 3 1 m/s 20 mph ^8.9 m/s 10 m 2

Signals SoI: SNR AoA Interferers discretes: Doppler

12.5 dB 65 deg

AoA Jammer: AoA Clutter: extent Nature of clutter: point scatterers

[400, - 8 0 0 , 1700, 1400, 325, -1650, 950, - 1 2 0 0 , - 1 2 5 , 145O]Hz [85, 120, 40, 35, 65, 120, 100, 65, 55, 125] deg 100 deg main beam separated by 0.06 deg in angle from each other

Geometry Range Grazing angle Clutter area (J0 Clutter RCS

30 km 10 deg 37 967 m 2 -26.2 dB 17.3 dBsm

where #3 $Q is the three dB beamwidth, AR the range resolution and grz the grazing angle. The radar cross section of the clutter, a c , can then be approximated by: Cf0 = Cr0- A

0

(12.78)

Using the parameters in Table 12.1, the computed clutter radar cross section (RCS) is 17.3 dB sm. The assumed target RCS is 10 m 2 , resulting in a per pulse signal-to-clutter ratio (SCR) of approximately —7 dB.

The received signals modelled in this simulation consist of the SoI, main beam clutter, multiple interferers, jammer and thermal noise. The clutter is modelled as point scatterers placed approximately every 0.06 degrees apart. The amplitude of the clutter points are modelled with a normal distribution about a mean that results in a signal-to-clutter ratio, SCR, of — 7.3 dB. Ten strong point scatterers are modelled in this simulation, based on the AoA and Doppler parameters in Table 12.1. Thermal noise generated in each receive channel is independent from channel to channel, and consists of a random phase and amplitude. This phase is uniformly distributed between 0 and lit. The resulting SNR is approximately 12.5 dB per receive channel per pulse. The jammer is modelled as a broadband noise signal that arrives from 100° in azimuth and covers all Doppler frequencies. Summing the power of the interfering sources received by the first channel and comparing it with the power of the SoI, the average input signal-to-interference-plus-noise ratio (SINR) is estimated to be — 32 dB. Since the algorithms developed in this chapter require a single space-time snapshot, only data from the range cell at 30 km, where the target resides, are generated. Given the size of the platform, and the cost and weight per channel, the number of receive channels is limited to ten. The cost of temporal DoF is not as expensive as spatial DoF. Initially, 16 coherent pulses are integrated and the processing performance is computed using the parameters in Table 12.1. Then the total number of DoF is enhanced by increasing the number of integrated coherent pulses, and the change in performance is noted. For the first case, N = 10 antennas and M = 16 pulses have been used in estimating the performance of the three D3LS algorithms, and is evaluated based upon the adaptive weight pattern and output SINR. The adaptive weight pattern is computed by taking the magnitude square of the discrete Fourier transform (DFT) of the resulting weight vector. In a similar manner, the output response pattern is computed using the weighted input data. For case I, only one constraint equation will be used to maintain gain on the SoI, and the steering vector (constraint) is aligned with the SoI vector. As will be seen from the results of case I, the forward and backward methods have very similar performance. Therefore, for the second case, N = 10 and M = 32, only the forward and forward-backward methods will be used. The three subcases that will be evaluated for case II are: (i) target aligned with the steering vector using a single constraint; (ii) target and steering vector misaligned using a single constraint; (iii) target and steering vector misaligned using multiple constraints.

12.6.2

Case I: single constraint space-time example

For the first case, Af = 10 and M = 16, the forward and backward methods utilise 7 spatial and 9 temporal DoF resulting in a product of the degrees of freedom (PDoF) of 63, and the forward-backward method employs 8 spatial and 9 temporal DoF, for a total of 72 PDoF. In this example we will look at the performance of each of the processors for the case where the SoI is aligned with the spatial and temporal steering vector, using just one constraint. The power spectrum of the input signals is shown in Figure 12.8. Here, the green circles indicate the locations of discretes, the green bar of circles is the main beam clutter and the blue circle locates the SoI. The

angle, deg

frequency, Hz

Figure 12.8

The power spectrum of the input signal2

adaptive weight and output patterns for the forward, backward and forward-backward methods are shown in Figures 12.9, 12.10, and 12.11, respectively. As seen in these three Figures, each of the processors performs well in generating nulls to mitigate discretes and the main beam clutter. The processors also place a deep null along the location of the jammer, at an AoA of 100°. While nulling these interferers, the system gain is maintained along the look direction, as can be seen by the strong response of the weight pattern centred on the blue circle. As mentioned above, the FW and BK methods employ 63 PDoF and the FB method utilises 72 PDoF. The improvement due to the increased PDoF is evident when comparing the weight patterns of Figures 12.9 and 12.10 with that of Figure 12.11. The forward-backward processor generates a deeper null, more in line with the clutter ridge, than does either the forward or backward processor. The performance of these processors can also be compared using the output signal-to-interference-plus-noise ratio (SINRoUt). We define the output signal-to-interference-plus-noise ratio SINRoUt as: (12.79) where A8 is the amplitude of the SoI and as is an estimate of the amplitude of the SoI. Using equation (12.79) for case I, the output SINR, averaged over 50 runs, is 40.5 dB 2

This figure is available in colour at www.ieee.org/mvieee/chl 2_figures

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.9

Output of the forward method2 a adaptive weight pattern b output response pattern

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.10

Output of the backward method2 a adaptive weight pattern b output response pattern

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.11

Output from the forward-backward method2 a adaptive weight pattern b output response pattern

forthe forward method, 40.7 dB forthe backward method and 43.0 dB forthe forwardbackward method. As mentioned earlier in this chapter, and shown here by the output SINR and adaptive weight patterns (Figures 12.9 and 12.10), the performances of the FW and the BK methods are very similar. Therefore, in the next section, we will focus on the comparison between the FW and FB methods, with the understanding that the BK method yields results comparable with that of the FW method.

12.6.3

Case II: multiple constraint space-time example

In this case the temporal DoF is increased to M = 32 and the performances of the forward and forward-backward processors are evaluated for the misaligned look-direction condition using various (numbers of) constraints. Each additional look-direction/Doppler constraint replaces a cancellation row. Therefore, maintaining adequate interference nulling while increasing the number of constraints requires an increase in the PDoF. In the first subcase the look direction is misaligned with the SoI and only one constraint is utilised. The SoI is located at an AoA of 63° with a Doppler frequency of 1278 Hz, and the system constraint is set at an AoA of 65° and a Doppler of 1378Hz. The adaptive weight pattern is shown in Figure 12.12 for the forward processor, and in Figure 12.13 for the forward-backward processor. In both Figures, the cyan circle indicates the location of the constraint, the dark blue circle indicates the location of the SoI and the dark blue rings identify the location where additional constraints will be implemented in the following subcases. With only one constraint, both processors treat the SoI as an interferer and attempt to null it, even though the SoI is arriving in the main beam of the radar (both spatially and temporally). The resulting output SINR is —33.7 dB for the forward processor and —39.3 dB for the forwardbackward (FB) processor. Once again, the additional degrees of freedom enable the FB processor to do a slightly better job of nulling the interference, which in this case includes the SoI. However, the performance of either method is less than desirable. The next subcase involves the misalignment of the SoI with the look-direction constraint, along with four additional constraints. Results for this five-constraint case for the FW method are shown in Figure 12.14. Once again the constraint locations are identified by the cyan coloured circles, and the dark blue circle marks the location of the SoI. The four additional constraints help keep the forward processor from placing a null on the SoI, through the main beam. Yet, with only five constraints, one at the centre and the others at four corners of a rectangle in the main beampattern, the adapted main beam weight pattern is protected, although slightly eroded along one edge. The resulting output SINR is —1.8 dB, 32 dB better than the single constraint case. This situation can be improved further by applying additional constraints. The results from the misaligned case using nine constraints in the FW method are shown in Figure 12.15. The four additional constraints help to maintain a more uniform response across the main beam, which keeps the processor from degrading the weight pattern along the edges. The output SINR for this case is 1.2 dB, a 3 dB improvement over the five-constraint case, and a 35 dB improvement over the single constraint case. Similar trends were obtained for the FB processor when going from one constraint to five constraints to nine constraints.

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.12

Results for the forward method, M = 32, steering vector misaligned, constraints2 = 1 a adaptive weight pattern b adaptive weight pattern, zoomed in

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.13

Results for the forward-backward method, M = 32, steering vector misaligned, constraints1 = 1 a adaptive weight pattern b adaptive weight pattern, zoomed in

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.14

Results for the forward method, M = 32, steering vector misaligned, number of constraints2 = 5 a adaptive weight pattern b adaptive weight pattern, zoomed in

angle, deg angle, deg

frequency, Hz

frequency, Hz

Figure 12.15

Results for the forward method, M = 32, steering vector misaligned number of constraints1 — 9 a adaptive weight pattern b adaptive weight pattern, zoomed in

12.7

Conclusions

In this chapter we have presented four deterministic direct data domain least-squares (D3LS) techniques for nulling interferers and extracting the signal of interest (SoI). The D 3 LS approaches generate adaptive weights on a snapshot-by-snapshot basis, eliminating the need for auxiliary training data sets. These techniques are capable of handling both coherent and non-coherent interferers, in a stationary or non-stationary environment. The four processors are the eigenvalue processor, and the three leastsquares methods: forward, backward and forward-backward procedures which may be implemented in real time on a signal processing chip. After describing the radar environment used to demonstrate these techniques, we developed the one-dimensional (space domain) implementation for each of the methods. We showed how to generate the system matrices using spatial samples and then solve for the adaptive weights, and estimate the amplitude of the desired signal. In order to maintain system gain of the SoI when the desired signal arrives slightly off the look direction, a method of setting multiple main beam constraints was presented. Next, we extended these techniques to the two-dimensional case (space-time domain) and identified how to select the proper weight dimensions to ensure a solution to the problem. Finally, an example related to a side-looking airborne radar is described. This example illustrates the challenges faced by an airborne radar and how, given the limited size and weight dimensions of a realistic system, the direct data domain leastsquares algorithms can assist in nulling interferers on a snapshot-by-snapshot basis. The greatest advantages of these procedures are that they yield four independent solutions using four different approaches. Hence, if the results between the methods are similar it provides greater confidence in the solution. The performance of the forward, backward and forward-backward methods has been evaluated using the output SINR and adaptive weight pattern for each of the methods. This has been done for both the SoI look-direction aligned and misaligned cases, as well as using various look-direction constraints (1,5 and 9) for both Doppler and angle of arrival. We showed that the D 3 LS algorithms perform extremely well for the aligned case. For the misaligned case it has been shown that unless multiple constraints are applied, the processors will treat the SoI as interference and attempt to null it. Increasing the number of constraints improved the processor's ability to maintain the system gain along the main beam, thereby improving the output SINR.

12.8 Ac a B as as (ib

List of variables clutter area (m2) vector containing amplitudes of coherent interferers system matrix of backward method complex amplitude of the SoI estimate of the as estimate of the SoI from the backward method

b C c d f § /r F FB /d G grz K K' L X M TV Na Nfa TVL TV0Ut Nt Nl T] p Amdesired Q fft ?k R r AR t tn O 0S S S' S" s O0 v

scale factor used in CGM arbitrary constant matrix of clutter, interference and thermal noise samples spacing between elements/channels order of the covariance matrix effective number of bits in data pulse repetition frequency system matrix for the forward method system matrix for the forward-backward method Doppler frequency of SoI number of data subarray positions grazing angle number of received signals, DoF related to one-dimensional FW and BK methods DoF for the one-dimensional FB method number of constraints wavelength (m) number of coherent pulses number of antenna elements number of spatial DoF - FW and BK methods number of spatial DoF - FB method maximum number of cancellation rows output interference plus noise number of temporal DoF - FW and BK methods number of temporal DoF - FB method number of range samples direction vectors used in C G M power of output interference and noise matrix of received samples with the SoI removed covariance matrix estimate of the covariance matrix number of range bins residual vector in C G M range resolution (m) time scale factor used in C G M angle of arrival of the various signals A o A of SoI matrix of samples of the SoI with unity amplitude matrix S with multiple spatial constraints matrix S with multiple space-time constraints vector containing snapshot of SoI clutter R C S ( m 2 ) steering vector

w Wb Wfb X x xr y yb yfb Z Z\ Z2 Z/i Z/2

weight vector - F W method weight vector - B K method weight vector - FB method matrix of snapshot samples containing all received signals snapshot of the voltages containing all components of the received signals snapshot of the received signals from rth range ring forcing function - F W method forcing function - B K method forcing function - FB method element to element phase shift due to SoI - one-dimensional method SoI spatial factor SoI temporal factor spatial factor due to /th constraint temporal factor due to /th constraint

References 1 SARKAR, T. K., WICKS, M. C , SALAZAR-PALMA, M., and BONNEAU, R.: 'Smart antennas' (John Wiley & Sons, New York, NY, 2003) 2 KLEMM, R.: 'Adaptive clutter suppression for airborne phased array radar', IEE Proc. F, Commun. Radar Signal Process., 1983,130, pp. 125-131 3 WARD, J.: 'Space time adaptive processing of airborne radar'. Lincoln Laboratory, Lexington, MA, Tech Report 1015, December 1994 4 ADVE, R. S. and SARKAR, T. K.: 'Compensation for the effects of mutual coupling on direct data domain adaptive algorithm', IEEE Trans. Antennas Propag., January 2000, 48, (1), pp. 86-94 5 APPLEBAUM, S. P.: 'Adaptive arrays'. Syracuse University Research Corp., dept SPL-769, June 1964 6 WIDROW, B., et al.: 'Adaptive antenna systems', Proc. IEEE, December 1967, 55, pp. 2143-2159 7 BRENNAN, L. E. and REED, L. S.: 'Theory of Adaptive Radar', IEEE Trans. Aerosp. Electron. Syst, March 1973, 9, pp. 237-252 8 REED, L. S., MALLETT, J. D., and BRENNAN, L. E.: 'Rapid convergence rate in adaptive arrays', IEEE Trans. Aerosp. Electron. Syst., November 1974, 10, pp. 853-863 9 WANG, H. and CAI, L.: 'On adaptive spatial-temporal processing for airborne surveillance radar systems', IEEE Trans. Aerosp. Electron. Syst., July 1994, 30, pp. 660-669 10 LUTHRA, A.: 'A solution to the adaptive nulling problem with a look-direction constraint in the presence of coherent jammers', IEEE Trans. Antennas Propag., May 1986, 34, pp. 702-710 11 SARKAR, T. K. and SANGRUJI, N.: 'An adaptive nulling system for a narrowband signal with a look direction constraint utilizing the conjugate gradient method', IEEE Trans. Antennas Propag, July 1989, 37, pp. 940-944

12 SCHNEIBLE, R.: 'A least squares approach for radar array adaptive nulling'. Doctoral dissertation, Syracuse University, May 1996 13 PARK, S.: 'Estimation of space-time parameters in non-homogeneous environments'. Doctoral dissertation, Syracuse University, August 2000 14 SARKAR, T. K., KOH, J., and ADVE, R., et al.: 'A pragmatic approach to adaptive antennas', IEEE Antennas Propag. Mag., April 2000, 42, (2), pp. 39-55 15 SARKAR, T. K., PARK, S., KOH, J., and SCHNEIBLE, R. A.: 4A deterministic least squares approach to adaptive antennas', Digit. Signal Process., 1996, 6, pp. 185-194 16 SARKAR, T. K., WANG, H., and PARK, S., et al.: 'A deterministic least squares approach to space time adaptive processing (STAP)', IEEE Trans. Antennas Propag., January 2001, 49, pp. 91-103 17 BROWN, R. and SARKAR, T. K.: 'Real time deconvolution utilizing the fast Fourier transform and the conjugate gradient method'. Presented at the 5th ASSP workshop on Spectral estimation and modelling, Rochester, NY, 1980 18 SARKAR, T. K.: 'Application of the conjugate gradient method to electromagnetics and signal analysis (Elsevier, Mass., 1991) 19 HAYKIN, S.: 'Adaptive filter theory', (Prentice-Hall, Englewood Cliffs, NJ, 1996, 3rd edn.) 20 JOHNSON, D. H. and DUDGEON, D. E.: 'Array signal processing' (PrenticeHall, Englewood Cliffs, NJ, 1993) 21 PARK, S. and SARKAR, T. K.: 'Prevention of signal cancellation in an adaptive nulling problem', Digit. Signal Process., 1998, 8, pp. 95-102 22 FAS Intelligence Programs website, www.fas.org/irp/program/collect/predator. html, created by John Pike, maintained by Steven Aftergood, updated Nov 6,2002 23 SKOLNIK, M. L: 'Introduction to radar systems' (McGraw-Hill Publishing, New York, NY, 1980)

Chapter 13

Robust techniques in space-time adaptive processing Keith E McDonald1 and Rick S. Blum2

13.1

Introduction

The function of most radar systems is to supply data about the presence and position of targets of interest [I]. A typical radar system emits an electromagnetic pulse and analyses the reflections of the transmitted energy. Ideally, the radar system will detect returns from range cells that contain targets and will reject returns from range cells that contain only background energy. A range cell represents a small block of physical space that is a certain distance (range) from the receiver. The radar checks for a target in each range cell of interest, typically referred to as a test cell. Range cells other than the test cell are called reference cells. The target signal competes with background energy that is potentially composed of a combination of thermal noise, clutter and jamming. Thermal noise is caused by the thermal agitation of electrons and is present in all real receivers. In this chapter, we consider radar arrays in which each antenna element has its own receiver. Therefore, the thermal noise at each antenna element is considered to be independent of the thermal noise at the other elements. Radar clutter is generically defined as the echoes from scatterers that are not deemed to be of tactical significance [115]. For airborne surveillance radar, the ground is the major source of clutter. Ground clutter is distributed in both angle and range. It is also spread in Doppler frequency due to the motion of the array platform. In airborne radars, ground clutter would ideally lie along a line in angle-Doppler

1

Keith McDonald's portion of this Chapter was funded by the MITRE Sponsored Research Task 51CCG-930-W3 (Sensor Technologies) Rick Blum's portion of this Chapter was partially supported by the Air Force Research Laboratory under Agreement No. F49620-03-1-0214

space. This spread occurs because the Doppler frequency of a reflection from a small patch of ground is proportional to the projection of the airplane velocity along the unit direction vector between the airplane and the small patch of ground [115]. Due to system and algorithm design imperfections, the line is typically widened into a structure called the clutter ridge. Jamming refers to the use of a signal to fool the radar or to intentionally limit its successful operation. Jamming has traditionally been associated with military scenarios, but the increase in public air traffic and the threat of terrorist activity has made jamming an important issue for commercial applications as well.

13.1.1 Initial development of space-time adaptive processing (STAP) algorithms In the detection process, a target return from the test cell competes with the thermal noise, jamming and clutter returns. Therefore, large amounts of energy from these components can significantly lower the probability of detection and/or raise the probability of false alarm. In order to mitigate the detrimental effects of this energy on the receiver, space-time adaptive processing (STAP) algorithms can be employed. The design and analysis of STAP algorithms for target detection have received significant attention from the radar and signal processing communities. STAP algorithms utilise the multiple receiving elements ('space') of a radar antenna array and multiple pulses ('time') to provide filtering of this energy in both the spatial and temporal domains [115]. 'Adaptive' refers to the fact that the filter weights are computed using information from the particular radar environment. The negative effects of noise (unless explicitly stated, hereafter noise is defined as clutter plus jamming plus thermal receiver noise) in the test cell are minimised through the calculation and subsequent application of a two-dimensional adaptive filter in the signal processing chain. Figure 13.1 provides visualisation of a typical STAP technique improving radar performance. Figure 13.1a depicts the signal energy received by a conventional radar that uses Doppler frequency selectivity. The dashed lines represent the filtering process. The clutter lies along a line, often called a ridge, with a larger amplitude produced by the mainlobe of the antenna pattern. As shown in Figure 13.1a, although the clutter from the mainlobe of the antenna response is cancelled via one-dimensional filtering, the sidelobe clutter remains. By allowing an extension of conventional radar processing to two dimensions, a STAP algorithm can produce a null along the entire clutter ridge, as shown in Figure 13.1b, if the clutter ridge location is exactly known. This two-dimensional filtering will improve the system's performance. Figure 13.1c shows a detection environment in which the radar reception is negatively impacted by a jammer. One common type ofjamming resembles thermal noise temporally, but looks like a point target in the spatial domain. In Figure 13.1 c, an attempt to filter out the jamming results in the filtering of the target response as well. Although still allowing the reception of the target signal, STAP provides an additional filtering dimension, as shown in Figure 13. Id, so that a properly placed null can limit the effects of jamming. This type of processing increases the probability of target detection in a clutter plus jamming environment.

mainlobe of antenna response

clutter

sidelobes of antenna response signal

clutter

signal

target spatial frequency

spatial frequency

jamming

signal

target

signal target

spatial frequency

Figure 13.1

spatial frequency

Conventional filtering versus STAP (adaptedfrom [204])

Much of the initial interest in adaptive processing is generated by the work presented in References 2 and 3. The automatic target detection processing proposed in References 2 and 3 attempts to minimise the power of the unwanted interference by maximising the signal-to-noise ratio (SNR) of the received process. The work in References 4, 5, 14 and 16 builds on the efforts of References 2 and 3 to design an automatic detection scheme, often called a test, with a constant false alarm rate (CFAR) characteristic known as the adaptive matched filter (AMF) test [4,5] or as the modified sample matrix inversion (SMI) algorithm [14,16]. Following References 2 and 3, the next most influential STAP research is likely Reference 6, where an alternative approach to that of References 2 and 3 is proposed to determine a CFAR processing scheme for the scenario in which the covariance matrix of the test cell is unknown. The approach in Reference 6 assumes that the statistics of the test cell data are the same as those of the data from the surrounding range cells and involves the derivation of a likelihood ratio test using maximum likelihood estimates. This particular methodology is termed the generalised likelihood ratio test (GLRT). Additional work concerning the GLRT test is presented in Reference 7 as well as in a number of recent papers including References 14 and 15. It is interesting to note that the matched filter discussed in References 2 and 3 appears as a portion of the GLRT test statistic, and that for large amounts of training data these two approaches are identical [15]. Recent work on adaptive algorithm design includes the introduction and development of the adaptive coherence estimator (ACE) test [188,178,212,213]. Expressions for the probability of false alarm and detection of the test are subsequently determined in Reference 213. The ACE test is especially attractive since it can be justified using the rigorous theory of invariant tests [223,219]. It is worth noting that the theory of invariant tests has been used to study STAP algorithms in the past [226,227] and, in fact, Reference 223 builds on the work in References 226 and 227 by adding an additional invariance to scale. Further, the relationship between the ACE and the

GLRT is carefully described in Reference 215. To summarise, the ACE test has been shown to be a generalised likelihood detector and has a derivation similar to that of Kelly's GLRT. However, the hypothesis testing procedure assumes that the scaling of the covariance matrix of the training data, relative to that of the test cell covariance matrix, is unknown [214,215]. Of course, this approximation will not necessarily agree with the true situation. There can be significantly more or less differences between the training and test cell data. A similar assumption, invariance to scale, leads to the invariance results in Reference 215, and thus, similar concerns can be raised. However, in general, the invariance properties and the relationship to the GLRT provide analytical justification that the ACE algorithm will work well for cases with slightly imperfect estimates, cases with limited training data and some cases with non-stationary training data. This support of the robustness of the ACE algorithm stimulates future investigations. Further, in a channel with time-varying gain or non-Gaussian statistics, the ACE algorithm has been shown to provide better performance than algorithms that do not possess the invariance properties [218]. The performance analysis of a set of adaptive matched subspace detectors, of which the ACE test is one, is discussed in Reference 216. A review of matched and adaptive detectors is supplemented by the work in Reference 217, and their relationship to other algorithms such as the GLRT and AMF is presented in Reference 218. Finally, the work presented in Reference 9 considers a large class of adaptive detection algorithms, of which the GLRT and the AMF are special cases. Furthermore, the ACE is a limiting case. Further work concerning the theory of invariance and its application to the hypothesis testing problem for adaptive arrays is given in References 226, 227,231 and 232. The adaptive array problem involves the search for a detection statistic that is complicated by the presence of the noise covariance matrix, a high-dimensional nuisance parameter. The authors discuss that the search for the detection statistic should be conducted over the set of detectors that satisfies the invariance criterion, and thus, one must only search for the detector with an optimality property over this set of detectors. A test that is invariant to the nuisance parameter (the unknown covariance matrix), since it is irrelevant to the decision, would thus be beneficial. Invariance tests do not distinguish between scenarios differing in their nuisance parameters. Two different cases are considered: one in which the noise covariance matrix is totally unknown and another in which a priori information about its structure exists. This invariance approach reduces the problem size since all invariance test statistics can be expressed in terms of a function (possibly vector-valued but having much fewer dimensions than the data) of the data called the maximal invariant. The distribution of the maximal invariant is parameterised by another low-dimensional function on the parameter space, and thus most of the nuisance parameters are removed. The use of the theory of invariance has also been implemented to validate the spatial stationarity assumption of the data in References 228 to 230. Invariance principles are used to ensure that the test has constant significance when the model is true by eliminating the dependence on the covariance matrix. The test decision depends only on the covariance matrix structure rather than on the actual covariance matrix.

13.1.2 Hypothesis testing problem To understand the target detection problem, it is essential to describe the method used to gather data at the receiver. Consider a uniform linear array composed of Af identical elements. The radar transmits a coherent burst of M pulses per coherent processing interval (CPI) at a constant pulse repetition frequency (PRF) /PR = l/T.A CPI is the time over which the waveforms are collected, and T is the pulse repetition interval (PRI). Thus, a CPI lasts MT time units. For each pulse transmitted, L samples are collected. Each sample corresponds to a return from a specific range from the receiver. The sampling scheme is shown in Figure 13.2a. With N receiver channels and M pulses, the number of received data samples per CPI is LMN. It is helpful to visualise the data set for a CPI as the data cube depicted in Figure 13.2b. The test cell data corresponds to one particular range cell and is illustrated by the shaded area of the data cube. Denote the observation corresponding to the nth receiving element and the mth pulse at the €th range sample (the range being examined) as 8n,m,n — 1 , . . . , N and m = 1,...,M. Let xm ^ e m e NxI vector of antenna element outputs for pulse m, so that / m = (<$i,m,..., &N,m)T? where T is the transpose operator. xm ^s called a spatial snapshot and corresponds to the data from

pulse 1

pulse 2

pulse 3

pulse M antenna 1

range sample 1 range sample 2 range sample L pulse 1

pulse 2

pulse 3

pulse M antenna 2

pulse 1

pulse 2

pulse 3

pulse M antenna TV

antenna element

MN samples for cell under test data .

Figure 13.2 a Data to be processed (adapted from [42]) b CPI data cube

the range cell under consideration and the rath pulse. The portion of the data cube that represents the Uh range cell is the MN x 1 vector Jt, where x is defined by stacking the spatial snapshot vectors as x = (/i, • . . , XM) T Now consider the use of Jt, called the space-time snapshot, to determine the presence or absence of a target in a particular range cell. The objective of the hypothesis testing problem is to determine which of the following is true for the range cell under consideration: x =n HQ: target absent x = KS + n H\: target present where n encompasses all of the noise contained in the test cell. The signal which can be added to the noise is o , where s is a unit length vector that is completely known (referred to as the signal vector) and /c is an unknown complex constant.

13.2

Real-world detection environments

In most of the adaptive radar detection algorithms, the filter weights are computed via a calculation that attempts to approximate a maximum likelihood estimate of the noise covariance matrix. However, the covariance matrix of the noise in the test cell is usually characterised using samples from the neighbouring reference cells. The set of vectors composed of the test cell and all of the reference cells is typically assumed to be an independent and identically distributed (HD) set of vectors. The noise in both the test and reference cells is usually assumed to have been generated from the same Gaussian distribution. The estimated covariance matrix generated from the reference cells is then used in the signal detection procedure. If the estimated covariance matrix accurately represents the covariance matrix of the test cell, the adaptive algorithm will usually perform well. Unfortunately, the non-ideal conditions of real-world detection environments can preclude the performance described in much of the theoretical STAP literature. Mismatches between the clutter statistics of the test cell and those used to design the adaptive processing scheme present one such difficulty that can result from these circumstances. This type of non-homogeneity occurs when the reference data statistically departs from the test cell data in the structure of its covariance matrix, thus violating the HD assumption critical to statistical estimation of the test cell covariance matrix [33]. Mismatches of this type have been observed in measured airborne radar data and can be intuitively justified. In airborne radar, the ground clutter in the reference cells is produced by reflections from portions of the ground different from those that produce the ground clutter in the test cell. Due to variations in terrain, weather and other factors (see discussion in the next section), the statistics of the ground clutter returns in the test cell can differ significantly from those of the reference cells. As an example, the United States Air Force Rome Laboratory has collected data under the multichannel airborne radar measurement (MCARM) programme in collaboration with Westinghouse Electric Corporation. The MCARM data has been shown to be non-homogeneous in range [33,35-38,42,43,45-49].

jammer

#-steering vector

jammer null

target s-signal (target) vector

clutter null

Figure 13.3

Steering vector mismatch (adapted from [115])

An additional difficulty with real-world covariance matrix estimation is a violation of the assumption that the reference cells are free of targets. In much of the literature, the target signal is assumed to be confined to the test cell so that the reference cells are free of signal components. Under practical circumstances, the assumption of targetfree reference data can be invalid [11,8,172-176,45^8]. For example, consider several targets with similar velocities that are spatially distributed in range. This configuration yields similar signals in both the test and reference cells. The use of a high range resolution (HRR) radar can produce scenarios such that even a single target can occupy tens of range cells3 [H]. When the reference cells contain signals, traditionally known as signal contamination, the estimate of the co variance matrix of the noise in the test cell can become inaccurate. Therefore, the performance of STAP algorithms can degrade significantly [126,45-48]. Steering vector mismatch is another type of mismatch that affects performance. It occurs when the radar antenna array forms a beam that is not pointed in the exact direction desired. The misalignment can also be caused by aircraft crabbing as well as antenna calibration errors. Certain algorithms can be less sensitive to steering vector mismatch than others [195-199,10,9]. Depending on the algorithm used, steering vector mismatch can have a negative impact on the radar signal processing. Steering vector mismatch is depicted in Figure 13.3.

3 The authors acknowledge that the regular resolution case is currently of more practical interest. Presently, the HRR radar mode is not used for target detection in any operational platforms known to the authors. However, with increases in observational resources and computational speed, it is feasible that in the future such a capability will exist

Jamming and the subsequent multipath signals that result from jamming (also known in the literature as hot clutter or terrain scattered interference (TSI)) are a major consideration, especially in hostile detection environments. Typically, the mitigation of a direct path jammer requires the application of only a one-dimensional spatial filter as depicted in Figure 13.Id. However, the jammer signal can reflect off of different terrain surfaces. Consequently, time-delayed versions of the signal with different angles of arrival are processed at the receiver. This type of interference requires a two-dimensional filtering procedure to reduce negative effects. Unfortunately, the filtering coefficients are significantly different from those required to cancel the ground clutter (also known as cold clutter in the TSI problem formulation) [206-208,141-143,77]. Even under ideal conditions of covariance matrix and steering vector match, the application of two-dimensional filters (or in the case of TSI, potentially two stages of two-dimensional filters or even three-dimensional filters) often presents a daunting challenge to the real-time computing community. The computational complexity necessary to conduct real-time adaptive processing quickly becomes challenging as the dimensionality of the filtering problem increases. Therefore, many reduced processing techniques (sometimes called partially-adaptive) have been introduced recently and evaluated under ideal detection conditions [201,58-61,211]. Under the conditions of covariance matrix and steering vector mismatch, many fully adaptive and partially adaptive STAP tests are not optimal and can suffer losses in performance. If not factored into the design of the algorithm, TSI presents additional challenges. Computational demands are often difficult to meet for the desired real-time target detection problem. In this chapter, the real-world difficulties that result in suboptimal STAP performance are analysed. Then, the evaluation of STAP performance under these non-ideal conditions is discussed and algorithm robustness is studied. We review the development of novel techniques that attempt to either circumvent or minimise the deleterious effects of non-homogeneous clutter, signal contamination, steering vector mismatch and TSI. Some of these algorithms provide an additional benefit of reduced computational complexity. We conclude that, for the case of detecting a target embedded in non-homogeneous clutter in the presence of antenna array errors and hostile jamming, the proper design of the adaptive signal processing algorithm, accounting for the unique detection environment and utilising intelligent exploitation of a variety of resources, can lead to significant performance enhancement over traditional methods. We recognise that due to the vast amount of literature on this topic, a fully comprehensive review and analysis is beyond the scope of a single chapter. Therefore, we summarise the efforts of major contributors with the hopes of introducing the reader to the significant topics of research as well as providing references for further investigation.

13.3

Non-homogeneity - causes and impact on performance

As discussed briefly in the introduction, the performance of most STAP algorithms is linked to the accuracy of the estimated covariance matrix of the noise in the test cell.

Clutter non-homogeneity leads to mismatch in structure, in terms of both amplitude and spectral characteristics, between the estimated and actual covariance matrices of the test cell data [34]. The non-homogeneous component causes a shift in the test statistic from the homogeneous case. Therefore, the filtering implemented by the adaptive processing suppresses the homogeneous component of the clutter, while passing the non-homogeneous components which are not nulled by the filtering [35,36]. It is precisely the non-homogeneous elements that impact the effort to discern between target and noise. Many different factors affect the covariance matrix estimation procedure and cause violation of the HD assumption needed for true maximum likelihood estimation of the test cell covariance matrix [34-36,38]. Therefore, Reference 38 divides the non-homogeneous causes into several categories. One category, amplitude nonhomogeneity, results from spatially varying clutter reflectivity, shadowing and clutter edges. Spatial variation of the clutter is caused by factors that include strong discretes such as point reflectors. Shadowing generally occurs due to terrain obscuration. Interference edges can yield a mix of reference cells, some with the same covariance matrix as the test cell, while others have a very different covariance matrix. Examples of clutter edges include littoral or rural-urban interfaces, or the boundary area of farmland and wooded hills [38]. As expected, performance degrades as the number of reference cells with a covariance matrix different from the test cell increases [12]. Depending on the test cell covariance matrix and the dominant reference cell type, the edge can produce an excessive false alarm rate, severe signal masking and inadequately nulled clutter. Various studies on the performance of STAP algorithms in the presence of clutter edges have been conducted. The work in Reference 14 discusses the performance of the AMF and GLRT. The AMF and GLRT are represented by the following test statistics: (13.2)

(13.3) where x is the test cell space-time snapshot, s is the steering vector, K is the number of reference cells used to calculate the covariance matrix estimate, t\ and T2 and are predefined thresholds and Qe is the estimated covariance matrix: (13.4) where the JC/ vectors are the reference cell space-time snapshots. Due to the additional term in the denominator that compensates the processor, the GLRT has been shown to be more robust in clutter edge environments than the AMF [14]. The amount of degradation is related to the clutter spectrum spread and the distance between the signal and clutter in the Doppler domain [15].

Spectral non-homogeneity is another category of non-homogeneity outlined in Reference 38. Spectral non-homogeneity stems from a host of effects jointly characterised as variable intrinsic clutter motion (ICM), which is caused by such factors as sea state or foliage moving in the wind. Variable velocity of the clutter causes a widening of the clutter ridge due to temporal decorrelation of clutter echoes, thus broadening the clutter spectrum [50]. The pulse-to-pulse decorrelation of the clutter alters the STAP covariance matrix [148]. Due to its similar impact, here we also discuss the effects of crabbing. Aircraft crabbing, which is airplane movement orthogonal to the antenna array, causes the clutter to spread off the diagonal line onto a curve, thus resulting in suboptimal cancellation [203,204]. Typically, the noise covariance matrix is represented by some large eigenvalues corresponding to the clutter/jamming and the remaining smaller eigenvalues corresponding to the thermal noise. The effects of ICM and crabbing cause an increase in the number of larger eigenvalues (the rank of the clutter/jamming covariance matrix). This might, for example, hinder the performance of reduced-rank STAP methods by causing them to inaccurately estimate rank. ICM and crabbing can also yield an adaptive filter response which produces uncancelled clutter or unintended signal cancellation. Under either scenario, the clutter ridge expansion causes degraded minimum detectable velocity (MDV) of targets. A distorted beampattern and high azimuth-Doppler sidelobes also result. Methods to raise the noise level to eliminate pattern distortion have been investigated [148]. The effects of ICM caused by wind-induced motion of trees and vegetation are studied in References 191, 192 and 194. This work conducts a statistical analysis on measured clutter data to consider appropriate modelling and then discusses detector performance. Errors caused by range walk, due to the platform motion during the interpulse sampling, can cause temporal decorrelation of space-time clutter echoes similar to that caused by ICM. Techniques exist to correct for difficulties caused by range walk [56]. Near-field scattering smears the clutter return off of the diagonal ridge into the entire azimuth-Doppler plane. Near-field obstacles include airplane structures, such as the wing. Consequently, the clutter cancellation in the presence of near-field obstacles will require more degrees of space-time freedom than are necessary in the absence of such obstacles. Increasing the number of elements to give more spatial degrees of freedom helps to alleviate this difficulty, since more spatial nulling can be achieved [202]. Increasing the number of temporal degrees of freedom is less effective in this case. A mathematical analysis of the effects of ICM, crabbing, near field obstacles and steering vector mismatch on the eigenvalues of the noise covariance matrix is given in Reference 204. Another type of non-homogeneity is clutter-to-thermal-noise ratio (CNR)dependent spectral mismatch. The non-homogeneity is caused by the CNR influence on spectral spreading mechanisms (timing jitter, scintillation, array errors and other decorrelation effects). The result is either an adaptive filter pattern that does not adequately encompass the primary data spectral width causing uncancelled clutter, or an overtly wide filter notch causing overnulling and a reduction in MDV The rank of the clutter covariance matrix tends to expand with greater CNR [38].

A violation of the homogeneous data assumption is caused by moving scatterers or multiple interfering targets, such as vehicles, aircraft, weather or chaff resident in either the primary or the secondary data. These scatterers cause angle-Doppler responses that appear spectrally distinct from the ground clutter power spectral density. The effects of moving interferers have been seen in several different multichannel airborne radar data collections. For example, it is difficult to avoid vehicles in the reference data set while surveying near a network of roads and highways. These non-stationary discretes can cause signal cancellation, distorted beam patterns and exhaustion of the degrees of freedom necessary to successfully perform clutter cancellation [38]. Non-stationarity may be due to radar geometry causing a range variation in clutter characteristics, and therefore has a certain predictability. Any configuration, with the exception of a side-looking array, has range-dependent Doppler. The effects of nonstationarity can be mitigated by reduced-rank methods, a range-dependent taper to adjust for the geometry or a covariance matrix taper. The work in Reference 38 illustrates the losses caused by each type of non-homogeneity.

13.3.1 Signal contamination Investigation of non-homogeneity due to signal contamination has seen significant growth in recent years, and warrants further attention. It has been observed that when the reference cells contain signals, the estimate of the noise in the test cell becomes inaccurate. If the false target has a stronger power than the primary target, the classical covariance matrix-based STAP approaches can attempt to cancel the primary target in favour of the false target [158]. Targets in the reference data cause the performance of many STAP algorithms to degrade significantly [126,199,172-176,158]. Therefore, robustness of traditional STAP algorithms in signal-contaminated environments is of particular interest, as are new methodologies used to mitigate the problem. The strength and direction of the contaminating signal play a major role in determining its effect on performance. As the signal strength increases, performance degrades substantially [12]. In Reference 126, the authors illustrate the deleterious effects of signal contamination on two detection schemes. Their work shows that performance can be significantly degraded when a strong signal contaminates only one reference cell. The impact of the contamination also depends critically on the angle between the desired and contaminating signals. Contamination close to the clutter frequency has much less of an effect on performance because it falls into the clutter notch. If the target and contaminating signals are orthogonal, the contamination has no influence. However, as the contaminating Doppler frequency approaches that of the desired signal (the angle and velocity difference between the desired and contaminated signal becomes smaller), the algorithmic performance decays further. These moving discretes can degrade the signal processing significantly, provided that the direction of arrival is within, or near, the 3 dB beamwidth of the mainbeam of the array [34,47,48]. Of course, as the number of reference cells increases, the effects of

distributed target STAP

traditional STAP test cell

antenna element

data cube

PRI

data cube antenna element

test cell

guard band cells reference cells

PRI

guard band cells reference cells

Figure 13.4 a traditional target detection b distributed target detection a contaminating signal can become averaged out. However, this technique increases the probability of yet another contaminating signal entering the estimate [126,14,15]. In an attempt to reduce the chance of signal contamination, some processing schemes omit the range cells adjacent to the test cell (known as guard bands) from the covariance matrix estimate (see Figure 13.4). However, the possibility of an inaccurate covariance matrix estimate increases as reference cells farther from the test cell are used in the estimation procedure. Additionally, implementing guard bands can result in the loss of potentially valuable signal information. In general, there are two approaches to combat the signal contamination problem. One attempt is to remove the signal contamination and present the processor with a more homogeneous data set, which is the topic of the next section. An alternative approach is to recognise that some of the signal contamination can be used to improve detection performance. This increased performance can occur if the goal of the system is to track a group of targets such as a convoy, or if the radar resolution is such that a single target is spread across several different range cells. We denote occurrences of signal contamination in range cells adjacent to, or within a few range cells of, the test cell as distributed target scenarios. Signal contamination by distributed targets occurs while attempting to detect targets in individual range cells as shown in Figure 13.4a. The alternative approach is to attempt to detect the presence of the target within an input data block (a set of test cells). This alternative technique is depicted in Figure 13.4b. The approach produces an effective integration gain of target returns that enhances the target set's detectability [127]. Hence, a presumed source of performance degradation (signal contamination) can actually be a source of performance enhancement. The penalty for grouping test cells into blocks is a loss in radar range resolution. STAP algorithms designed specifically for distributed target detection resembling the AMF and the GLRT, provided in the seminal work of Reference 8 and revisited in References 172 and 173, have attracted interest. The original STAP algorithms for

distributed target detection make the assumption of homogeneous range cell data in both algorithm design and performance analysis. Research on this topic is conducted in Reference 128, in which a detector called the modified generalised likelihood ratio test (MGLRT) is derived. Examples are provided that illustrate that the MGLRT can perform better than the GLRT. Unfortunately, the MGLRT does not have the desired CFAR characteristic exhibited by the GLRT. To address this concern, further work on the MGLRT is conducted in Reference 131, in which knowledge of the thermal noise is assumed to be known a priori. The approach results in a form that exhibits bounded CFAR characteristics and shows acceptable performance for range-distributed, Doppler-shifted targets. As previously discussed, the homogeneous data assumption can cause a system to perform poorly under real-world conditions. Adaptive detection for range distributed targets in partially homogeneous environments, in which the noise covariance matrices of the test and reference cells differ by a scale factor, is discussed in References 174 to 176. The derivation of the probability of detection and false alarm equations in References 172 to 176 involves conditional probability distribution functions (pdfs). A limiting aspect of these approaches stems from the fact that the conditioning is removed through a numerical expectation via a Monte Carlo technique rather than through analytical means. In many real-world scenarios, mismatches between the statistics of the test cells and those used to design the adaptive processing scheme are more complex than simple scalar differences. Complex mismatches have been observed in measured airborne radar data. Consequently, Reference 199 presents closed-form probability of false alarm and detection equations for an adaptive detection algorithm based on the AMF, for distributed target detection. The work in Reference 199 considers algorithm robustness under the real-world conditions of steering vector and covariance matrix mismatch and does not require the use of Monte Carlo techniques. Detection of distributed targets in noise environments that have non-Gaussian statistics is investigated in References 177 and 134. Such tools as those provided in References 6, 172 to 176 and 199 can prove valuable in discussing the robustness of algorithms in signal-contaminated, ideal/non-ideal detection conditions.

13.3.2 Non-homogeneity detection Although clutter non-homogeneity causes suboptimum performance by a STAP processor, this loss may be mitigated through intelligent approaches to covariance matrix estimation [38]. As discussed previously, individual targets in the reference data can significantly degrade performance. Two effects of the presence of targets in the STAP training set are a potentially significant target self-nulling and an overall degradation in clutter cancellation performance [114]. Outlier data vectors whose signal vector is highly correlated with the desired steering vector have the most impact [133]. The removal of these cells can provide substantial overall detection improvement [47,48,158]. One such removal approach is called non-homogeneity detection. In general, a non-homogeneity detector attempts to excise the reference cells that are least homogeneous from the group of cells to be used in the estimate

of the test cell noise covariance matrix. Frequently, the non-homogeneity detection schemes assume that the homogeneous clutter comprises the majority of cells to be tested. Computing the covariance matrix of the test cell from the identified 'most homogeneous' secondary data range cells ensures acceptable performance for the majority of the test cells [35,38]. The literature illustrates that proper identification and excision of non-homogeneous data lead to improved covariance estimation and a corresponding improvement in STAP detection. Some efforts have attempted to utilise a power-based technique to assess non-homogeneity through a calculation of the inner product of the return from each cell. It is assumed that the reference cells with the largest power are signal contaminated, and that their removal will leave the processor with a remaining set of homogeneous reference cells. Rather than removing those cells with large amounts of power, other approaches choose to estimate the test cell noise covariance matrix with the reference cells having the largest power. The efforts in References 91 and 94 illustrate such a technique called power-selected training (PST). The methodology produces a deeper filter notch than if the covariance matrix was formed by utilising the entire set of reference cells, thus ensuring that all of the clutter ridge is nulled. The work in Reference 114 extends PST to a more complete non-homogeneity detector in which both phase and power-selection criteria are utilised. The method first uses a phase-based criteria to remove strong moving targets from the training data. For a given Doppler, clutter comes from a common angle for all range cells over which STAP weights are formed. Thus, any significant deviation in angle indicates that the cell contains non-clutter returns. If the measured phase differs significantly from the expected phase of the clutter for a given Doppler bin, the cell is excluded. Subsequently, the reference data are put through PST to train on the clutter discretes. By using the strongest returns for each Doppler bin for training, STAP weights are based on the strongest clutter. Increased false alarms may result from target signals located in the sidelobes of the antenna beampattern. These extra target signals can be due to multiple detections of a single target, non-stationary jamming, etc. A technique has been presented for editing detections resulting from sidelobe targets and excising these detections [90]. The outputs of the normal detector are passed to an editor that examines the beamformer's outputs, the adaptive weights and the list of raw detections. This information is used to classify each raw detection as either from the radar mainlobe or sidelobe. Only the mainlobe detections are passed on to subsequent signal processing stages. Another type of non-homogeneity detection is discussed in Reference 89. In this work, the power of ground clutter at each range-Doppler cell is measured by forming a non-adaptive beam at each Doppler pointed at the azimuth at which ground clutter appears. The amount of nulling accomplished by the adaptive system is estimated, and then the measured power of the ground clutter and an estimate of nulling performance is used to adjust a threshold. This methodology attempts to account for any undernulled ground clutter discretes predicted to appear in the adaptive range-Doppler map. Other recent work on robust algorithms that mitigate the effects of signal contamination includes Reference 133. This work illustrates how the maximum likelihood

estimate of the number of discretes in the training data can combat the negative effects of non-homogeneity. Algorithms that censor outliers are derived using approximations to maximum likelihood estimates. A more robust AMF algorithm is devised by using these censoring algorithms to improve the covariance matrix estimate. A technique that uses the generalised inner product (GIP) as a method for nonhomogeneity detection has been introduced and analysed by the research community. The GIP can be utilised as a measure of the variability of the components available to estimate the covariance matrix of the noise in the test cell [33]. The GIP nonhomogeneity detector has been further validated through work which approaches the non-homogeneity problem via an eigenanalysis. This effort analyses edge effects due to rural-urban interfaces, stationary discretes and moving scatterers in the reference data and further justifies the GIP as a simple test of eigenstructure [43]. The work in Reference 158 introduces a variation of the GIP non-homogeneity detector to improve target detection in cases of non-homogeneous background with multiple targets in close proximity. This scheme involves the application of a weighting, termed an equalisation. For the pool of reference cells used in the noise covariance estimation procedure, the cells closer in range to the test cell are weighted with a higher probability of being homogeneous than those farther away, which are assumed to be less representative of the noise in the test cell. A statistical analysis of the GIP, in which its pdf is determined, is presented in Reference 159. This statistical effort introduces a GIP-based test for non-homogeneity detection that utilises a goodness-of-fit test on an empirically formed pdf. The work is extended to cases with very limited reference data in Reference 160, which implements the derived pdfs in the calculation of two techniques that use the nonhomogeneity detector for covariance estimation. Finally, the scheme is used to derive a non-homogeneity detector for non-Gaussian noise in which the clutter is modelled as a spherically invariant random process (SIRP) [161]. The statistics of the detector are determined with the aforementioned pdf employed. Overall, the literature has shown that for both simulated and measured data, the proper use of the GIP as a nonhomogeneity detector can yield a dramatic increase in signal processing robustness in non-homogeneous environments. Unfortunately, all aspects of the radar detection procedure are not factored into the GIP methodology. In Reference 71, the GIP is compared with the AMF detection statistic as a non-homogeneity detector. The authors suggest that the GIP nonjudiciously limits reference cell data supply since the measure is independent of the radar's steering vector. Therefore, the use of the GIP can remove reference cells that have no significant negative impact on the processing, such as those with contaminating signals far away in angle from the target signal. The study also shows that, under non-homogeneous conditions and non-ideal array environments (steering vector mismatch), the non-homogeneity detector might actually worsen the situation. Many non-homogeneity detectors do not specify how to detect targets in test cells which are deemed to be non-homogeneous. In Reference 72, a two-stage hybrid STAP algorithm is introduced to remove this limitation. The method first implements a deterministic algorithm to filter the uncorrelated interference. This type of interference includes discretes in the test cell coming from directions and velocities other

than the location that the radar is querying for a target. This deterministic technique only employs data from the test cell and is therefore known as a direct data domain algorithm. After this stage, which uses only one degree of freedom, a STAP algorithm is implemented. Classical STAP schemes would not place a null at the location of discretes in the test cell, since the test cell, and thus the discrete, is not incorporated in the estimation of the noise covariance matrix. This two-stage method nulls discretes other than the target in the test cell. The technique could be used in conjunction with one of the previously described non-homogeneity detection schemes to increase overall performance [74,76]. Further analysis on direct data domain techniques is presented in Reference 32. Non-homogeneity detection remains a very active topic in the adaptive algorithm community.

13.3.3 Knowledge-based signal processing Knowledge-based signal processing is a very general approach that attempts to use signal processing in an intelligent fashion by considering additional information and adapting the processing based on this supplementary knowledge. Non-homogeneity detection is one problem to which knowledge-based signal processing has been successfully applied. Experts in target detection have noticed that easily accessible databases can be employed in conjunction with adaptive algorithms to mitigate the difficulties caused by non-homogeneous reference cells. Sources include mapping data, communication link trackers and other information sources that can be utilised along with adaptive processing [44-48]. For example, using map data, the range cells that correspond to geographic locations with roads can be removed. This preventative action excises reference cells that can potentially contain moving targets and consequently avoids signal contamination of the covariance matrix estimate. Additionally, it is possible to prenull known jammers, so that the degrees of freedom used in the STAP processing are utilised only to mitigate the effects of clutter. The effort in Reference 70 suggests that reference cells can be removed from the covariance matrix estimation procedure if they do not have common codes that correspond to databases, such as United States Geological Survey Land Use and Land Cover codes that describe the earth based on a classification system. The area of knowledge-based signal processing has received special attention due to the initiation of the United States Defense Advanced Research Projects Associations's Knowledge Aided Sensor Signal Processing and Expert Reasoning Program (KASSPER).

13.3.4 Analysis of degraded performance due to non-homogeneity Since the non-ideal, real-world detection environment presents difficulties not taken into consideration in the derivation of classical STAP algorithms, the capability to accurately determine the robustness of these techniques to non-ideal conditions is extremely valuable. There has been a significant effort to quantify the robustness of a variety of algorithms, both via Monte Carlo analysis as well as through closed-form equations that predict performance. Towards this goal, a variety of clutter modelling techniques and software have been developed to model non-homogeneous clutter.

The efforts supplement examples of the effects of non-homogeneity and allow further understanding of the specific contributors in a controlled environment. Examples in the literature include Reference 34, which develops a composite model of clutter non-homogeniety that allows for amplitude and spectral fluctuation in angle-Doppler subspaces. Spectrally distinct forms of space-time clutter non-homogeniety due to moving scatterers are provided as well. More recently, the United States Air Force at Rome Laboratory has developed high fidelity clutter modelling software, entitled RL-STAP, that has received positive attention [120,121]. Concerning traditional algorithmic performance, the robustness of the AMF algorithm in non-homogeneous clutter environments has been discussed in Reference 15, but only under the assumption of the availability of an infinite number of reference cells. This essentially drives the variability of the estimate to zero, which reduces the problem to a known covariance matrix case. Performance changes due to non-homogeneous environments have also been studied using simulations in such References as 43 and 36, and have been analytically described in References 101 to 103. Other efforts include Reference 106, which is relevant since most of the algorithms discussed are front-end adaptive beamformers. Work has been done to study the effects of scaling mismatch between the true and estimated noise covariance matrix on algorithm performance [212]. Also, the performance of a general class of detection algorithms in non-homogeneous clutter environments with mismatched steering and signal vectors is provided in References 195 and 196. Upperbounds of performance in mismatched scenarios are given in Reference 198. The studies in References 101,195 and 196 are based on a similar assumption, called the generalised eigenrelation (GER) in Reference 101. Validation of the GER assumption is provided in References 105 and 109. A discussion of the effects of this assumption on derived results is given in Reference 196, and an interpretation of this constraint from an adaptive nulling perspective is given in Reference 105. This constraint is relaxed in References 197, 198 and 200, and results in closed-form equations for performance of a general class of STAP algorithms under steering vector and covariance matrix mismatch. To lessen the negative impact of non-homogeneity, some efforts have attempted to combine more than one classical STAP algorithm into a hybrid approach. For instance, References 101 to 103 discuss the adaptive sidelobe blanker (ASB) algorithm which stems from sequentially following the AMF test with the ACE test. The ASB performance is compared with other traditional algorithms in homogeneous environments in Reference 107. An in-depth analysis of the ASB in non-homogeneous environments, in relation to other traditional detectors like the ACE, AMF and GLRT, is presented in Reference 108. The work in Reference 103 presents two methods for attempting to choose the thresholds required for the ASB to operate successfully in non-homogeneous clutter. In comparison with other traditional algorithms, the ASB is shown to provide superior performance to the AMF test alone due to its ability to mitigate false alarms caused by clutter non-homogeneities. Although computationally more efficient, its performance is on par with the GLRT. The ACE algorithm seems to suffer more from lower mainlobe target sensitivity than does the hybrid algorithm.

13.4

Antenna array errors

Most adaptive algorithm designs do not account for antenna array errors and such oversight can degrade signal processing performance. Since channels differ generally in both elevation and azimuth patterns, even at a fixed frequency, the argument can be made that the array calibration difficulty has been underestimated [22,23]. This difference in element beampatterns is a cause of steering vector mismatch. Mismatch occurs when the antenna array forms a beam that is not pointed in the exact direction desired. Under the conditions of steering mismatch, most adaptive algorithms are not optimal and can suffer significant losses in performance. The effects of steering mismatch on the GLRT and AMF tests are studied in References 7 and 5, respectively. A study of the effects of steering mismatch on a larger class of STAP algorithms appears in References 9 and 104. In Reference 104, the author illustrates that the AMF is less sensitive to steering vector mismatch than the ASB or the GLRT. However, the AMF is more sensitive to targets in the sidelobes (resulting in a higher false alarm rate). In Reference 113, array misalignment is shown to reduce the radar's ability to detect slow moving targets. In the design of many STAP processing schemes, the main assumption is a linear array of equally spaced isotropic point sensors. Also, each element of the array is assumed to be exactly like every other element. In reality, the physical size of the sensor causes a reradiation of incident fields leading to mutual coupling between the array elements. In addition, the channels are mismatched as a result of manufacturing errors. The works of References 74 and 75 discuss the negative effects of steering vector mismatch as well as mutual coupling. The significance of these errors is discussed in Reference 79. The influence of channel effects on the estimated covariance matrix such as amplification and orthogonality errors in the in-phase and quadrature channels, delay errors and offset errors is discussed in Reference 233. Taking these real-world factors into consideration in the design of the signal processing can result in performance improvement [73]. For example, certain algorithms use a discrete Fourier transform (DFT) to transform spatial data to the angle domain. However, when channel mismatch and mutual coupling affect the spatial steering vectors, a DFT may not be the most appropriate transform. Much of the literature has ignored the difficulties that can occur due to aircraft motion causing a change in the pitch angles of the platform. Analysis in Reference 80 indicates that the pitch angle of the airplane can have a significant effect on algorithm performance. Many antenna arrays are mounted to the fuselage of the aircraft. The pitch angle of the aircraft causes the array axis to incline relative to the velocity vector. The angle of the array relative to the velocity vector causes a broadening of the clutter ridge leading to a decrease in the MDV of the radar as well as an increase in the false alarm rate due to the broader, shallower nulls of the resulting filter. The crab angle that results from aircraft movement yields similar problems, but does not seem to be as substantial a concern. The study in Reference 55 suggests the utilisation of Doppler compensation to negate the effects of wind drift leading to aircraft crabbing. Reference 80 describes the use of a Doppler warping technique as the compensation for the effects of array inclination. Doppler warping is based on

shifting the mainlobes of the Doppler filters as a function of range so that the shifting follows the movement of the clutter ridge over range caused by the pitch angle. Since the calculation needs completing only once every CPI, it is not computationally expensive. Another alternative to counter the effects of platform or interferer motion without constantly having to obtain and invert new covariance estimates is given in Reference 110, which describes a transform that spatially widens the nulls created by traditional adaptive processing algorithms. It is hoped that in the future, with improved antenna design and calibration techniques, the effects of steering vector mismatch and pitch angle will be reduced.

13.5

Deviation from Gaussian assumption

One of the fundamental design assumptions of many STAP algorithms is that the statistics of the clutter returns can be derived from a Gaussian distribution. Under certain conditions, the assumption may be invalid. For instance, Reference 125 argues that the increased resolution of modern radars has caused a deviation from the Gaussian assumption of the clutter. Arguments are presented for modelling the clutter as a Rayleigh mixture, and signal processors are derived according to Rayleigh mixture pdfs. Non-Gaussian statistics have been reported for scattered power from the ocean and terrain and have been confirmed by experimental data recorded from a high resolution airborne radar operating at low grazing angles [154], Therefore, a large effort has been placed on algorithm design for the detection of targets in non-Gaussian noise. The performance of the GLRT and the AMF algorithms, which are both designed under the Gaussian assumption, have been shown to provide nearly identical performance with Gaussian interference [13]. In this case, the more computationally complex GLRT implementation is often not required. However, as the interference becomes more non-Gaussian and heavy tailed (as in the cases of either Weibull or log-normal distributions), the GLRT shows more robustness than the AMF [13]. Discussion of robust estimators using a priori information about the covariance matrix structure is described in Reference 63. The robustness of algorithms to the deviation from the Gaussian assumption is confronted in References 99 and 100, in which adaptive detection and beamforming structures are analysed when the clutter is modelled by a complex multivariate elliptically contoured distribution. Classical studies on robust detection in non-Gaussian noise and the statistical theory of minimax robustness are also relevant [220]. Contributions to the design of receivers for operation in clutter that is modelled by a complex spherically invariant random vector of the circular class include References 153 and 154. The derived receiver is shown to be equivalent to a parametric adaptive matched filter as opposed to a data-dependent threshold detector. The noise covariance matrix is approximated by a parametric multichannel model in which the parameters are estimated from the secondary data. This approach is shown to exhibit better performance than the AMF in non-Gaussian noise. However, the test involves implicit knowledge of the clutter statistics. In References 155 and 156, the effort is

extended to derive a test that requires no a priori knowledge of the noise statistics. The methods presented in this work illustrate robustness with respect to variations in clutter texture power for compound Gaussian clutter and provide a potential approach for cases in which small amounts of reference cells are available to estimate the noise covariance matrix. This trait is important in non-homogeneous environments in which homogeneity only exists for several reference cells. An adaptive matched filter and covariance matrix estimation procedure for target detection in spherically invariant noise are developed in Reference 178. In References 162 to 165, the authors derive a new generalised likelihood ratio test for detecting a signal in unknown, strong, non-Gaussian, low-rank interference plus white Gaussian noise. The algorithm shows robustness to both steering vector mismatch and rank overestimate. Sea clutter is sometimes well modelled using a K distribution. In Reference 180, an algorithm is presented, based on second and higher-order cumulants, which is able to operate in an unknown interference environment by employing both non-parametric and parametric estimation techniques. Detection in clutter that can be modelled as an SIRP (in this case, correlated K distributed clutter) is presented in References 181 and 183, in which Kelly's GLRT is outperformed by a detection algorithm that exhibits CFAR behaviour with respect to the amplitude pdf parameters and to the correlation structure of the disturbance. Additional work on detection in compound Gaussian noise is accomplished in Reference 182, in which a new adaptive receiver has been proposed as a candidate for robust detection. This detector is CFAR with respect to texture statistics and is robust with respect to variations in clutter temporal correlation as well as to mismatch between the design and operating conditions. Further work on robust detection in compound Gaussian clutter plus thermal noise is conducted in References 192 and 193. A performance analysis in terms of the classical covariance matrix estimator as well as that proposed for nonGaussian environments (which is a normalised version of the previously described estimator) is provided in Reference 186. This type of covariance estimator is used in an adaptive linear quadratic (ALQ) detector which is designed to work against a background of correlated non-Gaussian clutter. In Reference 187, the ALQ and GLRT performance are illustrated when confronted with the problem of detection against (non-Gaussian) real sea clutter data. To combat the difficulties caused by the computational complexity of STAP algorithms designed for non-Gaussian environments, less computationally intense, suboptimum detection schemes based on a polynomial approximation of a data-dependent threshold have been suggested in References 184 and 185. It has been argued that sometimes the Gaussian assumption is only partially violated. For instance, radar returns from the ground can be described by a Gaussian distributed process, and the echoes from the sea may follow a K distributed random process. Therefore, in References 189 and 190, a detector is derived for this type of composite disturbance and is compared with those derived based on only Gaussian or K distributed clutter. This work formats the AMF adaptive array processing into the sidelobe canceller form and into the Gram-Schmidt cascaded canceller form similar to References 123 and 124. However, the sensitive least-squares method is replaced

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with a pseudo-median canceller that is much less sensitive to outliers in the data. Also, with no outliers and with an asymptotically large set of reference cells, this robust processor approaches the AMF. Additional work considers this processor cascaded with a reduced-rank scheme in Reference 130. In this effort, the optimal rank reduction capability from the multistage Wiener filter (described in the next section) is retained, and robustness to targets/outliers and non-stationary data are attributed to the pseudo-median canceller. Violation of both the Gaussian and homogeneous data assumptions is considered in Reference 129. The authors introduce a pseudo-median canceller as a robust adaptive array method which significantly reduces the deleterious effects of non-Gaussian, real-world interference with signal contamination on typical array performance metrics. This algorithm also offers protection against signal cancellation.

13.6

Jamming and terrain scattered interference

Jamming represents a significant source of degradation in the target detection procedure. Radar jamming can be caused by either a hostile enemy or unintentionally by a friendly emitter. There are several different modes of operation and sophistication for a jammer. Wideband jammers can be broadband noise-like signals with sufficient spectral width to ensure overlap with the frequency bands that the radar employs. Other jammers, known as intermittent or self-screening, are responsive in nature and attempt to adjust to changing radar frequency bands and signal characteristics [201]. Although jammers can significantly differ in their temporal behaviour, their spatial propagation (and hence the correlation of their signals between channels) is dependent only upon their angular location. Therefore, an exploitation of these spatial correlations, such as the spatial filter shown in Figure 13.Id, can result in significant performance improvement. To properly sample the jammer signal without clutter, the radar can utilise the 'quiet times' between CPIs or use an auxiliary channel. Other works advocate using the over-the-horizon mode of operation to obtain jammer samples without clutter [95]. Some authors attempt to exploit the jamming energy in the sidelobes to suppress a jammer that may be in the mainlobe [62]. Along with the signal received via the direct path between the sensing platform and the jammer, a more deleterious component often exists. The signal of the jammer can bounce off terrain, buildings and other structures before reaching the receiving radar. This reflected signal results in the reception of time-delayed jammer signals, known in the literature as multipath interference, hot clutter or terrain scattered interference (TSI). In regions where the earth is smooth, the multipath signal appears at the same azimuth as the direct path jamming. Both components can usually be cancelled via a single spatial filter, as in Figure 13. Id, or via the design of an antenna beampattern with a low sidelobe response. Unfortunately, in more variable terrain, the jammer signal is scattered off a large portion of ground, including the section covered by the mainbeam of the antenna pattern. The TSI entering the radar through the mainbeam can be considerably larger than that entering through the sidelobes. Thus, the TSI creates a jamming environment in which the interference is distributed spatially in

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with a pseudo-median canceller that is much less sensitive to outliers in the data. Also, with no outliers and with an asymptotically large set of reference cells, this robust processor approaches the AMF. Additional work considers this processor cascaded with a reduced-rank scheme in Reference 130. In this effort, the optimal rank reduction capability from the multistage Wiener filter (described in the next section) is retained, and robustness to targets/outliers and non-stationary data are attributed to the pseudo-median canceller. Violation of both the Gaussian and homogeneous data assumptions is considered in Reference 129. The authors introduce a pseudo-median canceller as a robust adaptive array method which significantly reduces the deleterious effects of non-Gaussian, real-world interference with signal contamination on typical array performance metrics. This algorithm also offers protection against signal cancellation.

13.6

Jamming and terrain scattered interference

Jamming represents a significant source of degradation in the target detection procedure. Radar jamming can be caused by either a hostile enemy or unintentionally by a friendly emitter. There are several different modes of operation and sophistication for a jammer. Wideband jammers can be broadband noise-like signals with sufficient spectral width to ensure overlap with the frequency bands that the radar employs. Other jammers, known as intermittent or self-screening, are responsive in nature and attempt to adjust to changing radar frequency bands and signal characteristics [201]. Although jammers can significantly differ in their temporal behaviour, their spatial propagation (and hence the correlation of their signals between channels) is dependent only upon their angular location. Therefore, an exploitation of these spatial correlations, such as the spatial filter shown in Figure 13.Id, can result in significant performance improvement. To properly sample the jammer signal without clutter, the radar can utilise the 'quiet times' between CPIs or use an auxiliary channel. Other works advocate using the over-the-horizon mode of operation to obtain jammer samples without clutter [95]. Some authors attempt to exploit the jamming energy in the sidelobes to suppress a jammer that may be in the mainlobe [62]. Along with the signal received via the direct path between the sensing platform and the jammer, a more deleterious component often exists. The signal of the jammer can bounce off terrain, buildings and other structures before reaching the receiving radar. This reflected signal results in the reception of time-delayed jammer signals, known in the literature as multipath interference, hot clutter or terrain scattered interference (TSI). In regions where the earth is smooth, the multipath signal appears at the same azimuth as the direct path jamming. Both components can usually be cancelled via a single spatial filter, as in Figure 13. Id, or via the design of an antenna beampattern with a low sidelobe response. Unfortunately, in more variable terrain, the jammer signal is scattered off a large portion of ground, including the section covered by the mainbeam of the antenna pattern. The TSI entering the radar through the mainbeam can be considerably larger than that entering through the sidelobes. Thus, the TSI creates a jamming environment in which the interference is distributed spatially in

azimuth throughout the mainbeam [208,61,77]. Placing a spatial null at a single azimuth in the mainbeam would not be effective, nor would placing nulls everywhere in the mainbeam succeed, since the target signal would also be cancelled. In the TSI problem formulation, the reflection of the radar transmission from the ground is specifically referred to as cold clutter, and the axis of the data cube given in Figure 13.4 denoting PRI and range are referred to as slow time and fast time, respectively. In this chapter, we use the term clutter to refer to cold clutter. TSI typically decorrelates across large time intervals (such as pulses) due to jammer motion. Thus processing across slow time, as done through the utilisation of range cells for the clutter-only scenario, is usually ineffective. However, the TSI will have a large temporal spread, containing significant correlation in fast time. Therefore, filters to ameliorate the problems of TSI must be designed to utilise the fast time correlation and have weights that are unfortunately quite different from those designed to combat the clutter [92,93]. The robustness of STAP algorithms in the presence of both direct path jamming and TSI is important in both military and commercial applications. This robustness is discussed in the following subsections. 13.6.1

Constraining detection

schemes

Designing and implementing constraints to combat the effects of jamming in the mainbeam, while maintaining acceptable performance levels, have recently received significant attention. Unconstrained adaptive algorithms can suppress the interfering signal by placing beampattern nulls at the jammer locations when interfering sources, such as TSI, are located in the mainbeam. However, the resultant beampattern can be distorted so that the desired target signal is cancelled along with the jamming. A potential solution involves the introduction of constraints to specifically prevent cancellation of the desired mainlobe signal while realising mitigation of the jamming [122-124]. The utilisation of multiple linear constraints is also investigated in References 150 and 151, in which adaptation is applied to pre-Doppler data without modification of the desired signal Doppler spectrum. The work in Reference 152 extends the linear constraints analysis to include both STAP in slow and fast time to mitigate both clutter and TSI. The work outlined in References 84 to 86 is combined and extended in References 87 and 88. The efforts confront the problem of TSI mitigation via a method of stochastic constraints and initially consider methods that require the use of range cells with only TSI and no clutter components [87]. However, attempts to remove this limitation are discussed in Reference 88. These publications address applications to the field of high-frequency over-the-horizon radar. The specific approach attempts to achieve TSI suppression while maintaining distortionless clutter post processing, a characteristic that is important when designing two-stage processors.

13.6.2

Two-stage processors

The technique of first using an adaptive spatial preprocessor for wideband noise jammer suppression followed by a STAP processor to promote clutter mitigation is

explored in works such as Reference 28. A frequency sideband, next to but distinct from the radar signal bandwidth, is used to obtain jammer-only training data for the presuppression processor. In some scenarios in which the reference data is limited, this approach has been shown to outperform the method of simultaneously nulling clutter and direct path jamming. When designing two-stage processing techniques, the impact of the first processor on the structure of the data should be taken into consideration for the design of the second stage [141,142,62]. The fact that two-stage algorithms mitigating TSI can modulate the clutter, and thereby degrade the performance of subsequent conventional STAP clutter cancellation techniques, is described in References 93, 92 and 95. In particular, References 92 and 96 present processors that attempt to cancel this modulated interference. The architecture modifies conventional post-Doppler STAP so that it has the freedom to place nulls in locations that will cancel modulated clutter as well as unmodulated clutter. To accomplish this task, the weights applied in the TSI canceller are analysed to determine how the clutter will be modulated. As previously mentioned, Reference 87 designs a two-stage processor utilising stochastic constraints, but reference cells with only hot clutter are required for the method's successful implementation. The work of Reference 88 attempts to remove this requirement through the use of a moving target indicator (MTI) filter to reject cold clutter for a few PRIs. The fast time samples within these PRIs are then used to determine the hot clutter filter for the first stage of the processing. The cold clutter filter is subsequently applied in the second stage. The success of this method depends on the ability of the MTI filter to remove the cold clutter. Other efforts on successfully cancelling jamming and clutter in two-stage processing schemes include References 209 and 210. A method of using antimodulation prefilters is described in Reference 97. This approach involves exploiting the structure of the weight modulation for TSI to construct special prefilters that largely eliminate modulated clutter. The basic approach cancels clutter of interest prior to TSI mitigation, so that when the remaining clutter is modulated, it lies below the thermal noise floor at its destination frequency. Also, the prefilter does not remove clutter from directions absent of modulation that will be cancelled in the STAP stage. Much of this work is summarised in Reference 98, which covers the effects of TSI modulation and clutter, methods to limit false alarms caused by TSI modulation and a variety of techniques to mitigate the effects of TSI and clutter in either two stages or simultaneously. A eountermeasure that is more sophisticated than wideband jamming is coherent repeater jamming, which mirrors the radar signal of interest. The jammer produces a coherent pulse train of the same carrier frequency, modulation type, pulse width and PRF as the radar system to be countered. It has a randomisation of the scan-to-scan range delay of the jammer pulse and an adjustable, linear initial phase progression over the pulse train. The coherent repeater thus produces target-like jamming signals with random range and Doppler. These countermeasures have been shown to degrade performance of algorithms developed under the assumption of homogeneity of the reference cells [21]. Since the jammer components are usually expected to be significantly stronger than targets, the aforementioned non-homogeneity detectors

represent a potential solution to this problem. Other techniques to combat coherent repeater jamming include spatial or temporal smoothing as well as filters designed to utilise constraints to decrease sensitivity to the presence of mainlobe coherent repeater jamming.

13.6.3

Three-dimensional STAP

In recognition of the limitations of two-stage approaches to mitigate the effects of jamming and TSI, some authors have realised that jamming, TSI and clutter environments can be dealt with concurrently in three-dimensional STAP schemes. These schemes are computationally complex and thus costly [201]. For example, References 206 and 207 consider a two-element adaptive canceller which includes both tapped delay lines and bandwidth partitioning. The ability of this system to cancel a direct jammer signal and its diffuse multipath is discussed under the assumption that the jammer signal is much stronger than any desired signal or system noise. Additional temporal degrees of freedom are achieved by using bandwidth partitioning (with a separate adaptive loop in each subband), an adaptive finite impulse response (FIR) filter, or a hybrid system that requires both bandwidth partitioning and adaptive FIR filters. An excellent development of three-dimensional STAP filtering is presented in Reference 208. This effort advocates the use of a processor with adaptive spatial degrees of freedom, as well as two different sets of adaptive temporal degrees of freedom. The two different sets of temporal taps required for each antenna element consist of one set spaced by the PRI, so as to cancel ground clutter, and another spaced by about one-half of the reciprocal of the radar bandwidth, so as to cancel the TSI. In References 141 and 142, the authors provide a three-dimensional STAP methodology for determining performance bounds of three-dimensional STAP when site-specific digital terrain data is used (clutter scattering coefficients are known a priori). In this way, future proposed TSI and clutter mitigation strategies can be compared with the bound presented to ascertain their absolute efficacy rather than only the relative performance. The real-time computing problem presented by two-dimensional STAP is only exacerbated by the addition of a third dimension. Therefore, Reference 143 explores the idea of reduced-rank three-dimensional STAP to reduce computational complexity, and provides a comparison of a multistage Wiener filter to a principal component method of rank reduction. Such methodologies are discussed in the next section. The technique presented in References 58 to 60 presents a compromise between two-dimensional and three-dimensional STAR The authors state that the majority of the clutter, jammer and TSI cancellation can be accomplished via a clutter filter. Further TSI suppression is achieved with a filter calculated by utilising the portion of the data cube corresponding to fast time and only one or a few elements of space and slow time. This technique is called extended beam-augmented STAR A further discussion of full-rank and reduced-rank methods for mitigating TSI is presented in Reference 61.

13.7

Reduction in computational complexity

STAP provides many interesting challenges to the real-time computing community. The computational rigours of computing the filter weights required for two-dimensional fully adaptive processing is demanding, and three-dimensional STAP computations are daunting at best. Thus there has been significant effort by the research community to investigate robust, suboptimal approaches that offer the benefit of a reduction in computational complexity. Several of these methodologies are discussed below.

13.7.1 Reduced-rank methods and covariance matrix tapers Reduced-rank, suboptimal adaptive detection schemes can provide robustness in nonideal detection environments. Reduced-rank methods have the benefit of reduced computational complexity as compared with fully adaptive algorithms. Reduced-rank processing exploits the low-rank nature of the clutter covariance matrix to reduce the required number of reference cells [38]. Since less data is needed, these algorithms can result in better performance than fully adaptive STAP in non-homogeneous environments. In addition, reduced-rank algorithms can be used in conjunction with other techniques such as non-homogeneity detectors to further enhance performance [37]. A general taxonomy of reduced-rank processors is provided in References 116 and 117. Reduced-rank approaches are also discussed in Reference 56. It is interesting to note that although reduced-rank methods have less desirable performance than full-rank methodologies in the case of a known clutter covariance matrix, they can outperform full-rank adaptive processing under non-ideal conditions. This limited performance of fully adaptive methods is due to errors resulting from the estimation process as well as the presence of thermal noise. These methodologies suppress estimation errors at the cost of a bias in the SNR. The net effect is a significant improvement for cases when the interference is truly low-rank [68]. Principal component inverse (PCI) is a modification of the AMF test that can provide a sufficient degree of adaptation with less training data than is required by fully adaptive processing. This technique uses a priori knowledge that, during the observation interval, the noise vectors are composed of a strong component and a background component (in which the strong component is well represented by a set of basis vectors that do not change over the adaptation interval). This principal components method chooses the largest eigenvectors corresponding to the largest estimated eigenvalues to generate an operator that reduces the rank of the data prior to adaptive processing, resulting in a low-rank representation of the full-rank space. PCI estimates the rank from data over the adaptation interval and assumes that the covariance has low rank [221,222]. It has been claimed that the reduced-rank methodology that maximises a cross-spectral metric (CSM) yields performance that is superior to that of PCI [57,64-67,147]. The CSM approach uses the estimated cross-spectral metric to

choose the eigenvectors and is formulated for a prescribed rank. Other modifications of the CSM are included in Reference 144. Rebuttal arguments are presented in References 221 and 222 to illustrate scenarios where PCI performs better than CSM, specifically in low-rank interference in which only a small number of reference cells are available. A third reduced-rank technique, called the eigencanceller, also uses the eigenstructure of the estimated test cell noise covariance matrix. The eigencanceller is an approach in which the weight vector is constrained to lie in the subspace composed of the small eigenvalues of the covariance matrix (corresponding mostly to the thermal noise), which is the subspace orthogonal to the dominant eigenvectors. The improved performance stems from using only the dominant eigenvectors in the formulation of the weight vector, whereas traditional methods use the smaller eigenvectors as well. The eigencanceller provides clutter and jammer cancellation, low variation and distortion of the beampattern, and is less computationally complex than fully adaptive algorithms. The spatial-only problem of mitigating jammer interference is considered in Reference 168. This spatial-only work is extended in Reference 169, in which two different space-time eigencanceller approaches are provided. In References 170 and 171, the eigencanceller is shown to be more robust than some traditional algorithms to estimation errors due to channel-to-channel mismatches (antenna responses are usually assumed to be identical, but due to manufacturing and calibration errors, always have some difference), pointing errors, limited number of reference cells or signal contamination. Another reduced-rank approach, discussed in References 118,119 and 69, results from a multistage representation of the Wiener filter. The Wiener filter is represented as a nested chain of Wiener filters. The reduced-rank approach differs from the fully adaptive approach by limiting the number of filters in the nested chain. One of the benefits of this approach is that it does not have the computational rigours involved in the eigendecomposition of the covariance matrix. In References 119 and 69, the authors demonstrate that under certain conditions the reduced-rank multistage Wiener filter may outperform such eigendecomposition approaches as the CSM (in minimum mean squared error and computational complexity). Some extended analysis of reduced-rank techniques is given in References 166 and 167. These studies consider the variation in performance due to estimation errors of the order of the filter, ICM, aircraft crabbing, the presence of jammers and the size of training data. Apparently, the multistage Wiener filter is less sensitive to the number of stages than is the eigencanceller, and the CSM is sensitive to problems with estimation of the rank of the clutter but is less sensitive to the effects of ICM. One disadvantage of the multistage Wiener filter model is that its order must be obtained a priori in an iterative fashion. Additional analysis of reduced-rank techniques is provided in Reference 144. We have noted that the non-ideal conditions of real-world detection environments can make an accurate calculation of the rank of the clutter covariance matrix difficult. Some of the resulting performance loss can be regained through the implementation of a covariance matrix taper (CMT). An important class of adapted pattern modification techniques is realised by the application of a conformal matrix taper to the original

covariance matrix. The work in Reference 145 reviews the concept of covariance matrix tapers and illustrates that previously introduced techniques to increase jammer notch width are special cases of covariance matrix tapers. CMTs provide diagonal loading benefits to minimise the effects of pattern distortion due to insufficient sample support and weight mismatch due to non-stationary interference. These CMTs can lower sidelobe levels and appropriately widen the notches meant to clear clutter ridges and jammers, thus making them more robust to errors in algorithm assumptions and limited sample support. In Reference 149, a CMT is combined with the principal components method to yield interference mitigation that restores the minimal sample support property, and yet does not suffer from undernulled interference. Such hybrid techniques can add robustness to alleviate the problems from which the principal components methods suffer in cases with severe eigenvalue spreading. This technique is studied further in Reference 146 in which the dominant eigenvectors are initially estimated by utilising a commensurately small sample support. Next, the remaining subdominant interference subspace is synthesised by applying a CMT. The dominant eigenvalues and eigenvectors are obtained in exactly the same manner as in the principal components method. However, to appropriately account for subspace leakage, a CMT is applied to synthesise the subdominant coloured noise space arising from the aforementioned modulation mechanisms. The white noise floor is introduced to properly set null depth and ensure non-singularity. Models of ICM and general knowledge of the environment and system can be used to generate the CMT. Conservative CMTs result in overnulling the clutter.

13.7.2

Techniques implementing limited reference cells

As previously described, the assumption that reference cells are homogeneous may only apply to a grouping of a few reference cells rather than the amount necessary to conduct fully adaptive STAP processing. Therefore, a variety of techniques has been developed to use a smaller number of reference cells, referred to as limited sample support, than that needed by fully adaptive STAR These processing schemes are suboptimal under ideal conditions, but exhibit a robustness in non-homogeneous environments not shared by traditional STAP architectures. One methodology designed for less samples than fully adaptive STAP is called spatial smoothing [205], or forward-backward averaging [111, 112]. This technique provides extra snapshots for covariance matrix estimation by forming subarrays of the spatial arrays. If the resulting loss in resolution produced by using a reduced number of elements at a time is acceptable, the covariance matrix can be smoothed. Therefore, subaperture averaging provides sufficient samples for estimating the covariance matrix in cases in which this calculation would not otherwise be possible and proves useful in sample-starved or non-homogeneous cases. The technique improves SNR and sidelobe levels and has been shown to be robust in the presence of antenna manifold errors. Other efforts, such as References 138 and 139, have explored the utility of smoothing to trade spatio-temporal aperture for increased effective sample support. These investigations show that we may obtain additional

sample support via subspatial and subtemporal aperture smoothing. The reduction in aperture can lead to a widening of the clutter ridge and hence an increase in MDV, but this can be an acceptable exchange for the increased number of samples. The use of diagonal loading to reduce the effects of limited reference cells has also received attention. As previously stated, the covariance matrix typically has several large eigenvalues corresponding to the clutter/jamming, with the remaining eigenvalues corresponding to the thermal noise. With very limited reference cell data, there is inadequate estimation of the thermal noise, and a large thermal noise eigenvalue spread occurs. As more training data is added, the estimation improves and the thermal noise eigenvalues converge to the correct values. If the eigenvalue spread is minimised, the adapted beam shape approaches the ideal pattern. It has been suggested that by increasing the effective value of the thermal noise eigenvalues by creating an artificial thermal noise level, or loading, limited training data performance can be improved. The large interference eigenvalues are minimally affected, but the eigenvalues well below the interference level are significantly increased. This technique can improve the adaptive sidelobe levels and enhance the shape of the mainbeam response when limited training data is available. A reduced null depth and hence an increase in residual interference is the consequence [83,47]. Further study illustrates that diagonal loading helps performance when signal contamination is the only source of non-homogeneity [48]. The work in Reference 140 attempts to determine the ideal scaling for the diagonal loading. It finds this scaling for the case when the number of reference cells used in the adaptive processing is larger than the number of eigenvalues corresponding to the clutter, and argues that this scaling should be set equal to zero when this number is less than the number of eigenvalues corresponding to the clutter. Another method for reducing the possibility of non-homogeneity is the implementation of a multifrequency system. This technique reduces the range from which reference cells need to be taken. It is hoped that the reference cells that are closer to the test cell in range will have a higher probability of homogeneity. In References 16 and 17, the authors suggest the use of frequency diversity. Multifrequency algorithms based on the AMF and the GLRT are developed. These schemes are shown to outperform traditional STAP algorithms in non-homogeneous environments in which a limited number of reference cells are available. Of course, such systems are inherently more complex and costly. An extension of this work is adopted by Reference 179, which proposes a multiband detector for targets in non-Gaussian noise modelled as compound Gaussian clutter. The approach presented in References 136 and 137 is also of interest for reducing the effects of non-homogeneity. Here, a multichannel adaptive whitening filter is obtained from an adaptive least-squares predictive transform signal modelling procedure. This whitening filter attempts to prewhiten the otherwise angle-Doppler correlated multichannel interference signal prior to detection. This type of processing has been shown to approach optimal processing for some cases with much smaller sample support and thus less computational burden. Another suboptimal approach to reduce the required sample support is described in Reference 78.

The subarraying approach in conjunction with a space-time least-squares FIR filter processor can significantly reduce complexity. A comparison of several of these techniques with optimal processing and reduction in computational requirements is presented in Reference 56. Finally, the approach in Reference 224 attempts to model the covariance matrix with only a few parameters as opposed to estimating the entire noise covariance matrix. If there are less parameters to estimate, the paper proposes that the algorithm requires less reference cells than that of a fully adaptive approach.

13.7.3 Low complexity approaches to STAP Real-time computing is one of the many challenges of target detection in complex environments. Reduced-rank methodologies provide less computational load than do fully adaptive techniques. Other approaches to reducing computational load are also of great interest. For example, References 18 to 20, 26 and 27 argue that applying STAP to the full data cube can be computationally wasteful and can result in poor performance when the data set used to estimate the noise covariance matrix is small. The authors suggest that the adaptivity should be applied only to those regions of the processing domain in which some detection improvement is 'necessary, possible, and probable'. The suggested technique is to perform a Fourier transform of the data. Based on some a priori knowledge of the types of interference or on the power distribution across the Doppler frequency bins, the Doppler domain is subsequently divided into flat regions and sharply changing regions. Only a few angular bins close to the transmitted energy are analysed, and all Doppler bins are considered because the target Doppler frequency shift is unknown to the processor. For the flat or nearly flat spectrum regions, the conventional windowed DFT or FIR filter bank has achieved nearly optimum performance. The sharply changing regions denote areas that could result in improved detection performance via adaptive algorithm application. This technique is called the joint-domain localised generalised likelihood ratio (JDL-GLR) processor. A much cited reference concerning suboptimal STAP techniques with reduced complexity is Reference 211. This paper outlines the so-called factored approach, in which Doppler filtering is initially performed on each array element, followed by spatial filtering in each separate Doppler channel. This separation, or factoring, of the temporal and spatial filtering functions leads to considerable computational savings at the expense of some performance loss. The technique advocates a data transform into the space-frequency domain, in which the transformed signals are less correlated. It is this reduction in correlation which promotes an approximation of the covariance matrix which can result in techniques to simplify the normally demanding inversion of the sample covariance matrix. Along with a restriction of the allowed target Doppler, these algorithms can result in good performance with a dramatic reduction in computational load [211]. Work on the factored approach with robustness to limited reference cell data is provided in Reference 225. The work in Reference 157 presents the parametric adaptive matched filter methodology. This method is based on approximating the interference spectrum

with a multichannel autoregressive model of low order. The fact that a low-order autoregressive model provides an accurate representation of simulated and measured interference for a wide variety of system and scenario conditions leads to reduced computational requirements. The modelling fidelity is attained using a small fraction of the amount of reference cells required for fully adaptive STAR It also seems to be tolerant of the presence of targets in the secondary data for both small and large secondary data set sizes. However, this method does not have the desirable CFAR property. Another method for reducing computational load [81] considers how the samples for the covariance matrix estimate are gathered and exactly how that matrix is calculated. Typically, reference cells surround the test cell, with a few guard bands between the test and reference cells. The methods given in Reference 81 illustrate how different training strategies can limit the computational burden. Such methods include using the same weights for a variety of primary data cells. Another method is called the sliding hole, in which a covariance matrix is estimated once using all of the range gates in a range segment. The target range gate and guard bands are removed (the hole). Then the hole slides through the training region. A method for achieving a reduction in computational load is also given in Reference 135, in which the expanding block structure of the covariance matrix is exploited to update the covariance matrix inverse as each pulse is received. The block augmented matrix inversion algorithm results in significant computational gain, and the residual information in the incoming pulse is used to generate an adaptive stopping criterion that addresses the sample data support problem. Several adaptive algorithm techniques have been developed which use only several spatial channels and thus produce a reduction in computational complexity [116,117]. Alternatively, References 41 and 42 discuss a low complexity STAP scheme called adaptive displaced phase centred antenna (DPCA). It is shown that this technique performs well under certain conditions of covariance matrix mismatch and can potentially outperform fully adaptive STAR The technique is based on a pulse cancelling structure, which allows the scheme to cancel clutter with high correlation across several pulses even if this correlation is not present in the training data. This structure can be interpreted as changing the steering vector in the approach [2,3]. One weakness of this scheme is that there are stringent requirements for its application [47]. The problem with non-adaptive DPCA is that it is sensitive to antenna errors and requires the platform velocity to be known well enough to adjust the interpulse period [204]. The work in References 24, 25 and 31 considers a type of STAP denoted as sum/difference STAR An antenna array can be divided into two sections, each with its own beampattern. The sum beam is formed by adding the result of these two subarrays, and the difference beam is formed by subtracting the two returns. Therefore, sum/difference STAP utilises only the sum and difference channels of the radar. It requires a much smaller sample support than do other approaches and seems to perform well in non-homogeneous clutter. Additionally, this technique uses fewer channels than other techniques so that system calibration, which can be costly, is less difficult. The sum and difference channels are processed both spatially and

temporally to minimise clutter effects while preserving the predesigned gain on the desired target signal. Additionally, this technique can be implemented in conjunction with the presuppression of wideband noise jammers using spatially adaptive presuppression [29,30]. The authors argue that the clutter suppression in this twostage processor is almost as successful as full STAP, even though the beampattern is changed before STAP is applied. In Reference 132, a technique is developed which shows faster convergence (and thus less need for sample support) than traditional methods. A maximum likelihood solution under the assumption that the input interference is Gaussian is derived for a structured covariance matrix that has the form of the identity matrix plus an unknown, positive, semidefinite Hermitian matrix. This work is subsequently leveraged to develop a fast converging detector for distributed targets. Other methods that have reduced computational load (and potentially less sample support) and therefore may be more applicable in current computing conditions are provided in References 50 to 54, 82, 39, 40 and 128.

13.8

Conclusions

Many STAP algorithms have been designed under the assumption of ideal conditions in the operational detection environment. Unfortunately, factors such as clutter nonhomogeneity, signal contamination, antenna array errors, jamming and TSI are often present in the real world. To combat the deleterious effects that such components have on radar systems, a variety of techniques have been developed to either circumvent or ameliorate the impact of such conditions. Some of these algorithms introduce a level of complexity greater than that required by the previously proposed schemes, and other techniques offer the benefit of a reduction in computing resources. It is crucial for the commercial user as well as the warfighter to understand the robustness of these algorithms in such challenging environments. Here we have presented a brief review of several traditional STAP schemes as well as an explanation of the complex detection environments in which many radar signal processors must operate. A review of the deterioration of the performance of algorithms, as well as the benefits and drawbacks of newly proposed adaptive processing approaches, has been provided. We discussed that an engineering trade off exists between the design of the algorithm and its computational difficulties in real-world implementation. In the future, we look forward to the potential marriage of several of the presented signal processing techniques to provide overall system robustness and real-time target detection in real-world, non-ideal detection environments. References 1 CURTIS SCHLEHER, D.: 4MTI and pulsed Doppler radar' (Artech House, Inc., Norwood, MA, 1991) 2 BRENNAN, L. E. and REED, I. S.: Theory of adaptive radar', IEEE Trans. Aerosp. Electron. Syst9 March 1973, 9, (2), pp. 237-252

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122 GERLACH, K.: 'Implementation and convergence considerations of a linearly constrained adaptive array', IEEE Trans. Aerosp. Electron. Syst., March 1990, 26, pp. 263-272 123 GERLACH, K. and KRETSCHIMER, F. E.: 'Convergence properties of Gram-Schmidt and SMI adaptive algorithms', IEEE Trans. Aerosp. Electron. Syst, January 1990, 26, pp. 44-56 124 GERLACH, K. and KRETSCHMER, F. E.: 'Convergence properties of Gram-Schmidt and SMI adaptive algorithms: Part II', IEEE Trans. Aerosp. Electron. Syst., January 1991, 27, pp. 83-91 125 SANGSTON, K. J. and GERLACH, K. R.: 'Coherent detection of radar targets in a non-Gaussian background', IEEE Trans. Aerosp. Electron. Syst., April 1994, 30, pp. 330-340 126 GERLACH, K.: 'The effects of signal contamination on two adaptive detectors', IEEE Trans. Aerosp. Electron. Syst., January 1995, 31, pp.297-309 127 GERLACH, K.: 'Adaptive detection of range distributed targets', IEEE Trans. Signal Process., July 1999, 47, pp. 1844-1851 128 STEINER, M. and GERLACH, K.: 'Fast-converging maximum-likelihood interference cancellation'. Proceedings of the IEEE Radar conference, Dallas, TX, May 11-14, 1998, pp. 117-122 129 PICCIOLO, M. L. and GERLACH, K.: 'Fast converging robust adaptive arrays'. Proceedings of the 2001 IEEE Radar conference, Atlanta, GA, May 1-5, 2001, pp. 216-221 130 PICCIOLO, M. L., GERLACH, K., and GOLDSTEIN, J. S.: 'An adaptive multistage median cascaded canceller'. Proceedings of the IEEE Radar conference, Long Beach, CA, April 22-25, 2002, pp. 218-325 131 GERLACH, K. and STEINER, M. J.: 'Fast converging adaptive detection of Doppler-shifted, range-distributed targets', IEEE Trans. Signal Process., September 2000, 48, pp. 2686-2690 132 STEINER, M. and GERLACH, K.: 'Fast converging adaptive processor of a structured covariance matrix', IEEE Trans. Aerosp. Electron. Syst., October 2002, 36, pp. 1115-1126 133 GERLACH, K.: 'Outlier resistant adaptive matched filtering', IEEE Trans. Aerosp. Electron. Syst., July 2002, 38, pp. 885-901 134 GERLACH, K.: 'Spatially distributed target detection in non-Gaussian clutter', IEEE Trans. Aerosp. Electron. Syst., July 1999, 35, pp. 926-934 135 PILLAI, S. U., LEE, W. C , and GUERCI, J.: 'Multichannel space-time adaptive processing'. Record of the twenty-ninth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, October 30-November 1, 1995, 2, pp. 1183-1186 136 GUERCI, J. R., PILLAI, S. U., and KIM, Y. L.: 'Efficient space-time adaptive processing for airborne MTI-mode radar'. Proceedings of the IEEE international conference on Acoustics, speech and signal processing, Atlanta, GA, May 7-10, 1996, 5, pp. 2781-2784

137 GUERCI, J. R. and FERIA, E. H.: 'Application of a least squares predictive transform modeling methodology to space-time adaptive array processing', IEEE Trans. Signal Process., July 1996, 44, pp. 1825-1833 138 PILLAI, S. U., GUERCI, J. R., andKIM, Y. L.: 'Generalized forward/backward subaperture smoothing techniques for sample starved STAP'. Proceedings of the IEEE international conference on Acoustics, speech, and signal processing, Seattle, WA, May 12-15, 4, pp. 2501-2504 139 PILLAI, S. U., KIM, Y. L., and GUERCI, J. R.: 'Generalized forward/backward subaperture smoothing techniques for sample starved STAP', IEEE Trans. Signal Process., December 2000, 48, (12), pp. 3569-3574 140 KIM, Y. L., PILLAI, S. U., and GUERCI, J. R.: 'Optimal loading factor for minimal sample support space-time adaptive radar'. Proceedings of the 1998 international conference on Acoustics, speech, and signal processing, Seattle, WA, May 12-15, 1998, 4, pp. 2505-2508 141 TECHAU, P. M., GUERCI, J. R., SLOCUMB, T. H., and GRIFFITHS, L. J.: 'Performance bounds for interference mitigation in radar systems'. Record of the IEEE Radar conference, Waltham, MA, April 20-22, 1999, pp. 12-17 142 TECHAU, P. M., GUERCI, J. R., SLOCUMB, T. H., and GRIFFITHS, L. J.: 'Performance bounds for hot and cold clutter mitigation', IEEE Trans. Aerosp. Electron. Syst, October 1999, 35, pp. 1253-1265 143 GUERCI, J. R., GOLDSTEIN, J. S., ZULCH, P. A., and REED, I. S.: 'Optimal reduced-rank 3D STAP for joint hot and cold clutter mitigation'. Record of the IEEE Radar conference, Waltham, MA, April 20-22, 1999, pp. 119-124 144 GUERCI, J. R., GOLDSTEIN, J. S., and REED, I. S.: 'Optimal and adaptive reduced-rank STAP', IEEE Trans. Aerosp. Electron. Syst., April 2000, 36, pp. 647-663 145 GUERCI, J. R.: 'Theory and application of covariance matrix tapers for robust adaptive beamforming', IEEE Trans. Signal Process., April 1999, 47, pp. 977-985 146 GUERCI, J. R. and BERGIN, J. S.: 'Rapid adaptation in subspace leakage environments via covariance matrix tapering'. Record of the thirty-fourth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, October 29-November 1, 2000,1, pp. 283-286 147 GOLDSTEIN, J. S., GUERCI, J. R., and REED, I. S.: 'Advanced concepts in STAP'. Record of the IEEE international Radar conference, Alexandria, VA, May 7-12, 2000, pp. 699-704 148 TECHAU, P. M., BERGIN, J. S., and GUERCI, J. R.: 'Effects of internal clutter motion on STAP in a heterogeneous environment'. Proceedings of the IEEE Radar conference, Atlanta, GA, May 1-3, 2001, pp. 204-209 149 GUERCI, J. R. and BERGIN, J. S.: 'Principal components, covariance matrix tapers, and the subspace leakage problem', IEEE Trans. Aerosp. Electron. Syst, January 2002, 38, pp. 152-162 150 GRIFFITHS, L. J.: 'Linear constraints in hot clutter cancellation'. Proceedings of the IEEE international conference on Acoustics, speech, and signal processing (ICASSP), Atlanta, GA, May 7-10, 1996, 2, pp. 1181-1184

151 GRIFFITHS, L. J.: 'Multiple-pulse STAP adaptation prior to radar Doppler processing'. Record of the thirtieth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, November 3-6, 1996,1, pp. 394-398 152 GRIFFITHS, L. J.: 'Linear constraints in pre-Doppler STAP processing'. Proceedings of the IEEE international conference on Acoustics, speech, and signal processing, Munich, Germany, April 21-24, 1997, 5, pp. 3481-3484 153 RANGASWAMY, M. and MICHELS, J. H.: 'A parametric multichannel detection algorithm for correlated non-Gaussian random processes'. Proceedings of the IEEE national Radar conference, Syracuse, NY, May 13-15, 1997, pp. 349-354 154 RANGASWAMY, M. and MICHELS, J. H.: 'Adaptive signal processing in non-Gaussian noise backgrounds'. Proceedings of the ninth Signal processing workshop on Statistical signal and array processing, Portland, OR, September 14-16, 1998, pp. 53-56 155 MICHELS, J. H., RANGASWAMY, M., and HIMED, B.: 'Performance of STAP tests in compound-Gaussian clutter'. Proceedings of the IEEE Sensor array and multichannel signal processing workshop, Cambridge, MA, March 16-17, 2000, pp. 250-255 156 MICHELS, J. H., HIMED, B., and RANGASWAMY, M.: 'Evaluation of the normalized parametric adaptive matched filter STAP test in airborne radar clutter'. Record of the IEEE international Radar conference, Alexandria, VA, May 7-12, 2000, pp. 169-11A 157 ROMAN, J. R., RANGASWAMY, M., DAVIS, D. W., ZHANG, Q., HIMED, B., and MICHELS, J. H.: 'Parametric adaptive matched filter for airborne radar applications', IEEE Trans. Aerosp. Electron. Syst, April 2000, 36, pp. 677-692 158 HIMED, B., SALAMA, Y, and MICHELS, J. H.: 'Improved detection of close proximity targets using two-step NHD'. Record of the IEEE international Radar conference, Alexandria, VA, May 7-12, 2000, pp. 781-786 159 RANGASWAMY, M., HIMED, B., and MICHELS, J. H.: 'Statistical analysis of the nonhomogeneity detector'. Proceedings of the thirty-fourth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, October 29-November 1, 2000, 2, pp. 1117-1121 160 RANGASWAMY,M., HIMED, B., andMICHELS, J. H.: 'Performanceanalysis of the nonhomogeneity detector for STAP applications'. Proceedings of the IEEE Radar conference, Atlanta, GA, May 1-5, 2001, pp. 193-197 161 RANGASWAMY, M., MICHELS, J. H., and HIMED, B.: 'Statistical analysis of the nonhomogeneity detector for non-Gaussian interference backgrounds'. Proceedings of the IEEE Radar conference, Long Beach, CA, April 22-25, 2002, pp. 304-310 162 KIRSTEINS, I. P. and RANGASWAMY, M.: 'Approximate CFAR signal detection in strong low rank non-Gaussian interference'. Proceedings of the IEEE conference and exhibition OCEANS, Providence, RI, September 11-14, 2000, 2, pp. 1305-1308 163 KIRSTEINS, I. P. and RANGASWAMY, M.: 'Approximate CFAR signal detection in strong low rank non-Gaussian interference'. Proceedings of the tenth

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177 CONTE5 E. and DE MAIO, A.: 'A CFAR detection of distributed targets in non-Gaussian disturbance', IEEE Trans. Aerosp. Electron. SySt., April 2002, 38, pp. 612-621 178 CONTE, E., LOPS, M. and RICCI, G.: 'Adaptive matched filter detection in spherically invariant noise', IEEE Signal Process. Lett., August 1996, 3, pp. 248-250 179 LOMBARDO, P., PASTINA, D., and BUCCIARELLI, T.: 6A multiband GLRTLQ algorithm for the coherent radar detection against compound-Gaussian clutter'. Proceedings of Radar 97, Edinburgh, UK, October 14-16, 1997, pp.576-580 180 GINI, R: 'A cumulant-based adaptive technique for coherent radar detection in a mixture of K-distributed clutter and Gaussian disturbance', IEEE Trans. Signal Process., June 1997, 45, pp. 1507-1519 181 GINI, R, GRECO, M. V., SANGSTON, K. J., and FARINA, A.: 'Coherent adaptive radar detection in non-Gaussian clutter'. Record of the thirty-first Asilomar conference on Signals, systems and computers, Pacific Grove, CA, November 2-5, 1997,1, pp. 225-259 182 CONTE, E., LOPS, M., and RICCI, G.: 'Adaptive detection schemes in compound-Gaussian clutter', IEEE Trans. Aerosp. Electron. SySt., October 1998, 34, pp. 1058-1069 183 GINI, F. and GRECO, M. V.: ' Suboptimum approach to adaptive coherent radar detection in compound-Gaussian clutter', IEEE Trans. Aerosp. Electron. Syst., July 1999, 35, pp. 1095-1104 184 GINI, R, GRECO, M. V., and FARINA, A.: 'Clairvoyant and adaptive signal detection in non-Gaussian clutter: a data dependent threshold interpretation', IEEE Trans. Signal Process., June 1999, 47, pp. 1522-1531 185 SANGSTON, K. J., GINI, R, GRECO, M. V., and FARINA, A.: 'Structures for radar detection in compound Gaussian clutter', IEEE Trans. Aerosp. Electron. Syst, April 1999, 35, pp. 445-458 186 GINI, F. and MICHELS, J. H.: 'Performance analysis of two covariance matrix estimators in compound-Gaussian clutter', IEE Proc, Radar Sonar Navig., June 1999,146, pp. 133-140 187 GINI, R, GRECO, M. V, GIANI, M., and VERRAZZANI, L.: 'Performance analysis of two adaptive radar detectors against non-Gaussian real sea clutter data', IEEE Trans. Aerosp. Electron. Syst, October 2000, 36, pp.1429-1439 188 CONTE, E., LOPS, M., and RICCI, G.: 'Asymptotically optimum radar detection in compound-Gaussian clutter', IEEE Trans. Aerosp. Electron. Syst, April 1995, 31, (2), pp. 617-625 189 GINI, R, GRECO, M. V., FARINA, A., and LOMBARDO, R: 'Optimum and mismatched detection against A'-distributed plus Gaussian clutter', IEEE Trans. Aerosp. Electron. Syst, July 1998, 34, pp. 860-876 190 GINI, R, GRECO, M. V., FARINA, A., and LOMBARDO, P.: 'Note on optimum and mismatched detection against ^-distributed plus Gaussian clutter', IEEE Trans. Aerosp. Electron. Syst, January 2001, 37, pp. 296-297

191 GRECO, M., GINI, R, FARINA, A., and BILLINGSLEY, J. B.: 'Validation of windblown radar ground clutter spectral shape', IEEE Trans. Aerosp. Electron. Syst9 April 2001, 37, pp. 538-548 192 LOMBARDO, R, GRECO, M., GINI, R, FARINA, A., and BILLINGSLEY, J. B.: 'Impact of clutter spectra on radar performance prediction', IEEE Trans. Aerosp. Electron. Syst., July 2001, 37, pp. 1022-1038 193 GRECO, M., GINI, R, and DIANI, M.: 'Robust CFAR detection of random signals in compound-Gaussian clutter plus thermal noise', IEE Proc, Radar Sonar Navig., August 2001,148, pp. 227-232 194 GRECO, M., GINI, R, FARINA, A., and BILLINGSLEY, J. B.: 'Analysis of clutter cancellation in the presence of measured L-band radar ground clutter data'. Record of the IEEE international Radar conference, Alexandria, VA, May 7-12, 2000, pp. 422-427 195 McDONALD, K. R and BLUM, R. S.: 'Analytical analysis of STAP algorithms for cases with mismatched steering and clutter statistics'. Record of the IEEE Radar conference, Waltham, MA, April 20-22, 1999, pp. 267-272 196 BLUM, R. S. and McDONALD, K. R: 'Analysis of STAP algorithms for cases with mismatched steering and clutter statistics', IEEE Trans. Signal Process., February 2000, 48, pp. 301-310 197 McDONALD, K. P. and BLUM, R. S.: 'Performance characterization of STAP algorithms with mismatched steering and clutter statistics'. Record of the thirtyfourth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, October 29-November 1, 2000,1, pp. 646-650 198 McDONALD, K. P. and BLUM, R. S.: 'Exact performance of STAP algorithms with mismatched steering and clutter statistics', IEEE Trans. Signal Process., October 2000, 48, (10), pp. 2750-2763 199 McDONALD, K. P. and BLUM, R. S.: 'Performance characterization of spacetime adaptive processing algorithms for distributed target detection in non-ideal environments'. Proceedings of the IEEE Radar conference, Long Beach, CA, April 22-25, 2002, pp. 298-303 200 McDONALD, K. P.: 'Models and analysis of antenna array signal processing systems'. Ph.D. dissertation, Lehigh University, Bethlehem, PA, 2000 201 BRIEMIE, E.: 'Aspects of adaptive beamforming with an AESA radar with subarray architecture'. Proceedings of the IEE colloquium on Electronic beam steering, London, UK, October 28, 1998, 4, pp. 1-9 202 BABU, B. N. S., TORRES, J. A., and GUELLA, T. P.: 'Impact of nearfield scattering on multichannel airborne radar measurements (MCARM)'. Proceedings of the IEEE national Radar conference, Syracuse, NY, May 13-15, 1997, pp. 227-231 203 BARILE, E. C , FANTE, R. L., GUELLA, T. P., and TORRES, J. A.: 'Performance of space-time adaptive airborne radar'. Record of the IEEE national Radar conference, Lynnfield, MA, April 20-22, 1993, pp. 173-175 204 BARILE, E. C , FANTE, R. L., and TORRES, J. A.: 'Some limitations on the effectiveness of airborne adaptive radar', IEEE Trans. Aerosp. Electron. Syst., October 1992, 28, pp. 1015-1032

205 FANTE, R. L., BARILE, E. C , andGUELLA, R. P.: 'Cluttercovariance smoothing by subaperture averaging', IEEE Trans. Aerosp. Electron. Syst, July 1994, 30, pp. 941-945 206 FANTE, R. L.: 'Cancellation of specular and diffuse jammer multipath using a hybrid adaptive array'. Proceedings of the IEEE national Radar conference, Los Angeles, CA, March 12-13, 1991, pp. 54-57 207 FANTE, R. L.: 'Cancellation of specular and diffuse jammer multipath using a hybrid adaptive array', IEEE Trans. Aerosp. Electron. SySt., September 1991, 27, pp. 823-837 208 FANTE, R. L. and TORRES, J. A.: 'Cancellation of diffuse jammer multipath by an airborne adaptive radar', IEEE Trans. Aerosp. Electron. Syst, April 1995, 31, pp. 805-820 209 DAVIS, R. M., FANTE, R. L., CROSBY, W. J., and BALLA, R. J.: 'A maximum likelihood beamspace processor for improved search and track'. Record of the IEEE international Radar conference, Alexandria, VA, May 7-12, 2000, pp. 590-595 210 DAVIS, R. M., FANTE, R. L., CROSBY, W. J., andBALLA, R. J.: 'Amaximum likelihood beamspace processor for improved search and track', IEEE Trans. Antennas Propag, July 2001, 49, pp. 1043-1053 211 DiPIETRO, R. C.: 'Extended factored space-time processing for airborne radar systems'. Record of the twenty-sixth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, October 26-28, 1992, pp. 425-^30 212 SCHARF, L. L. and McWHORTER, L. T.: 'Adaptive matched subspace detectors and adaptive coherence estimators'. Proceedings of the 1996 thirtieth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, November 3-6, 1996, l,pp. 1114-1117 213 McWHORTER, L. T., SCHARF, L. L., and GRIFFITHS, L. J.: 'Adaptive coherence estimation for radar signal processing'. Record of the thirtieth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, November 3-6, 1996,1, pp. 536-540 214 KRAUT, S. and SCHARF, L. L.: 'The CFAR adaptive subspace detector is a scale-invariant GLRT'. Proceedings of the ninth IEEE Signal processing workshop on Statistical signal and array processing, Portland, OR, September 14-16, 1998, pp. 57-60 215 KRAUT, S. and SCHARF, L. L.: 'The CFAR adaptive subspace detector is a scale-invariant GLRT', IEEE Trans. Signal Process., September 1999, 47, pp.2538-2541 216 KRAUT, S., McWHORTER, L. T., and SCHARF, L. L.: 'A canonical representation for distributions of adaptive matched subspace detectors'. Record of the thirty-first Asilomar conference on Signals, systems and computers, Pacific Grove, CA, November 2-5, 1997, 2, pp. 1331-1335 217 SCHARF, L. L., KRAUT, S., and McCLOUD, M. L.: 'A review of matched and adaptive subspace detectors'. Proceedings of the IEEE Adaptive systems for signal processing, communications, and control symposium (AS-SPCC), Lake Louise, Alta., Canada, October 1-4, 2000, pp. 82-86

218 KRAUT, S., SCHARF, L. L., and McWHORTER, L. T.: 'Adaptive subspace detectors', IEEE Trans. Signal Process., January 2001, 49, pp. 1-16 219 SCHARF, L. L.: 'Statistical signal processing: detection, estimation, and time series analysis' (Addison-Wesley Publishing Company, Massachusetts, 1990) 220 KASSAM, S. A. and POOR, H. V.: 'Robust techniques for signal processing: a survey', Proc. IEEE, March 1985, 73, (3), pp. 433-481 221 BREBURGER, B. E. and TUFTS, D. W.: 'Rapidly adaptive signal detection using the principal components inverse (PCI) method'. Record of the thirty-first Asilomar conference on Signals, systems and computers, Pacific Grove, CA, November 2-5, 1997,1, pp. 765-769 222 FREBURGER, B. E. and TUFTS, D. W.: 'Case study of principal component inverse and cross spectral metric for low rank interference adaptation'. Proceedings of the IEEE international conference on Acoustics, speech, and signal processing, Seattle, WA, May 12-15, 1998, 4, pp. 1977-1980 223 KRAUT, S. and KROLIK, J.: 'Applications of maximal invariance to the ACE detection problem'. Proceedings of the thirty-fourth Asilomar conference on Signals, systems and computers, Pacific Grove, CA, October 29-November 1, 2000, pp. 417-420 224 LIN, X. and BLUM, R. S.: 'Robust STAP algorithms using prior knowledge for airborne radar applications', Signal Process., 1999, 79, pp. 273-287 225 WARD, J. and STEINHARDT, A. O.: 'Multiwindow post-Doppler spacetime adaptive processing'. Proceedings of the IEEE seventh Signal processing workshop on Statistical signal and array processing, June 1994, pp. 26-29 226 BOSE, S. and STEINHARDT, A. O.: 'A maximal invariant framework for adaptive detection with arrays'. Proceedings of the IEEE international conference on Acoustics, speech, and signal processing (ICASSP), San Francisco, CA, March 23-26, 1992, 5, pp. 357-360 227 BOSE, S. and STEINHARDT, A. O.: 'A maximal invariant framework for adaptive detection with structured and unstructured covariance matrices', IEEE Trans. Signal Process., September 1995, 43, (9), pp. 2164-2175 228 BOSE, S. and STEINHARDT, A. O.: 'Invariant tests for spatial stationarity using covariance structure'. Proceedings of the IEEE international conference on Acoustics, speech, and signal processing, Minneapolis, MN, April 27-30, 1993, 4, pp. 41-44 229 BOSE, S. and STEINHARDT, A. O.: 'Invariant tests for spatial stationarity using covariance structure'. Proceedings of the IEEE seventh Signal processing workshop on Statistical signal and array processing, June 26-29, 1994, pp. 101-104 230 BOSE, S. and STEINHARDT, A. O.: 'Invariant tests for spatial stationarity using covariance structure', IEEE Trans. Signal Process., June 1996, 44, pp. 1523-1533 231 BOSE, S. and STEINHARDT, A. O.: 'Optimum array detector for a weak signal in unknown noise', IEEE Trans. Aerosp. Electron. SySt., July 1996, 32, (3), pp. 911-922

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232 BOSE, S. and STEINHARDT, A. 0.: 'Adaptive array detection of uncertain rank one waveforms', IEEE Trans. Signal Process., November 1996, 44, (11), pp. 2801-2809 233 NICKEL, U.: 'On the influence of channel errors on array signal processing methods'. Proceedings of the AEU, 1993, 47, (4), pp. 209-219

embedded power of TSD, dB

Previous Page

direction of arrival from target boresight, deg

trial number

Plate 1 Asymptotic SINR loss due to target-like signal corrupting covariance matrix estimate ([13], © 2001 IEEE)

velocity, m/s

Plate 2

SINR loss for each individual Poisson trial when using finite training data corrupted by TSD

trial number

velocity, m/s

Plate 3

SINR loss using homogeneous clutter-plus-noise training data (no TSD), 200 different Poisson trials

Doppler, Hz Doppler, Hz

angle, degrees

angle, degrees

Plate 4

Comparison ofactual (top) and simulated (bottom) MCARMMVDR spectra

Doppler, Hz

slant range, km

Plate 5

Range-Doppler map and sum-channel power versus range curve for MCARM575 data

Plate 6

DelMarVa site-specific radar cross section map including grazing angle dependence ([30], O 2003 IEEE)

velocity, m/s

site specific model

slant range, km

Plate 7 Site-specific synthetic range-Doppler velocity map ([30], © 2003 IEEE)

velocity, m/s

bald earth model

slant range, km

Simulated range-Doppler velocity map using bald earth model ([30], © 2003 IEEE)

depth, m

Plate 8

range, km

Conventional range-depth ambiguity surface with vertical array for a source at 9.06 km and depth of 76 m. The colour scale is the cross correlation coefficient

depth, m

Plate 9

range, km

Plate 10

Conventional range-depth ambiguity surface of the single-element preDoppler MFPfor source at 76 m towed at 2.5 m/s from 9.18 km. The colour scale is the cross correlation coefficient

relative speed, m/s

range, km

Plate 11 Conventional range-speed ambiguity surface of the single-element preDoppler MFPfor source at 76 m towed at 2.5 m/s from 9.18 km. The colour scale is the cross correlation coefficient

own-ship noise

bottom bounce

Hap 'innunzB

Sap 'irjnuiize time, s

time, s

target

responses at 10° azimuth

SP 'I3A3J

Sap 'mnunzB

own-ship bottom bounce target

time, s

Plate 12

time, s

Single-Line BTRs of each signal component are shown in the top two and the bottom-left panels. The beam-time responses at 10° azimuth are shown in the bottom-right panel. The colour scales are in dB

bottom bounce

azimuth, deg

azimuth, deg

own-ship noise

Doppler, Hz

Doppler, Hz target

responses at 10° azimuth

level, dB

azimuth, deg

own-ship bottom bounce target

Doppler, Hz

Doppler, Hz

Plate 13

Single-Line Doppler-/Azimuth responses with a 256-sec integration time of each signal component are shown in the top two and the bottom-left panels. The Doppler responses at 10° azimuth are shown in the bottom-right panel. The colour scales are in dB adaptive plane-wave (10°)

time, min

time, min

conventional plane wave (10°)

Doppler, Hz

level, dB

time, min

Doppler, Hz

adaptive PW(I line) adaptive MFP(I line) adaptive MFP (4_line_vert) Doppler, Hz

Plate 14

time, min

Single-line spectrogramsfor the conventional plane wave and the adaptive plane-wave beamforming at 10° azimuth are shown in the top two panels. The single-line adaptive MFP track-cell spectrogram is shown in the bottom-left panel. The bottom-right panel shows the time histories of the peak responses over Doppler for single-line adaptive plane wave beamforming, single-line adaptive MFP, and 4-vertical-line adaptive MFP The colour scales are in dB

single line, adaptive MFP

time, min

time, min

single line, conventional MFP

range, km

range, km 4_line_vertical, adaptive MFP

time, min

time, min

4_line_sequential, adaptive MFP

range, km

range, km

Plate 15 Array-size dependence of MFP range tracking at target depth and target speed. Target was from 10 km moving in to 6.5 km after 25 min. The top two panels show the track-cell-gram for the single-line conventional MFP and the single-line adaptive MFP The bottom-left panel shows the 4-sequential-line adaptive MFP track-cell-gram. The bottom-right panel shows the 4-vertical-line adaptive MFP track-cell-gram. The colour scales are in dB depth = 60 m

time, min

time, min

depth= 10 m

range, km

range, km

depth= 180 m

time, min

time, min

depth = 90 m

range, km

range, km

Plate 16 Depth discrimination of 4-vertical-line adaptive MFP range tracking at target speed. The top two panels show MFP track-cell-grams at depths of 10 m and 60 m. The bottom two panels show MFP track-cell-grams at depths of 90 m and 180 m. Target was at 90 m in depth and was from 10 km moving in to 6.5 km after 25 min. The colour scales are in dB

speed = 1 m/s

time, min

time, min

speed = 3 m/s

range, km

range, km

speed=—3 m/s

time, min

time, min

speed = - 1 m/s

range, km

Plate 17

range, km

Speed discrimination of 4-vertical-line adaptive MFP range tracking at target depth. The top two panels show MFP track-cell-grams at speeds of 3 m/s and 1 m/s moving toward the array. The bottom two panels show MFP track-cell-grams at speeds ofl m/s and 3 m/s moving away from the array. Target was moving toward the array at 3 m/s and was from 10 km moving in to 6.5 km after 25 min. The colour scales are in dB jammer

cost/!,

Plate 18

Space-frequency characteristic of reverberation and illustration of signal processing. BF = beamformer, DFT= discrete Fourier transform, B ~ ^/Tpuhe- The green curves in plot A and B correspond to the output of the ideal processing with no jammer present. The blue curves in plot C and D refer to the output of the realistic processing with no jammer present. The read lines in all plots correspond to a situation with a jammer

pixel

test data

pixel transformed data (radon domain)

filtered data (radon domain)

0 (degrees)

6 (degrees)

pixel

filtered data

pixel

Plate 19

Illustration of echogram image enhancement. First row: test data image, second row/left side: transformed test data image (Radon domain), second row/right side: filtered transformed test data image (Radon domain), third row: filtered test data image

frequency, Hz

frequency, Hz

distance, m

distance, m

Dependence of Doppler spectrum on range; left side: conventional processing, right side: fully adaptive STAP

Plate 21

Dependence of Doppler spectrum on direction; left side: conventional processing, right side: fully adaptive STAP

frequency, Hz

Plate 20

frequency, Hz

spectrum stave 20, data piece 48, ping 15, 512 FFT

distance, m

Plate 22

Spectrum of a single sensor signal of VLFACTAS

frequency, Hz

frequency, Hz

distance, m

Left side: conventional analysis of sea data; right side: post-Doppler adaptive beamforming of MFACTAS data. In both figures a target can be seen at a distance of about 100Om and a frequency of about 5340Hz

frequency, Hz

frequency, Hz

Plate 23

distance, m

Plate 24

distance, m

distance, m

Left side: conventional analysis of sea data; right side: post-Doppler adaptive beamforming of VLFA CTAS data. The strong signal at afrequency of about 990Hz and a distance of about 12 km is due to a transponder at the target position. This transponder signal is suppressed on the righthand side because the window length M is adapted to the pulse length, see Section 19.2.2

range, kyd

beam number

Original echogram image with target at about 21.5 kyd and beam 8 to beam 15

range, kyd

Plate 25

beam number

Plate 26

Filtered echogram image of Figure 19.18

range, kyd

range, kyd

test statistic

ping number

Left side: echogram of one beam with about 20 sounding periods; right side: test statistic of the proposed detection scheme

range, kyd

Plate 27

beam number

Plate 28

Test statistic of all beams

velocity, m/s

depth, m

location, m

Plate 29

Synthetic data example, diffraction events have not been modelled a near-offset section extracted from the prestack data b common midpoint gather for midpoint location 2000 m

location, km

time, s

time, s

location, km

Plate 30

location, km

time, s

time, s

location, km

Synthetic data example a coherence section b emergence angle section c section of the radius of curvature 7?NIP — ^NIP d section of curvature K^ associated with stacked section shown in Figure 21.6

velocity, m/s

depth, m depth, m

velocity, m/s

location, m

depth, m

velocity, m/s

location, m

location, m

Plate 31

Inversion of the CRS wavefield attributes a reconstructed smooth model with reconstructed normal rays b reconstructed smooth model with reconstructed reflector elements c reconstructed reflector elements superimposed onto the true blocky model

Section B

Miscellaneous space-time processing applications

Part V

Ground target tracking with STAP radar

Chapter 14

Ground target tracking with STAP radar: the sensor Richard Klemm

14.1

Introduction

In this chapter we deal with the problem of tracking moving objects on or close to the ground by means of an airborne or space-based radar. A typical airborne radar geometry is assumed in the numerical examples. However, the results and insights apply as well to spaceborne geometries. Specifically, we focus on the sensor aspects, i.e. we discuss what kind of information a suitable radar can contribute to the tracking function. The material contained in this chapter has been presented at a RTO SET symposium [H]. Ground targets have to compete with strong ground clutter returns. Therefore, clutter rejection is an indispensable feature of the tracking radar. If the radar is carried by a moving platform (e.g. airplane, satellite) the clutter Doppler spectrum is spread out proportional to the radar velocity. Therefore, low Doppler targets may fall into the clutter band which makes them difficult to detect by conventional (temporal only) MTI techniques. As is well known from the existing literature (see for instance the list of references given in Reference 6) space-time adaptive processing (STAP) techniques can solve the problem of detecting slow moving targets within the clutter Doppler band. STAP processors implicitly compensate for the radar platform motion. The principle of STAP including a comparison with conventional (temporal or spatial only) techniques is illustrated in Reference 6, p. 111. For the subsequent calculations, if not denoted otherwise, a notional radar characterised by the parameters listed in Table 14.1 has been assumed. 14.2

Properties of the STAP radar sensor

In this section we briefly summarise some properties of STAP radar as far as required for the subsequent analysis.

Table 14.1 Radar parameters Platform velocity Range Number of elements Wavelength Element spacing Number of processed echoes ^^Nyquist Clutter-to-noise ratio, single element, single pulse Signal-to-noise ratio, single element, single pulse ASEP, spatial dimension ASEP, temporal dimension

Up = 240 m/s 10 km N = 24 X = 0.03 m X/2 (Nyquist) 24 32000Hz 20 dB — 10 dB K = 5 L= 5

14.2.1 Processing techniques 14.2.1.1 The optimum STAP processor [6, Chapter 4] Let us assume an array of Af receive elements and a transmit antenna which transmits M coherent pulses. The (N x 1) vector xm includes the echo signals received by the individual receive elements due to the mth transmitted pulse. Let us introduce vector notation for the following quantities:

(14.1) where c, s, n and j denote the space-time vectors of clutter, target signal, noise and jamming, x is the vector of actual received data which may contain clutter, noise, jamming and a target signal. The optimum processor is given by: w opt = / Q - 1 S = JzD + - 1 D- 1 S

(14.2)

where

(14.3)

is the space-time eovariance matrix of clutter, noise and jamming and s is the space-time signal replica (or matched filter). Since Q is positive definite it can be factorised into two matrices, D* and D. The term D*" 1 in equation (14.2) whitens clutter and jamming in space and time. The term D" 1 equalises the matched filter s for the distortion of a received target signal in x. The efficiency of any processor can be judged by calculating the improvement factor (IF) which becomes for the optimum processor: (14.4) In Figures 14.1 and 14.2 the optimum improvement factor has been depicted for a sidelooking (sensors aligned with flight path) and a forward-looking (sensor arranged in across-flight direction) linear array versus the look angle cp and the normalised Doppler frequency F (clutter + noise, no jamming). As can be seen, there is a clutter trench along the diagonal for the side-looking (SL) and a semicircular trench for the forwardlooking (FL) case. In a certain look direction one obtains a cross section through such a plot which results in an IF as shown by the upper curve of Figure 14.3. Here 0 dB means the theoretical optimum IF, i.e. the maximum achievable IF determined by the receiver noise limitation. It is obvious that the optimum curve reaches the theoretical limit everywhere, except for a narrow clutter notch. Slow targets show up close to the

Figure 14.1 Improvement factor (side-looking array)

versus normalised Doppler and azimuth

Figure 14.2

Improvementfactor versus normalised Doppler and azimuth (forwardlooking array)

clutter notch. Therefore, the width of the clutter notch determines the detectability of a slow target and is, therefore, referred to as minimum detectable velocity (MDV). Since the covariance matrix Q includes the spatial and temporal characteristics of the interference, all processing schemes derived from equation (14.12) are adaptive. For large numbers of sensors TV and radar pulses M the optimum processor cannot be realised for several reasons (computational load, arithmetic accuracy and lack in training data support). However, it may serve as a reference for judging the interference suppression performance of suboptimum architectures with real-time processing capability. 14.2.1.2 The potential of STAP In Figure 14.3 we find a comparison between optimum STAP processing, conventional (beamforming plus temporal only) MTI processing, and just beamforming plus Doppler filtering (no interference suppression). The three curves demonstrate clearly the potential of STAR At the clutter Doppler frequency associated with the look direction (here 45°) we find the clutter notch which is very narrow for the optimum processor, but much broader for one-dimensional processors. There is also considerable sidelobe ripple in the passbands of the one-dimensional processors.

optimum processing beamformer adaptive temporal filter beamformer plus Doppler filter

Figure 14.3

The potential of space-time adaptive processing

14.2.1.3 Subspace STAP processing A wide class of suboptimum processing architectures can be derived from the optimum processor by applying a certain linear transform T to the received data. Such transforms may be space-time by nature [6, Chapters 5, 9] or spatial only [6, Chapter 6]. Then we obtain for the various vector quantities: c T = T*c,

DT = T*n,

sT = T*s,

j T - T*j,

x T - T*x,

Q x = T*QT (14.5)

and the improvement factor becomes in the transformed domain:

(14.6)

The transform matrix T determines the architecture of the individual suboptimum processor. The ASEP (auxiliary sensor/echo processor [6, pp. 278-282]) uses a symmetric sidelobe canceller architecture in both the spatial and temporal dimension. The

transform is given by:

(14.7)

where Wf\ • • • WfM are weights of a Doppler filter at frequency / . The spatial submatrices have the form:

(14.8)

where b\ • • • bs are beamformer weights. The clutter suppression performance of a processor based on the transform of equations (14.7) and (14.8) W1 = Q^1ST

(14.9)

approximates the optimum processor of equation (14.2) very closely, and the computational burden is strongly reduced. The results presented below will be based on this processor.

14.2.2 A rray properties Linear equispaced arrays are particularly useful for the design of suboptimum processing architectures because they can be subdivided into subarrays with uniform shape so that the clutter spectra received by all the subarrays are identical except for a spatial phase due to the subarray displacement. Subtracting, for instance, two identical spectra from each other results in an effective clutter suppression. For more details see Reference 6, Chapter 6, and Reference 8.

14.2.2.1 Side-looking linear array Side-looking arrays are being used for surveillance radar applications (e.g. Joint-STARS, [3]; SOSTAR [5]) and synthetic aperture radar (SAR). A side-looking linear array has a particular aptitude for STAP: the clutter Doppler frequency is range independent. This has a number of advantages pertaining to the realisation of an adaptive GMTI system: 1 STAP filtering is range independent 2 range-ambiguous clutter returns have all the same Doppler frequency and, therefore, produce only one clutter notch 3 there are enough HD data samples to train the adaptive STAP algorithm. Of course, the clutter Doppler is independent of range only as long as the array is aligned with the flight path. If the radar carrier is crabbing to some extent, for instance due to cross-wind drift, some range dependence will occur, and the aforementioned advantages are no longer valid. For more details see References 6 and 10. 14.2.2.2 Forward-looking linear array A typical application of forward-looking arrays is the nose radar of a fighter aircraft (e.g. AMSAR [I]). All the advantageous properties of the side-looking array mentioned above do not apply for the forward-looking array. 14.2.2.3 Omnidirectional arrays The use of omnidirectional arrays has been suggested in Chapter 5 [8] of this volume for use with airborne surveillance radar. Omnidirectional arrays are based on omnidirectional dipole elements as have been used in the crow's nest antenna [18]. 14.2.2.4 Sensor directivity patterns In the following calculations we assume that the array elements have directivity patterns of the form: D((p) = 0.5(1 + cos(2((? -
(14.10)

D(O) = 0.5(1 + cos(2(<9 - #L)))

(14.11)

where the angles #?L and #L denote the look direction of the array element (across-flight for side-looking, in-flight for forward-looking arrays).

14.2.3 Summary of the data output provided by the STAP radar During the tracking process the radar provides a number of data to the tracker: (i) Detection probability The detection probability is associated with the signalto-noise-plus-interference ratio at the output of the radar. (ii) Estimates of the angular target direction and velocity We assume that the radar produces bias-free estimates of these quantities.

(iii)

(iv)

Variances of the target parameters It is assumed that the parameter estimates given by the radar are optimum. Therefore, the Cramer-Rao bounds are used as variances, Variance of the range estimate It is assumed that the variance of the range interval is defined by the response of the conventional matched filter. The standard deviation of the range estimate then becomes:

where Br is the relative system bandwidth.

14.3

The scenario

We will now consider a radar target scenario as depicted in Figure 14.4. The radar platform moves at the height hp in the x-direction at speed vp. The target is supposed to move on a straight line at zero height in the direction cpc (course angle). It is further assumed that the radar revisits the target in uniform intervals.

14.3.1 SNIR and Pd of a moving target Let us assume that the course angle is 0, i.e. the target and the radar are moving on parallel straight paths, see Figure 14.5. Furthermore, we assume a linear side-looking array. Figure 14.6 shows the target (thick) and the radial clutter velocities at the target position. At the beginning both velocities are positive which reflects the dominating

target positions O=O)

radar positions

Figure 14.4

The tracking scenario (perspective view)

target positions

radar positions

Figure 14.5

Radar and target on parallel flight paths (top view)

revisits

Figure 14.6

Radial target (thick) and clutter velocities versus revisit time (target velocity lOm/s)

revisits

Figure 14.7

Radial target (thick) and clutter velocities versus revisit time (target velocity 100 m/s)

impact of the approaching radar carrier. The radial target velocity is slightly lower than the clutter velocity, because the target moves in the same direction at t>t = 10 m/s. After about 75 revisits we notice that the velocity curves are crossing each other. This is the point where the radar overtakes the target. At this moment the radial target velocity is zero. In Figure 14.7 a faster target (vt = 100 m/s) has been assumed. Accordingly, the point, where the radar overtakes the target and the radial velocity is zero, occurs at a later time. By comparing Figures 14.6 and 14.7 we find that for the slow target the velocity curve runs very closely to the clutter velocity curve whereas the faster target produces a velocity curve that is much better separated from the clutter curve. We find for the slow target a sliding intersection whereas the higher target velocity in Figure 14.7 results in a very distinct intersection. Let us now consider the associated signal-to-noise-plus-interference ratio (SNIR) curves shown in Figure 14.8. The upper row shows the slow target (a for white noise and b for clutter plus noise) and the lower row shows the SNIR for the 100 m/s target. The left column shows the SNIR for the clutter-free case, the right for clutter plus noise. There are two reasons for the maximum at the point where the aircraft overtakes the target (Figures 14.8a and c). First, this is the point of closest approach, therefore, due to the radar range equation the signal power becomes maximum. Second, the

Figure 14.8

SNIR versus revisit time, with and without clutter a no clutter, vt = lOm/s b CNR = 2OdB, vt = lOm/s c no clutter, vt = lOOm/s d CNR = 2OdB, vt = lOOm/s

SNIR curve is modulated by the directivity patterns of the antenna elements, both on transmit and receive, whose maximum point is at 90° (across-flight). When the radial target velocity is zero the processor cannot distinguish between target and clutter. In this case, the target is suppressed which results in a clutter notch in the SNIR curve. For the faster target the clutter notch occurs at a later time. 14.3.1.1 Impact of SNR Figures 14.9 and 14.10 illustrate the effect of signal-to-noise ratio (SNR) on the SNIR and the associated probability of detection. The higher the SNR, the narrower the clutter notch becomes. In Figures 14.11 and 14.12 the Cramer-Rao bound for the estimates of angle and radial velocity has been plotted. It can be seen clearly that the standard deviation of the angle and velocity estimates decreases with increasing SNR. This means also that the sensor directivity patterns and the range law according to the radar equation lead to increased standard deviations of both estimates at the beginning and the end of the tracking scene (left and right in Figures 14.11 and 14.12). In the position of the clutter notch the CRB curves show a local maximum.

SNIR, dB

revisits

Figure 14.9

SNIR versus revisit time: impact of the signal-to-noise ratio. SNR/dB = o -10; * O; x 10

revisits

Figure 14.10

Pd versus revisit time. SNR/dB = o -10; * 0; x 10

standard deviation, azimuth, deg

revisits

CRB of azimuth estimate versus revisit time. SNR/dB = o —10; * 0; x 10

standard deviation, velocity, m/s

Figure 14.11

revisits

Figure 14.12

CRB of velocity estimate versus revisit time. SNRJdB = o —10; * 0; x 70

14.3.1.2 Target stop If the ground target stops it behaves like ground clutter and, therefore, is suppressed by the STAP processor. A numerical example is shown in Figures 14.13 and 14.14. The target stops between the revisits 95 and 105. The corresponding clutter notch can be seen clearly. Because of the rectangular shape of this notch the associated probability of detection is nearly independent of the SNR.

14.3.2 System aspects

SNIR, dB

14.3.2.1 System dimensions The width of the clutter notch is a measure for the detectability of slow targets. It depends on the system dimensions (N — number of array elements, M = number of coherent echoes). Larger spatial and temporal apertures result in higher resolution capability and, hence, in narrower clutter notches. This well known fact can be observed in Figure 14.15. On the other hand, the width of the clutter notch depends as well as on the geometry of the tracking scene. For instance, the larger the distance between radar and target, the broader the clutter notch becomes when observed during the revisits. In Figure 14.16 one can notice that (except for the different levels of SNIR) the clutter notches of all four curves have about the same width which indicates that for the given

revisits

Figure 14.13

SNIR versus revisit time. SNR/dB = o -10; * 0; x 10

revisits

Pd versus revisit time. SNR/dB = o —10; * O; x 10

SNIR, dB

Figure 14.14

azimuth, deg

Figure 14.15

Influence of the system dimensions, SNIR versus azimuth (SNR = OdB). N = M = o 6; * 12; x 24; + 48

SNIR, dB

revisits

Figure 14.16

Influence of the system dimensions, SNIR versus revisit time (SNR = OdB). N = M = o 6; * 12; x 24; + 48

scenario the clutter notch is determined by the varying target radar geometry rather than by the system dimensions Af and M. 14.3.2.2

Comparison of STAP with one-dimensional clutter suppression techniques In this section we discuss the advantages of space-time processing over conventional MTI techniques (beamformer plus adaptive temporal clutter filter; beamformer plus Doppler filter bank, i.e. no clutter filtering). The upper curve in Figure 14.17 shows the SNIR versus revisit time as achieved by the ASEP processor defined by equations (14.7) and (14.8). Again, the effects of the radar range equation and the sensor directivity pattern as well as the clutter notches due to tangential motion and target stop can be noticed. The two lower curves show the SNIR versus revisit time of a beamformer/adaptive temporal clutter filter (middle) and a beamformer cascaded with a Doppler filter bank (lower curve). Dramatic losses can be recognised which are reflected in serious losses in detection probability (Figure 14.18). The superiority of space-time clutter rejection over one-dimensional processing techniques is obvious. 14.3.2.3 Forward-looking array Forward-looking radar plays an important role in fighter aircraft. Compared with a side-looking array, the aperture is now turned by 90° and the directivity patterns of the array elements are directed in a forward direction.

ASEP, ,Y= M= 24, K=5,L=5 beamformer plus adaptive temporal filter beamformer plus Doppler filter

revisits

Figure 14.17

SNlR for suboptimum processing

In the scenario shown in Figure 14.5 a remarkable coincidence can be observed. At the time the radar passes the target the radial velocity is zero so that a clutter notch occurs. At the same time the target meets the null of the array element pattern which points in the flight direction. After the radar passed the target is in the back of the antenna and, therefore, invisible. Figure 14.19 shows a numerical example where the cut-off of the SNIR curves can be seen clearly. Let us consider an alternative scenario as shown in Figure 14.20. The aircraft is moving the same as before, however, the target is now crossing its way. At the time the target meets the x-axis the flight direction is perpendicular to the radar look direction so that a clutter notch can be expected. As can be seen in Figure 14.21 the SNIR increases as the radar approaches the target (left to right). This increase is caused by the range law only. There is almost no influence of the directivity pattern of the array elements because the target is in the main look direction of the radar. For a very slow target (a) the clutter notch is smeared out so that no real notch can be recognised. The faster the target is the more distinct the clutter notch becomes. At the end of the scene the radar carrier flies over the target so that subsequent echoes will arrive in the back of the antenna, the SNIR breaks off. The detection probability curves in Figure 14.22 reveal that a slow target with SNR = 0 dB will be detected relatively late because it spends all its time in the

Pd versus revisit time for suboptimum processing a ASEP processing b Beamformer plus adaptive temporal filter c Beamformer plus Doppler filter

SNIR, dB

Figure 14.18

revisits

Figure 14.19

SNIR versus revisit time. SNR/dB = o -10; * 0; x 10

radar positions y = 0,z = hp

target positions

y = D,z=0

Figure 14.20

Radar and target on crossing flight paths (top view)

Figure 14.21

SNIR for the scenario in Figure 14.20 (SNR = 0dB) a Vt = lOm/s b ut = 20m/s c Vt = 30m/s d ut = 120m/s

Figure 14.22

Pdfor the scenario in Figure 14.20 (SNR = OdB) a Vt = lOm/s b ut=20m/s c ut — 30m/s d V1 = 120 m/s

broad clutter notch. Faster targets appear outside the clutter notch and can be detected earlier. The clutter notch is reflected in the detection probability curves.

14.4

Degrading effects

In this section the influence of some radar parameters on the tracking performance of a STAP radar is analysed.

14.4.1 Bandwidth effects In all previous examples it was assumed that the radar is narrowband which means that the system bandwidth has no influence on the spatial resolution capability of the array and, hence, on the width of the clutter notch. Let us consider again a slow target (10 m/s). The velocity curve (thick line in Figure 14.6) runs very closely to the clutter notch (thin line in Figure 14.6). If the clutter is broadened due to the system bandwidth (see grey line in Figure 14.23) the target line will be partly or even totally submerged in the clutter notch. This effect is illustrated in Figure 14.24. The six curves have been calculated for different system

revisits

Figure 14.23

Radial target (black) and clutter (grey) velocities versus revisit time (target velocity lOm/s)

revisits

Figure 14.24

Impact of the system bandwidth (SNIR versus revisits, SL); relative bandwidth [%]; o 10, * 3, x 1,+0.3,DOJ, > 0.03

Figure 14.25

Impact of the system bandwidth (PD versus revisits, SL); relative bandwidth [%] a 1 b 0.3 c 0.1 d 0.03

bandwidths. As can be seen, the SNIR achieved by the STAP processor is lower the larger the bandwidth. It can further be noticed that the difference between the two upper curves (Br = 0.1% and 0.03%) is much smaller than the difference between the lower curves. This indicates that the clutter notch has become so narrow that the slow target shows up in the passband. Figure 14.25 shows the associated detection probabilities for the upper four curves.

14.4.2 Doppler ambiguities The choice of the PRF determines the amount of ambiguity. By choosing a low PRF range ambiguities can be avoided, however, by taking more Doppler ambiguities into account. As a consequence one encounters ambiguous clutter notches (blind velocities). If the PRF is smaller than half the clutter Doppler bandwidth, ambiguous clutter notches will show up inside the clutter Doppler band. Figure 14.26 shows SNIR curves for four different target velocities. If the target velocity is smaller than the unambiguous Doppler range no ambiguous clutter notches occur (a and b). For target velocities larger than the unambiguous Doppler range ambiguous clutter notches show up (c and d).

Figure 14.26

Impact of Doppler ambiguities (SNIR versus revisits, SL, PRF 2000Hz, N = 96); vtgt [m/s] a 5 b 20 c 60 d 120

This can be further illustrated by Figure 14.27. The velocities of a target and the primary clutter velocities are plotted as thick lines. The thin lines denote the ambiguous clutter notches. Notice that there are three intersections between the target line and the ambiguous clutter lines which correspond to the positions of the clutter notches in Figure 14.26c. Target lines for very slow targets (Figures 14.26a,b) would run between the ambiguous clutter lines without crossing them. In Figure 14.28 we consider the same scenario, however, for a radar with a forward-looking array antenna. As we know from Figure 14.19, the primary clutter notch coincides with the null in the directivity pattern of the array element. Therefore, the primary clutter notch should show up at the point where the curves break off. Nevertheless, if the target velocity is high enough, some ambiguous clutter notches can be seen in Figures 14.28c,d.

14.43

Range ambiguities

In the medium PRF mode both Doppler and range ambiguities occur. The clutter returns are then the superposition of several arrivals coming from different directions. For a side-looking array clutter returns are range independent, which means that all

revisits

Figure 14.27

Target and ambiguous clutter velocities (SL, PRF 2000Hz, vt = 60 m/s, N = 96)

Figure 14.28

Impact of Doppler ambiguities (SNIR versus revisits, FL, PRF 2000Hz); vt [m/s] a 5, b 20, c 60, d 120

arrivals contribute to the same clutter notch. For any other array configuration, e.g. a forward-looking array, the ambiguous arrivals exhibit different Doppler frequencies which results in additional clutter notches. Usually one finds two main clutter notches, the primary one and another one caused by all the ambiguous arrivals. Figure 14.29 shows the same scene as Figure 14.28, however with range ambiguities included. It can be noticed that both plots show roughly the same results. Obviously there is no additional degradation of the SNIR curves due to range ambiguities when the radar is used in a track mode. Let us try to explain this behaviour. The track can start only if a detection has been made, that means, a target has shown up in the passband of the STAP filter, i.e. outside of a clutter notch. From one revisit to the next both the target and the radar move a certain distance. Therefore, the radar target geometry changes slightly in angle, range and Doppler. The trajectory of the clutter notch can be considered to be a hyperplane in the three-dimensional (angle, range and Doppler) clutter cube. For the chosen scenarios the target course through the three-dimensional clutter volume was such that no crossing of a clutter notch happened. It is not clear at this time whether this behaviour occurs in all possible radar target scenes, or if there are certain scenarios where the target falls into an ambiguous clutter notch.

Figure 14.29

Doppler and range ambiguities (SNIR versus revisits, FL, PRF 2000Hz); vt [m/s] a 5 b 20 c 60 d 120

14.4.4 STAP radar under jamming conditions In this section we discuss the effect of jamming in addition to clutter. We assume standoff jammers radiating white noise on the radar; the scenario under consideration is shown in Figure 14.30. Up to four stationary jammers are located on a straight line parallel to the flight paths of radar and target. Let us first consider the effect of jamming only. In Figure 14.31 we find SNIR curves for different jammer configurations achieved by the ASEP processor used in the previous examples. Recall from Table 14.1 that a processor with five spatial and five temporal channels has been assumed. We notice jammer notches (1, 2 or 4 jammers) showing up clearly. The case of a very slow target (ut = lOm/s) is under consideration. The performance degrades considerably, however, if additional clutter is involved (Figure 14.32). If the jammer direction is well separated from the clutter notch (a) one finds two distinct notches, where the one close to revisit 50 is due to clutter. If the jammer direction comes closer to the direction associated with the clutter notch then both notches are amalgamated to one bigger clutter-plus-jammer notch (b). In Figure 14.31c we have the case of two jammers separated from the clutter notch. In the case of four jammers the processor runs out of degrees of freedom. Recall that a STAP processor with five antenna channels has only four spatial degrees of freedom (for broadband jammers only the spatial degrees of freedom are relevant). Therefore, it fails to cancel four jammers plus clutter so that the SNIR breaks down and the target will no longer be detected (see the corresponding P& curves in Figure 14.33).

stationary jammers

target positions

radar positions

Figure 14.30

Radar, target and jamming scenario (top view)

Figure 14.31 Jamming only (vt = Wm/s) a jammer #1, b jammer Wl, c jammers #1 & #3, d 4 jammers

Figure 14.32

Clutter and jamming, SNIR versus revisits (vt = lOm/s) a jammer #1, b jammer #2, c jammers #1 & #3, d 4 jammers

Figure 14.33

Clutter and jamming, Pd versus revisits (vt = Wm/s) a jammer #1 b jammer #2 c jammers #1 & #3 d 4 jammers

The SNIR curves in Figure 14.34 are based on a faster target (ut — 100 m/s). The jammer configurations are the same as in Figure 14.32. As can be seen there are no jammer notches visible. The reason is as follows: a jammer notch occurs only if the target and jammer directions coincide, i.e. if jammer, target and radar are roughly (within the resolution capability of the array) located on a straight line. In this case the jammer hits the radar main beam. If the target is very slow there is a high probability that this geometry occurs. For faster target motion it becomes much less likely to encounter this configuration. Therefore, in Figure 14.34 the jammers radiate only in the sidelobes of the array antenna. Notice that the results in Figure 14.34 are still based on STAP processing. Without adaptive processing the SNIR would be considerably lower so that targets would hardly be detected. Notice again the degradation in SNIR in Figure 14.34d due to a lack of spatial degrees of freedom of the STAP radar.

14.5

Issues in convoy tracking

Modern down-look radars have a GMTI mode for detecting moving targets and an imaging mode for target recognition. The imaging mode usually makes use of

Figure 14.34

Clutter and jamming, SNIR versus revisits (vt = lOOm/s) a jammer #1 b jammer #2 c jammers #1 & #3 d 4 jammers

the SAR principle to achieve high geometric resolution (state of the art: less than 10 x 10 cm [2]). With this kind of resolution the application is confined to selected targets or small scenes of interest. The purpose of the GMTI mode is to detect moving targets over a wide area and, therefore, it has to operate at a much lower resolution than the imaging mode. The range resolution can be determined by choosing the system bandwidth, and the azimuth resolution is given by the antenna beamwidth. For fine target location superresolution techniques such as monopulse may be applied. Such techniques have to incorporate interference suppression which leads to the concept of adaptive monopulse [14]. There is to our knowledge no information on convoy tracking available. In the following we develop a few preliminary ideas on this topic which might serve as a guideline for future in-depth research.

14.5.1

Convoy detection by range-only information

In the following considerations on convoy tracking we assume an HRR mode (high range resolution) which means that the members of a convoy can be resolved in range

Figure 14.35 Apparent target length during tracking; Course angle a 0° b 45° c 90° d 135° but not in azimuth. This requires that the range resolution cell is chosen to be equal to or smaller than the target size. The task is to decide on whether a moving convoy is contained in the observed scene or not, and to estimate the velocity vector of the target motion. Such a decision may initialise the imaging mode for finer analysis of the observed scene, for instance, for classification of the individual members of a convoy. Measurement of the apparent length of observed targets by HRR radar has been successfully proven [15]. Including the apparent target length in the tracking algorithm may result in a more precise estimate of the convoy length. In Figure 14.35 some histories of the target length are plotted. It is assumed that the convoy forms a straight line. The angle between the horizontal and the convoy is referred to as the course angle cpc, see Figure 14.4. The four sub figures show the apparent length measured by range-only measurements (full length = 100) during the tracking process for four different course angles. A maximum indicates that the convoy axis is aligned with the radar look direction whereas it becomes zero if the convoy is perpendicular to the look direction.

14.5.2

Convoy detection by azimuth variance analysis

Let us assume again a number of vehicles arranged in a straight line (convoy axis). As stated before, no length measurement can be made from range profiles if the convoy

axis is perpendicular to the radar look direction. However, in this case, the whole convoy shows up in a single range cell (assuming that the range resolution is chosen accordingly). The radar observes, therefore, a target extended in azimuth. The idea is to exploit this extension to detect a convoy by interpretation of the radar response. A localisation technique such as monopulse, which is based on a point target model, will produce a higher variance of the azimuth estimate for an extended target than for a point target. It can be expected, therefore, that the variance of the azimuth estimate runs like the length curves in Figure 14.35, however, upside down. It should be noted that in this procedure the mismatch between the radar sensor (monopulse angle estimator matched to a point target) and a target with geometrical extension is exploited. Therefore, the Cramer-Rao bound cannot be used as an indicator because it is based on perfect match between radar and target. Let us now consider two typical scenarios. 14.5.2.1 Convoy in march formation If the convoy is in march formation (see Figure 14.36) the maximum of the azimuth variance will occur at the time the convoy moves perpendicular to the radar look direction. At this point, the apparent azimuthal width observed by the radar is maximum. In a clutter-free zone this might give a significant increase of the azimuth variance which can be interpreted as 'convoy present'. If the scene is, however, buried in ground clutter this maximum occurs exactly in the position of the clutter notch because the radial target velocity is zero.

convoy positions

radar positions

Figure 14.36

Radar-target scenario (march formation)

It is well known from earlier publications (References 16 and 17, and in some detail in Reference 6, Chapter 14) that the variance has a maximum at the location of the clutter notch. It follows that for a convoy in march formation, according to Figure 14.36, the maximum of the azimuth variance due to the extension of a target coincides with the maximum due to the clutter notch. The increase in variance due to target extension will probably be masked by the increase due to clutter. 14.5.2.2 Convoy in echelon formation If the motion of the convoy is perpendicular to the convoy axis (see, for example, Figure 14.37), the radial target velocity is maximum when the convoy axis is perpendicular to the radar look direction. Therefore, for this scenario an increase in azimuthal variance due to the geometrical extension of a target can be expected to be separable from the clutter effects. Some preliminary calculations have shown that the variance of the adaptive monopulse due to an extended target will be significant only if the radar can resolve the vehicles in the convoy. 14.5.2.3 Matching the response of extended targets The well known conventional beamformer is matched to a plane wave radiated from a point source in the far field. Instead, one might design a generalised beamformer which is matched to an extended target. Such a beamformer has been used, for instance, to improve the location accuracy of acoustic arrays in shallow water [9]. Similar techniques have been proposed to make an adaptive array robust against

convoy positions

radar positions

Figure 14.37

Radar-target scenario (echelon formation)

wideband interference [4]. In both applications the targets appear to be spread in angle. Application of such a technique would require that the array beamwidth is narrower than the extended target. This requirement is somewhat contradictory to our starting point which was the identification of a convoy with a radar with high range but low azimuth resolution. It remains an open issue.

14.6

Summary

In this chapter the perspectives of STAP radar for application to ground target tracking have been pointed out. The role of the radial target velocity was put forward. If the radial velocity becomes zero, as in the case of tangential motion or a stop, the STAP processor cannot distinguish between the target and the ground clutter and, therefore, suppresses the target. Several degrading effects have been discussed, including the influence of a large bandwidth, jamming and radar ambiguities. A few preliminary thoughts on tracking convoys of vehicles have been added which might be useful to inspire further research in this field. In the next chapter (Chapter 15), W. Koch will analyse the impact of the presented results on ground target tracking.

References 1 ALBAREL, G., TANNER, J. S., and UHLMANN, M.: 'The trinational AMSAR programme: CAR active antenna architecture'. Radar'97, 14-16 October 1997, Edinburgh, Scotland, pp. 344-347 2 BRENNER, A. R. and ENDER, J. H. G.: 'First experimental results achieved with the new very wideband SAR system PAMIR'. Proceedings of EUSAR 2002, 4-6 June 2002, Cologne, Germany, pp. 81-86 3 COVAULT, C : 'Joint-Stars patrols Bosnia', Aviation Week & Space Technology, February 1996, pp. 44-49 4 GERSHMAN, A. B., SEREBRYAKV, G. V., and BOEHME, J. F.: 'Constrained Hung-Turner adaptive beam-forming algorithm with additional robustness to wideband and moving jammers', IEEE Trans. Antennas Propag., March 1996, 44, (3), pp. 361-367 5 HOOGEBOOM, P., et al.: 4SOSTAR-X, a high performance radar demonstrator for airborne ground surveillance'. Proceedings of EUSAR 2000,23-25 May 2000, Munich, Germany, pp. 825-827 (VDE Publishers) 6 KLEMM, R.: 'Principles of space-time adaptive processing' (IEE Publishing, London, UK, 2002, 2nd edn.) 7 KLEMM, R.: 'Effect of ambiguities on GMTI radar'. Proceedings of IEE Radar 2002, Edinburgh, Scotland, 2002, pp. 148-152 8 KLEMM, R.: 'STAP with omnidirectional antenna arrays', in KLEMM, R. (Ed.): 'Applications of space-time adaptive processing', this volume

9 KLEMM, R.: 'Low-error bearing estimation in shallow water', IEEE Trans. Aerosp. Electron. Syst., July 1982, 18, (4), pp. 352-357 10 KLEMM, R.: 'Effect of aircraft crabbing on sidelooking STAP radar'. EUSAR 2002, Cologne, Germany, 4-6 June 2002 11 KLEMM, R.: 'Ground target tracking with STAP radar: the sensor'. RTO SET symposium fall 2003, Budapest, Hungary, 15-17 October 2003 12 KOCH, W. and KLEMM, R.: 'Ground target tracking with STAP radar', IEE Proc, Radar Sonar Navig, June 2001, 148, (3), pp. 173-185 13 KOCH, W.: 'Effect of Doppler ambiguities on GMTI tracking'. Proceedings of IEE Radar 2002, Edinburgh, Scotland, 2002, pp. 153-157 14 NICKEL, U.: 'Performance of corrected adaptive monopulse estimation', IEE Proc, Radar Sonar Navig., February 1999,146, (1), pp. 17-24 15 SCHILLER, J.: private communication 16 WARD, J.: 'Cramer-Rao bounds for target angle and Doppler estimation with space-time adaptive processing radar'. Proceedings of 29th ASILOMAR conference on Signals, systems and computers, 30 October-2 November 1995, pp.1198-1203 17 WARD, J.: 'Maximum likelihood angle and velocity estimation with space-time adaptive processing radar'. Proceedings of ASILOMAR 96, 4-6 November 1996 18 WILDEN, H. and ENDER, J.: 'The crow's nest antenna - experimental results'. IEEE international Radar conference, Arlington, VA, 1990, pp. 280-285

Chapter 15

Ground target tracking with STAP radar: selected tracking aspects Wolfgang Koch

15.1

Introduction

Ground surveillance aims at near real-time production of a dynamic ground picture. This task comprises track extraction and track maintenance of single ground moving vehicles and convoys, mobile weapon systems or military equipment, as well as lowflying targets such as helicopters. As ground target tracking is a challenging problem, all available information sources must be exploited, i.e. the sensor data themselves as well as background knowledge about the sensor performance and the underlying scenario. For long-range, wide-area, all-weather and all-day ground surveillance operating at high data update rates, GMTI radar proves to be the sensor system of choice (GMTI: ground moving target indication). By using airborne sensor platforms in stand-off ground surveillance applications the effect of topographical screening is alleviated, thus extending the sensors' field of view. Chapter 14 is devoted to characteristic problems of signal processing related to GMTI tracking with STAP radar, whereas here we discuss selected tracking aspects, which arise from using more appropriate sensor models and exploiting background information. The following topics are of particular interest: (i) Doppler blindness Even after platform motion compensation by using STAP techniques [9], ground moving vehicles can well be masked by the clutter notch of the sensor. This physical phenomenon directly results from the low-Doppler characteristics of ground moving vehicles and causes interfering fading effects that seriously affect track accuracy and track continuity. Unless appropriately handled, Doppler blindness can cause serious problems, which seem to be even more difficult in the presence of Doppler ambiguities.

(ii) Road map information Even military targets usually move on road networks, whose topographical coordinates are known in many cases. Digitised topographical road maps, such as those provided by geographical information systems (GIS), should therefore enter into the target tracking and sensor data fusion process. The problem of battlefield surveillance is not considered here (i.e. transition to off-road targets). (iii) Sensor data fusion Since a single GMTI sensor on a moving airborne platform can record the situation of interest merely over short periods of time, sensor data fusion proves to be of particular importance. The data processing and fusion algorithms used for ground surveillance are closely related to the statistical, logical and combinatorial methods applied to air surveillance. A GMTI radar sensor produces measurements of kinematical target parameters and possibly false returns caused by residual clutter, for example. In addition, measurement errors and sensor parameters are provided. The resulting tracks represent the kinematical states of the targets along the corresponding accuracies and as such are prerequisites for producing a ground picture. We here discuss track maintenance for well separated vehicles for the case of low residual clutter. The effect of road map information and data fusion is addressed. Track initiation/extraction and track termination [7, 8] are not serious problems in the situations considered here. References 11 and 12 are standard references explicitly devoted to the problem of GMTI tracking with single and multiple sensors. For particular aspects see References 1 and 21. The impact of sensor modelling on GMTI tracking is discussed in References 19, 15 and 17. For a more general introduction into modern target tracking algorithms see References 3, 2 and 14.

15.1.1 Discussion of an idealised scenario Assuming a flat earth, Figure 15.1 shows an idealised scenario with two airborne GMTI sensors observing a ground vehicle moving with constant speed (15m/s = 54 km/h) parallel to the x-axis for most of the time. This situation is typical of standoff or gap-filling ground surveillance missions. In the second half of the observation period over Atmax = 25min the target stops for 7min. Then it speeds up again reaching its initial velocity. Finally, the target leaves the field of view of sensor 2. In Table 15.1 selected sensor and platform parameters are summarised. hp, vp denote the constant height and speed of the sensor platforms over ground; Ar, Acp are the range and azimuth regions covered by each sensor during observation. Here and in the following the azimuth angle