RF Architectures & Digital Signal Processing Aspects of Digital Wireless Transceivers
Mohamed K. Nezami, Ph.D., KI4CUA © 2003 I
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers Mohamed K. Nezami, Ph.D., KI4CUA
[email protected] Chapter 1 : RF Architectures and Digital Wireless Transceivers 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16
Heterodyne and Homodyne Receiver Architectures 1-2 Image Rejection Receiver Architectures 1-6 Principle of Digital Down Converter and Sample Decimation 1-8 Alternatives to Direct Conversion Receivers 1-12 Quadrature Mismatch and Image Rejection 1-13 Impact of the Quadrature Imbalances on the Receiver Bit Error Rate 1-16 Time Variant and static (time invariant) DC Offsets 1-17 DC Offset Removal Algorithms 1-18 Tone-Aided Quadrature Imbalance Estimation and Compensation 1-19 Frequency Domain Quadrature Imbalance Estimation and Compensation 1-20 Preambles Quadrature Imbalance Estimation and Compensation 1-25 Quadrature Imbalance Estimation and Correction in Image Rejection Receiver 1-28 Digital Transmitter Architectures 1-30 Quadrature Imbalances in Digital Transmitters 1-32 Multi-Carrier Transmitter Architectures 1-37 References 1-41
Chapter 2 : Introduction to Coherent Demodulation 2.1 2.2 2.3 2.4 2.5 2.6 2.6.1
Order of Synchronization in Digital Receivers 2-2 Impact of Symbol Timing Coherency on Demodulation 2-3 Illustration Example : Synchronization presence form IF to baseband Impact of Carrier Coherency on Demodulation 2-5 Classification Based on Implementation Approaches 2-13 Modeling Channel Impairments and Synchronization Errors 2-18 Model for Symbol Generation, Pulse Shaping, and Match Filtering
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2.6.2 2.6.3 2.7
Model for Intentional Synchronization Error Insertion2-26 Model for Additive White Gaussian Noise Channel 2-27 References 2-30
Chapter 3: Feedback Carrier Synchronization Systems 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.4 3.5 3.6 3.7 3.8 3.8.1 3.9 3.10 3.11 3.11.1 3.11.2 3.11.3 3.12 3.13 3.14 3.14.1 3.14.2 3.14.3 3.15 3.16
Introduction to Feedback Synchronization 3-2 Principle of Analog Phase Locked Loops 3-6 Second order phase locked loop 3-7 Second Order Loop Dynamic Behavior 3-9 Digital phase lock loop Principle 3-11 Loop Filter Digital Implementation 3-13 Illustration Example: Simulated PLL in Fading Channels 3-15 BPSK Carrier Recovery using Squaring Loops 3-20 Principle QPSK Carrier Recovery using Quadrupling Loops 3-22 Minimum-Shift Keying (MSK) Carrier Recovery using Squaring Loops 3-23 Carrier Recovery using Costas Loops 3-26 Digital Costas BPSK Carrier Recovery Loop 3-28 Digital Costas QPSK Carrier Recovery Loop 3-30 Digital Tanlock carrier recovery loops 3-32 Carrier Phase Lock Indicators 3-38 Automatic Frequency Control Loops (AFC) 3-39 Discrete Fourier Transform based AFC 3-41 Dual BandPass Filter AFC method 3-42 Dual NCO AFC System 3-43 Combining AFC with carrier phase tracking Loops 3-46 Carrier Frequency Lock Indicators 3-46 Carrier Frequency Acquisition and Phase Tracking Modem Example 3-47 Acquisition Loop 3-48 Carrier Tracking Loop 3-50 Carrier Tracking Digital Loop Filter Design 3-51 Dealing with carrier phase ambiguity 3-55 Reference 3-57
Chapter 4: Feedback Symbol Timing Synchronization and Automatic Gain Control 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.3 4.4 4.5 4.6 4.7 4.7.1 4.7.2 4.7.3 4.8
Mid-Phase Integration Symbol Timing Detector 4-2 Mid-phase integration Symbol Timing Detector for DBPSK signals Early-Late Gate Symbol Timing Detector 4-5 Muller-Muller (M&M) Symbol Timing Detector 4-8 Gardner BPSK Symbol Timing Detector 4-10 Gardner QPSK Symbol Timing Detector 4-11 Timing Correction Interpolator Filters 4-13 Symbol Timing Loop Filter Design 4-18 Symbol timing lock Indicator 4-19 Preamble-Aided Symbol Timing Synchronization 4-20 Symbol Timing Synchronization using Unique Word 4-21 Automatic Gain Control (AGC) Loops 4-23 Linear Signal Magnitude Based AGC Algorithm 4-23 Log Signal Magnitude Based AGC Algorithm 4-26 Exponential Based AGC Algorithm 4-28 REFERENCES 4-29
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Chapter 5: Introduction to Feedorward Synchronization 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
Feedforward Symbol Timing Recovery 5-2 Feedforward Carrier Frequency Offset Recovery 5-2 Feedforward Carrier Phase Recovery 5-3 Maximum Likelihood Principle 5-3 Maximum Likelihood Estimation Lower Bounds 5-7 Synchronization Error Impact on Receiver BER Performance 5-10 Equivalence Nature Between FF and FB Synchronization Systems References 5-17
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Chapter 6: Feedforward Symbol Timing Synchronization Algorithms and Mitigation of Fading Impact on Receiver 6.1 6.2 6.3 6.4 6.5 6.6 6.6.1 6.7 6.8 6.9 6.10 6.11 6.12 6.12.1 6.12.2 6.13 6.14 6.15 6.16 6.16.1 6.16.2 6.16.3 6.16.4 6.16.4.1 6.16.4.2 6.16.4.3 6.17 6.18
ML Feedforward Synchronization Principle 6-3 Variances of Feedforward Symbol Timing Estimator Output 6-5 ML-Based Symbol Timing Estimation Algorithms 6-5 Data and Decision Aided Symbol Timing Estimation Algorithms Spectral Line NDA Symbol Timing Estimation Algorithm 6-8 DFT-Based NDA Symbol Timing Estimation 6-10 Removing Dependency on Frequency Offset 6-12 Impact of Nonlinearity Type on Feedforward Symbol Timing Estimation Impact of Roll-off Factor on Feedforward Symbol Timing Estimation Feedforward Symbol Timing Correction using Interpolators 6-18 Performance of NDA Symbol Timing Estimation in AWGN 6-22 Performance of NDA Timing Algorithms in Presence of Frequency Offset Impact of Fading Channels on Feedforward Symbol Timing synchronization Flat Fading Channel Model 6-26 Impact of LCR and AFD on symbol timing estimates 6-32 Frequency Selective Mobile Channel Model 6-42 Impact of Fading on Feedforward NDA Symbol Timing Estimation Schemes for Improving NDA Symbol Timing in Fading Channels Schemes for Improving NDA Symbol Timing in Fading Channels Optimizing the Observation Interval Based on Fading Frequency Overlapping Observation Intervals 6-56 Estimate Post Processing (SPP) 6-57 Fly wheeling through Fading Durations 6-58 Performance in Flat Fading Channels 6-59 Performance in Selective Fading Channels 6-60 Performance in Rician Fading Channels 6-62 Illustrating Example: Non-Data Aided Symbol Timing offset estimation References 6-68
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6-13 6-16
6-25 6-26
6-48 6-50 6-54 6-55
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Chapter 7: Feedforward Carrier Frequency and Carrier Phase Offsets Estimation Algorithms 7.1 7.2 7.3 7.4 7.5 7.5.1
Introduction to Feedforward Carrier Recovery 7-4 Problems Associated with Feedback Carrier Recovery Schemes Principle of Open Loop Feedforward Carrier Recovery 7-7 Estimating the Error Variance and Lower bounds 7-8 Feedforward Frequency Estimation Algorithms 7-9 The M-Power NDA Frequency Offset Estimation Algorithm 7-9
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7.5.2 7.5.3 7.5.4 7.5.5 7.5.7 7.5.8 7.6 7.6.1 7.6.2 7.7 7.7.1 7.7.2 7.8 7.9 7.10 7.11 7.12 7.12.1 7.12.2 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21
Viterbi NDA Frequency Offset Estimation Algorithm 7-11 DFT-based NDA Frequency Offset Estimation Algorithm 7-12 Window Enhanced DFT-based NDA Frequency Estimation Algorithm 7-16 Symbol Auto-Correlation Based Frequency Estimation Algorithms 7-18 Frequency Offset Estimation Using Adaptive Digital Filter 7-23 Frequency Estimation using the Linear Least Square Curve fit method 7-23 Feedforward Phase Estimation Algorithms 7-25 The M-Power NDA Carrier Phase Estimation Algorithm 7-25 Viterbi NDA Carrier Phase Estimation Algorithm 7-27 Extension of M-Power NDA Phase Carrier Estimator to QAM 7-33 Removing MQAM PSK Modulations 7-33 Simulated M-Power 16QAM Carrier Phase Estimation Performance 7-35 Proposed Scheme for NDA Carrier Recovery for 16-QAM Modulations 7-38 Effects of Frequency Residual on Phase Estimation 7-40 Phase Ambiguity in M-Power Carrier Phase Estimation 7-43 Cycle Slipping Issues 7-43 Impact of fading on M-power NDA FF phase estimators 7-44 Estimated Phase Error Variance in AWGN Channels 7-44 Estimated Phase Error Variance in Fading Channels 7-47 Illustration Example: Computer Simulation of Data-aided Frequency Estimation 7-49 Illustrative Example: Computer Simulated Data-aided Carrier Phase Estimation 7-58 Illustrating Example: Non-Data Aided Carrier Frequency Offset Estimation Algorithms 7-59 Illustration Example: NDA Carrier Phase Estimation Algorithms 7-64 Illustrating Example: Phase Estimation Ambiguity Estimation 7-68 Illustrating Example: Carrier Frequency Offset Estimation to CPFSK signals 7-70 Illustrating Example: Carrier Frequency Offset Estimation to CPFSK signals 7-71 Illustrating Example: Diversity Combining of Synchronization Parameters 7-73 References 7-75
Chapter 8 : Carrier Acquisition and Carrier Tracking for Burst TDMA Satellite and Mobile Radio Receivers 8.1 8.2 8.3 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.5 8.5.1 8.5.2 8.5.3 8.6 8.6.1 8.6.2 8.6.3 8.7 8.7.1 8.7.2 8.7.3 8.7.4 8.7.5 8.8
Preamble-based Carrier Recovery Techniques for Satellite Receivers Sources of Carrier Frequency Offset in Satellite Systems 8-4 Impact of Satellite Channel on Carrier Recovery 8-5 Conventional Burst Satellite Carrier Acquisition and Tracking 8-6 Frequency Offset Error detector 8-7 Phase Error Detector 8-8 Loop Filter Design 8-13 Simulations Performance 8-14 Estimating and Tracking Carriers with Doppler Rate of Change Third Order Loops Feedback Doppler Rate of Change Estimator Feedforward Doppler Rate of Change Estimator 8-18 Least Square Based Doppler Rate of Change Estimator 8-19 NDA Feedback Carrier Recovery Scheme 8-19 Details of the Acquisition and Tracking Algorithm 8-22 Performance Studies 8-23 Adaptive State Machine Based Carrier Recovery Scheme 8-27 DFT-aided Carrier Recovery Scheme 8-30 Details of the Algorithm 8-30 Coarse Acquisition Loop (DFT-aided Open Loop) 8-33 DFT Peak Search Refinement 8-36 DFT Based Fine Acquisition Loop (FF Loop) 8-38 Simulations Performance 8-38 NDA Extension of the DFT-aided Open Loop Algorithm 8-44
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8.9 8.10 8.10.1 8.10.2 8.10.3 8.11 8.11.1 8.11.2 8.11.3 8.12
Dual-Chirp Tone Aided Carrier Acquisition Scheme and Tracking Practical DSP Implementation Issues 8-51 Complex Magnitude Approximation 8-51 Complex Division Approximation 8-52 Complex ATAN Approximation 8-53 Illustrating Example: Carrier, symbol timing, and channel tracking Illustrating Example: Carrier phase racking 8-57 Illustrating Example: Channel Gain Tracking (AGC) 8-61 Illustrating Example: Symbol Timing Tracking 8-61 Reference 8-62
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Chapter 9: Synchronization in Spread Spectrum Communication Systems 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10
Principle of DS-SS PN Code Acquisition and Tracking 9-1 PN-Code Acquisition Algorithm 9-4 Probability of Acquisition Detection and Probability False Alarm PN-Code Tracking algorithm 9-7 Tau-Dither PN Tracking Loop 9-14 Feedforward PN Code Synchronization Algorithm 9-15 Pilot-aided PN Code Synchronization Algorithm 9-18 Decision-Directed PN Code Synchronization Algorithm 9-20 Frequency-Hopped Speared Spectrum Synchronization 9-21 References 9-27
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Chapter 10: Synchronization in Orthogonal Frequency Division Multiplexing (OFDM) Systems 10.1 Introduction 10-1 10.2 OFDM Modulator 10-4 10.3 OFDM Demodulator 10-5 10.4 Guard Interval Insertion 10-7 10.5 OFDM System Parameter Design 10-8 10.6 Impact of Frequency Offset Synchronization Errors on OFDM Reception 10.7 Impact of Symbol Timing Synchronization on OFDM Reception 10-16 10.8 Impact of Sampling Time Synchronization Errors on OFDM Reception 10-17 10.9 SNR Degradation Due to Carrier Frequency Offsets 10-18 10.10 SNR Degradation Due to Sampling Frequency Offsets 10-19 10.11 SNR Degradation due to Carrier Phase Noise Offsets 10-20 10.12 OFDM Synchronization Algorithms 10-24 10.13 Synchronization Algorithm using Cyclic Prefix 10-25 10.13.1 Maximum Likelihood Synchronization using Cyclic Prefix 10-26 10.13.2 Another Variant of Maximum Likelihood Synchronization Scheme 10-29 10.14 Pilot-Aided OFDM Synchronization Algorithms 10-30 10.15 Dual Pilot Tone Synchronization Method 10-34 10.16 Compensation of Carrier Frequency Offset 10-35 10.17 Illustrative Example 10-36 10.18 Reference 10-47
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Chapter 11: Techniques for Acquiring and Tracking Signals with Efficient Modulations
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11. 11.1.1 11.1.2 11.2 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.4 11.5 11.6 11.7 11.8 11.8.1 11.9 11.10
1 Concatenated Turbo Coded Signals 11-2 Turbo Encoder 11-3 Turbo Decoder 11-3 Carrier and Symbol Timing Acquisition and Tracking of Turbo Coded Signals 11-5 Tentative Decision-aided Turbo Coded Signal Carrier Phase Tracking 11-5 Non-Data-Aided Turbo Coded Tracking 11-7 Pilot-Aided Tracking 11-8 Iterative Turbo Coded Channel Gain and Noise Variance Estimation 11-14 Extrinsic Information-Based Turbo Coded Signal Carrier and symbol Tracking 11-17 Continuous Phase Modulated Signals 11-21 Acquiring CPM Signals Using MSK Preamble 11-23 Principle of DFT-Based MSK Preamble Acquisition 11-26 Interpreting the DFT Complex Preamble Spectrum 11-28 DFT Preamble SNR Performance 11-28 Preamble Acquisition Estimates Accuracy and Estimates Lower Bounds 11-31 Preamble Tone Estimate Refinement Using DFT Bin Interpolation 11-33 Preamble Tone Estimate Refinement Using DFT Bin Interpolation 11-35 Preamble Probability of Detection and Probability of False Alarm 11-36 Tracking of Continuous Phase Modulated (CPM) Signals 11-40 CPM Tracking Loop Parameter Design 11-43 Acquiring and Tracking Trellis-Coded MPSK Signals 11-45 References: 11-53
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Chapter 1 RF Architectures Digital Wireless Transceivers The design and manufacturing of wireless radio frequency (RF) transceivers has developed rapidly in recent ten yeas due to rapid development of RF integrated circuits and the evolution of high-speed digital signal processors (DSP). Such high speed signal processors, in conjunction with the development of high resolution analog to digital converters and digital to analog converters, has made it possible for RF designers to digitize higher intermediate frequencies, thus reducing the RF section and enhancing the overall performance of the RF section. Figure 1-1 illustrates a block diagram of a modern digital wireless receiver. Here the RF analog section is limited to a single front end, while most of the baseband and IF processing is carried out digitally. This chapter will detail modern methods involved in the RF to IF section, while the rest of the book is dedicated to digital signal processing and the decoding of the transmitted data.
In this chapter, we detail important issues of modern software-defined wireless transceiver RF system architectures and the signal processing algorithms involved in the RF t IF section. The baseband demodulation and synchronization in particular are covered in chapters 2 through 11. Figure 1-2 lists both floating point and fixed point DSP chips currently employed in wireless receivers [19]. The fixed-point processors operate at higher processing rates than floating point processors; however, these higher rates are at the expense of complexity in programming. The Texas Instrument current C64x DSP chip products are being produced with special purpose instructions for complex decoding algorithms, such as Viterbi and Turbo decoder, which greatly reduce complexity and make the design of wireless receivers easier.
Antenna
RF-to-IF section Antenna interface module
IF-to-Baseband section
IF section
• Low Noise Amplifier • Band preselection filters • RF to IF frequency conversion • Synthesized RF local oscillators • AGC
• IF filtering • AGC • RF to IF frequency conversion • A/D
• DDC • Baseband channel Filtering •AGC • Synchronization • Decoding
Digital Control
Figure 1- 1: Detailed block diagram of digital wireless radio receiver. Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Figure 1- 2: Texas Instrument’s current state of the art DSP processing power (as of year 2003).
1.1 Heterodyne and Homodyne Receiver Architectures Software defined radios (SDR) are one of the solutions for realizing a multi-mode [20], multi-band radio terminal. Currently, becasue of their high performance and ease of implementation, most tactical receivers are implemented using the conventional heterodyne architecture. The heterodyne architecture requires the use of high-Q radio frequency (RF) and intermediate frequency (IF) circuitry and additional local oscillators (LO). This circuitry increases the cost, size, and power consumption of these receivers and prohibits them from being integrated or packaged in small form factors. Alternatively, direct-conversion receivers (DCR) are one class of receivers that promises superior performance in power consumption, size, and cost over the conventional heterodyne architectures. However, the use of direct-conversion receiver architectures has been limited due to several design-related issues [1]. DCR limitations are include LO leakage, DC offset, I/Q imbalances, flicker noise (1/f noise), 2nd-order inter-modulation and the necessity of using high dynamic range analog-to-digital (A/D) converters with a very linear front end. While there are many approaches to minimizing these issues in hardware, there is great interest in implementing the estimation and compensation for these hardware impairments using digital signal processing (DSP). This circumvents production problems, such as component tolerances, aging, and labor by reducing the interaction between the DSP and RF subsystems, which allow for full automation of production and testing. This, therefore, allows a true re-configurable terminal (i.e., softwaredefined radio).
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
LNA
ADC Preselection Filter
IR Filter
LO1
Channel Select Filter
DDC
LO2
Figure 1- 3: Conventional Heterodyne Receiver Architecture
IF21 140Mhz
IF2 29Mhz
ISM signal 915 Mhz A/D & DSP
1055Mhz
169Mhz
Figure 1- 4: Heterodyne receiver example used in the 900Mhz ISM wireless local area networks (WLAN).
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Demodulator
The oldest and most commonly used receiver architecture is the heterodyne receiver shown in Figure 1-3. The high-Q parts, such as the first IF and the image rejection (IR) filters in the front end, are the devices that pose the most problems to integration. The received RF signal in Figure 1-3 is first filtered and then amplified using a low noise amplifier (LNA). Then the signal is down converted to a first IF. The first IF is again down converted to another lower IF frequency that is well within sampling and the dynamic range of the A/D converter used. The A/D samples rate is typically at frequencies of 10-50 MHz, and passes the digital samples to a digital down converter (DDC). The DDC is used to shift the digital IF to baseband. Then, decimation and filtering are performed to yield the final number of samples-per-symbols for final demodulation and bit decoding. Figures 1-4 through 1-6 illustrate examples of heterodyne receiver architecture for various commercial systems.
RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
IF21 135Mhz
IF2 15.42Mhz
GPS signal 1.57542Ghz DSP
1.44Ghz
120Mhz
Figure 1- 5: Heterodyne receiver example used for the global positioning satellite (GPS). IF21 100Mhz
IF2 450khz
PCS signal 1930 Mhz A/D & DSP
2030Mhz
99.55Mhz
Figure 1- 6: Heterodyne receiver example used for the personal communication system (PCS)
The main characteristics of receivers built using the heterodyne architecture are their high selectivity and high image rejection. These desirable characteristics come as a result of the use of high-Q filters and the double conversion scheme. The disadvantage of this architecture is that, since it uses high-Q devices and a double conversion scheme (in some cases three-stage conversion), it is difficult to integrate in a single chip or small set of chips. Furthermore, because of the double conversion and since most of the receiver gain is concentrated at RF stages, it consumes considerable direct current (DC) power, prohibiting the use of this receiver in mobile battery-operated applications for long periods of time. Furthermore, since the second IF conversion is usually fixed and matched to a channel bandwidth, this prohibits the use of variable data rates and forces the designer to use large bandwidths to match the highest rate. As a result of using the same fixed bandwidth that is matched to the highest data rate for lower data rates, the radio will be less sensitive, at the lower data rates, and more susceptible to interference and blockage. The main advantage of the architecture illustrated in Figure 1-3 is that there is no DC offsets and no quadrature mismatches between the real part of the baseband (I) and the imaginary part of the baseband (Q), since it is constructed digitally using the DDC. Overall, the heterodyne receiver enjoys high popularity among receiver designers of non-battery operated systems since its fundamentals are very well understood and there are no significant challenges in its use of current technology. To summarize, conventional heterodyne receiver architecture is limited by: • •
large size greater power consumption
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
• • •
difficulty of integration fixed data bandwidth not scalable for multi-rates and multi-waveforms.
The limitations of the heterodyne receiver have prompted designers to adopt the homodyne receiver architecture. The homodyne receiver shown in Figure 1-7 uses the direct conversion principle to eliminate the image filter and IF stages, with the result that it consumes less DC power and is capable of into a single chip. The down conversion is carried out using a local oscillator that has the same frequency as that of the incoming carrier being received. The architecture has most of its signal gain concentrated in the baseband and not the RF section, which reduces the power consumption and eases the circuit design. I A/D
BPF
LNA 0
90
AGC
LPF
cos(2πf ct + ψ (t ) + θ )
DSP
Q
A/D
cos(2πf ct )
Figure 1- 7: Homodyne Receiver Architecture Furthermore, since there is no pre-selection at RF or IF frequencies, the channel pre-selection and most of the Automatic Gain Control (AGC) are implemented at baseband, further reducing the complexity of the receiver RF section. The baseband pre-selection is carried out using tunable, switched capacitor filters that vary the bandwidth, based on the desired data rate. Despite these advantages over heterodyne receivers, direct conversion receivers using the architecture in Figure 1-7 have several serious design challenges that, in some instances, make them less favorable. The design problems arise as a result of the quadrature mismatches that are caused by the separate baseband processing of the imaginary and real parts of the down converted signal. These mismatches regenerate the image interference and can severely limit the use of this receiver architecture. The mismatch also distorts and rotates the digital modulation constellation, thus causing bits to be wrongly detected. Another problem arises from the local oscillator frequency being equal to the received signal frequency and the finite isolation between the mixer LO-port and RF-port of the mixer. This results in a large DC offset being created by the LO self-mixing. Another problem is the flicker noise, which is associated with the semiconductor devices used in both RF and baseband sections. This noise is distributed exponentially from DC to a few hundreds of kilohertz, which is usually where the baseband information is carried. Fortunately, Quadrature mismatches, DC offsets, and flicker noise all can be minimized or eliminated by using a receiver architecture that deviates slightly from the architecture in Figure 1-7 in conjunction with baseband DSP algorithms that estimate, track and null these impairments. Before these techniques are detailed, the image rejection receiver architectures, which is another variant of DCR, is described.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
1.2 Image Rejection Receiver Architectures
Image rejection mixers have been successfully utilized in limited applications as a compromise between heterodyne and DCRs. There are two types of image rejection receivers. Both types use trigonometric identities (phasing) to eliminate the image, and thus remove the need for the large, bulky front-end image rejection filters while retaining a small IF section. The first image rejection architecture is the Weaver receiver shown in Figure 1-8. Here, assuming that the two down converted quadrature channels are matched in quadrature phase and equal in amplitude, the additional conversion and quadrature summing stage will yield an image (solid spectrum) that is out of phase at the summing junction, and a desired signal (shaded spectrum) that is in phase. When both are combined, the image is completely canceled and the desired signal is doubled. The second image rejection architecture is the Hartley receiver shown in Figure 1-9. The Hartley receiver is similar in principle to the Weaver architecture, except that it cancels the image using passive 90o phase shift instead of frequency translation. This removes the need for an additional stage of frequency conversion. Since the 90o phase shift is implemented using resistor and capacitor [3], its exact quadrature match will only be maintained over a narrow band, limiting the receiver frequency range. Both Hartley and Weaver architectures suffer from quadrature imbalances and the DC offset when the final IF is not zero. Nevertheless, both Hartley and Weaver architectures have been successfully used in commercial receivers [3, 4]. Hz
0
IF1
Hz
BPF
LNA 0
90
90
0
LPF A/D
DDC
cos(2πf ct + ψ (t ) + θ ) Hz
0
Fixed LO1 cos(2πf LO1t )
LO2 cos(2πf LO 2t )
Hz
0
Figure 1- 8: Weaver Receiver Architecture.
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Demodulator
0
RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
IF1
yu ,d (t ) + yu , IM (t ) 90o
BPF
LNA
+
DSPDemodulator
90 0 cos(2πf rf t )
-
yb,d (t ) + yb, IM (t )
cos(2π (f rf − f LO )t )
cos(2πf LOt )
Figure 1- 9: the Hartley image reject receiver The following derivation illustrates the principle of image elimination in Hartley type image rejection receivers. Assume that the desired signal for the upper branch containing the 90-degree phase shifter in Figure (1-9) is given by y u ,d (t ) = cos( 2πf rf t ) sin( 2πf LO t ) =
1 {sin(2π [ f rf + f LO ]t ) − sin(2π [ f rf − f LO ]t )} 2
(1-1)
and that image signal is given by y u , IM (t ) = cos( 2πf IM t ) sin( 2πf LO t ) =
1 {sin( 2π [ f IM + f LO ]t ) − sin(2π [ f IM − f LO ]t )} 2
(1-2)
After the 90-degree phase shift, and using sin( x − 90 o ) = − cos( x) , the desired signal in (1-2) becomes, 1 ~ y u ,d (t ) = {− cos( 2π [ f rf + f LO ]t ) + cos( 2π [ f rf − f LO ]t )} 2
(1-3)
and the image signal becomes, 1 y u , IM (t ) = {− cos( 2π [ f IM + f LO ]t ) + cos( 2π [ f IM − f LO ]t )} 2
(1-4)
Likewise, quadrature channel, the desired signal in the bottom branch is given by y b ,d (t ) = cos( 2πf rf t ) cos( 2πf LO t ) =
1 {cos(2π [ f rf + f LO ]t ) + cos(2π [ f rf − f LO ]t )} 2
(1-5)
and the image is given by y b, IM (t ) = cos( 2πf IM t ) cos( 2πf LO t ) =
Dr. Mohamed Khalid Nezami © 2003
1 {cos( 2π [ f IM + f LO ]t ) + cos( 2π [ f IM − f LO ]t )} 2 1-7
(1-6)
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
After summing both branches as illustrated in Figure 1-9, the resultant signal is give by y out (t ) = y u ,d (t ) + y u , IM (t ) − y b ,d (t ) − y b, IM (t ) = cos( 2π [ f rf − f LO ]t ) = cos( 2πf IF t )
(1-7)
Thus, by assuming there are no quadrature mismatches between the two channels in Figure 1-9, the 90-degree phasing phase shift results in the removal of the image and the yield of only the desired IF signal. The 90degree shift is implemented as an analog circuit using resistor-capacitor passive networks [4], which can result in some problems due to aging, temperature variation, and narrow band response. Figure 1-10 shows one example of an integrated Hartley receiver chip [21]. The integration is one approach to obtaining a repeatable tight tolerance of the 90-degree phase shifted branch.
Figure 1- 10: An example of integrated Hartley receiver chip (Maxim., California, USA).
1.3 Principle of Digital Down Converter and Sample Decimation
One of the most essential parts of a modern RF receiver is the digital down converter that translates the over sampled digital IF frequency to baseband and then decimates the signal to a lower sample rate. Figure 1-11 shows a block diagram of a common digital down converter system. Here the sampled IF signal gets sorted into odd and even indexed samples. These samples are then passed to the numerical complex multiplier that forms the baseband by complex multiplication, assuming there is no frequency mismatch between NCO and the received carrier frequencyThe resulting in-phase and quadrature phase branches will be a true baseband signal (center frequency is zero Hz). The quadrature input channels are at a high sample rate (typically higher than 4).By decimating the samples to a lower number of samples per symbols (1/8,1/16,1/32,…etc.), the subsequent DSP algorithms consume no more than the needed effort for demodulation computations, yet the signal information contents are still retained.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
n:odd ⎛ 2π ⎞ cos ⎜ n ⎝ N ⎠
RF
IF
A/D
CIC ↓
FIR
I
CIC ↓
FIR
Q
NCO ⎛ 2π ⎞ − sin ⎜ n ⎝ N ⎠
n:even
Digital Downconverter
÷2
fs Figure 1- 11: Representation of digital down converter (DDC) Mathematically, the DDC uses an in-phase NCO signal given by,
⎛ 2πf IF ⎞ i NCO (n ) = cos⎜⎜ n ⎟⎟ f ⎝ s ⎠ and a quadrature signal given by ⎛ 2πf IF ⎞ q NCO (n ) = − sin⎜⎜ n ⎟⎟ f ⎝ s ⎠
(1-8)
(1-9)
fs in Figure 1-11 is the f IF oversampling factor. One way that has commonly been implemented to simplify the DDC computational process is to use a sample rate that is a multiple of 4 times the IF frequency. Substituting f s = 4 f IF in (1-8) and (1-9), the NCO in-phase signal is reduced to, ⎛ π ⎞ ⎧ 0 n , n : odd i NCO (n ) cos⎜ n ⎟ = ⎨ (1-10) ⎝ 2 ⎠ ⎩(− 1) 2 , n : even When the ratio of the DDC sample rate to the intermediate frequency as N =
and the quadrature NCO signal is reduced to
⎛ π ⎞ ⎧ 0 n +1 , n : even q NCO (n ) = − sin⎜ n ⎟ = ⎨ ⎝ 2 ⎠ ⎩(− 1) 2 , n : odd Dr. Mohamed Khalid Nezami © 2003
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By inspection of (1-10) and (1-11), the NCO sinusoidal signal will take on the value of only (0, -1, or +1). This is very beneficial since now the multiplication in Figure 1-11 is realized by sample index multiplexing and inverters. Thus the DDC in Figure 1-11 becomes a free multiplication process. To illustrate this, assume that the received IF samples are fed to the DDC in Figure 1-11 with a sampling rate that is four times the IF frequency and are given by (1-12) x (n ) = {x 0 , x1 , x 2 , x3 , x 4 , x5 , x6 , x 7, ...} with performing the in-phase NCO down conversion according to (1-10) is given by, i NCO (n ) = {1,0,−1,0,1,0,−1,0,...}
(1-13)
the digitally down converted in-phase samples (baseband) are then give by I (n )i NCO (n ) = {x0 ,0,− x2 ,0, x 4 ,0,− x6 ,0,...}
(1-14)
Which only utilized even index selection, filing zero for odd indexed samples, and an inversion process according to (1-13). Likewise, for the quadrature phase branch, the NCO sequence is given by q NCO (n ) = {0,−1,0,+1,0,−1,0,+1,...}
(1-15)
As a result, the digitally down converted quadrature samples (baseband) are then give by Q (n )q NCO (n ) = {0,− x1 ,0, x3 ,0,− x5 ,0, x7 ,...}
(1-16)
A commonly used commercial DDC chip is the AD6620 available from Analog Devices (USA, MA) [13] and the Intersil (USA, FL) HSP50016 [14]. For the Analog Devices AD6620, the frequency translation stage is accomplished with a 32-bit complex NCO. Following the frequency translation is a fixed coefficient, high speed decimating filter that reduces the sample rate by a programmable ratio between 2 and 16 using a second order, cascaded integrator comb FIR filter (CIC2). Following the CIC2 a second stage of fixed-coefficient decimating filter is implemented using a fifth order decimator filter (CIC5), which further reduces the sample rate by a programmable ratio from 1 to 32. Clearly, the job of the digital down converter involves frequency translation to baseband and decimation of the highly oversampled baseband signal sufficient for the consequent DSP algorithms (carrier recovery, AGC, and decoding). The decimators in conjunction with the FIR LPF in (111) define the baseband bandwidth (i.e., channel preselection) and reduce the unnecessary sample rate as shown in Figure 1-12. Clearly here the decimation ratio (M) can be used to control the null-to-null frequency f bandwidth, which is given by f null = s . The frequency response of the CIC filter is given by [15]: M
N
⎡ ⎛ f ⎞⎤ ⎢ sin ⎜⎜ Mπ ⎟⎟ ⎥ fs ⎠⎥ , H( f ) ∝ ⎢ ⎝ ⎢ ⎛ f ⎞ ⎥ ⎢ sin ⎜⎜ π ⎟⎟ ⎥ ⎝ f s ⎠ ⎦⎥ ⎣⎢
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0≤
1 f ≤ fs M
1-10
(1-17)
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Where M is the CIC decimation ratio and N is the CIC filter order. Figure 1-12 shows a fifth order and second order CIC frequency response for a few decimation ratios. Notice that the decimator has a notch at a frequency f 1 that is equal to , which is of great benefit to receivers since this is one of the stages at which channel = fs M pre-selection can be defined. One way to utilize this notch frequency is to place the desired signal image or any other undesired spur at the notch frequency. 0 0
M=4 M=8 M=16
-20
-30
-60
-50
-100
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
M=8 M=4
-80
-0.2
M=16
-40
-40
-60 -0.25
M=4,8,16 CIC RESPONSE 5th ORDER
-20 Normalized gain -dB
-10
M=4,8,16 CIC RESPONSE 2ND ORDER
-120 -0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Normalized frequency -f/fs
Figure 1- 12: Frequency response for the 2nd order CIC with decimation M =4,8, and 16 (left), and the 5th order CIC filter with decimation M =4,8, and 16 (right).
The LPF in Figure 1-11 is necessary to attenuate the side lobes resulting from the decimation filters that otherwise cause severe adjacent channel interference (ACI). As an illustrative example, assume that the downconverter in Figure 1-11 has a modulated IF=100 MHz with a bandwidth of 1.2288Mbps. The sample rate used by the DSP demodulation and bit decoder is normally at least four times the bit/chip rate. Using an arbitrary choice for the decimation ratio prior to the decimation and FIR filtering in Figure 1-11, let us assume that the overall over sampling factor was 40, hence the sample rate from the A/D is f s = 49.152Msps. For the sampled 100Mhz IF signal, the resulting IF signal spectrum is folded back in the first Nyquist zone. As a result, the DDC will have a digital intermediate frequency of
f DIF = 100Mhz − 2(49.152) = 1.696Mhz The required four times the chip rate samples and FIR filtering can be acheived by using two sequential decimation stages, first a second order CIC decimator using M=4, followed by a 5th order CIC decimator using M=10. If a decimation ratio with M=5 in the first CIC filter is used, it will result in a frequency notch at f null = 49.152 / 5 = 9.825Mhz Then if a decimation with M=2 is used in the second decimator, a second notch placed at f DIF = 9.825 / 2 = 4.9125Mhz results. Clearly, a cleaver way of assigning the M value is to use it to control the location of the notch frequency to reject sampling clocks and known fixed interferences. Since both of the CIC filters in Figure 1-11 have broad filter passband when using small decimation ratios, the image and adjacent channels rejection that pass through the hardware filtering (IF filter) can only be filtered Dr. Mohamed Khalid Nezami © 2003
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effectively using the channel final FIR programmable filter, which is normally designed with high order on the order of 64 to 256 taps. Such filters can offer up to 60 dB of attenuation of adjacent channels [14] , which is a requirement from most cellular and short range radio receivers .
1.4 Alternatives to Direct Conversion Receivers
One way to circumvent the problems associated with the conventional Homodyne architecture (see figure 1-7) is to use a “near zero IF” receiver architecture, commonly known as a low IF receiver. This architecture translates the RF signal to a low IF frequency that is just slightly mis-tuned to the baseband so that DC offsets and flicker noise are avoided (see figure 1-13). This architecture retains the benefits of the homodyne architecture. The low IF is usually one to a few channels. For instance, for Global System Mobile (GSM) handsets, the low IF is 200 to 300 kHz. One advantage of this architecture is that, since the fractional bandwidth of the low IF band pass is large, it is possible to implement it with low-Q components. In figure 113, the sampled low IF is brought to baseband using an additional digital down converter. The adjacent channels present at the input of the A/D are the images of the desired channel. The complex down converter (see figure 1-13) acts like a Weaver mixer, since it unfolds and cancels the image at the complex summing junction. Another attractive feature of the low IF architecture in Figure 1-13 is that the phase locked loop (PLL) used for the first local oscillator can be used as a coarse oscillator and the digital down converter would then fine-tune the narrow channel to baseband. This simplifies the design of the first LO since it can facilitates the use of high reference frequency in the PLL, which increases the loop switching time and broadens the loop bandwidth, resulting in better phase noise performance. One of the main drawbacks, however, is that this architecture suffers from quadrature mismatches since the quadrature low IF is digitized and filtered in two separate analog channels. Also, since the image and interference are present at the A/D input the sensitivity of the receiver will be reduced if not filtered properly.. For this reason, the A/D used in low IF receivers uses a high dynamic range (more bits and faster sampling rate) to enable the receiver to cope with the presence of adjacent channel and interferences. cos(2πf LO 2t )
BPF 1
A/D BPF
LNA 0
90
LPF
sin( 2πf LO 2t )
AGC
Q
BPF
Demodulator/Decoder
I cos( 2πf c t + ψ (t ) + θ )
A/D cos( 2πf LO 2t )
90o splitter
cos( 2πf LO1t )
NCO
cos( 2πf LO1t )
Figure 1- 13: Low-IF and Receiver Architecture.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
I
cos(2πf ct + ψ (t ) + θ )
BPF 1
LNA 0
90
sin(2πf LO 2t )
LPF
A/D
BPF
A/D
BPF
AGC
-
Q
Demodulator
cos(2πf LO 2t )
cos(2πf LO 2t )
90o splitter
Fixed LO
cos(2πf LO1t )
cos(2πf LO1t )
Figure 1- 14: Double Conversion Wideband IF Receiver Architecture. Another deviation from the Homodyne receiver that is also suitable for integration is the double conversion wideband IF receiver shown in Figure 1-14. Here the first mixing is performed using a fixed local oscillator that block down converts all of the desired channels to a wideband IF. The down conversion is carried out while retaining the individual channel positions relative to each other, then the desired channel is quadrature down converted again to baseband using a secondary complex analog mixing stage and a fine-tuned oscillator to yield only the desired channel at baseband. Since the first oscillator is not running at the carrier frequency, there will be no DC offset, nor flicker noise problem. Since the first oscillator is fixed, it can be constructed using a highly stable and high-Q Surface-Acoustical Wave (SAW) filter yielding low phase noise. Since the second LO is running at a low frequency, its phase noise is also well controlled . However, unlike the receiver in Figure 1-13, there are two quadrature mismatch contributors present for this architecture because there are two quadrature analog conversion stages. 1.5 Quadrature Mismatch and Image Rejection
To analyze the impact of quadrature mismatches in the receivers shown in Figure 1-13 and 1-14, we first derive the mathematical representation of the image that arises due to the quadrature amplitude and phase mismatches. Assume that the RF signal received using the architecture shown in Figure 1-13 is given by: rrf (t ) = I (t ) cos( 2πf rf t ) − Q (t ) sin( 2πf rf t )
(1-18)
and that the quadrature local oscillator used in Figure 1-13 has a quadrature amplitude mismatch of α , and a phase mismatch of θ and is given by rLO (t ) = cos(2πf LO t ) + jα sin(2πf LO t + θ )
(1-19)
then the quadrature down conversion in Figure 1-13 is carried out by multiplying equation (1-18) and (1-19). That is, ⎡ e j 2πf LO t + e − j 2πf LO t e j (2πf LO t +θ ) − e − j (2πf LO t +θ ) ⎤ rrf (t ) rLO (t ) = rrf (t ) ⎢ + αj ⎥ 2 2j ⎦ ⎣
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(1-20)
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Clearly, with no quadrature mismatches, or α = 1 and θ = 0 , Equation (1-20) yields a maximum amplitude complex signal with the real part of y I (t ) = I (t ) and an imaginary part of y Q (t ) = jQ (t ) . However, with the quadrature imbalances present, the constellation representing the trace of the digital samples for the complex baseband in (1-20) will deviate from its optimal shape as shown in Figure 1-15 for an amplitude mismatch of α =.01 and an angle mismatch of θ =30 degrees. As a result of this mismatch, the image rejection in the receiver will be degraded. Further manipulations of (1-20) yield the following relationship,
[
(
)
(
1 rrf (t )rLO (t ) = rrf (t ) e j 2πf LOt 1 + αe jθ + e − j 2πf LOt 1 − αe − jθ 2
)]
(1-21)
The magnitude of the image rejection ratio (IRR) can be derived by taking the ratio of the magnitude of the resultant image signal (negative frequency) in (1-21) to the desired signal (positive frequency). That is,
IRR =
1 − αe− jθ (1-22)
1 + αe+ jθ
Which is further reduced to IRR =
1 + α 2 + 2α cosθ 1 + α 2 − 2α cosθ
(1-23)
2
2 10% mismatch
1
1
0
0
Volts
Imaginary
30-deg mismatch
-1
-2 -2
-1
-1
0
1
-2
2
1.05v 0.95V 0
50
100
150
200
Sample #
Real
Figure 1- 15: BPSK Constellation Due to Quadrature Mismatches Caused by θ =30o and α = 0.1 . One way to model quadrature mismatch impact on digital modulation using the receiver in Figure 1-13 is carried out by computer modeling, using the model in Figure 1-16.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
θ
cos(2πf RF t + ) 2
1+
α 2
I
Q 1−
θ
sin( 2πf RF t − ) 2
α 2
Figure 1- 16: Model for Quadrature Imbalances for Low IF Architecture in Figure 1- 5.
The derivation of the IRR in Equation (23) is appropriate for mismatches of a single stage analog conversion such as those present in Figure 1- 7 and Figure 1-13. For double conversion architectures, such as the image rejection receiver shown in Figure 1-8 or the double conversion wideband IF receiver in Figure 1-14, Equation (23) has to be modified to include quadrature mismatches from the second quadrature analog down conversion. Figure 1-17 illustrates the quadrature imbalance model for the dual conversion receiver architectures of Figure 1-9 and Figure 1-14. The image rejection ratio in this case is given by 1 + α 2 + 2α cos(θ1 + θ 2 ) IRR = 1 + α 2 − 2α cos(θ1 + θ 2 )
(1-24)
where θ 1 + θ 2 is the sum of mismatches of the first and second stages of quadrature down conversion, and α is the overall quadrature amplitude imbalances.
θ
cos( 2πf if 1t + ) 2
θ
θ
cos(2πf if 2t + ) 2
LPF
LPF
LPF
LPF
sin( 2πf if 1t − ) 2
θ
sin( 2πf if 2t − ) 2
1+
α 2 I
Q 1−
α 2
Figure 1- 17: Quadrature imbalances for wideband double conversion IF receiver. Figure 1-18 shows a plot of the input rejection obtained in the receiver structures discussed above as a function of moderate mismatches that are commonly experienced with commercially available quadrature two-way down converters. Figure 1-18 shows that for IRR=35 dB, the maximum phase and amplitude mismatches have to be less than 2o and 2% respectively. Dr. Mohamed Khalid Nezami © 2003
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-15 10-deg.
-20
5-deg. -25 2-deg. IR-dB
-30
-35 1-deg. -40
-45 0.5-deg. -50
0
0.01
0.02
0.03
0.04 0.05 0.06 Amplitude Mismatch
0.07
0.08
0.09
0.1
Figure 1- 18: Image Rejection Versus Quadrature Mismatches.
1.6 Impact of the Quadrature Imbalances on the Receiver Bit Error Rate
The impact of the I/Q imbalances on the receiver performance can be evaluated using the degradation in BER performance due to signal loss and symbol rotation. The standard probability of bit error (Pb) of QPSK signals having quadrature amplitude mismatch is computed by altering the energy per bit (Eb ) by the amplitude loss due to mismatch [16] and given by Pb ,α ≈
α 1 ⎡ ⎛⎜ 2 E b (1 + ) 2 ⎢Q⎜ 2 ⎢⎣ ⎝ N o 2
⎞ ⎛ 2Eb α ⎟ + Q⎜ (1 − ) 2 ⎟ ⎜ N 2 o ⎠ ⎝
⎞⎤ ⎟⎥ ⎟ ⎠⎥⎦
(1-25)
Figure 1-19 (left) shows a plot of the BER deterioration of QPSK signal due to an amplitude mismatch of 10%, 20%, 30%, 40%, and 50 %. Notice that for 20% amplitude mismatch, which is a very typical of low cost quadrature down converters, the BER degradation due to the quadrature imbalance is less than 1.0 dB.
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10
10
10
Pe
10
10
10
10
-1
10
-2
10
-3
10 I/Q gain Mismatch
-4
Pe
0% 10% 20% 30% 50%
-5
-6
10 10
10 -7
2
4
6
8
10
12
10
Eb / N o
-1
-2
-3
-4
I/Q Phase Mismatch o 20 15o 10o 5o 0o
-5
-6
-7
4
2
6
10
8
12
Eb / N o
Figure 1- 19: effect of gain and phase mismatch on QPSK bit error rate.
The quadrature phase mismatch impact on the BER can be found in a similar approach. Assuming that the mismatch angle is small, the small angle assumption can be used. The net effect of the mismatch is then θ considered as a reduction in the bit energy Eb and is proportional to sin . Thus by using the approximation of 2 θ θ sin ≈ the BER degradation due to phase mismatch is given by, 2 2 Pb ,θ ≈
⎛ 2 Eb 1 ⎡ ⎛⎜ 2 Eb θ ⎞ θ ⎞⎤ (1 + ) 2 ⎟ + Q⎜ (1 − ) 2 ⎟⎥ ⎢Q⎜ ⎜ N 2 ⎣⎢ ⎝ N o 2 ⎟⎠ 2 ⎟⎠⎦⎥ o ⎝
(1-26)
Figure 1- 19 (right) illustrates the BER degradation of QPSK signals due to quadrature phase imbalance of 0o, 5o, 10o, 15o, and 20o. The figure shows that to guarantee a degradation of less than 0.5 dB in the receiver, the quadrature branch phase imbalance must be less than 5 degrees, which is also a typical specification of low cost quadrature down converters. 1.7 Time Variant and static (time invariant) DC Offsets
The presence of non-time varying DC offsets, due to LO leakage from the mixer LO port shown in Figure 1-7, is critical to the receiver performance. The leakage problems are due to finite mixer LO-to-RF isolation, substrate radiation, and capacitive and magnetic LO coupling. The LO self-mixing can be illustrated by squaring a sinusoidal LO signal, that is: 2 ( ALO cos 2πf LO t )2 = ALO
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2 2 + ALO 2 cos 4πf LO t
1-17
(1-27)
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which clearly creates a DC term and a high order term corresponding to twice the LO frequency. The DC term often can add millivolts into the receiver’s input, while the desired received RF signal is microvolts, thus causing a carrier-to-interference ratio of up to C/I= -30 dB. The worst scenario is when the LO leaks into the input of the low noise amplifier (LNA) input, producing an even stronger DC offset due to the high gain of the LNA, which then saturates the subsequent A/D. A DC offset can also result from a strong in-band interferer, which, when leaking from the LNA output into the LO port of the mixer, will cause self-mixing and thus produce a DC offset in the baseband section. The second order inter-modulations between adjacent channels, due to non-linearity’s in the baseband section of the receivers shown in Figure 1-13 and Figure 1-14, also produce DC offsets. To illustrate this, assume that the second order inter-modulations can be represented by a polynomial having three nonlinear terms ℑ( x ) = a1 x + a 2 x 2 + ... . , with the input signal given by rrf (t ) = Arf cos 2πf ch1t into the non-linearity where f ch1 is the desired channel. In this case, the non-linearity output will contain the desired channel signal, a DC term, and higher order terms given by
ℑ(rrf (t ) ) = a 2 Arf2 / 2 + a1 Arf cos 2πf ch1t + a2 Arf2 / 2 cos 4πf ch1t
(1-28)
where the ratio of the term a1 / a2 = IIP2 in (1-28) is the second order input intercept point [1]. To minimize the higher order terms and the DC term in direct conversion receivers due to this in-band interference, the ratio of a1 / a2 has to be maximized (i.e., use higher IIP2 baseband devices). The second order non-linearity also causes in-band interference created by the presence of adjacent channels. To illustrate this, consider the input signal to the direct conversion receiver to be given by: rrf (t ) = cos 2πf ch1t + cos 2πf ch 2 t
(1-29)
where f ch1 and f ch 2 are two adjacent channels. After this signal passes through a second order non-linearity, terms having the sum and difference of f ch1 and f ch 2 are created. Since the IF is zero, the different terms of the sinusoidal with frequency of f ch1 and f ch 2 will create an in-band interference, while the sum will create another interference term that is interference for the third adjacent channel. This again emphasizes the importance of the second order inter-modulation of the baseband section of the receiver. One way to avoid second order effects is to use balanced differential circuit components (indicated by the double lines of Figure 1-7), high IIP2 devices, good frontend-to-baseband isolation, and good printed circuit board (PCB) layout. A time variant DC offset is caused by the presence of Doppler frequency associated with the reception of the reradiated LO as the radio physically moves (a moving car or walking pedestrian) or as the object moves. This DC offset will have a frequency spread and thus only DSP methods are able to successfully estimate, track and then cancel the frequency spread. The frequency spread is dependent on the relative velocity with respect to the stationery or moving emitted LO from the receiver frontend port (i.e., antenna). 1.8 DC Offset Removal Algorithms
One way to remove static (non-time varying) DC offsets is by using high pass filter (HPF) after the down conversion. This is practical howver, only if the modulation does not contain any significant spectral information at or near DC [10, 12], such as Frequency Shift Keying (FSK) waveforms, or wideband modulations like spread spectrum. For example, [12] showed that less than 0.2 dB of bit error rate deterioration was present when using an HPF with a cut-off frequency on the order of 0.1% of the spread spectrum chip rate. For TDMA applications, the use of HPFs, which use large capacitors may cause slow settling times in the receiver circuits. ahigh BER at the start of the TDMA burst will result.. Furthermore, use of large capacitors Dr. Mohamed Khalid Nezami © 2003
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may prohibit integrating the receiver into a chip because of the area required to physical layout the capacitor. An alternative to using a HPF example was used in [10,11], where the DC offsets are removed in the TDMA systems using a capacitor that stores the DC charge during the TDMA burst off period. Then the DC is measured quickly before the burst is on again, and the measured value is subtracted in the analog section during the on time of the TDMA burst. Figure 1-20 shows another variant of the same receiver, where the DC offset is estimated using the discrete Fourier transform (DFT) of an input calibration tone. Figure 1-21 shows one proposed DC removal algorithm that detects the most significant bit (MSB) of the A/D to indicate saturation due to large DC offset. When the DC offset does not cause A/D saturation, an algorithm in the baseband DSP section estimates the mean of the quadrature samples resulting from test tone injected in the frontend of the receiver, which correspond to the DC value. The DC measured value is then subtracted by the feedback loop in the analog section before the A/D. The time variant DC offset is removed using DSP algorithms in the baseband section using a feed-forward loop to the A/D output. Figure 1-20 shows one commercial receiver direct conversion chip that incorporate a similar idea for the DC removal. DC removal loop
Figure 1- 20: The Motorola MC13760 direct conversion chip (Motorola, Arizona, USA).
1.9 Tone-Aided Quadrature Imbalance Estimation and Compensation
One of the commonly used techniques to estimate the quadrature mismatches in DCRs has been the use of calibration tones injected at the front-end of the receiver. By measuring both image and desired parts of the calibration signal levels, an adjustment is derived that is fed back to de-rotate and to adjust the amplitude of the quadrature path relative to the in-phase path. Figure 1-21 shows one such implementation that is used with the low IF receiver architecture from Figure 1-13 [5]. Assume that the quadrature LO signal, including the imbalances is given by rLO (t ) = (1 + ∆ ) cos(2πf LO t + θ ) + j (1 − ∆ ) sin(2πf LO t − θ )
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The phase mismatch is estimated by injecting a tone at a frequency that is centered in the received channel. Image and desired components are then measured at baseband and are used to estimate the imbalance correction parameters β and α shown in Figure 1- 21. The quadrature phase imbalance is estimated by:
⎛ I 2 d Q2 i − I 2 i Q2 d 2 2 ⎝ I 2 d + Q2 d
θˆ = 2 tan −1 ⎜⎜
⎞ ⎟ ⎟ ⎠
(1-31)
and the quadrature amplitude mismatch is estimated by:
⎛ I I + Q2 d Q2 i ∆ˆ = 2⎜⎜ 2 d 22i 2 ⎝ I 2 d + Q2 d
⎞ ⎟ ⎟ ⎠
(1-32)
where I 2 d = II − QQ , and Q2 d = IQ + QI are the measured real and imaginary parts of the desired calibrating tone levels, and I 2i = II + QQ and Q2i = IQ − QI are the real and imaginary parts of the measured calibrating image signal levels. Using the estimates in (1-31) and (1-32), the amplitude imbalance compensation, as shown in Figure 1-21 is given by,
α=
1
(1-33)
(1 − ∆ˆ ) cos θˆ
and the imbalance compensation for the angle is given by:
β = − tan θˆ
(1-34)
With − 5 o ≤ θ ≤ +5 o , and − 10% ≤ ∆ ≤ +10% , the algorithm in (33) and (34) achieved IIR of more than 50dB.
1.10
Frequency Domain Quadrature Imbalance Estimation and Compensation
Another approach is to use the Discrete Fourier Transform (DFT) coefficients of the calibration signal to estimate the quadrature mismatches and the DC offset as shown in Figure 1-22. Here the received calibration tone is sampled and then processed through a DFT algorithm from which bins are manipulated to derive DC and quadrature imbalances. To illustrate this method, assume that the calibration tone quadrature down converted low IF including the DC offset is given by:
and
y I (t ) = (1 + α )I (t ) cos( 2πf if t ) + idc
(1-35)
y Q (t ) = Q (t ) sin( 2πf if t ) + q dc
(1-36)
where f if is the low IF calibration tone frequency, idc and qdc are the DC offsets associated with the real and imaginary parts of the sampled low IF signal. The test tone at f = f if is then sampled by a frequency such that f if = f s 4 , which results in the following quadrature signal:
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y I (n) = (1 + α ) cos(π
n + ϕ ) + idc 2
(1-37)
and y Q (n) = sin(π
n n + ϕ ) + q dc 2
(1-38)
where n is the sample number, and ϕ is an arbitrary phase shift. The samples in Equation (1-37) and (1-38) are then processed using a 4-point DFT is given by k =3
S (k ) = ∑ s (n)e
−j
2πnk 4
(1-39)
n =0
where, s ( n) = y I ( n) + y Q (n) . Expanding Equation (1-39), the 4-bins of S (0) , S (1) , S (2) and S (3) are then exploited to extract DC and the imbalance estimates. The first complex bin, S (0) represent an estimate of the DC offset associated with both real and imaginary parts of the calibration tone signal in (1-29). That is: 1 iˆdc = Re{S (0)} 4 1 qˆ dc = Im{S (0)} 4
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
DC offset removal
N −1
1∑ N k= 0
-
iˆdc
idc (t ) - ˆ
A/D-MSB
cos( 2πf LO2t )
Q1 I 1
A/D
cos (2π f t + ψ (t ) + θ ) c
Implemented in DSP
~ I1
II
c
BPF 1
LNA 0
90
LPF
qˆdc
qˆdc ( t)
AGC
-
cos (2π f CALt )
Q1
-
QQ
β
+
IQ
-
~2 Q
BPF
yQ
A/D
α CalibratingTones
yI BPF
I1
cos( 2πf LO1t )
Compensation network
+
QI
cos( 2 πf LO2t )
sin(2πf LO2t )
DSP Demodulator
DAC
DDS- (LO2)
yI yQ
Figure 1- 21: Proposed Quadrature and DC Baseband Compensation in Time Domain.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
cos( 2πf LO 2t )
A/D
cos( 2πf c t + ψ (t ) + θ )
BPF 1
~ I1
-
-
I1
LNA 0
90
LPF
iˆdc (t )
QQ IQ
Q1
-
+
-
~ Q2
yQ BPF
A/D
qˆ dc (t )
cos( 2πf LO1t ) D/A
QI
+
-
Calibrating Tones
yI BPF
AGC
qˆ dc
cos(2πf CAL t )
Implemented in DSP II
P (z ) E ( z ) + 1
Compensation network
cos( 2πf LO 2t )
sin( 2πf LO 2t )
DDS- (LO2)
4pt-DFT
yI yQ
Figure 1- 22: Proposed Quadrature and DC Baseband Compensation in Frequency Domain.
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Demodulator/Decoder
iˆdc
RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Further manipulations of the other three bins lead to the quadrature imbalance compensation coefficients shown in Figure 1-22, ⎧ 2S (3) ⎫ E = − Re⎨ ∗ (1-42) ⎬ S ( 1 ) S ( 3 ) + ⎭ ⎩ and ⎧ 2S (3) ⎫ P = − Im⎨ ∗ (1-43) ⎬ S ( 1 ) S ( 3 ) + ⎭ ⎩ Using the imbalance error estimates obtained in Equations (1-42) and (1-43), the in-phase compensated received signal is given by: yˆ I (n) = (1 + E ) y I (n) − iˆdc (1-44) and the quadrature phase compensated received signal is given by: yˆ Q ( n) = Py I ( n) + y Q ( n) − qˆ dc
(1-45)
A computer program was used to simulate the system in Figure 1-22 and then execute the algorithm derived in Equations (1-44) and (1-45). Figure 1-23 shows the performance of this algorithm for four cases of quadrature imbalances marked at the top of each figure. From the results in Figure 1-23, the algorithm seems to estimate amplitude mismatches up to 80% with small angles mismatches. However, for large angle mismatches, on the order of 10o, the algorithm loses its effectiveness. (20%,5o )
1 0.5 Q
(80%,5o )
1 0.5
I/Qmismatched
Q
0
0 original & corrected
-0.5 -1 -1
-0.5
-0.5
0 I
0.5
-1 -1
1
(20%,10o )
1
1
0.5 Q
0 0.5 I (20%,22.5o )
1
0.5
0
Q
-0.5 -1 -1
-0.5
0 -0.5
-0.5
0 I
0.5
-1 -1
1
-0.5
0
0.5
1
I
Figure 1- 23: Performance of the DFT-based Quadrature Mismatches and DC Estimation Algorithm.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
One of the drawbacks of both algorithms in Figure 1-21 and Figure 1- 22 is that their imbalance estimates are only accurate at a single frequency, which is the calibration tone that is usually chosen to be the middle of the channel. For wideband channels (more than 10 kHz), neither algorithm is optimal. One way to broaden the performance of these algorithms for wideband channels is to use multiple calibration tones. The compensation network is then changed from a single constant value multiplication to a frequency-dependent compensation digital network that will perform quadrature imbalance compensation at multiple frequency offsets. The calibration tones can be chosen as fractions of the Nyquist bandwidth such as,
fi = [
fs 2 fs 3 fs 4 fs , , , ] 10 10 10 10
(1-46)
which produces a set of correction coefficients given by Pi and E i , where i =1, 2, 3, 4. These estimates are then used to synthesize a digital, finite impulse response filter given by:
(
∏ k4 =1,k ≠i z + z −1 − 2 cos 2πf k
4
E ( z) = ∑ i =1
∏
4 k =1, k ≠ i k
and
(
i =1
Ei f i f s − 2 cos 2πf k )
∏ k4 =1,k ≠i z + z −1 − 2 cos 2πf k
4
P( z ) = ∑
(2 cos 2π
∏
4 k =1, k ≠ i k
(2 cos 2π
)
(1-47)
)
Pi f i f s − 2 cos 2πf k )
(1-48)
The compensation outputs of (1-47) and (1-48) are then used for correction as shown in the system in Figure 122.
1.11
Preambles Quadrature Imbalance Estimation and Compensation
Another major disadvantage of both algorithms in Figure 1-21 and Figure 1-22 is that they require additional hardware to implement the calibrating tone loopback. Also, calibration cannot take place without halting normal receiver operations. Figure 1-24 illustrates one method for imbalance corrections without use of calibration tones and with estimation of imbalances and correction carried out during normal reception.
yI I
α z
−1
-
µ
γ
Q
µ
(. )
z −1
-
(. )
yQ
Figure 1- 24: Toneless Quadrature Imbalance Correction.
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The principle of I/Q imbalances estimation and correction in Figure 1-24 is based on the orthogonality between the I and Q signals. Since the imaginary and real parts of a perfectly balanced MPSK modulated signal are orthogonal, if there are any quadrature imbalances, this orthogonality is distorted. Based on this principle, a simple approach is to estimate amplitude mismatches from the autocorrelation function of the imaginary and real parts of the modulated Mary-PSK signal, as shown in Figure 1-24. The estimates are arrived at by iteration using an adaptive noise canceller filter with the least mean square error method [6]. The quadrature-corrected signal is given by y Q = (αI + Q )β
(1-49)
where the angle correction β is found using the error measure of cross product given by I ( n)Q ( n) . This can be recursively estimated using the following algorithm,
β (n + 1) = β (n) + µ {I (n) − Q(n) }
(1-50)
and the amplitude correction signal α is estimated using the error measure of I (n) − Q(n) and is given by
α (n + 1) = α (n) − µ {I (n)Q(n)}
(1-51)
where 0 ≤ µ ≤ 1 is a convergence factor. As an example, Figure 1-25 shows both phase and gain mismatches caused by quadrature imbalances. Figure 1-26 shows the performance of the algorithm described by equation 1-50 and 1-51 showing an amplitude imbalance of 60% compensation converges in less than 200 samples as seen in Figure 1-26. For the angle compensation, Figure 1-27 illustrates the performance of (1-50) using a convergence factor of 0.001.
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Figure 1- 25: Constellation QPSK with perfect Gain and phase imbalance (left), 60% gain imbalance (right), and phase imbalance of 30 degrees (bottom). 2
real
1 0 -1 -2
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
2
imag
1 0 -1 -2
Sample Number
Figure 1- 26: The real and imaginary of the QPSK signal before, during, and after gain imbalance correction.
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0
-0.1
Phase Mismatch Correction
µ=0.001; -0.2
-0.3
-0.4
-0.5
-0.6
0
0.5
1
1.5
2
Sample Number
2.5
3 4
x 10
Figure 1- 27: Correction signal for phase imbalance of 30 degrees (0.53 radians) and a convergence factor of 0.001. 1.12
Quadrature Imbalance Estimation and Correction in Image Rejection Receiver
Most of the techniques discussed above can be extended to correcting the quadrature imbalances of the image rejection receivers illustrated in Figure 1-8 and Figure 1-9. Figure 1-28 shows a proposed architecture to correct the Hartley receiver. The receiver is based on a digital metric that measures the quadrature mismatch error between the upper and lower branches and then suppresses the image as shown. Assuming that the input RF signal to the quadrature down converter in Figure 1-28 is given by
{
}
{
Re s (t )e j 2πf c t + Re I (t )e j 2πf c t
}
(1-52)
Assuming that the signal that enhances the desired signal (upper branch) is given by Ae = h1 s ( n) + h2 I ∗ (n)
where I ∗ (n) is the sampled image and s (n) is the sampled desired input signal . Consider that the lower branch enhances the image Be = h2∗ s ( n) + h1∗ I ∗ ( n) , where h1 and h2 are coefficients representing the quadrature imbalances [4], and are given by h1 = 1 2 + 1 2 α cosθ + j 1 2α sin θ
(1-53)
h2 = 1 2 − 1 2 α cosθ + j 1 2α sin θ
(1-54)
and
where α is the quadrature amplitude imbalance and θ is the quadrature phase imbalance. Thus, the image rejection ratio (IRR) is given by IRR = 20 log10 ( h1 h2 ) . To maximize this rejection, the digital processor
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shown in Figure 1-28 contains a correlation coefficient calculator that measures the mismatch by calculating a modified correlation coefficient ρ given by:
{
E Ae Be∗
ρ=
}
(1-55)
E{ Ae } + E{ Be } 2
2
Substituting the values of Ae and Be in Equation (55) yields
h1 h2
ρ=
h1 + h2 2
(1-56)
2
which is totally dependent on the quadrature mismatches. The correlation coefficient in (50) will be maximized if E {Ae Be∗ } is maximized, which indicates a total correlation between Ae and Be . The image component within the desired signal, Ae , can be completely removed by first multiplying Be by the correlation coefficient ρ and then subtracting the result from the desired signal Ae . That is y out = Ae − ρBe
(1-57)
Substituting the values of Ae , Be , and ρ into Equation (57), the output signal is then given by: y out =
h1 h1
2
h1 + h2 2
2
s ( n) +
h2 h2
2
h1 + h2 2
2
I ∗ ( n)
(1-58)
which shows that the compensation algorithm attenuated the signals very little but suppressed the image I (n) by the factor
h2 h2
2
h1 + h2 2
2
.
IF1
Ae
+
Implemented in DSP
+
BPF
LNA 0
+
ρ
90
LPF
-
cos( 2π f ct + ψ ( t ) + θ ) _ -90o
+
A/D
Be
cos( 2π f LO1t )
Demodulator/Decoder
A/D
yout
LO1
Figure 1- 28: Proposed Scheme for compensating Quadrature imbalances for Hartley Image Rejection Receiver.
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1.13
Digital Transmitter Architectures
Before detailing different transmitter architectures, we will illustrate the principle behind digitally upconverting a baseband signal to an intermediate frequency or directly onto the final transmitter carrier. The process of digital upconversion is simply the opposite of the digital down conversion process shown in Figure 1-11. Here the baseband samples are faithfully translated from a frequency of zero (baseband) to either an intermediate IF or directly to the carrier frequency using a quadrature NCO in conjunction with a complex multiplier as shown in Figure 1-29. The output signal from the digital up converter (DUC) shown in Figure 1- 29 is given by
⎛ 2πf IF sTX (n ) = I (n) cos⎜⎜ ⎝ fs
⎞ ⎛ 2πf IF n ⎟⎟ − Q(n) sin⎜⎜ ⎠ ⎝ fs
⎞ n ⎟⎟ ⎠
(1-59)
where I (n) and Q (n) are the baseband samples to be transmitted. For the case of f s = 4 f IF the NCO in-phase signal is reduced to ⎛π ⎞ ⎛π ⎞ sTX (n ) = I (n) cos⎜ n ⎟ − Q (n) sin ⎜ n ⎟ ⎝2 ⎠ ⎝2 ⎠
(1-60)
Which can be implemented using a multiplier free process by cyclically inverting or zeroing samples according to the sequences discussed in the DDC in Figure 1-11. In a practical system the baseband samples are upsampled and shaped to define the spectral bandwidth of the transmitted data. As an illustrative example, consider a baseband data rate of 64kbps processed for upconversion using the system in Figure 1-29. The data is split into even and odd indexed streams and then fed to I and Q., The I and Q are at a data rate of 32kbps. Two baseband filters are then used to shape the spectrum of the data streams. Normally these filters are implemented using a root raised-cosine filter (RCF) [16] with a specific roll-off value that is specified to control the signal bandwidth. For this example the filter sampling frequency is 10 times the data rate, so the sample rate out of the shaping filter in Figure 1-29 is 320ksps. If the IF needs to be somewhere in the region of 5 to 20 MHz, then an interpolation factor of M=12 can be used, resulting in an IF frequency of 3.84 MHz.
I
Filter
INT ↑I
n:odd ⎛ 2π ⎞ n⎟ cos⎜ ⎝ N ⎠
IF
NCO
Q
⎛ 2π − sin ⎜ ⎝ N
Filter
INT Q ↑
+
DAC
⎞ n⎟ ⎠
n:even
Figure 1- 29: Representation of digital up converter (DUC).
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Commercially, there are several DUC chips that are available. The Graychip (recently becoming part of Texas Instruments) GC4116 is a quad up converter chip that uses three serial stages of processing of the baseband data before it is multiplied by the NCO. The first stage of baseband processing is a 63-tap programmable coefficient interpolate by 2 FIR filter [19]. The second stage of processing is an interpolate by 2 CIC filter. The third stage is a CIC interpolator with a programmable interpolation ratio from 8 to 1448. This gives an overall interpolation from 32 to 5792. The chip is designed to run at up to 100 MSPS. A rival chip is the AD6623 from Analog Devices [13This chip also uses three stages of baseband processing before the samples are multiplied by the NCO. The first stage is a programmable FIR that can be implemented as a root Raised-Cosine filter that interpolates the input samples by M=2. The second stage is a 5th order CIC with an interpolation factor from 1 to 32. The last stage of processing is a 2nd order CIC interpolator filter with interpolation factor from 1 to 4096.
In addition to the DUC, there are three classes of transmitter architectures that are currently in use, namely Figure 1-30 through Figure 1-32. One very popular transmitter architecture that is often used is the IFmodulation up-conversion transmitter topology shown in Figure 1-30. Here, after the baseband signal is digitally upconverted to an IF, the IF filter at the output of the quadrature modulator rejects the harmonics of the IF signal. The signal is then analog up converted again to the final RF carrier and then amplified and fed to the antenna. This architecture is large in size and expensive, and does not allow full transmitter integration into a single chip since both IF and the up converter filters are large. However, since the quadrature modulation is performed at a low IF frequency, it is easier to handle quadrature imbalances, filtering and amplitude control resulting in accurate modulation constellations. Usually, the TX harmonic filter is required to have a harmonic rejection of 50-60dB. The use of filtering early in the IF stages will relax the harmonic filter specifications.
One way to reduce the cost and size of the topology in Figure 1-30 is to remove the IF stage by using a single up conversion stage using the direct-modulation architecture shown in Figure 1-31. Here the baseband signal is directly modulated onto the carrier. Since the LO is running at the carrier frequency, the finite isolation between the output of the power amplifier (PA) and the modulator pose a serious problem causing injection pulling. Therefore, to avoid the PA pulling problem, it is recommended that the local oscillator be operating at twice the RF carrier frequency, then a divide by two circuit is used inside the quadrature mixer chip to obtain the correct LO frequency. Another unattractive feature of this topology is that the PA output power dynamic range is limited by the carrier “feed through”, so an alternative method for controlling the PA drive has to be used. Nevertheless, since there is no IF filter used, this transmitter topology can be integrated in small packages and the final harmonic filter would still be a separate module.
To circumvent the use of the large harmonic filter at the output of the PA in the transmitters of Figure 1-30 and Figure 1-31, the Offset-PLL transmitter shown in Figure 1-32 can be used. This architecture is also popular for cases in which the same local oscillator used with the offset-PLL is also used as the main LO for the receiver. The attractive feature of this architecture is that the PLL acts as a narrow BPF that filters harmonic frequencies, thus the bulky high Q harmonic filter used in the previous transmitter topologies is not needed. Also using the offset as a receiver LOreduces the radio’s complexity and cost. The other advantage over the heterodyne architectures in figure 1-30 and figure 1-31 is that the output is protected from the high noise figure of the offset mixer. This eliminates the need for the large harmonic filter at the output of the transmitter. The offset-PLL topology may suffer from other spurious sources such as the interaction between various oscillators and mixers [17]. The Offset PLL transmitter is limited to phase Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
modulated signals and can not be used with QAM signals. Integrated polar modulators have recently gained some popularity [18]. 1.14
Quadrature Imbalances in Digital Transmitters
Quadrature imbalances are also present in the transmitter topologies shown in Figure 1-30 and Figure 1-31. Both topologies use an analog quadrature mixer that has some quadrature mismatches and can vary over frequency. These problems are not associated with the topology in Figure 1-32. As an example, the commercial part RF2484 device from RF Micro Devices (USA, NC) has a quadrature mismatch of 1 degree and 2% of amplitude mismatch. If this transmitter is used to transmit constant envelope signals such as continuous phase frequency shift keying (CPFSK) and continuous phase modulation (CPM), the quadrature imbalances in the transmitter will result in an undesirable peak-to-average power ratio (PAR).
Figure 1-33 shows the simulated PAR as a result of transmitting constant envelope CPM signal through a quadrature modulator with amplitude quadrature imbalance up to 70% and 20 degrees. Clearly, the transmitter topology in Figure 1-32 is favored over those in Figure 1-30 and Figure 1- 31because it offers the capability of suppressing harmonics without a bulky harmonic filter and can modulate constant envelope signals faithfully. To illustrate the quadrature mismatch and DC offset impact on digital modulators, assume that the baseband signal from a digital modulator such as those detailed earlier is given by, y (t ) = I (t ) cos(2πf LO t ) + Q (t ) sin(2πf LO t )
(1-61)
where the baseband signal terms in (1-61) have some small DC offset, that is I (t ) = i (t ) cos(2πf m t ) + dc m and Q(t ) = q(t )(2πf m t ) + dc m , and let the LO signal also contain some DC offset. Furthermore, consider that the LO signal with DC offset and quadrature imbalances is given by y LO (t ) = cos(2πf LO t ) + dc LO + α sin(2πf LO t + θ ) + dc LO
(1-62)
Substituting equation 1-62 into 1-61, the quadrature modulator output in both Figure 1-30 and Figure 1-32 architectures is given by y (t ) = (i (t ) + dc m )(cos(2πf LO t ) + dc LO ) + α (q(t ) + dc m )(sin(2πf LO t + θ ) + dc LO )
(1-62)
Expanding (1-62), the resulting signal output of the quadrature modulator is, y (t ) =
1 [i(t ) + αq(t ) cos(θ )]cos(2π ( f m − f LO )t ) + 1 [i(t )q(t )α sin (θ )]sin (2π ( f m − f LO )t ) + 2 2 1 1 [i(t ) − αq(t ) cos(θ )]cos(2π ( f m + f LO )t ) + [i(t )q(t )α sin (θ )]sin (2π ( f m + f LO )t ) + 2 2 (1-63) dc m cos(2πf LO t ) + dc mα sin (2πf LO t ) cos(θ ) + dc mα cos(2πf LO t )sin (θ ) + dc LO i (t ) cos(2πf m t ) + αq (t )dc LO sin (2πf m t ) +
dc m dc LO + αdc m dc LO
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Equation 1-63 contains the desired transmitter term cos(2π ( f m + f LO )t ) , the undesired lower side band cos(2π ( f m − f LO )t ) ,
the LO leakage cos(2πf LO t ) , the baseband or passband leakage sin (2πf m t ) and several DC terms. The DC terms and the modulation baseband leakage can be filter by a high pass filter (AC coupling). The ratio of lower sideband frequency f m − f LO to the upper sideband frequency f m + f LO is known as the sideband suppression. With that, the desired signal is given by
yd (t ) =
1 [1 + αq(t ) cos(θ )]cos(2π ( f m − f LO )t ) + 1 [α sin (θ )]sin (2π ( f m − f LO )t ) 2 2
(1-64)
remove the common factor ½, the envelope of (1-64) is given by
y d (t ) = [1 + α cos(θ )] + [α sin (θ )] = 1 + 2α cos(θ ) + α 2 cos 2 (θ ) + α 2 sin 2 (θ ) 2
2
(1-65)
Utilizing the trigonometric formula of cos 2 (θ ) + sin 2 (θ ) = 1 in (1-65),
y d (t ) = [1 + α cos(θ )] + [α sin (θ )] = 1 + 2α cos(θ ) + α 2 2
2
(1-66)
Similarly, the undesired term is given by 1 1 yud (t ) = [1 − α cos(θ )]cos(2π ( f m + f LO )t ) + [α sin (θ )]sin (2π ( f m + f LO )t ) 2 2 which has an envelope given by, y d (t ) = [1 − α cos(θ )] + [α sin(θ )] = 1 − 2α cos(θ ) + α 2 2
2
(1-67)
(1-68)
As a result, the sideband suppression or equivalently, the image rejection ratio (IRR) is given by, LSB 1 + 2α cos(θ ) + α 2 = 10 log10 USB 1 − 2α cos(θ ) + α 2
(1-69)
Clearly the relationship in (1-69) is familiar since it is the same formula derived for the image rejection ratio derived for the receiver in Equation (1-23). As expected, the quadrature downconverter and upconverter suffer the same effect in the presence of I/Q imbalances. The LO leakage to the desired signal ratio in (1-63) is given by dc 2 + 2αdc m dc LO sin (θ ) + α 2 dc m2 LO = 10 log10 m USB 1 + 2α cos(θ ) + α 2 / 4
(
)
(1-70)
The relationship in (1-70) shows that LO leakage is a function of quadrature imbalances and how much DC offset is present at the LO and modulation ports ( that is dc m and dc LO ), It is also observed in (1-70) that if the DC offset term leaking through the baseband is zero, dc m = 0, the LO term is very minimal. Also, if the DC
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
offset term of the quadrature phase mismatch is zero, dc LO = 0, the LO power output is dominated by the magnitude of dc m . Clearly the baseband port and the LPF used in the quadrature mixers shown in the transmitter topologies shown in Figure 1-30 and Figure 1-31 must be carefully designed to minimize the LO leakage and quadrature mismatch, both of which contribute to the creation of image components in the transmitter.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
LPF 90
fTX
f IF
0
PA Modulator filter
LPF
TX harmonic filter
fTX − f IF
f IF
Figure 1- 30: IF-Modulation /Up-Conversion Transmitter.
LPF f TX
f TX
/2
90
0
PA
LPF 2 f TX
2xLO
Figure 1- 31: Direct-Modulation Transmitter using a single up conversion stage.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
f IF
Loop Filter
BPF
VCO
fTX
Offset mixer
fTX − f offset = f IF
BPF
f TX ± f offset
~ RX-IF
f offset
From RX-Antenna
Figure 1- 32: Offset-PLL Transmitter.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
3
2.5
dB
2
1.5 20 deg 1 10 deg 5 deg 0.5 0 deg 0
0.1
0.2
0.3 0.4 Amplitude mis ma tc h
0.5
0.6
Figure 1- 33: Peak to Average Ratio with I/Q mismatches with 0,5,10, and 20 degree of phase mismatch.
1.15
Multi-Carrier Transmitter Architectures
Figure 1-34 shows a conventional multicarrier system. The carriers are combined after the power amplifiers. This architecture has several disadvantages. First, the individual carrier branches require individual tuning. Second, the hybrid power channel combiner has loss associated with it, requiring high power to compensate for this loss. Finally, N carrier systems require 2N DAC devices, N analog vector modulators, N bandpass filters and N power amplifiers., These requirements result in a large hardware system that requires higher DC power consumption,and the generation of extra heat. Figure 1-35 shows an alternative to using the analog system. Here the system generates the quadrature modulated individual carriers (sub channels) in the digital domain and combines the carriers in the digital domain. Next, the carriers are converted to an analog signal using a single DAC device and then amplified by one common amplifier. This saves enormous amounts of analog components, many of which require lengthy production tuning. Consequently, an expensive and tedious part of the manufacturing process is eliminated. A single linear amplifier replaces the conventional high-level combination of individual amplifiers. Also, the power losses in a hybrid combiner are avoided. By not using an analog quadrature modulator, which removes the difficulties in adjusting the DC offset and quadrature imbalances, the system eliminates a large source of error vector magnitude (EVM).
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The main drawback of the architecture in figure 1-35 is that the system requires the use of high linearity wideband upconversion and linearized power amplifier because all of the N carriers are passing through the same PA. As an illustrative example, consider that the system in Figure 1-35 is used to transmit 16 multicarriers, each transmitting a QPSK modulated signal at 4.096Mbps with an oversampling factor of 16. The total sample rate out of each carrier branch in Figure 1-35 is, ⎛ 4.096 ⎞ (1-71) Fclock = 16 sps⎜ Msps ⎟ = 32.768MHz ⎝ 2 ⎠ To keep images from aliasing, the output frequency of the DDS used for the NCOM is usually kept at 40% of BW FClock . This means that Fclock ≥ , where BW = ( N ch ∆f + FIF ) is the total bandwidth occupied by the 0 .4 multicarriers. The 4.096Mbps modulation is assumed to use 5MHz of bandwidth. The FIF frequency can be placed at zero Hz; however, this may result in distorting the lower edge of the first channel. By moving the first channel up 5MHz, the occupied bandwidth is now from 5MHz to 90 MHz. The total bandwidth is then given by BW = (16 x5Mhz + 5Mhz ) = 85Mhz and the DDS clock will be Fclock = 85 / 0.4 = 212.5Mhz . Since the overall interpolation required is 16, half band filters can be used to keep the system complexity at a minimum. For the multicarrier system in figure 1- 28, the D/A converter can exhibit an amplitude frequency whose distortion is dependent on the final output spectrum. This amplitude distortion is a low pass filter function given by
H( f ) ≥
sin(π f Fclock ) π f Fclock
(1-72)
For Fclock = 212.5MHz , at the upper edge of 85 MHz, the total roll-off loss due to the sin(x)/x amplitude attenuation is -2.42dB. One method to compensate for this distortion is the inverse sin(x)/x equalizer [15]. Using a 32-bit DDS for generating the numerical local oscillator signal, with FClock = 212.5Mhz , the NCO is
Fclock , where NCOword is the programmable phase 2 32 NCOword = 231 and vary from NCOword = 0 to
programmed using the relations of FTX _ Ch = NCOword register value in the DDS which can FTX _ Ch1 = {5Mhz ,10 Mhz ,15Mhz ,.......85Mhz}.
In this chapter we have surveyed in detail most of the RF subcomponents used in various topologies of radio design. Next we will examine the radio receiver digital baseband section, where demodulation and decoding take place.
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DAC/LPF fTX 1
LO-3
90
0
PA
0
PA
0
PA
DAC/LPF fTX 2
LO-2
90
DAC/LPF
Channel Combineer
Wideband Matrix of multi user data
DAC/LPF
DAC/LPF fTX
LO-N
90
DAC/LPF
Figure 1- 34: Conventional semi digital multicarrier transmitter architecture Dr. Mohamed Khalid Nezami © 2003
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Wideband Matrix of multi user data
RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Filter
Interpolator 1
Interpolator 2
NCOM 1
Filter
Interpolator 1
Interpolator 2
NCOM 2
f ch1
digital adder
+
f ch 2
+
Sin(x)/x D/A
fIF
RF Upconverter
Final LO
fTX
Filter
Interpolator 1
Interpolator 2
NCOM N
f chN
+
Figure 1- 35: All digital Multicarrier transmitter.
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PA
1.16
References
1. Won Namgoog, and Teresa H. Meng, “Direct-Conversion RF Receiver Design,” IEEE Trans. Commun. Vol. 49 No. 3, pp. 518 – 529, March 2001. 2. Behzad Razavi, “Design Considerations for Direct-Conversion Receivers,” IEEE Trans. Circuits and Systems-II, vol. 44 No. 6, pp. 428 – 435, June 1997. 3. Wu, S.; Razavi, B., “A 900-MHz/1.8-GHz CMOS receiver for dual-band applications,”, IEEE Journal of Solid-State Circuits, Volume: 33 Issue: 12, Dec. 1998, pp. 2178 –2185. 4. Chun-Chyuan Chen and Chia-Chi Huang, “ On the architecture and performance of a hybrid image rejection receiver,”, IEEE Trans. Commun. Vol. 19 No. 6, pp. 1029 – 1040, June 2001. 5. Jack Glas, “ Digital I/Q imbalance compensation in a low-IF receiver,”, IEEE. 6. Fred Harris, “Digital filter equalizer of analog gain and phase mismatches in I-Q receivers,”, IEEE. 7. Kong-pang, Jose Franca, and Azeredo-Leme, “Wideband digital correction of I and Q mismatch in quadrature radio receivers,” 8. Tien-Yow, Staurt Golden, and Naiel Askar, “A spectral correction algorithm for I-Q channel imbalance problem,”, IEEE. 9. Mikko Vakama, Markku Renfors, and Visa Koivunen, “Compensation of frequency-selective I/Q imbalance in wideband receivers: models and algorithms,”, IEEE 10. Hiroshi Tsurumi, Miyuki Soeya, Hiroshi Yoshida, Takafumi Yamaji, Hiroshi Tanimotot, and Yasuo Suzuki, “ system-level compensation approach to overcome signal saturation, DC offset, and 2nd-order nonlinear distortion in linear direct conversion receiver,”, IEICE Trans. Elctron., Vol. E82-C, No. 5, May 1999. 11. Seiichi Sampi, and Kamilo Feher, “Adaptive DC-offset compensation algorithm for burst mode operated direct conversion receivers,”, IEEE. 12. J. H. Mikkelsen, T.E. Kolding, T. Larsen, T. Klingenbrunn, K.I. Pedersen, and P. Morgensen, “Feasibility study of DC offset filtering for UTRA-FDD/WCDMA direct-conversion receiver,”, IEEE 1999. 13. www.analog.com 14. www.intersil.com 15. Alan V. Oppenheim, Ronald W. Schafer , Discrete-Time Signal Processing, Prentice Hall Signal Processing Series, 1999. 16. Bernard Sklar, Digital Communications: Fundamentals and Applications, 2nd Edition, 1999. 17. Jeff Dekosky, Fred Martin, and Jeff Rollman, "Offset PLL Analysis can cut spurious levels," RF& Microwaves, Nov. 1999.
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18. www.Tropian.com. 19. www.ti.com 20. Mohamed K. Nezami, “Performance Assessment of Baseband Algorithms for Direct Conversion Tactical Software Defined Receivers: I/Q Imbalance Correction, Image Rejection, DC Removal, and Channelization,”, IEEE Milcom Conf. , Nov. 2002.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Chapter 2 Introduction to Coherent Demodulation Wireless receivers process signals that bear information as well as disturbances caused by the transmitter/receiver circuits and channel impairments such as fading, interference, and additive white Gaussian noise (AWGN). Usually, the receiver knows only some statistical properties of the signal and disturbances. From these statistical properties and using a finite observation of the received signal, the receiver is able to estimate the transmitted data symbols. The receiver makes decisions on the received data using locally generated symbol clock and carrier oscillator, both of which are not referenced to the actual versions used to generate the data at the transmitter. The receiver has to estimate the offset between locally generated carrier and symbol clock to those used at the transmitter. Clock mismatches are labeled as symbol timing jitters, while local carrier mismatches are labeled either as phase rotation errors or frequency offset errors. Symbol timing synchronization is the process in which the receiver estimates the offset between the locally generated symbol clock at the receiver and the actual symbol clock used at the transmitter local clock. The receiver then uses the offset estimate to correct the free running local clock. This clock, unless matched to the transmitter clock, will cause the receiver symbol decision circuitry to sample the symbols at the wrong instance, resulting in detection errors. Carrier phase recovery is the process in which the receiver estimates the offset between the local oscillator phase and the actual phase of the transmitted carrier. Carrier frequency offset recovery is the process of estimating the offset between the frequency drift/change of the local oscillator and the actual (received) carrier frequency transmitted at the transmitter. For burst type of TDMA transmissions, frame synchronization is also needed. Typically, the frame is detected using uniquely coded words that are inserted at the beginning and end of each burst [1,2,3,4]. 2.1 Order of Synchronization in Digital Receivers Ideally, symbol timing errors, carrier frequency offset and carrier phase rotation have to be jointly estimated. Due to the unmanageable complexity, practical schemes employ a sequential estimation approach when the initial offsets are within acceptable limits. Notice that accurate phase estimation requires prior knowledge of symbol timing and frequency offset estimates, while frequency offset estimation requires prior knowledge of symbol timing. Thus joint phase-frequency estimation is not possible. However, the frequency and symbol timing errors can be estimated jointly, as will be shown in the next chapters.
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Practically, the flow of the synchronization process is carried out as a series of processes, as shown in Figure 2-1. The process is implemented in the following sequence of estimations: First the digital baseband is obtained using an analog-to-digital (A/D) converter whose sample timing frequency is free running,. This introduces symbol timing errors because it is physically impossible for the A/D clock to match the symbol clock rate used at the transmitter. After the signal is properly filtered and maintained at a proper level, timing offset is estimated first and corrected, followed by frequency offsets and phase rotational offsets estimation and correction. Normally frequency-offset estimation (and correction) assumes a signal with negligible timing errors, while phase estimation (and correction) assumes negligible frequency variation during the estimation interval. Though some frequency estimation schemes exist that do not require symboltiming information – which we will discuss in the coming chapters - these algorithms may not be as optimal. Nevertheless, they provide reasonably close estimates during initial carrier search and acquisition.
A/D
Analog Baseband sugnal
Symbol Timing recovery
Carrier recovery
Data detection
Binary data
Digitized Baseband signal
Figure 2-1: Flow of carrier ands symbol timing recovery.
2.2 Impact of Symbol Timing Coherency on Demodulation Although the symbol rate is typically known to the receiver, what is not known is when to sample the received signal, and thus knowing the starting sample of a symbol (i.e., symbol boundary). Sampling at the wrong instance and then integrating and dumping will cause interference and thus reduction in the bit error rate (BER). Figure 2-2 illustrates the impact of symbol timing error on impact of BPSK symbol timing error on BER. The simulations show that with an error of 20% of the symbol timing period, the BER degradation is 3dB at BER of 10-3.
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10
-1
τ = 0.3T
Bit-Error-rate
10
10
-2
τ = 0.2T
-3
τ = 0.1T 10
10
10
-4
Ideal BPSK
-5
-6
0
1
2
3
4
5 S NR, dB
6
7
8
9
10
Figure 2- 2: Impact of symbol timing error (error=10%, 20%, and 30% of symbol interval T) on BPSK demodulations.
2.3 Illustration Example : Synchronization presence form IF to baseband Use the digital IF receiver frontend in Figure 1-11 to illustrate mathematically the impact of carrier offsets and symbol timing on the received signal. Figure E2-1 illustrates the envisioned digital signal processing of the front-end IF section preceding the proposed synchronization and channel estimation algorithms. Here the input RF signal is given by r (t ) = g (t )a(t )e − j 2π ( f c + ∆f +θ c +φ )t + n(t ) where f c is the carrier frequency, ∆f is the unknown frequency offsets due to Doppler and other TX-RX uncertainties, θ c is the carrier arbitrary phase due to the channel, ϕ is the M-ary phase modulations that are CPFSK for MUOS, a (t ) is the fast fading channel gain due to multipath fading, and g (t ) is the slow channel gain due to blockage or propagation losses. This signal is first down converted to an intermediate frequency (IF) that is within the range of the analog to digital converter (A/D). With current commercial A/Ds, this is of the order of a 100MHz. As a result, the digitized IF signal is given by y k = g k a k e − j (2π (∆f + f DIF )Ts +θ k +φk ) + nk
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where f DIF is the digital IF frequency that results from a Nyquist region choice, Ts is the A/D sampling time that is N-times the symbol rate received ( Ts = T N ), and the rest of the terms correspond to the discrete sampled representation of the parameters associated with the received signal that has to be estimated and corrected. Following the A/D is an additional stage of digital frequency translation that will attempt to bring the digital IF frequency down to baseband that is given by y k = g k a k e − j (2π∆f ( kTs +τ ) +θ k +φk ) + nk
For long term signal changes and to keep the A/D input signal constant and centered, an automatic gain controller (AGC) is used. The digital baseband samples from the digital tuner (DDC) are used to compute the loss in the signal drive into the A/D. After the AGC, the signal in (3) contains the undesired and unknown parameters that prohibit demodulation and detection of the transmitted information. These impairments are due to synchronization errors and channel distortion: •
Carrier frequency offset, ∆f .
•
Carrier phase offset, θ c .
•
Fast fading channel gain, a k .
•
Slow fading channel gain, g k .
•
Symbol timing offset, τ .
This final signal is also at unnecessary multiple samples per symbols, since only a fractional ratio of samples per symbols (N=4, 8, 16, 32 ..etc) are commonly needed by the subsequent following demodulation algorithms. The process of reducing the sample rate is carried out using digital decimation filters implemented using cascaded integrator comb (CIC) filters that are free of complex multiplication. The frequency response of these filters is usually loose, and thus offers no pre-selection or filtering of adjacent channel interference (ACI). Since the decimator filters do not offer any kind of noise shaping or ACI filtering, a high order (256, or 128) tap FIR filter is usually required after decimation and before passing the final signal to the following acquisition and tracking algorithms.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers r (t ) RF section
IF section
X
A/D
e − j 2π ( f − f )t IF ~ 100 Mhz c
1 Ts ~ 100Mhz fs =
IF
CIC ↓
DDC
e 2πf
N
NCO kTs
y (kTs )
FIR
y (kTs ) = g k ak e j (2π (kT +τ )∆f +θ +ϕ ) + nk s
where Ts =
k
T N
AGC
E2-1: Illustration of sample signal processing used before acquiring, tracking and demodulating the signal.
2.4 Impact of Carrier Coherency on Demodulation The impact of carrier phase and frequency offset on the received signal can be illustrated by the following. Assume that the received signal intermediate frequency (IF) in AWGN channel is given by
s (t ) = 2 P cos(2πf ct + φ (t ) + θ (t ) ) + n(t )
(2-1)
where P is the power level of the received down converted IF signal, f c is the carrier frequency in Hz, θ (t ) is the carrier time varying phase offsets, and φ (t ) is the baseband symbols represented as phase modulations. For M-PSK modulations, φ (t ) is given by
φ (t ) =
2πm M
(2-2)
where m = 0,1,2,.....M − 1 , and for BPSK M=2, and for QPSK M=4. The first thing that the received signal undergoes in a typical digital received is its down conversion to baseband by mixing with a local oscillator that has a frequency that is as close as possible to this frequency but has a random varying phase. The down conversion process is shown in Figure 2-3. The quadrature and in-phase signals of the local oscillator is are given by − sin(2πf LO t + θˆ) and cos(2πf LO t + θˆ) respectively. The resultant baseband quadrature at the output of the low pass filter (LPF) is given by
(
)
(2-3)
(
)
(2-4)
i(t ) =
P cos 2π∆ft + φ (t ) + θ (t ) − θˆ(t ) + ni (t ) 2
q(t ) =
P sin 2π∆ft + φ (t ) + θ (t ) − θˆ(t ) + nq (t ) 2
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
where ∆f = f c − f LO carrier frequency offset due to the mismatch between the local oscillator frequency and the carrier frequency due to various reasons that will be discussed later and that the carrier phase offset is given by the value of θ (t ) − θˆ(t ) ??? . The presence of the carrier phase and frequency offsets cause the MPSK signal to deviate from its optimal constellation and thus produce bit errors due to the time varying rotation caused by the phase and frequency offsets. i (t )
LPF
cos(2πf LO t + θˆ) s (t )
Baseband Signal processing
Local oscillator
Detected data
− sin( 2πf LO t + θˆ) LPF
q (t )
Figure 2- 3: Basic Quadrature detection of MPSK signal.
The BER for QPSK signals as a function of carrier phase offset is given by ⎛ 2 Eb Pe = Q⎜⎜ (cosθ − sin θ ) No ⎝
⎞ ⎛ ⎟ + Q⎜ (cosθ + sin θ ) 2 Eb ⎟ ⎜ No ⎠ ⎝
⎞ ⎟ ⎟ ⎠
(2-5)
⎛ 2 Eb ⎞ ⎟ with perfect synchronization, where Q(x ) is the error compare to Pe = 2Q⎜ ⎜ N ⎟ o ⎠ ⎝ function. The BER deterioration in (2-5) due to carrier offset can be interpreted as power implementation loss (distance loss) given by
Dθ = 20 log10 (cos θ − sin θ )
(2-6)
Likewise, for the BER degradation for QPSK signals is given by ⎛ 2 Eb Pe = Q⎜⎜ cos θ No ⎝
⎞ ⎟ ⎟ ⎠
(2-7)
and the power implementation loss due to the synchronizer is given by Dθ = 20 log10 (cos θ )
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Figure 2-4 shows Equation (2-6) and Equation (2-8) plotted for the range of 0o ≤ θ ≤ 40 o . As an example, a carrier phase error of 15 degrees causes ~2.75 dB loss for QPSK signals, and only 0.3 dB loss for BPSK signals.
5 4.5 QPSK
4 3.5
Loss-dB
3 2.5 2 BPSK
1.5 1 0.5 0
0
5
10
15 20 25 Carrier phase offset -deg
30
35
40
Figure 2- 4: Implementation loss due to carrier phase offset for BPSK and QPSK.
To further illustrate the effect of carrier and frequency offsets on MPSK, consider the 2PSK signal (BPSK). Here, if the carrier has no phase and no frequency offsets, the energy will be totally concentrated in the in-phase channel as shown in Figure 2-5. The baseband signal in (2-3) without phase offset is now given by,
i(t ) =
P cos(φ (t ) ) + ni (t ) 2
(2-9)
and a quadrature signal in (2-4) is given by
q (t ) =
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P sin (φ (t ) ) + nq (t ) 2
(2-10)
2-7
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Since the 2-PSK symbols are given by φ (t ) = πm , with m=0 or 1, the baseband signal is then i(t ) ∈ {+ 1,−1} and q(t ) = 0 ( or only residual AWGN) as shown in figure 2-5 and hence has the undistorted constellation shown in Figure 2-6. However, when considering carrier phase offsets, the quadrature part will no longer be zero., As a result of the carrier phase offsets, the constellation of BPSK will look like that shown in Figure 2-7 and also as illustrated by the constellation in Figure 2-8.
The impact of frequency also can be interpreted in term of phase. For a given frequency offset ∆f , the error in degrees that this frequency offset inflict on the symbols is given by
θ ∆f = ∆fT 360 where T is the symbol time duration. From this, the error per sample can be calculated as,
θ ∆f =
∆f 360 fs
(2-11)
For instance, if the frequency offset associated with an IF 70 Mhz signal is ∆f = 100khz , the phase rotation per sample due to this frequency offset is given by 105 θ ∆f = 360 = 0.5o / sample . For a symbol that is over sampled by 4 samples/symbol, 7 7.10 the total phase error rotation per symbol is almost 2 degrees. In addition to the phase rotations, the offset also results in a drop in the matched filter power output given by
D∆f
⎛ sin(2π∆fT ) ⎞ ⎟⎟ = ⎜⎜ 2 π fT ∆ ⎠ ⎝
2
(2-12)
where T is the symbol rate. Such loss translates directly into BER deterioration. For instance, with 10 kbps transmission rates, and a 1 kHz carrier frequency offset, the loss due to the frequency offset is ~0.6 dB loss in Eb/No link budget [5].
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2
real
1 0 -1 -2
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
2
imag
1 0 -1 -2
Figure 2- 5: Ideal BPSK baseband signal received with no frequency offsets, ∆f = 0 and no carrier phase offsets, θ (t ) − θˆ(t ) = 0 . BPSK
1.5 1
Quadrature
0.5 0 -0.5 -1 -1.5
-1.5
-1
-0.5
0 0.5 In-Phase
1
1.5
Figure 2- 6: Ideal BPSK baseband signal constellation when received with no frequency offsets, ∆f = 0 and no carrier phase offsets, θ (t ) − θˆ(t ) = 0 .
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2
real
1 0 -1 -2
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
2
imag
1 0 -1 -2
Figure 2- 7: Ideal BPSK baseband signal received with no frequency offsets, ∆f = 0 and a carrier phase offsets, θ (t ) − θˆ(t ) = 10 o . BPSK with phase shift
1.5 1
Quadrature
0.5 0 -0.5 -1 -1.5
-1.5
-1
-0.5
0 0.5 In-Phase
1
1.5
Figure 2- 8: Ideal BPSK baseband signal constellation received with no frequency offsets ∆f = 0 , but with a carrier phase offset of θ (t ) − θˆ(t ) = 10 o .
Figure 2-9 shows the BPSK shown in Figure 2-5 after having been subjected to frequency offset of ∆f = Rb / 100 , where Rb is the data rate. Notice the quadrature part that is normally only noise has gained a significant portion of the signal, since the carrier offset rotates the constellation from 0 and 180 degree by 2π∆fkT either clockwise or counter
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clockwise depending on the sign of ∆f relative to the carrier frequency, which in this case is the baseband frequency (i.e. 0 Hz). The impact also can be illustrated by inspecting the constellation shown in Figure 2-10 for an offset of ∆fT = 10 −2 and Figure 2-11 for a smaller offset of ∆fT = 10 −3 , which shows clearly that for large frequency offsets (relative to the data rate 1/T) the familiar BPSK does not resemble any BPSK constellation as that shown in Figure 2-6. 2
real
1 0 -1 -2
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2
imag
1 0 -1 -2
Figure 2- 9: Ideal BPSK baseband signal constellation received with no frequency offsets, ∆f = Rb / 100 and a carrier phase offsets, θ (t ) − θˆ(t ) = 0 . BPSK with frequency shift
1.5 1
Quadrature
0.5 0 -0.5 -1 -1.5
-1.5
-1
-0.5
0 0.5 In-Phase
1
1.5
Figure 2- 10: Ideal BPSK baseband signal constellation received with frequency offsets of ∆f = Rb / 100 and a carrier phase offsets, θ (t ) − θˆ(t ) = 0 . Dr. Mohamed Khalid Nezami © 2003
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1.5 1 0.5 0 -0.5 -1 -1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 2- 11: Ideal BPSK baseband signal constellation received with frequency offsets of ∆f = Rb / 1000 and a carrier phase offsets, θ (t ) − θˆ(t ) = 0 .
The same impact is also experienced by QPSK signals as shown in Figure 2-12 for QPSK baseband signal with phase offset θ (t ) − θˆ(t ) = 10 o and a frequency offset of ∆f = 0 . Again, with large frequency offsets, the QPASK signal will rotate clockwise as shown in Figure 2-13 for a frequency offset of ∆f = Rb / 100 . Notice that with the presence of carrier frequency offset, BPSK signals and QPSK signals are indistinguishable. QPSK with phase shift
1.5 1
Quadrature
0.5 0 -0.5 -1 -1.5
-1.5
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-1
-0.5
0 0.5 In-Phase
1
1.5
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Figure 2- 12: Ideal QPSK baseband signal constellation received with no frequency offsets, ∆f = 0 and a carrier phase offsets, θ (t ) − θˆ(t ) = 10 o . QPSK with frequency shift 2 1.5
Quadrature
1 0.5 0 -0.5 -1 -1.5 -2 -2
-1
0 In-Phase
1
2
Figure 2- 13: Ideal BPSK baseband signal constellation received with no frequency offsets, ∆f = Rb / 100 and a carrier phase offsets, θ (t ) − θˆ(t ) = 0 .
Next we detail the classification of algorithms that are used to resolve and correct both carrier frequencies so that the decoding is carried out properly. 2.5 Classification Based on Implementation Approaches
From the synchronization implementation point of view, synchronization circuits can be divided into two broad categories, viz., •
Feedback (FB) synchronization
•
Feedforward (FF) synchronization.
The feedback synchronization systems shown in Figure 2-14 involve the use of some form of phase locked loop (PLL) [5]. Although very popular and well understood, the emergence of burst type transmissions, that have become available with fast digital signal processors means that designs which are based on robust feedforward topologies are becoming prevalent. The reasons for departing from feedback topologies for bursty type time division multiplexing access (TDMA) are their slow algorithmic convergence and their inability to be implemented fully in the digital domain. An example of a feedback scheme includes the well known Costas loop used for carrier recovery in satellite and point-to-point systems [6,7]. Costas and PLL based carrier synchronization schemes typically have loop bandwidths ( BL ) on the order of one thousandth of the symbol rate Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
for optimal carrier phase tracking. As loop-settling times are proportional to 1 / BL , such loops require significant time to achieve stable synchronization. Thus for TDMA transmissions with data rates of a few hundred kbps, a duration of few hundred symbols would be required to achieve synchronization. Typically, feedback clock recovery schemes use a nonlinearity (such as a diode rectifier) to obtain a spectral line from the received random binary data, which is then fed into a PLL for tracking [1]. This technique is affected by the clock frequency deviation caused by the narrow loop bandwidth used and the increase in noise as a result of the nonlinearity employed. For frequency offset recovery, wide bandwidth AFC loops [8] are used for large frequency offset correction (Figure 2-14). Often, AFC loops are used in conjunction with a feed forward synchronizer [8]. Since AFC uses wide loop bandwidth to achieve fast response, its tracking operation is inaccurate, so a carrier phase tracking is still needs to be implemented as a separate entity in addition to the AFC loop that proceeds it . Feedforward (FF) synchronization systems shown in Figure 2-15 perform digital openloop estimation of unknown random parameters (timing, phase, and frequency offset) from the incoming matched filter samples. One of the attractive characteristics of this method is the absence of hang up and cycle slip [9,10] problems since there is no feedback (estimates are updated block by block). Cycle slips dodge conventional PLL and Costas loops. The absence of cycle slipping and hang-ups allows rapid acquisition of short burst type signaling. Furthermore, digital signal processing (DSP) implementation of FF synchronizers can make the receiver design more flexible and less expensive. Further, in FF systems, there is no feedback path from the synchronizer to the analog front-end of the receiver, which allows modular design and independent testing, resulting in rapid product development. Based on the ways of decoded data, feedforward synchronization algorithms are also further classified into the following types: •
Data-Aided (DA) feedforward synchronization systems
•
Decision-Aided (DD) feedforward synchronization systems
•
Non-Data Aided (NDA) feedforward synchronization systems
Data-Aided (DA) systems: Data aided synchronizers [11,12,13] are implemented by incorporating known preamble or amble symbols that are used by the feedforward loop to aid in the estimation. The disadvantage of DA systems is that they require an overhead transmission for the preamble/ample symbols, which is not desirable for short duration TDMA bursts, since it reduces the spectral efficiency. Decision-Directed (DD) systems: Decision-directed synchronizers [5,13] use an estimate of the data and not the true data extracted from the transmitted preamble. The
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performance of these algorithms is only optimal at high SNRwhere its performance approaches the performance of DA when the signal-to-noise ratio (SNR) is high. Non-Data Aided (NDA) systems: Neither data nor any decision on data is used in this technique [14,15] for obtaining the estimates of synchronization parameters. Instead, it averages over the data to obtain reliable estimates. At low SNR, NDA is the only available preamble-less rapid technique to estimate phase, frequency, and symbol timing. The main disadvantage of the NDA system is that it degrades heavily once the received signal has been distorted by multipath fading. Still, at low SNR it operates more reliably than its DD/DA counterparts. DA synchronizers perform well under fadingbecasue they rely on a training sequence. Again, this can work reliably only at high SNR since the synchronizer performance is based on the correctness of the detected training preamble.
Both feedback and Feedforward synchronization systems will be detailed in subsequent chapters. Next, however, we detail the methods employed for modeling synchronization systems using computer tools to study them and verify their algorithms. The models are used in conjunction with Monte-Carlo simulation approaches to study the impact of receiver and transmission impairments on the synchronization algorithms that are detailed throughout the book.
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Antenna
External AGC loop
Loop Filter
D/A
RF/IF
AGC Loop
Q(kT) Interpolator
X
I (kT )
τˆ
e NCO
(
− j 2π∆fˆkT +θˆk
)
X
Carrier tracking Loop
+
Symbol detection/decoding Phase offset error
θˆk
2π πkTf ∆fˆ
Loop Filter
Frequency offset error
Loop Filter
Symbol timing error
Symbol Timing Loop Loop Filter
Figure 2- 14: Feedback Synchronization system
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RF IN
Timing estimation and correction
Frequency estimation and correction
I (k ) RF section
Interpolator/ Decimator
Matched filter
Phase estimation and correction x
x
Decoder/ Receiver Display
Q(k ) τˆ Timing estimator
∆fˆ ∆f
Timing Post Processing
Frequency estimator
Phase post processing
Phase estimator
θˆ
Frame estimation
Figure 2- 15: Feed forward Synchronization system.
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2.6 Modeling Channel Impairments and Synchronization Errors
In order to investigate the performance of the synchronization algorithms overviewed in this chapter, a computer model for the digital transceiver is developed for generating the symbols, matched filter, shaping filter, and transmission channels for fading and AWGN shown in Figure 2-16. This model has provisions for injecting synchronization errors into the transmitted signal for testing purposes. The criteria for evaluating the synchronized algorithms are carried out using the mean and variance of the estimated parameter from the received signal. One way to achieve this is to introduce synchronization errors, then, since the introduced synchronization errors are known prior, this task is simplified(?) This will be the method by which synchronization performance is evaluated throughout the coming chapters. Figure 2-16 shows a computer model for the digital burst transmitter for generating the QAM symbols, matched filter, shaping filter, and transmission channels for fading and AWGN. The input variables to the model are a set of parameters given by
{θ TX , ∆f TX ,θ RX , ∆f RX , v, n(kT ),α , ε }
(2-13)
where {θ TX , θ RX } are phase errors given in radians introduced at the transmitter and receiver respectively, v is the velocity of the moving mobile receiver given in meters/sec, ε is the timing error introduced at the receiver by non-synchronous sampling given in fractions of a symbol interval εT, where 0<ε<1 , {∆f TX , ∆f RX } are the frequency errors introduced by the transmitter and receiver given in Hz, n(kT ) is AWGN, and α is the pulse shaping Roll-off factor. The performance of synchronization algorithms is evaluated based on the variances of estimated parameter, namely θˆ, ∆ˆ f ,τˆ .
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α SRC
data
Burst Format
M-QAM mapping
SRC
∆fTx
θTx
n(kT )
TX-synch. error insertion
+
Synch. error insertion
+ α ε
HT ( f )
fc v
∆f Rx θ Rx RX-synch. error insertion SRC
HR( f )
SRC
Raleigh fading channel Model
var[
εˆT − T T
]
var[ ∆fˆ − ∆f ]
var[ θˆ − θ ]
Data
Timing Recovery and Correction
Frequency Estimation and correction
Phase estimation and correction
εˆ
∆fˆ
θˆ
DECODER
Figure 2- 16: Baseband computer model for analyzing time/frequency/phase synchronization errors for M-QAM transceiver system . Dr. Mohamed Khalid Nezami © 2003
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2.6.1
Model for Symbol Generation, Pulse Shaping, and Match Filtering
Symbols are generated by random bits that are streamed into a symbol mapper, J bits at a time to create one of the possible 2 J symbols. The symbols are then shaped by a shapingfilter to constrain the spectral shape of the transmitted signal to a specified bandwidth within the allocated channel. Assuming that the symbols are Nyquist shaped, the receiver matched-filter can be implemented as a square root raised cosine filter (SRC) [16]. The model adds noise, propagation channel impairments, and frequency and phase offsets at the transmitter output, while symbol timing errors are added at the receiver input. Both transmitter filter H T ( f ) and matched filter H R ( f ) are implemented as a raised cosine filter equally split between the transmitter and the receiver. That is
H ( f ) = H R ( f )H T ( f )
(2-14)
where H T ( f ) and H R ( f ) are the transfer functions of transmit and receive filter respectively. It follows that H R ( f ) and H T ( f ) have the same amplitude characteristic, H R ( f ) = HT ( f ) = H( f )
(2-15)
One class of Nyquist functions which is extensively used to implement both filters is the raised-cosine-rolloff, ⎧ T π ⎪⎪ ⎤ ⎡ H ( f ) = ⎨T cos 2 ⎢ ( 2 fT − 1 + α )⎥ ⎦ ⎣ 4α ⎪ 0 ⎩⎪
1−α 2T 1−α 1+α ≤ f ≤ 2T 2T otherwise f ≤
(2-16)
where α is restricted to 0 ≤ α ≤ 1 and called the Roll-off or access-bandwidth factor. This filter also has the characteristics that satisfy the Nyquist criteria for no ISI. The inverse Fourier transform of H ( f ) is given by
h (t ) =
sin(πt T ) cos(απt T ) (πt T )(1 − 4α 2 t 2 T 2 )
(2-17)
Both forms of h(t ) or H ( f ) can be used in the simulation model to form the combined filtering effects of transmitter pulse shaping and receiver matched filter. However, in the model of Figure 2-16, it is necessary to implement pulse shaping and matched filtering separately as SRC, so that symbol timing errors can be inserted at the receiver by altering the coefficients of the impulse response of the receiver SRC alone which has a frequency given by
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⎧ T ⎪⎪ π ⎡ ⎤ H ( f ) = ⎨ T cos⎢ ( 2 fT − 1 + α )⎥ ⎣ 4α ⎦ ⎪ 0 ⎪⎩
1−α 2T 1−α 1+α ≤ f ≤ 2T 2T otherwise f ≤
(2-18)
Most digital receivers use four samples per symbol factor, the number of taps used in implementing these filters is based on the criteria of channel spectral shape, and minimization of ISI. In the model of Figure 2-16, for example, a 32-tap FIR filter was used to extend the impulse response convolution over at least 8 symbols, that is four symbols on each side of the center tap coefficient. After the symbols have been filtered using the transmitter filter H T ( f ) , they are packed into TDMA frames. Typically each TDMA frame is dedicated to a particular user by the use of a unique word (UW) assigned to that user. The frame also may include synchronization symbols, such as an alternating sequence of zeros and ones; however, for the present study no dedicated synchronization symbols are needed. Figure 2-17 shows the computer model used to illustrate the constructed TDMA frame at the transmitter. Here the serially transmitted data is first converted from serial stream to a two a parallel (S/P) symbol formation. Each symbol is then filtered using the SRC filter and then modulated onto the final carrier. Both inphase (I) and the quadrature phase (Q) branches are summed and filtered using bandpass filter (BPSF) to reject all out- of- band harmonics.
Modulator 01 I
Bits
cos(2πfc t )
RRCF R SRC
S/P
0110
BPF
-
Q
To RF TX
SRC RRCF 10 sin (2πf c t )
Figure 2- 17: MPSK Baseband modulation system. At the receiver, synchronization observation interval can be any part of the burst, whether it is data or control symbols. Also, the observation interval can be accumulated from a number of adjacent subintervals from adjacent bursts as shown in the hypothetical TDMA frame structure depicted in Figure 2-18 .Here each message is made of a synchronization preamble used in the case of data aided synchronization, a unique word that is often used to resolve phase ambiguity or to identify users, and the payload data transmitted to that specific user. Notice that the bits used for preamble and UW are an Dr. Mohamed Khalid Nezami © 2003
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overhead that reduces the TDMA transmission efficiency. Figure 2-19 shows an actual commercially used burst structure. Here the frame starts with a preamble sequence, then a unique word is used to indicate the start of a message (SOM), then a data payload packet is followed by an error check code and flush bits for the forward convolutional encoder. Synchr. Preamble
Senders address UW
Receivers address UW
Guard symbols
Message Data, Voice, …etc
TDMA BURST One observation interval of L-symbols
Symbol #1
Symbol #L
Figure 2- 18: A typical TDMA burst format and preamble structure.
User data Modulation type Start of Message Used for synchronization
Fields required for proper decoding
Figure 2- 19: A practical TDMA burst format showing the preamble and other overhead symbols. Figure 2-20 presents generation of the 16-QAM signal simulations using a RCF (2-16) with a Roll-off factor of α=0.75 using the model in Figure 2-17. The figure provides I-Q waveform constellation before and after being shaped by channel filtering. The impulse and frequency response of the transmit raised-cosine filter is shown in Figure 2-21. Figure 2-22 shows the power spectral densities of relevant signals showing the signal at various points throughout the model developed in Figure 2-16 for symbolrate of 10,000 symbols per seconds (sps). Indicated in Figure 2-22(a) is the transmitted signal spectral Dr. Mohamed Khalid Nezami © 2003
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density that indicates spectral null at the data rate of 10 kHz. The received signal after being passed through the matched filter is shown in Figure 2-22(b). The effect of absolute value nonlinearity on PSD (this will be detailed in the following chapters) is shown in Figure 2-22(c), which indicates the presence of the symbol rate of 10 kHz to be recovered from the received signal. As a result of the nonlinearity used, a strong DC is present. This DC term was removed by a simple high pass FIR filter with poles at +1 and –1 as shown in Figure 2-22(d). The effect of a smaller Roll-off factor (e.g., 0.35) is depicted in Figure 2-23 (a-d). It may be noted that limiting the bandwidth of the symbols results in the weakening of the spectral line corresponding to the symbol rate as shown in Figure 223(d). 3
2
1
0
-1
-2
-3 -3
-2
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3
2
1
0 Q
-1
-2
-3 -4 -4
-3
-2
1
0
-1
2
3
4
I
Figure 2- 20: 16-QAM constellation before (top) and after (bottom) pulse shape filtering using the SRC filter. 10
0.7
0
0.6
-10
0.5
-20
0.4 -30
0.3 -40
0.2 -50
0.1
-60
0
-70
-0.1
-80 0
0.5
1
Hz
0
2
1.5
4
x 10
10
20 Coef. Index
30
Figure 2- 21: RC filter impulse and frequency response, Roll-off = 0.75.
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20
-20 -40 -60
-20 -40 -60
-80
-80 0
0.5
1
1.5
2 x 10
0
0.5
1
1.5
4
2 x 10
20
4
20 DC- term
C)
DC-LPF-OUT
NLN-OUT
b)
0
a)
MF-OUT
TX-OUT
0
0
-20
-40
d)
0
-20
-40 0
0.5
1 Fre q.
1.5
2 x 10
0
0.5
1 Fre q.
4
1.5
2 x 10
4
Figure 2- 22: PSDs of the outputs (see model figure 2-16) at (a) Tx (b) MF with Roll-off = 0.75 (c) ABS nonlinearity and (d) HPF (DC removing filter). 20
20
-20
-40
-60
-20 -40 -60
-80
-80
0
0.5
1
1.5
2 x 10
0
0.5
1
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1.5
4
x 10
20
4
20
DC-LPF-OUT
C)
NLN-OUT
b)
0
a)
MF-OUT
TX-OUT
0
0
-20
-40
d)
0
-20
-40
0
0.5
1 Freq.
1.5
2 x 10
0
4
0.5
1 Fre q.
1.5
2 x 10
4
Figure 2- 23: PSDs of the outputs at (a) Tx (b) MF with Roll-off = 0.35 (c) ABS nonlinearity and (d) HPF (DC removing filter).
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2.6.2
Model for Intentional Synchronization Error Insertion
To investigate synchronization errors and further test the performance of the synchronization algorithms, intentional timing error ε and carrier frequency and phase reference errors have to be inserted in the transceiver as shown in Figure 2-16. However, symbol timing errors εT usually happen at the receiver sides. Since these errors are associated with the local clock of the Analog-to-Digital converter (A/D), they must be inserted either in the receiver matched filter or after it. The insertion of both carrier frequency and carrier phase offsets involves a complex multiplication of either the 2π∆fkT +θ
k transmitted or receiver signal by the phasor of e ± j , where k is the symbol or sample index. Symbol timing insertion is a process that is also identical to correction symbol timing errors once an estimate of the error has been obtained using symbol timing synchronizers. One way timing offsets can be introduced into the MF output samples is by either shifting the MF impulse response by a timing offset equal to εT, or resampling the MF output samples using an interpolator to obtain the sample values at the instants (k+ε)T.
The resampling scheme can be implemented by a FIR interpolator filter whose coefficients are generated based on LaGrange polynomial [17,18]. For simulation experiments, shifting the MF filter impulse response is preferred because of its simplicity. To illustrate this, a 32-tap FIR filter, with coefficients of [h(0):h(+31)], that extends over the duration of [-4T:+4T], is used as the MF. Figure 2-24(a) shows h(15) and h(16) at equal magnitude, where the difference of h(16)-h(15) is zero, indicating no symbol timing error. However, if there is symbol timing error at the receiver, the coefficients h(16) and h(15) will move to new locations as shown in Figure 2-24(b) , where h(16)-h(15) will give a value that roughly is proportional to the fractional symbol timing error ε . Figure 2-24(c) shows a comparison between the case ε =0 and ε =0.25.
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1 a) timing error = 0
0.5 0 0
10
20
30
40
1 b) timing error =0.25 T
0.5 0 0
10
20
30
40
1 c) superposition of (a) and (b)
0.5
h(-16)
h(15)
0 0
10
20
30
40
Figure 2- 24: Shifted MF impulse responses for introducing intentional symbol timing errors.
2.6.3
Model for Additive White Gaussian Noise Channel
For simulating the addition of AWGN to the baseband signal in the model of Figure 2-16, the real and imaginary of the signal are generated using random voltage signals and not powers. An additive white Gaussian noise is then generated using random number generators and then added to the quadrature samples to emulate AWGN channel. The inphase digital samples are given by z I (kTs ) = I (kT ) + δ nη I (kT ) and the quadrature digital samples are given by z Q ( kTs ) = Q ( kT ) + δ nη Q ( kT )
where δ n is an attenuation factor used to control the level of the noise added to emulate
Eb . The variable δ n is then calculated as follows. First, energy per No bit ( Eb ) and noise power density ( N o ) are related by
variation in SNR or
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Eb =
S power Rb
[W.T/bit]
(2-19)
where Rb is the data rate in bits per seconds, S power is the signal power per symbol. The noise power in the baseband signal is then given by, Nb =
N power Rs
[W/Hz]
(2-20)
where N power is the noise power per symbol, and Rs is the symbol rate. Taking the ratios of (2-19) to (2-20) , that is
S power Rs Eb = N o N power Rb
(2-20)
After manipulations of (2-20), the noise power which is varied in the model of Figure 2E 16 to emulate variations in b or SNR, is given by No
N power =
S power Rs Eb N o Rb
(2-21)
Eb given in dBs, Equation (2-21) is calculated in the simulations using the No following, with
N power =
S power ⎛ Rs ⎜ ⎛ Eb N o ⎞ ⎜ ⎜ ⎟ ⎝ Rb 10 ⎝ 10 ⎠
⎞ ⎟⎟ ⎠
(2-22)
Since both quadrature samples of the baseband signal z(kTs) are generated using voltage values and not power values, the noise power calculations in Equation (2-22) have to be converted to voltages when adding the Gaussiamn generated in-phase and quadrature noise densities to the baseband signal samples. That is
δn =
1 N power = 2
1 S power 2 ⎛⎜ Eb10N o ⎞⎟ ⎠ 10 ⎝
⎛ Rs ⎜⎜ ⎝ Rb
⎞ ⎟⎟ ⎠
(2-23)
For BPSK signals, where each bit is one symbol, Rs = Rb , Equation (2-23) becomes
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δ n , BPSK =
1 S power 2 ⎛⎜ Eb10N o ⎞⎟ ⎠ 10 ⎝
(2-24)
Now to represent this further in terms of signal-to-noise ratio (SNR), we use the simple fact that
Eb ⎛ S ⎞ Bn =⎜ ⎟ N o ⎝ N ⎠ Rb
(2-23)
Bn is the ratio of noise bandwidth to the bit rate. Very often people mistakenly Rb use Rs instead of Rb which will result in 3dB error for 4-ary signals such as QPSK. The computation of S power in (2-23) for complex baseband signal z (kT ) having a record of N
where
samples, is given by,
S power =
1 N
N −1
∑ z (kT )z
*
(kT )
(2-24)
k =0
where z ( kT ) z * ( kT ) Notice that the noise addition and calculations have to be performed for both in-phase and quadrature phase independently, i.e. the power of the in-phase 1 N −1 branch is S power ,Q = ∑ Q 2 (kT ) and the power in the quadrature branch is N k =0 N −1 1 S power , I = ∑ I 2 (kT ) . N k =0 In simulations, the noise generation (2-24) that is based on a given desired
Eb is carried No
using the procedure shown in Figure 2-25 (?)
EbNo_dB = 6 % range of Eb/No=6dB in dB Ns = 4; % number of samples per symbol (2,4,8,….etc) M = 8; % order of modulation (M-ary, 2 for BPSK, 4 for QPSK) code_rate = 2/3; % code rate (rate ½ , 1/3 ..etc) EbNo= 10^(EbNo_dB/10); %Convert Eb/No from dB to linear domain, then % compute standard deviation Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers std_dev=sqrt(Ns/(2*EbNo*code_rate*log2(M))); % generate noise using randn(x) function: noise = std_dev(mm)*(randn(length(r),1)+j*randn(length(r),1)); % add AWGN noise to the basebandsignal r_noisy_signal=z+noise; % z is baseband with Ns sample per symbols
Figure 2- 25: AWGN simulation routine.
2.7 References
1. Chorng Shue, Y. Huang, and C. Huang, “Joint symbol, frame, and carrier synchronization for Eureka 147 DAB system”, Proceedings of Vehicular Technology Conf., 1997, pp. 693-697. 2. Si-Ming, and D. Dodds, “Frame Synchronization probability calculation including effects of pattern bifix”, IEEE Commun. Letters, vol. 45, No. Y, pp. 100-102, Feb 2000. 3. J. Beek, M. Sandell, M. Isaksson, and P. Borjesson, “Low-complex frame synchronization in OFDM systems” Lulea University of Technology, Sweden, 1997. 4. Chorng Shue, Y. Huang, and C. Huang, “Joint symbol, Frame, and Carrier synchronization for Eureka 147 DAB system”, Proceedings of ICC, 1997. 5. Heinrich Meyr and Gerd Ascheid, Synchronization in Digital Communications Volume-1, New York, Wiley, 1989. 6. Mordechi Rennert and Ben Zion Bobrovsky, “Designing second order Costas loops and PLL's to track Doppler shift-analysis and optimization” Proceedings of GLOBECOM, 1995. 7. J.P. Costas, “Synchronous Communications,” Proceedings of the IRE, pp. 17131718, Dec 1956. 8. F.D. Natali, “AFC tracking Algorithms”, IEEE Trans. Commun, vol. 32, No. 8, Aug 1984, pp. 935-947. 9. Geert de Jonghe and marc Moeneclaey, “Cycle slipping behavior of NDA feedforward carrier synchronization for time-varying frequency-nonselective fading channels” Proceedings of GLOBECOM-1995, pp. 350-354. 10. G. Jonghe, and Marc Moeneclaey, “The effect of the averaging filter on the cycle slipping of NDA feedforward carrier synchronizers for MPSK,” Proceedings of the ICC, pp. 0365-0369, Aug 1992.
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11. Umberto Mengali and M. Morelli, “Data-Aided frequency Estimation for burst digital transmission” IEEE Trans. Commun., vol. 43, No. 1, pp. 23-25, Jan 1997. 12. K. Geothals, and M. Moeneclaey, “PSK symbol aided synchronization of MLoriented data-aided algorithms for nonselective fading channels”, IEEE Trans. Commun., vol. COM-46, No. 2/3/4, Feb 1995. 13. Umberto Mengali and M. Morelli, “Data-Aided frequency Estimation for burst digital transmission” IEEE Trans. Commun., vol. 43, No. 1, pp. 23-25, Jan 1997. 14. Estathiou, and H. Aghvami, “Preamble-less non-decision aided (NDA) feedforward synchronization techniques for 16-QAM TDMA demodulators,” Proceedings of ICC, 1998, pp. 1090-1094. 15. Marc Moeneclaey and Geert de Jongh, “Tracking performance comparison of two Feedforward ML-oriented carrier-independent NDA symbol” IEEE Trans. Commun., vol. 40, no. 9, pp. 1423-1425, Sept 1992. 16. Theodore S. Rappaport, Wireless Communications, Principle and Practice, Prentice Hall, NY, 1996, chapters 1-2. 17. Jean Armstrong, and David Strickland, “Symbol timing using samples and interpolation”, IEEE Trans. Commun., vol. 41, No. 2, pp. 318-321, Feb 1993. 18. Tony Kirke, “Interpolation, resampling, and structures for digital receivers”, Communication System Design Magazine, pp. 43-49, Jul 1998.
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Chapter 3
Feedback Carrier Synchronization Systems
This chapter discusses the principles and application of feedback carrier synchronization systems in modern digital wireless receivers. We first derive the mathematical principles of digital and analog phase locked loops, and then introduce various forms of carrier phase tracking and carrier frequency acquisition loops. The objective is not to complicate these systems with tedious mathematical derivation, but to introduce their principle of operations and show their implementation, then compare their performance and their implementation complexity. Methods for deriving lock indicators from the loops are also illustrated. Finally we give an example illustrating the design and analyses of a typical synchronization feedback loop using simulations.
3.1 Introduction to Feedback Synchronization The classical approaches for closed loop carrier phase and carrier frequency synchronization are very diverse. However, phase locked loops (PLL) represent the core of any of these systems. The principles of PLLs date to their first appearance in the 1920s. Since then, and through the rapid development of digital signal processing, PLLs have become a part of many electronic and mechanical systems. PLLs can be implemented fully in analog, partially analog, partially digital, or fully digital. Systems that are totally analog are being phased out and are becoming less common. Systems that are partially digital involve processing the error signal using phase detector as a sensor and then the correction is fed back from the digital section to the analog section using digital-to-analog converters (DAC). Figure 3-1 illustrates one implementation example that is partial analog and partial digital. Here both voltage controlled oscillator (VCO) and the voltage controlled crystal oscillator (VCXO) are used in conjunction with an A/Ds, DACs and a digital signal processor to form the closed feedback loop for correcting both carrier offsets and symbol timing clock errors. In Figure 3-1, the signal is first down converted from radio frequency (RF) to an intermediate low frequency (IF) that is sampled using the A/D, at a sample rate controlled by a VCXO controlled by the digital section. The local oscillator used to downconvert the signal is initially set to a nominal frequency as close as possible to what is thought to be the carrier frequency offsetby the IF frequency, with its phase and frequency controlled by the error signal that Dr. Mohamed Khalid Nezami © 2003
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is manipulated in the DSP section. The loops in Figure 3-1 are commonly used; however, since their operation involves feedback from digital to analog, they are becoming less popular becasue there is a desir to implement both timing and carrier recovery loops totally in the DSP section. This increases the reliability of these systems and allows both RF and DSP sections to be tested and manufactured separately.
Symbol timing error correction signal
Symbol timing circuit
Tunable voltage controlled crystal clock
RF section
A/D
Demodulator DSP Bit decoder
exp(-j2π fc t)
~ Voltage controlled local oscillator
Carrier offset correction signal
Carrier recovery circuit
Figure 3- 1: Semi analog, semi digital carrier and symbol timing synchronization system.
Figure 3-2 shows a carrier and symbol timing synchronization system that is fully implemented in digital and is very attractive for use in software defined radio receivers (SDR). The receiver here has four loops, two synchronization loops used to acquire and track the carrier phase and frequency offsets separately, a loop for symbol timing tracking, and a loop for automatic gain control (AGC). Both carrier phase and carrier frequency errors are sensed digitally before the error signal is processed using a loop filter. The result is then programmed in the numerically controlled oscillator (NCO) that is used to perform a final frequency translation from the near baseband to true baseband. Symbol timing errors ( τ ) are detected also in digital domain. A correction signal is then used to program an interpolator filter that is used to resample the samples of the near baseband signal to a rate that matches the rate of symbols used at the transmitter. Finally the AGC feedback loop, which is an essential part of the synchronization loops, is used to keep the signal level at an appropriate level for the various loops. Notice that the AGC is made of two loops. The first inner loop tries to adjust the signal level to a fixed reference digitally in the baseband section; however, if this is not achieved, the loop will feed an overflow or underflow indicator to the analog section via the DAC to adjust the RF signal level in the front end of the receiver to bring its level to a range that is within the abilities of the inner AGC loop. The AGC loop algorithms will be examined in more detail in Dr. Mohamed Khalid Nezami © 2003
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chapter 4. Figure 3-2 also shows lock indicators that are usually derived by comparing the carrier and symbol timing error signal against a minimum error threshold, and then are used to notify the user and various parts of the receiver to start the demodulation process. If for any reason the loops fail during tracking, these flags will request a reacquisition of carrier, symbol timing, and AGC. At the heart of the three synchronization loops in Figure 3-2 is a phase lock loop (PLL). The loop operation and performance are presented next. The reader may refer to books such as [1,3,4,5,16] for further mathematical analysis of these loops. For our purposes, however, we will only focus on the essential design equations that are needed for successful implementation of both frequency and phase carrier acquisition and tracking by practicing engineers and researchers. The closed loop timing adjustment will be examined in more detail in chapter 4. In this chapter we mainly focus on carrier acquisition and tracking loops. First we derive the second order analog loop, then using classical z-transform techniques, we derive the equivalent second order digital locked loop that is for DSP implementation. Then we will detail various common carrier recovery loops, explain their theory of operation, and review their performance limitations.
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Antenna
External AGC loop
RF/IF
Loop Filter
D/A
AGC Loop
Q( t )
I (t )
A/D
↓
AA
decimator
Interpolator
X
ˆ ˆ e − j (2π∆fkT+θ )
Symbol detection/decoding
X
k
τˆ
CO N NCO
Carrier tracking Loop
Phase offset error
θˆk
+
2ππ∆∆fˆkT fˆ
Loop Filter
Frequency offset error
Loop Filter
Symbol timing error
Symbol Timing Loop Loop Filter
τˆ
Lock indicators
Symbol timing: Acquired/Tracked
∆fˆ
θˆk
Carrier: acquired /tracked
Figure 3- 2: Block diagram of a digital wireless receiver emphasizing the baseband sample processing, AGC, and carrier and symbol timing feedback synchronization loops.
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3.2 Principle of Analog Phase Locked Loops
Figure 3-3 shows the simple principle of analog phase lock loop operation. Here the input signals are assumed to be a continuous wave sinusoidal signal with a constant frequency wc = 2πf c and arbitrary phase shift θ . That is s (t ) = cos( wct + θ )
(3-1)
This signal s (t ) is then down converted to baseband using a local oscillator that has a nominal frequency set as close as possible to w = 2πf , but has an arbitrary phase θˆ . c
c
That is
v(t ) = − sin(wc t + θˆ)
(3-2)
The error signal is then derived by e(t ) = s (t )v (t )
(3-3)
Assuming that there is no frequency mismatch between the local VCO signal (3-2) and the incoming signal frequency (3-1), the error signal output is given by
e(t ) = − cos(wc t + θ ). sin(wc t + θˆ)
(3-4)
which is then expanded using trigonometric formulas [16] to
e(t ) = −0.5 sin(2wc t + θ + θˆ) + 0.5 sin(θ − θˆ)
(3-5)
This error signal contains a double frequency term 2wc , which is filtered using the loop filterwhich has a low pass frequency response. As a result, the phase detector (multiplier) output is given by e(t ) = 0.5 sin(θ − θˆ)
(3-6)
It will be further shown that when the phase error is small (i.e., true when the loop is in the tracking mode), that the phase error signal sin(θ − θˆ) → θ − θˆ , which means that the loop will vary the phase of the local oscillator until the error signal e(t ) is zero, that is θ = θˆ , once that is achieved, the carrier tracking loop is said to locked to the phase of the incoming received signal. The carrier phase recovery is implemented using a phase lock loop structure as illustrated in Figure 3-3. The loop is made of a phase controlled
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( θˆ ) local oscillator in conjunction with a phase detector that produces the error signal that indicates the difference between the local oscillator carrier phase, and a loop filter used to filter the error so that the local oscillator phase perturbation is dynamically stable.The loop filter used in Figure 3-3 is one of the main controlling design factors in the speed and accuracy of the phase estimation and will be detailed further in the coming sections.
e(t )
s (t )
Loop Filter
VCO
v(t )
Figure 3- 3: Simple illustration of carrier phase locked loop
3.2.1
Second order phase locked loop
Figure 3-4 shows the linear representation of the analog second order phase locked loop using the Laplace transform [16]. The VCO is modeled as an analog (continuous time) Ko perfect integrator with a transfer function of , where K o is the VCO gain s (radians/volts). One metric describing the loop dynamics and used to obtain the loop parameter given a desired dynamic response is the input-to-output ratio of the loop, which is known as the loop transfer function. The transfer function describes the ratio of the phase of the VCO to the phase of the incoming signal and is given by H PLL ( s ) =
ˆ (s) K d K o F ( s) Θ = Θ( s ) s + K d K i F ( s )
(3-7)
where K d is the phase detector gain constant (radians/volts). For a second order phase locked loop, the filter used in the loop (loop filter) is commonly a first order filter with a transfer function given by
K2 (3-8) s where K1 is a proportional constant, and K 2 is the integral constant which both impacts the speed and noise performance of the loop. Substituting the VCO and the loop filter F ( s) = K1 +
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transfer functions into the PLL transfer function in Equation (3-7), results in the loop transfer function as a function of the loop variables. That is
H PLL ( s) =
K d K i K1 s + K d K o K 2 s + K d K o K1 s + K d K o K 2
(3-9)
2
Equation (3-9) shows that the tracking performance of the loop is controlled by the VCO gain, the phase detector gain, and the loop filter constants. For a given desired performance (noise bandwidth of the loop and speed of convergence) to obtain an optimal set of the loop variables, the PLL transfer function (3-9) is equated to a standard second order control loop transfer function that has been evaluated and tabulated in various sources [16,17] and given by H ( s) =
2ς wn s + wn2 s 2 + 2ς wn s + wn2
(3-10)
By equating the canonical second order loop transfer function in (3-10) to the second order phase locked loop transfer function in (3-9), the second order PLL dynamic parameter’s natural frequency wn and the damping factor in (3-10) are obtained as a function of the loop filter coefficients and the VCO and phase detector gains. That is wn = K d K i K 2
(3-11)
and
ς=
K1 2
Ko K p
(3-12)
K2
Equation (3-11) and (3-12) are all that is needed to implement these loops using either analog loops or digital signal processing loops.
Θ(s )
Φ (s )
+ -
Kd
F (s )
ˆ (s) Φ Ko s Figure 3- 4: Linear phase locked loop model. The loop natural frequency is further used to obtain the loop noise bandwidth given by
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Bn =
∞
1 H ( 0)
2
∫ H (w) dw
(3-13)
−∞
For the second order loop, the loop bandwidth is simply [16], ⎛ 1 B L = πBn ⎜⎜ ς + 4ς ⎝
⎞ ⎟⎟ ⎠
(3-14)
wn . This loop bandwidth dictates the 2π signal to noise ratio in the loop itself, which is also an important measure of the loop behavior. The loop SNR ratio is given by where Bn is the loop natural bandwidth B n =
SNRL =
SNRi Bi 2 BL
(3-15)
or in logarithmically relations is given by
⎛B SNRL (dB) = SNRi (dB) + 10 log10 ⎜⎜ i ⎝ BL
⎞ ⎟⎟ − 3dB ⎠
(3-16)
where Bi is the bandwidth of the signal being tracked (input signal). By inspection of Equation (3-16), one realizes that the carrier recovery loop enhances the SNR of the input signal by the ratio of the loop bandwidth to the bandwidth of the input signal itself.
3.2.2
Second Order Loop Dynamic Behavior
One of the most important dynamic behaviors of the carrier recovery loop is its ability to acquire and track either frequency offset or phase offset within an acceptable time. The acquisition time of the PLL is defined as the time required for the PLL to go from initial frequency offset ∆f to phase lock where θ = θˆ . The process for acquisition is performed first with the loop reducing the frequency offset ∆f to zero. Once frequency lock is achieved, an additional time period (lock time) is required to reduce the loop phase error to an acceptable level. So the lock time of the loop is given by the sum of the frequency lock time and the phase lock time. That is t lock = t f _ lock + t ph _ lock
(3-17)
The frequency lock time (acquisition time) is inversely proportional to the cube of the loop bandwidth [16] and is proportional to the square of the offset frequency and given by, 3-8
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t f _ lock ≈ 4
(∆f )2
(3-18)
B L3
The phase lock loop, however, is only proportional to the loop bandwidth and is given by t ph _ lock ≈
1 .3 BL
(3-19)
Another important measure of the carrier recovery loops is how large a frequency offset they can acquire for a given loop bandwidth. This range is commonly referred to as the ‘pull-in’ frequency range and is proportional to both the damping factor and the loop bandwidth, and is given by ∆f ≤ 2 B L 2πς
(3-20)
The loop bandwidth also controls another measure of the loop behavior, which is the variance of the tracked phase, and is commonly known as the phase jitter. Phase jitter at the output of the PLL is given by [1,5,8,16]
σ θ2 =
BLT [rad 2 ] Es N o
(3-21)
Further more, the loop may occasionally get hit by what is known as cycle slips, which is a major source of carrier recovery failures, specially during frequency offset error (i.e., fading and Doppler) and low SNR. This measure was derived by Viterbi in [4] for SNRL > 10dB and is given by
τ cs =
π 2 BL
e SNRL
(3-22)
Notice that the average time between cycle slips τ cs
is inversely proportional to the
loop bandwidth. For high SNR values, the cycle slip diminishes and τ cs is assumed to be in terms of days or many hours. For a practical modem design, the initial loop design parameters can now be derived simply using equations (3-14) through (3-21), assuming that the following receiver operating parameters are specified, o Maximum frequency offset ∆f . o Desired time to lock t lock . o Desired Bit-Error-Rate (BER) at a specific E s N o .
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o Desired average time between cycle slips τ cs . o Desired Output carrier jitter σ θ2 . o Phase detector gain. o VCO gain.
3.2.3
Digital phase lock loop Principle
Most modern receivers are implemented in digital domain, so the loop in Figure 3-3 is only good as an analytical model from which derive the initial design parameters of the digital loop are guessed. For DSP based closed loop implementation, the analog loop is converted using the classical Z-transform, resulting in a digital equivalence for the analog loop [17]. Like the S-domain Laplace transform for contnous (analog) signals, the Zdomain is equivalently used for sampled signals (discrete signals). The difference between the analog loop and the digital loops is that the VCO is now implemented using a numerical controlled oscillator (NCO), the phase detector is implemented using complex multipliers instead of analog multipliers, and the loop filter is a perfect filtersince the proportionality and integration is perfectly performed using numerical equations that are independent of manufacturing tolerances, aging, and temperature variations. The ultimate objective is then to implement the digital equivalency of the analog PLL in Figure 3-5 while maintaining the desired dynamic behavior that is derived in the continuous time loop. In the next section, we derive the digital phase locked loop (DPLL) representation.
One easy way to transform the transfer function of the loop in Equation 3-9 to a digital 2 1 − z −1 domain equivalent is by the bilinear Z-transform method which substitutes s = T 1 − z −1 in Equation 3-9. Avoiding some of the lengthy derivations [1], and noting that the VCO in Figure 3-4 is substituted for by a numerical controlled oscillator implemented as a perfect integrator with a transfer function given by (??)
N ( z) =
Ko 1 + z −1
(3-23)
The overall transfer function of the digital phase locked loop (DPLL) in Figure 3-5 ,relating the input to output phase response, is given by H ( z) =
K d F ( z) N ( z) 1 + F ( z ) N ( z ) z −1
(3-24)
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Using a first order loop digital filter F ( z ) in (3-24), the transfer function becomes the well-known standard of a second order discrete control loop given by H ( z) =
K d K o (g 0 z −1 + g1 z −2 ) 1 + (K d K o g 0 − 2 )z −1 + (K d K o g1 + 1)z − 2
(3-25)
where g0 =
1 Kd Ko
2 ⎛ ⎞ 2(wn T ) − 8 ⎜ ⎟ + 2 2 ⎜ 4(w T ) + ςw T + 4 ⎟ n n ⎝ ⎠
(3-26)
g1 =
1 Kd Ko
⎛ (wn T )2 − 4ςwn T + 4 ⎞ ⎜ ⎟ ⎜ 4 (w T )2 + ς w T + 4 − 1 ⎟ n n ⎝ ⎠
(3-27)
and
where Ko is the NCO gain or radians sec-1 units-1 in the digital domain, Kd is the phase detector gain in volts/ radians in the analog domain. Next we present three different loop filter implementations. These filters all have the same order and are mathematically equivalent. However some topologies may be preferable to others depending on the way the software implementation is carried out. Loop Filter
Phase detector model
Θ(z )
Φ (z )
+
Kd
F (z )
-
z
NCO model
N (z )
Ki 1 − z −1 ˆ ( z) Θ
−1
Delay appropriate to make the system casual
Figure 3- 5: Discrete time phase locked loop model.
3.2.4
Loop Filter Digital Implementation
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The order of the loop filter is determined by the highest (negative) power of z in the denominator of the transfer function (3-25). Most of the loop filters used are some variant of the loop shown in Figure 3-6. Since the filter is a first order filter, and because of the additional pole introduced by the numerically controlled oscillator (NCO), the overall DPLL will be a second-order type system. The loop filter in the PLL system functions as follows. The phase detector output, x(z) is multiplied by the proportional gain Kp in the upper arm. Then the phase detector output in the lower arm is multiplied by the integral gain constant Ki . The result is then fed to an integrator consisting of an adder and a unit delay register. The final output y(z) is then the sum of both branches, which is fed to the NCO to update the phase value.
Kp
x ( z)
+
z −1
y (z)
+
Ki Figure 3- 6: Second order loop filter topology -I. Using discrete time domain representation of the filter’s frequency response function, the loop filter in Figure 3-6 has a transfer function given by, −1 y ( z ) K p + (K i − K p )z = x( z ) 1 − z −1
(3-28)
When implemented using DSP instructed chips, , the output of the loop filter using (3-28) is given by, y(n) = Kp * x(n) + Ki * x(n-1) + y(n-1) - Kp * x(n-1)
The proportional constant K p and integral constant K i are given by [16]
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⎛ ⎞ ⎜ ⎟ ⎟ Twn2 ⎜ 1 Kp = ⎜1 + ⎟ 2K o K d ⎜ ⎛ Twn ⎞ ⎟ ⎜ ⎟ tan⎜ ⎟⎟ ⎜ ⎝ 4ς ⎠ ⎠ ⎝
(3-29)
Twn2 Ki = KoKd
(3-30)
and
Another first-order loop is shown in Figure 3-7. This loop filter is similar to the first loop filter in Figure 3-6, except that the output of the integrator in the lower arm is taken from the adder instead of from the register. The loop is implemented in software as y(n) = Kp * x(n) + Ki * x(n) + y(n-1) - Kp * x(n-1)
Kp
x ( z)
Ki
y (z)
+
+
z −1
Figure 3- 7 : Loop filter topology II.
The loop transfer function now is given by −1 y ( z ) (K i + K p ) − K p z = x( z ) 1 − z −1
(3-31)
Both proportional and integral constants are also computed using equation (3-29) and (330). A third variation of the loop filter is shown in Figure 3-8,
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Kp
z −1
x ( z)
+
Ki
y (z)
+
z −1
Figure 3- 8: Loop filter topology III.
The transfer function relating the phase detector output to the NCO phase correction signal is given by −1 y( z) K p + K i z (3-32) = x( z ) 1 − z −1 Where the constant K p and integral constant K i are calculated using (3-29) and (3-30). The loop in software is given by y(n) = Kp * x(n) + Ki * x(n-1) + y(n-1)
Next we will introduce various forms of carrier recovery using both analog and digital implementations.
3.3 Illustration Example: Simulated PLL in Fading Channels Use computer simulations to implement a second order type PLL showing impact of fading on the PLL tracking operation: The program below illustrates the creation of a PLL that acquires and tracks a complex signal at 1khz in both AWGN and Fading. % PLL illustration using MATLAB % Mohamed K. Nezami % 2002
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clear all close all % define initial phase offset and % incoming CW frequency and % sampling frequency theta = 60*pi/180; f=1e3; fs=100e3; % create the real and imaginary parts of a CW non-modulated.. % carrier to be tracked. k=1:1:1000; delf=f/20; cpx1=exp(j*(2*pi*k*(f+delf)/fs+theta))+.01*(rand(1,1000)+j*rand(1,1000)); % ** % initilize PLL loop phi_hat(1)=30; e(1)=0; phd_output(1)=0; nco(1)=0 % define loop filter parameters kp=0.15; % proportional constant ki=0.1; % Integrator constant % PLL implementation for n=2:length(cpx1) nco(n)=conj(exp(j*(2*pi*n*f/fs+phi_hat(n-1)))); % Compute NCO phd_output(n)=imag(cpx1(n)*nco(n)); % complex multiply NCO x input e(n)=e(n-1)+(kp+ki)*phd_output(n)-ki*phd_output(n-1); % Filter integrator some(n)=(kp+ki)*phd_output(n)-ki*phd_output(n-1); phi_hat(n)=phi_hat(n-1)+e(n); % update NCO end;
% ************************************************* % plot waveforms index_stop=200; figure subplot(211),plot(1:index_stop, phd_output(1:index_stop)),ylabel('Ph. Det.') subplot(212),plot(1:index_stop, phi_hat(1:index_stop)*180/pi),ylabel('Est. Phs.') figure, index_stop=200; subplot(211),plot(1:index_stop, real(nco(1:index_stop)),1:index_stop, real(cpx1(1:index_stop))), ylabel('RE-PLL')
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subplot(212),plot(1:index_stop, imag(nco(1:index_stop)),1:index_stop, imag(cpx1(1:index_stop))), ylabel('IM-PLL')
1 Initial Phase = 60 degrees Input frequency=1kHz Sampling Frequency=100kHz
Ph. Det.
0.5 0 -0.5 -1
0
20
40
60
80
100
120
140
160
180
200
160
180
200
Est. Phs. degrees
150 Frequency Offset=50Hz 100
50 Frequency Offset=0Hz 0
0
20
40
60
80
100 120 Sample #
140
Figure E3- 1: PLL phase error and loop filter output signals with and without frequency offsets of 50Hz and an initial phase offset 60-degrees. 1.5
RE-PLL
1
Incoming signalreal part
NCO signal
0.5 0 -0.5 -1
0
20
40
60
80
100
1.5 NCO signal
IM-PLL
1
120
140
160
180
200
180
200
Incoming signalimaginary part
0.5 0 -0.5 -1
0
20
40
60
80
100 120 Sample #
140
160
Figure E3- 2: Comparison of the NCO signal versus the incoming signal showing tracking operation. Figure E3-3 through Figure E3-6 illustrate the fading impact of the operation of the PLL. . Here a phase locked loop (PLL) is used to acquire and track a CW preamble of a
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waveform that is corrupted with noise and fading as shown in Figure E3-3 (bottom). The figure shows that for moderate slow fading, the PLL can track the carrier, however during deep fades (20dB -30dB), the PLL is presented only with AWGN that perturb the loop to random phase (see Figure E3-4) corrections, which often drive it into cycle slipping (see Figure E3-5) that causes a burst of data errors. Figure E3-6 illustrates another draw back to these conventional loops. Here sudden noise and fading cause the loop dynamics to propagate for a finite time into future data symbols, and often may even drive the loop into a state of no return as illustrated. These drawbacks and the fact that the burst type systems leave only one option as will be seen in the coming chapters, and that is to use block based estimation and feedforward corrections.(??) 6
RE-P LL
4 2 0 -2 -4
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
6 4 2 0 -2 -4
Figure E3- 3: Phase lock loop operation with a short disturbance due to fading and noise. 4
P h. Det.
2 0 -2 -4 -6
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
300
Es t. P hs .
200 100 0 -100
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Figure E3- 4: Phase error signal (top) and phase correction for 60-degree carrier error (no cycle slips).
4
P h. Det.
2 0 -2 -4 -6
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
500
Es t. P hs .
0 -500 -1000 -1500
Figure E3- 5: Phase error signal (top) and phase correction for 60-degree carrier error (with cycle slips). 4
P h. De t.
2 0 -2 -4
0
500
1000
1500
2000
2500
3000
100
Es t. P hs .
80 60 40 20
0
200
400
600
800
1000
1200
1400
1600
1800
FIgure E3- 6: Phase error signal (top) and phase correction for 60-degree carrier error (with cycle slips and high variance).
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3.4 BPSK Carrier Recovery using Squaring Loops One method of tracking the carrier in BPSK systems is the square loop system shown in Figure 3-9. This system is popular in analog since the squaring operation is produced by passing the signal through a switching diode. The principal of operation of this loop is obtained by applying direct squaring of the input signal s (t )
s (t ) = 2 P d (t ) cos(2πf c t + θ )
(3-33)
where d (t ) is BPSK symbol baseband representation, and cos(2πf c t + θ ) is the local oscillator signal. The squaring of (3-33) yields the signal given by s 2 (t ) = Pd 2 (t ) cos( 4πf c t + 2θ )
(3-34)
With d 2 (t ) = 1 for BPSK systems, a discrete spectral line then appears at f = 2 f c which is filtered by the bandpass filter whose bandwidth need only range over the carrier uncertainty of f = 2 f c as shown in figure 3-9. This is then applied to a PLL tuned to f = 2 f which yields a phase estimate θ = 2θˆ as shown in Figure 3-9. The local c
reference may be divided by two to provide − sin(2πf c t + θˆ) , which is then used to down convert the input signal properly to baseband as shown in Figure 3-9. The squaring loop structure can also be extended for use with higher order MPSK signals by passing the received signal through M-power nonlinearity. For instance, to utilize this scheme with QPSK signals, the input signal is passed through a 4th power that removes the modulations and produce a spectral line at four times the received carrier as shown in Figure 3-11. We will look at this in detail later.
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s(t )
BPF f=fc
(x)2
BPF f=2fc
Loop Filter e(t )
− sin( 4πf c t + 2θˆ)
VCO
÷2 − sin( 2πf c t + θˆ)
LPF
Data decoder
Figure 3- 9: Squaring Loop for BPSK. Figure 3-10 shows a 10kbps BPSK signal at baseband after having been filtered using a zonal filter and then passed through the squaring function. Clearly the figure shows the appearance of the spectral lines corresponding to the symbol rate of 10kHz, since fc=0 Hz. The squaring operation results in noise cross noise terms that degrade the loop performance compared to PLL. The squaring loss is given by L SNR = 1 +
1 2 SNRi
(3-35)
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80
|BPSK|
Power Spectrum Magnitude (dB)
60
2
40
20
0
-20
-40
-60
0
0.2
0.4
0.6
0.8
1 1.2 Frequency
1.4
1.6
1.8
2 x 10
4
Figure 3- 10: Spectral line extraction from BPSK baseband signal using square law nonlinearity, which can be used with the PLL system in Figure 3-9.
3.5 Principle QPSK Carrier Recovery using Quadrupling Loops The fourth power loop shown in Figure 3-11 causes the QPSK signal modulation to be cancelled and hence results in the presence of a spectral line signal corresponding to the carrier four-times, which is then acquired and tracked. That is
[
]
8 A4 cos 4 (2πt + φ ) = A4 cos(8πt + 4φ )
(3-36)
The loop in Figure 3-11 avoids having to divide by four by running the PLL at four times the received carrier. Both fourth power and the squaring loop can be implemented at baseband or at low IF frequency in the digital domain. Figure 3-12 shows 10kbps QPSK signals at baseband after having been filtered using a zonal filter and then passed through a quadrupling function. Clearly the figure shows the appearance of two spectral lines at harmonics at the symbol rate of 10kHz. The quadrupling operation enhances the noise by a factor given by LSNR = 1 +
9 6 3 + + 2 SNRi SNRi SNRi3
(3-37)
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A sin( 2πf ct s + θ )
I LPF cos( 2πf c t s + θˆ)
Q LPF
− sin( 2πf c t s + θˆ )
− 45 0
+ 45 0
cos(8πfc t s + θ ' )
8(
BPF
)4
Loop Filter
cos(8πf c t s + θ ' ' )
VCO
8(
90 0
)4
Figure 3- 11: Block diagram of a QPSK carrier recovery loop using quadrupler nonlinearity.
80
Power Spectrum Magnitude (dB)
60
|QPSK|
4
40
20
0
-20
-40
-60
0
0.5
1
1.5
2
2.5 Frequency
3
3.5
4
4.5
5 x 10
4
Figure 3- 12: Spectral line extraction from baseband QPSK using fourth law nonlinearity.
3.6 Minimum-Shift Keying (MSK) Carrier Recovery using Squaring Loops Squaring loops for BPSK can be extended for use with minimum shift keying (MSK) signals, which can also be thought of as being some form of binary modulations. Figure 3-13 shows the non linear processing of MSK signals to extract a spectral line from
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which one can obtain carrier and symbol timing recovery. Assuming that the MSK modulated received signal is given by s (t ) = sin( wt + φ (t ))
(3-38)
with +/-1 as MSK symbols, the phase φ (t ) φ (t ) increases (symbol +1) or decreases (symbol -1) by either φ (t ) = +2πf1 or φ (t ) = −2πf1 , where f 1 = 1 / 4T , and the difference between the two tones is what is known as the modulation indices, that is h = 2π ( f 2 − f1 )T = 2 f1T . So MSK tones due to +/- 1 as a symbol are separated by the data rate, so h = 1 / 2 , and thus the phase modulation φ (t ) is an odd multiple of π / 2 for odd symbol, and an even multiple of π / 2 for even symbol. This periodical phase jump now can be exploited to create ways to cancel the modulations and hence achieve symbol timing and carrier tracking.
By squaring this signal the modulations are canceled and a spectral line appears at 1 Hz, which then is filtered as shown in Figure 3-13. The upper PLL or f = 2 fc ± 2T 1 while the lower BPSF is tuned to the narrow BPSF is tuned to the tone at f = 2 f c + 2T 1 , after which both are mixed to produce one tone at the symbol rate tone at f = 2 f c − 2T (1/T) and another at four times the carrier frequency. So the upper branch is then used for symbol timing recovery, while the lower branch is used to track the carrier frequency. 1 f = 2 fc + 2T s (t )
BPF
(x)2
f = 1/ T BPF
Recovered symbol timing
BPF
Recovered carrier (x4)
PLL/ BPF
f = 4 fc
PLL/ BPF f = 2 fc −
1 2T
Figure 3- 13: Block diagram of MSK carrier recovery using squaring loop.
The system in Figure 3-13 can also be implemented in baseband. Figure 3-15 shows an example of extracting the symbol-timing clock at half the data rate for MSK using squarer and a BPF. This 10kbps MSK signal was first passed through a squaring
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1 , or f = ±5khz . This scheme is easier 2T to implement since the carrier is null, or the loop is implemented at baseband. Now for the lower branch of the system shown in Figure 3-13, the filter will only contain DC power if there is no carrier offset., But if there is carrier offset, it will contain a spectral line at exactly the frequency offset, which then can be filtered first as shown in Figure 315, then estimated using various means that will be detailed in the coming chapters. For symbol timing recovery, the upper branch will still contain a spectral line at half the symbol rate, which then can also be extracted using direct methods. We will explore these methods in subsequent chapters.
function, producing the spectral line at f = 0 ±
MSK 2 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -2
-1.5
-1
-0.5
0 Hz
0.5
1
1.5
2 x 10
4
Figure 3- 14: MSK signal after being squared, Rb=10kbps. -10 -15 -20 -25 -30 -35 -40 -45 -50 2000
3000
4000
5000
6000
7000
8000
9000
Hz
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Figure 3- 15: The squared MSK signal after filtering with BPF at the data rate with center frequency at 1/2T, or 5khz. Another implementation of MSK recovering a spectral line at the data rate from which carrier recovery can be obtained is shown in Figure 3-16. Here the angle of each pair of MSK complex samples is extracted by the arctangent function. Since the MSK uses h=1/2 for frequency modulation indices, squaring the obtained angles will remove the modulations, but it will also produce a signal that is at half the symbol rate. Doubling the angle again and then passing it through a cos(4θ k ) nonlinearity produces a spectral line at the symbol rate as shown in Figure 3-17.
qk DDC & filtering
A/D
tan
−1
(qk
θk
ik )
cos(4θ k )
Extracted Spectral line at Symbol rate
ik
Figure 3- 16: MSK closed loop carrier recovery synchronization.
10
0
-10 dB
-20
-30
-40
-50 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
F/Rs
Figure 3- 17: Spectrum of the recovered signal using the system shown in Figure 3-16.
3.7 Carrier Recovery using Costas Loops The previous systems that are based on modulation removal using a non-linear process all experience SNR loss. The following scheme avoids using nonlinearity use and still achieves carrier acquisition and tracking using phase locked loops. Assuming the received input signal is given by
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s (t ) = 2 P d (t ) cos(2πf c t + θ )
(3-39)
where d (t ) is the binary data, and where d (t ) ∈ {− 1,+1}, f c is the modulated carrier that we wish to track. The objective of the loop here is to cancel the modulations, so that the loop can operate on tracking a continuous signal without having to use non-linear modulation removal. The first thing is to mix the input signal with a replica of it. This is accomplished by using a voltage adjusted oscillator running at the same frequency with an arbitrary phase as shown in Figure 3-18. The quadrature branch of the loop after the low pass filter is given by
q(t ) =
P d (t ) sin(θ − θˆ) 2
(3-40)
and the in-phase branch after the low pass filter is given by
i (t ) =
P d (t ) cos(θ − θˆ) 2
(3-41)
Both signals contain the data modulations, which can be canceled by the generation of the phase error signal given by, e(t ) = i (t ) q (t )
(3-42)
Substituting (3-40) and (3-41), the multiplication in (3-36) is reduced to e( t ) = i ( t ) q ( t ) = P 2 P d (t ) cos(θ − θˆ) sin(θ − θˆ) = sin(2 θ − θˆ ) 2 4
(
)
(3-43)
This is surprisingly simple and only dependent on the carrier phase offset given by θ − θˆ . For small error signal (i.e., during tracking), the small angle approximation can be used, sin( 2 θ − θˆ ) ≈ 2 θ − θˆ and all design intuition derived above for PLLs systems in Figure 3-4 and Figure 3-5 can be applied to the Costas loop shown in Figure 3-8. When the loop is locked, the in-phase branch is proportional to i (t ) ∝ d (t ) , which reduces to cos( 2 θ − θˆ ) = 1 , so the transmitted data can then be recovered totally from the in-phase
(
(
) (
)
)
branch.
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Data detector
LPF cos( 2πf ct + θˆ )
i(t )
s (t )
Loop Filter
VCO
e(t ) q (t )
− sin( 2πf c t + θˆ) LPF
Figure 3- 18: Costas loop for BPSK signals.
3.8 Digital Costas BPSK Carrier Recovery Loop For modern communication receivers implemented using digital signal processing, the Costas loop in Figure 3-18 is completely implemented in discrete time. Figure 3-19 shows a direct digital implementation of the BPSK Costas loop in Figure 3-18. Here the received signal is first digitized and then mixed with a local quadrature NCO signal at the same frequency. As a result, if the incoming signal s (t ) has an exact frequency of the NCO, the output of the inphase multiplication branch after it has passed through proper sample decimation and low pass filtering (to reject the high frequency components) is given by
ik =
P d k cos(φ k ) 2
(3-44)
and the quadrature phase branch is given by
qk =
P d k sin(φ k ) 2
(3-45)
Where the carrier phase error at t=kT is given by φ k = θ k − θˆk . The phase error is obtained by performing the following multiplication, ek = sign{ik }q k
(3-46)
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Assuming no frequency offset, and small carrier phase error, thus P P ik → d k cos(0) = d k , which means that the in-phase branch is actually the data. 2 2 Substituting the value of the quadrature output into the phase error signal in Equation (346), yields an error signal given by,
ek =
P sign{d k }sin(φ k ) d k 2
(3-47)
For BPSK modulations, d k ∈ {+ 1,−1}, which results in sign{d k }d k = +1 , the phase error signal in (3-47) is reduced to ek ≈ sin(φ k )
(3-48)
This is surprisingly also this is the same error signal weobtained by using the phase locked loop in Figure 3-18. However, unlike the analog Costas loops used in Figures 318, the error signal for digital Costas loop is performed without squaring as is the case of the phase error signal in Equation (3-43) due to the feasibility of implementing the function sign{ik }. As a result, the loop has to become a decision-aided (what?)instead of relying on canceling the modulations using a squarer, which enhances the noise by 3dB.
FIR LPF
1 N
N
∑ k =1
↓
i(kT )
dk
sign{ik } = d k
cos( 2πf c kTs + θˆ)
dk s (t ) A/D
s ( kTs )
ek
Loop Filter
NCO
qk
− sin( 2πf c kTs + θˆ ) FIR LPF Running at sampling rate
1 N
Ts
3-28
N
∑ k =1
↓
q(kT )
Running at symbol rate
T
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Figure 3- 19 : Figure BPSK Costas loop.
3.8.1
Digital Costas QPSK Carrier Recovery Loop
The QPSK Costas loop can also be derived in a manner similar to what was employed with BPSK. A modulated QPSK signal in discrete time can be represented by
s (t ) = 2 P I k cos(2πf c kT + θ k ) − 2 PQk sin(2πf c kT + θ k )
(3-49)
With the data symbols given by Q(t ), I (t ) ∈ {+ 1,−1} . The first step in tracking the phase of the modulated signal in Equation (3-49) is to down convert it to baseband using mixing with a quadrature (complex) NCO whose frequency is nominally at f NCO = f c , but which has an arbitrary phase as shown in Figure 3-20. The in-phase NCO signal given by
K NCO cos(2πf NCO kTs + θˆ)
(3-50)
and the quadrature signal given by
− K NCO sin(2πf NCO kTs + θˆ)
(3-51)
where K NCO is the NCO gain. After the low pass filtering in Figure 3-18, the in-phase and quadrature signals are given by
ik =
P P K NCO I k cos(θ k − θˆk ) − K NCO Qk sin(θ k − θˆk ) 2 2
(3-52)
qk =
P P K NCO I k sin(θ k − θˆk ) + K NCO Qk cos(θ k − θˆk ) 2 2
(3-53)
The error is then derived as follow, ek = ik sign(q k ) − q k sign(ik ) ek =
(3-54)
P K NCO I k Qk cos(θ k − θˆk ) + 2
P − K NCO I k Qk cos(θ k − θˆk ) + 2
P K NCO I k2 sin(θ k − θˆk ) 2
(3-55)
P K NCO Qk2 sin(θ k − θˆk ) 2
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Again, this takes advantage of the small angle assumption when the loop is in tracking mode, that is sin(θ k − θˆk ) → θ k − θˆk and cos(θ k − θˆk ) → 1 . As a result, the phase detector error output signal is given by
ek =
[
P K NCO sin(θ k − θˆk ) I k2 + Qk2 2
[
]
(3-56)
]
Since for QPSK signal, I k2 + Qk2 = 2 , the error signal is reduced to,
ek = 2 P K NCO sin(θ k − θˆk )
(3-57)
Again the error signal closely resembles the phase error signal from a standard PLL operating on CW signals. sign{qk }
FIR LPF
1 N
N
∑ k =1
↓
sign{ik }
cos( 2πf c kTs + θˆ) s (t ) A/D
s ( kTs )
dˆi ,k
i(kT )
ek
Loop Filter
NCO
-
+
− sin( 2πf c kTs + θˆ ) FIR LPF Running at sampling rate
1 N
N
∑ k =1
Ts
↓
q (kT ) Running at symbol rate
dˆq ,k
T
Figure 3- 20: Figure QPSK Costas loop.
3.9 Digital Tanlock carrier recovery loops
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Digital tanlock loops have been around for decades; however, they were not practical until the availability of digital signal processors because they need the computation of the angle of a complex vector. As the name implies, the tanlock carrier recovery loop relies on explicit tan −1 (.) operation to yield the carrier phase error signal. Thus no need for the conventional PLL/Costas loop sinθ ≈ θ approximation. As a result, the use of tan −1 (.) function gives an accurate phase estimate during acquisition and during tracking due to the wide linear range of the arctangent function. Figure 3-21 shows one implementation of the tanlock-tracking loop for M-PSK signal. The multiplication of the arctangent output by M is used to remove the modulations similar to the methods used in Figure 3-9 and 3-11. In figure 3-20, the carrier phase error is estimated from the modulated signal using the function of
ek = M tan −1 (
ik ) qk
(3-58)
The value of M required for modulation cancellation is given by ⎧2 BPSK M =⎨ ⎩4 QPSK
(3-59)
Another apparent advantage of using arctangent phase detector can be observed from the fact that the carrier phase error is detected by taking the ratio of the imaginary to the real part of the received sample. As a result, the error function and thus the phase estimates are not effected by amplitude modulations (AM) or fading. However, the multiplication with M produces an M fold phase ambiguity. This ambiguity can be removed using the mod{2π } function. Phase ambiguities are resolved using a special field in the preamble sent (??) and detailed in chapter 7. Both the arctangent and the modulus 2π are simple to implement using current DSPs. The arctangent is very often implemented using table look up methods (TLU), which only requires memory. The module 2π can also be implemented using logic functions, such as the Exclusive ‘OR’ gate or table look up.
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FIR LPF
∫
Tb
(.)
i (kTb )
cos(2πfc kTs + θˆ) s (t ) A/D
s (kTs )
Loop Filter
NCO
ek
-
− sin(2πf ckTs + θˆ) FIR LPF
tan− 1 ( )
mod 2π M
∫
Tb
(.)
Running at sampling rateT s
q ( kTb )
Running at bit rate
Tb
Figure 3- 21 : Arctangent based loop.
Figure 3-22 shows one example of implementation of the arctangent loop in Figure 3-21. Here the loop uses the loop filter given in Figure 3-6. The mathematical implementation of this loop starts with computing the NCO phase error updates given by
φˆk +1 = φˆk + s k
(3-60)
s k = φ k• + k1ek
(3-61)
Where
and the error ek is given by
φ k•+1 = φ k• + k1 k 2 ek
(3-62)
A good starting value for the loop parameters is thus a damping factor equal to R ς = 0.707 , and a natural frequency given by f n = s . The loop proportional constant 100 is computed using the results derived in 3-29. That is
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2πf n (3-63) Rs and the integral constant is computed using the result obtained in 3-30. That is k p = 2ς
⎛ 2πf n k i = ⎜⎜ ⎝ Rs
LPF
∫
Ts
⎞ ⎟⎟ ⎠
2
(3-64)
φˆk (.)
Digital Signal Processing
ik
kTs
To demodulator
Kp
s (t )
NCO
M . tan − 1 ( mod (2 π )
φˆk
) φ k
Loop filter
x
ek
+
sk
Ki
+
-
φˆk
x
φk•
+ z −1
+ LPF
∫
Ts
qk
(.)
kTs
z −1 NCO
To demodulator
Figure 3- 22: Non data-aided arctangent based AFC loop.
Another implementation of the arctangent carrier recovery loop is shown in Figure 3-23. The implementation assumes that the input signal has no modulations (i.e., CW). Assuming that the received sinusoidal input signal (this can be achieved using a CW preamble by sending a modulation symbols that are all marks), the resultant in-phase signals in figure 3-22 are given by ik = A sin (2πf o t k + θ k ) and the quadrature signal is given by
(3-65)
q k = A cos(2πf o t k + θ k )
(3-66)
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where f o is the free running frequency of the digital controlled oscillator (DCO). The interval of the DCO between t k and t k −1 is controlled by the correction signal c k −1 shown 1 in Figure 23. That is Tk = − c k −1 , and c k = F ( z )ek . Where F (z ) given by fo kp (3-67) F ( z) = ki + 1 − z −1 The carrier frequency and phase offsets are given by
θ k = 2π∆f + θ o
(3-68)
where the ∆f = f − f o is the initial carrier frequency offset and θ o is the initial carrier phase offset. Now the total time t k up to kth sampling instant is given by k −1
t k = kTo − ∑ c( j )
(3-69)
j =0
where To =
1 . Substituting (3-69) into (3-75) and (3-66), the in-phase signal is given by fo k −1 ⎞ ⎛ ⎡ ⎤ i k = A sin ⎜ 2πf o ⎢kTo − ∑ c ( j )⎥ + θ k ⎟ ⎟ ⎜ j =0 ⎣ ⎦ ⎠ ⎝
(3-70)
and the quadrature signal is given by k −1 ⎞ ⎛ ⎡ ⎤ q k = A cos⎜ 2πf o ⎢kTo − ∑ c ( j ) ⎥ + θ k ⎟ ⎟ ⎜ j =0 ⎣ ⎦ ⎠ ⎝
(3-71)
Further manipulations reduces (3-70) and (3-71) to k −1 ⎛ ⎞ ik = A sin⎜⎜θ k − 2πf o ∑ c( j ) ⎟⎟ j =0 ⎝ ⎠
(3-72)
k −1 ⎛ ⎞ q k = A cos⎜⎜θ k − 2πf o ∑ c( j ) ⎟⎟ j =0 ⎝ ⎠
(3-73)
and
k −1
Defining the carrier phase error as φ k = θ k − 2πf o ∑ c( j ) , and then substituting in (3-72) j =0
and (3-73) . The in-phase and quadrature signals are reduced to, ik = A sin (φ k )
(3-74)
and
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q k = A cos(φ k )
(3-75)
The carrier phase error is then extracted using the arctangent of the ratio of (3-74) and (375). That is
⎛q ⎞ ek = tan −1 ⎜⎜ k ⎟⎟ ⎝ ik ⎠
(3-76)
The error signal is then processed by ek = F (ek ) to detect and remove an ambiguity, where
F ( x ) = −π + {(φ + π ) mod(2π )}
(3-77)
for instance, if φ = 2π , then F (3π ) = −π + {(3π + π ) mod(2π )} = π , and so the process function keeps the angle within the range of − π ≤ φ ≤ +π . The capture range for the DTL is given by ∆f ≤
4 fo
(3-78)
kp ⎞ ⎛ k i ⎜⎜ 2 + ⎟⎟ ki ⎠ ⎝
Notice that if the loop is a first order loop, k p = 0 , the capture range of this loop is less than ∆f ≤
2 fo ki
(3-79)
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F (z) = k p +
To demodulator
ik
sampler
tk
Loop filter
tan
Fixed NCO
s (t )
(i k / q k ) (2 π )
ki x
−1
mod
ki 1 − z −1
ek
kp + x
+ z −1
To demodulator
qk
sampler
ck
tk DCO
Figure 3- 23: Digital tanlock loop.
3.10
Carrier Phase Lock Indicators
The carrier and symbol timing recovery loops in Figure 2 also included a lock indicator. There are logic signals that are used in the receiver to indicate successful synchronization, and thus can be used as a green light for the receiver to start the demodulation process. For BPSK loops, during tracking, all of the signal energy should be contained in the real part of the signal as shown in Figure 2-5. One way to derive a lock indicator signal is then to average the difference of I k − Qk over a reasonable interval and then compare it against a threshold Γ . That is,
(
)
⎧ L −1 I k − Qk ≥ Γ lock ⎪ LDθ = ⎨∑ k =0 unlock ⎪⎩ else
(3-80)
The threshold ( Γ ) is obtained based on minimum tolerable phase offset. One way to avoid a false lock indication is to pass the difference I k − Qk through a low pass filter (or moving average filter) and then compare the output value to the threshold. That is
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(
y k = c1 y k −1 + c 2 I k − Qk
)
(3-81)
where y k is the output of the low pass filter, and the constants c1 and c2 are the filter coefficients. These coefficients are designed based on the desired speed of convergence and the desired final variance of the lock estimator around a desired mean value in AWGN.
For QPSK signals, since the energy is distributed in both I and Q channels, one obvious way to derive a lock indicator is by taking the difference between the absolute value of both branches and compare it with a threshold (ideally it should be zero). That is
⎧M ⎛ ⎪ ⎜ I k − Qk ⎞⎟ ≤ Γ , lock LDθ = ⎨∑ ⎠ k =1 ⎝ , unlock ⎪⎩ else
(3-82)
Another lock detection algorithm that has lower computational load is given by M ⎛ I +Q − Q − I k k k k LDθ = ∑ ⎜⎜ − ( I k − Qk 2 k =1 ⎝
3.11
⎞
) ⎟⎟
(3-83)
⎠
Automatic Frequency Control Loops (AFC)
The carrier phase recovery loops discussed above assume that frequency offset presence during carrier phase acquisition and tracking loops is negligible. However, in most practical situations, there is a carrier frequency offset that has to be acquired before tracking the phase. Small frequency offset can be acquired using carrier phase lock loops that have a loop bandwidth that is at least of the order of the offset. But if the offset is too large, the carrier phase lock loops discussed above have to be aided by AFC algorithms to acquire this frequency offset so that the carrier phase can be tracked with optimal phase variance. AFC becomes especially important in time division multiple access (TDMA) systems where the bursts are too short for conventional carrier phase only loops to converge without the aid of AFC loops. Since AFC loops are non-coherent loops, the notion of jitter or phase variance is not applicable since there is no tracking involved. The main criteria for designing these loops is based on their frequency offset capture range and their acquisition time. One of the simplest AFC loops is based on the frequency difference detector. This detector resembles digital frequency modulation (FM) detectors. To illustrate this detector, assume that the received complex signal input into the quadrature demodulator shown in Figure 3-29 is given by
s r (t ) = I (t ) cos(2πf r t + θ r ) + jQ(t ) sin(2πf r t + θ r )
3-37
(3-84)
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where f r and θ r are the frequency and phase associated with the received signal. This signal is downconverted by mixing with a free running complex local oscillator (CNCO) given by
y L (t ) = cos(2πf L t + θ L ) − j sin(2πf L t + θ L )
(3-85)
Let the resultant in-phase and quadrature phase samples produced be,
and
I (k ) = Re{y L (t ) s r (t )} = cos(2π∆fkT + θ )
(3-86)
Q(k ) = Im{Re{y L (t ) s r (t )}} = sin(2π∆fkT + θ )
(3-87)
where ∆f = f L − f r is the carrier frequency offset and θ = θ L − θ r is the carrier phase error. In order to perform correct data detection, the receiver must estimate and remove ∆f and θ from the received signal. To do this, the receiver generates an error signal proportional to the magnitude of ∆f and θ .
Re{ }
I( t)
x -
z −1
Q( t- ∆ T)
z −1
I( t- ∆ T)
Im{ }
sr ( t )
Re{ }
I( t) z −1
y L (t )
z −1
rc (t )
⎛ r (t ) ⎞ tan −1⎜⎜ c ⎟⎟ 2πT ⎝ rc (t ) ⎠
x
Q( t)
x
+
1
∆fˆ
x I( t- ∆ T)
+
Q( t- ∆ T)
Im{ }
rs (t )
x Q( t)
Figure 3- 24: Arctangent based frequency-offset detector. The cross product terms in Figure 3-24 are is given by rc (t ) = Q(t ) I (t − ∆T ) − I (t )Q(t − ∆T )
(3-89)
which further reduce to rc (t ) = sin(2π∆fτ 1 )
(3-90)
The dot product terms are given by
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rs (t ) = I (t ) I (t − ∆T ) + Q(t )Q(t − ∆T )
(3-91)
which is further reduced to rc (t ) = cos(2π∆fτ 1 )
(3-92)
Where τ 1 is the time delay of the double-product frequency discriminator. The frequency offset is then given by
∆fˆ =
⎛ r (t ) ⎞ tan −1 ⎜⎜ c ⎟⎟ 2πτ 1 ⎝ rs (t ) ⎠ 1
(3-93)
Very common in the delay is a single symbol. The frequency offset is then given by
∆fˆ =
⎛ r (t ) ⎞ 1 tan −1 ⎜⎜ c ⎟⎟ 2πT ⎝ rs (t ) ⎠
(3-94)
Because of the characteristics of the arctangent function used in the frequency estimate 0.1 . The detector in Figure 3(3-94), the estimates are only accurate for offsets of ∆fˆ ≤ T 24 can be used in conjunction with any of the carrier phase recovery loops illustrated earlier to aid in the acquisition of large frequency offsets. 3.11.1 Discrete Fourier Transform based AFC
The AFC detector presented in Figure 3-24 works only on continuous wave signals (i.e., no modulations), so it is appropriate for systems with preambles that contain no modulations (all marks). Another method that is based on the Discrete Fourier Transform (DFT) is shown in Figure 3-25. The frequency offset associated with the received signal is then estimated by the difference of two adjacent DFT bins that are placed equally on both sides of the baseband (or IF) bin [12]. That is ∆fˆ = P1 − P−1
(3-95)
where a measure of frequency error is computed using Pk = Rk2 + M k2 (see Figure 3-25), which is given by
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Pk = 1 M
1 M
M −1
⎡
n=0
⎣
⎛ knπ ⎞ ⎛ knπ ⎟ + Q (n) sin ⎜ M ⎠ ⎝ M
∑ ⎢ I (n) cos⎜⎝
⎡ ⎛ knπ ⎞ ⎛ knπ ∑ ⎢Q(n) cos⎜ M ⎟ − I (n) sin ⎜ M ⎝ ⎠ ⎝ n=0 ⎣
M −1
⎞⎤ ⎟⎥ + ⎠⎦
(3-96)
⎞⎤ ⎟⎥ ⎠⎦
The capture frequency range of this algorithm was found to be ∆f ≤
In
Rk
FIR LPF
( )2
1 . 2T
P1
cos(2π fc kTs + θˆ) A/D
s( kTs )
DFT
NCO
-
− sin(2πf ckTs + θˆ) Mk
FIR LPF
Qn
( )2
+
P−1
Loop Filter
Figure 3- 25: DFT based AFC system.
3.11.2 Dual BandPass Filter AFC method
Another extension of the AFC loop based on the DFT in figure 3-25 is to replace the DFT bins with bandpass filters as shown in Figure 3-27. Here
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x
BPF
f =
2
1 2T
s (t ) A/D
-
e −2πf
ˆIF kT
NCO
x
BPF
f =−
+
2
1 2T
Loop Filter
Figure 3- 26: AFC using the principle of dual BPF automatic frequency control.
The principle of the frequency-offset detector shown in Figure 3-26 in AFC loops is shown in Figure 3-27. Here the frequency automatic control is obtained using the energy difference between the levels detected from two filters placed equally on both sides of the expected center frequency of the received signal. The difference in the DC component detected in the upper and lower filter output is a direct indication of the frequency. This signal is then used as an error signal that steers the NCO to track the exact frequency of the incoming signal.
S( f ) Lower filter
−
1 2T
∆f
Upper filter
+
0
1 2T
Figure 3- 27: Principle of dual BPF based automatic frequency control.
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3.11.3 Dual NCO AFC System
Another variant of the loop shown in Figure 3-26 is shown in Figure 3-28. Here, by running a dual NCOs, one is running at f = − f IF + ∆f i and the other one is running at f = f IF + ∆f i . where ∆f i is a tunable value of frequency that ranges over 0 ≤ ∆f i ≤ ∆f max . Both NCOs are then used to down convert the signal by multiplying the incoming signal with both NCOs, after which the result of multiplication is passed through a low pass filter and accumulated over the duration of one symbol. Then the accumulator power is measured in both channels and subtracted. If the received signal s (t ) is centered exactly at f = f IF , the power out of each branch over the duration of a single symbol (N-samples) will be equal and hence the loop acquires the frequency offset. If the received signal is offset from its nominal IF frequency, then the difference in power is measured between the two branches where the difference and its sign are used as the direction and the amount by which the NCO will be corrected. This process is repeated until the power difference converges to zero and the loop achieves lock.
N −1
In
1 N
∑=
x
2
k 0
NCO
s (t ) A/D
f = f IF + ∆f
s(kTs )
f = − f IF + ∆f
NCO
∑
+
-
N −1
Qn
1 N
∑ k =0
x
2
Figure 3- 28: An alternative to the dual BPF approach for automatic frequency control.
Figure 3-29 illustrates another form of AFC loops that is suitable for use with BPSK modulations. The principle is that the frequency offset can be detected after the received signal is passed through an angle doublers(??) that basically removes the modulations.Thus the loops will acquire the resultant CW signal that has a frequency
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equal to the frequency offset associated with the received signal. Consider that the received modulated BPSK signal in Figure 3-29 is given by
s( kTs ) = d k A cos(2πf IF kTs + θ k )
(3-97)
Where f IF is the intermediate frequency, and d k is the transmitted data symbols, d k ∈ {+ 1,−1}. After quadrature down conversion, the resultant quadrature signals are given by
and
ik = d k A cos(2π∆fkTs + θ k )
(3-98)
q k = d k A sin (2π∆fkTs + θ k )
(3-99)
where ∆f is the frequency offset due to the mismatch between the local oscillator frequency and the frequency of the received signal. This frequency error then can be estimated by forming what is known as the cross product and dot product terms from the baseband complex signal similar to Figure 3-24. Where the dot product term is given by x k = i k2 − q k2 = A 2 cos(2π∆fkTs + 2θ k )
(3-100)
and the dot product term is given by y k = i k .q k =
1 2 A sin (2π∆fkTs + 2θ k ) 2
(3-101)
Clearly the data modulation has been removed by the squaring. Now a frequency error signal can be derived using Equations (3-100) and (3-101), ek = x k −1 y k − x k y k −1 =
1 4 A sin (2π∆fkTs ) 2
(3-102)
1 , then the 10T small angle approximation ( sin (2π∆fkTs ) ≈ 2π∆fkTs can be used. Thus the frequency error signal is then given by
If the frequency offset is small relative to the data rate (1/T) such hat ∆f ≤
e k = A 4π∆fkT
(3-103)
Clearly, this system has a narrower frequency estimation (capture) range because of the use of the small angle approximation, which is the case with all PLLs based on sin(θ ) ≈ θ .
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FIR LPF
ik
( )2
xk
+
z−1
-
cos( 2πf c kTs + θˆ )
ek
s(t ) A/D
NCO
+
-
s(kTs )
− sin( 2πf c kTs + θˆ )
FIR 2 LPF q ( ) k
yk
z −1
Loop Filter
Figure 3- 29: Block diagram of discrete implementation of Automatic frequency control loop.
3.12 Combining AFC with carrier phase tracking Loops Since carrier phase recovery loops have narrow loop bandwidth that does not allow them to operate when the signal has large frequency offset, one way to cope with this situation is to construct a double loop as shown in Figure 3-30. Here, the AFC acquisition time is proportional to t acq ∝ 1 B L while the carrier recovery loops acquisition time - such as the Costas loop detailed earlier - is proportional to t acq ∝ 1 BL3 . A reasonable capture range of carrier recovery loops is usually limited to frequency offsets on the order of ∆f ≤
Rb 100
Rb [1,5,12]. In chapter 8 10 several loops of this type will be illustrated for satellite and mobile receivers.
, while the capture range of AFC loops is on the order of ∆f ≤
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FIR LPF
ik
cos(2πf c kTs + θˆ)
Frequency Error detector
s (t )
NCO
A/D
Phase Error detector
s (kTs )
− sin( 2πf ckTs + θˆ) FIR qk LPF
+
Phase Phase Loop Loop Filter Filter Frequency Loop Filter
eθ e∆f
Figure 3- 30: Combing AFC with carrier phase tracking loops.
3.13
Carrier Frequency Lock Indicators
Under steady state conditions and in AWGN channels, the frequency error signal is proportional to the cross product I k I k −1 + Qk Qk −1 . One way to implement a frequency lock indicator is by averaging (accumulating) the cross product over a properly chosen interval and then comparing it to a properly chosen threshold Γ chosen based on the maximum allowable frequency offset. This is
LD∆f
⎧ L −1 ⎪ (I k I k −1 + Qk Qk −1 ) ≥ Γ , lock = ⎨∑ k =0 , unlock ⎪⎩ else
(3-104)
Next we present one example of applying modern digital signal processing to enable both carrier frequency offset and carrier phase tracking for a typical medium rate modem using the scheme detailed earlier.
3.14 Carrier Frequency Acquisition and Phase Tracking Modem Example
Here we present one practical example for acquiring and tracking both the carrier and symbol timing offsets for a wireless digital receiver detailed in Figure 3-31. The carrier
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tracking loop is carried out in two stages. First , carrier frequency acquisition is carried out using a transmitted preamble and then, second, the carrier phase is tracked using a decision-aided Phase locked loop operating on the rest of the frame transmitted symbols.
cos(2π ( f c + ∆f )t + φ (t ) + θ c (t ) )
Timing Estimation
εT RF
IF
ADC ADC DDC
θk
ak e j
Baseband Interpolation DSP
Turbo Decoder Decoder
z fixed NCO LO
cos(2πf LO t + θ LO (t ) )
DFT-based acquisition
e
⎛ ˆ ⎜ − j 2π ∆f +θˆk ⎜ fs ⎝
∗
a k −d
NCO
θˆk
kT + θ 2 π∆fˆ DFT DFT
−d
LF
⎞ ⎟ ⎟ ⎠
Figure 3- 31 : An example of acquisition and tracking in digital receiver.
The receiver illustrated in Figure 3-31 first down converts the digitized IF to near baseband. Then, once the samples are streaming into the DSP engine, acquisition and tracking algorithms are engaged. The frequency and symbol timing of the initial large offsets are estimated and fed to the buffered signal. Once the offsets are within a reasonable range of the tracing loops, both decision-directed symbol timing and decisiondirected carrier tracking are engaged, as illustrated using the feedback path in Figure 331. Signal (???) is made of the RF section that down convert the RF received signal to an IF frequency. The digital section includes an analog-to-digital converter, digital down converter, two decision-aided feedback tracking loops [15], and one feedforward acquisition based on using discrete Fourier techniques [14]. This chapter in only concerned with carrier tracking. Symbol tracking will be covered in chapter 4. Next we separately illustrate both algorithms in detail. 3.14.1 Acquisition Loop
Since the initial frequency offset ∆f is beyond the capture range of the narrow loop bandwidth tracking loop, we employ a feed forward loop that uses the characteristics of the preamble transmitted that contains a spectral line at one-fourth of the symbol rate to estimate the initial large frequency offset using DFT periodogram methods [19]. The idea here is that the TDMA frame contains an a priori known preamble made of an alternating sequence of 11001100 as shown in Figure 3-32. The preamble 1100 causes the received
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baseband to be a square wave with a frequency that is ¼ of the symbol rate since it takes four symbols (bits) (0011) to repeat. Without frequency offset, the received preamble baseband signal spectrum contains three DFT lines. One spectral line will be at DC and two will be at a frequency of f = ± 1 4T as shown in Figure 3-33. If the transmitted signal is subjected to a frequency offset, these three spectral lines will be shifted by an amount that is equal in magnitude and direction that corresponds to exactly the frequency offset associated with the received carrier. Figure 3-33 illustrates as an example, the 28000sps preamble signal after being processed using the DFT acquisition block shown in Figure 3-31 with carrier frequency offset of ∆f = 1200 Hz. The acquisition loops operate totally during the preamble as follows:
•
• • •
•
Based on the desired loop bandwidth of the tracking loops chose a DFT frequency ∆f resolution f DFT = max for acquisition of carrier offset. 2 Capture 1024 (n+1) baseband samples, where n=0,1,3,5,….. f Perform N DFT = s DFT points. ∆f max Sort the N DFT bins and locate the bins that correspond to the three tones at 1 f = 0 ± ∆f , and f = ± + ∆f . 4T The initial carrier frequency estimates ∆f is estimated from the location of the middle bin after having been miss located from its location at 0 Hz.
•
The initial carrier phase estimates θˆ DFT is equal to the difference of the middle bin will be detailed later in coming sections.
•
Phase and either one of the two upper or lower bins shown in Figure 3-33.
•
Feed forward both estimates to the NCO shown in Figure 3-31 to perform multiplication of the baseband signal by the term
⎛ exp ⎜⎜ − j 2π ⎝
∆fˆDFT
fs
⎞ +θˆDFT ⎟⎟ ⎠
, which
corresponds to the phase and frequency offset correction to bring the received signal to near baseband.
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Imag QPSK 182 Preamble
Real QPSK 182 Preamble
RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
2 1 0 -1 -2
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
2 1 0 -1 -2
Sample #
Figure 3- 32: TDMA frame showing the alternating sequence preamble on the real part of the signal.
1 0.9
Symbol Rate =28000bps ∆f=1200Hz
Center at f=0Hz
0.8
Center at f=0Hz
Shifted by 1200 Hz
DFT of Frame
0.7 0.6 0.5 Lowe tone at =-1/4Ts Hz
0.4
Upper tone at =1/4Ts Hz
0.3 0.2 0.1 0 -6000
-4000
-2000
0
2000
4000
6000
8000
Frequency (Hz
Figure 3- 33: spectral density comparison between the alternating sequence preamble with 1200Hz carrier frequency offset .
3.14.2 Carrier Tracking Loop
The carrier phase tracking loop shown in Figure 3-34 is based on decision-aided loop detailed in [15]. The carrier phase error signal is detected using the imaginary part of the cross correlation between the data decision and baseband signal that resulted in that decision-making. That is
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( )
eθ , k = − Im dˆk∗ zk
(3-105)
Where dˆ k∗ is the conjugate of the decision symbol, and z k is the baseband signal. To illustrate the validity of the phase error signal from equation 3-105, realize that the matched filter that resulted in the decision of dˆk is given by zk = e jθ k . Substituting this in the phase error detector in Equation (3-105) results in,
(
eθ , k = − Im d k∗ zk e − jθ k ˆ
)
(3-106)
(
eθ , k = − Im d k∗d k e − jθ k e − jθ k ˆ
(
)
(3-107)
)
(
ˆ eθ ,k = − Im d k∗d k e − jθ k e − jθ k = sin θ k − θˆk
)
(3-108)
where d k∗d k = 1 . Using the small angle approximation in (3-108), the phase error signal is reduced to sin θ k − θˆk ≈ θ k − θˆk .
(
)
Next we illustrate the design principles of the tracking loop that processes the error signal in (3-108) and then feeds to the NCO. This filter plays a major rule in the stability and quality of the carrier tracking. 3.14.3 Carrier Tracking Digital Loop Filter Design
The tracking loop in Figure 3-31is implemented using a second order type feedback loop. This loop filterhas the property that its steady state error converges to zero for both phase and frequency. Standard type-2 loop filters inside a carrier tracking loops are modeled by Figure 3-34 using a Lead-Lag controller principle, where a direct gain Kp path and an integrated path Ki are used as a filter frequency offsets and a phase accumulator.
θi (k ) = 2π∆fkTs+ θo θi (k )
+ -
θˆr (k )
θerr (k ) = θ i (k ) − θˆr (k ) x Loop filter Kp z
−1
+
+
+ z
x
−1
Ki
Figure 3- 34: Linear model for the MLE carrier tracking.
The loop transfer function was derived earlier and is repeated here,
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Ki ) K p + Ki θˆr ( z ) = K p + Ki θ i ( z) z 2 − 2(1 − ) z + (1 − K p ) 2 K p + Ki (z −
(3-109)
The loop filter parameters Kp and Ki are designed so that they minimize the phase and offset errors while yielding minimum acquisition time. The design procedure of this loop filter is as follow: 1. The natural frequency of the loop is given by,
2 BL ξ + 1 4ξ 2. Using the computed wn and the standard second order canonical control equation, the integral constant of the loop filter is computed by, wn =
K i = 4(wnTs 2)
2
3. The proportional constant is given by, K p = 4ξ (wn Ts 2 )
4. where BL is the noise loop bandwidth, ξ is the second order-damping factor, and Ts is the symbol rate (i.e., loop iteration rate). 5. The acquisition time of the loop [16] is given by,
t acq = 4(∆f 2 BL3 ) 6. The phase lock time (assuming no frequency offset [16]) is approximated by t l ock = 1.3 B L
7. The minimum carrier tracking phase error variance is given by,
σ θ2 =
BLT [rad2] Es N o
8. where E s N o is the symbol energy to noise spectral density. Figure 3-35 shows the variance as a function of both E s N o and the normalized loop filter
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bandwidth. This figure can be used to obtain initial design values for the loop filter bandwidth. 9. The effective loop signal-to- noise ratio ( SNRL ) is related to both the input signal to the loop’s SNR and the loop bandwidth and given by,
SNR L = SNRi + 10 log10 ( Bi 2 BL )
10 Es/No=-4
9
Tracking phase STD error-deg
8 Es/No=-2
7 6
Es/No=0
5 Es/No=2 4
Es/No=4
3
Es/No=6
2
Es/No=8 Es/No=1 Es/No=1
1 0
0
0.001
0.002
0.003 0.004 0.005 0.006 0.007 BlT- Normalized Loop Bandiwdth
0.008
0.009
0.01
Figure 3- 35: Carrier phase tracking error standard deviation versus Es/No.
The listed relations can be used to obtain the initial values of the loop filter coefficients, and then used as an initial design that is then fine tuned using computer simulations. For the highest QPSK with rate of Rs = 28000 symbols-per-second with the desire to operate the loop optimally at Es/No=8 dB for BER of 10-5, while allowing a tracking phase of σ θ = 2 o in a channel noise bandwidth of Bi = 25kHz. (??) Using the relation between the tracking phase variance, Es/No, the normalized loop bandwidth can be computer. That is
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BLT ⇒ B L T = 0.0077 , or BL = 0.0077(28000) = 196Hz , using a damping factor Es N o 2BL wn = = 370 rad/sec. ξ = 0.707 , the loop natural frequency is ξ + 1 4ξ
σ θ2 = of
K i = 4(wn Ts 2) ⇒ K i = 0.0001754 , K p = 4ξ (wn Ts 2 ) ⇒ K p = 0.00467 . With a worst 2
(
case frequency offset error of ∆f max = 100 Hz, t acq = 4 ∆f
2
B L3 ) ⇒ t acq = 0.005 seconds, or
148 symbols to capture a frequency offset of 100 Hz. For the acquisition loop, using ∆f max = 100 as the worst DFT initial carrier frequency estimation error, the required number of DFT points is computed by 28000 ∆f DFT = ∆f max = 100 Hz . N DFT = = 560 bins. The nearest 2N for this value is ∆f DFT 2 N DFT = 1024 which further reduces the initial frequency offset error to 28000 ∆f DFT = = 27 Hz . 1024 Figure 3-36 shows the loop transient response using computer simulations for two different initial frequency offsets. Clearly once the loop presented with the initial DFT estimates, it cycle slips until eventually it pulls in and converges. The lock time t l ock and acquisition time t acq shown in Figure 3-36 closely is in agreement with the theoretical values derived earlier. The figure shows that when the frequency offset is doubled, the lock time t l ock is also doubled (700 symbols versus 1400 symbols).
1
∆f1
0.8 Phase error signal
∆f1 = 2∆f 2 0.6 0.4 0.2 0 -0.2 -0.4
0
200
400
600
800
1000 1200 symbol
1400
1600
1800
2000
Figure 3- 36 : Decision-aided frequency and phase tracking for two different offsets. 3-52
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To evaluate the loop further, a Monte-Carlo analysis is used to derive the final carrier phase tracking error variance compared to the ideal variance obtained by the modified Cramer-Rao bound (MCRB) that will be derived in subsequentchapters [13]. The CRBis an ideal figure of merit that is used for comparison. This bound is a theoretical bound that is used as a benchmark to evaluate tracking loops. Figure 3-37 shows the variance of the decision-aided carrier tracking loop in Figure 3-31. One way to interpret the results of Figure 3-37 is to check the loop simulation results against the predicted theoretical results. For instance, the loop has a tracking variance of σ θ2 = 8 x10 −4 [rad2] at Es N o = 8dB . This shows that the loop has a mean standard error of 1.6 degrees. From
BLT 0.0077 = = 12 x10 −4 , or by inspection 6.31 Es N o from Figure 3-37, both show that the loop has an expected variance of 2 degrees for BL T = 0.0077 , which is very close. the theoretical loop variance given by σ θ2 =
10
Variance of Phase Estimates vs. MCRB
10
10
10
10
10
10
10
1
0
σ θ2
-1
-2
-3
-4
MCRB -5
-6
0
5
10
15 Eb/No -dB
20
25
30
Figure 3- 37: decision-aided phase tracking variance versus Cramer-Rao lower bound.
3.15 Dealing with carrier phase ambiguity One method that has often been usedhto resolve phase ambiguities is based on resolving the rotational phase ambiguity using the recovered priory known unique word sent after 3-53
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the carrier recovery preamble field and before the data as shown in Figure 3-38. The unique word (UW) field contains a known bit pattern that is detected after carrier recovery. If the carrier recovery has undergone phase ambiguity then, after demodulations of the UW and a compariosn between the pattern of data with the UW field bits stored (priory), the receiver is able to determine phase ambiguity and compensate for it before decoding the following data field.
CW or 101 CR sequence
Unique word
data
Figure 3- 38: TDMA frame showing the use of unique word insertion in the preamble to resolve carrier phase ambiguities. For instance, with QPSK modulations with 90 degree ambiguity, if ambiguity takes place the constellation will rotate by four discrete values, 90 degree, -90 degree, 180 degree, or 270 degrees. For QPSK modulations, Figure 3-39 and Table 3-1 show the outcome possibilities of the detected UW after having been through the four possible ambiguities. For instance, if the UW used was given by the bit pattern of UW= [10 01 01 10], and after reception of the UW and assuming that the PLL used for carrier tracking slipped into a +90 degree ambiguity, the resulting UW would then be UW=[00 11 11 00], and likewise for the other three combination of ambiguities. Ambiguity compensation can be incorporated as a phase offset value for the NC, or can be included as part of the data decoder using look up tables, where every symbol is mapped to its corresponding correct unambiguous symbol.
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θˆ = θ + π 2
00
01
θˆ = θ − π 2
θˆ = θ + π
10
11
θˆ = θ + 3π 2 Figure 3- 39: Illustration of ambiguity effects on QPSK symbols.
Symbol 00 01 11 10
+90o 01 11 10 00
-90 o 10 00 01 11
180 o 11 10 00 01
270 o 10 00 01 11
Table 3- 1: Mapping values for QPSK symbols with phase ambiguities of +90 o, -90 o, 180 o, and 270 o.
3.16
Reference
1. C. Bingham, The Theory and Practice of Modem Design, John Wiley & Sons, NY, 1988. 2. Steven Leeland, “Digital signal processing in satellite modem design,” Communication System Design Magazine, pp. 21-29, June 1998. 3. R.E. Ziemer and R.L. Peterson, Introduction to Digital Communication, MacMillan Publishing Co., NY, 1992. 4. J Viterbi and J.K. Omura, Principles of Digital Communication and Coding, McGraw-Hill, NY, 1979.
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5. J.J. Stiffler, Theory of Synchronous Communications, New Jersey, Prentice-Hall, 1971. 6. Tri T. Ha, Digital Satellite Communications, New York McCraw-Hill, 1990. 7. K. Cartwright, “Fourth power phase estimation with alternative two-dimensional odd-bit constellations,” IEEE Commun. Letters, pp. 199-201, Jun 2000. 8. L.E. Franks, “Carrier and bit synchronization in data communication - a tutorial review” IEEE Trans. Commun., vol. Com-28, No. 8, pp. 1107-112, Aug 1980. 9. William Cowley, “Phase and frequency estimation for PSK packets: bounds and algorithms” IEEE Trans. Commun., vol. 44, no. 1, pp. 26-28, Jan 1996. 10. J. Balodis, “Laboratory comparison of tanlock and phase lock receivers”, National Telemetry Conf., 1964. 11. Mileant and S. Hinedi, “Lock detection in Costas Loops” IEEE Trans Commun., Letters, vol. 40, No. 3, Mar 1992, pp. 480-483. 12. F.D. Natali, “AFC tracking Algorithms”, IEEE Trans. Commun, vol. 32, No. 8, Aug 1984, pp. 935-947. 13. Tony Kirke, “Interpolation, resampling, and structures for digital receivers”, Communication System Design Magazine, pp. 43-49, Jul 1998. 14. H. So, Y. Chan, O. Ma, and P. Ching, “Comparison of Various Periodograms for sinusoid detection and Frequency Estimation,”, IEEE Trans. Aerospace and Electronics systems, Vol. 35, No. 3, July 1999. 15. Myung Kim, Jin Ho Kim, Yoon Song, Ji Jung, Jong Chae, and Hwang Lee, “Design and analysis of decision-directed carrier recovery for High-Speed Satellite communications,” IECE trans. Commun., Vol E81-B, No. 12, December 1998. 16. F.M. Gardner, Phase techniques, 2nd edition, John-Wiley, NY, 1979. 17. Boza Porat, A Course in Digital Signal Processing, John Wiley, NY, 1997. 18. Richard Lambert, A real time GMSK modem will all digital symbol synchronization, Master degree thesis, Texas A&M University, 1998. 19. Mohamed Nezami, “Acquiring and tracking MIL-STD-188-181B signals”, Milcom 2002.
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Chapter 4
Feedback Symbol Timing Synchronization and Automatic Gain Control
Symbol timing recovery is as important as carrier recovery. Chapter 3 focused on carrier recovery; in this chapter, we introduce symbol timing detectors and their utilization in a symbol timing feedback system. Figure 3-1 showed semi-analog semi- digital symbol timing alignment. This system involves the use of a digital to analog converter (DAC) that is used to tune the crystal oscillator. The crystal oscillator is affected by both manufacturing tolerances and environmental variations, and thus is less favored in modern wireless receivers that are produced in high volumes with minimal RF circuits tuning..
Figure 4-1 shows a better implementation of a symbol timing loop that is fully implemented in digital domain, where the clock that generates the sampling frequency of the A/D is freely running at the nominal sample rate of T N ± τ , N is the over sampling factor (number of samples per symbol), and ± τ is the symbol timing error which is given by 0 ≤ τ ≤ T . The correction of this timing error is carried out using digital resembling filter that resamples the matched filter output z (t ) at the optimal symbol timing instance using mathematical interpolations, thus the matched filter becomes z (t ± τ ) . (??) The next section will detail several symbol timing error detectors algorithms used in conjunction with the symbol timing loop illustrated in Figure 4-1.
s (t )
A/D t=k
Matched Filter
Interpolator
T ±τ N
e − jθ ( t ) ˆ
Loop Filter
DAC
Dr. Mohamed Khalid Nezami © 2003
s (kT )
ek
Timing error detector
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Figure 4- 1: Block diagram of the symbol timing error detection and correction loop.
4.1.1
Mid-Phase Integration Symbol Timing Detector
One way to derive timing offset error measure in coherent BPSK demodulators involves mid-bit integrations. When the middle phase shown in Figure 4-2 is integrated from the middle of one symbol period to the middle of the next symbol (assuming the symbols are not equal), the resulting values has a signal component that is proportional to the error in local symbol timing whenever the modulation polarity changes. If the sample instance of the symbol was exactly in the middle of both symbols, the integration would have the same signal but opposite polarities on both sides of the middle phase sample. The error measure is only valid whenever there is a symbol transition (+1 to –1, or –1 to +1), or d k ≠ d k −1 , where d k and d k −1 , are the hard decision values of the most recent two symbols.
previuos symbol
I k −3
current symbol
I k −1
I k −2
Ik
Mid-symbol
Figure 4- 2: Mid-Phase Timing Error Detector
As illustrated in Figure 4-2, this timing detector requires two samplings per symbol. Let the subscript k demote the samples obtained at the kth sampling time and j be the subscript denoting the mid-bit period extending from the middle symbol of the previous symbol to the middle of the current symbol as shown in Figure 4-2. The mid-bit BPSK I-channel integration is then given by
I j = I k −2 + I k −1
(4-1)
Assuming a transition bit or dˆk ≠ dˆk −1 (i.e., the two consecutive symbols are different), where dˆ k = sign( I k −1 + I k ) and dˆ k −1 = sign( I k −2 + I k −3 ) , then if the symbol timing
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error is in the middle of the symbol (i.e., no symbol timing error) it can be seen that the symbol timing error can be indicated by I j which is also I k −2 + I k −1 = 0 .
When the symbol timing is either later or earlier than the middle phase represented by I j ? will only have to be corrected by the polarity of the error. To automate the error detection, we incorporate a polarity based on the following bit hard decision by,
and that
mˆ i = sign(I k −1 + I k )
(4-2)
mˆ i −1 = sign(I k −3 + I k −2 )
(4-3)
Now the mid phase timing error detector is then generalized as ⎧mˆ I , mˆ i−1mˆ i = −1 ek = ⎨ i j ⎩ 0 , mˆ i−1mˆ i = +1
(4-4)
Equation (4-4) means that the two adjacent symbols are multiplied, if their sign (positive or negative, where sign(x) = +/-1) is the same, then the error signal is not updated since that means that there is was no symbol transition (10 or 01), and the detector will wait until there is a symbol that is different than the current symbol at hand. Figure 4-3 shows the graphical block diagram of this timing detector applied for the in-phase channel, which is the case for BPSK modulations.
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I k−3
I k−2
sign (
I k −1
Ik
sign (
)
)
dˆ i
dˆi −1
Ij sign(
)
ek
Figure 4- 3: BPSK mid-bit symbol timing error detector.
Notice that this timing detector requires at least 2 symbols and a minimum of four samples to work (or two samples per symbol). Notice also that this timing detector is impacted by the carrier phase offsets. For it to work perfectly, carrier recovery must be achieved with minimum phase residuals using the schemes detailed in Chapter 3 and shown in Figure 4-1. The effect of carrier phase on this error signal is given by ek = mˆ i I j cos θ .
4.1.2
Mid-phase integration Symbol Timing Detector for DBPSK signals
One way to make the error detector in Figure 4-3 carrier independent is to use DBPSK modulations so that cos(θ ) = 0 . However, the error detector will now be degraded if there
ˆ i I j cos (θ k − θ k −1 ) . Figure 4-4 shows an is any carrier frequency offsets given by ek = m
illustration of the mid-phase timing error detector for DBPSK modulations. channel mid-bit integration in Figure 4-4 is given by
Q j = Qk −2 + Qk −1 which is the same as
Ij
The Q
(4-5)
given in equation 4-1. Using the in-phase and quadrature phase
samples in Figure 4-4.(?) For the in-phase channel, we obtain Dr. Mohamed Khalid Nezami © 2003
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I i = I k −1 + I k
(4-6)
and for the quadrature phase,
Qi = Qk −1 + Qk
(4-7)
One way to eliminate the carrier frequency dependency on the timing error signal is to perform the following,
⎧ I I + Qi Q j , mˆ i < 0 ek = ⎨ i j 0 , mˆ i > 0 ⎩
(4-8)
ˆ i is the DBPSK demodulator data decision and is given by where m
mˆ i = I i−1I i + Qi−1Qi
(4-9)
Again, symbol-timing loops that are based on the error signal in (4-8) will only update the loop error signal if the cross product of the adjacent symbols is not positive.
I k −3
I k −2
I k −1
Ik
I jIi
ek
Q j Qi
Q k −3
Q k −2
Qk −1
dˆk = Ii Ii −1 + Qi Qi −1 < 0
Qk
Figure 4- 4: DBPSK symbol timing error detector
4.1.3
Early-Late Gate Symbol Timing Detector
The Early-Late Gate timing error detector generates its error by using the samples that are earlier and later compared to the ideal sampling point. This timing error requires at least three samples per symbol. The difference in the amplitude of the samples of both early Dr. Mohamed Khalid Nezami © 2003
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and late is the error signal as shown in Figure 4-5 and the middle sample (on time sample) is the actual symbol-timing clock. That is
ek = y e , k − y l , k
(4-10)
On-time sample
Late-sample
5
Early -sample
So for the case illustrated in Figure 4-5, the timing error signal is given by ek = 3 − 3 = 0 . Consider now the case where the symbol sample is sampling the symbol a little early as shown in Figure 4-6, the timing error signal is then given by ek = 3.3 − 2.7 = 0.6 . Furthermore, consider also the scenario in Figure 4-7, where the sample clock of the symbols is fast. Here the timing error signal is given by ek = 2.7 − 3.3 = −0.6 . Figure 4-8 illustrates how a symbol timing recovery loop that uses the early-late timing algorithm converges to the optimal symbol timing. Here an alternating sequence of 1010101 is used to train the timing loop. The error detector starts at a late instance and gradually the loop steers the symbol-sampling clock to sample the bits at the middle of the symbol.
3 one symbol
τe
τl
τ ot
Figure 4- 5: Early-Late timing error detector with ideal timing (early sample =late sample).
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Late-sample
On-time sample
Early -sample
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5 3.3 2.7 one symbol
τe
τl
τ ot
Late-sample
On-time sample
Early -sample
Figure 4- 6: Early-Late timing error detector with early timing error offset.
5 3.3 2.7 one symbol
τe
τl
τ ot
Figure 4- 7: Early-Late timing error detector early timing error offset.
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Input signal
0.8
Early-Late timing Error
0.6
Symbol timing
0.4 0.2 0 -0.2 -0.4 -0.6 0
50
100
150
200
250
300
350
Sample #
Figure 4- 8: Symbol timing recovery based on the early-late timing algorithm.
4.1.4
Muller-Muller (M&M) Symbol Timing Detector
The M&M symbol timing detector [5] uses only one sample per symbol. The timing error is given by
ek = yk dˆk −1 − yk −1dˆk
(4-11)
Consider the scenario shown in Figure 4-9 where the data decision
dˆk −1 = −1 ,
and
dˆk = +1
that are obtained by taking the sign of the sample, causing the symbol timing error (4-11) that is given by ek = (− .5)(+ 1) − (0.8)(− 1) = +0.3 . Consider another scenario (late) shown in Figure 4-10. Here the symbol timing error is given b ek = (− 0.8)(+ 1) − (0.5)(− 1) = −0.3 . As with the mid-phase integration error detector in Equation (4-4), the M&M timing detector algorithm is also sensitive to carrier phase offsets.
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yk −1 = 0.8
previous symbol
current symbol
dk
dk − 1
yk = −0.5
Figure 4- 9: Sampling is slower than the optimal instance.
y k −1 = 0.5 previous symbol
current symbol
d k −1
dk
yk = −.8
Figure 4- 10: Sampling is faster than the optimal instance.
Illustrative example: Figure E4-1 illustrate the use of midphase or what is also known as zero crossing timing detector. As an illustration, assuming 10 samples per symbol, the timing error detector is implemented as follow, N=10 % sps for i=ns+1:length(I) tau(i)=I(i-N/2)*(sign(I(i))-sign(I(i-N))); end;
Notice that when there is no symbol transition, that is I(n)=I(n-1), there is no timing update, as illustrated in the figure. The correct symbol timing will be N/2 samples from the zero crossing point.
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1.5
Symbol timing
Timing Error
1
Input signal 0.5
0
-0.5
-1
-1.5 1300
1350
1400
1450
1500
Sample #
Figure E4-1: Illustration of the zero crossing symbol timing detector discussed above with 10 samples per symbols for shaped BPSK. 4.1.5 Gardner BPSK Symbol Timing Detector
The Gardner symbol timing detector algorithm is probably the most widely used symbol timing error detector in MPSK modulations. This detector uses two samples per symbol and has the advantage of being insensitive to carrier phase errors [3]. Using a complex signal representation of the symbols, the timing error is given by
ek = ( y k − y k − 2 ) y k −1
(4-12)
y k , y k − 2 , and y k −1 are shown in Figure 4-11. The time offset between y k , and y k − 2 is one symbol period. The time between y k and y k −1 is a half where the samples
symbol as shown in Figure 4-11. Notice that if there is no data transition, the value of y k − y k − 2 is zero, thus the Gardner timing does not update the timing correction loop unless there is a data transition. The data transition y k − y k − 2 also automatically supplies the polarity and the slope of the error to steer the timing correction in the right direction. That is ⎧− 1, y k = 0 , y k −2 = 1 y k − y k −2 = ⎨ (4-13) ⎩ + 1, y k = 1 , y k − 2 = 0 Consider the situation of the received samples shown in Figure 4-11., Here the timing error using the Gardner detector in Equation (4-12) is ek = 0.2(− 0.8 − 0.8) = −0.32 . Likewise, if the sampling clock as that shown in Figure 4-12, the Gardner timing detector is ek = 0.0(− 0.8 − 0.8) = 0 , indicating correct sampling instance, since the error is zero. Dr. Mohamed Khalid Nezami © 2003
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yk −2 = 0.8
previous symbol
current symbol
dk −1
dk yk−1 = 0.2 T 2
yk = −0.8
T
Figure 4- 11: Gardner timing error detector
y k −2 = 0.8
previous symbol
current symbol
dk −1
dk yk −1 = 0.0
T 2
yk = −0.8
T
Figure 4- 12: Gardner timing error detector with no timing error.
4.1.6
Gardner QPSK Symbol Timing Detector
For QPSK modulations, the Gardner error-timing detector in Equation (4-12) is given by e k = (I k − I k − 2 )I k −1 + (Q k − Q k − 2 )Q k −1
(4-14)
1 1 Where I k −1 = I (t − T ) , and Q k −1 = Q (t − T ) , in the continuous time domain. The 2 2 timing error detector in equation (4-14) can be physically understood as follows: The timing error detector samples the quadrature (complex) signal between the strobe times, if there is a data transition, ( (I k − I k − 2 ) ≠ 0 , (Q k − Q k − 2 ) ≠ 0 , The average mid-way samples will be zero in the absence of a timing error, otherwise, the timing error is a nonDr. Mohamed Khalid Nezami © 2003
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zero midway sample and the slope necessary for the error steering mechanism is supplied by the difference between the strobes, (Q k − Q k − 2 ) , and (I k − I k − 2 ) . The product between this slope and the mid-way sample provides the proper error magnitude and direction. Figure 4-13 illustrates both the acquisition and tracking of an error timing loop. The loop filter used in Figure 4-13 is the same topology as those illustrated with carrier phase tracking in Chapter 3.
Although the Gardner timing detector in (4-14) is independent of carrier phase, it is still affected by frequency offset. As a result of this frequency error, the symbols will have an additional phase rotation that is given by ∆θ =
Rs 3600 fs
(4-15)
Rs is the ratio of symbol rate to sample rate. Assume that a 2400 symbols per fs second wireless radio utilizing the Gardner timing detector has a residual carrier error of 120 120Hz. The phase rotations due to the 120Hz frequency offset is ∆θ = 360 0 = 18 0 . 2400 This phase shift associated with the symbols causes the timing error signal to be reduced 180 by the factor cos = 0.987 . For a larger frequency offset of 400 Hz, the loss is 2 60 cos = 0.5 or 3 dB, which is severe for this case. A rule of thump, if the normalized 2 frequency offset is of the order of ∆fT = 10 −2 , is that the degradation of the Gardner detector is considered negligible. where
Figure 4-13 shows QPSK symbol timing detection and tracking using the Gardner timing detector in conjunction with the loop filter derived earlier for carrier recovery in Chapter 3 (see figure 3-6) The interpolator filters are used as a way of correcting the symboltiming instances by sample interpolation methods (i.e., resampling). The interpolator used here can be either a Farrow filter or simply a sinc interpolator function. In Chapter 6 we will detail the use of Farrow filters for symbol timing correction. Here we will detail the use of the raise-cosine-filter (RCF) which was introduced in Chapter 2 as a matched filter that is also used to correct the timing sampling instance as shown in Figure 4-13.
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1 / 2T sign(x)
IF/BB
I ADC
interpolator
I
Base Band filtering
Q interpolator
Q
sign(x)
Ki
4/T
Kp System Clock
1/ T
Sign(x<1)=-1 Sign(x>1)=+1
Figure 4- 13: BPSK symbol timing tracking using the Gardner symbol timing detector and interpolator.
4.2 Timing Correction Interpolator Filters
This section presents a method and procedure for using a RCF pulse sinc interpolator as a timing error corrector. The RCF was introduced in Chapter 2, where the pulse shaped function that has a shape controlled by the roll-off ( α ) factor is given by
⎡ sin (πx T ) ⎤ ⎡ cos(απx T ) ⎤ c( x) = ⎢ ⎥⎢ 2⎥ ⎣ πx T ⎦ ⎣1 − (απx T ) ⎦
(4-16)
where x=-N_taps/ 2+1:1:N_taps/ 2 . To introduce timing shift in the signal that is being fed to the RCF matched filter, we deviate the variable x by x = x+fract_de l . So the fractional delay of the sampling clock is given by
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(4-17)
fract_del= del/N_phas es
where N_phases is the desired number of resolution steps in fractions of a symbol, most commonly the value of N_phases is 8, or 32. The variable del is the delay using a range that is given by -(N_phases-1)/N_phases):1/N_phases: (N_phases-1)/N_phases The correction mechanism using the RCF interpolator in (4-16) is done as follows. First, we decide the fractional steps of correction that are to be performed. The interpolator coefficients are then found by evaluating (4-16) for every one of these steps, and the result is then stored in a table look up (TLU). Using the timing error signal from the Gardner detector that is quantized to the closest value in the TLU, the coefficients that correspond to correction are used as coefficients for the FIR matched filter, which is then used to filter the data. For example, assume that the Gardner timing error in Figure 4-13 estimated the error to be 0 of a symbol (no error), using a 6th order FIR RCF pulse interpolator in (4-16) with N_taps = 8 , or fract_del = 1 / 8 , and roll-off=0.35 then evaluating Equation (4-16) with del=0, Figure 4-14 shows a plot of the impulse response of the filter, which shows that the coefficients are all zero except the one at t= 0, or at the symbol rate. Figure 4-15 shows the impulse response of the RCF interpolator in Equation (4-16) with del=-1/8. Notice here that the middle coefficient has the largest value and corresponds to the value of x=-1/8 as shown in the listing of the coefficient table (listed on the figure). Figures 4-16 through Figure 4- 18 shows the coefficients of the interpolating filter as corresponding to the timing correction being –2/8, –3/8, and –4/8. Figure 4-19 shows the interpolator coefficients as a function of the timing error signal for symbols with an oversampling factor of 8. In modem usage the interpolator in Equation (4-16) is not computed on line, since that might be computationally expensive. Instead, the interpolator coefficients are evaluated and stored in memory and the correction scheme is implemented by a table look up every time the timing error is updated.
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1.2
de,l=0/8 1 X TAP -3.0000 0.0000 -2.0000 -0.0000 -1.0000 0.0000 0 1.0000 1.0000 -0.0000 2.0000 -0.0000 3.0000 0.0000 4.0000 -0.0000
0.8
0.6
0.4
0.2
0
-0.2 -3
-2
-1
0
1
2
3
4
Figure 4- 14: coefficients for delay =0/8.
1.2
del=-1/8
1
-3.1250 -0.0099 -2.1250 0.0328 -1.1250 -0.0934 -0.1250 0.9728 0.8750 0.1274 1.8750 -0.0424 2.8750 0.0139 3.8750 -0.0022
0.8
0.6
0.4
0.2
0
-0.2 -4
-3
-2
-1
0
1
2
3
4
Figure 4- 15: coefficients for delay =-1/8.
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del =-2/8 1
-3.2500 -0.0151 -2.2500 0.0531 -1.2500 -0.1499 -0.2500 0.8939 0.7500 0.2812 1.7500 -0.0889 2.7500 0.0300 3.7500 -0.0057
0.8
0.6
0.4
0.2
0
-0.2 -4
-3
-2
-1
0
1
2
3
4
Figure 4- 16: coefficients for delay =-2/8. 0.8 0.7
del =-3/8
0.6
-3.3750 -0.0160 -2.3750 0.0606 -1.3750 -0.1711 -0.3750 0.7717 0.6250 0.4498 1.6250 -0.1320 2.6250 0.0456 3.6250 -0.0099
0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -4
-3
-2
-1
0
1
2
3
4
Figure 4- 17: coefficients for delay =-3/8.
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del =-4/8 1
-3.5000 -0.0138 -2.5000 0.0570 -1.5000 -0.1624 -0.5000 0.6186 0.5000 0.6186 1.5000 -0.1624 2.5000 0.0570 3.5000 -0.0138
0.8
0.6
0.4
0.2
0
-0.2 -4
-3
-2
-1
0
1
2
3
4
Figure 4- 18: coefficients for delay =-4/8.
delay -1.0000 -0.8750 -0.7500 -0.6250 -0.5000 -0.3750 -0.2500 -0.1250 0 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750
C0 C1 0.0000 -0.0000 0.0139 -0.0424 0.0300 -0.0889 0.0456 -0.1320 0.0570 -0.1624 0.0606 -0.1711 0.0531 -0.1499 0.0328 -0.0934 -0.0000 0.0000 -0.0424 0.1274 -0.0889 0.2812 -0.1320 0.4498 -0.1624 0.6186 -0.1711 0.7717 -0.1499 0.8939 -0.0934 0.9728
C2 0.0000 0.1274 0.2812 0.4498 0.6186 0.7717 0.8939 0.9728 1.0000 0.9728 0.8939 0.7717 0.6186 0.4498 0.2812 0.1274
C3 C4 1.0000 0.0000 0.9728 -0.0934 0.8939 -0.1499 0.7717 -0.1711 0.6186 -0.1624 0.4498 -0.1320 0.2812 -0.0889 0.1274 -0.0424 -0.0000 -0.0000 -0.0934 0.0328 -0.1499 0.0531 -0.1711 0.0606 -0.1624 0.0570 -0.1320 0.0456 -0.0889 0.0300 -0.0424 0.0139
C5 -0.0000 0.0328 0.0531 0.0606 0.0570 0.0456 0.0300 0.0139 0.0000 -0.0099 -0.0151 -0.0160 -0.0138 -0.0099 -0.0057 -0.0022
Figure 4- 19: Tap coefficients a 6th order FIR RCF pulse interpolator using roll-off=0.35 with timing resolution of 1/8T.
Another method of symbol timing correction is obtained using the size delay line shown in Figure 4-20, which uses a single weight interpolator. Here the output is given by y ( n) = x ( n) + Bx ( n − 1)
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This interpolator has a transfer function given by y ( n) 1 = x(n) 1 + Bz −1 and the filter’s group delay (correction) is given by B=
(2τ − 1) cos(2π
f fs )±
(1 − 2τ )2 cos 2 (2π 2(1 − τ )
f f s ) + 4τ (1 − τ )
(4-19)
As an example, for f f s =0.156, τ =0.6T, substituting these values in the weight of the delay line in Figure 4-20, gives the result of B=1.38.
x ( n)
y( n)
+
z −1
B Figure 4- 20: Delay line for timing error correction.
4.3 Symbol Timing Loop Filter Design
In figure 4-13, the timing error was passed through a loop filter that is similar to those derived in Chapter 3 for carrier recovery loops. However, since the symbol timing recovery is performed after precise frequency recovery, there is no need for the filter to have the integrator constant, and thus the loop can be a first order system, that is K p = 0 and so the error signal fed to the interpolator becomes,
xk +1 = xk + K p ek
(4-20)
Figure 4-21 shows a comparison between the adjustment signal out of the proportional loop filter ( x k +1 ) for a value of K p = 0.25 and K p = 0.025 . Clearly the tradeoff is speed of convergence versus variance of the final error estimates.
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1
1
0.8
0.8
0.6
0.6 Error
1.2
Error
1.2
0.4
0.4
0.2
0.2
0
0
-0.2
0
20
40
60 80 Normalized time
100
120
-0.2
140
0
20
40
60 80 Normalized time
100
120
140
Figure 4- 21: Comparison between the adjustment (Equation 4-20) signal out of the proportional loop filter ( x k +1 ) for a value of K p = 0.25 (Left) and K p = 0.025 (Right).
The loop bandwidth of the loop based on the recursive equation in (4-20) is given by
BLT =
K p Kd
2(2 − K p K d )
where K d is the slope of the error time detector (see figure 4-21) when the timing error is zero.
4.4 Symbol timing lock Indicator
The lock indicator is a useful tool for monitoring the receiver performance and also is an indicator that is used to initiate the decoder operation. One symbol-timing indicator is given by ⎡ ⎛ ⎞⎤ LDτ = Re ⎢ x 3 ⎜⎜ x ∗ 5 − x ∗ 1 ⎟⎟⎥ k+ k+ k+ ⎥ 4 ⎠⎦ ⎣⎢ 4 ⎝ 4
(4-21)
where x ∗ is the complex sample of the symbol x . This lock indicator in (4-21) requires four samples per symbols to operate. For systems with two samples per symbols, the following lock indicator can also be used,
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⎡ ⎤ LDτ = Re ⎢ x k x k∗ − x ∗ 1 x 1 ⎥ k− k− 2 2⎦ ⎣
(4-22)
Figure 4-22 shows the lock indicator in (4-22) and the Gardner timing error detector (414) is applied to the loop shown in Figure 4-13. Notice that the lock detector is a maximum value when the timing error detector is zero. Gardner timing error detector
Averaged detector output
Lock indicator
-T/2
+T/2 Normalized Timing error (del/T)
Figure 4- 22: Lock indicator and Gardner timing error detector.
4.5 Preamble-Aided Symbol Timing Synchronization
Figure 4-23 shows an example of a digital receiver that corrects timing based on a timing error detector based on the correlation between a local replica of a preamble and the same preamble that is transmitted through the channel. The receiver captures a set of samples that contains a transmitted preamble (CW tone) that is also alternating sequence of +11+1-1… as shown in Figure 4-24. The maximum correlation value directly indicates the proper sample shift in order to line up the received preamble with the local preamble that is then used to control the sample index, so that the receiver symbols sampler strobe either advances or retards by an amount that is equal to the error. The error signal is usually filtered using a low pass filter to yield a proper signal that can be compared with a threshold detector to indicate valid alignment as shown in the Figure 4-24 (bottom). This technique requires that the received signal is oversampled by a factor larger than 8 (usually), and does not involve any feedback.
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Interpolator Matched Filter
CNCO Phase Phase rotation Correction
Down Sampling
Carrier Phase Phase Estimation Estimation
Symbol
Extract Training
Decoder
Synchronization Offset
estimation
Figure 4- 23:carrier and symbol timing Synchronization scheme using training sequence.
2 0 -2
Received preamble 0
1000
2000
3000
4000
5000
6000
7000
2 0
Preamble reference -2 2
0
1000
2000
3000
4000
2
0
1000
2000
3000
4000
7000
5000
6000
7000
Low pass Cross correlation result
1 0
6000
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Figure 4- 24: Preamble based start of a symbol estimation using 32 samples/symbol.
4.6 Symbol Timing Synchronization using Unique Word
Another method of extracting symbol-timing alignment at the receiver is based on the characteristics of a unique word (UW) sent to mark the beginning of a symbol. Figure 425 shows the autocorrelation characteristics of a UW sequence given by UW= [1
1 -1 -1
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UW= [1
1 -1
-1
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Figure 4- 25: Autocorrelation of the unique word used to mark symbol start.
Although this unique word here is chosen at random, there are some sequences that have better autocorrelation shape, such as the Barker code or Gold codes. However for our demonstration here, it is sufficient to assume that the autocorrelation in Figure 4-25 is good enough for illustrative purposes. Figure 4-26 shows symbol timing error using a signal containing the alternating sequence Symbol timing error using the transmitted UW as illustrated in Figure 4-25. (??) The upper graph shows the received signal containing the preamble at the beginning, then an alternating sequence that can be used for carrier recovery of automatic gain control, and then the UW that is shown by the arrow indicator. The received signal is cross correlated with a local replica of the UW, and the output is checked against a correlation score (threshold), as indicated in Figure 4-26. The peak corresponds to the sample index used as an error signal to feed to the receiver sampler.
The cross-correlation of any two sequences
xn
(received preamble) and
y n (local UW) is given by
(
)
(
)
(4-23)
y n∗ , m ≥ 0
(4-24)
E xn yn∗−m = Rxy (m) = E xn+m yn∗ where
Rxy (m) =
N − m −1
∑x n =0
n+m
The relation in Figure 4-24 points out that by trying different time-shifts in steps of 1 sample at a time while observing the cross-correlation magnitude, the symbol timing can L
be found. Given that the unique word training sequence (UW) is c( k ) k =1 , and that the search window is bounded by t ∈ [t start − tend ] , The symbol timing is then found by Dr. Mohamed Khalid Nezami © 2003
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searching for the time instance that maximizes the cross-correlation between a UW sequence c (k ) and the received set of samples r (k ) . That is L ⎫ ⎧ t sampling = arg ⎨ max r ( k − kN + t sampling ) * c (k ) ⎬ ∑ ⎭ ⎩t sampling ∈[t start −tend ] k =1
(4-25)
where the sampling time is given by t s = NT . 1
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Figure 4- 26: The waveform illustrating the unique word detection using a crosscorrelation method.
4.7 Automatic Gain Control (AGC) Loops
Figure 3-2 illustrated the necessity of using AGC to obtain a proper signal for synchronization. The AGC operates by sensing the matched filter output in conjunction with a user specified signal level at the desired setting (reference) , from which it derives the AGC error signal. The error signal is then filtered in a loop filter producing an AGC correction update that is fed to a multiplier that scales the samples out of the tracking bufferjust before it enters the decoder. There are two types of AGC loops One is based on a linear relation between the received signal and a desired reference and the other is based on the logarithmic of the received signal against a reference. The next sections point out the differences and illustrate both loops using computer simulations. 4.7.1
Linear Signal Magnitude Based AGC Algorithm
The linear magnitude based AGC is shown in Figure 4-27. Here the loop computes the signal level at the output of the AGC, next it compares it to a fixed reference, then the Dr. Mohamed Khalid Nezami © 2003
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error signal is scaled by the factor α, which is less than 1 and controls the steady state response.
y (n) = A(n) x(n)
x(n)
X level Estimate
A(n + 1)
A(n) z
−1
+
X
α
+
-
R
Figure 4- 27: Linear implementation of AGC loop filter.
One of the fundamental problems with this classical algorithm is that the steady state response (settling time) of the AGC loop when responding to a step in the input signal which is controlled not only by α, but also by the input signal level change, which gives inconsistent attack and release times, and results in deterioration of the receiver performance.(??) The impact t of the input signal level on the AGC control signal can be illustrated mathematically as follow. The output signal of the AGC loop shown above in Figure 4-27 is given by y ( n) = A( n) x (n)
(4-26)
where A(n) : the AGC control signal.
A(n + 1) = A(n ) + α [R − A( n ) x( n ) ]
(4-27)
Which is then expanded to,
A(n + 1) = A(n )[1 − α x(n ) ] + α R
(4-28)
To illustrate the sudden jump in received signal level, assume that the input signal x(n) is a unit step with an amplitude of c , or x (n) = cu ( n) . The control signal (gain value) is then given by,
A(n + 1) = A(n )[1 − α c] + α R
(4-29)
Using standard difference equation, the steady state response of the loop can be shown as
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[
]
R n 1 − (1 − α c ) u ( n ) c which has a steady state time response of A( n ) =
(4-30)
R
− t R A(t ) ∝ (1 − e αc ) c
(4-31)
R , which is a desirable result. But c 1 unfortunately, the loop time constant is proportional to , which is a function of the αc step in the signal magnitude. This shows that if the signal jumps by a small magnitudethe AGC loop will have a large time constant, which will take a long time to steady. Likewise, if the signal jumps by a large magnitude, the time constant will be small, resulting in overshoot and the possibility of oscillation. This phenomenon is shown in Figure 4-28. For the small signal jump during sample numbers of 300 to 500, it can be shown clearly how slow the system responds. In contrast, for large signal jumps, such as that taking place during samples 1100 to 1300and 700 to 900, we see that the loop has a very small time constant. A violent overshoot results. Figure 4-29 shows the same algorithm applied to a baseband signal of 10 kbps of BPSK waveform. .
This response has a steady state value of
AGC input
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Figure 4- 28: Simulations of the linear AGC loop with low IF and α=0.2.
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AGC input
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Figure 4- 29: Simulations of the linear AGC loop with Binary signal and α=0.2.
4.7.2
Log Signal Magnitude Based AGC Algorithm
The inherent problems shown in Figure 4-29 are avoided by using an alternative approach that weights the correction signal term by the logarithmic ratio of the input to the reference as oppose to the difference between them. Figure 4-30 shows the AGC loop of figure 4-27 being modified to use the log function of the output level.
x(n)
y (n) = A( n) x( n) X
log{A( n)}
level Estimate
log{A(n + 1)}
exp(.) z
−1
+
X
α
+
-
Log e (.)
Loge (R )
Figure 4- 30: The Logarithmic magnitude based AGC loop. Here the gain output signal is given by
log{A(n + 1)} = log{A(n)} + α [log{R} − log{ A(n) x(n) }]
(4-32)
which is then expanded to, Dr. Mohamed Khalid Nezami © 2003
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log{A(n + 1)} = log{A(n)}[1 − α ] − α [log{ x(n) } − log{R}] and thus,
log{A(n + 1)} = log{A(n)}[1 − α ] − α log{ x(n) / R}
Again, with x(n) as a unit step with amplitude of c , or x (n) = cu ( n) , the gain signal is then given by,
log{A(n + 1)} = log{A(n)}[1 − α ] − α log{c / R}
(4-33)
which is further simplified to,
[
]
log{A( n)} = − log{c / R}1 − (1 − α ) u ( n) n
(4-34)
R , which is again a desirable value. However, c unlike the system shown in Figure 4-27, the final value, or steady state of the loop shown 1 and hence is independent of the input signal level. in Figure 4-30, is proportional to
This shows that the steady state value is
α This results in a consistent loop time constant. Figure 4-31 shows the simulation of the loop in Figure 4-29. Notice that the AGC control voltage (bottom plot), unlike that in Figure 4-28, has a consistent rise and fall times, which are not a function of the input magnitude jump. . AGC input
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-2 4 2 0 -2
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Figure 4- 31: Simulation of the Log based AGC loop with 10 kbps BPSK and α=0.2.
4.7.3
Exponential Based AGC Algorithm
Another loop that also presents a similar response to that shown in Figure 4-31 is one where the correction signal is passed through the exponential function. This algorithm does not utilize the Log(x) function and only attenuates the control voltage using a nonlinear function that de-emphasizes small signal jump impact on the loop dynamics. The algorithm is applied as follow: (4-35)
y(n+1) = x(n)*exp(a(n));
where y(n) is the AGC output, x(n) is the AGC input, and a(n) is the AGC control voltage. The error signal is then formed by, (4-36)
e(n) = r-abs(y(n))
where r is the reference level. Using both (4-35) and (4-36) the gain control is given by, a(n+1) = a(n) + α*e(n);
(4-37)
where α is the convergence factor that is less than one. The nonlinearity used here is the exp(a(n)) function. Figure 4-32 and Figure 4-33 shows the algorithm (4-37) performance with α=0.2 for a 60kHz CW and BPSK signals respectively.
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-2 10
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Figure 4- 32: AGC performance of the algorithm discussed in with low IF of 10 kHz [1]. AGC input
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Figure 4- 33: AGC performance of the algorithm discussed in with 10kbps BPSK waveform.
4.8 REFERENCES
1. S. Haykin, Communication Systems, Wiley, NY, 1994. 2. Gardner, F. Phaselock Techniques. New York, NY: John Wiley, 1966. 3. F. M. Gardner, “A BPSK/QPSK Timing Error Detector for Sampled Receivers,” IEEE Transactions on Communications, vol. COM-34, pp. 423-429, May 1986. 4. D. N. Godard, “Passband Timing Recovery in an All-digital Modem Receiver,” IEEE Transactions on Communications, vol. COM-26, pp. 517-523, May 1978. 5. K. H. Mueller and M. S. Muller, “Timing Recovery in Digital Synchronous Data Receivers,” IEEE Transactions on Communications, vol. COM-24, pp. 516-531, May 1976. 6. Jean Armstrong, and David Strickland, “Symbol timing using samples and interpolation”, IEEE Trans. Commun., vol. 41, No. 2, pp. 318-321, Feb 1993. Dr. Mohamed Khalid Nezami © 2003
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7. Jussi Vesma, Markku Renfors, Jukka Rinne, “Comparison of efficient interpolation techniques for symbol timing recovery”, Proceedings of GLOBECOM, 1996.
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Chapter 5 Introduction to Feedorward Synchronization In this chapter we introduce several feedforward synchronization algorithm. Figure 5-1 presents the schematic of a typical wireless digital radio receiver with the relevant blocks for feedforward synchronization processing. The order of synchronization can be as that illustrated in the figure where symbol timing is obtained first, then -since carrier frequency offset and carrier phase offset operate on a single sample per symbol - they are performed after. Frequency offsets is estimated and corrected first since phase estimations require no frequency offset present. The Figure also indicates the frame estimator, which is usually based on use of unique words or other markers within the transmitted data. Gain estimation is estimated at the last stage to normalize the symbol samples for the decoder.
RF IN
Timing estimation and correction
I (k ) RF section
Frequency estimation and correction Interpolator/ Decimator
Matched filter
Phase estimation and correction x
x
Q (k )
Channel gain estimation
τˆ Timing estimator
Timing Post Processing
Frequency estimator
∆fˆ Phase estimator
Phase post processing
δˆ
θˆ
Frame estimation
Figure 5- 1: Digital wireless radio schematic emphasizing synchronization blocks.
5.1 Feedforward Symbol Timing Recovery The asynchronously oversampled baseband signal is stored on a burst-by-burst basis; a block of samples extending over a finite duration (a fraction of burst duration) is then buffered for processing by the symbol timing algorithm. The algorithm estimates the Dr. Mohamed Khalid Nezami © 2003 5-1
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symbol timing error offset ( τˆ ), which is used to correct the sampling instances. The correction process can be performed either by correcting the A/D variable sample clock using a feedback scheme, or by the open-loop digital resampling method shown in Figure 5-1. In the digital resampling method, the synchronous values are estimated by interpolating the non-synchronized samples using an interpolator filter. A decimating filter is then used to lower the sample rate.
5.2 Feedforward Carrier Frequency Offset Recovery During the second step, a block of the decimated complex samples from the symbol timing correction interpolator/decimator network is buffered and presented to a frequency estimation algorithm, which in turn estimates an initial offset frequency ( ∆fˆ ) used by a frequency compensation network. The frequency correction can be implemented either by offsetting the variable complex local oscillator by an offset of (- ∆fˆ ), or by using an open loop method that numerically rotates the samples using a CNCO and a complex multiplier as shown in Figure 5-1. As the frequency estimator of Figure 5-1 can only cope with a narrow range of frequency offsets, an automatic frequency control (AFC) circuit is needed for large frequency offsets to perform the initial carrier acquisition [88]. 5.3 Feedforward Carrier Phase Recovery The processed samples with minimal symbol error, and frequency offsets, are then subjected to carrier phase error estimation and correction. The phase estimation process in Figure 5-1 is similar to the frequency offset recovery; the phase estimates ( θˆ ) are used by a CNCO in conjunction with a complex multiplier to undo the undesired phase rotation associated with the received baseband samples. Figure 5-1 also depicts other signal processing blocks associated with synchronization, such as the channel gain ( δˆ ) estimator, the AFC circuit, the frame synchronizer and the receiver decoder. The channel gain estimator estimates and equalizes signal variation of the received signal due to fast fading. The automatic frequency control (AFC) circuit acquires the signal [88] when the frequency offsets are larger than the capture range of the baseband digital carrier offset estimator, especially during the initial receiver powerup stages. Frame synchronizer is used to estimate the beginning and end of the TDMA burst. The receiver decoder is the block where Viterbi decoding and other decoding or data formatting routines are implemented. 5.4 Maximum Likelihood Principle The feedforward digital synchronization parameters (ε , ∆f , θ ) in Figure 5-1 are estimated using the principle of maximum likelihood (ML). This yields an estimate of the single desired synchronization parameter, which could be symbol timing, phase offset, or
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frequency offset. To illustrate the use of the maximum likelihood estimation method, we use the following example. Given N observations of the parameter θ in a received (observed) signal r corrupted by additive white Gaussian noise (AWGN) n , one has ri = θ + ni
(5-1)
where i = 1,2,3,..., N with ni representing independent, zero-mean Gaussian variables with identical variance σ n2 . The conditional probability of the received signal r given the parameter θ , is given by N
p (r / θ ) = ∏ i =1
⎛ (ri − θ )2 ⎞ ⎟ exp⎜⎜ − 2 ⎟ 2 σ 2π σ n n ⎝ ⎠ 1
(5-2)
The ML estimate of θ is obtained by the maximizing p(r / θ ) with respect to θ . Thus
⎤ d ln( p(r / θ )) N ⎡⎛ 1 N ⎞ = 2 ⎢⎜ ∑ ri ⎟ − θ ⎥ = 0 dθ σ n ⎣⎝ N i =1 ⎠ ⎦
(5-3)
which yields the following equality, ⎛1 ⎜ ⎝N
⎞
N
∑ r − θ ⎟⎠ = 0
(5-4)
i
i =1
which gives the estimate
θˆ(r ) =
1 N ∑ ri N i =1
(5-5)
This serves as the basis for recovering synchronization parameters using the ML principle. The variance of the estimates is bounded by lower theoretical bounds such as Cramer-Rao bound (CRB) [5-12]. CRB serves as a benchmark to check the accuracy of estimation and will be used throughout the book. To illustrate that the ML can also be used with feedback systems, let us present the derivation of symbol timing estimation and correction using feedback [1,2,3] shown in Figure 5-3. Given the ML function of the MF output λ (θ ,τ ) as a function of phase and symbol timing error by
λ (θ ,τ ) =
+∞
∫ [y(t − εT )e
jθ
]
2
− p(t ) dt
(5-6)
−∞
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Where y(t) is the received signal and the functions p(t) is a set representing the trial over the possible transmitted symbols. This can also be considered a training sequence. 2 Expanding the inner term of the integral, [ y (t ) − p (t )] y 2 (t ) − 2 y (t ) p (t ) − p 2 (t ) , and removing all correlated parts yields the MLE equation that only contains the y (t ) p (t ) terms [2,3].
λ (θ ,τ ) =
+∞
∫ [y (t − τ )e
jθ
]
2
p(t ) dt
(5-7)
−∞
The ML function that is dependent on symbol timing and carrier phase (5-7) now is a correlation between the basis ML set p(t) and the received signal y(t) that has synchronization offsets. Now to perform ML, we need to maximize this ML function (57), which means we correlate the received signal with the whole set of ML entries, then use the one entry among the set that resulted in maximum correlation value, which is a measure of which waveform was sent and then what value of symbol timing and phase offset, since maximum correlation means the maximum magnitude caused by the use of variable that caused result in the most probable match between y(t) and a locally generated basis function p(t, ⎛ ⎜ ⎝
⎞ ⎟ ⎠
λ (θ , εT ) = max⎜ ∫ [y (t − εT )e jθ p(t )] dt ⎟ εT ,θ
2
(5-8)
It turned out that for AWGN channels in which y(t) is only corrupted by AWGN, this can be simplified to ⎛ ⎜ ⎝
⎞ ⎟ ⎠
λ (θ , εT ) = max⎜ Re ∫ [y (t − εT )e jθ p(t )] dt ⎟ ε ,θ
2
(5-9)
To estimate (θ , εT ) we use the principle of decision aided ML synchronization loop [1]. The phase estimates are given by ⎛ ⎞ 2 d ⎜ Re ∫ y (t − εT )e jθ p(t ) dt ⎟ ⎜ ⎟ dλ (θ , εT ) ⎝ ⎠ =0 = dθ dθ
[
]
and the timing symbol errors will be given by ⎛ ⎞ 2 d ⎜ Re ∫ y (t − εT )e jθ p(t ) dt ⎟ ⎜ ⎟ dλ (θ , εT ) ⎝ ⎠ =0 = dεT dθ
[
]
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One way to obtain a timing error (εT ) is by using the early-late Gate correlator, which approximates the derivative above as dy y E − y L ≈ dτ τ E − τ L
(5-9)
where y L and y L are the early and late samples as shown in Figure 5-2. Clearly the differentiation of the ML function with respect to the timing error (εT ) had produced the earlier introduced early-late time detector that was used in Chapter 4.
dy y E − y L ≈ dτ τ E − τ L
yk
yE
yL
k
TE
TL
Figure 5- 2: Early-Late-Gate timing error detector derived from the ML principle
The Early-Late ML-based timing technique involves delaying the ML input signal in a buffer, then performing correlation of the current tentative decision symbol of pˆ (t ) with each one of the early and late samples of the MLE inputs as shown in Figure 5-3. This can be mathematically represented by ⎡ T + 12 T ⎤ T − 12 T 2 2 ⎢ ⎥ jθ jθ ˆ ˆ − Re y ( t T ) e p ( t ) dt y ( t T ) e p ( t ) dt ε − − ε ∫ ∫ ⎢ ⎥ 1 1 ⎥⎦ 0− 2 T dλ (θ , εT ) ⎢⎣ 0+ 2 T = dτ 2T
[
]
[
]
(5-10)
where T is the symbol timing period. The estimation procedure in (5-10) is illustrated in Figure 5-3. Here the ML principle of estimating the transmitted symbol issued first, then two consecutive samples of the input samples to the ML estimator are used to implement Dr. Mohamed Khalid Nezami © 2003 5-5
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the symbol timing error detector in (5-9). This error signal is then filtered using a loop filter similar to those detailed in Chapters 3 and 4. The filtered error signal is then passed to the analog to digital converter to correct its sampling instance.
Loop Filter
Z-1 Timing Generator
∑
−
∑
+
Σ
pˆ (t )
y(t)
Maxim um Likelihood Estimator
A/D
Data Decision
NCO
Figure 5- 3: MLE symbol timing Tracking implementation
5.5 Maximum Likelihood Estimation Lower Bounds Assuming that θˆ is the unbiased estimate in (5-1), a lower bound to the variance of the error in estimation θˆ(r ) − θ was derived by Cramer and Rao [11] to be CRB (θ ) =
1
(5-11)
⎧⎪ ⎡ d ln[ p (r / θ )]⎤ 2 ⎫⎪ E r ⎨⎢ ⎥⎦ ⎬⎪ dθ ⎪⎩ ⎣ ⎭
and equivalently represented as
CRB(θ ) =
1
⎧ d ln[ p (r / θ )]⎫ Er ⎨ ⎬ dθ 2 ⎩ ⎭ 2
[
≤ var θˆ(r ) − θ
]
(5-12)
. is the expectation of the enclosed argument with respect to the subscript where Er {} variable and p(r / θ ) is the conditional probability density function on r for a given value θ . The performance of different practical estimators is evaluated by comparing the estimated variance against this theoretical CRB. Dr. Mohamed Khalid Nezami © 2003 5-6
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In the case of multiple parameter estimation, the parameter set may be expanded into desired parameter θ and the undesired parameter u, then p (r / θ ) can be written as +∞
p(r / θ ) = ∫ p(r / u,θ )p(u )du
(5-13)
−∞
where p(r / u,θ ) is the conditional probability density function of r given u and θ . The evaluation of CRB using equation 5-11 and 5-13 is difficult, as the integral in equation 513 cannot be carried out analytically in most cases. To circumvent this problem, a modified version of CRB, viz., the Modified Cramer-Rao bound (MCRB) is given by [10]. MCRB(θ ) =
1
(5-14)
⎧⎪⎛ d ln[ p(r / u,θ )] ⎞ 2 ⎫⎪ E r ,u ⎨⎜ ⎟ ⎬ dθ ⎪⎩⎝ ⎠ ⎪⎭
where E r ,u { } is the expectation of the enclosed quantity with respect to the vector u containing the undesired variables. Although MCRB in (5-14) has the same structure as the CRB given in equation 5-11, it is easier to evaluate. As an example, the evaluation of the MCRB of the synchronization parameters associated with a phase-modulated signals in AWGN channel is demonstrated. Let the signal be given by
s(t ) = e j (2π∆f (t −t0 )+θ ) ∑ c k g (t − kT − εT )
(5-15)
k
where ∆f is an associated carrier frequency offset error, θ is the carrier phase offset, T is the symbol clock, {c k } are the complex data symbols, g (t ) is the pulse shape, t0 is an initial timing shift, and εT is the fractional symbol timing error. To compute MCRB for separate estimation of the synchronization parameters {∆f ,θ , εT } of the received signal s (t ) given in equation 5-19, first the vector containing the unwanted parameters is designated. It is assumed that {∆f ,θ , εT } are constant within the observation period, εT is uniformly distributed between {0 ≤ εT ≤ T }, and {ck } are zero-mean independent random variables. For estimating the lower bound of the carrier frequency offset, say MCRB(∆f ) , the unwanted parameters u ∆f are chosen as {θ , εT , c k }. Similarly, for phase
rotation lower bound MCRB(θ ) , the unwanted parameters uθ are {∆f , εT , c k }, and for symbol timing MCRB(εT ) , the unwanted parameters uεT are {∆f , εθ , c k } . The following example illustrates deriving the MCRB for carrier phase variance.
Assuming that the received (observed) signal in AWGN (5-15) is given by
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r (t ) = s(t ) + n(t )
(5-16)
where n(t ) is a complex-valued AWGN with a two-sided power spectral density 2N 0 .
Using (1-16), the conditional probability p(r / u,θ ) in equation 5-13 is replaced by the ML function [10] given by ⎧⎪ 1 l(θ , u ) = exp ⎨− ⎪⎩ 2 N 0
∫ L
⎫⎪ 2 r (t ) − s (t ) dt ⎬ ⎪⎭
(5-17)
where L is the observation interval. Equation 1-14 becomes MCRB(θ ) =
1
(5-18)
⎧⎪⎛ d ln[l(u,θ )] ⎞ 2 ⎫⎪ E n ,u ⎨⎜ ⎟ ⎬ dθ ⎪⎩⎝ ⎠ ⎪⎭
Here E n ,u {} . is the expectation over Gaussian noise and unwanted parameters. Substitution of equation (5-17) in equation (5-18) yields the phase estimate Cramer-Rao bound for the signal in (5-16) MCRB(θ ) =
N0
(5-19)
⎧⎪ ds (t ) 2 ⎫⎪ Euθ ⎨∫ dt ⎬ ⎪⎩ L dθ ⎪⎭
where the unwanted parameters vector is uθ = {∆f , εT , c k } . The integrand in equation (519) for the phase modulated signal (5-15) is given by
∫ L
ds(t ) 2 dt = ∫ m(t ) dt dθ L 2
(5-20)
where m(t ) = ∑ c k g (t − kT − εT )
(5-21)
k
The integration of the message m(t) in the integral of 5-20, is simplified to,
∫ m(t )
2
dt = 2 Es L
(5-22)
L
where E s is the energy per symbol. Finally, using equation 5-19, the MCRB of the phase estimate is obtained as
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MCRB (θ ) =
1 2 L( E s N 0 )
(5-23)
Clearly, lowering the MCRB(θ ) in 5-23 results in better carrier phase estimates (i.e., lower variance), which can be obtained either by increasing the operating symbol energy to noise ratio or by using longer observation intervals (i.e., by averaging over more symbols in equation 5-19). This lower bound can be used as a benchmark reference for practical implementation of synchronization algorithms. It can also be used to predict the synchronization limited performance range, based on the specifications of an overall link budget for the receiver. For instance, the minimum phase variance that can be achieved with an observation interval of 100 symbols and E s N 0 of 10 dB is 1 MCRB = = 5 x10 − 4 rad 2 . This value can be used to specify the smallest bit 2(100)(10) error rate degradation (also known as implementation loss) allowed due to the insertion of a non-ideal phase synchronizer into the digital receiver. The lower bound for joint estimation (i.e., phase, frequency, and symbol timing) is derived in [12,13,14,15] using the Fisher information matrix. If the estimated parameters are independent (i.e., each synchronization parameter can be estimated independent of the other synchronization parameters), the Fisher matrix will contain only diagonal elements, which then reduce to the values of CRB for each individual synchronization parameter. The CRB derived in equation 5-23 is only applicable for AWGN channels When considering other channel impairments such as fading and shadowing, the lower bound will rise [7].
5.6 Synchronization Error Impact on Receiver BER Performance
For high-speed multi-level modulation receivers, symbol rate clock is held at a very tight tolerance in terms of drift and variation. Typically the receiver clock is specified with a tolerance less than 0.1 to 0.2 parts per million (PPM) over a wide operating temperature range and years of service. Such tight tolerance makes clock tracking and correction a necessity. For instance, a practical QPSK receiver with 35% access bandwidth using a 15% symbol timing clock error, (typical frequency uncertainty of off-the-shelf crystal oscillator), will cause 1.5 dB of link deterioration for symbol-error-rate (SER) of 10-6; a 10% symbol timing error for 16-QAM signals will cause more than 1.7 dB of deterioration for the same SER. The presence of carrier frequency offsets of ∆f results in a drop of matched filter magnitude according to sinc2( 2π∆fT ), where T is the symbol rate. Such loss translates directly into BER deterioration [12]. For instance, with 10 kbps transmission, a 1 kHz carrier offset will cause 0.6 dB loss in Eb/No link budget. Frequency offsets are caused by Dr. Mohamed Khalid Nezami © 2003 5-9
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oscillator drifts in the transmitter and receiver local oscillators, as well as by the Doppler shift due to the relative movement of the mobile receiver with respect to the transmitter. Table 5-1 lists typical specified maximum frequency offsets for common wireless systems. For typical satellite and PCS voice transmission rates of 2400 bps or 9600 bps, the receiver may experience frequency offsets as large as the data rate being received. This is considered to be one of the critical issues in the design of TDMA receivers and is addressed in Chapter 8 and 9. Carrier phase rotations are caused by channel phase distortion or by the associated transmitter/receiver circuit component, such as phase noise in the local oscillators, phase distortion in the IF high Roll-off filters, and I/Q imbalances. The rotation causes the data symbols to rotate and move toward neighboring symbols, resulting in detection errors. For QPSK and 16-QAM signals, a 10o rotation of the carrier phase respectively causes a 1.7 dB and 2.0 dB reduction in receiver Es/No link budget. To incorporate the effect of synchronization errors into the overall transmission SNR link budget, and hence into the receiver design specifications, the variance of the synchronizer output estimates need to be related to the bit-error-rate (BER). BER degradation due to synchronization errors is defined as the increase in Eb/No (in dB) that one needs to obtain the same system BER performance when perfect synchronization occurs. This loss is also known as synchronizer implementation loss.
Transmission Type
Maximum Offset
Frequency Carrier Frequency
WIRELESS LAN
200 kHz
5000 MHz
MILSAT-UHF
2.4 kHz
200-400 MHz
PCS/CELL
200 Hz
800 MHz
PCS/CELL
300 kHz
1800-1900 MHz
LEO-SAT
62 kHz
2400 MHz
Table 5- 1: Frequency offsets experienced in various transmission systems. For carrier phase errors that are relatively small enough for their distribution to be Gaussian (as happens in a well-designed system), the BER degradation due to carrier phase error (in dB) [13] is given by
()
⎞ ⎛E Dθ = 4.3⎜⎜ s + 1⎟⎟ var θˆ ⎠ ⎝ N0
(5-24)
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and the BER degradation due to timing errors is given by
⎛ 2E s DT = 4.3⎜⎜ g ' ' (0)T 2 + N0 ⎝
∑ (g ' (kT )T ) k
2
⎞ ⎟⎟ var(εˆ ) ⎠
(5-25)
where εˆ is the estimated symbol timing, the term var(εˆ ) is the variance of the symbol timing error that is normalized by the symbol rate T, θˆ is the estimated carrier phase, and
g' (kT ) and g ' ' (kT ) are the first and second time derivatives of the transmitted pulse shape of the combined matched and bandwidth shaping filters. Moeneclaey in [13] computed the terms g' (kT ) and g ' ' (kT ) for a system using square-root cosine pulse with 1+α a bandwidth equal to Hz, where T is the symbol rate and α is the Roll-off value 2T used. Using the parameters in equation (5-25) (?) can be simplified to ⎛ E ⎞ DT = 4.3⎜⎜ A + s B ⎟⎟ var(εˆ ) N0 ⎠ ⎝
(5-26)
where, A and B are a functions of the Roll-off factor used. Table 5-2 lists A and B for commonly used Roll-off values.
Roll-off ( α ) 1.00 0.80 0.50 0.35
A 5.2 4.2 3.8 3.5
B 0.2 0.4 0.5 0.8
Table 5- 2: Values for A and B of (5-26) as a function of Roll-off factor used.
Using (5-26), the modified symbol error rate (SER) that includes the additional amount E of s to make up for synchronization errors for M-ary signals is given by N0 1 ⎞ ⎛⎜ 3 log 2 M ⎛ Pe = 4⎜1 − ⎟Q M ⎠ ⎜ ( M − 1) ⎝ ⎝
D ⎛ Eb ⎞⎞ 10 ⎟ ⎟ ⎜ − 10 ⎜N ⎟⎟ ⎝ o ⎠⎠
(5-27)
Using equations 5-24 through 5-27 the impact of symbol and carrier phase variances on the overall SER is depicted in Figures 5-4 to 5-8 for QAM and QPSK signals. It can be seen that with the QPSK receiver that has a carrier synchronizer with a error deviation of
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Es of 5 dB to retain the same SER performance as Pe = 10 −6 N0 E (Figure 5-4); for 16 QAM, a phase error of 5o requires an increase in s of more than N0 1.7 dB to retain the same SER (Figure 5-5).
15o requires an increase in
Symbol timing error impact on SER in equation (5-26) depends on the Roll-off factor of E the channel spectrum shaping filter and the operating s . A QPSK system using 35% of N0 access bandwidth and a symbol error of 10% of the symbol rate requires an increase in Es of 1.2 dB to retain the same SER performance (Figure 5-6). The impact is less for N0 larger access bandwidth, for instance using 80% access bandwidth, the loss is reduced to less than 1.0 dB (Figure 5-7). For QAM signals, the deterioration due to symbol timing error is even greater. For symbol timing errors of 15% of the symbol rate, SER is Es increases (Figure 5-8). The bounded by irreducible SER floor of 1.5x10-4 as N0 explanation is that the timing errors generate inter-symbol-interference (ISI), which, in turn, produces decision errors even in the absence of noise. Synchronization implementation losses can be expressed in terms of signal-to-noise ratio (SNR) and carrier-to-noise ratio (CNR) by using the following relations
SNR =
Eb log 2 M N0
(5-28)
CNR =
Eb + 10 log10 Rb N0
(5-29)
and
where Rb is the bit rate. By closely examining Figures 5-4 through 5-8, the following observations can be made: •
Es and variance of a synchronization parameter, larger constellation N0 E modulations (which need higher s to achieve a given SER) give rise to larger SER N0 degradation (i.e., QAM is more vulnerable than QPSK).
For given
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•
SER degradation as a result of symbol timing error depends on the access channel bandwidth used (i.e., Roll-off factor and SNR).
•
SER performance as a function of symbol timing varianc, is bounded by an irreducible floor even in the absence of noise.
The observations underline the severe adverse impact of synchronization errors on the receiver performance. Care must therefore be taken in designing and specifying synchronization parameters for such receivers. 10
0
QPSK
10
Pe
10
10
10
10
10
10
-1
-2
-3
-4
-5
No carrier errors
δ θ = 5o δ θ = 10o δ θ = 15o
-6
-7
2
4
8
6
10
12
14
Es/No
Figure 5- 4: Impact of carrier phase error on QPSK SER, δ θ = [0o,5o,8o,10o, 15o].
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10
10
16QAM
-1
-2
Pe
10
0
10
10
δ θ = 10 o
-3
δ θ = 5o
-4
No carrier errors
10
10
-5
-6
2
4
6
8
10
12
14
16
E s/N o
Figure 5- 5: Impact of carrier phase error on 16QAM SER, δ θ = [0o, 5o, 10o]. 0
10
QPSK 35% access Bandwidth
-1
10
-2
10
Pe
-3
10
-4
10
δ T = 0.2T
-5
10
δ T = 0.15T
-6
10
δ T = 0.10T
-7
10
2
4
6
8
10
12
14
Es/No
Figure 5- 6: Impact of symbol error on QPSK, δ T =[0T .1T .15T .20T], Roll-off=0. 35.
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0
10
QPSK 80% access Bandwidth
-1
10
-2
10
Pe
-3
10
-4
10
δ T = 0.2T
-5
10
-6
10
δ T = 0.10T
δ T = 0.15T
-7
10
10
8
6
4
2
Figure 1-5: QPSK SER with Symol Error (0
Es/No
.1
12
14
.15 .20 )T
Figure 5- 7: Impact of symbol error on QPSK systems, δ T =[0T .1T .15T .20T], Roll-off=0.80. 0
10
16QAM 35% access Bandwidth
-1
10
-2
10
Pe
-3
10
δ T = 0.15T
-4
10
δ T = 0.10T
-5
10
-6
10
-7
10
4
6
8
12
10
14
16
18
Figure 1-6: 16QAM SER with Symol Error (0.1 0.15)T Es/No
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Figure 5- 8: Impact of symbol error on 16QAM, δ T =[0T .1T .15T], Roll-off=0.35.
5.7 Equivalence Nature Between FF and FB Synchronization Systems
There is an equivalent relationship between FF and FB synchronization systems [1], which helps in utilizing some of the already available analytical methods of FB in analyzing FF loops. For example, the observation interval (L) in FF synchronizers is related to the loop bandwidth of the PLL synchronizer as 1 = 2 B LT L
(5-30)
where T is the symbol rate. For example, a PLL synchronizer with loop bandwidth of BL = 50 Hz used for carrier recovery in a receiver and operating with a data rate of 10 ksps ( T = 10 −4 ), has the same performance as a FF synchronizer using an observation interval of L = 100 symbols. Thus the design parameters of FF synchronizers can be chosen based on specifications that are derived from a feedback synchronizer. Using equation 5-30, both loops will have a minimum non-approachable phase estimate variance of δ θ2 = 5 x10 −4 , which will cause a minimum SER implementation loss of 0.25 dB according to equation 5-24. 5.8
References
1. L.E. Franks, “Carrier and bit synchronization in data communication - a tutorial review” IEEE Trans. Commun., vol. Com-28, No. 8, pp. 1107-112, Aug 1980. 2. Classen, H. Meyr, and P. Sehier, “Maximum likelihood open loop carrier synchronizer for digital radio”, Proceedings of ICC, 1993. 3. Pooi Yuen Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations” IEEE Trans. Commun., vol. 34, No. 6, pp. 520-527, Sept 1986. 4. Classen, H. Meyr, and P. Sehier, “Maximum likelihood open loop carrier synchronizer for digital radio”, Proceedings of ICC, 1993. 5. Surat White and Norman Beaulien, “On the application of Cramer-Rao and detection theory bounds to mean square error of symbol timing recovery” IEEE Trans. Commun. vol. 40, No. 10, pp. 1635-1643, Oct. 1992.
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6. Fulvio Gini, Marco Luise, and Ruggero Reggiannini, “Cramer-Rao bounds in the parametric estimation of fading radio transmission channels,” IEEE Trans, Commun. vol. 46, No. 10, pp. 1390-1398, October 1998. 7. R. Reggiannini, “A fundamental lower bound to the performance of phase estimators over Rician-fading channels,” IEEE Trans. Commun. vol. 45, No. 7, pp. 775-778, July 1997. 8. K.C. Ho, “Modified CRLB on the modulation parameters of PSK signal” Proceedings of ICC 1999, pp. 782-785. 9. Ruggero Reggiannini, “A fundamental bound to the performance of phase estimate over Rician-fading channels” IEEE Trans. Commun. vol. 45, No. 7, pp. 775-778, July 1997. 10. Aldo D'Andrea, Umberto Mengali, and Ruggero Reggiannini, “The modified Cramer-Rao bound and its applications to synchronization problems” IEEE Trans. Commun. vol. 42, No. 2/3/4, pp. 1391-1399, Feb/Mar/Apr 1994. 11. E. Dilaveroglu, “Simple expression for worst and best case Cramer-Rao bounds for amplitude and phase estimation of low frequency sinusoid,” Electronics Letters, vol. 35, No. 3, pp. 206-208, Feb 1999. 12. Norman Beaulieu, and Staurt White, “A lower bound on the mean square error of a symbol timing recovery for NRZ rectangular signals,” IEEE Trans. Commun. vol. 43, No. 7, pp. 2183-2183, July 1995. 13. Bucket, and Marc Moencely, “Effect of random carrier phase and timing errors on the detection of narrowband M-PSK and Bandlimited DS/SS M-PSK signals”, IEEE Trans. Commun., vol. 43, No. 2/3/4, Feb/Mar/Apr 1995.
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Chapter 6 Feedforward Symbol Timing Synchronization Algorithms and Mitigation of Fading Impact on Receiver This chapter derives the algorithmic details of prominent schemes for estimating feedforward symbol timing. First, we derive the maximum likelihood principle for timing synchronization and then we derive both data aided (DA) and non-data aided (NDA) symbol timing synchronization algorithms. Estimated timing parameters using both algorithms are then compared to the modified Cramer-Rao bound (MCRB) derived in Chapter 5. For receivers operating in mobile channels, we use fading channel simulation to study the performance of the discussed algorithms, then present methods for reducing their deterioration due to Doppler frequency offset and fading signal levels. 6.1 ML Feedforward Synchronization Principle Assuming that the received signal has no frequency offset, symbol timing ( ε ) and carrier phase offset ( θ ) can be jointly estimated using maximum likelihood (ML) criteria. The relevant ML function is given by P( r f / θ , ε ) = ∑ P( a ) P( r f / a,θ , ε )
(6-1)
a
where P ( r f / θ , ε ) is the conditional probability of receiving the matched filter (MF) signal r f on the random variable phase θ and symbol timing error ε . To estimate carrier phase separately, the dependency of the ML function in (6-1) is eliminated by averaging over ε . That is, ⎡ ⎤ P ( rf / θ ) = ∫ ⎢∑ P( a ) P( r f / a,θ , ε ) P(ε )⎥ dε ⎣a ⎦
(6-2)
Similarly, for estimating symbol timing separately, the ML function is averaged over the phase term θ to remove the dependency of the ML on carrier phase. That is ⎡ ⎤ P( r f / ε ) = ∫ ⎢∑ P( a ) P( r f / a,θ , ε ) P(θ )⎥ dθ ⎣a ⎦
(6-3)
Generally equations 6-2 and 6-3 do not have closed-form solutions. So, one must resort to approximations of the ML function to yield reasonable estimates.
The received signal r f ( kTs ) , sampled at the input of the matched filter [1],is given by Dr. Mohamed Khalid Nezami © 2003
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r f ( kTs ) = ∑ a n g (kTs − nT − ε 0T )e jθ + n(kTs ) A
(6-4)
n
where Ts is the sampling time and T is the symbol duration, g (kTs − nT − εT0 ) is the convolution of the channel response with the pre-filter impulse response having an associated delay of ε 0T and a phase
shift of e jθ , an is the complex data symbol, A is a scaling factor, and n (kTs ) A is a scaled version of AWGN. Defining a matched filter whose impulse response g MF (kTs ) is matched to the received signal, the MF output is written as z (nT + εT ) = ∑ r f (kTs )g MF (nT + εT − kTs )
(6-5)
k
By substituting equation (6-31), one obtains
⎡⎡ ⎤ n(kTs ) ⎤ z (nT + εT ) = ∑ ⎢ ⎢∑ a m g (kTs − mT − ε oT )e jθ + g MF (nT + εT − kTs )⎥ ⎥ A ⎦ k ⎣⎣ m ⎦
(6-6)
The matched filter output is used as the input to the synchronizer as shown in Figure 5-1. Using equation (6-6), a conditional probability of the received signal on the synchronization parameters and hence the ML equation can be obtained. This ML equation is then appropriately averaged for the estimation of individual synchronization parameter. The normalized likelihood function is then given by [1]
[
]
⎧ 1 ⎛ L −1 ⎞⎫ 2 P (rf / a,θ , ε ) ∝ exp ⎨− 2 ⎜ ∑ an − 2 Re an∗ zn (nT + εT ) e − jθ ⎟⎬ ⎠⎭ ⎩ σ n ⎝ n=0 −1
⎛E ⎞ where σ = ⎜⎜ s ⎟⎟ . The term ⎝ N0 ⎠ 2 n
L −1
discarded because
∑ n=0
L −1
L −1
∑
an
{
an
n=0
an → ∑ E 2
n=0
2
2
{
}
(6-7)
is independent of synchronization parameters and can be
}= constant. Approximated ML equation based on use of (6-
7) is given by,
⎧ 2 ⎤⎫ ⎡ L −1 l(a,θ , ε ) = exp ⎨− 2 Re ⎢∑ an∗ zn (ε )e − jθ ⎥ ⎬ ⎦⎭ ⎣n =0 ⎩ σn
(6-8)
Equation 6-8 can be used to estimate any synchronization parameter individually by eliminating the unwanted parameters through approximation and averaging per (6-2) and (6-3). Although (6-8) is based on the presence of only timing and phase synchronization errors, it could have easily been modified to include a frequency offset. In the ensuing derivation of the synchronization algorithms based on equation 6-8, the following assumptions are made: •
Ideal channel and no propagation effects are present.
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•
The parameters to be estimated are assumed to be either constants or slowly varying within the estimation interval.
•
The data pulses are Nyquist shaped.
•
The matched filter has a symmetric frequency response within the bandwidth of the received signal.
•
The combination of channel response and the pre-selector has a flat frequency response.
•
No inter symbol interference (ISI) is present.
6.2 Variances of Feedforward Symbol Timing Estimator Output
The MCRB for symbol timing was defined in [2]. For symbols that are shaped by raised cosine filter with Roll-off factor α , the MCRB for symbol timing estimation is given by, ⎛ ⎞ ⎟ T − εˆT ⎤ 1 ⎛ 1 ⎞⎜ 1 ⎟ ≤ var ⎡ ⎜⎜ ⎟⎟⎜ MCRB (εˆ ) = ⎢⎣ T ⎥⎦ 2 L ⎝ Es N 0 ⎠⎜ α 2π 2 + 1 π 2 − 8α 2 ⎟ ⎜ ⎟ 3 ⎝ ⎠
(6-9)
where Es/No is the operating symbol energy to noise ratio, and L is the estimation interval. Figure 6-1 plots (6-9) for symbol timing estimates as a function of observation interval L and Es/No. Figure 2-16 shows that the lower bound decreases infinitely by increasing Es/No. However, with the practical symbol timing synchronizer, variances reach an irreducible value for high Es N 0 ( E s N 0 ≥ 20dB ), where the variances become independent of Es N 0 and inversely proportional to the observation length L. The degradation is due to the self-noise generated by nonlinear processing of the matched filter output while extracting the symbol timing signal. For moderate Es N 0 ( E s N 0 ≥ 10dB ) the variance decreases inversely with the observation interval L and Es N 0 .
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10
-2
10
-4
MCRB
10
-6
L=50
10
L=100 L=500
-8
10
-10
10
5
0
10
15
25
20
30
35
40
45
50
Eb/No
Figure 6- 1: MCRB for symbol timing vs. Eb/N0 for observation intervals L=50, 100, and 500, with Rolloff = 0.75. 6.3 ML-Based Symbol Timing Estimation Algorithms Classification of ML-based synchronization emerges from the way data dependency in (6-8) is eliminated. When the data sequence is known, for example, a preamble sequence {a0 } is sent during initial TDMA burst (data aided (DA)) acquisition, only one term of the sum in (6-8) remains.) Hence, the estimation of ε or θ using (6-8) reduces to maximizing the likelihood function P (r f / a = a 0 ,θ , ε ) by ⎫ max (P (rf / a = a0 ,θ , ε ))⎬ ⎩ θ ,ε ⎭
(θ , ε )DA = arg ⎧⎨
(6-10)
If an estimate of {a0 } is used in equation (6-8), we speak of decision-directed (DD) synchronization algorithms; thus the estimates of ε or θ using (6-8) are obtained by, ⎫ max (P (rf / a = aˆ ,θ , ε ))⎬ ⎩ θ ,ε ⎭
(θ , ε )DD = arg ⎧⎨
(6-11)
where aˆ is an estimate of a . For high SNR, aˆ → a , DA and DD algorithms have the same performance. If the operation used in (6-8) to obtain estimates of ε or θ is averaged, we speak of Nondata aided algorithms (NDA).
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6.4 Data and Decision Aided Symbol Timing Estimation Algorithms With the aid of the detected preamble symbols given by aˆ , equation (6-8) is modified to represent the decision aided ML estimator,
⎧ 2 ⎡ L −1 ⎤⎫ l(aˆ , θ , ε ) = exp ⎨− 2 Re⎢∑ aˆ n∗ z n (ε )e − jθ ⎥ ⎬ ⎣ n =0 ⎦⎭ ⎩ σn
(6-12)
where n is the symbol number, and L is the observation length in symbols. Now, equation (6-12) can be reconfigured to yield a separate estimation process for phase and timing. To this end, define a new variable L −1
µ (ε ) = ∑ aˆ n∗ z n (ε )
(6-13)
n =0
which can be represented in the polar form as
µ (ε ) = µ (ε ) e j arg µ (ε )
(6-14)
Substituting equation (6-14), the final DA ML-equation becomes
[
]
⎫ ⎧ 2 ˆ l(aˆ , θ , ε ) = exp ⎨− 2 Re e − j (θ −arg µ (ε )) µ (ε ) ⎬ ⎭ ⎩ σn
(6-15)
Now DA symbol timing estimates is obtained by maximizing equation (6-15) over both phase and timing errors. That is
{
}
max{l( aˆ ,θ , ε )} = max{ µ (ε ) Re e − j (θ −arg µ (ε )) } θ ,ε
θ ,ε
(6-16)
By inspection of (6-16), timing estimate εˆ can be obtained independent of phase information by maximizing µ (ε ) , because e jθ = 1 . Thus εˆ is given by
⎛
L −1
⎞
n =0
⎠
εˆ = arg⎜⎜ max ∑ aˆ n∗ z n (ε ) ⎟⎟ ⎝
ε
(6-17)
The operation of the DA symbol timing estimation algorithm in (6-17) is illustrated below. Consider a PCS communication receiver having preamble symbols that consists of an alternating sequence of ones and zeros, or a = [1,0,1,0,1,0,.....1,0] , and assume that the received MF one-zero pattern sequence ( aˆ ) is the one given in Figure 6-2. Applying (6-17), Figure 6-2 shows the resultant signal based on the sampleby - sample operation using the DA algorithm in (6-17). By maximizing the result over ε , it becomes apparent that there is an error that is equal to 12 to 13 samples in lining up the two sequences, which correspond to the optimal symbol sampling instances of the received signal sequences, before entering the data decision detector. Dr. Mohamed Khalid Nezami © 2003
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2
5 4
1.5
3 1 2 1 volts
Volts
0.5 0
0 -1
-0.5
-2 -1 -3 -1.5 -2
-4
0
200
400
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800 1000 sample #
1200
1400
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-5
0
1000
2000
3000 4000 Sample #
5000
6000
7000
Figure 6- 2: Data aided preamble (left), and the received noisy preamble sequence for DA/DD symbol timing recovery. 5 4.5 4 3.5
Volts
3 2.5 2 1.5 1 0.5 0 2000
2200
2400
2600 sample #
2800
3000
Figure 6- 3: Output of Equation (6-17) indicating maximum values at optimal timing instances. 6.5 Spectral Line NDA Symbol Timing Estimation Algorithm The algorithm presented in the previous section is based on using data preambles, which must be detected reliably by the receiver prior to obtaining synchronization. To obtain an estimate that is independent of preamble, the ML function of equation (6-12) (with no frequency-offset error) is maximized over εT by eliminating its dependency on θ phase and data {a}. This is achieved by employing a magnitude nonlinear operation F (.) on the (cyclostationary) MF output signal z ( kT + εT ) . Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
The effect of the nonlinearity is to produce a spectral component at the symbol rate, from which symbol timing estimates are derived. Thus the resultant ML equation can be shown to be [1,15,17] L −1
l(ε ) = ∑ F ( z (kT + εT ) )
(6-18)
k =0
where kT is the sampling instant, and F ( z (kT ) ) is an appropriate nonlinearity performed on the MF output z(kT). An example of such nonlinearity is the magnitude square law detector. Substituting this nonlinearity into equation (6-18) one gets L −1
l(ε ) = ∑ z (kT + εT )
2
(6-19)
k =0
It can be shown that z (kT − εT ) is a cyclostationary signal [1], implying that it contains symbol periodicity statistics within reasonable observation interval. The symbol rate clock is estimated by solving for the value of ε , which maximizes equation (6-19) over the interval of L symbols and the timing offset range of [− T 2, T 2] . According to Poisson's theory [1,229,241], any cyclostationary signal can be represented by a Fourier.Applying this using the ML in (6-19) produces, 2
L −1
z (kT + εT ) = 2
∑C e
k =− L
j 2πkε
(6-20)
k
The Fourier coefficients C k are random variables defined by 1
Ck = ∫ z (kT + εT ) e − j 2πkε dε 2
(6-21)
0
Maximizing equation (6-19) now is equivalent to maximizing its Fourier series. That is ⎛ L −1 ⎞ ⎛ ⎞ max⎜ ∑ Ck e j 2πkε ⎟ = max⎜ C0 + 2 Re C1e j 2πε + ∑ 2 Re Ck e j 2πkε ⎟ k ≥2 ⎝ k =0 ⎠ ⎝ ⎠
{
}
{
}
(6-22)
Observing that C0 is not a function of timing estimate, and treating the sum of the higher order terms
∑ 2 Re{C e k ≥2
k
j 2πkε
} as disturbances, the maximization is only carried out over the second term coefficients
C1 . Thus optimum symbol timing estimate εˆ is obtained as
εˆ = −
T arg (C1 ) 2π
(6-23)
where arg(C1 ) is the phase angle of C1 , which is defined as,
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(6-24)
Generalizing this to any nonlinearity F ( z (kT ) ) , the non-data aided timing algorithm (NDA) estimate is given by
εˆ =
2π −j k⎤ ⎡ L −1 −T arg ⎢∑ F ( z (kT ) )e N ⎥ 2π ⎣ k =0 ⎦
(6-25)
where N is the ratio of samples to symbols (over-sampling factor). For N=4, equation (6-25) simplifies −j
2π k 4
is either ± 1 or ± − 1 . This simplifies implementation with to a multiplication free form because e DSP processors and hence is commonly employed in commercial systems. Because of F ( z (kT ) ) in (6-25), the carrier frequency offsets of the received signal do not have any effect on symbol timing estimation. This can be easily proven for PSK signals. Consider the output of a BPSK modulator given by, sT ( t ) =
∞
∑a
n
g T (t − nT )
(6-26)
n = −∞
where { a n } are the transmitted BPSK data symbols. The signal after frequency offset insertion can be represented by sT ( t ) =
∞
∑a
n
g T (t − nT )e − j 2π∆ft
(6-27)
n = −∞
when performing symbol timing estimation using (6-250) using the nonlinearity z (kT ) yields, 2
sT (t − εT ) = 2
2
∞
∑a g n
T
(t − nT − εT )e
− j 2π∆ft
(6-28)
n = −∞
=
∞
∑ a h(t − nT − εT )
2
n
n = −∞
which is independent of the frequency offset. Hence, timing estimates in (6-25) are independent of frequency offsets.
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6.6 DFT-Based NDA Symbol Timing Estimation Due to the magnitude nonlinearity operation z (kT − εT ) in (6-25), any frequency offset associated with
the matched filter signal z (kT ) would be lost, as
z (kT − εT )e − j 2π∆f = z (kT − εT ) ˆ
(6-29)
To preserve frequency offset information, other means of achieving data independence in equation (618) can be used. One way is to use z (t ) z ∗ (t − T ) instead of z (t ) in equation (6-25). This nonlinearity is referred to as the delay-conjugate-multiply nonlinearity (DELCONJ). Using DELCONJ non-linearity, timing estimates can be extracted from the MF-signal z(t ) by performing the differential operator z (t )z ∗ (t − T ) , which will extract line spectra at the symbol rate. Unlike the component created by using the magnitude nonlinearities detailed above, the output of this nonlinearity is a complex signal, whose DFT representation has a DC term and two strong spectral components. Since the DELCONJ nonlinearity output is affected by carrier frequency offsets, this effect makes it possible to explore both spectral terms to obtain an estimate of symbol timing and frequency offsets as joint estimation. Simulations indicate that the algorithm has higher sensitivity to fading and frequency offset residuals. The symbol timing estimates is also noisier, since the DELCONJ nonlinearity is viewed as a squaring operator that enhances the background noise by causing doubling of the noise spectrum. Assuming that the matched filter output signal in Figure 6-6 is given by ⎡ ⎤ z (t ) = ⎢m(t )e jθt ∑ a n g (t − nT ) + n(t )⎥ e j 2π∆ft n ⎣ ⎦
(6-30)
where m(t )e jθt is the channel gain with varying phase, n( t ) is AWGN, ∆f is the carrier frequency offset, g (t − nT ) is the pulse shape of the combined transmitter and receiver matched filters, and {a n } are the complex data symbols. Assuming that the channel changes little between symbols (i.e., negligible channel phase and gain variation), the output of the DELCONJ nonlinearity is given by z (t )z ∗ (t − T ) = m( t )e jθt e j 2π∆fkT σ a2 ∑ g (t − nT )g ∗ (t − T − nT ) + G (.) 2
(6-31)
n
where G (.) is a non-periodic function not of interest and σ a2 = E
{
an
2
} is the averaged data.
Following the analysis of equations (6-20) and (6-31), z (t )z ∗ (t − T ) contains a spectral component at the symbol rate f = ±1 / T , whose phase is used to estimate both ∆f and optimum sampling instants
kT + εT . The term m(t )e jθt is a narrowband process (i.e., variation is low compared to symbol rate), which causes smearing of the recovered symbol rate signal from its nominal frequency Its analysis is presented in the subsequence chapters. Ignoring the DC term and treating the 2nd and higher order terms in equation (6-30) as disturbances, the two coefficients ( C ±1 ) are defined as Dr. Mohamed Khalid Nezami © 2003
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C −1 = ∑ z (kT )z
∗
((k − 1)T )e
−j
2πkn N
(6-32)
k
and C +1 = ∑ z (kT )z ∗ ((k − 1)T )e
j
2πnk N
(6-33)
k
When no frequency offset is present and time dispersion of the channel is small, the terms C−1 and C+1 can be used to obtain symbol-timing information. That is
εˆ =
2πkn ⎛⎡ −j ⎤⎞ −T arg⎜ ⎢∑ z (kT )z ∗ ((k − 1)T )e N ⎥ ⎟ ⎟ ⎜ k 2π ⎦⎠ ⎝⎣
(6-34)
A quick inspection of the algorithm (6-34) reveals that it is identical to (6-25), except that it uses a DELCONJ nonlinearity instead of absolute value based nonlinearities. When frequency offset is present, the frequency estimation process has to be performed to correct the signal prior to symbol timing estimation. Scott and Olasz [23] derived an algorithm for recovering frequency offsets ∆f independent of symbol timing given by ∆f =
1 arg{C −1C +1 } 4πT
(6-35)
Figure 6-4 shows one proposed implementation of this algorithm, which yields a frequency offset and symbol timing estimates based on equations (6-34) and (6-35) respectively. After the offset ∆f in (6-35) is corrected (see Figure 6-4), the symbol timing offset is obtained as shown in Figure 6-4 by computing
εˆ =
Dr. Mohamed Khalid Nezami © 2003
(
−T ˆ arg C −1C +1e − j 2π∆fkT 2π
)
6-10
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e
Matched filter
+j
2πk N
X
∑( )
X
∑( )
L
z (kT )
BPF
X
Received Signal Z
−1
()
z ((k − 1)T )
L
∗
e
Complex Signal Real Signal
−j
2πk N
C+1
C−1 X
1 arg ( 4πT
∆f
)
e− j2π∆f
εˆT
−T arg( 2π
)
X
Figure 6- 4: DFT-based algorithm for symbol and frequency offset estimation. Notice that the algorithm is an implementation of the Fourier transform of z (t )z ∗ (t − T ) evaluated at a single frequency bin of ± 1 / T corresponding to the symbol rate. Using the computer model in Chapter 2, the algorithm will not yield acceptable performance unless the matched filter output signal is postprocessed by a bandpass filter. The filter bandwidth depends on the Roll-off factor ( α ) . The filter should be designed with a center frequency at the spectral component of interest in z ( kT ) , which contributes to the periodic component of interest in z ∗ (( k − 1)T ) z (kT ) . Roughly, the BPF should have a center frequency of 1 / 2T Hz. This is not squaring since the DELCONJ can be viewed as a squarer; components at 1/2T will be placed at 1/T. 6.6.1
Removing Dependency on Frequency Offset
During the analyses of the DELCONJ timing algorithm in (6-34), it became apparent that the symbol timing error offsets were estimated in a similar fashion to (6-25), yet independent of frequency information. By taking the DFT of the DELCONJ nonlinearity output ( ℑ{z ∗ (( k − 1)T ) z (kT )} ) on a burst-by-burst basis - but evaluated over a range of bins that extend over the frequency offset range near f = ±1 / T (designated as C+1 and C−1 in equations (6-32) and (6-33)) - symbol timing estimate can then be obtained in the frequency domain and independent of frequency offset. If frequency offset is present, −1 +1 both C+1 and C−1 components are miss-located at fˆC−1 = + ∆f and fˆC+1 = + ∆f . By taking the T T difference of the phases ( φC+1 − φC−1 ), where φC −1 = −2πεfˆC +1 + 2π∆fε + θ and φC +1 = 2πεfˆC +1 + 2π∆fε + θ , then substituting the values of fˆ and fˆ , one obtains, C +1
φC = −2πε ( +1
C −1
+1 + ∆f ) + 2π∆fε + θ T
(6-36)
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φC = −2πε ( −1
−1 + ∆f ) + 2π∆fε + θ T
(6-37)
Taking the difference of (6-36) and (6-37), the symbol timing error ε is obtained as
ε=
1 2π
⎛ φC +1 − φC −1 ⎜⎜ 2 ⎝
⎞ ⎟⎟ ⎠
(6-38)
which is independent of frequency offset ∆f present in the MF output.
6.7 Impact of Nonlinearity Type on Feedforward Symbol Timing Estimation
The effectiveness of the nonlinearity used in NDA symbol timing estimation under various channel conditions is often reflected by: •
Strength of the PSD peak at symbol rate,
•
Noise enhancement and ‘Self-Noise’ generation and
•
Frequency offset effects.
The nonlinearity F ( z (kT + εT ) ) in (6-25) and z (t )z ∗ (t − T ) in (6-34) are chosen to extract the periodical spectral line, from which symbol timing estimates are obtained. Several other nonlinearities were reported in the literature. As an example, the square-law rectifier (SLR) given by F ( z (kT ) ) = z I (kT ) + jzQ (kT )
2
(6-39)
Another nonlinearity example is the absolute-value rectifier (AVR) given by F ( z (kT ) ) = z I (kT ) + jzQ (kT )
(6-40)
where z I (kT ) and z Q (kT ) are the in-phase and quadrature phase components of the MF output z (kT ) . It has been shown that the AVR nonlinearity provides preferable jitter performance for both static and Rayleigh channels. Similar results were reported by D’Andrea and Mengali in [24]. In addition, another nonlineairty was investigated using fourth-law rectifier (FLR) nonlinearity defined as F ( z (kT ) ) = z I (kT ) + jzQ (kT )
4
(6-41)
which has excellent performance for systems operating with small Roll-off factors of less than 0.20. Another magnitude-based nonlinearity for QPSK systems given by
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
F ( z (kT ) ) = z I (kT ) + zQ (kT )
(6-42)
This nonlinearity uses three absolute operations which save a number of computational steps over (642). Figure 6-5 through Figure 6-9 present the PSD plots for the output of the non-linearity z (kT ) , z (kT ) , z (kT ) , and z (kT )z ∗ (( k − 1)T ) for Roll-off values of 0.75 and 0.35 and a 16QAM signal with 2
4
10ksps. The figure indicates that among the nonlinearities considered, the AVR nonlinearity provides the largest SNR of the symbol rate component as indicated by the spectral peak at the symbol rate. Furthermore, these simulations confirm that the background interference power depends on the type of nonlinearity and the Roll-off value used. Higher Roll-off factors yield higher SNR. 50
Power Spectrum Magnitude (dB)
40 30
|z(kT)|
DC- term
roll-off=0.75 16QAM
20 10 0 -10 -20 -30 0
0.2
0.4
0.6
0.8
1 1.2 Frequency
1.4
1.6
1.8
2 x 10
4
Figure 6- 5: PSD of AVR nonlinearity with Roll-off = 0.75. 50 40 30 DC- term
|z(kT)|
20
roll-off=0.35 16QAM
10 0 -10 -20 -30 -40
0
0.2
0.4
0.6
0.8
1
1.2
Frequency
1.4
1.6
1.8
2 4
x 10
Figure 6- 6: PSD of AVR nonlinearity with Roll-off = 0.35. Dr. Mohamed Khalid Nezami © 2003
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DC- term
|z(kT)|2 30 roll-off=0.75 16QAM
20 10 0 -10 -20 -30 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2 4
Frequency
x 10
Figure 6- 7: PSD of SLR nonlinearity with Roll-off = 0.75. 10
0
Delconj z(kT)z*((k-1)T)
DC- term
roll-off=.75 16QAM
-10
-20
-30
-40
-50
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency
1.6
1.8
2 4
x 10
Figure 6- 8: PSD of DELCONJ nonlinearity with Roll-off = 0.75.
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|z(kT)|4
40
roll-off=0.75 16QAM
30 20 10 0 -10 -20 -30 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency
1.6
1.8
2 4
x 10
Figure 6- 9: PSD of FLR nonlinearity with Roll-off = 0.75.
6.8 Impact of Roll-off Factor on Feedforward Symbol Timing Estimation
The magnitude of the spectral line obtained by the nonlinearity operation is strongly dependent on the Roll-off factor used. Let q(t) be the raised-cosine shaped signal at MF, and then assume that the transmission channel is flat and has a constant gain of γ ch . The representation of q(t) after propagating 1 through the channel γ ch (t ) is represented in the frequency domain by the DFT at spectral lines f = ± 2T that are relevant for symbol timing after squaring or using the AVR. Q( f +
α 1 πfT ⎤ T⎡ ) = ⎢1 − sin( )⎥ for 0 ≤ f ≤ 2T 2⎣ α ⎦ 2T
(6-43)
Q( f −
T⎡ πfT ⎤ 1 α ) = ⎢1 + sin( )⎥ for 0 ≤ f ≤ α ⎦ 2T 2⎣ 2T
(6-44)
and
Using (6-43,44), the signal power out of the nonlinearity as a function of Roll-off factor is given by [23]
A( f ) =
2σ a2 γ ch T
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f=
α 2T
∫
f =0
Q( f +
1 1 )Q ( f − ) cos( 2πfT )df 2T 2T
(6-45)
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Performing the integration in (6-45) for different Roll-off factors, the impact of the pulse Roll-off on the magnitude of the periodic signal is analyzed. Equation (6-45) shows that the spectral line strength is proportional to: •
Symbol variance σ a2 .
•
Channel gain γ ch .
•
Gain of Q ( f ) .
•
Roll-off factor α .
Using the M-QAM simulation model in chapter 2, a strip of random 8000 symbols are used to study the effect of Roll-off factor on the strength of the recovered symbol rate signal. Studies are conducted using 2 4 nonlinearities of z (kT ) , z (kT ) , z (kT ) , and z (kT )z ∗ (( k − 1)T ) for 16-QAM and 4-QAM (QPSK). Figure 6-10 shows the amplitude of the symbol rate signal recovered using the indicated nonlinearities as a function of Roll-off factor for 16-QAM. For reference, the analytical expression of (6-45) is also plotted (Scott [23]). The figure shows that AVR has the best performance. The figure also shows that SLR and DELCONJ have the same performance, while FLR nonlinearity is the worst. Figure 6-11 presents the results of the study conducted for QPSK. Both figures indicate a slight discrepancy between the analytical results of (6-45) and the experimental results obtained by simulations. This discrepancy is due to the difference in the implementation of the RCF using FIR techniques, and the use of a constant bandwidth when measuring the spectral line power that is a function of Roll-off factor. Nevertheless, it is used to observe the impact of roll-off factor on the strength and quality of the extracted spectral line. 1
0.9
16-QAM
Relative Spectral Peak streangth
0.8
0.7
Sscott [127]
0.6
0.5 0.4
0.3
AVR
0.2
FLR SLR & DELCONJ
0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Roll-Off Factor
Figure 6- 10: Relative spectral peak strength vs. Roll-off for 16-QAM.
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0.9
QPSK
Relative Spectral Peak streangth
0.8 Sscott [127]
0.7
0.6
0.5
AVR
0.4
0.3 SLR & DELCONJ
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Roll-Off Factor
0.6
0.7
0.8
0.9
1
Figure 6- 11: Relative spectral peak strength vs. Roll-off for QPSK.
6.9 Feedforward Symbol Timing Correction using Interpolators After the symbol timing information εˆ is obtained, interpolator filters use this to generate the sample strobe instant that corresponding (??) to the maximum eye opening. The corrected re-sampled signal is then used for data decision after both carrier phase and offset have been estimated as shown in Figure 512. The interpolator is implemented using FIR interpolating filters [263,256], whose coefficients are adapted according to the estimated timing error. rf (kT )
Re {.}
MF
Im {.}
Timing estimators
Interpolator
Delay
εˆ
I
Decimator
Q
Estimate postprocessing
Figure 6- 12: Computer model for symbol timing estimation and correction. Gardner [26,27] reported the performance of second and fourth degree polynomials for MPSK modems. The interpolation performance is measured in terms of timing jitter, BER deterioration, and pulse distortion (amplitude and frequency). Gardner showed that using a perfect synchronizer and an oversampling factor of at least 2.5 linear interpolator introduces only 0.1 dB of BER deterioration for MPSK. Phase and amplitude distortion by interpolators can be compensated for by using several techniques reported in [27]. Implementation of this polynomial filter takes two steps. First, for the determined polynomial order, the variable FIR filter coefficients are computed online (or by table look-up) based on the symbol timing error estimate ( εˆ ) update. Then, the corrected sequence is obtained as the output of the interpolating filter. The polynomial coefficients h(n) are obtained using LaGrange interpolation [27] formulas given by Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers N
h(n ) = ∏ k =0 k ≠n
εˆ − k
(6-46)
n−k
where, N+1=L is the length of the filter, and εˆ is the fractional timing error. For linear interpolation, where N=1 and L=2, the interpolator FIR coefficients are given by
h(0) = 1 − εˆ
(6-47)
h(1) = εˆ
(6-48)
With samples into the interpolating filter at T/N, the interpolated data point xint (k ) is given by x int ( k ) = x ( kT + εT ) = h (0)x ( kT ) + h (1)x ( kT − T )
(6-49)
x int = (1 − εˆ )x ( kT ) + εˆx ( kT − T )
(6-50)
which reduces to
Using (6-50), the system function of the interpolator filter is given by X int (z ) = (1 − εˆ )z −1 + εˆ X (z )
(6-51)
Figure 6-13 shows an implementation of (6-51) using FIR topology, where εˆ is fed from symbol timing estimator.
x(kT )
Resampling at (k + εˆ)T
L-symbol delay z
Symbol timing recovery
εˆ
-
−1
+
x((k + εˆ)T ) X
+
Figure 6- 13: First order interpolator for symbol timing correction. Higher order polynomials can be developed accordingly; for example, the interpolator samples using a cubic polynomial is given by xint (k ) = h(3) x(k − 3) + h(2) x(k − 2) + h(1) x(k − 1) + h(0) x(k )
(6-52)
where the coefficients are given by [1] Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
h(0 ) = −
1 (εˆ − 1)(εˆ − 2)(εˆ − 3) 6
(6-53)
1 h(1) = − εˆ (εˆ − 2 )(εˆ − 3) 2
(6-54)
1 h(2 ) = − εˆ(εˆ − 1)(εˆ − 3) 2
(6-55)
h(3) =
(6-56)
1 εˆ (εˆ − 1)(εˆ − 2 ) 6
One possible implementation of the cubic interpolator for symbol error correction according to (6-52) is shown in Figure 6-14. Though the cubic interpolator has higher computational load than the linear interpolator, the interpolated output sample is more accurate. Cubic interpolator has flatter frequency response as it uses four points to derive the new sample. The polynomial interpolating filters tend to induce waveform distortion due to their nonlinear phase characteristics and amplitude attenuation. This is in addition to the Gibbs phenomenon of deterioration of magnitude response close to the Nyquist frequencies. Amplitude distortion results directly in receiver SNR deterioration, while frequency and phase distortion result in inaccurate symbol re-sampling that lead to SNR reduction. x(kT) 1/6
0 z
+
-1/6
1/2
z
+
-1
z
-1/6
x
εˆ
+
+
1/2
−1
z
+
1
−1
−1
z
+ x
0
−1
-1/2
z
+
z
+ z
−1
−1
+
1
−1
z
−1
z
+
1/2
−1
+
−1
z
+
-1/2
0
−1
−1
+
-1/3
z
+ x
++
0
x((k + εˆ)T)
Figure 6- 14: Cubic interpolator for symbol timing correction.
Figure 6-15 shows phase delay responses of the LaGrange interpolators with linear and cubic 1 polynomials. When εˆ is closer to 0 or ± the delay response is flatter. Figure 6-16 presents amplitude 2 versus frequency and delay distortion responses as a function of εˆ for the linear interpolator, and confirms that filters provide excellent approximation at low frequencies. However, the approximation bandwidth grows as εˆ increases. It is usually a good design practice to use an input signal that is at least over-sampled ten times into these interpolators. Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
0.5
0.4
0.3
0.2
0.1
ε = 0.5 ε = 0.47368 ε = 0.44737 ε = 0.42105 ε = 0.39474 ε = 0.36842 ε = 0.34211 ε = 0.31579 ε = 0.28947 ε = 0.26316 ε = 0.23684 ε = 0.21053 ε = 0.18421 ε = 0.15789 ε = 0.13158 ε = 0.10526 ε = 0.078947 ε = 0.052632 ε = 0.026316
Normalized PHASE DELAY ε
Normalized PHASE DELAY ε
0.6 FIR Linear Interpolator
0.5
0.4
0.3
0.2
0.1
0
-0.1 0
FIR Cubic Interpolator
ε= 0.5 ε = 0.47368 ε = 0.44737 ε = 0.42105 ε = 0.39474 ε = 0.36842 ε = 0.34211 ε = 0.31579 ε = 0.28947 ε = 0.26316 ε = 0.23684 ε = 0.21053 ε = 0.18421 ε = 0.15789 ε = 0.13158 ε = 0.10526 ε = 0.078947 ε = 0.052632 ε = 0.026316
0
0.2
0.4 0.6 0.8 NORMALIZED FREQ.
1
0
0.2
0.4 0.6 0.8 NORMALIZED FREQ.
1
Figure 6- 15: Variation of interpolator delay responses with ε. 1
ε =.5
0.9
MAGNITUDE RESPONSE
0.8
X int ( jw, ε ) X ( jw, ε )
0.7 0.6
Linear Interpolator steps of ε =0.0260
0.5 0.4 0.3 0.2 0.1
ε =.026 0
0
0.2
0.4 0.6 NORMALIZED FREQ.
0.8
1
Figure 6- 16: Variation of amplitude response of linear interpolator with ε. Based on the simulation work by Gardner [26,27], several desirable features for cubic interpolators are identified: •
Cubic interpolator frequency response features broad nulls that are centered at the harmonics of the sampling frequency (Fs, 2Fs, …, nFs), which coincides with spectral images of the input sample sequence. Therefore stopband attenuation is concentrated where it is needed most. The nulls also are much wider than those of linear interpolator.
•
The main lobe of the cubic interpolator is relatively flat over a frequency range wider than that of linear interpolators. The attenuation at half the symbol frequency is only 0.6 dB [26].
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•
Sidelobes were down by –30 dB improving receiver selectivity.
•
The magnitude response never exceeds unity when the delay is near half the filter length, thus avoiding the need for normalization networks.
Lakso [28] reported a detailed comparison of several polynomials based on FIR and all pass filters. Zhang [29] proposed a different implementation of the cubic interpolator, which yields 1 dB improvement in receiver BER performance. Further, the DC gain of the interpolator is independent of symbol timing error, and the even symmetry of the coefficients that allow simpler DSP implementation. Kirke reported a family of higher order interpolators suitable for use in variable-rate receivers [30]. Another approach is to combine a symbol timing interpolator filter with a decimating filter [31]. 6.10
Performance of NDA Symbol Timing Estimation in AWGN
The model in Chapter 2 was used to test the ML symbol-timing estimator of (6-25) and (6-34) for different nonlinearities and under different channel conditions. For experimentation, an intentional symbol timing error with ε = 0.25 is inserted into M-QAM symbol stream using the error insertion network shown in Figure 2-24 A Roll-off factor of 0.75 and observation interval of 400 samples (or L=100 symbols) were used. The NDA timing estimates (as per (6-25)) for the two cases of Eb N o = 40 dB and Eb N o = 6 dB, are presented in Figures 6-17a and 6-17b for a strip of 4000 samples (or 1000 symbols). Figure 6-17c illustrate the symbol timing estimation for smaller roll-off of α =0.35. a) Roll-off=0.75 Eb/No =40 dB
0.4 0.3 0.2 0.1 0
500
1000
1500
2000
2500
0.4
3000
3500
4000
b) Roll-off=0.75 Eb/No =6 dB
0.3 0.2 0.1 0
500
1000
1500
2000
2500
3000
3500
4000
c) Roll-off=0.35 Eb/No =40 dB
0.4 0.3 0.2 0.1
0
500
1000
1500
2000
2500
3000
3500
4000
Index sample
Figure 6- 17: NDA timing estimates as function of Eb/No and Roll-off.
Based on the results of Figure 6-17, it is clear that the mean value of the estimates converges to the inserted timing error (estimator is unbiased). The accuracy of the estimation improves with Eb N o , while the accuracy of the estimation improves with wider bandwidth (larger Roll-off).
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Through exhaustive simulations the timing algorithm performance is characterized using the statistical means and variances of the estimators. Figure 6-18 presents the behavior of the timing estimator variance with Eb N o for different nonlinearities. The results correspond to a Roll-off of 0.75, observation interval of L=100 symbols and a stretch of 8000 symbols. These variance plots can be used to predict the overall BER degradation of the receiver as a result of the non-ideal synchronizer, as indicated in Chapter 5. -1
10
-2
10
|z(kT)|4 -3
Variance
10
z(kT)| z*((k-1)T)|
-4
10
|z(kT)|2 -5
10
|z(kT)|
-6
10
0
10
20
30
40
Eb/No
Figure 6- 18: Timing estimate variances vs. Eb/No for different nonlinearities with Roll-off = 0.75 and L = 100. The following inferences can be made from the variance plots in Figure 6-18: •
Absolute value nonlinearity yields the most accurate estimation, followed by square law nonlinearity and DELCONJ nonlinearity. The results are in accordance with those presented in Figure 2-26.
•
Estimates are unbiased.
•
The variance improves with Eb N o up to Eb N o = 25 dB.
•
The variance approaches an irreducible value after Eb N o =25 dB.
•
Similar performance is observed for DELCONJ and square law magnitude non-linearity, which is in accordance with the results shown in Figure 6-10 and Figure 6-11.
The experiments in Figure 18 repeated symbol timing estimates for different observation lengths and Roll-off factors. Figure 6-19 shows the variance performance of the absolute value nonlinearity for Rolloff factors of 0.75 and 0.35. As predicted, the performance degrades for smaller Roll-off factors. Figure 6-20 shows the variance performance for observation lengths of L=50, L=100, and L=500 symbols showing that the variance improves with L, which is in accordance with (6-36). Dr. Mohamed Khalid Nezami © 2003
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Figure 6-21 shows the symbol timing estimation comparison between QPSK and 16-QAM. The figure indicates that QPSK and 16-QAM NDA symbol timing recovery have similar performance for moderate Eb N o values. However, for large Eb N o values, QPSK symbol timing estimation outperforms 16QAM. The reason lies in the nonlinearity generated self-noise [1], as QPSK has less self-noise than 16QAM signalsbecause it has fewer constellation points and smaller envelope variation. 10
Variance
10
10
-1
-2
-3
α=0.35 10
-4
α=0.75 10
-5
0
5
10
15
20
25
30
35
40
Eb/No
Figure 6- 19: Timing estimate variances vs. Eb/No for AVR nonlinearity with Roll-off = 0.75 , 0.35, and L = 100. -1
10
-2
10
-3
Variance
10
-4
10
L=50 -5
L=100
10
L=500
-6
10
0
10
20
30
40
50
Eb/No
Figure 6- 20: Timing estimate variances vs. Eb/N0 for AVR nonlinearity with Roll-off = 0.75, and L = 50, 100, and 500.
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-1
QPSK/16QAM 10
Variance
10
10
-2
-3
-4
QPSK
10
-5
16-QAm
10
-6
0
10
20
30
40
Eb/No
Figure 6- 21: Timing estimate variances vs. Eb/N0 for QPSK vs. 16 QAM with Roll-off = 0.75, and L = 100. 6.11
Performance of NDA Timing Algorithms in Presence of Frequency Offset
NDA algorithms that use absolute value nonlinearities (AVR, SLR, and FLR) in (6-25) are immune to frequency offset in the matched filter output as demonstrated by (6-29). However, NDA algorithms that use the DELCONJ nonlinearity (6-34) degrade with frequency offsets present in the matched filter output. Figure 2-22 shows the timing error variance performance for recovering a 1/4T intentional symbol error inserted into a strip of 8000 symbols at 10ksps that has –1000 Hz frequency offset (10% clock drift). The results show that the algorithm has higher variance as a result of the frequency offset. This impact limits the use of DELCONJ-based NDA algorithms to cases where the frequency offset is accurately estimated and corrected prior to symbol timing recovery. To use this estimate, the system in Figure 6-4 is used where this frequency is estimated using the algorithm in (6-35) before using the time estimator (6-34). -1
10
16QAM Roll-off =0.75
-2
10
DELCONJ, with freq. offset
-3
variance
10
-4
10
DELCONJ, without freq. offset
-5
10
-6
10
MCRB
-7
10
0
10
20
30
40
Eb /No dB Dr. Mohamed Khalid Nezami © 2003
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Figure 6- 22: Effect of frequency offset on DELCONJ based estimates with Roll-off = 0.75, and L = 100.
6.12
Impact of Fading Channels on Feedforward Symbol Timing synchronization
A Flat fading channel model is introduced here to characterize the performance of symbol timing (6-25) and (6-35). Due to reflections, scattering and diffraction of the transmitted signal in a mobile channel, multiple versions of the transmitted signal arrive at the receiver with different amplitudes and phases. This time dispersion of the transmitted signal results in either flat or frequency selective fading. In this section, we consider the impact of flat fading only on symbol timing recovery.
6.12.1 Flat Fading Channel Model Flat fading channel has a constant gain and linear phase response over a bandwidth greater than that of the transmitted signal [16-22]. In a flat fading channel, the spectral characteristics of the transmitted signal are preserved, but the received signal strength changes in magnitude with time. By considering that the mobile receiver is moving at a velocity of v and the incoming signal arriving at an angle ϕ with respect to the motion of the mobile unit, the received signal can be represented as S r (t ) = Re{ Ae j ( 2πf c t − βx cos ϕ ) }
(6-57)
where β is the wave number ( = 2π / λ , where λ is the wavelength), x is the displacement, A is an amplitude constant and f c is the carrier frequency of the transmitter. Equation (6-57) is simplified as S r (t ) = Re{ Ae
v j 2π ( f c − cos ϕ ) t
λ
} = Re{ Ae j 2π ( f c − f d ) t }
(6-58)
where f d is the Doppler frequency shift due to the mobile receiver’s relative movement, fd =
v cos ϕ
λ
⎛v⎞ = ⎜ ⎟ f c cos ϕ ⎝c⎠
(6-59)
where c is the velocity of light. Since the multipath component is composed of many field reflections in different directions, the received signal can be modeled by the following equation assuming k-reflected waves arriving at the receiver. k
S r (t ) = Re{∑ Ai e
v j ( 2πf c + cos ϕ i +φi ) t
λ
}
(6-60)
i =1
where ϕ i is the direction of the ith wave arrival, Ai is a complex random variable with zero mean and unity variance and φi is a random phase angle uniformly distributed between 0 to 2π . Equation (6-60) can be decomposed into in-phase and quadrature in-phase signals given by, Dr. Mohamed Khalid Nezami © 2003
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S I (t ) = ∑ Ai cos( 2πΨi + φi )
(6-61)
i =1
and k
S Q (t ) = ∑ Ai sin( 2πΨi + φi )
(6-62)
i =1
where Ψi = βv cos(ϕ i ) / 2π . Both SQ (t ) and S I (t ) can be used to simulate Rayleigh fading channel. To speed up simulations and use less computational resources, the Rayleigh fading channel is implemented at baseband using MATLAB™ tools as shown in Figure 6-23.
H( f )
Random generator Q
H( f )
rI (kT )
Complex multiplication
Random generator I
fd
I (kT )
rQ (kT )
Q(kT )
Figure 6- 23: Baseband Rayleigh fading channel model for evaluating synchronization algorithms. Here, two independent complex Gaussian white noise generators are passed through two Doppler filters. The Doppler filters have a frequency response given by 2 ⎧ ⎪ A / πf d 1 − ⎛⎜ f ⎞⎟ ⎜f ⎟ H(f ) = ⎨ ⎝ d⎠ ⎪ 0 ⎩
for for
f ≤ fd f ≥ fd
(6-63)
where f d is the Doppler frequency shift induced by the relative velocity and angle of arrival. Figure 624 shows the magnitude response of (6-63) for fading channel using 900 MHz carrier, and v=80 km/hr ( f d = 66 Hz). Figure 6-25 shows waveforms of both channel envelope and induced phase at typical vehicular speeds of 80 km/hr at 900 MHz. The model in Figure 6-23 will be used throughout this work to evaluate the performance of synchronization algorithms.
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00 -10
H
(f )
Magnitude ( dB)
-20 -30 -40 -50 -60 -70 -80 100
150
200
250
Frequency (Hz)
Figure 6- 24: Magnitude response of the Doppler filter with fd=66 Hz. 10
4 rI (kT ) + jrQ (kT )
0
2
-5
1
-10
0
-15
-1
-20
-2
-25
-3
-30
0
5000
arg{rI (kT ) + jrQ (kT )}
3
rad.
dB
5
10000
-4
0
5000
10000
Sample
Sample
Figure 6- 25: Envelope and phase variation of fading channel model with v=80 km/hr and fc=900 MHz.
One way to characterize fading is by its level crossing rate and the average duration of fade. It was found that these parameters have most of the impact on synchronization. Level crossing rate is generally defined as the expected rate at which the received RF signal envelope given in (6-61) and (6-62) crosses a specified level γ s in the positive direction [6], and given by ∞
ℵγ s = ∫ R ′p(γ s , γ ′)dγ ′
(6-64)
0
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where γ ′ is the time derivative of γ , p (γ s , γ ′) is a joint PDF of γ and γ ′ for γ = γ s . It can be shown that the level-crossing rate ℵγ s for a signal received by a vertical monopole antenna is given by ⎛ γ ⎞ ⎛ γ2 ℵγ s = 2π f d ⎜ s ⎟ exp⎜ − s 2 ⎜ 2σ ⎜ 2σ ⎟ µ µ ⎠ ⎝ ⎝
⎞ ⎟ ⎟ ⎠
(6-65)
where σ µ is the average power of the received faded signal, f d is the Doppler frequency and γ s and γ ′ are statistically independent parameters with a joint probability distribution function (PDF) given by p (γ s , γ ′) = p (γ s ) p (γ ′)
(6-66)
The PDF of γ ′ is Gaussian and given by ⎛ ⎛ γ′ p (γ ′) = exp⎜ − ⎜ ⎜ ⎜ 2πf d σ µ 2π π f d σ µ ⎝ ⎝ 1
⎞ ⎟ ⎟ ⎠
2
⎞ ⎟ ⎟ ⎠
(6-67)
Defining a threshold ρ from (6-67), given by
ρ=
γs πσ µ
(6-68)
By substituting equation (6-68) into (6-65), the level crossing rate (LCR) is simplified to ℵ ρ = 2π f d ρe − ρ
2
(6-69)
The LCR in (6-69) is a function of Doppler frequency and the threshold developed in (6-68), both of which were found to impact the symbol estimation process in (6-25). For Rician fading channels, the lower LCR [6] is given by, ℵρ = 2π ( K R + 1) f d ρe −[ ρ
2
( K R +1) + K R ]
I 0 ( 2 K R ρ 2( K R + 1) )
(6-70)
where the Rician factor ( K R ). Figure 6-26 shows a comparison of LCR for Rayleigh fading channel having a Doppler frequency of 66 Hz and a Rician channels with Rician factors K R = –12, –10, and –6 dB.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers 80 70 60 Rayleigh L C R
Fd=66 Hz
50 40
K R = 6dB
30
K R = 10dB
20 10
K R = 12dB
0 -10 -50
-40
-30 -20 -10 Received signal level relative to rms (dB)
0
10
Figure 6- 26: A comparison between LCR for Rayleigh and Rician channels. The value of (6-70) in conjunction with a pre-determined threshold level given by (6-68) are used in a series of proposed algorithms to improve symbol timing estimates in (6-25). Another parameter that impacts the estimates in (6-50) is the average fade duration (AFD) as shown in. AFD is defined as the average time spent while the signal power level being less than the threshold ρ given in (6-68) and is given by, ⎛ 1 fade duration = ⎜ ⎜ ℵγ ⎝ s
⎞ ⎟ p (γ ≤ γ s ) ⎟ ⎠
(6-71)
where . indicates timing averaging operation, and γs
p(γ ≤ γ s ) = ∫ p(γ )dγ
(6-72)
0
where the PDF in (6-72) can be defined as ⎛ γ2 P (γ ≤ γ s ) = 1 − exp⎜ − s 2 ⎜ 2σ µ ⎝
⎞ ⎟ ⎟ ⎠
(6-73)
Substituting (6-73) into (6-71), the average duration that the received signal spends below a given threshold is defined as
τR =
⎡ ⎛ γ2 ⎞ ⎤ ⎢exp⎜⎜ s 2 ⎟⎟ − 1⎥ ⎞ γ s ⎟ ⎢⎣ ⎝ 2σ µ ⎠ ⎥⎦ fd 2σ µ ⎟⎠
1 ⎛ 2π ⎜ ⎜ ⎝
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By inspection of (6-74), it is clear that the AFD ( τ R ) is a function of LCR and Doppler frequency and can further be simplified to,
eρ − 1 τR = 2π f d ρ 2
(6-75)
Figure 6-27 presents the level crossing rate as a function of Doppler frequency for received signal levels of –30 dB to 0 dB below the r.m.s. value of a carrier of 900 MHz with Doppler frequencies of 33, 66, 100, and 277 Hz. For instance, using a threshold of 10 dB below r.m.s level, a 900 MHz carrier having 100 Hz Doppler will have an average of 80 crossings/second that the signal exceeds that level. Figure 628 illustrates AFD as a function of the threshold level and Doppler frequency. Using computer simulations (see Figure 6-23), Figure 6-29 shows both theoretical LCR and AFD results based on both versus the simulated values using large sequence of fading samples. The figures illustrate the validity in the model, since the AFD and LCR theoretical and simulated values in Figure 6-28 have a high degree of agreement. Next we present the impact of both LCR and AFD on symbol timing estimate, which will produce some ideas on how to combat this sever impact.
Crossing rate (/sec)
10
10
10
3
fd=33Hz fd =66Hz fd =100Hz fd =277Hz
2
1
0
10 -30
-25
-20 -15 -10 -5 Avg. received signal envelope (dB)
0
Figure 6- 27: LCR as a function of Doppler frequency and the received RF signal levels. received RF signal levels with 900 MHz and 120km/hr.
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-1
Avg. fade duration (sec)
fd =33Hz fd =66Hz
10
10
10
10
fd =100Hz fd =277Hz
-2
-3
-4
-5
-30
-25
-20 -15 -10 -5 Avg. received signal envelope (dB)
0
Figure 6- 28: AFD as a function of Doppler frequency and received RF signal levels.
Rayleigh Fading Envelope Level Crossing Rate
3
Rayleigh Fading Envelope Average Fade Duration
-1
10
10
Calculated Simulated
2
-2
10
10 AFD (seconds)
LCR (crossings per second)
Calculated Simulated
1
-3
10
10
0
10 -30
-4
-25
-20
-15
-10 ρ (dB)
-5
0
5
10
10
-30
-25
-20
-15
-10
-5
0
5
ρ (dB)
Figure 6- 29: Theoretical versus simulated LCR and AFD for a Doppler frequency of 100 Hz.
6.12.2 Impact of LCR and AFD on symbol timing estimates
Using the computer model developed in Chapter 2, Figure 6-29 shows fading envelope for two Doppler frequencies of 100 Hz and 200 Hz for a carrier frequency of 900 MHz. Notice how both LCR and ADF are proportional to the Doppler frequency. It is observed Dr. Mohamed Khalid Nezami © 2003
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10
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from the figure that both how often LCR is and length of AFD highly impact the accuracy and final variance of the estimates. (???) With slow fading channel (small Doppler frequency), the AFD durations are large and can span a relatively long interval compared to the estimator block length, which causes a sever impact on the estimates. With fast channels, there are more rapid LCR; however, the AFD for a specified threshold are not very long when compared with the estimator block length. Table 6-1 and Table 6-2 list AFD and LCR statistics of the typical 900/2400 MHz PCS wireless system used throughout this chapter. Based on these values, one can assess the impact of LCR and AFD parameters on symbol timing recovery. A mobile receiver operating with 900 MHz carrier and velocity of 80 km/hr will experience an average LCR (ℵρ=-10dB) of 52.8 crossings/sec and an average duration of fade (τR ) of 2 msec. Thus at a velocity of 80 km/hr and data rate of 10,000 symbols/s, the duration of fade lasts 20 symbols on the average, while the fade crossings occur on an average of every 189 symbols. Thus, on average, after every 3.8 estimation intervals the signal level will be 10 dB below its r.m.s. value for about 20 symbols. Using L=50 symbols in (6-25), this indicates that one out of every four estimates of εˆ is likely to suffer due to fading at 80 km/hr. 10 5 0
Relative power dB
-5 -10 -15 -20 -25 fd=100Hz
-30
fd =200 Hz
-35 -40
0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
Figure 6- 30: LCR and AFD for 900 MHz carrier and Doppler of 100 and 200 Hz. Further, we observe that at a velocity of 120 km/hr and data rate of 10,000 symbols/sec, the duration of fade lasts 13 symbols on average, while the fade crossings occur at an average rate of every 126 symbols for 900 MHz systems. This means that, on average, after every 2.53 observation intervals the signal level will be 10 dB below its r.m.s. value for about 13 symbols. This indicates that one out of every three estimates is likely to suffer due to fading at 120 km/hr.
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Mobile
Doppler @
velocity
AFD @
LCR @
900 MHz
900 MHz
AFD @ LCR @ Doppler @ 2400 2400 MHz 2400 MHz MHz
900 MHz 120 km/hr
100 Hz
1.3 msec
79.2/sec
267 Hz
0.5 msec
211/sec
80 km/hr
66 Hz
2.0 msec
52.8/sec
178 Hz
0.75 msec
140/sec
40 km/hr
33 Hz
4.0 msec
26.4/sec
89 Hz
1.5 msec
70/sec
Table 6- 1: Fade statistics for 900/2400 MHz and ρ = −10dB (in sec.).
velocity
AFD @
Km/hr
900 MHz
LCR @ AFD @
LCR @
2400 MHz
L=50
L=50
120
13.2 symbols
2.53L
126.14 symbols
5 symbols
0.95L
211.4 symbols
80
19.9 symbols
3.78L
189.23symbols
7.5 symbols
1.41L
140 symbols
40
40symbols 7.569 L
378 symbols
15 symbols
2.83L
70 symbols
Table 6- 2: Fade statistics in term of symbol duration for observation of L=50 symbols.
6.13
Frequency Selective Mobile Channel Model
By combining an additional path with a delay in the Rayleigh propagation model discussed earlier as shown in Figure 6-31, causes the direct path to sum out of phase at some frequencies, which will make the channel a frequency selective channel. The frequency selectivity is observed by the fact that when monitoring the attenuation of the signal through the channel, some frequency offsets of the signal spectrum will be attenuatedmore than other offsets. This usually happens when the spread delay of the indirect path is longer than the transmitted symbol time. The mathematical representation of a frequency selective channel is given by its impulse response, h(t ) = a d exp( jφ1 )δ (t ) + a s exp( jφ2 )δ (t − τ )
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where a d and a s are the magnitude of the direct and delayed paths, and τ is the spread delay associated with the indirect oath which causes the channel to be selective. The channel time response in (6-76) can be represented using its baseband form in frequency domain by
H ( f ) = 1 − be − j 2π ( f )τ
(6-77)
The channel in (6-77) will have a minimum and maximum gain and hence produce frequency selectivity. The maximum channel gain is when minimum channel gain occurs when
e − j 2π ( f )τ = 1 , and a
e − j 2π ( f )τ = −1 . This will take
place every f =
1
τ
1− b ⎞ Hz. The fade depth is given by 20 log10 ⎛⎜ ⎟. ⎝1+ b ⎠
As an example, let us assume that b=0.9, and τ =1 microsecond, the channel frequency response in Equation (6-77) is then given by H ( f ) = 1 − 0.9e − j 2π ( f )τ
(6-78)
This channel will vary from +5.5dB and –20dB and have a notch every 1Mhz.
AWGN
αs From Transmitter
++
++
To Receiver
αd τ Delay
Figure 6- 31: Two ray frequency selective channel model.
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These delayed paths in (6-78) may occur due to scattering from nearby buildings, trees, or hilly terrain. The r.m.s. delay spread values ( τ ) are of the order of 1 to 3 microseconds for typical urban 900 MHz PCS systems, 0.1 to 1 microseconds for suburban terrain, and 6 to 7 microseconds for rural mountains. For urban areas, where most of the communication activity occurs, a delay profile of 10 microseconds is a common value for a worst case performance evaluation [32]. This value will be used to represent the worst case of distortion caused by fading used in our simulations. Table 6-3 lists the standard spread delay profile used to test GSM systems, and Table 6-4 lists common worst case spread delay profiles for other typical North American PCS/Cellular systems.
Path
Delay (τ) in
Power ratio
number
µ-seconds
(dB)
1
0
-10
2
0.1
-8
3
0.3
-6
4
0.5
-4
5
0.7
0
6
1.0
0
7
1.3
-4
8
15
-8
9
15.2
-9
10
15.7
-10
11
17.2
-12
12
20
-14
Table 6- 3: GSM delay profile used for experimentation. System type Micro-cellular
Transmitter coverage range 200 m
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Cellular
10 km
10
Land Mobile Radio
100 km
15
Table 6- 4: North American delay profile used for experimentation.
Assuming a BPSK transmitted signal through this channel with data rate of of Rb = 50kbps and assuming the spread delay of τ = 50 µs , (the coherent bandwidth is obtained using the definition of [32] and is given by, Bcoherence =
1 5τ rms
= 4khz
(6-79)
Since the bandwidth of the BPSK signal is Bcsignal = 100 khz , which is much wider than the coherence bandwidth in (6-79), the channel is considered to be a frequency selective channel (not flat), since Bcoherence << Bcsignal . Substituting the spread delay value in the channel frequency response in (6-78), the channel will have a maximum gain given by max = 20 * log(1 + .9) = 5.5dB and a minimum at min = 20 * log(1 − .9) = −20dB . Figure 6-32 shows a BPSK with 50kbps being transmitted through the channel model in Figure 6-32 where the maximum and minimum of the channel are observed and the notch location causes deep fading at frequencies that null at multiples of frequencies equal to 1 f = = 20khz from the center of the baseband (or carrier frequency) in both τ rms directions as shown in Figure 6-32. Figure 6-33 shows the real part of the signal transmitted in the channel with the impact of the fading on it.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Rs=50kbps, Spread Delay=50us null at 1/50us=20 khz 0
-10
-20
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-50
-2.5
-2
-1.5
-1
-0.5
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x 10
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Figure 6- 32: Frequency response of the frequency selective channel overlapped on a transmitted BPSK signal through it. 1
volts
0.5
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Figure 6- 33: Envelope of the received signal showing the signal fading effect. Figure 6-34 shows the binary transmitted symbols of Figure 6-33 with a close look at the impact of the frequency selective channel fading on the symbols. Clearly the figure illustrates the ISI caused by the channel distortion. If we make the spread delay small Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
compared to the symbol period, or equivalently make the data rate small compared to the frequency notches of the channel at 20khz, the channel will not be selective anymore and will become a flat fading channel as shown in Figure 6-35. Here no ISI remains on the symbols since there is amplitude distortion but no frequency distortion Next we use the model in Figure 6-23 and 6-31 to study the impact of fading on symbol timing estimators (6-25) and (6-50), and then recommend some methods by which we can minimize this sever impact.
2 Transmitted Received
1.5 1
Amplitude
0.5 0 -0.5 -1 -1.5 -2
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Time (seconds)
0.01
Figure 6- 34: Frequency selective fading impact on transmitted binary pulses.
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2 1 0 -1 -2
Rs=1200bps
6000
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Figure 6- 35: Flat fading channel impact on transmitted binary symbols.
6.14
Impact of Fading on Feedforward NDA Symbol Timing Estimation
Symbol timing algorithms based on equations (6-25) and (6-34) are optimal only under AWGN conditions. Hence the performance is degraded in a multipath fading environment. To analyze their performance, the computer fading models discussed earlier were used to simulate the fading channel effects on the transmitted signal used by the NDA timing algorithms in (6-25) and (6-34). The experiment was carried out for the fading experienced by a mobile receiver with a speed of 80 km/hr at PCS carrier frequency of 900 MHz. This corresponds to a Doppler frequency of 66 Hz. Figure 6-36 relates the deviation of symbol timing estimates to the fading envelope for a stretch of 4000 16-QAM samples (or 1000 symbols) of the received signal. The following observations are made by examining Figure 6-36: •
Inaccurate estimates mainly occur during relatively deep fades (10 dB below r.m.s. value) that are sizably longer than the estimation interval of 50 symbols.
•
Fade crossings occur at an average rate of the Doppler cycle (6.67 ms).
•
Fade cycles last an average of 400 samples (100 symbols), which are two observation intervals.
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As a result of deep fade cycles, one out of every four symbol estimates is likely to be severely affected by the fading process while using L=50 symbols. 0 .4
(a)
0.25T
With AWGN
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(b)
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|Z(kT)| 0 500
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Figure 6- 36: Symbol timing estimates (a) with AWGN (b) with fade; (c) fade envelope, and (d) absolute value nonlinearity output. Repeating the experiment for a larger number of symbols, Figure 6-37 presents the variances of symbol timing estimation for Rayleigh fading channels. Figure 6-37a shows the variance for a transmitted signal of 900 MHz with Doppler frequencies of 66 Hz, and 100 Hz. Figure 6-37b shows the same results for a higher carrier frequency of 2400 MHz, and a Doppler frequency of 150 Hz, 200 Hz. Both figures indicate the severe adverse impact of Rayleigh fading channel on NDA symbol timing estimates. For instance, a system operating at 900 MHz with a Doppler frequency of 100 Hz has an irreducible symbol timing error variance at Eb N o ≥ 15 dB, a total of 10 dB degradation relative to the case with no fading effects. Similar results are observed with 2400 MHz systems with a Doppler frequency of 200 Hz.
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6-49
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers -1
10
variance
10
10
MCRB
(b)
-2
10
-3
100Hz 10
-1
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(a)
variance
10
-4
66Hz
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-2
-3
200Hz 10
-4
150Hz
no-fade 10
no-fade
-5
10
20 Eb/No
30
40
10
-5
10
20 Eb/No
30
40
Figure 6- 37: Effect of fading on symbol timing estimate variances for carriers at 900 MHz ad 2400 MHz for AVR nonlinearity with Roll-off = 0.75 and L = 50. Following on these observations, the next section proposes methods to minimize multipath fading effect on symbol timing estimationby utilizing the fading statisticsderived in (6-69) and (6-75).
6.15
Schemes for Improving NDA Symbol Timing in Fading Channels
In the previous section, we have observed how the NDA symbol timing algorithms of (625) performance in fading channels and AWGN. It was shown that the algorithm degrades heavily due to multipath fading. In this section, we develop methods to estimate fading channel characteristics from the received envelope, which will be used to implement schemes that will attempt to reduce multipath fading effects on symbol timing recovery. Several existing symbol timing schemes are presented for improving symbol timing recovery in the mobile channel. Furthermore, two new schemes are developed that rely on using the developed fading characteristics, namelyaverage fade duration (AFD) and level crossing rate (LCR) to improve symbol timing estimates. The new schemes are: •
Smoothed post processing (SPP) estimates,
•
Fade envelope weighted (FEW) estimates
and
First we develop the algorithms mathematically; second we present their implementation; and third we compare their performance using computer simulation experiments. To test and evaluate these algorithms against several scenarios of multipath fading, we revisit the computer model developed in Figure 6-23, and Figure 6-31 for Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
evaluating the proposed symbol timing algorithmsThe NDA symbol timing estimation algorithm was derived in (6-25), and is reintroduced here as
εˆ =
2π −j k⎤ ⎡ LN −1 −T arg ⎢ ∑ F ( z (kT ) )e N ⎥ 2π ⎣ k =0 ⎦
(6-80)
The estimate in equation (6-80) is optimal under AWGN conditions only. However, in a multipath fading environment, the performance of the estimator is degraded. To analyze the performance under flat fading conditions, we need to modify (6-80) to include the multipath (Rayleigh) fading envelope. To assess the impact of fading on (6-80) we derive the fading modified non-linearity output F ( z (kT ) ) , and then illustrate its impact on the timing estimate εˆ . The received infaded phase and quadrature phase signal at the input to the quadrature demodulator is given by, r (t ) = I (t ) cos( 2πf c t ) − Q(t ) sin( 2πf c t )
(6-81)
where k
I (t ) = ∑ Ai cos(ϕ i )
(6-82)
i =1
and k
Q (t ) = ∑ Ai sin(ϕ i )
(6-83)
i =1
Both I (t ) and Q ( t ) have zero mean and identical variances given by
σ µ2 =
1 2 E{ Ai } ϕ 2 i
(6-84)
The envelope of the received complex signal in (6-81) is given by a (t ) = I 2 (t ) + Q 2 (t )
(6-85)
where Q(t) and I(t) are the real and imaginary components of the received complex MF signal resulting from the summation of the received i-paths in (6-82) and the quadrature path in (6-83). It can be shown that (6-85) has a Rayleigh probability density function. That is
P (a ) =
a
σ µ2
exp(− a 2 2σ µ2 )
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where σ µ2 is the variance of the Gaussian distributed I(t) and Q(t). If the received signal contains a significant amount of power due to a direct line-of-sight path d ( t ) , the Rician model is used instead of (6-86) according to the model in Figure 631. Adding d ( t ) to the faded signal received (6-85), the envelope will have a probability distribution function defined as P(a ) =
a
σ µ2
e
(
)
− a 2 + d 2 2σ µ2
⎛ ad ⎞ I0 ⎜ 2 ⎟ ⎜σ ⎟ ⎝ µ⎠
(6-87)
⎛ ad ⎞ where I 0 ⎜ 2 ⎟ is the modified Bessel function of the first kind and σ µ2 is the variance of ⎜σ ⎟ ⎝ µ⎠ the two quadrature components of I(t) and Q(t). For cases with d ( t ) = 0 , the Rician distribution in (6-87) becomes Rayleigh distribution; for cases with d (t ) = ∞ , the distribution will be a normal Gaussian distribution (AWGN channel). The ratio of the direct path power to the Rayleigh fading power is an important parameter that will be used in assessing symbol time algorithms.Known as the Rician factor, this ratio is defined as KR =
d2
(6-88)
σ µ2
Furthermore, to complete a characterization of the channel, the following parameter is defined to indicate the SNR of the received signal while propagating through the fading channel
γ=
d 2 + 2σ µ2
(6-89)
σ n2
If there is no direct signal path present, equation (6-89) will be reduce to that of a Rayleigh fading channel, where the average SNR is given by,
γ=
2σ µ2
(6-90)
σ n2
To quantify symbol timing performance as a function of γ , K R , and σ µ2 , we define the unfaded baseband MF signal samples in AWGN as a result of receiving the faded signal as z u (kT ) = z I (kT ) + jz Q (kT ) + n (kT )
(6-91)
where n(kT ) is a complex AWGN. We define the multipath envelope of (6-91) as Dr. Mohamed Khalid Nezami © 2003
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F (kT ) = ℑ I (kT ) + jℑ Q (kT )
(6-92)
Following manipulations of equations (6-91) and (6-92) into the timing estimates in (625) using a SLD nonlinearity, the symbol timing estimates under fading can be obtained as
{
E z (kT )
2
}= E{ℑ (kT ) + ℑ 2 I
2 Q
(kT )}∑ g 2 (kT − nT − εT ) + σ ℑ2
(6-93)
n
where σ ℑ2 is the variance of η~ (kT ) , and η~ (kT ) = n (kT )(ℑ I (kT ) + jℑ Q (kT )) Using Poisson's sum formula in equation (6-93), the resultant signal can be shown to have a spectral line at the frequency of ± 1 T . However, unlike estimates in equation (625), this spectral peak is spread by the fading envelope power given by
σ µ2 = E {ℑ 2I + ℑQ2 }
(6-94)
Similar analyses can be used for symbol timing estimators that are based on DELCONJ nonlinearity (6-50), given
εˆ =
2πkn ⎛⎡ −j ⎤⎞ −T ∗ ⎜ arg ⎢∑ z (kT )z ((k − 1)T )e N ⎥ ⎟ ⎜⎢ k 2π ⎥⎦ ⎟⎠ ⎝⎣
(6-95)
when including the fading effects, the DELCONJ nonlinearity is given by
z (t )z ∗ (t − T ) = c(t ) σ a2 ∑ g (t − nT )g ∗ (t − T − nT ) + G (.) 2
(6-96)
n
where c(t ) is the fading channel envelope, G (.) is non-periodic function which is not of interest and σ a2 = E
{
an
2
} is the averaged data. By substituting (6-96) into (6-95), the
effect of fading channel envelope c(t ) on symbol estimates in (6-95) becomes apparent; fading causes amplitude modulation and smearing of the symbols timing spectral line.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers 50 Intentional error of 1/4T
40
estimator output
30 20 r.m.s reference 0 dB
10 0 -10 -20
Received signal envelope
-30 -40 Eb/No=11 dB -50
0
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Figure 6- 38: Fading effects on symbol timing estimates. The effects of fading on timing estimates can be revealed through simulations shown in Figure 6-38. The figure relates the quality of timing estimates to the fading attenuation of the transmitted signal. As anticipated, the narrow attenuation levels (‘dips’) of the received signal do not affect the estimation process if the duration of fades is short compared to the observation length (50 symbols in this case). Thus the figure confirms the evidence of a relationship between fade characteristics (duration and fade level crossings) and instances of bad symbol timing estimates. This relationship can be engaged to reveal and motivate new schemes to improve the estimates through a priori knowledge of the channel estimated parameters such as LCR and AFD.
6.16
Schemes for Improving NDA Symbol Timing in Fading Channels
Several methods had been reported in the literature for improving symbol timing synchronization in fading channels. Chuang in [33] proposed a method to derive an optimal loop bandwidth of a PLL-based symbol synchronizer. The optimal loop bandwidth is based on the worst Doppler frequency experienced by the receiver. Sampi in [34] derived a method for estimating the fade envelope of the received M-QAM signal, which is then used in conjunction with several baseband algorithms to undo the distortion induced by the channel. Song in [34] proposed the use of a transmitted GSM preamble sequence to extract DA-symbol timing by use of a cross-correlator and a smoothing algorithm to enhance the performance in fading. The smoothing algorithm is carried out by a first-order filter with an asymmetric step size. Moeneclaey proposed a scheme [36] that incorporates an estimate of the channel transfer function into the estimation in (6-25). Methods used for estimating the fading channel vary in effectiveness and are complex because they require a preamble and the use of large computational resources.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
To keep the receiver implementation complexity low, we propose four schemes to improve the estimation process while the signal is in fade. The first two methods are based on choosing optimal observation intervals and interval averaging, while the latter two schemes utilize estimates of AFD and LCR to enhance the estimate in (6-25) without adding any additional complexity to the receiver. 6.16.1 Optimizing the Observation Interval Based on Fading Frequency Based on the results in Tables 6-3 and 6-4, one simple approach is to improve the estimate by using an observation interval that is much longer than the average fade duration. The improvement is illustrated in Figures 6-39, 6-40, and 6-41 using three different observation intervals. For the case with L=20 symbols and a Doppler frequency of 66 Hz (Figure 6-39), the AFD is 2 msec or 80 symbols. By having AFD period equal to 80 symbols, four adjacent estimation intervals (L=20) are contained in one AFD period as shown Figure 6-39. As a result, the estimates in (6-50) are severely degraded during instances that contain fades with periods that extend over a significant portion of the estimates interval (L). If the observation interval is increased to 50 symbols (Figure 6-40), AFD will extend only over 2.5 observation intervals.This provides some improvement but still degrades the estimates as shown in Figure 6-40. If the observation interval is chosen to be longer than AFD, such as the case of L=300 symbols ( Figure 6-41), the estimates are minimally affected by fading as shown. In practice, changing the observation interval is a viable approach except in situations where the allowed observation interval is too short, or where the channel parameters such as frequency offset, symbol timing, and phase offset may no longer be held constants during such long interval. 0.4
10 dB, L=20
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Figure 6- 39: Symbol timing estimates for observation interval of 20 symbols and Eb/No = 10 dB. Dr. Mohamed Khalid Nezami © 2003
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0.4
10 dB, L=50
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Figure 6- 40: Symbol timing estimates for observation interval of 50 symbols and Eb/No = 10 dB. 0.4
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Figure 6- 41: Symbol timing estimates for observation interval of 300 symbols and Eb/No = 10 dB. 6.16.2 Overlapping Observation Intervals
With the subinterval averaging method, the estimation interval (L) is subdivided into finer intervals, the symbol estimate εˆi are obtained for each sub-interval, and the overall estimate ε~ is given by
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
ε~ =
1 N
N
∑ εˆ
(6-97)
i
i
where N is the number of subintervals.
6.16.3 Estimate Post Processing (SPP) A direct way to improve the estimates is by smoothed post processing (SPP) as shown in Figure 6-42. Here the average fade duration is used to control the effective time constant of the smoothing filter. The timing estimates using this scheme are given by
εˆk ( SPP ) =
1 P −1 ∑ bPεˆk − p P p =0
(6-98)
where εˆ k is symbol timing estimation obtained via equation (6-98), {b p } are the coefficients of the smoothing FIR filter, and P is the filter order which is dictated by the average fade duration.
Post smoothing estimates r (t )
kT N Matched Filter
z(kT N )
F(
)
x(kT )
NDA estimator
εˆk (NDA )
P −1
∑
p =0
b P εˆ k − p
εˆk (SPP )
()
Moving Average
yk
selector
ρ Threshold
sign(γ k − ρ )
NDA estimator
εˆk (FEW )
gk flag
γk
Block Average
Fading envelope weighted estimates Figure 6- 42: Schematic of the proposed schemes for symbol timing estimation.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
6.16.4 Fly wheeling through Fading Durations
The second implementation scheme is shown in Figure 6-42. Here the RF channel fade envelope is extracted from the matched filter envelope and is used to exclude the estimates while the signal is in deep fade. As illustrated in Figure 6-43, the envelope detection algorithm tracks the envelope changes due to RF signal variation, while rejecting the variations due to the QAM modulation.
5 Fade envelope
0
level (dB)
-5 -10 -15 -20
Block averaging Received MF signal
-25 -30
0
500
1000 samples
1500
2000
Figure 6- 43: Illustration of RF fade envelope extraction. The scheme does not suffer from large computational needs or the convergence problems usually associated with adaptive channel estimation techniques. As illustrated in Figure 6-42, the MF output envelope samples ( z (kT ) ) are filtered to obtain an estimate of the fade envelope. This envelope is then compared against a threshold obtained using (6-68). Samples selected from nonlinearity output are unaffected by the deep fade. The NDA timing is then estimated using the selected samples as per equation (6-25). As the data envelope varies much faster, the fade envelope can be estimated with a simple moving average (MA) low-pass filter of equal coefficients. The filtered output is given by, P −1
yi = ∑ xi − k
(6-99)
k =0
where P is the filter order. Another approach to extract the RF envelope is to use a Savitzky-Golay smoothing filter [37]. This filter is a low-pass filter with coefficients that Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
are computed by least-squares fitting a polynomial in a moving window, centered on each MF sample, so the new value will be the zeroth (k=0) coefficient of the polynomial. The filter coefficients for a specified polynomial degree and window width are computed independent of MF samples and stored in a common block. The filter smoothes out the data variations and noise but retains variations due to fading. In our simulations, the moving average filter method was preferred for its simplicity and ease of implementation. In order to trace the narrow deep-fade nulls that are of the order of τ R = 2 msec or 80 samples, the filter length P in (6-99) is chosen to be about τ R 10 or 8 samples. To ensure reasonable stretches of selected samples, the moving average output yk is further averaged over small non-overlapping blocks to obtain γ k . So as not to fill in the deep fades, we chose a block length that was half of worst fading duration length, which works out to be 10 symbols in 80 km/hr case. The gating signal gk which equals sign(yk - γ k ), is then used to exclude nonlinearity output xk affected by deep fade. 6.16.4.1 Performance in Flat Fading Channels The fading channel model developed earlier (Figure 6-33) was used to. evaluate the performance of the proposed schemes in Figure 6-42 The fading channel model is modified to include a second path that has an arrival spread delay (two-ray model) developed in (6-31). Finally, the same model is modified to include a line-of-sight component, to implement Rician channel cases as in (6-87) The signals |z(kT)|, yk , γ k and gk in the proposed schemes are plotted in Figures 6-44(a)(c). Figures 6-44(d)-(f) compare the timing estimates obtained by conventional NDA and the proposed schemes described above for a transmission with 80 km/hr fading at Eb/No=10 dB for 6000 received MF samples. The simulation model introduced an intentional timing error equal to T/4. Notice that the outlier estimates of Figure 6-44 (d) corresponding to deep fade are eliminated in the estimates of the new schemes shown in Figures 6-44 (e)-(f). Based on exhaustive simulations, Figure 6-45 presents the variance of the timing estimates as a function of Eb/No, along with the modified Cramer-Rao lower bound (MCRB). The proposed algorithms appreciably improve the estimates, especially for moderate Eb/No less than 15 dB. Due to the averaging process, the performance of the SSP algorithm is better than that of the FEW scheme. However, the increased delay in convergence may not be practical for systems using short TDMA bursts [13,14,15].
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers 3 z(kT)
1
0
εˆk ( NDA)
(d)
0.3
/T
volts
(a)
2
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/T
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(f)
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/T
G ate
2
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0.1
1000 2000 Samp le Number
3000
2000 4000 Samp le Number
6000
Figure 6- 44: Signals and estimates associated with the proposed schemes, Doppler of 66 Hz, ρ=-10 dB, and Eb/No=10 dB. -1
variance of symbol estimates
10
-2
10
FEW-algorithm SPP -algorithm
-3
10
with fade
-4
10
without fade
-5
10
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-6
10
5
10
15 20 Eb/No
25
30
Figure 6- 45: Estimate variance versus Eb/No for the proposed algorithms, Doppler of 66 Hz, and ρ=-10 dB. 6.16.4.2 Performance in Selective Fading Channels The performance of the proposed algorithms were studied in frequency selective channels using the 2-ray model defined in Figure 6-31. The delay spread was 10 microseconds.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
2
(d) εˆk ( NDA)
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(a)
/T
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/T
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/T
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4000
1000 2000 3000 Sam p le Number
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Figure 6- 46: Signals and estimates associated with the proposed schemes operating in 2 ray fading channel model, Doppler frequency of 66 Hz, Eb/No=10dB, delay spread=10 µs, and ρ=-10 dB. Figures 6-46(a-f) present the performance results of both SSP and FEW algorithms for two paths Raleigh signals. As evident from Figure 6-46b and 6-46c, the additional path has resulted in an increase in the crossing rate and narrowed the deeper nulls, thus increasing the duty cycle of the gating signal. Performing the experiment for the whole range of Eb/N0 yields the results shown in Figure 6-47. The variances of both algorithms are smaller than those obtained using the standard ML symbol timing estimation algorithm. 10
-1
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no-fade
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Figure 6- 47: Variance performance of the proposed schemes for the 2 ray path fading channel, fd=66 Hz, and delay spreads=10 µs. Dr. Mohamed Khalid Nezami © 2003
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6.16.4.3 Performance in Rician Fading Channels The experiment of the previous section was repeated for a MF signal that has a Rayleigh path, and a direct path with an associated spread delay and a specified Rician factor of 10 dB. Figure 6-48 presents the estimation results. Figure 6-48 (b) indicates fewer deep fade nulls due to the presence of the direct component and a lower LCR than anticipated according to Figure 6-48. Timing estimates for both FEW and SSP algorithms consistently show improved performance. For signals that have Rician fading characteristics, the proposed algorithm performance approached the performance of an AWGN channel. Figure 6-49 presents the variance of symbol timing estimates versus Eb/N0 for a case of 66 Hz Doppler frequency and 10 dB Rician factor.
2
(d)
0.3
z(kT)
(a)
/T
volts
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0
εˆk ( NDA)
0.2
0.1
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10 (b)
-10
γk
-20 C(kT)
-30
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(e)
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dB
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1000
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1
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(f)
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G ate
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-1
3000
4000
εˆk ( FEW )
0.2
0.1
1000 2000 3000 Sam p le Number
4000
1000 2000 3000 Sam p le Number
4000
Figure 6- 48: Performance of the proposed algorithms in Rician fading channel with Fd=66 Hz, Eb/No=10 dB, delay spread =10 µs, KR=10 dB. 10
10
-1
FEW MCRB
-2
No Post Processing
10
no-fade
-3
Variance
SPP 10
10
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-4
-5
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-7
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Eb/No
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Figure 6- 49: Variance performance the proposed schemes for Rician fading channel, with Doppler frequency of 66 Hz, delay spread=10 µs, and KR=10 dB.
6.17 Illustrating Example: Non-Data Aided Symbol Timing offset estimation and correction The example illustrates the baseband digital signal processing algorithms involved in Continuous Phase Frequency Shift Keying (CPFSK) and QPSK signals acquiring and tracking of synchronization parameters. Simulations of symbol timing offset and correction, and automatic symbol rate detection, are also simulated. Figure E6-1 illustrates a fully digital feedforward block based synchronization system. For the symbol timing estimation process, it is assumed that the frequency offset estimation and correction has already been achieved. It was shown earlier in the chapter that by processing the baseband signal y k corresponding to an oversampled random stream of MPSK or M-ary CPFSK using the nonlinearity y k
2
produces a spectral line at
1 2 .(??) By utilizing spectral analysis algorithms on y k , the T symbol clock spectral line can be located and used for auto bauding and its phase is related to the symbol timing offset ( τ ) by the following relation,
the symbol rate, f =
2πτ T
θˆ =
(E6-1)
Thus by evaluating the argument of y k
2
, that is equal to
2πτ , symbol timing offset is T
estimated. That’s is
τ=
T 2 arg( y k ) 2π
(E6-2)
The result in (E6-2) is based on a single sample. To get more reliable estimate of ( τ ), (E6-2) is averaged over an observation interval made of L-symbols (or NL samples). Using the standard definition of DFT, the phase shift of the single bin (spectral line at 1/T) from which the timing offset is estimated is given by
τ =−
NL −1 T 2 −j arg ∑ y k e 2π k =0
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(E6-3)
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where arg( x ) = tan −1 (x ) , and since − π ≤ arg( x ) ≤ +π , the estimation in (E6-3) can only sense errors bounded by − T 2 ≤ τ ≤ + T 2 . One way to improve the estimates in (E6-3) against AWGN and ACI is to prefilter y k prior to using it in (E6-3) with a bandpass filter 1 2 , and then post filter y k with another bandpass that has a center frequency at f = 2T 1 filter with a center frequency at f = . It was also shown that the jitter of the final T estimates is completely removed if the prefilter is symmetric. Figure E6-2 illustrates the processing of a 9600sps QPSK signal using the algorithm in (E6-3). The resulting time domain symbol rate signal (right graph) conventionally was tracked using phase locked loops, which still can be carried out here instead of DFT computations in (E6-3).
y ( kTs ) = g k a k e
z
y 2( kTs )
⎛ ⎛ T ⎞ ⎞ j ⎜⎜ 2π ⎜ k +ε ∆f +θ k +ϕ k ⎝ ⎝ N ⎠ ⎠
− LTs
Symbol offset and rate estimation/correction
y1(Ts )
z
− LTs
1 gˆ k
z
y3(kT ) y 4( kT )
− LTs
INT
z
− LT
− τˆ
e − j 2π∆fkTs
τˆ
gˆ ∆ fˆ
e − jθ k ˆ
θˆ
( ( ) ) y 4 k = a k e j δθ +2πTδf k +δτ +ϕ k
decoder (
e − j δθ k + 2πkTδf k
)
1 aˆ k
dk
z −1 error
NCO
AGLF
CLF
Figure E6- 1: Baseband Acquisition and tracking including the non-Data aided symbol timing synchronization.
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10 Fc=1/2T 0
Fc =1/T
0.8
-10
0.6
-20
0.4 0.2
dB
volts
-30 -40
0 -0.2
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2 Hz
2.5
3
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250
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300
350
400
4
x 10
450 500 sample #
550
600
650
700
Figure E6- 2: Symbol rate spectral line extracted using (3) for QPSK signal with 9600sps (left), and the resultant time domain extracted symbol clock (right). For M-ary CPFSK signals, the same estimation algorithm in (E6-3) is valid. Figure E6-3 through Figure E6-5 illustrate the procedure detailed above for MSK with h=1/2, 4-ary CPFSK with h=1/4, and 8-ary CPFSK with h=1/8, respectively. In all of these cases, the spectral line corresponding to the symbol rate has the same SNR, indicating that the self noise generated from the squaring nonlinearity is equal for all of the waveforms considered. 10 2-ary CPFSK
5
Pre-filter
Post filter
0.1 Final signal at Rs=9600
0
0.05
-5 dB
0 -10 -15
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8000 Hz
10000
12000
14000
16000
280
300
320
340
360 time
380
400
420
440
Figure E6- 3: Generation of symbol timing clock from MSK signal using the nonlinearity 2 yk .
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10
dB
0.1
4-ary CPFSK, h=1/4 Rs=9600sps
5
0.08
0
0.06
-5
0.04 0.02
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8000 Hz
10000
12000
14000
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500 520 time
540
560
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600
Figure E6- 4: Generation of symbol timing clock from 4-ary CPFSK, h=1/4 signal using 2 the nonlinearity y k . 10 8-ary CPFSK Rs=9600sps h=1/8
5
0.08 0.06
0 0.04
dB
-5
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-25 -30
-0.08 -0.1
0
2000
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8000 Hz
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14000
16000
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300
350
400
450
Figure E6- 5: Generation of symbol timing clock from 8-ary CPFSK, h=1/8 signal using 2 the nonlinearity y k .
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y (kTs ) = g k sk e z
− LTs
⎞ ⎛ ⎛ T ⎞ j ⎜⎜ 2π ⎜ k +ε ∆f +θ k ⎠ ⎝ ⎝ N ⎠
Symbol offset and rate estimation/correction
y3(kT )
y 2(kTs )
y1(kTs ) z
− LTs
z e
1 gˆ k
− LTs
− j 2π∆fkTs
y 4(kT )
z − LT
INT
− τˆ
aˆkM
εˆ
e
− jθˆk
θˆ
∆fˆ ak
ak
ak
ak
( ( )) y 4k = ∆ak sk e j δθ + 2πTδf k +δτ
rk
dˆ j
Remove pilots & INT-1
z − LT ⎛ 2 ∗⎞ ⎜⎜ 2 cˆk ⎝ σˆ n ⎠
Turbo decoder
λ(qi )
(q)
F (x) Iterative Channel & noise variance estimator
yˆ
(q) k
INT & re-insert pilots
yp
xˆi( q ) ⎛ λ( q ) ⎞ soft decision, xˆi( q ) = tanh⎜⎜ i ⎝ 2 ⎠
Hard decision, xˆi( q ) = sign(λ(i q ) )
Figure E6- 6: Data aided symbol timing acquisition. Unlike the estimates in (3) however, the data-aided estimates based in (5) are not valid when a frequency offset ∆f is present. This requires that frequency offsets are estimated and removed prior to (5), as illustrated in Figure 6. Using LT-symbols of received preamble, the data aided estimate in (5) is given by
2πk
NL −1 −j T ˆ τ = −4 arg ∑ y k e 4 N e − 2π∆fkT 2π k =0
(6)
where ∆fˆ is the estimated frequency offset.
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6.18 References
1. H. Meyr and G. Ascheid, Synchronization in Digital Communications, John Wiley & Sons, NY, 1990. 2. Surat White and Norman Beaulien, “On the application of Cramer-Rao and detection theory bounds to mean square error of symbol timing recovery” IEEE Trans. Commun. vol. 40, No. 10, pp. 1635-1643, Oct. 1992. 3. D. Efstathio and A.H. Aghavami, “Preamble-less Non-decision-aided (NDA) techniques for 16-QAM carrier phase recovery and gain error correction, for burst transmissions” IEEE Trans. Vehicular Technology, pp. 673-684, May 1998. 4. Estathiou, and H. Aghvami, “Preamble-less non-decision aided (NDA) feedforward synchronization techniques for 16-QAM TDMA demodulators,” Proceedings of ICC, 1998, pp. 1090-1094. 5. W. Weber, III, “Performance of phase-locked loop in the performance of fading communication channels,” IEEE Trans. Commun., vol. COM-24, No. 5, May 1976. 6. T. Rappaport, K. Blankenship, and H. Xu, “Propagation and Radio system design issues in mobile radio systems for the GloMO project,” www.DARPA.com 7. T. Eyceoz, A. Duel-Hallen, and H. Hallen, “Using the physics of the fast fading to improve performance for mobile radio channels,” www.eos.ncsu.edu. 8. W. Bodtmann, H. Arnold, “Fade duration statistics of Rayleigh-distributed wave,” IEEE Trans. Commun., vol. COM-30, No. 3, pp. 549-553, March 1982. 9. S. Sampi and T. Sunaga, “Rayleigh fading compensation for QAM in Land mobile radio communications,” IEEE Trans. Vehicular Tech., vol. 42, No. 2, May 1993. 10. J. Chuang, “The effects of multipath delay spread spectrum on timing recovery,” IEEE Trans. Vehicular Tech., vol. VT-35, No. 3, pp. 135-140, August 1987. 11. S. Slimane, “An improved PSK scheme for fading channels,” Royal Inst. of Tech., Sweden. 12. K. Geothals, and M. Moeneclaey, “PSK symbol synchronization performance of ML-oriented data aided algorithms for nonselective fading channels”, IEEE Trans. Commun., vol. 43, No. 2/3/4, pp. 767-772, Feb/Mar/Apr 1995.\ 13. M. Nezami and R. Sudhakar, “New schemes for 16-QAM symbol recovery” EUROCOMM 2000 Münich, Germany, 17-19 May 2000.
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14. M. Nezami and R. Sudhakar, “M-QAM digital symbol timing synchronization in flat Rayleigh fading channels” PRMIC, Osaka Japan, Nov 1999. 15. M. Nezami, “Non-linear M-QAM digital symbol timing synchronization algorithm suited for wireless handheld radios”. Third international conference on computational aspects and their applications in electrical engineering, Amman, 19-20 Oct 1999. 16. F. Gini, and G. Giannakis, “Frequency offset and symbol timing recovery in flatfading channels: A cyclostationary approach,” IEEE Trans. Commun., vol. COM46, No. 3, pp. 400-411, Mar 1998. 17. K. Geothals, and M. Moeneclaey, “PSK symbol aided synchronization of MLoriented data-aided algorithms for nonselective fading channels”, IEEE Trans. Commun., vol. COM-46, No. 2/3/4, Feb 1995. 18. H. Song, “Timing synchronization using the reliability check and smoothing algorithm in the fading channels”, IEICE Trans. Commun., Vol. E82-B, No. 4, pp. 664-668, Apr 1999. 19. W. Weber, “Performance of phase-locked loops in the presence of fading communication channels”, IEEE Trans. Commun., vol. COM-24, No. 5, Feb 1976. 20. V. Fung. and T. Rappaport, “Bit error simulation for pi/4 DQPSK mobile radio communications using two-ray and measurement-based impulse response models”, IEEE Trans. Selected Areas in Communications., vol. 11, No. 3, pp. 393-405, Apr 1993. 21. S. Sampi, and T. Sunaga, “Rayleigh fading compensation for QAM in land mobile radio communications”, IEEE Trans. Commun., vol. 42, No. 2, pp. 137147, May 1993. 22. J. Chuang, “The effects of multipath delay spread on timing recovery”, IEEE Trans. Vehicular Tech., vol. VT-35, No. 3, Aug 1987. 23. K. Scott, E. Olasz, “Simultaneous clock phase and frequency offset estimation,” IEEE Trans. Commun. vol. 43, No. 7, pp. 2263-2270, Jul 1995. 24. N. D'Andrea and U. Mengali, “A simulation study of clock recovery in QPSK and 9QPRS systems” IEEE Trans. Commun., vol. 33, No. 10, pp. 1139-1142, Oct 1985. 25. Jussi Vesma, Markku Renfors, Jukka Rinne, “Comparison of efficient interpolation techniques for symbol timing recovery”, Proceedings of GLOBECOM, 1996.
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26. F.M. Gardner, “Interpolation in digital modems-part I” IEEE Trans. Commun., vol. 14, No 6, pp. 501-507, May 1993. 27. F.M. Gardner, “Interpolation in digital modems-part II” IEEE Trans. Commun., vol. 14, No 6, pp. 998-1008, Jun 1993. 28. Timo Lakso, Vesa Valimaki, Matti Karjalainen, and Unto Laine, “Splitting the unit delay: Tools for fractional delay filter design”, IEEE Signal Proc. Magazine, pp. 30-60, Jan 1996. 29. H. Zhang, “Interpolator for all-digital receivers”, Electronics Letters, vol. 33, No. 4, pp. 261-262, Feb 1997. 30. Tony Kirke, “Interpolation, resampling, and structures for digital receivers”, Communication System Design Magazine, pp. 43-49, Jul 1998. 31. Real time Implementation of symbol http://yake.ecn.purdue.edu/~cominfo.
timing
recovery,
available
at
32. Theodore S. Rappaport, Wireless Communications, Principle and Practice, Prentice Hall, NY, 1996, chapters 1-2. 33. J. Chuang, “The effects of multipath delay spread on timing recovery”, IEEE Trans. Vehicular Tech., vol. VT-35, No. 3, Aug 1987. 34. S. Sampi, and T. Sunaga, “Rayleigh fading compensation for QAM in land mobile radio communications”, IEEE Trans. Commun., vol. 42, No. 2, pp. 137147, May 1993. 35. H. Song, “Timing synchronization using the reliability check and smoothing algorithm in the fading channels”, IEICE Trans. Commun., Vol. E82-B, No. 4, pp. 664-668, Apr 1999. 36. Geothals, and Marc Moeneclaey, “PSK symbol synchronization performance of ML-oriented data aided algorithms for nonselective fading channels,” IEEE Trans. Commun., vol. 43, No. 2/3/4, Feb/Mar/Apr 1995. 37. Boza Porat, A Course in Digital Signal Processing, John Wiley, NY, 1997.
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Chapter 7 Feedforward Carrier Frequency and Carrier Phase Offsets Estimation Algorithms The scope of this chapter is to present the details and performance of the open loop carrier recovery schemes that are performed as shown in Figure 7-1. Several schemes for frequency offset estimation and carrier phase estimation are investigated. The first set of algorithms relies on the use of maximum likelihood theory, where an M-power nonlinearity is used to remove the data modulations from the matched filter (MF) samples, M being the symmetry angle of the signal constellation [5]. The second set of algorithms utilizes the DFT spectral estimation algorithm to estimate a frequency bin that represents the carrier frequency offset. The third set of algorithms is based on the samples auto-correlation function that extracts carrier offset information from the phase jumps between consecutive matched filter samples. The algorithms performances evaluation is carried out using Monte-Carlo simulation procedure. Wherever possible, the computer model developed in Chapter 2 is used to test and evaluate these algorithms for QPSK and 16-QAM modulations. The performance evaluation is represented by the mean and variance of the estimates, and frequency or phase capture-range. The evaluation is also based on variables like channel parameters, bandwidth of the received baseband signal, complexity of implementation, and performance against SNR and fading.
Phase estimation and correction
Offset estimation and correction Matched filter
z−L
x
z−L
received signal with frequency phase and offset errors
x
e − jθ
ˆ
Frequency acquisition/ NCO
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− j 2π∆fˆ
Carrier Phase tracking/NCO
received signal without frequency phase and offset errors
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Figure 7- 1: Overall Feedforward Baseband Carrier Synchronization System.
For both frequency offsets and the carrier phase estimation algorithms investigated, the dependency of the MF samples on modulations is removed by raising the signal samples to M power nonlinearity. While this works well for QPSK (M=4) and BPSK modulations (M=2), it will be shown that the Mpower nonlinearity alone does not work well for removing M-QAM modulations. For this reason, mathematical analysis and simulations will be used to investigate the performance of several existing nonlinearity schemes that attempt to improve data removal in QAM modulations. Based on this investigation, an alternative to the existing schemes is introduced . The scheme is aimed at improving the removal of 16-QAM modulations through a simple approach of passing only symbols (±3,±3) to the M-power nonlinearity. The symbols (±3,±3) are chosen for their high energy level that yields an optimal synchronizer SNR, and because it represents a sub-QPSK constellation that represents an optimal constellation for using the M=4 nonlinearity. As a result of using M-power nonlinearity, phase estimates are given with 2π M ambiguity [6]. Several schemes are presented that deal with resolving this ambiguity. As a result of using a scheme that can remove ambiguity l, especially while operating at low SNR and fading channels, a phenomenon known as cycle slipping occurs. A brief introduction and a survey of the issue of cycle slipping are presented, and the relationship of average timing slip to the estimated phase error variance is derived. Phase and carrier offset estimates deteriorate dramatically during the presence of fading. An assessment study is also carried out to quantify this degradation by deriving a relationship between phase estimate error variance and fade statistics represented by fade levels, Doppler frequency, and Rician factor. The joint probability density function (PDF) of the signal amplitude and phase error estimates for both Rayleigh and Rician channels is derived. This PDF is then used to derive the final variance degradation that ultimately leads to formulating the receiver BER degradation as a function of fade statistics, particularly for the carrier phase recovery algorithms. 7.1 Introduction to Feedforward Carrier Recovery The conventional way of dealing with frequency offset has been through use of automatic frequency control loops (AFC), which has the purpose of tracking large frequency offsets. Such schemes are detailed in [7]. Though AFC loops are accurate, their acquisition time may be too long since their implementation involves feedback. This makes the AFC loops unsuitable for use in systems with data transmitted in short bursts or packets, or in applications where a fast re-acquisition after deep fade or shadowing is required. There are several reported schemes that alter the conventional AFC loops to achieve faster response [7,8];however, most share a great deal of implementation complexity and may involve the use of analog circuitry. ML-based data or decision-aided carrier phase and frequency offset FF open loops are suitable for systems with a training sequence and operating well at high SNR. The basic idea of DA/DD carrier recovery is that modulations can be removed from signal samples making use of differential decisions of the received preamble sequence. Phase rotation is then estimated by measuring phase rotations between consecutive samples, while frequency offset is estimated by measuring the differential phase rotations between consecutive samples. Although DD/DA schemes work well, their use requires the use of preamble symbols resulting in spectral efficiency reduction, especially for short burst systems. A Dr. Mohamed Khalid Nezami © 2003
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worthwhile improvement in this efficiency will result if this preamble sequence is eliminated through use of Non-Data aided (NDA) carrier recovery algorithms. 7.2 Problems Associated with Feedback Carrier Recovery Schemes Carrier phase errors at the receiver can result in rotation of the received symbols, resulting in degradation of the bit error rate and ultimately degradation in the receiver sensitivity. The BER deterioration as a result of phase mismatches is given by Dθ = 4.3(1 + E s N 0 )σ θ2 [dB]
(7-1)
where E s N 0 is the symbol energy to noise ratio; σ θ2 is the phase error variance of the synchronizer estimates. For second order AFC Costas loop [1,7], the phase error variance is given by
σ θ2 =
BLT [rad2] Es N 0
(7-2)
where BL is the one-sided loop bandwidth and T is the symbol rate (loop iteration rate). Acquiring and tracking carriers using feedback loops for short-burst TDMA signals can prove to be a difficult task. The difficulty can be seen by examining the loop acquisition and lock time as a function of loop bandwidth and the frequency offset [9]. That is given by, t acq
π 2 (2π∆f ) 2 2π = + [sec] 3 2πB L 16ζ (2πB L )
(7-3)
here ζ is the classical second order type loop damping factor and ∆f is the frequency offset to be estimated (acquired) and tracked. To achieve faster acquisition in (7-3), the loop bandwidth has to be increased; while doing so, the phase error variance increases as seen in (7-2) and ultimately leads to increase in BER as seen in (7-1). The term on the right-hand side of (7-3) is the tracking time, while the term on the left-hand side is the “pull-in” time. To illustrate the loop bandwidth, acquisition timing and BER degradation tradeoffs, assume a frequency offset due to Doppler of 1 kHz and ζ = 0.707. Figure 72 shows a plot of t acq in (7-3), the phase variance (7-2) and BER degradation (7-1) as a function of the loop bandwidth B L .
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10
-1
10
Acquisition time [sec] Degradation of BER [dB]
-2
10
Phase variance [rad2] -3
10
-4
10
1
10
2
10 One sided Loop bandwidth [Hz]
3
10
Figure 7- 2: Loop Bandwidth Versus Carrier Phase Variance, Acquisition Time, and BER Deterioration. As the loop bandwidth becomes narrowerto achieve lower BER degradation by having lower σ θ2 , the acquisition time becomes impractical for typical short TDMA bursts [10]. On the other hand, if the loop bandwidth is widened to achieve faster acquisition, both phase error variance and BER deterioration become large. For most TDMA modems with data rates of up to 56 kbps, the Feedback loop bandwidth is typically of the order of 50-5 Hz, This will have a maximum phase error variance of σ θ2 =10-3 rad2 (a standard deviation of σ θ =1.8105o), which results in a BER degradation of less than 0.1 dB, which is acceptable. However, with such low loop bandwidthacquisition time may be on the order of few seconds, which may not be appropriate forburst systems. Several methods have been used to increase acquisition time for feedback loops [11]. One method involves tuning the receiver initially with a wider loop bandwidth to be able to pull in large frequency offsets, and then once frequency acquisition occurs, the loop bandwidth is narrowed gradually to track phase errors. A second approach uses a fixed loop bandwidth, but the frequency oscillator (NCO) of the receiver down converter is swept over the uncertainty frequency offset region ( ± ∆f max ) at a sufficiently low rate to enable the narrow loop to lock [9]. Other non-conventional feedback type techniques involve the use of inband tone for carrier acquisition and tracking. This method reduces the spectral efficiency of the channel, introduces spurs and interference, and requires complex analog signal processing. The pilot symbol aided carrier recovery has also been widely used in wireless systems. This method involves periodically inserting a known symbol, which is then used in conjunction with sliding a correlator at the receiver to extract carrier phase and carrier frequency offset estimates. These systems require high SNR and use overhead symbols, hence reducing the spectral efficiency of the channel. Another recent approach has been to use DFT-aided acquisition. Here, the DFT algorithm determines the initial frequency offset in a relatively short processing time and performs correction using an open loop method, yielding small residual offset that is within the pull-in range of conventional narrowband Costas loop. While this method works, it requires large computational resources and may cause false lock during operation in high interference channels. Dr. Mohamed Khalid Nezami © 2003
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In summary, the feedback carrier acquisition and tracking schemes are inexpensive and well established but have a smaller frequency capture range and a longer acquisition time at low SNR, thus making them unsuitable for short burst TDMA systems. On the other hand, open loop systems achieve faster response, are easier to implement for burst TDMA systems and do not require preamble training when implemented as NDA. 7.3 Principle of Open Loop Feedforward Carrier Recovery Including frequency offset parameter in the ML equation defined earlier in Chapter 6, the matched filter output is given by [1] z (εˆ, Ω ) = e − j 2π∆fkT
+∞
∑ r (kT )g (nT − kT )e
k = −∞
f
s
MF
− j 2π∆f ( kTs − nT )
s
(7-4)
+∞
∑ r (kT )g (nT − kT ) is the output of a frequency-translated MF. For small
where the expression
k = −∞
f
s
MF
s
frequency offset we approximate this frequency translated MF term with its non-translated version. The joint phase and frequency ML equation is then derived using equation (7-4) in conjunction with a specified known preamble sequence of {a(kT )} [151, 180, 206],
{
}
⎛ ⎧ L−1 ∗ ⎫⎞ ˆ ˆ ˆ ⎜ θ , ∆f = arg⎜ max ⎨∑ a (kT )e − jθ e − j 2πkT∆f z ( kT ) ⎬ ⎟⎟ ⎭⎠ ⎝ ∆f ,θ ⎩ k =0
(7-5)
Equation (7-5) is a two dimensional ML function. To separate the joint phase and frequency ML function, a new variable Y (∆f ) is introduced
( )
L −1
Y ∆fˆ = ∑ a ∗ (kT )e − j 2π∆fkT z (kT )
(7-6)
k =0
By substituting the polar form of Y (∆f ) in (7-5), the ML-function becomes
{θˆ, ∆fˆ }= arg⎛⎜⎝ max( Y (2π∆f ) Re{e
(
(
− j θ + arg Y ( ∆fˆ )
∆f ,θ
)
) )} ⎞ ⎟ ⎠
(7-7)
Using the fact that Y (∆fˆ ) is not a function of phase and that the maximization of Re{e − j (θ + arg (Y ( ∆f ) ))} is ˆ
trivial, the frequency offset can be estimated as ∆fˆ =
1 ⎧ L −1 ⎫ arg ⎨∑ bk a(kT )a ∗ ((k − 1)T z ((k − 1)T ) z ∗ (kT )⎬ 2πT ⎩ k =0 ⎭
(
)
(7-8)
where bk is a smoothing function defined as [176]
bk =
⎤ 1 ⎡ L2 − 1 − k (k + 1)⎥ ⎢ 2⎣ 4 ⎦
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By substituting the estimated frequency offset value ∆fˆ in (7-7), the carrier phase estimates are obtained as ⎛ L −1
⎞
θˆ = arg{max⎜ ∑ a ∗ (kT )e − j 2 ∆fkT z (kT ) ⎟} θ
ˆ
⎝ n =0
⎠
(7-10)
Thus, in order to obtain phase estimates using the algorithm in (7-10), frequency offset has to be obtained first using (7-8). Notice that the frequency estimate in (7-8) is a weighted average over the phase increments of two successive MF samples z ((k − 1)T ) z ∗ (kT ) . The weighting function (window) is at its maximum at the center of the estimation interval and decreases quadratically toward the boundaries of the observation window. The maximum phase increments that can be uniquely estimated using (7-8) defines the maximum usable frequency estimation range of this algorithm. This is obtained by making the argument in (7-8) equal to π , which results in the frequency estimation capture range of (7-8) defined as ∆f max ≤
1 2T
(7-11)
To perform frequency correction according to the system in Figure 7-1, the estimates of ∆fˆ and θˆ are further fed into a direct digital complex synthesizer (CNCO) to synthesize signals of exp{− j 2π∆fˆt} and exp{− jθˆt} for frequency offset and phase correction.
7.4 Estimating the Error Variance and Lower bounds Both frequency offset and phase rotations estimates are bounded by a form of the modified Carmer-Rao bound derived in Chapter 5. This bound is used as a benchmark check and a comparison reference for analyzing and comparing different estimators. The MCRB defined in Chapter 5 for frequency and phase offset estimates are restated as
( )
MCRB ∆fˆ =
⎛ 1 ⎞ 3T ⎟ ⎜ 3 ⎜ 2π (LT ) ⎝ Es N 0 ⎟⎠ 2
(7-12)
and
()
1 ⎛ 1 ⎞ ⎟ ⎜ MCRB θˆ = 2 L ⎜⎝ Es N 0 ⎟⎠
(7-13)
Both bounds are functions of the observation window length (L) and the ratio of Es N 0 . To illustrate the use of these bounds, assume that the carrier phase recovery estimate in (7-10) is obtained using L = 100 symbols and Es N 0 = 10 dB. Using (7-13), the minimum variance bounded by the nonapproachable MCRB of the phase error variance is 5x10-4 rad2. If a conventional Costas feedback loop is used instead of the open loop estimator, carrier phase error variance of 5x10-4 requires a bandwidth of 50 Hz, which is practical for most TDMA burst systems. In practice, the loop bandwidth must be much smaller than 50 Hz, as the practical synchronizer error variance never approaches the MCRB. If the Dr. Mohamed Khalid Nezami © 2003
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same transmission parameters are used for the carrier frequency offset estimator, then the minimum frequency estimate bound is MCRB ∆fˆ T 2 = 15.2 x10 − 6 (i.e., 15.2 PPM for 10ksps).
( )
7.5 Feedforward Frequency Estimation Algorithms A carrier frequency offset recovery system accomplishes two basic functions, (a) it derives an estimate of the carrier frequency offset ∆fˆ , and (b) it compensates for this offset by counter-rotating the received signal z (kT ) (as shown in Figure 7-1) at an angular velocity of 2π∆fˆt . Most open loop FF estimation algorithms have a maximum frequency estimation range on the order of 10% of the transmitted symbol rate (i.e., ∆fˆ ≤ 1 /(10T ) ), which is of practical significance to modern communication receivers. If z (kT ) has a frequency offset that is larger than 1 /(10T ) , an AFC loop has to precede [88] the FF algorithm. In Chapter 8, we present a new scheme that improves the FF estimation algorithms to have a range that is on the order of the transmitted data symbol rate ( ∆fˆ ≅ 1 / T ). The frequency estimate in (78) performs data modulation removal using the available preamble sequence by performing the differential multiplication of (a ( kT )a ∗ (( k − 1)T ) . The preamble may not be available for most short burst TDMA systems. Therefore, it is desirable to obtain frequency estimates without use of preamble. These non-data aided (NDA) algorithms remove data dependency by processing the MF signal with a nonlinearity as in Chapter 3. By doing so, the NDA technique eliminates the overhead required for the preambles, thus yielding high channel throughput. 7.5.1
The M-Power NDA Frequency Offset Estimation Algorithm
The M-power non-linear frequency offset estimation algorithm is the most commonly used technique in satellite and cellular receivers. To derive the mathematical representation, we represent the demodulated MF baseband signal as
z(t ) = e j ( 2π∆ft+θ (t )+φ ( t )) + n(t )
(7-14)
where n (t ) is a complex AWGN with real and imaginary parts that are independent with identical variance of σ n2 = [N 0 / 2 E s ] and where E s is the energy per information symbol. Here ∆f is the
frequency offset to be estimated, φ (t ) is a phase term corresponding to the transmitted symbol, and θ (t ) is the phase term that includes carrier phase error and phase rotation introduced by the propagation channel, which needs to be estimated. In order to perform frequency offset estimates using (7-14), the signal z(t) is first passed through a differential detector to remove the slow varying phase components of θ (t ) . That is
z (kT ) z ∗ ((k − 1)T ) = exp{ j (2π∆fkT − 2π∆f ((k − 1)T ) + θ (kT ) − θ ((k − 1)T ) + φ (kT ) − φ ((k − 1)T )) + n′(kT )} (7-15) '
where n ( kT ) is statistically equivalent to n(kT ) . As the carrier frequency variation from symbol to symbol is assumed to be negligible, the differential detection in (7-15) causes cancellation of the term Dr. Mohamed Khalid Nezami © 2003
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j (φ ( kT ) −φ (( k −1)T ))
unity. The symbol differential phase term e can ∆f , making the term e now be removed by processing the signal through an M-power nonlinearity [12], where M is the number of phase states in the modulation constellation. Applying the M-power non-linear operation to the signal in (7-15), its output is
y (kT ) = e jM ( 2π∆fkT ) + n ' (kT )
(7-16)
The signal in (7-16) is a noisy sine wave that has a frequency corresponding to M-times the final carrier offset. For example, the QPSK modulation removal is achieved by using M=4 in (7-16), as π
e
4 j ( ± n) 4
= 1 for n=0,1,2,3..
(7-17)
For more complex modulations such as QAM, (7-17) may not yield complete modulation removal because of residual amplitude and phase modulation. This tends to degrade the performance of this algorithm. More details on this issue are presented in later sections. Let the kth complex sample of the differential baseband signal in (7-16) have a real part Re( y k ) = I k and an imaginary part Im( y k ) = Qk . If the current MF sample is y k = I k + jQk and the previous sample is y * k −1 = I k −1 − jQk −1 , the resultant frequency error detector is,
y k y k∗−1 = ( I k −1 I k + Qk −1Qk ) + j ( I k −1Qk − I k Qk −1 )
(7-18)
The frequency error detector in (7-18) contains commonly known terms The first is known as the dot product term I k −1 I k + Qk −1Qk = cos(2πM∆fT ) ; the second is known as the cross product term j ( I k −1Qk − I k Qk −1 ) = sin(2πM∆fT ) . Further simplification of (7-18) using the dot and cross terms leads to an initial frequency acquisition estimate given by
2πM∆fT = arg{ y k y k∗−1}
(7-19)
. = tan −1 (.) . To reduce random variations of (7-19) caused by AWGN and phase residuals where arg{} from data modulations, it is further averaged over L-symbols (observation interval) yielding a final NDA carrier offset estimator algorithm given by ⎧ L (I k −1Qk − I k Qk −1 ) ⎫⎪ ∑ ⎪ 1 ⎪ ⎪ tan −1 ⎨ kL=1 ∆fˆ = ⎬ 2πMT ⎪ ∑ (I k −1 I k + Qk −1Qk ) ⎪ ⎪⎭ ⎪⎩ k =1
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1 Hz, or roughly 10% of the transmitted 2 MT data rate for QPSK and QAM modulations. For example, a 10 ksps QPSK satellite modem using M=4 is able to cope with frequency Doppler as high as ∆f = ±1250 Hz.
This algorithm has a maximum capture range of ∆f ≤ ±
The algorithm (7-20) can be fully implemented by using available low cost digital signal processors or by use of field programmable gate arrays (FPGAs) [13]. Figure 7-3 shows one possible implementation of this algorithm using DSP techniques. Symbol Timing
Re{x}
Ik
Qk I k −1 − Qk −1 I k
x
Non-linearity z
[x]
M
MF/DEC
−1
z
Baseband signal
-
x
⎧4 , QPSK M =⎨ ⎩2 , BPSK
Im{x}
L
x 1 arg{.} 2πMT
Qk Re{x}
∑( )
+
−1
Ik
∆fˆ
x z z
−1
∑( )
+
L
−1
Im{x}
I k I k −1 + Qk −1Qk
x Qk
Figure 7- 3: Digital Implementation of M-power NDA Frequency Offset Estimation Algorithm. 7.5.2
Viterbi NDA Frequency Offset Estimation Algorithm
Another variant of the NDA nonlinear algorithm in (7-20) was developed by Viterbi [14]. This algorithm employs two separate nonlinearity operators, one operating on the argument using M arg (z k z k∗ −1 ) and the other operating on the magnitude of the differential detector output using
( z z ) . Applying both nonlinearities to (7-18), the frequency offset estimates becomes k
∗ k −1
l
∆fˆ =
(
)
l ∗ ⎧ L −1 ⎫ 1 arg ⎨∑ d m z k z k∗−1 e jM (arg( zk zk −1 ) ) ⎬ 2πMT ⎩0 ⎭
(7-21)
Here d m is a smoothing filter intended (a) to provide robustness while operating at low SNR and (b) to counteract slow signal variations that may be present due to multipath fading attenuation. Figure 7-4 shows one possible implementation of this algorithm using DSP techniques. In our simulation experiments, the smoothing filter is implemented as a first order low-pass filter represented by H ( z) =
1
(7-22)
1 − µz −1
where µ is a constant such that 0 ≤ µ ≤ 1 . The estimates in (7-20) and (7-21) have a capture frequency offset range of ∆f ≤ ± Rs 2 M . There has been some work reported by [27] that used polar coordinates Dr. Mohamed Khalid Nezami © 2003
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mathematical techniques (CRODIC) to further reduce the computational complexity of (7-21). It was shown [27] that such an approach reduces the computational load by a factor of 2. Matched Filter
BPF
Received Signal
M arg(zk )
-
z −1
x
+
zk Rectangular to Polar
Polar to Rectangular
(zk )l
zk
X
z −1
y
Smoothing filter
x
Smoothing filter
arg{y, x}
1 2πMT
∆fˆ
Figure 7- 4: Digital Implementation of the Viterbi NDA Frequency Estimation Algorithm. 7.5.3
DFT-based NDA Frequency Offset Estimation Algorithm
Another technique for estimating the carrier frequency offset without preamble (NDA) is to use the DFT algorithm [47]. Unlike the algorithms discussed in (7-20) and (7-21), this algorithm does not require symbol-timing estimates and can be cascaded with the symbol timing estimation in Figure (6-4). Figure 7-5 shows a digital implementation of this algorithm.
e
+j
2πk N
X
∑( )
X
∑( )
L
z (kT )
Matched filter
BPF
X
Received Signal Z
( )∗
−1
z (( k − 1)T )
e
Complex Signal Real Signal
2πk −j N
L
C +1
C−1 X
∆f
1 arg ( 4πT
)
Figure 7- 5: Digital Implementation of the DFT-based NDA Frequency Estimation Algorithm. The frequency-offset of this algorithm is given by Dr. Mohamed Khalid Nezami © 2003
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∆fˆ =
1 ⎛ k =L ⎞ arg⎜ ∑ [C−1 (k )C+1 (k )]⎟ 4πT ⎝ k −0 ⎠
(7-23)
where
C −1 (k ) = ∑ z (k )z (k − N )e
j
2πn N
C +1 (k ) = ∑ z (k )z ∗ (k − N )e
j
−2πn N
∗
(7-24)
L
and (7-25)
L
1 Hz, where N is the number of NT samples per symbols. For our simulation experiment, N is 4 samples per symbols, yielding a capture 1 range of ∆f ≤ , which is twice the range obtained using (7-20) or (7-21). Studies indicate [15] that 4T the algorithm (7-23) will not yield acceptable performance, unless the matched filter signal is post filtered using a narrow bandpass-shaping filter. The purpose of this shaping filter is to reduce the variance of estimates, but it results in reducing the capture range of the algorithm. For our simulations, we varied the filter bandwidth depending on the Roll-off factor ( α ) of the MF signal. The filter is centered at the spectral component of interest in z (kT ) that contributes to the periodic component of
The estimate in (7-23) has a maximum capture range of ∆f ≤
interest in z ∗ (( k − 4)T ) z (kT ) . The simulations used are a BPF center frequency of 1 /( 2T ) Hz and a bandwidth of α / T Hz.
Experimental Results: The performance of the estimators in (7-20), (7-21) and (7-23) were evaluated using simulations. To this end, the estimator capture range, as well as the estimator mean and error variance, is determined as a function of E s / N o . A strip of 16000 random QPSK and 16-QAM shaped symbols is transmitted using the model in Figure 2-16. The symbols are then filtered using a matched filter with a specified Roll-off value. The estimators in (7-20), (7-21), and (7-23) were then invoked in parallel. Their estimates are passed to the model in Figure 2-16, which then performed calculations of error variance, mean estimates and capture range as a function of Roll-off value, E s / N o , observation length L, and the intentional frequency offset introduced ranging from –5000 Hz to +5000 Hz for the rate 10ksps. Figure 7-6 shows the capture range of the algorithms in (7-20), (7-21), and (7-23) for QPSK symbols running at 10 ksps sampled at 40 ksps (N=4) and observation length of L=100 symbols, E s / N o = 10dB , and a Roll-off factor of 0.75. The frequency offset estimates in (7-20) and (7-21) are unbiased in the capture range of ± 1.25 kHz, while the estimates of (7-23) is biased but has a wider capture range of ± 2.5 kHz. The bias is removed by using a BPF with 3 dB edges at 5 kHz and 15 kHz. Figure 7-7 shows the results with the BPF. Though the estimates have no bias, their estimation capture range is reduced from ± 2.5 kHz to ± 700 Hz. Figure 7-8 shows the estimator performance for a smaller Roll-off factor of 0.35. The bias for the estimator in (7-23) worsens as Roll-off factors get smaller (because of signals with Dr. Mohamed Khalid Nezami © 2003
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narrower access bandwidth), while the estimators in (7-20) and (7-21) maintained no bias and the same capture range. Figure 7-9 plots the estimated frequency error variance against E s / N o . Because of the bias in the estimation in (7-23), the variance approaches an irreducible variance floor of σ ∆2f T 2 = 2 x10 −4 at
E s / N o =12 dB. The estimators in (7-20) and (7-21) have better performance, exhibiting a variance floor of σ ∆2f T 2 = 3x10 −7 at E s / N o = 20 dB. 5000 L=100 Es/No=10 dB Roll-Off=0.75 QPSK BPF none l=M =4
Mean of estimated frequency error (Hz)
4000
3000
2000
DFT -based
1000
0
-1000 MP and Viterbi
-2000
-3000
-4000
-5000 -5000 -4000 -3000 -2000 -1000
0
1000
2000
3000
4000
5000
Introduced frequency error (Hz)
Figure 7- 6: Capture Range (in Hz) of the Algorithms in (7-20), (7-21), and (7-23). 2000 L=100 Es/No=10 dB Roll-Off=0.75 QPSK BPF: 5-15 kHz l=M =4
Mean of estimated frequency error (Hz)
1500
1000
DFT -based
500 0
-500 MP and Viterbi
-1000 -1500 -2000 -2000
-1500
-1000
0
-500
500
1000
1500
2000
Introduced frequency error (Hz)
Figure 7- 7: Capture Range (in Hz) of the Algorithms in (7-20), (7-21) and Post-processed (7-23).
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Mean of estimates (Hz)
2000
1000
DFT -based
0
MP and Viterbi
-1000
-2000
-3000 -3000
-2000
-1000 0 1000 Freq. introduced offset (Hz)
2000
3000
Figure 7- 8: Capture Range (in Hz) of the Algorithms in (7-20), (7-21), and (7-23) for Roll-off value 0.35. 10
variance of estimate error
10
-2
-3
DFT based 10
10
-4
L=100 Roll-Off=0.75 QPSK BPF: 5-15 Khz µ=0.3
-5
∆f=500 Hz l=M =4
10
-6
MP/Viterbi
10
-7
5
10
15
20
25
30
Eb/No (dB)
Figure 7- 9: Variance of Frequency Estimates for (7-20), (7-21), and (7-23). 7.5.4
Window Enhanced DFT-based NDA Frequency Estimation Algorithm
One way to improve the estimates in (7-23) is by using windowing technique to convert (7-23) into weighted sums of phase differences. Kay [15-22] introduced two variants of this technique; the first variant has frequency estimates given by ∆fˆ =
1 2πT
∑ w(k ) arg{z(k ) z (k − 1)} L
∗
(7-26)
k =1
while the second variant has frequency estimates given by Dr. Mohamed Khalid Nezami © 2003
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∆fˆ =
1 ⎫ ⎧L arg ⎨∑ w(k )z ( k ) z ∗ ( k − 1) ⎬ 2πT ⎭ ⎩ k =1
(7-27)
where the smoothing window function is defined as
w(k ) =
6k (L − k ) L L2 − 1
(
)
, k=1,2,…L-1
(7-28)
Luise & W [23] introduced a modification of the algorithm (7-27) that produces a frequency estimate given by
∆fˆ =
1 ⎧ L −1 ⎫ arg ⎨∑ w( k )e j arg {z (k )}− j arg {z (k −1)} ⎬ 2πT ⎩ k =0 ⎭
(7-29)
Several approaches are used to simplify the windowing function in (7-26) and (7-27) [23]. One simplification uses a trapezoidal window in the weighted estimates, which introduced only a 0.5 dB of relative variance degradation while reducing their computational complexity. Another technique has been suggested to improved the Kay algorithm by 3 dB. This technique was to use the sample splitting function given by, a(k ) =
z (k ) + z (k − 1) 2
(7-29)
where z (k ) is the MF output given by, z (k ) = Ae j (2π∆fk +θ ) + nk
substituting z (k ) into (7-29), a (k ) = Ae
⎛ ⎛ 1⎞ ⎞ j ⎜⎜ 2π∆f ⎜ k − ⎟ +θ ⎟⎟ ⎝ 2⎠ ⎠ ⎝
⎛ ⎛ 2π∆f ⎜⎜ cos⎜ ⎝ ⎝ 2
' ' ⎞ nk + nk −1 ⎞ ⎟⎟ ⎟+ 2 ⎠ ⎠
' ' By inspection, it becomes apparent that the variance of nk + nk −1 is now 3dB less than the original noise
2 n k . Hence the threshold (the point at which the estimator performance meets the CRB) of this
modification of the Kay algorithm is lowered by 3dB. The final form of this estimator is given by ∆fˆ =
1 2πT
∑ w(k ) arg{a(k )a L −1
∗
}
(k − 1)
k =2
Experimental Results: The experiments described earlier were repeated for the algorithms in (7-26) and (7-27). The performance S-curves are presented in Figure 7-10. The figure shows that the estimators Dr. Mohamed Khalid Nezami © 2003
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are unbiased and the capture range for the estimator in (7-26) is the same as those of (7-20) and (7-21). However, the estimation range of (7-27) is smaller than that of (7-23). Figure 7-11 plots the estimators mean frequency estimates against E s / N o , for a frequency offset of -500 Hz. The estimators of (7-27) and (7-20) converge to the actual offset at E s / N o =4 dB while the estimates have an error of -100 Hz for E s / N o ≥ 15dB .
1500 L=100 QPSK M=4 Roll-Off=0.75 Es/No =10 dB
Mean of Estimated Freq. Offset Hz
1000
500
0 Sum(w(I)arg(I)
-500 -4
8
x 10
7 6 5
-1000
4 3 2 1 0 0
-1500 -1500
-1000
200
400
600
800
1000
1200
1400
1600
1800
2000
-500 0 500 Introduced Freq. Offset Hz
1000
1500
Figure 7- 10: Capture Range of Frequency Estimates Using the Algorithms in (7-26) and (7-27). 0 L=100 QPSK M=4 Roll-Off=0.75 ∆f=-500 Hz
Mean of Estimated Freq. Offset Hz
-100
-200
-300
-400
-500
-600 MP -700 0
5
10
15 20 E /N dB b
25
30
35
o
Figure 7- 11: Mean frequency estimates of the algorithms in (7-26) and (7-27).
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7.5.5
Symbol Auto-Correlation Based Frequency Estimation Algorithms
Several algorithms have been employed which utilize the sample-to-sample correlation function that involve more computational complexity. Assuming a MF output signal defined by z (k ) = c k e j ( 2π∆fkT +θ ) + n( k ) , where n(k ) is a complex AWGN, and {c k } are the transmitted complex data
symbols. (??)Since z (k ) depends on {c k } (data dependency). The data symbols (i.e., modulation) is removed from z(k) to isolate the synchronization parameter by the fact that p c k cˆ k∗ ≈ c k
2
= 1 , where cˆk
is a locally stored reference symbol (i.e., any arbitrary locally known data reference at the receiver) to remove modulations by multiplying z (k ) by {c k∗ } as shown in Figure 7-12. That is z (k )c k = c k cˆ k∗ e j ( 2π∆fkT +θ ) + n( k )cˆ k∗
(7-30)
where n(k )cˆ k∗ is statistically equivalent to n(k ) . Thus, the resultant of the preamble multiplication (correlations) by the MF samples (7-30) is also a noisy sinewave similar to that derived in (7-16). However, unlike that derived in (7-16), there has not been any use of M-order nonlinearity; therefore, it is expected that for frequency offset estimates using (7-30) it will not suffer noise enhancement due to the use of the high order nonlinearity used in (7-16) to remove the data modulations. Furthermore, the estimates out of (7-30) will not be M-fold This removes the need for ambiguity resolution. One way to obtain a frequency offset estimation based on the use of (7-30) is achieved by exploiting the samples correlation function given by R (m ) =
1 L −1 ∑ z (k )z ∗ (k − m ) , 0 ≤ m ≤ N − 1 L − m k =m
(7-31)
Here m is the auto-correlation lag and L is the observation interval. It can be shown that arg{R (m )} ∝ [2πm∆fkT + γ (m)] , where γ (m ) is a non-relevant term. The frequency offset estimations are then obtained using one of the three algorithms described below. Luise and Reggiannini Estimation Algorithm [24]: Here, the frequency offset estimates are related to the symbol correlation function (7-31) as
∆fˆ =
1 2π T
N
∑ w (m )[arg R ( m ) − arg R ( m − 1)]
(7-32)
m =1
where the window function is given by
w(k ) =
3[(L − m )(L − m + 1) − N (L − N ))] N (4 N 2 − 6 NL + 3L2 − 1)
(7-33)
and N ≤ L 2 . Fitz Estimation Algorithm[25]: Here the carrier frequency offsets are given by
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∆fˆ =
2 πN (N + 1)T
with a capture range of ∆f =
N
∑ arg{R(m )}
(7-34)
m =1
1 . 2 NT
L&R Estimation Algorithm [26]: Here, the frequency estimates are given by ∆fˆ =
1 ⎧N ⎫ arg ⎨∑ R (m )⎬ π (N + 1)T ⎩m =1 ⎭
with a capture range of ∆f =
(7-35)
1 . NT
It was shown in [28] that the estimates in (7-32), (7-34) and (7-35) achieve minimum frequency error variance for N= L / 2 which coincides with the MCRB in (7-12). The digital implementation of the algorithm in (7-35) is presented in Figure 7-12. The preamble symbols {c k } are stored locally or fed back from the soft decision data decoder, and then used to perform data modulation removal. The algorithm in (7-35) can also be used in conjunction with an M-power nonlinearity to implement NDA scheme. Figure 7-13 shows one proposed implementation of this approach.
Matched filter
X
(.) ∗
z −1
Received Signal
z −1
1 L −1 X
1 L−2
z
−1
1 L−N X
X
+
+
c k∗
X
Refence Preamble N
∑ R (m) 1
1 arg{.} ( N + 1)πT
∆fˆ
Figure 7- 12: Digital Implementation of the Algorithm in (7-35).
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Matched filter
F ( z ( kT ) )
∗
(.)
z
Received Signal
−1
1 L −1
z
X
−1
1 L−2
z
X
−1
1 L−N
X
+
+
X
N
∑ R(m) 1
1 arg{.} ( N + 1)πT
∆fˆ
Figure 7- 13: Modification of (7-35) to an NDA Algorithm. Experimental Results: The performance of the algorithms in (7-32), (7-34), and (7-35) using QPSK modulated signals were evaluated by Mengali and Morelli [24]. Figure 7-14 shows a comparison of the frequency error variance performance against E b N o with parameters N =64, L=128 symbols, and a Roll-off factor of 0.5. From an implementation point of view, it is observed that as N decreases in (7-32), 7-34), and (7-35), the computational load diminishes but the estimation accuracy degrades. Decreasing N also results in narrowing the estimation range. Riggiannini [28] implemented the estimation algorithm in (7-35) for a GSM burst receiver operating with data rate of 270.8 kbps, where an estimation range of ∆f = ±10 kHz was achieved in an acquisition time of less than 200 symbols at
Es =10 dB N0
For some applications, from an implementation point of view, the estimators of (7-32), (7-34), and (735) involve calculation of the auto-correlation function R(m ) , which is a computationally intensive task requiring ( 2 N − L − 1) L / 2 complex multiplication and ( 2 N − L − 3) L / 2 additions. There have been attempts to reduce this complexity through use of scalable VLSI technology and systolic VLSI implementation [30].
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Normalized variance
L&Walgorithm
10-8
MCRB
10-9 L&Rand Fitz
10-10 0
2
4
6
8
10
12
14
16
18
20
Eb/No-dB
Figure 7- 14: Performance of the Algorithms in (7-32), (7-34), and (7-35) Using QPSK Modulated Signals[28].
7.5.6 Carrier Frequency Offset Estimation Using Diversity Systems This estimator was based on using (7-35) for a digital diversity receiver [29] operating in wideband frequency selective multipath fading channel. The diversity schemes achieved better variance as a result of the increase of SNR. The main advantage of this algorithm is that it does not require channel state information (channel estimates), rather it requires knowledge of the auto-correlation function of the channel, which is given by
ρ (τ ) = E{hc (t + τ )hc ∗ (t )}
(7-36)
where hc (t ) is the channel impulse function, and τ is a specific delay spread. The channel autocorrelation function gives a measure of the rapidity of the channel variation due to multipath fading. The channel auto-correlation is ρ (τ ) = 1 , or 100% correlation when τ =0 and quickly fades to zero as τ becomes large. Hebley frequency estimates [29] are given by ∆fˆ =
L −1 L −1 ⎫ ⎧N 1 arg ⎨∑ R (m )∑∑ ρ (l1 , l 2 )c k∗ − l1 c k − m − l2 ⎬ π (N + 1)T l1 l 2 ⎭ ⎩ k =1
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where ρ (τ ) = E{hc∗ (l1 ) hc (l 2 )} , where hc (l1 ) and hc (l 2 ) is the channel impulse function obtained at two spatially separated receivers, located at l1 and l 2 . The algorithm in (7-37) was tested with two to four branches of a diversity system, and the estimator was found to be unbiased over a wide range of carrier offsets. It was found to be insensitive to channel delay spread and symbol timing errors. The effect of diversity on the estimator extended the estimation capture range from 5% to 10% of the symbol range with four diversity branches, and a relative variance reduction of 12 dB.
7.5.7
Frequency Offset Estimation Using Adaptive Digital Filter
The Algorithm shown in Figure 7-15, is used with DPSK modulations, here assuming that the baseband signal x(n) is given by x( n) = d n e j 2π∆fT
(7-38)
⎧ 0 , '0' For DPSK the data bit d n = e j (θ n −θ n −1 ) , and (θ n − θ n −1 ) = ⎨ . ⎩π , '1' The input to the decision device is given by
y (n) = x(n)Wˆ n∗
(7-39)
where the adaptive processor complex weight Wˆ n∗ is the estimates of e j 2π∆fT given by
Wˆ n∗ = (1 − α )Wˆ n∗−1 + αx n∗−1 dˆ n −1
(7-40)
where 0 ≤ α ≤ 1 is the convergence factor. xn t = nT
s(t )
Wˆn
LPF
e
−1
z
j 2 πf c t
( )∗
( )∗
dˆn yn
Figure 7- 15: Frequency Offset Detection and Compensation Using Adaptive Processor 7.5.8
Frequency Estimation using the Linear Least Square Curve fit method
If the phase of the received carrier is available, the data points of the phase can be modeled as a line equation from which the slope can be extracted to yield a frequency offset. This can be illustrated as follows. Assume that the phase of the received CW signal is given by
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The estimate of the frequency offset can be estimated by dφ (t ) = 2π∆f dt
(7-42)
and thus the frequency offset is obtained using ∆f =
1 ⎛ dφ (t ) ⎞ ⎟ ⎜ 2π ⎝ dt ⎠
(7-43)
So the frequency offset is simply estimated by extracting the slope of the phase time varying function of the received carrier curve. The slope is then found by the linear regression (curve fitting) to obtain the straight line φˆ(t ) = 2π∆fˆt + θˆ(t ) , which is closest to the measured phase line φ (t ) = 2π∆ft + θ (t ) by
(
)
2 minimizing the metric φ (t ) − φˆ(t ) . recursively by N
The least squares estimates of ∆fˆ and θˆ are then obtained
[
(
ek = ∑ φ (k ) − 2π∆fˆkT + θˆ k =1
)]
2
(7-44)
Where N is the number of phase estimates obtained. The minimum of ek in Equation (7-44) can be found de de by setting k = 0 and k = 0 , then solving for fˆ , dθˆ dfˆ N
∆fˆ =
N
Realizing that
∑k = k =1
1 2πT
N
N
N ∑ kθ k − ∑ k ∑ θ k k =1
k =1
k =1
⎛ ⎞ N∑k 2 − ⎜∑k ⎟ k =1 ⎝ k =1 ⎠ N
N ( N + 1) , and 2
N
N
∑k
2
=
k =1
2
(7-45)
N ( N + 1)(2 N + 1) , the estimates in Equation (7-45) is 6
simplified to 1 ∆fˆ = 2πT
N
∑ ( Ak + B )θ k =1
(7-46)
k
where
A=
12 NT ( N − 1)( N + 1)
(7-47)
and
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B=
−6 NT ( N − 1)
(7-48)
One disadvantage of this method is that during fading the phase estimates may change erratically, and hence produce bad frequency offset corrections. This can be avoided by smoothing the frequency estimates obtained in Equation (7-46) by using the following procedure:
•
For the first observation of N phase samples, ∆fˆavg (0) = ∆fˆ (0 ) .
•
For the second observation of N phase samples, ∆fˆavg (1) =
•
For all subsequent observations (bursts), the smoothed estimates is given by ∆fˆavg (n ) =
∆fˆavg (0) + ∆fˆ (1) 2
7 ˆ 1 ∆f avg (n − 1) + ∆fˆavg (n ) . 8 8
.
(7-49)
7.6 Feedforward Phase Estimation Algorithms After frequency offset estimation and correction, the matched filter output still has an arbitrary phase error that has to be estimated and compensated (refer to Figure 7-1). The phase estimation and correction process is similar to the frequency offset estimation and correction process, except that the input to the carrier phase estimator is the MF samples and not the differential samples defined by (7-15) used in frequency estimation. 7.6.1
The M-Power NDA Carrier Phase Estimation Algorithm
The phase error is estimated by calculating the average rotation of the constellation points. It is assumed that the signal is sampled once per symbol at the instants of maximum eye opening and that the input has no frequency offset residual from the prior frequency estimation algorithm. By processing the MF samples using a M-power nonlinearity similar to that applied in (7-16), the data modulation is removed, resulting in an output that contains only phase terms that correspond to the carrier phase. This phase term can be estimated using ML-based algorithms similar to those derived for frequency offset estimation [31]. The phase estimate is given by
⎧ L Im z (kT ) M ∑ ⎪ 1 ⎪ θˆ = arg ⎨ k L=0 M ⎪ ∑ Re z (kT ) M ⎪⎩ k =0
{
{
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}⎫⎪⎪ ⎬ ⎪ ⎪⎭
}
(7-50)
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By constraining the arg{ } function in (7-50) to within ± π , the phase estimates capture range is π π obtained as − . Figure 7-16 shows the digital implementation of this algorithm. The ≤ θˆ ≤ M M algorithm in (7-50) deteriorates in the presence frequency offset. The phase variation is worst at the edge of the observation interval θ = 2π∆fLT . However, if the observation interval (L) is kept relatively short, such phase variation can be assumed negligible. Re{ z M (kT )}
From matched filter
z(kT )
[z (kT )]M
Complex to Polar
1 arg ∑ ( M L
)
θˆ
Im{z M (kT )}
Figure 7- 16: Digital implementation of the M-power phase estimation algorithm. To reduce the phase error variance of (7-50) further while operating in fading channels, or in presence of residual frequency offsets, we propose smoothing the estimates of (7-50) using a lowpass filter. Figure 7-17 shows a first order recursive post-processing filter used for this purpose. The new modified phase estimates (7-50) after post-processing are given by
~
θ k = µθˆk + (1 − µ )θˆk −1
(7-51)
where 0 ≤ µ ≤ 1 is the convergence factor. The value of µ is used to control the time constant of this filter to reduce the error variance of the phase estimates and hence to counteract fade effects on the estimator. The bandwidth of this smoother is given by BWSPP =
(1 − µ ) 2T
(7-52)
For example, if the estimator is used in a mobile channel with a worst case Doppler frequency of 100 Hz, the filter bandwidth is chosen to be less than 100 Hz, which requires µ = 0.98 . Figure 7-18 shows the reduction of phase error estimation variance as a function of the smoothing factor µ . Receiver front end
zk
{ }
real z kM
1 L
k = L −1
∑
x
k =0
θˆ
~
arg{ x , y }
{ }
imag z kM
1 k = L −1 ∑ L k =0
y
θ
+ 1− µ
z
−1
µ
Figure 7- 17: Carrier Phases Estimate Post-processing.
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18
Relative variance reduction - dB
16
14 +
12
1− µ
z −1
10
µ
8
6
4
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Smoothing factor µ
Figure 7- 18: Theoretical Noise Variance Reduction Due to Using Post-processing. 7.6.2 Viterbi NDA Carrier Phase Estimation Algorithm
Viterbi and Viterbi [14] proposed a scheme that is a variant of the M-power algorithm in (7-50). It uses two nonlinearities; one for the magnitude and other for the argument of the MF samples, to yield better estimates for some specific transmission schemes. The Viterbi-Viterbi carrier phase estimate is given by
{ z(kT ) { z(kT )
⎧ L −1 ⎪⎪ ∑ Im 1 ˆ θ = arg ⎨ L0−1 M ⎪ ∑ Re ⎪⎩ 0
} }
⎫ e jM arg{ z (kT )} ⎪ ⎪ ⎬ l jM arg{ z ( kT )} ⎪ e ⎪⎭
l
(7-53)
where l is the order of the magnitude nonlinearity chosen based on the modulation type. Figure 7-19 shows one typical implementation of this algorithm. From matched filter
(.)
arg() .
l
Polar to Rectangular
Z I (kT ) ZQ (kT )
1 arg ∑ ( M L
)
θˆ
X
M
Figure 7- 19: Digital Implementation of the Viterbi Carrier Phase Estimation Algorithm.
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The nonlinearity l controls the contribution of higher-order noise terms to the estimator error variance E and has the most effect at low and moderate s . An optimum order of nonlinearity that minimizes the N0 E phase estimation error variance, depends on s and the residual frequency offset. The following limits N0 were obtained for optimum nonlinearity while using PSK modulations with no frequency offset present. F ( z (kT ) )Optimal
⎧⎪ z (kT ) M , =⎨ ⎪⎩ z (kT ) ,
Es N 0 → 0 Es N 0 → ∞
(7-54)
In another study, Aghvami [32] proved that F ( z ( kT ) ) = z ( kT ) (i.e., M=8) is an optimal nonlinearity 8
for QPSK modulations. Two other variants of the algorithm (7-53) were reported in the literature; the first estimator is given by 1 ⎡ 1 L −1 ⎤ 2 r (t ) k arg{z ( kT ) M }⎥ ∑ ⎢ M ⎣ L k =0 ⎦ θˆ = L −1 1⎡ 2⎤ r (t ) k ⎥ ∑ ⎢ L ⎣ k =0 ⎦
(7-55)
Here r (t ) is the MF output sample. The second variant is given by ⎫ ⎧ 1 L −1 j z (k ) M ⎪ ⎪ ∑ Im e ⎪ −1 ⎪ L k = 0 ˆ θ = tan ⎨ L −1 ⎬ ⎪ 1 ⎡ Re e j z ( k ) M ⎤ ⎪ ⎥⎪ ⎪⎩ L ⎢⎣∑ k =0 ⎦⎭
{ {
} }
(7-56)
Es =12 dB, the performance of the estimators in (7-55) and N0 (7-56) is comparable to the Viterbi estimator in (7-50). Table 7-1 presents a comparison of the computational complexity for three-phase estimation algorithms discussed above.
It is shown that for QPSK modulations at
For the Viterbi algorithm in (7-50), the complexity can be reduced by 2 using polar implementation and an averaging scheme that does not involve multiplication. A similar implementation of (7-53) is reported in [33] where the algorithm performance in fading channel is also improved by employing a diversity selection scheme based on a quality measure derived as part of the estimator.
Estimator M-power(7-50) algorithm Viterbi (7-53) algorithm Ragheff (7-55) algorithm Ragheff (7-56) algorithm Dr. Mohamed Khalid Nezami © 2003
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Real Addition 2(L-1) (L-1) 2(L-1) 2(L-1)
7-25
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Table 7- 1: Comparison of the Computational Complexity of Phase Estimators. Experimental Results: Using the computer model in Chapter 2 and a strip of 16000 random QPSK symbols running at 10 ksps and sampled at 40 ksps (N=4), an observation length of L=100 symbols, and Roll-off factor of 0.75, the estimators in (7-50) and (7-53) are invoked to estimate an intentional phase error of 10o. Figure 7-20 shows the mean phase estimates for both (7-50) and (7-53) using M= l =4, which indicate similar performance for both algorithms. The estimates achieve convergence for Es Es ≥ 5dB . Figure 7-21 plots the estimated error variance against , which shows that both N0 N0 E algorithms have an irreducible error variance of σ θ2 = 1x10 −6 [rad2] for s > 20 dB. N0
The same experiments were repeated using 16000 16-QAM symbols. Figure 7-22 shows the mean estimates variance against a sweep of intentional phase errors of − 60 o ≤ θ ≤ +60 o . The estimators are unbiased and have a capture range of − 45o ≤ θˆ ≤ +45o .
Estimated phase offset deg.
-10 L=100 Roll-Off=0.75 QPSK θ=10 deg. l=M =4
-10.5
-11
-11.5 0
10
20
30 Eb/No (dB)
40
50
Figure 7- 20: Mean Phase Estimates for the Estimators in (7-38) and (7-41) for QPSK Modulation.
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10
L=100 Roll-Off=0.75 QPSK θ =10 deg. l=M =4
-3
variance of phase error
10
-4
10
-5
10
-6
10
-7
10
0
10
20
30 Eb/No (dB)
Figure 7- 21: Variance of Phase Error Estimates Against
40
50
Es for the Estimators in (7-38) and (7-41) for N0
QPSK Modulation
60 L=100 Roll-off=0.75 16QAM M=4 Eb/No=10dB
Mean of estimates
40
20
0
-20
-40
-60 -60
-40
-20 0 20 Introduced ph. Error deg.
40
60
Figure 7- 22: Capture Range of Phase Estimators in (7-38) and (7-41) for 16 QAM. Both estimators were then invoked to estimate an intentional phase error of 10o. Figure 7-23 shows the mean estimates of (7-50) and (7-53) using M= l =4. Both algorithms have similar performance for 16E QAM and the estimates converge for s ≥ 11dB . Figure 7-24 plots the estimated error variance against N0
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Es for the same setup, where both algorithms have a final error variance of σ θ2 = 1.5 x10 −3 [rad2]. N0 Figure 23 illustrates 16QAM phase estimates with a phase error 10-degrees. Figure 24 shows the E variance of estimates of the same error as a function of s . N0
Mean of Phase Est. error deg.
-3 -4 L=100 Roll-Off=0.75 16QAM θ=10 deg. l=M =4
-5 -6 -7 -8 -9 -10 -11 0
5
10
15
20 25 Eb/No (dB)
30
35
40
Figure 7- 23: Mean Estimates Using (7-50) and (7-53) for 16-QAM Modulation -1
Variance of Est. error
10
L=100 Roll-Off=0.75 16QAM θ=10 deg. l=M =4
-2
10
-3
10
0
5
10
15
20 25 Eb/No (dB)
30
35
40
Figure 7- 24: Variance of Estimates Using (7-50) and (7-53) for 16-QAM Modulation Dr. Mohamed Khalid Nezami © 2003
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To study the impact of the post-processing smoother in Figure 7-17, the simulation experiments in Figure 7-20 were repeated with the modified estimator shown in Figure 7-17. Figures 7-25 and 7-26 present Monte-Carlo simulations of the phase estimates obtained by the modified algorithm in (7-50) using the convergence factor µ =1 and µ =0.997. Notice in Figure 7-26 that the estimator variance using the post processing smoother is greatly improved over that of the results with µ =1.
deg.
θ=10 deg L=50 Roll-Off=0.35 µ=1 QPSK
(dB)
Figure 7- 25: Phase Estimates Using the Non-modified M-power Phase Estimator Algorithm in (7-38).
estimated carrier phase offset deg.
5 θ=10 deg L=50 Roll-Off=0.35 µ=0.997 QPSK
0
-5
-10
-15
-20
-25 0
5
10
15 Es/No (dB)
20
25
30
Figure 7- 26: Phase Estimates Using the modified M-power Phase Estimator Algorithm in (7-39).
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7.7 Extension of M-Power NDA Phase Carrier Estimator to QAM
This section generalizes the M-power nonlinear operation for M-ary modulations. Noise enhancement due to the non-linear operation is explained and derived. Furthermore, several schemes are detailed that attempt to reduce the noise enhancement problem that takes place after using the M-power multiplication process. 7.7.1
Removing MQAM PSK Modulations
With MPSK signals, the M-powering results in complete removal of data modulation, thus yielding accurate estimates of phase and frequency offset. For example, with BPSK symbols given by c k = a k + jbk , where {a k , bk } ∈ {+1, -1}, the symbol modulations are c k = e j ( m +1)π e jθ , where m=0,1. The data dependency can be removed by powering the samples to M=2, which results in [c k ]2 = e j ( m +1) 2π e j 2θ or [ck ]2 = e j 2θ . Note that phase shifts due to data modulation has been removed 1 2 completely, thus the phase estimates are given by θˆ = arg{[z ( k )] } , where z ( k ) is the modulated 2 j
π
baseband signal. Similarly for QPSK signals with symbols c k = e e 4
j(m)
π 2
e jθ , where m=0,1,2,3,
powering the signal samples by M=4 yields [c k ] = e j 4θ , and the phase estimates are obtained as 1 θˆ = arg{[z ( k )]4 } . Likewise, for 8-PSK modulations (i.e., GSM EDGE type modulations), the EDGE 4 modulations are completely removed by powering the EDGE signal samples by M=8, yielding the phase 1 8 estimates given by θˆ = arg{[z ( k )] } [34]. 8 4
With QAM modulated signal, the process is a little different. For instance, with 16-QAM modulated signals given by z ( kT ) = c k e j ( 2π∆fkT +θ ) + n( kT )
where the transmitted symbols are given by ck = a k + jbk , with {a k , bk } ∈ {±1,±3} and n ( kT ) is a
5 ⎛T ⎞ T ⎜⎜ ⎟⎟ , where is the ratio of symbol E s / N o ⎝ Ts ⎠ Ts rate to sample rate, the powering to M=4, can not remove completely the modulations as revealed by 4 examining the nonlinearity output [z ( kT )] , where with QAM symbols, this is given by,
complex AWGN; the variance of n(kT) is given by σ n2 =
( ) ( ) ( )
⎧ 2 4 e j 4 (2π∆fkT +θ ) ⎪ 4 [z(k )]4 = ⎪⎨ 10 e j 4 (2π∆fkT +θ +ϕ ) ⎪ 18 4 e j 4 (2π∆fkT +θ ) ⎪⎩
c k ∈ {± 1} others
(7-57)
c k ∈ {± 3}
⎛ 1⎞ The term ϕ = tan −1 ⎜ ± ⎟ is a residual phase error with a polarity dependent on the symbols transmitted. ⎝ 3⎠ Notice that (7-57) represents a sine wave modulated both in amplitude and phase. As the estimates of Dr. Mohamed Khalid Nezami © 2003
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∆f and θ are hidden in the arg{} of [z ( kT )] , it is clear that the presence of the phase modulation term ⎛ 1⎞ tan −1 ⎜ ± ⎟ complicates the carrier recovery in 16-QAM system. This distortion results mostly from the ⎝ 3⎠ 4
( )
phase-modulated term of 10 e j 4 (2π∆fkT +θ +ϕ ) because the other two terms of [z ( kT )] in (7-57) contain only amplitude modulations with a peak-to-average energy of 9.0 dB This does not introduce any phase distortion to carrier recovery because amplitude modulations are transparent to the M-power phase or frequency estimators when implemented using the arctangent function. 4
4
As a result of the inability to remove modulations completely, the use of M=4 nonlinearity for 16-QAM is not optimal. One way to alleviate the distortion in (7-57) is to use separate nonlinearities for amplitude (l-th powering) and Mth powering of the angle, which is equal to 4. Thus the MF samples are l
given by F ( z ( kT ) ) = z ( kT ) e
4 arg{ z ( kT )}
(7-58)
The relationship in (7-58) is the same as the algorithm developed in (7-21) and (7-53). The magnitude
( )
nonlinearity l is chosen such that it will emphasize the constellation point 18 e j 4 (2π∆fkT +θ ) , which has the highest energy in the constellation and has no distortion. A modified nonlinearity is employed so as to de-emphasize the other two terms in (7-57) to make the output close to a pure sine wave. This can be used to obtain reliable estimates of carrier phase and carrier frequency offsets. The nonlinearity is defined by l
0 , C k ∈ {± 1} ⎧ ⎪ F ( z ( kT ) ) ≅ ⎨ 0 , others l j 4 ( 2π∆fkT +θ ) ⎪ 18 e , C k ∈ {± 3} ⎩
( )
(7-59)
However, the performance of the system of (7-59) may be inadequate for high data rate 16-QAM systems operating with low SNR, as it suffers from two drawbacks: •
For short 16-QAM TDMA bursts it is probable that only a few symbols of the type c k ∈ {± 3} are transmitted in a single burst, which causes F ( z ( kT ) ) to be dominated by AWGN noise and the estimates in (7-21) and (7-53) to fail.
•
The use of high order nonlinearity l in (7-59) enhances the background noise, which is generated by the cross product terms of signal and noise in the powering operation [1]. Therefore, it would be beneficial to use an alternative scheme to remove QAM modulations without excessively enhancing the background noise. Following in this path, a new scheme is proposed in [35] for improving carrier recovery of QAM modulated signals by creating a special constellation that has carrier recovery error variance of the same order as that of QPSK systems. The price paid is only a slight increase in the constellation average energy and in the peak to average energy ratio (about 2 dB increase). However the scheme is not practically feasible, considering the huge infrastructure investment that has already been deployed using the conventional QAM constellation.
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7.7.2
Simulated M-Power 16QAM Carrier Phase Estimation Performance
Performance comparisons of the frequency offset estimators with different order of magnitude nonlinearity in (7-21) was achieved using computer simulations. Experiments used the model developed in Figure 2-16 and a strip of 16000 random 16-QAM symbols running at 10 ksps and sampled at 40 ksps (N=4) with an observation length of L=100 symbols and Roll-off factor of 0.75 and a frequency offset of +500Hz. The estimators in (7-21) and (7-22) were invoked sequentially to estimate phase and frequency offsets using M=4, l =4, l =8, and l =16. Figure 7-27 presents the frequency offset error variance of the carrier frequency offset estimator in (7-21) for several values of l . The nonlinearity order of l =16 provides the best performance as expected by (7-21). The experiment was repeated for different Roll-off values corresponding to the access bandwidth range of 20%-90%. Figure 7-28 shows E the performance of the three nonlinearity orders against Roll-off factor with s of 40dB. Again, the N0 nonlinearity with order of l =16 is the optimum one for the Roll-off access bandwidth range of 40% to 90%. The performance of l =8 and l =16 are similar for Roll-off range of 20% to 40%. -2
Variance of Estimation error
10
L=100 Roll-Off=0.75 16QAM BPF 5-15 Khz M=4 µ=0.3 ∆f=500 Hz -3
10
l=4
l =8
-4
10
l = 16
0
10
20
30 Eb/No (dB)
40
50
Figure 7- 27: Performance of 16 QAM Frequency Offset Estimator as a function of
Es for Three N0
Different Nonlinearity Orders with Roll-off of 0.75.
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10
Variance of Freq. est. error
l=4
-3
10
L=100 16QAM M=4 µ=0.997 ∆f=500 Hz Eb/No=40dB l =8
-4
10
l = 16
0.2
0.3
0.4
0.5 0.6 Roll-off value
0.7
0.8
0.9
Figure 7- 28: Performance of 16-QAM Frequency Offset Estimator as a Function of Roll-off for Three E Different Nonlinearity Orders with s = 40 dB. N0 The simulation experiments were repeated with the carrier phase estimator in (7-53) as a function of l . Figure 7-29 presents the phase error variance obtained for several values of l in (7-53). The optimal E E nonlinearity seems to be dependent on s . For moderate values of s , the nonlinearity of l =4 is N0 N0 E optimal, while for large s the optimal nonlinearity has order of l =16. As shown by Figure 7-30, the N0 E inferences are consistent throughout all Roll-off values for moderate s . N0 -1
Variance of Phs. Est. error
10
L=100 Roll-Off=0.75 16QAM M=4 θ=10 deg. -2
10
l=4
-3
10
l=8
l = 16 -4
10
0
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5
10
15
20 25 Eb/No (dB)
30
35
40
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Figure 7- 29: Performance of 16-QAM Phase Estimator as a Function of
Es for Three Different N0
Nonlinearity Orders with Roll-off of 0.75. -1
10
Variance of phs. est. error
L=100 16QAM M=4 θ=10 deg. Eb/No=10 dB
l=4
-2
10
l=8
l = 16
-3
10
0.1
0.2
0.3
0.4
0.5 0.6 Roll-off value
0.7
0.8
0.9
Figure 7- 30: Performance of 16-QAM Phase Estimator as a Function of Roll-off for Three Different E Nonlinearity Orders with s = 10 dB. N0
Es using Roll-off values in the access bandwidth range of N0 20% to 90%. Figure 7-31 confirms that l =16 is the optimal nonlinearity for signals with access bandwidth larger than 50%. The experiments were repeated for high
l=4
Variance of phs. est. error
-3
10
l=8 l = 16
-4
10
0.2
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L=100 16QAM M=4 θ=10 deg. Eb/No=40 dB
0.3
0.4
0.5 0.6 Roll-off value
0.7
0.8
0.9
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Figure 7- 31: Performance of 16-QAM Phase Estimator as a Function of Roll-off for Three Different E Nonlinearity Orders with s = 40 dB. N0 Next we propose a new method by which to remove the 16QAM modulations using sub-constellation partitioning. The idea is to use outer and inner square sub-constellation.
7.8 Proposed Scheme for NDA Carrier Recovery for 16-QAM Modulations
Based on the analyses and inferences of the previous sections, it was clear that a scheme can be devised to improve the16-QAM phase and frequency estimates by manipulating the constellation, such that only symbols that belong to certain sections of the constellation are allowed to be processed by the frequency or phase estimator. Figure 7-32 shows a partition of the 16-QAM constellation plane. Notice that there are two sub-QPSK constellations with a total of eight signal points. The first sub-QPSK consists of symbols corresponding to ck = {( +1,+1), ( −1,+1), ( −1,−1), ( +1,−1)} with normalized amplitude of c k = 2 .The
second
sub-QPSK
constellation
consists
of
the
symbols
of
ck = {( +3,+3), ( −3,+3), ( −3,−3), ( +3,−3)} with higher normalized amplitude of c k = 18 . The remaining symbols within the dashed region are devoid of any quadrature symmetry, and therefore represent the symbols that generate noise for the M=4 nonlinearity as shown in Figure 7-33. The idea for the proposed phase/frequency estimator algorithm is to distinguish between points from the sub-constellation and those from the excluded region as shown in Figure 7-34. If the received symbols lie within the subconstellation labeled Sub-QPSK 1, the M-power estimator is activated for regular QPSK using a low order magnitude nonlinearity with M=4 and l =1, otherwise consider the current phase/frequency estimates as the last estimate, or an interpolated estimate between the last estimate and n-estimates from previous intervals. By this approach, the noise enhancement problem is circumvented without any additional complexity. Note that the symbols corresponding to ck = {( +3,+3), ( −3,+3), ( −3,−3), ( +3,−3)} are easier to distinguish against other constellation points due to their higher energy. The schematic is shown in Figure 7-34. Here, the symbol discriminator (or a threshold detector) inhibits the estimator each time the signal power exceeds c k = 18 . The new carrier phase estimate is given by ⎧1 ⎧ L −1 j 4 arg ( z k ) ⎫ arg ⎨∑ z k e ⎬ , z ≥ 18 ⎪ 4 ⎩0 ⎭ k ⎪ ˆ θk = ⎨ ⎪ θˆk −1 , z k ≤ 18 ⎪ ⎩
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Sub-QPSK 1
ck = 18
3
2
Sub-QPSK 2 1
Q= Im{z k }
ck = 2 0
-1
-2
Excluded region -3
-4 -4
-3
-2
-1
0
1
2
3
4
I = Re{z k } Figure 7- 32: 16-QAM Constellations Diagram Illustrating the Excluded Region for the Proposed Carrier Recovery Algorithm Figure 7- 33: QAM Signal Through the 4th Power. A number of issues are made about the proposed scheme: • • •
Fading may degrade the scheme by distorting the symbol energy level relative to the fixed threshold. For short bursts, it is probable that only a few QAM symbols may be sent that lie in the second sub-QPSK constellation of c k = {(3,3), (−3,3), (−3,−3), (3,−3)} . The scheme has good performance for TDMA systems that utilize preamble (DD/DA), since the preamble can be made to include the QAM symbols ck = {( +3,+3), ( −3,+3), ( −3,−3), ( +3,−3)}. This can be sent periodically to assure presence of sub-QPSK symbols for each burst.
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Wait for next symbol
No L=?
Full Output L- symbols to the estimator
Received Signal
z(kT )
Matched filter
L- symbol Buffer
I 2 + Q2
≥ 18
Normalize
Wait for next symbol,
No
θˆk = θˆk −1
For phase estimation remove this block
4 arg(zk )
Yes
z −1
x
-
+
Rectangular to Polar
Polar to Rectangular
zk
⎧ Rs ⎪⎪ scale = ⎨ 2πM1 ⎪ ⎪⎩ M
for
(zk )4
z −1
∆fˆ
for θˆ
∆fˆ ,θˆ
scale
X
y
Smoothing filter
x
Smoothing filter
arg{y, x}
Figure 7- 34: Schematic of the Proposed Algorithm. 7.9 Effects of Frequency Residual on Phase Estimation
Optimal performance of the carrier phase recovery using the algorithms discussed above are optimal only for AWGN channels The phase estimator input samples have no frequency offsets associated with it. However, in practical situations, there is always non-zero frequency residual error caused by the inaccuracies of the frequency-offset algorithm used [36, 37]. As a result of such frequency-offsets, the phase estimate is no longer a constant within one observation interval, but is biased by a phase error that is a linear function of the index of symbols used during the observation interval. This bias is worst at the edge of the estimation interval (k=L) and is equal to θ (k ) = 2π∆fTL degrees. For a given frequency offset, there is an optimal phase estimation interval ( Loptimal ) that results in minimum phase error variance given by [36] ⎡ 1.165 ⎤ Loptimal = int ⎢ ⎥ ⎣ Mπ∆fT ⎦
(7-60)
Table 7-2 presents the results of one experiment conducted by [37] using QPSK modulations using the M-power NDA algorithm in (7-50). The operating SNR is 10 dB and the frequency residuals were ∆fT = 1x10 −3 and ∆fT = 2 x10 −3 . Notice that the listed simulation values of Loptimal highly agree with the computed (7-60).
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∆fT
2
2
var{ θˆ − θ }
L (arbitrary)
var{ θˆ − θ }
∆fT = 1x10 −3 100
3o
200
5o
∆fT = 2 x10 −3 50
6o
100
10o
Loptimal
Table 7- 2: Optimum Observation Length for NDA in the Presence of Frequency Residual Error [37]. From Table 7-2 and [36], one can infer that phase recovery in the presence of frequency offset can not be improved by simply increasing L infinitely, as its is conventionally believed according to the MCRB (7-13) for AWGN channels. For a given frequency offset, there is a singular observation length for FF systems that yield optimal phase recovery performance. Mengali [37] derived a phase estimate error variance that coincides with the modified Cramer-Rao bound and that include the effects of frequency offset residual errors is given by, var( θˆ − θ ) =
1 2 Lρ (∆fT ) E s / N o
(7-61)
where
ρ ( ∆fT ) =
sin(π∆fTL) L sin(π∆fT )
(7-62)
For the case where no frequency offsets are present, ρ ( ∆fT ) =1, and (7-61) becomes the MCRB in (713). By maximizing (7-62) with respect to L the optimal window ( Loptimal ) can be derived as ⎧ sin(π∆fTL) ⎫ Loptimal = max ⎨ ⎬ L ⎩ L sin(π∆fT ) ⎭
(7-63)
Figure 7-36 shows plots (7-63) as a function of L for several values of ∆fT . The figure shows that for a normalized offset of ∆fT = 1x10 −3 , Loptimal is of the order of 100 symbols This agrees closely with the computed value obtained using (7-60).
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Loptimal = max (.) L
∆fT = 1x10−3
∆fT = 3x10
∆fT = 4 x10−3 0
20
40
60
∆fT = 2 x10−3
−3
80
100
120
140
160
180
200
L − symbols
Figure 7- 35: Optimal Observation Length for Phase Estimator, with Normalized Frequency Offsets Specified at ∆fT =0.0010, ∆fT =0.0020, ∆fT = 0.0030, and ∆fT =0.0040. Figure 7-37 plots (7-61), (7-62) and (7-13) as a function of ∆fT using L=100 symbols and E s / N o =10dB. Using the results in Figure 7-37, it can be determined that with a normalized frequency offset of ∆fT = 5 x10 −3 ( ∆f =50 Hz for 10 ksps), there is a phase error standard deviation of σ θ = 1.6 0 . This error results in a negligible BER degradation of 0.04 dB as per (7-1). However, if the offset increases to ∆fT = 9 x10 −3 (90 Hz for 10 ksps), the phase error standard deviation increases to σ θ = 4 0 , resulting in a large BER degradation that is ten-times (0.3 dB). 0
10
ρ ( ∆fT ) -1
10
L = 100 Symbols Es / N 0 = 10dB
-2
10
σ θ2 (∆fT )
σ θ2 (∆fT = 5 x10 −3 )
-3
10
σ θ2 (∆fT = 0) = MCRB -4
10
0
1
2
3
4
∆fT
5
6
7
9
8
-3
x 10
Figure 7- 36: The Relationship between (7-61), (7-62) and (7-15) as a Function of ∆fT Using L=100 Symbols and E s / N o =10 dB. Clearly, a maximum frequency residual tolerance can be specified for carrier phase recovery algorithms, where this frequency offset is based on the maximum phase error variance σ θ2 as a function of frequency error. For example, with the time duration of a one-burst interval of LT seconds, and for any Dr. Mohamed Khalid Nezami © 2003
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given minimum tolerable phase variation of σ θ , min , the maximum frequency offset is ∆f max =
σ θ ,min . 2πLT
Thus for a 10 Mbps receiver with L=100 symbols and σ θ , min = 1.8 degrees, the frequency stability required during the time of L=100 symbols is ∆f max = ±500 Hz.
7.10
Phase Ambiguity in M-Power Carrier Phase Estimation
The NDA algorithms developed in the previous sections were all based on the evaluation of the function arg(.) . As a result, the M-power carrier synchronizer of (7-21) cannot distinguish between an angle θ and an angle of θ + k 2 π M , where k = ±1,±2 ± 3..... . Thus, the estimator has an infinite number of stable operating points. To avoid this ambiguity problem, the phase estimates have to be limited within π . Common solutions can be categorized into three approaches. The first the valid range of θ ≤ M approach is to employ differentially encoding, which results in 3 dB of SNR loss. A second approach is to utilize unique words within a preamble [38]. That was detailed in Chapter 3. A third approach is to post-process the estimates using the feedback network shown in Figure 7-38, which is highly suitable for NDA algorithms at low SNR. The network uses negative feedback in conjunction with an estimate limiter that restricts the phase estimates from slipping into neighboring equilibrium by checking the difference between each two consecutive phase estimate. The modified phase estimate is given by ~
~
~
θ k = θ k −1 + mod{[θˆk − θ k −1 ],
2π } M
(7-64)
The drawback of this network is that it introduces convergence delay and may introduce cycle slipping into an otherwise feedforward synchronization system at low SNR [39].
θˆ
+-
−π M
~
θ
+
π M
z−1
Figure 7- 37: Post-processing Scheme to Resolve Phase Ambiguity and to Perform Smoothing. 7.11
Cycle Slipping Issues
Cycle-slip phenomenon can occur in NDA carrier phase estimates due to thermal noise, fading, interference, or frequency offset. To illustrate the phenomenon, assume that the phase θ to be estimated is a constant ∈ [− π M ≤ θ ≤ π M ] . Typically, the NDA FF estimate of θˆ exhibits small fluctuation about the correct phase θ . Occasionally, due to noise effects, the FF phase estimate may momentarily Dr. Mohamed Khalid Nezami © 2003
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leave the interval [− π M ≤ θ ≤ π M ] before returning to the ambient value of θ . Cycle slipping in feedback synchronizer is analyzed in terms of the mean time between timing slips [39]. For FB synchronizers (PLL and Costas loops), the mean of timing slips E{t slip } is defined as 2 BL E{t slip } =
π 2
exp(
P2 2πσ θ2
)
(7-65)
where BL is the loop bandwidth, σ θ2 is the estimated phase error variance, and P is the phase π , [39]. The estimates in (7-65) are highly discriminator period. For QPSK modulations, P = 2 dependent on the estimated phase error variance of the synchronizer that is also dependent on the channel dynamics and SNR. Channel degradations such as fading result in degradation of the synchronizer estimate, hence causing an increase in the estimate of (7-65). To assess the effects of fading on carrier recovery algorithms, one needs to develop a relationship between the estimated error variance and channel fade dynamic parameters, such as Doppler frequency, average SNR, and Rician factor. Such a relation is critical in optimizing the design of carrier recovery algorithms, and can be used by system designers to predict worst-scenario performance of the receiver. 7.12
Impact of fading on M-power NDA FF phase estimators
An important performance monitor of the carrier recovery algorithms discussed in this chapter is their ability to operate reliably in fading channels. In fading channels, the estimation error variance increases in proportion to the degree of channel fade experienced by the receiver. In addition, the cycle slipping increases per (7-65). In such cases, the MCRB described in (7-12) and (7-13) can no longer be considered valid representation of the minimum variance. One way to assess this deterioration is by deriving the relationship between the phase estimates of the carrier recovery algorithm and the fading channel dynamics represented by Doppler frequency, average SNR, and Rician factor. This relationship is obtained by formulating the joint probability density function of the received faded signal amplitude and estimated phase error. To validate this approach, the phase error variance is first derived for AWGN channels, which must coincide with the MCRB in (7-13). After validating the approach, it is extended to derive the same variance for Rician fading channels. The derived variance is then evaluated for the lower limits of the Rician factor (phase error variance for AWGN), and for the upper limits (variance for Rayleigh fading channels) when the Rician factor is zero. 7.12.1 Estimated Phase Error Variance in AWGN Channels
The phase error variance is used to determine the bit error deterioration contributed by the imperfect operation of carrier phase estimator as indicated in (7-1) through simulations. By using random bursts in conjunction with intentionally introduced phase errors (Figure 2-29), the phase estimate variance can then be calculated using hundreds of bursts at each SNR value. This variance is lower bounded by MCRB (7-13). Analytically, the variance can be determined from the joint probability density function (PDF), amplitude and the phase error of the received signal. The amplitude distribution is Gaussian for AWGN, and either Rayleigh or Rician distributed for multipath signals. To develop the phase error density function, define the received signal as r (t ) = s (t ) + n (t ) Dr. Mohamed Khalid Nezami © 2003
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where s (t ) is the information signal and n(t ) is a complex white Gaussian noise with zero mean and variance σ n2 . For phase modulated signals such as QPSK and BPSK, the transmitted information signal
can be represented by s(t ) = a(t )e j ( 2πf c t +θ (t )) ; where a (t ) is the information to be transmitted, f c is the operating carrier frequency and θ (t ) is the phase due to carrier rotation that has to be estimated for an AWGN channel. Assuming that the signal envelope and carrier phase are constant within the observation interval (L). The quadrature signal of the M-power estimator in Figure 7-17 is given by 1 N
x=
N −1
∑ real ( skM ) , y = k =0
1 N
N −1
∑ imag ( s
M k
(7-67)
)
k =0
Without any loss of generality, it can be assumed that θ = 00 and the error in carrier phase estimation is ϕ = θˆ − 0 , or ϕ = θˆ . The envelope of the received signal fed into the phase estimation algorithm (7-50) is given by R = x 2 + y 2 and an estimation error given by ϕ . The joint density function of the amplitude and phase error is given by P ( R, ϕ ) =
⎡ R 2 γR cos ϕ ⎤ e −γ / 2 R + exp ⎢− ⎥ 2 σn ⎦ 2πσ n2 ⎣ 2σ n
(7-68)
where σ n2 is the noise variance in (7-66). The average signal-to-noise ratio γ in (7-68) is given by, γ =
a2 / 2
(7-69)
σ n2
Equation (7-68) can be integrated over R to derive the phase error density function. That is p (ϕ ) =
∞ ⎡ R 2 γR cos ϕ ⎤ e −γ / 2 + R. exp ⎢ − dR 2 σ n ⎥⎦ 2πσ n2 ∫0 ⎣ 2σ n
(7-70)
By including the observation interval L in (7-50), equation (7-70) is evaluated as p (ϕ )L =
e − Lγ 2π
{ 1+
(
4πLγ cos ϕ exp(γL cos 2 ϕ ).Q − 2πLγ cos ϕ
)}
(7-71)
The subscript L indicates that the calculation is performed using L numbers of observed symbols. Figure 7-39 shows this phase density function evaluated for an AWGN channel using several values of γ of 0, 3, 6, 9 and 15 dB for L=1. The figure shows that the error distribution changes linearly with SNR. Figure 7-40 presents the effect of observation interval on the phase error variance with γ = 15 dB. The phase error density function improves (gets narrower) as L and γ increase. With L=1, the standard deviation of the recovered phase error is σ θ = 5.7 o , while for L=100 deviation is lowered to σ θ = 1.72 o .
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers 2.5 γ=15 dB
AWGN
2
12 dB
1.5
p (ϕ , R )
θ
9dB
1 6 dB
3 dB
0.5 0 dB
0 -2
-1
-1.5
-0.5
0
1.5
1
0.5
2
Figure 7- 38: Phase Density Function Evaluated for an AWGN Channel Using Several Values of γ of 0, 3, 6, 9 and 15 dB for L=1. 3
2.5
2
1.5
1 L=1000
L=1
0.5
L=100
L=10
0 -1
-0.8
-0.6
-0.4
0.2 0 -0.2 phase error (rad)
0.4
0.6
0.8
1
Figure 7- 39: Effect of Larger Observation Interval on the Phase Error Variance. The phase error variance at the output of the estimator in (7-50) can then be obtained by numerically integrating (7-75). That is π
σ ϕ2 = ∫ ϕ 2 p (ϕ )dϕ
(7-72)
−π
By evaluating (7-72), the estimated error variance of (7-50) for L symbols in AWGN is given by
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σ ϕ2 =
1 2γL
(7-73)
This variance coincides with the Cramer-Rao lower bound (7-13). Next we use the same approach to derive the phase error variance for fading channels. 7.12.2 Estimated Phase Error Variance in Fading Channels
For Rayleigh and Rician channels, the phase estimator in (7-50) can no longer deal with a constant phase during a single estimation interval; instead, the amplitude of the output of the nonlinearity will have either Rayleigh or Rician distribution envelope. The received signal in equation (7-66) is modified to include the fading impact. That is
r (t ) = a (t )e j ( 2πfct +θ (t )) + χ (t ) + n(t )
(7-74)
where a(t )e j ( 2πf c t +θ (t )) is the direct path signal, and χ (t ) is the diffused complex signal at the receiver having real and imaginary parts with zero mean and a variance σ x2 . The Rician factor is defined as the ratio of direct received signal power to the diffused total power at the receiver and is given by
K=
a2 / 2
(7-75)
σ x2
Since the phase is still uniformly distributed for Rician and Rayleigh channels, (7-74) can be rewritten as:
r ( t ) = R ( t ) e j ( 2 π f c t + ϑ ( t )) + n ( t )
(7-76)
where ϑ (t ) is the phase to be estimated and distributed in [ −π , π ) , and R(t ) = a 2 (t ) + χ (t ) 2 is the overall baseband envelope of the received signal. This envelope is either Rayleigh or Rician distribution. Since Rayleigh distribution is a special case of Rician distribution, that is distributed according to, p( R) =
⎡ R 2 + a 2 ⎤ ⎛ Ra ⎞ ⎟ exp ⎢− ⎥I0 ⎜ σ x2 2σ x2 ⎦ ⎜⎝ σ x2 ⎟⎠ ⎣ R
(7-77)
where I 0 (.) is the zeroth-order modified Bessel function of the first kind. Redefining SNR to account for the extra energy in the received signal due to the direct path, one gets, γx =
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a 2 / 2 + σ x2
(7-78)
σ n2
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σ x2 For a small Rician factor (K), (7-78) reduces to γ x = 2 , which is the SNR of the Rayleigh channel. For σn large K factor, (7-78) reduces to γ x =
a2 / 2
σ n2
which is the SNR for Gaussian channel derived in (7-75).
Hence the evaluation of estimated phases error density and phase error variance for Rician channels leads to a general solution that includes variance values for AWGN and Rayleigh channels. The phase error density function of the estimated phase error for the Rayleigh channel [40] is given by P (ϕ )L =
γ x L cos ϕ 1 + 2 2π (1 + γ x L sin ϕ ) 2π (1 + γ x L sin 2 ϕ )3 2
⎛π γ x L cos ϕ ⎞⎟ ⎜ + tan −1 ⎜2 (1 + γ x L sin 2 ϕ )1 2 ⎟⎠ ⎝
(7-79)
Likewise, the variance of the phase estimates for the Rician channel is given by σ ϕ2 =
{
(
)
}
π ∞ ⎡ y2 ⎛ ⎛ 2 K ( K + 1) ⎞ 2 2 K + 1 ⎞⎤ K + 1 −K ⎟dϕdy ⎟⎟⎥ 1 + y 2π cos ϕ e ( y 2 )cos ϕ [1 − Q ( y cos ϕ )] I 0 ⎜ y ⎜⎜1 + e ∫ ∫ ϕ 2 y exp⎢− ⎜ πγ x L γ x L ⎠⎦ γ x L ⎟⎠ θ =0 y =0 ⎣ 2 ⎝ ⎝
(7-80) where y = L R . For a given set of σ n2 , γ x , L, and K, the variance in (7-80) can only be solved using σn
numerical methods. Figure 7-41 shows a plot of this variance for several values of Rician factor K using L=10 [40]. For large values of the Rician factor (K→ ∞ ), where most of the received signal is due to direct line-of-site, the transmission channel becomes an AWGN channel and the variance reduces to the MCRB in (7-13). On the other hand, for small values of the Rician factor (K→ 0), where most of the received signal is due to diffused paths, the variance becomes an upper bound for phase estimates in Rayleigh channels. From the figure, one can predict the performance of carrier phase tracking in Rayleigh, Rician, and AWGN. This can supply a benchmark reference from which worst case estimates can be supplied for the design of receiver links operating in faded channels. To illustrate the use of (780) and Figure (7-41), assume that a mobile receiver is operating in AWGN channel with γ = 30 dB, the phase error variance obtained using the carrier phase estimator in (7-50) will have a variance that is on the order of 4.5 x10 −5 , or 0.38 o , which causes a BER degradation of 0.2 dB. If the mobile receiver operates in a Rician fading channel with K=3 dB instead, the estimator error variance will degrade to 0.9 o which causes the receiver BER to degrade by 1.3dB. Furthermore, if the channel Rician factor gets large, or becomes a Rayleigh fading channel, the estimated phase error variance is 7 x10 −4 , or a standad deviation of 1.9 o and hence cause the receiver BER to degrade by 4.3 dB.
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10
0
-1
Variance of Phase estimates error
K=0, Rayleigh 10
-2
K=10dB ~ AWGN
10
10
10
10
-3
-4
-5
-6
0
5
10
15
20
γ - dB
25
30
35
40
Figure 7- 40: The Variance of Estimated Phase Error for Several Values of Rician Factor K (dB).
7.13 Illustration Example: Computer Simulation of Data-aided Carrier Frequency Offset Estimation Algorithm
Generate a minimum shift-keying (MSK) signal with symbol rate of 9600 symbols per seconds, and then illustrate carrier frequency recovery using the various algorithms detailed in this chapter: If data preamble or ample symbols were available and known a prior to reception, then the modulation removal can be carried out using a conjugated complex version of this preamble that is generated locally. Then this conjugate replica is multiplied by the received signal that contains the same preamble symbols, thus removing the modulations and leaving the single with only carrier, channel, and symbol timing impairments that are to be estimated. Figure E7-1 illustrates this process. Here the preamble replica known prior is the sequence {a k∗ } .
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y (kTs ) = g k sk e z
− LT
⎛ ⎞ ⎛ T ⎞ j ⎜⎜ 2π ⎜ k +ε ⎟ ∆f +θ k ⎟⎟ ⎝ N ⎠ ⎝ ⎠
s
z
− LT
s
z e
1 gˆ k
− j 2π ∆ fkT
j (δθ + 2π T δf (k + δτ ) y 4k = ∆ak sk e
y3(kT )
y 2(kTs )
y1(kTs )
− LT
s
INT
s
z
− LT
− τˆ
εˆ
aˆ kM
e
− jθˆ
k
θˆ
∆fˆ ∗
∗
ak
∗
ak a
ak
∗ k
Figure E7- 1: Acquisition of Frequency Offset Using Data-aided AlgorithmAssuming that the received near baseband that includes symbol and carrier impairments during part or the entire preamble {a k } is given by y k = ck g k e j (2π∆f (kT +τ )+θ ) + nk
(E7-1)
where the symbols ak ∈{ck } , ck = eϕ k represents phase modulated symbols, and g k is the channel gain. The first step taken to estimate the synchronization parameters {τ , θ , ∆f } is to get rid of the phase modulation ck = eϕ k . Knowing that the signal yk contains part or all (if we are lucky) of the preamble or amble symbols the data modulations are removed by multiplying yk by a locally generated version of the preamble {a k∗ } . That is
z k = y k ak∗ = ck ak∗ g k e j (2π∆f (kT +τ )+θ ) + nk ak∗
(E7-2)
If the frame is known accurately, then the samples corresponding to yk during the preamble can only be separated. Then the process performed in (E7-2) will utilize the fact that ck a k∗ ≈ 1 , and thus the processed signal is now given by z k = g k e j (2π∆f (kT +τ )+θ ) + nk a k∗
(E7-3)
Now as a result of using the reference data ck in (E7-2), the term n k a k∗ contains much less variance than
what takes place when using non-linear processing ( y k + nk ) to remove modulations Also notice that the noise cross signal terms are no longer present. This is the fundamental reason why data-aided techniques will always converge to the Cramer-Rao bound, unlike the behavior of non-data algorithms. M
Now the signal in (E7-3) contains four errors that have to be recovered and removed (equalized), {g , τ , θ , ∆f } . Similar to the case with non-data aided, this multi- variable system can be separated. The first separation is based on the use of differential processing which removes symbol timing and carrier phase dependency in (E7-3). That is Dr. Mohamed Khalid Nezami © 2003
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(
)(
zk zk∗ −1 = e j (2π∆f (kT +τ )+θ ) + nk' e − j (2π∆f ((k −1)T +τ )+θ ) + nk'
)
(E7-4)
Considering that the de-coupled parameters {τ , θ } do not change from symbol to symbol, the differentially processed signal in (E7-4) is now given by, z k z k∗ −1 = e j 2π∆fT + nk''
(E7-5)
The signal in (E7-5) is now a sine-wave signal containing a frequency of ∆f . To estimate this frequency, the easiest way to do this is by averaging the entire preamble period, and then estimating the frequency by, ∆fˆ =
L −1 1 arg ∑ z k z k∗−1 2πT k =0
(E7-6)
1 1 , which is much larger than the ≤ ∆fˆ ≤ + 2T 2T range associated with the non-data estimate algorithms detailed in memo 3.
The estimates in (E7-6) now have a valid range that is −
There are two other algorithms that have been popular and outperform the estimates in (E7-6). These are the Fitz algorithm [1,3] and the Luis-Regianinni algorithm [2]. The frequency offset estimates based on the Fitz algorithm is given by, ∆fˆ =
1 πN (N + 1)T
N
∑ arg{R(m)}
(E7-7)
m =1
where 1 ≤ m ≤ N , and the condition on N is made based on the maximum frequency offset N ≤
1 . 2∆fT
The inner term in (E7-7) is the sample correlation function and is given by R ( m) =
1 L −1 z k z k∗−m ∑ L − m k =m
(E7-8)
where 1 ≤ m ≤ L − 1 . The L&R [2] frequency offset and is given by, ∆fˆ =
N 1 arg ∑ R (m) π ( N + 1)T m =1
(E7-9)
The value for N in (E7-9) is chosen based on an optimal criterion given by N = L / 2 . Notice that the difference in the estimate obtained using (E7-7) and the estimate using (E7-9) is in the averaging 1 1 for (E7-7) and ∆f ≤ for (E7-9). placement and in the frequency estimation range that is ∆f ≤ 2 NT NT Dr. Mohamed Khalid Nezami © 2003
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Also notice that the differential processing z k z k∗−1 (E7-5) to de-couple ∆f from θ ,τ was carried out over two consecutive samples or symbols, and also could have been performed on multiple symbols or samples, as illustrated in frequency estimator shown in Figure E7-1. The choice of how many samples or symbols to perform (E7-5) is totally dependent on the frequency maximum offset that is known, and on the maximum differential time that would still make phase and symbol offsets almost constants. The frequency estimation using z k z k∗−1 in (E7-6), (E7-7), and (E7-9) may not be the most optimal way to estimate the frequency offset ∆f from y k in (E7-1) after having removed the modulations using the local symbol reference. Noise data interaction and the noise associated with z k and z k∗−1 undergo a z k z k∗−1 is computed.
squaring process once
This squaring results in SNR degradation, and thus The product z k z k∗−1 produces the
increases the variance of the final frequency offset estimates. following terms,
(
)(
zk zk∗ = g k e j (2π∆f (kT +τ )+θ ) + ak∗ nk g k −1e j (2π∆f (( k −1)T +τ )+θ ) + ak∗ −1nk −1
)
∗
(E7-10)
Ignoring both timing and phase offsets for now, the signal with the frequency offset is given by
(
)(
z k z k∗ = g k e j 2π∆fkT + a k∗ nk −1 g k −1e j 2π∆fkT e j 2π∆fT + a k∗−1nk −1
)
∗
(E7-11)
Reducing inner product terms further, the modulation-striped signal in (E7-2) with the frequency offset is given by, ∗
j 2π∆fT
∗ k k
z z = g k −1 g k e 123 disturbance
⎞ ⎛ ak∗ nk ⎞⎛ ak∗ −1nk −1 ⎟ ⎜⎜1 + ⎟⎜1 + j 2π∆fkT ⎟⎜ j 2π∆fT j 2π∆fkT ⎟ g k −1e e ke ⎝ ⎠ ⎝14g4 ⎠ 4444424444444 3
(E7-12)
disturbance
The signal containing the frequency offset in (E7-12) now contains two sources of disturbances. The first comes as a result of the channel interaction with the signal, g k −1 g k , while the second comes from a k∗−1 nk −1 a k∗ nk . Both terms are disturbances that will increase g k −1e j 2π∆fT e j 2π∆fkT g k e j 2π∆fkT the variance of the final estimates of ∆fˆ .
the noise-data interaction,
To overcome this problem, we propose the use of the DFT processing on (E7-2) without having to resort to phase differentiation with respect to time that results in the disturbance terms in (E7-12). The method by which the frequency offset is extracted using DFT (i.e., implemented as FFT) is detailed in [7]. Using a proper notation, the DFT of the signal in (E7-2) is given by N −1
Z (k ) = ∑ z[n]e
− j 2π
k n N
, 0 ≤ k ≤ N −1
(E7-13)
n =0
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where z[ n] ≡ z k = g k e j (2π∆f (kT +τ )+θ ) + nk ak∗ . Substituting this in (E7-13), the N −1
(
Z (k ) = ∑ g k e n =0
j ( 2π∆f ( nT +τ )+θ )
∗ k
)
+ nk a e
− j 2π
k n N
(E7-14)
where the notation kT was replaced with nT . Expanding the inner terms of (E7-14) , the DFT of the modulation stripped signal (E7-2) is given by
N −1
Z (k ) = ∑ g k e
k ⎞ ⎛ j ⎜ 2π∆fnT + 2π∆fτ +θ − 2π n ⎟ N ⎠ ⎝
n =0
N −1
− j 2π
k n N
+ ∑ nk a e n =0 1 442443 ∗ k
(E7-15)
disturbance
The frequency offset ∆f now can be estimated by searching the magnitude of the spectrum of the DFT bins in (E7-5) for the maximum peak. That is N −1
∆fˆ = max ∑ g k e ∆f
n =0
k ⎞ ⎛ j ⎜ 2π∆fnT + 2π∆fτ +θ − 2π n ⎟ N ⎠ ⎝
N −1
− j 2π
k
N + ∑ nk ak∗ e n =0 1 442443 n
(E7-16)
disturbance
Now the symbol timing τ and carrier frequency θ is coupled and can not be estimated as a joint. If symbol timing offset is known, then the carrier phase is estimated by, ⎛ k ⎞ ⎛ k ⎜ N −1 N −1 j ⎜ 2π∆fnT + 2π∆fτ +θ − 2π n ⎟ − j 2π n N ⎝ ⎠ N ˆ + ∑ nk a k∗ e θ = arg⎜ max ∑ g k e ⎜ ∆f n = 0 n =0 1 442443 ⎜ disturbance ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
(E7-17)
Both carrier frequency and phase offsets can be estimated from (E7-2) by processing the signal through a DFT. Then by locating the maximum energy bin, its frequency corresponds to an estimate of the carrier frequency offset, and its phase corresponds to the carrier phase offsets. This algorithm has smaller disturbance, as shown in (E7-17), than when computing the frequency offset from z k z k∗−1 . Also realize that the phase estimation is also estimated from the same operation taken to obtain estimates of ∆f , unlike when the carrier phase offset can only be estimated after ∆f corrections and again after modulation removal. Another thing to notice here is that a joint estimation of both symbol timing and carrier offset estimates could have been obtained using the DFT in (E7-15), if the frequency offset were known beforehand. Figures E7-2 through Figure E7-8 illustrate the simulated performance of the algorithm in (E7-6) and (E7-16) in AWGN channel. In all of the figures, the the DFT is a better way to estimate the frequency offset from the signal that had its phase modulations removed. Figure E7-2 illustrates a test MSK signal at 9600sps , having a frequency offset of ∆f = −375 Hz , the reference random sequence used as the preamble, or pilots if you desire. Figure E7-3 illustrates the signal after modulation removal using the Dr. Mohamed Khalid Nezami © 2003
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reference known symbols, with the frequency offset evident in it as shown in the DFT of Figure E7-4. Figure E7-5 shows the estimation range achieved when using both algorithms, which verifies the 1 1 4 1 1 , where ± predicted analytical range of − ≤ ∆fˆ ≤ + =± = ± 9600 = ±19200 Hz . Figure 2T 2T 2T 2Ts 2 E7-6 illustrates the quantization error taking a place because of finite DFT resolution. Figure E7-7 and Figure E7-8 illustrate the performance of both algorithms (E7-6 & E7-16) as a function of Eb N o . The frequency offset was set to -1000Hz, and the SNR was varied. The DFT was 1024 bins wide, and the PN reference symbols were 300 for Figure E7-7 and E7-5 symbols for Figure E7-8. In both figures, the over sampling factor was 4 samples per symbols. The figures clearly show that using the DFT as an estimator after data removal is the most optimal. In the following memos, we will present more results on both algorithms, especially in reference to fading channels. Real part of RX and preamble PN sequence
4
I
2 0 -2 -4
0
50
100
150
200
250
300
350
400
450
Imaginary part of RX and preamble PN sequence 4
Q
2 0 -2 -4
0
50
100
150
200 250 sample #
300
350
400
450
Figure E7- 2: A Burst Signal with 1000 Symbols and 100 Preamble PN Sequence, Over Sampled at 4 Samples per Symbols, with ∆f = −375 Hz , and θ = 10 o , Eb N o = 8dB MSK Signal with h=1/2 and 9600sps.
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I
2 0 -2 -4
0
50
100
150
200
250
300
350
400
450
0
50
100
150
200 250 Sample #
300
350
400
450
4
Q
2 0 -2 -4
Figure E7- 3: Recovered Signal Containing Frequency Offset and Phases Removal Using z k z k∗−1 , Over Sampled at 4 Samples per Symbols, with ∆f = −375 Hz , and θ = 10 o , Eb N o = 8dB MSK Signal with h=1/2 and 9600sps.
4
2 1.5
4
corr
x 10
2 Rs=9600sps N=4 samples/symbol fs=38400
1.5 Estimated frequency offset -Hz
Estimated freq. offset - Hz
1 0.5 0 -0.5
-fs/2=19200 Hz
-1 -1.5 -2 -2
DFT
x 10
1 0.5 0 -0.5 -1 -1.5
-1.5
-1
-0.5 0 0.5 Intentional freq. offset - Hz
1
1.5
-2 -2
2 4
x 10
-1.5
-1
-0.5 0 0.5 Intentional frequency offset -Hz
1
1.5
2 4
x 10
Figure E7- 4: Estimated Range for both Algorithms, E b N o = 100dB , 9600sps; Over Sampling Factor is 4 Samples per Symbols, DFT has 4096 bins.
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10 DFT, 4096 bins
arg(z(t)*z(t-ts)
Estimation error -Hz
5
0
-5
-10
-15 -1.5
-1
-0.5 0 0.5 Intentional frequency offset -Hz
1 4
x 10
Figure E7- 5: Estimated Error of Both Algorithms, E b N o = 100dB , 9600sps; Over Sampling Factor is 4 Samples per Symbols, and DFT has 4096 bins, Blue is Algorithm (E7-6).
4000 3000
arg z(t)z*(t-ts)
Estimated offset-Hz
2000
PN = 300 symbols Ns=4 samles/symbol DFT=1024 bins Fo ff=375 Hz (exact bin)
1000 0 -1000
DFT
-2000 -3000 -4000 -10
-5
0
5
10
15
Eb /No
Figure E7- 6: Performance of the Frequency Offset Estimation Algorithms, DFT (E7-16) and the Phase Differentiation Algorithm in (E7-6). The PN sequence preamble was 300 symbols, over sampled by 4, and the frequency offset was fixed to 375 Hz (exact DFT bin).
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2
x 10
Frequency offset =-1000Hz PN=5 symbols Ns=4 samples/symbol
Estimated Frequency- Hz
1.5
1
DFT algorithm (16)
0.5
0
-0.5 phase diff. Algorithm (6) -1
-1.5 -10
-5
0
5
10
15
Eb /No -dB
Figure E7- 7: Performance of the Frequency Offset Estimation Algorithms, DFT (E7-16) and the Phase Differentiation Algorithm in (E7-6). The PN sequence preamble was 5 symbols, over sampled by 4, and the frequency offset was fixed to -1000 Hz (not exact DFT bin).
Standard deviation from introduced offset -Hz
6000 Freq-Off=300Hz L=50 symbols Ns=4 samples/symbol
5000
MSK, h=1/2 4000 Phase method differentiation method DPH
3000
2000
1000
DFT method
0
-1000 -6
-4
-2
0
2 4 Eb /No -dB
6
8
10
12
Figure E7- 8: Standard Deviation of the Estimates Using both Algorithms in (E7-6) and (E7-16), Illustrating That the DFT Based Estimates Perform Much Better Than the Differential Phase estimation Method
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500 PHD method Phase differentiation method
450
Freq.-Off=300Hz L=50 Symbols Ns=4 samples/symbol
Mean of estimates - Hz
400 350 300 250 200
DFT method
150 100 50 0 -6
-4
-2
0
2
4
6
8
10
12
Eb /No
Figure E7- 9: Mean of Estimates of 300Hz Offset Using Both Algorithms in (E7-6) and (E7-16), Illustrating That the DFT Based Estimates Perform Much Better Than the Differential Phase Estimation Method.
7.14 Illustrative Example: Computer Simulated Data-aided Carrier Phase Offset Estimation Algorithm
Using the figure illustrated in E7-1, discuss and simulate data-aided carrier phase estimation for the MSK signal used. Having corrected for the frequency offset ∆f in (E7-1), the carrier phase offset θ can be extracted from the same signal after removing the modulations, assuming that the frequency offset and symbol timing estimation and correction were performed accurately. The corrected baseband signal (E7-1) is now given by, y3k e − j 2π∆fkT = ck g k e jθ + nk e − j 2π∆fkT ˆ
ˆ
(E7-16)
To estimate and remove carrier phase rotations in (E7-16), the modulations ck have to be removed first. That is y3k ak∗ e − j 2π∆fkT = ak∗ ck g k e jθ + nk ak∗ e − j 2π∆fkT ˆ
ˆ
(E7-17)
Assuming that the disturbance terms nk ak∗ e − j 2π∆fkT are still some form of AWGN, the signal in (E7-17) is now reduced to, ˆ
y3k ak∗ e − j 2π∆fkT = g k e jθ + nk' ˆ
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The signal in (E7-18) now contains only carrier phase rotations, and thus estimation of this phase is obtained by, L −1
L −1
θˆ = arg ∑ y3 k a k∗ e − j 2π∆fkT = arg ∑ e − jθ = θ ˆ
k =0
(E7-19)
k =0
The estimate in (E7-19) is bounded by the − π ≤ arg( x) ≤ +π , which is much larger than the estimated range of the M-power non-data algorithms illustrated previously in memo 3.
20
STD and CRB of estimated phase -degrees
18
L=50 symbols Ns=4 samples/symbol
16
Phase-Off=10o
14 12 10 8 6 4 2 0 -10
σ θ ,CRB -5
0
5
10
15
Eb /No -dB
Figure E7- 10: Standard Deviation of DA Phase Estimates Using Both Algorithms in (E7-19).
7.15
Illustrating Example: Non-Data Aided Carrier Frequency Offset Estimation Algorithms
Using the figure illustrated in E7-11, discuss and simulate non-data aided carrier phase and frequency offset estimation for the MSK signal used. Non-data aided acquisition algorithms are those that can estimate and compensate symbol timing, channel gains, and carrier offsets without the use of either a priori such as preamble or unique words, Dr. Mohamed Khalid Nezami © 2003
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nor use any data decisions. These algorithms utilize nonlinear processing to strip the data modulations from the received signals leaving it with only channel impairments, such as channel fading gain, symbol timing, and carrier offsets. While these algorithms require no transmission of overhead, which is attractive from a transmission efficiency point of view, they tend to be noisy due to the noise enhancement arising from the use of non-linear processing.
y ( kTs ) = g k sk e
⎛ ⎛ T ⎞ ⎞ j ⎜⎜ 2π ⎜ k +ε ⎟ ∆f +θ k ⎟⎟ ⎝ ⎝ N ⎠ ⎠
y1(kTs )
z −LTs
y 2( kTs ) z − LTs
1 gˆ k
z e
y 3( kT )
− LTs
− j 2π∆fkTs
z
INT
− LT
− τˆ
εˆ
aˆ kM
y 4( kT )
e
− jθˆk
y 4 k = ∆ a k sk e j (δθ + 2πTδf (k +δτ ))
θˆ
∆fˆ
Carrier Phase Offset estimation/correction
Carrier Frequency Offset estimation/correction
Figure Figure E7- 11: Illustration of NDA Carrier Recovery Algorithms. Figure E7-11 illustrates two stages of carrier acquisition using NDA algorithms. The first algorithm estimates and corrects the frequency offsets associated with the input near baseband signal, while the second stage estimates the carrier offsets and corrects for them. Realize that frequency offset corrections can be carried out prior to timing, and thus can be performed at the sample rate or at multiple samples per symbols, while carrier phase estimation is carried out at the symbol rate. This is advantageous because most of the time the maximum frequency offset is known beforehand and thus the frequency estimation algorithm can decimate the number of samples per symbol to an appropriate rate that can accommodate this maximum frequency offset, and thus reduce the computational load of the algorithm. For frequency estimates, while ignoring channel gain, the input near baseband signal in AWGN channel is given by y1(kTs ) = e j (2π∆f ( kTs +τ )+θ k +φk ) + nk
(E7-1)
where e − j (2π∆f ( kTs +τ )+θ k ) is a phasor representing the symbol timing as well as the carrier frequency and phase offsets. Realize that the frequency estimation and correction can be carried out without the symbol timing offset error information τ k and without knowledge of the carrier phase θ k . This is true since the frequency estimation process (as will be detailed later) is based on differential sample cross-correlation functions that eliminates symbol timing and phase dependency in (E7-1). Frequency offset correction is then achieved by multiplying the signal in (E7-1) by a phasor that conjugates the frequency shift at the sample rate. That is y 2(kTs ) = y1(kTs )e − j 2π∆fkTs = ak e j (−2π∆fτ +θ k +φk ) + nk ˆ
(E7-2)
Even though the correction was done at the sample rate in (E7-2), the estimation of ∆f itself does not have to be carried out at the sample rate, since the maximum offset that ∆f max is known prior.
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To estimate the initial frequency offset ∆f in (E7-2), the phase modulations as well as symbol timing and carrier rotations have to be removed first. Realize that symbol timing and phase rotations from one sample to another (or multiple of them) over a reasonable interval are constants or slowly varying, that is τ k ≈ τ k −1 and θ k ≈ θ k −1 . This assumption will able us to cancel both parameters from (E7-2) by differential detection (sample cross-correlation), then the carrier phase modulations will be removed by nonlinear processing as described in [1,2,3,4], assuming that the carrier phase modulations is represented by s (kTs )e − jφk . The M-nonlinear processing of (E7-2) to remove MPSK modulations is carried out by equation (E7-3),
(y
[
y k∗−1 ) = s(kT ) s ((k − 1)T ) * (e j (2π∆f (kT +τ )+θ ) + nk )(e j (2π∆f ((k −1)T +τ )+θ ) + nk ) M
k
]
∗ M
(E7-3)
Expanding the inner terms, (E7-3) is reduced to,
(y
y k∗−1 ) = (s (kT ) s ((k − 1)T ) * e j (2π∆fkT + 2π∆fτ +θ − 2π∆fkT + 2π∆fT −τ 2π∆f −θ ) + n k' ) M
k
M
(E7-4)
Canceling terms in (E7-4), the nonlinearly processed signal that is fed into the frequency estimator is given by
(y
k
)
M
y k∗−1
(
= s ( kT ) s (( k − 1)T ) * e j 2π∆fT + n k'
)
M
(E7-5)
Equation (E7-5) now contains only terms that correspond to differential symbol detection and the Mpowering of the near baseband signal, which is better represented by (E7-6),
(y
k
y k∗−1
)
(
M
= s k s k∗−1
)
M
e j 2πM∆fT + n k''
(E7-6)
For MPSK, the term (sk sk∗ −1 ) = 1 , hence the nonlinearly processed signal in (E7-6), is now given by, M
(y
k
y k∗−1
)
M
= e j 2πM∆fT + n k''
(E7-7)
This is a sine wave signal with a fundamental frequency of M∆f with some variance due to the nonlinear processing of the noise. This frequency can then be estimated using spectral analysis algorithms (DFT) or by simply estimating the phase that is associated with (E7-7), that is ∆f ≈
M 1 arg ( y k y k∗−1 ) 2πMT
(E7-8)
To reduce the impact of the variance of this estimate due to the nonlinear noise processing in (E7-7), this estimate can be averaged over a finite interval of symbols. That is ∆fˆ =
L −1 M 1 arg ∑ ( y k y k∗−1 ) 2πMT k =0
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The estimate in (E7-9) is bounded by the non-ambiguous region of the arctangent function, that is − 1 2 MT ≤ ∆fˆ ≤ + 1 2 MT . Scatter plot 2
1.5
1.5
1
1
0.5
0.5 Quadrature
Quadrature
Scatter plot 2
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2 -2
-1
0 In-Phase
1
-2 -2
2
-1
0 In-Phase
1
12
14
2
Figure E7- 12: Frequency Offset of 1200 Hz with 9600sps QPSK. 5000 4000
Frequency Estimates-Hz
3000 2000 1000 0 -1000 -2000 -3000 -4000 -5000
0
2
4
6 8 Eb /No -dB
10
Figure E7- 13: Estimates Using (E7-9) with Offsets of 0Hz and 1200Hz, 9600sps with N=4, L=1000 Symbols.
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2500
L=25 symbols N=4 samples/symbols
Estimation error STD -Hz
2000
1500
1000
500
0
4
5
6
7
8
9 10 Eb /No -dB
11
12
13
14
15
Figure E7- 14: Estimation Variance and Mean Error (top) as a Function of Eb N o using L=25 Symbols (N=4), CRB is indicated (bottom)
5000 4000
Frequency Estimates -Hz
3000 2000 1000 0 -1000 -2000 -3000 -4000 -5000 -6000
-4000
-2000 0 2000 Frequency Introduced Error-Hz
4000
6000
Figure E7- 15: Estimate Range Using (E7-9) , 9600sps with N=4, L=1000 Symbols, and Eb N o = 10dB and Eb N o = 5dB . Dr. Mohamed Khalid Nezami © 2003
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7.16
Illustration Example: NDA Carrier Phase Estimation Algorithms
The signal, after it has been corrected with initial carrier frequency and symbol timing offsets, is given by, y3(kTs ) = e j (θ k +φk ) + nk
(E7-1)
where e j (θ k +φk ) is a phasor representing the carrier phase offsets and the phase modulations. To obtain a reliable estimate of θ k the phase modulations e j (φk ) have to be removed. One way to achieve this is to pass the baseband signal through a nonlinear M-power operator that cancels the modulations. That is
( y3(kTs ) )M
(
= e j (θ k +φk ) + nk
)
M
(E7-2)
which is then represented by, y kM = e jM (θ k +φk ) + nkM
(E7-3)
It will be shown using some example that for MPSK modulations, e jM (φk ) = 1 , and thus the modulation removal from (E7-1), leaving the signal only with the carrier phase shift ,that is y kM = e jMθ k + nkM
(E7-4)
However, the noise term nkM that has been enhanced by the non-linear processing is a major disturbance of these estimates, and will prove to be the limiting factor in using these algorithms at low SNR especially for Turbo coded signals. Using (E7-4), it is clear that the carrier phase estimates are now estimated by,
θˆ =
L −1 1 arg ∑ y kM M k =0
(E7-4)
where M is the symmetry angle of the modulation constellations (M=2 for BPSK/MSK, and 4 for QPSK). Again, since the arctangent function has a non-ambiguous region only over − π ≤ arg( x ) ≤ +π , the estimation in (4) is bounded by −
π
π ≤ θˆ ≤ − M M
(E7-5)
To cope with the M-fold ambiguity in the phase estimates, it is possible to use differential decoding (which is not an option for us), or using unique words (UW). The use of UW will be considered here as a viable option to extend the range of these estimators. The estimates in (4) are also lower bounded by the Cramer-Rao bound [1] that is given by Dr. Mohamed Khalid Nezami © 2003
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σ θ2ˆ > CRB (θˆ) =
1 2 L(E s / N o )
(E7-6)
To illustrate the estimates in (E7-4) using MPSK, we take as example BPSK and QPSK carrier phase estimation. Assuming that the received BPSK baseband signal and carrier phase offsets in AWGN channel is given by x k = c k e jθ + n k
(E7-7)
where the BPSK symbols are given by c k = e j ( m +1)π , where m=0 for Binary '0' and m=0 for Binary '1'. Substituting this in (E7-7), yields,
xk = e j ( m +1)π e jθ k + nk
(E7-8)
Using the estimation algorithm in (E7-4) with M=2 for BPSK, the carrier phase estimates using a block of L-symbols is given by
θˆ = arg ∑ (e j ( m +1)π e jθ + nk ) 1 2
L −1
k
2
(E7-9)
k =0
Realizing that e 2 j ( m +1)π e 2 jθ k = e 2 jθ k in (E7-10), and then expanding the inner terms in, the carrier phase estimates are given by L −1
⎛
⎞
θˆ = arg ∑ ⎜ e 2 jθ + 2e j ( m+1)π e jθ nk + nk2 ⎟ 1 2
k =0
⎜ ⎝
k
k
14442444 3⎟ noise ⎠
(E7-10)
This estimate is then limited to the non-ambiguous region (E7-5) of − 90 o ≥ θˆ ≤ +90o . Similarly, For QPSK symbols c k = e given by xk = e
j(
2 m +1 )π 4
j(
2 m +1 )π 4
, where m=0,1,2,3, the baseband signal in AWGN with phase rotations is
e jθ k + n k
(E7-11)
Performing the carrier estimation algorithm in (E7-4) over (E7-11), the carrier phase estimates over Lsymbols is given by, π 2 m +1 L −1 ⎛ −j ⎞ j( )π 1 jθ k 4 ˆ ⎜ θ = arg ∑ ⎜ (e e + nk )e 4 ⎟⎟ 4 k =0 ⎝ ⎠
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(E7-12)
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π
The extra phasor e 4 in (E7-12) was added to rotate the standard QPSK constellation by 45-degrees so that the angle corresponding to the carrier frequency is actually at the base reference of the first quadrant. Expanding the inner terms in (E7-12), and using the fact that e j 2πm e 4 jθ = e 4 jθ (i.e., e jmπ is the same regardless of the value of m) the estimates are then given by 4
L −1 ⎛ ⎛ j ( 2 m +1 )π L −1 ⎞ − jπ ⎞ 1 1 θˆ = arg ∑ ⎜ ⎜⎜ e 4 e jθ k + nk ⎟⎟e 4 ⎟ = arg ∑ e 4 jθ k + n' ⎟ ⎜ 4 4 k =0 ⎝ ⎝ k =0 ⎠ ⎠
(E7-13)
Again, the estimates in (E7-13) are bounded by the non-ambiguous region of the arctangent function used in (4), − 45o ≥ θˆ ≤ +45o . The following figures illustrate computer simulations of the algorithms derived in (E7-13) for QPSK signals in AWGN channels. The simulations were carried out for QPSK and MSK modulations using 9600sps as an example. Factors such as frequency offset, SNR ratio, and observation lengths were used to compare the performances of both algorithms. Figure E7-16 illustrates the algorithm (E7-13) capture range when operating on QPSK signals at Eb N o = 4dB and with an observation length of 100 symbols (10 milliseconds).
Figure E7-17 illustrates the accuracy of the estimator in (E7-13) as a function of Eb N o for an observation interval (L) of 100 symbols (10 milliseconds). The figure indicates that at Eb N o =5dB, the standard deviation in estimation error is less than 3-degrees compared with 1-degree predicted by the CRB (indicated by blue line). Figure E7-18 illustrates the same simulation experiment, but with a larger observation length of 500 symbols (50 milliseconds). The algorithm yields much more accurate estimates and converges to the CRB lower bound earlier. It can be seen that at Eb N o =5dB the standard deviation error is less than 1-degree, compared with 0.5 degree using the CRB.
Figure E7-18 illustrate the impact of frequency offset on the algorithm in (E7-13). It can be seen that for this algorithm to operate properly the frequency offset must be accurately estimated and corrected prior to phase estimation and correction. Figure 6 indicates that at Eb N o =10dB the carrier phase estimates obtained using the algorithm in (4) will have an error of ± 5 o when presented with a normalized residual frequency offset error of ∆fT = ±1x10 −4 from the previous frequency estimation algorithms.
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50 L=100 symbols N=4 samples/symbols Eb /No =4dB
40
Estimated phase error - degrees
30 20 10 0 -10 -20 -30 -40 -50 -50
-40
-30
-20 -10 0 10 20 Introduced phase error - degrees
30
40
50
Figure E7- 16: QPSK Carrier Phase Estimation Range for Eb N o = 4dB and an Observation Interval (L) of 100 Symbols. 14 L=100 symbols N=4
12
sqrt(var) - degrees
10
8
6
4
2
0
0
5
10
15
Eb /No
Figure E7- 17 Estimation Phase Error (top) and Variance (bottom) as a Function of Eb N o for an Observation Interval (L) of 100 Symbols.
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3.5 L=500 symbols Ns=4 samples/symbol
3
sqrt(var)
2.5
2
1.5
1
0.5
0
0
5
10
15
Eb /No
Figure 1: Estimation Phase error (top) and Variance (bottom) as a Function of Eb N o for an Observation Interval (L) of 500 Symbols. 3.5 L=500 symbols Ns=4 samples/symbol
3
Norm. Frequency offset: 1E-4
sqrt(var)
2.5
2
1.5
1
0.5
0
0
5
10
15
Eb/No
Figure E7- 18: QPSK Estimated Phase Variance and CRB (indicated by blue line) Comparison with Estimates Using a Signal with a Frequency Offset of ∆fT = 0 and ∆fT = 1x10 −4 ; the Observation Length (L) is 500 symbols. 7.17
Illustrating Example: Phase Estimation Ambiguity Estimation
Illustrate the use of unique words as a way to resolve ambiguities with NDA phase estimators. Dr. Mohamed Khalid Nezami © 2003
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2π Since the estimate in (E7-13) will always be θˆ = θ + m , where m=0, 1, 2, …., corresponding to the M phase ambiguous region number, the estimator can never tell weather its current estimate is in the valid m-region. To validate the value of m using the estimator (E7-13), a transmitted unique word can be used to establish this reference by comparing it with a local replica that is known to the receiver. Assuming that the UW samples from the transmitted signal are given by
x k = u k e jθ + n k
(E7-14)
Where u k are samples of the transmitted UW, where 0 ≤ k ≤ LUW − 1 .
These samples are correlated
∗ k
with the local conjugated replica of the UW symbols c ,
(
y k = ck∗ u k e jθ + nk
)
(E7- 15)
resulting in removal of these symbols from (E7- 14).
2π With the estimate from (E7-13) being θˆ = θ + m (with m being unknown), the correlation in (E7M 15) is then given by
(
y k = (u k e + nk ) c e jθ
∗ − jθˆk k
) = (u e k
jθ
⎛ ∗ − j ⎛⎜ θ + 2Mπ m ⎞⎟ ⎞ ⎠⎟ + nk )⎜ ck e ⎝ ⎟ ⎜ ⎠ ⎝
(E7- 16)
This is then reduced to, yk = e
−j
2π m M
+ n k'
(E7- 17)
The signal in (E7-17) is represents the baseband signal, after having its modulation symbol corresponding to the UW removed using (E7-15) and also after having been corrected with the estimated carrier phase offsets estimated using (E7-4). This signal now will contain a phase shift that corresponds directly to the m-ambiguity, and this can be guessed now using a maximum likelihood (ML) approach. That ML based estimate of the ambiguity region (m) is then given by Dr. Mohamed Khalid Nezami © 2003
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⎛ ⎧ + j 2π mˆ ⎫ mˆ = arg ⎜ max Re⎨ y k e M ⎬ ⎜ mˆ ⎩ ⎭ ⎝
⎞ ⎟ ⎟ ⎠
(E7-18)
Now with this technique, the received carrier does not have to be constrained in initial offset to the π π range of − ≤ θ ≤ + anymore, and thus the algorithm in (E7-13) can be used without any M M ambiguities.
7.18
Illustrating Example: Carrier Frequency Offset Estimation to CPFSK signals
Illustrate the use of the NDA frequency estimation algorithm used in Figure E7-11 for use with MSK modulated signals. While the estimates in (E7-13) are strictly applicable to MPSK signals, it can be extended to M-ary CPFSK signals. Realizing that for MSK signals, the differential phase between two trellis phase states is always given by
ϕ (kT ,α ) − ϕ (kT − T ,α ) = ±
π
(E7-19)
2
Removal of the MSK modulations will be carried out by a similar nonlinear processing as in (E7-13); however, the order of the non-linearity is now 2P, where the denominator of the modulation index is 1 . Thus the near baseband CPFSK signal is given by (h), where h = P
y k2 P (2 E s T ) e j 2 P (2π∆fk +θ ) + nk 2P
(E7-20)
which again is a simple a sine wave signal with a frequency of 2 P∆f . Thus the estimated frequency offset is obtained by,
∆fˆ =
L −1 2P 1 arg ∑ ( y k y k∗−1 ) 4πPT k =0
(E7-21)
Figure E7-2 illustrates this nonlinear processing using a 9600sps MSK signal, where as a result of a frequency offset of ∆f = 1kHz , the created spectral tone is at ∆fˆ = 4kHz .
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20 15
Power Spectrum Magnitude (dB)
10 5 0 -5 -10 -15 -20 -25 -30
0
0.5
1
1.5
2 Frequency
2.5
3
3.5
4 4
x 10
Figure E7- 19: Spectrum of ( y k y k∗−1 ) for MSK Signal with ∆f = 1kHz . 2P
7.19
Illustrating Example: Carrier Frequency Offset Estimation to CPFSK signals
Illustrate the use of the NDA phase estimation algorithm used in Figure E7-11 for use with MSK modulated signals. The extension of the algorithm (E7-9) to MSK modulated signals can be performed in a manner similar
to that detailed above for MPSK signals. Here passing the basenabd signal through ( y k ) removes the modulations, resulting in a signal that contains only phase rotations. Let the CPFSK modulation index K be given by h = , by taking the baseband CPFSK to the power 2 P . In this way, the CPFSK P modulations are removed. 2P
P
y
2P k
⎛ 2E ⎞ = ⎜ s ⎟ e j 2 Pθ + nk ⎝ T ⎠
(E7-22)
which now can be used for obtaining the phase estimates given by
θˆ =
L −1 1 arg ∑ yk2 P 2P k =0
(E7-23)
π π . Figure E7-20 and Figure E7-21 illustrate The estimate θˆ in (E7-2) is also bounded by − ≤ θˆ ≤ 2P 2P the carrier phase estimation variance as a function of Eb N o for an observation interval of 300 symbols Dr. Mohamed Khalid Nezami © 2003
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(30 milliseconds) and 1000 symbols (100 milliseconds)), respectively. The estimation error standard deviation at Eb N o =5dB shows an error of 10 o for L=300 symbols compare to 2 o of CRB, and an error of 9 o for L=1000 symbols. 12
10
L=300 symbols
sqrt(var)
8
6
4
2
0
5
10
15
20
Eb /No
Figure E7- 20: MSK Carrier Phase Error (top) and Variance as a Function of Eb N o for 2P=4 and an Observation Interval of L=300 Symbols.
9 8 L=1000 symbols 7
sqrt(var)
6 5 4 3 2 1 0
5
10
15
20
Eb /No-dB
Figure E7- 21: MSK Carrier Phase Error (top) and Variance as a Function of Eb N o for 2P=4 and an Observation Interval of L=1000 Symbols. Dr. Mohamed Khalid Nezami © 2003
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7.20
Illustrating Example: Diversity Combining of Synchronization Parameters
Illustrate how estimation algorithms detailed above can be incorporated in a diversity receiver to increase the reliability during fading by co-phasing multipel signals that are being received using antennas that are spatially separated and have independent fading. The use of these algorithms in the pre-detection proposed three branches diversity system is illustrated in Figure E7-22. The usefulness of the algorithms becomes apparent by observing what happens to the received signals in time and frequency as illustrated in Figure E7-23 and Figure 7-24, respectively. Figure E7-23 illustrates the frequency selective fading impact on a nine adjacent narrowband communication system, while Figure 7-24 illustrates the impact for fading on a preamble that is sent for data-aiding synchronization. In the following analysis we show how co-phasing multiple of antennas can actually avoid impact for fading on the data being transmitted through use of the synchronization algorithms detailed above.
Tree Antenna system Analog Section
Delay FIFO
RF Receiver r1
r2
Delay FIFO
Correction R1’
r3
R2’
Final Correction and co-phasing
R3’
Fine Timing offset, Frequency offset, Phase offset, and gain Block estimators
Coarse Timing offset, Frequency offset, Phase offset, and gain Block estimators
Figure E7- 22: Proposed Feedforward Diversity Combining
0
0
-10
-10
-20
-20 dB
10
dB
10
-30
-30
-40
-40
-50
-50
-60 -1.5
-1
-0.5
0 Hz
0.5
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-60 -1.5
5
x 10
-1
-0.5
0 Hz
0.5
1
1.5 5
x 10
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Figure E7- 23: Indicated Frequency Fading for Frequency Selective Channel with Delay at 9600sps and fs=307200sps Impact on 8-ary CPFSK in a 300kHz Bandwidth , and Then More Paths. 1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
0
1000
2000
3000
4000
5000 time
6000
7000
8000
-1.5
9000 10000
0
1000
2000
3000
4000
5000 time
6000
7000
8000
9000 10000
Figure E7- 24: Impact of Fading on the Preamble of the Signal. The propagating channel response experienced by the sampled baseband signal is typically made of amplitude gain a k , phase shift θ k , and a frequency offset ∆f k , where k is the sample index. Using these parameters, the received signals in Figure 1 are given by ⎡ x1k ⎤ ⎢ x2 ⎥ + a2k e a3 k e ⎢ k⎥ ⎢⎣ x3k ⎥⎦ (E7-24) ⎡ n1k ⎤ a1k e j (θ 1k + 2πk∆f 1k T ) a 2 k e j (θ 2k + 2πk∆f 2k T ) a3k e j (θ 3k + 2πk∆f 3k T ) ⎢⎢n 2 k ⎥⎥ ⎢⎣ n3k ⎥⎦
⎡ r1k ⎤ ⎢r 2 ⎥ = a1 e j (θ 1k + 2πk∆f 1k T ) k ⎢ k⎥ ⎢⎣ r 3k ⎥⎦
[
j (θ 2 k + 2πk∆f 2 k T )
j (θ 3k + 2πk∆f 3k T )
[
]
]
where {x1k , x 2 k , x3 k } is the phase modulated signal sent, {n1k , n 2 k , n3k } is the corresponding AWGN in the three branches, {a1k , a 2 k , a3 k }are the discrete representation of the three channel gains, {θ 1k ,θ 2 k ,θ 3k }are the phase rotations associated with each branch, and {∆f 1k , ∆f 2 k , ∆f 3k } are the rk is then obtained frequency offsets associated with each branch. The maximally ratio combined output ~ using a set of optimal complex weights that are obtained using a coarse and fine estimators. That is
[
~ rk = hˆ1∗ k
hˆ2∗ k
hˆ3∗ k
]
⎡ r1k ⎤ ⎢r 2 ⎥ ⎢ k⎥ ⎢⎣ r 3k ⎥⎦
(E7-25)
where the complex optimal weights of each branch are given by Dr. Mohamed Khalid Nezami © 2003
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(E7-26)
where the notion of xˆ indicates the estimate of x . The coarse and fine estimates for obtaining the channel gain and phase, and the frequency offsets, are based on either data-aided (DA) schemes such as those details in or on non-data aided (NDA) schemes. .
7.21
References
1 Steve Gardner, “Burst modem design, part 1,” Communication System Design Magazine, pp. 37-44, Aug 1999. 2 Heinrich Myer and Gerd Ascheid, Synchronization in Digital Communications Volume-1, New York, Wiley, 1989. 3 J. Chuang, and N. Sollenberger, “Burst coherent detection with robust frequency and timing estimation for portable radio communications,” Proceedings of ICC, 1988, pp. 26.1.1-26.1.6. 4 C. Couvreur, Y. Bresler, “Modeling and estimation of Doppler-shifted Gaussian random process,” IEEE workshop on Statistical Signal and Array Processing, Greece, Jun 24-26, 1996. 5 Abbas Aghamohamadi, Henrich Meyr, and Gerd Ascheid, “A new Method for Phase Synchronization and Auto gain Control of Linearly Modulated Signals on Frequency-Flat Fading Channels,” IEEE Trans. Commun. vol. 39, No. 1, pp. 25-29, January 1991. 6 M. Morelli, A. D’Andrea, and Mengali, “Frequency ambiguity resolution in OFDM systems,” IEEE Commun. Lett., vol. 4, No. 4, pp. 134-136, Apr 2000. 7 F.D. Natali, “AFC tracking Algorithms,” IEEE Trans. Commun, vol. 32, No. 8, Aug 1984, pp. 935947. 8 Said Moridi and Hikmet Sari, “Analysis of four decision-feedback recovery loops in the presence of inter-symbolic interference,” IEEE Trans. Commun. vol. Com-33, no. 6, pp. 543-550, Jun 1985. 9 M. Nezami, “Phase Lock Loop Designs, making it easy with Pspice Simulations,” Communications Magazine, pp. 143-153, Apr 1995. 10 Steven Leeland, “Digital signal processing in satellite modem design,” Communication System Design Magazine, pp. 21-29, June 1998. Dr. Mohamed Khalid Nezami © 2003
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11 J.J. Stiffler, Theory of Synchronous Communications, New Jersey, Prentice-Hall, 1971. 12 S. Bellini, C. Molinari, and G. Tartara “Digital frequency estimation in burst mode QPSK transmission,” IEEE Trans. Commun., vol. 38, no. 7, pp. 959-961, Jul 1990. 13 Thad Genrich, “BPSK demodulator/bit synchronizer FPGA implementation & benchmarks,” ICSPAT 1999. 14 Andrew Viterbi and Audrey Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Commun., vol. IT-29, No. 4, pp. 543-551, Jul 1983. 15 M. Nezami and R. Sudhakar, “M-QAM digital symbol timing synchronization in flat Rayleigh fading channels,” PRMIC, Osaka, Japan, Nov 1999. 16 M.K. Nezami and J. Bard, “Preamble-less carrier recovery in fading channels,” Milcom2000, Los Angeles, CA, Oct 2000. 17 M. Nezami and R. Sudhakar, “New schemes for 16-QAM symbol recovery,” EUROCOMM 2000 Münich, Germany, 17-19 May 2000. 18 M. Nezami and R. Sudhakar, “M-QAM digital symbol timing synchronization in flat Rayleigh fading channels,” PRMIC, Osaka Japan, Nov 1999. 19 M. Nezami, “DSP algorithms for carrier offset estimation and correction,” ICSPAT, Orlando, FL, USA, Nov 1999. 20 M. Nezami and H. Otum, “Fine tuning frequency offset errors in M-QAM digital burst receivers using DSP techniques,” Third international conference on computational aspects and their applications in electrical engineering, Amman, 19-20 Oct 1999. 21 M. Nezami, “Non-linear M-QAM digital symbol timing synchronization algorithm suited for wireless handheld radios,” Third international conference on computational aspects and their applications in electrical engineering, Amman, 19-20 Oct 1999. 22 M. Nezami, “Digital synchronization algorithms for wireless burst QAM receivers,” Wireless Symposium, San Jose, CA/USA, Feb 1999. 23 M. Fitz, “Further results in the fast estimation of a single frequency,” IEEE Trans. Commun., vol. 42, No. 2/3/4, pp. 862-864, Feb/Mar/Apr 1994. 24 Umberto Mengali and M. Morelli, “Data-Aided frequency Estimation for burst digital transmission,” IEEE Trans. Commun., vol. 43, No. 1, pp. 23-25, Jan 1997. 25 M. Fitz, “Further results in the fast estimation of a single frequency,” IEEE Trans. Commun., vol. 42, No. 2/3/4, pp. 862-864, Feb/Mar/Apr 1994. 26 Umberto Mengali and M. Morelli, “Data-Aided frequency Estimation for burst digital transmission,” IEEE Trans. Commun., vol. 43, No. 1, pp. 23-25, Jan 1997. Dr. Mohamed Khalid Nezami © 2003
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27 C. Morlet, M. Boucheret, and I. Buret, “Low complexity Carrier-phase estimator suited to on-board implementation,” IEEE Trans. Commun., vol. 48, No. 9, pp.1451-1454, Sept 2000. 28 Marco Luise and R. Reggiannini, “Carrier frequency recovery in all-digital modems for burst-mode transmissions,” IEEE Trans. Commun., vol. 43, no. 2/2/4, pp. 1169-1178, Apr 1995. 29 M. Hebley, “The effect of diversity on a burst-mode carrier frequency estimator in the frequencyselective multipath channel,” IEEE Trans. Commun., vol. 46, No. 4, pp. 553-559, Apr 1998. 30 Jiang, W. Ting, F. Verahami, R. Richmond, and J. Baras, “A carrier frequency estimation method of MPSK signals and its systolic VLSI implementation,” NASA publication, www.isr.umd.edu/CSHCN/ 31 Andrew Viterbi and Audrey Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Commun., vol. IT-29, No. 4, pp. 543-551, Jul 1983. 32 Estathiou, and H. Aghvami, “Preamble-less non-decision aided (NDA) feedforward synchronization techniques for 16-QAM TDMA demodulators,” Proceedings of ICC, 1998, pp. 1090-1094. 33 Nelson R. Sollenberger and Justin C. Chang, “Low-overhead symbol timing carrier recovery for TDMA portable radio systems,” IEEE Trans. Commun., vol. 38, No. 10, pp. 1886-1892, Sept 1984. 34 R. Gaudenzi, and V. Vanghi, “All-digital carrier phase and clock timing recovery for 8PSK,” Proceedings of Globecomm, pp. 12.3.1-12.3.5, Jan 1991. 35 K. Cartwright, “Fourth power phase estimation with alternative two-dimensional odd-bit constellations,” IEEE Commun. Letters, pp. 199-201, Jun 2000. 36 Jonghe, Marc Moeneclaey, “Optimal averaging filter length of the Viterbi & Viterbi carrier synchronizer for a given frequency offset,”Proceedings of Globecomm, 1994, pp. 1363-1368. 37 Daffara, and J. Lamour, “Comparison between digital phase recovery techniques in the presence of a frequency shift,” Proceedings of ICC, pp. 493-497, Feb 1994. 38 M. Morelli, A. D’Andrea, and Mengali, “Frequency ambiguity resolution in OFDM systems,” IEEE Commun. Lett., vol. 4, No. 4, pp. 134-136, Apr 2000. 39 Geert de Jonghe and marc Moeneclaey, “Cycle slipping behavior of NDA feedforward carrier synchronization for time-varying frequency-nonselective fading channels,” Proceedings of GLOBECOM-1995, pp. 350-354. 40 R. Reggiannini, “A fundamental lower bound to the performance of phase estimators over Ricianfading channels,” IEEE Trans. Commun. vol. 45, No. 7, pp. 775-778, July 1997.
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Chapter 8 Carrier Acquisition and Carrier Tracking for Burst TDMA Satellite and Mobile Radio Receivers In this chapter, we present various burst type synchronization algorithms. These are intended for use with satellite or land mobile radios. They include modeling, analysis, and the simulation results of preamble-less and preamble assisted carrier synchronization algorithms for burst signals subject to additive white Gaussian noise (AWGN) and severe Doppler frequency shifts that are relatively large compared to the data rate being received. The chapter details conventional feedback carrier recovery systems that rely on preamble or a training sequence and non data-aided algorithm that require no preamble.
8.1 Preamble-based Carrier Recovery Techniques for Satellite Receivers
In TDMA, multiple users share the same channel by using the bandwidth for discrete intervals of time slots. Only one user can access the channel at any instant in time. A network controller for each user frame assigns time slots. Each user terminal has a unique carrier phase and frequency offset resulting in unpredictable carrier changes from message to message. Currently, to aid acquisition, each message or frame has a preamble or training sequence. The preamble is transmitted as the initial part of each communications burst. The preamble typically provides information necessary for signal acquisition and synchronization, coding information, a unique word that is associated with each individual receiver used for phase ambiguity resolution. In addition, the preamble has a field that indicates the data coding rate being sent and the modulation type. The preamble format usually consists of a continuous wave (CW) followed by a dot pattern (alternating or repeating sequence of 1's and 0's) as shown in Figure 8-1 [4]. Conventionally, the CW portion of the preamble is generated with constant data on both the I and Q signals (real and imaginary part of the received signal), while the dot pattern portion of the preamble consists of alternating data on the I channel (I = 0 1 0 1 . . .) and an opposite alternating phase on the Q channel (Q= 101010…). The CW sequence (all 0’s) creates a CW tone that is commonly used by receivers to recover frequency offsets and tracking using some form of feedback loops, such as Costas loops. The 1-0 pattern creates a dot pattern that is also used by the receiver for bit and carrier synchronization (see data aided symbol timing algorithms discussed in Chapter 6). Since these Dr. Mohamed Khalid Nezami © 2003
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overhead bits of the preamble reduce the signaling efficiency, it is desirable that such preamble be as short as possible, yet be long enough to allow carrier synchronization with minimal final phase errors.
2
1.5
1
( )
± 00o
0.5
0
-0.5
(
-1
± 180 o
-1.5
-2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
) 2
Figure 8- 1: Typical satellite burst TDMA frame. 8.2 Sources of Carrier Frequency Offset in Satellite Systems Satellite signal impairments are mostly due to the propagation channel effects and the transmitter/receiver circuitry of both the ground stations and the satellite transponder. Frequency offsets in mobile communication terminals are experienced due to factors such as oscillator frequency-uncertainty, oscillators drift, and Doppler effects arising from vehicular motion with respect to the satellite. Depending on the carrier frequency and the relative velocity between satellite and ground receiver, such frequency offsets can vary from a few hundred Hz up to several hundred kHz. Such large carrier offsets result in increased BER due to reduced receiver sensitivity and phase rotation in the received symbols. The Doppler frequency shift of the received signal is given by fd = fc
v (t ) cos(α (t )) c
(8-1)
where v (t ) is the relative velocity of the receiver terminal with respect to the satellite in meters/sec, f c is the carrier frequency in Hz, and α (t ) is the angle between the relative velocity vector and the signal propagation direction in degrees (satellite elevation angle). Figure 8-2 shows the Doppler frequency variations as a function of the angle α (t ) for a GEO system operating with f c =2 GHz, Dr. Mohamed Khalid Nezami © 2003
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for an aircraft traveling at 1000 km/hr. The Doppler frequency is minimum (at α = 90o) when the satellite is passing overhead, and is a maximum ( f d = ± 1800 Hz) when the satellite is over the horizon (at α = 0o and 180o). LEO satellites are located at heights of 10,000 to 20,000 km above the equator [6], and have a relative velocity of 1500 m/s. If operating at 2 GHz, the Doppler shifts ( f d ) can be as high as 10 kHz, and the Doppler frequency shifts can change by up to 250 Hz/seconds [6]. This complicates the carrier frequency offset estimation algorithm, especially when it uses long observation intervals (for typical moderate burst lengths). It is required that such algorithms account for this continuous change in frequency. Contrary to LEOs, GEO satellites are located at height of 36,000 km and do not have any relative motion relative to a fixed ground terminal. However, due to vehicle velocities up to 100 km/hr, the ground mobile receivers may experience Doppler shifts of up to 138 Hz for L-band signals. For aircraft traveling at speeds of 1000 km/hr, the Doppler frequency shift can be as high as 1800 Hz for 2 GHz systems as shown in Figure 8-2. 2000 GEO- 2 GHz, Aircraft speed at 1000km/hr
1500 1000
Doppler in Hz
overhead
500 0 -500 -1000 -1500 -2000 0
20
40
60
80
100 angle
120
140
160
180
Figure 8- 2: GEO Satellite-to-aircraft Doppler Profiles at 2 GHz. In addition to Doppler shifts, frequency offsets are also introduced by the tolerances of the local oscillator frequencies in the receiver terminal as well as the satellite transponder translator. For instance, if the tolerance of the receiver local oscillator is specified at 0.2 PPM for a carrier at 2 GHz, the introduced frequency offset will be 400 Hz. More information on frequency offset system distribution and analysis techniques for multiple access satellite receivers can be found in [7]. 8.3 Impact of Satellite Channel on Carrier Recovery The effect of received signal level on phase estimates was analyzed in Chapter 7, which predicted the minimum achievable variance of phase error estimates for Rician, Rayleigh and AWGN channels per (7-80). With satellite receivers, signal variations are caused by shadowing, multipath fading, and space attenuation, which impact the carrier recovery process. Here we briefly summarize the most important aspects of satellite channel effects that may impact the carrier recovery operation. Dr. Mohamed Khalid Nezami © 2003
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Land Mobile Satellite Receivers: Typically most mobile satellite terminals on the ground can experience up to 20-dB of signal level variation. The degree of fade at any particular location on earth depends on the surrounding geography and the satellite elevation angle. Shadow fading can make the signal vary by as much as 13-20 dB. For UHF bands, fading due to mountains can vary between 2 to 5 dB at elevation of 45°, and 2 to 8 dB at an elevation of 30°. Fades due to roadside tree effect have an exponential statistical distribution, which achieves 3 dB, 2 dB, and 1 dB values with probabilities of 1%, 10% and 40%.
Most mobile satellite receivers, regardless where they operate, have an average duration of fade (AFD) of 40-50 msec with the worst duration of fade of 300 msec. To illustrate the impact on carrier synchronization, consider the example of the 10,000 sps system with AFD is 40-50 msec where fade duration extends over 500 symbols of the received signal. To retain good phase or frequency estimates, the observation interval must be several times greater than the AFD of 500 symbols, otherwise some anti-fading techniques or diversity combining schemes will have to be used [8]. Airborne Satellite Receivers: For aircraft moving at typical velocities of 1000 km/hr, signal reception is dominated by the diffused signal that results from reflections off the earth surface below. Depending on the earth surface roughness, these reflections can have large spread delay relative to the direct path, resulting in a very large ISI distortion. Hence the channel can be modeled as a frequency selective Rician type channel with typical Rician factors of 10-15 dB and a fading bandwidth of 30-100 Hz. Maritime Mobile Receivers: For Maritime receivers, most of the received signal is due to multipath reflections. The calm sea acts like a mirror, resulting in large coherent radio path at the receiver. A non-negligible Rayleigh type signal will also be present due to the independent distribution of points at sea surface. So maritime channels are considered a Rician channel, with typical Rician factors around 7 dB and a fading bandwidth of 1 Hz. A particular problem occurs on shipswhere shadowing can take place by the ship structure that is usually circumvented by the use of Quad diversity combiners [8].
8.4 Conventional Burst Satellite Carrier Acquisition and Tracking In continuous TDMA satellite modems such as home satellite receivers, or news gathering mobile terminals, the user can afford to wait a few seconds for initialization during which the receiver goes through an acquisition and phase tracking process using slow narrow band PLL or Costas loop systems. In contrast, with burst satellite modems, the user data burst consumes only a fraction of the overall time frame ( < 20%); hence long acquisition times contribute an unacceptable level of overhead to the system and substantially reduce the transmission capacity. Thus, burst modems require a special acquisition process that will quickly estimate the carrier frequency offsets and phase to perform corrections. Their narrow capture range that is proportional to the loop bandwidth used also limits the use of PLL or Costas loop.
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The CW-Costas loop can only be used during the CW portion of the preamble shown in Figure 8-1. To extend the operation of feedback carrier recovery loops to acquire and track the received carrier in the presence of modulations during the message portion of the burst, the conventional Costas loop has to be modified. Such loops derive phase/frequency error signal based on the transition and data decision of both I and Q channels. The decision aided phase/frequency error is sometimes called polarity type error detector as opposed to the conventional ‘sine’ phase detector that is used in CW unmodulated systems. Consider the received complex signal input into the quadrature demodulator given by
s r (t ) = I (t ) cos(2πf r t + θ r ) − Q(t ) cos(2πf r t + θ r )
(8-2)
where f r and θ r are the frequency and phase associated with the received signal. This signal is downconverted by mixing with a free running complex local oscillator (CNCO) given by
y L (t ) = cos(2πf L t + θ L ) − cos(2πf L t + θ L )
(8-3)
As a result, the baseband in-phase and quadrature phase samples produced are given by,
y I (k ) = 1. cos(2π∆fkT + θ )
(8-4)
y Q ( k ) = 1. sin( 2π∆fkT + θ )
(8-5)
and
where ∆f = f L − f r is the carrier frequency offset and θ = θ L − θ r is the carrier phase error. In order to perform correct data detection, the receiver must estimate and remove ∆f and θ from the received signal. To do this, the receiver generates an error signal proportional to the magnitude of ∆f and θ . Unlike the FF algorithms developed in Chapter 7, this error signal is not equal to the final offset (immediate estimate); instead the feedback error signal is used to steer a freely running CNCO. One common method used in satellite receivers is to acquire the carrier (frequency offset estimation) during the CW portion of the preamble and then carrier tracking (phase estimation) during the rest of the burst that extends over the UW and the transmitted message. 8.4.1
Frequency Offset Error detector
During initial stages of receiver operation, the incoming carrier in (8-2) is received with a unknown frequency offset; hence phase tracking or data detection cannot be performed unless the frequency offset ∆f is removed first. This frequency offset causes power drops in the MF output, and large unprecedented symbol rotation. To acquire and lock the local oscillator signal in (8-3) to the incoming carrier, a frequency offset signal detector in conjunction with a closed correction loop is used to sense the amount and direction of this offset. The process of rapid frequency estimate and correction is known as automatic frequency control (AFC). The loop uses a frequency error detector Dr. Mohamed Khalid Nezami © 2003
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to bring the initial frequency error ∆f close to zero or to a small offset value that is tolerable by the decoding section. AFC loops alone will have a phase error that randomly walks around as AWGN noise perturbs it. Since the AFC loop is not responsive to phase at all, there is no coherent phase reference being produced. Frequency error signals proportional to ∆f can be derived in a fashion similar to the estimators used in Figure 8-3. The error is generated by correlating the received MF sample z (k ) and the conjugate of the previous MF sample z ∗ (k − 1) , where the frequency-offset error is given by
z k z k∗ −1 = [I k I k −1 + Q k Q k −1 ] + j [Q k I k −1 − Q k −1 I k ] 1442443 1442443
{
Re z k z k∗ −1
}
Re{x}
{
Im z k z k∗ −1
Ik
(8-6)
}
x -
z −1
+
x
Qk I k −1 − Qk −1I k
z −1 z (k ) exp[− j 2πkTfCNCO + jθ k ]
Im{x}
x Qk
Re{x}
+
Ik
e∆fˆ
x z −1 z
+
−1
Im{x}
I k I k −1 + Qk −1Qk
x Qk
Figure 8- 3: Frequency Error Detector for CW Signal. The error detector in (8-6) assumes a CW MF signal, so this error signal is valid to operate only during the CW portion of the preamble in Figure 8-1. If the frequency-offset recovery is extended over portions of the preamble with data modulations, an M-power nonlinearity can be used to remove modulations and produce a CW signal with nominal frequency of M∆f . Hence, (8-6) produces an initial frequency offset error signal proportional to M folds of the actual offset ∆f . Once this offset is estimated and the incoming signal is corrected, phase tracking can start using another phase detector similar to Figure 3-30. 8.4.2
Phase Error Detector
After carrier frequency offset detection using (8-6) and correction, the normalized complex MF signal can now be represented by an in-phase sample of I k = cos(θ k ) and a quadrature phase sample of Q k = sin(θ k ) , where θ is the phase error to be tracked. If carrier phase tracking is performed during the CW portion of the preamble as in Figure 8-1, the phase error θ k → 0 , hence θ k ≈ sin(θ k ) Dr. Mohamed Khalid Nezami © 2003
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and cos(θ k ) = 1 . Using the small angle approximation, the error is then θ k = Im{z ( k )} . However, these results will not hold if the recovery process is carried out over samples that contain data modulations. For instance, with BPSK, the received MF samples are either the pair of { I k = cos(θ k + 0 0 ) , Q k = sin(θ k + 0 0 ) } or the pair of { I k = cos(θ k + 180 0 ) for Binary ‘1’, and then Q k = sin(θ k + 180 0 ) } for Binary ‘0’. Using the assumption that θ k → 0 , the carrier phase error
signal for modulated BPSK is given by θ k = Q k for the first pair and θ k = −Q k for the second pair. Clearly the negative sign of the error is coupled with the sign of the I channel. Therefore, the sign of the I channel can be used to generalize a formula for the carrier phase error associated with BPSK modulated signals. That is
⎧+ Q k eθ , k = ⎨ ⎩− Q k
I >0
(8-7)
I <0
or simply eθ , k = Q k sign( I k )
(8-8)
The error signal in (8-8) indicates that all of the signal energy should be in the I channel when the BPSK signal is perfectly synchronized θ k ~0. That is after the sign of I channel distinguishes the polarity of the error signal. Figure 8-4 shows the carrier phase error function (8-8) as it is seen on the constellation diagram. The figure shows one particular scenario with which the carrier phase resulted in rotating the BPSK constellation counter-clock by θ k degrees. The figure further illustrates the geometrical interpretation of carrier phase error. imag{z k } = Q
sign ( I ) = +1 θ k ∝ +Qk
Nominal Binary ‘0’ or 180o
θk θk
received Binary ‘0’ or 180o
Phase error counterclockwise directions Binary ‘1’ or 0o real{z k } = I
sign ( I ) = −1 θ k ∝ −Qk
Figure 8- 4: BPSK Carrier Phase Error Extraction.
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For QPSK and OQPSK modulated signals, the phase error detector must exhibit four-fold symmetry about the four possible phase angles of the MF samples. Given that the in-phase and quadrature phase QPSK, MF samples at the output of the quadrature demodulator having unknown phase error θ k is given by I ( k ) = cos(θ k +
π
π
+ m ) , for m = 0,1,2,3 4 2
(8-9)
and Q k = sin(θ k +
π
π
+ m ) , for m = 0,1,2,3 4 2
(8-10)
Here the phase error signal can be obtained for each constellation point separately. Consider the case for m = 0 , where the in-phase sample is given by Ik =
2 [cos(θ k ) − sin(θ k )] 2
(8-11)
and the quadrature phase sample is given by Qk =
2 [cos(θ k ) − sin(θ k )] 2
(8-12)
By assuming that θ k is small (i.e., the loop is in tracking mode), the carrier phase error signal can be generated by taking the difference of (8-11) and (8-12). That is
θk =
Qk − I k
(8-13)
2
Likewise, the error signal can be extended for the rest of the QPSK symbols. Table 8-1 lists the error signal for m = 1,2,3 . m
π 4
0
+m
π
IQ
θk
11
Qk − I k
2
π 4
1 2 3
3
5
7
π
01
4
π
00
4
π
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2 Qk + I k 2 Qk − I k
2 Qk + I k 2
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Table 8- 1: Phase Error Signal for QPSK Modulated Signals.
Pictorially, Figure 8-5 shows the carrier phase error derivation for the four possible QPSK symbols that may be received when tracking the modulated carrier. Ideally, when Q k and I k are equal, then the carrier error is minimal. However, the sign of I k and sign of Qk can be used to derive and discriminate the direction of the error signal as illustrated in table 8-1. By inspection of the entries for the entire QPSK symbols in Table (8-1) and Figure 8-5, it becomes clear that the error detector can be generalized by
θ k = Q k sign( I k ) − I k sign(Q k )
(8-14)
For the case where the incoming carrier is modulated by a pair of alternating sequence of QPSK symbols as in Figure 8-1, where the symbols are alternating pairs with 180 degrees of phase difference, for instance { I k = 11 , Q k = 00 }, the carrier phase error is given by
θ k = sign(I k )
Qk − I k 2
(8-15)
Similarly for the preamble pair of { I k = 01 , Q k = 01 }, the carrier phase error is given by
θ k = sign(I k )
Qk + I k
(8-16)
2
Based on the error signals derived above, one common conventional implementation of carrier recovery system for QPSK and BPSK satellite modems is shown in Figure 8-6. Here the digitized IF samples are translated down to baseband by complex numerically controlled oscillator (CNCO) and a complex multiplier. The baseband samples are then passed to the MF, which is implemented using an interpolator filter in conjunction with a decimator. The interpolator filter is a FIR filter whose coefficients are controlled by an off-line symbol recovery algorithm as described in Chapters 4 and 6. The in-phase (I) and quadrature (Q) MF outputs are then input to a carrier error signal detector as described above. The frequency error detector and phase-offset detector are used in conjunction with a common loop filter to complete the feedback control system. The loop filter can be programmed during initial power up of the receiver to establish carrier frequency offset recovery. Once the offset is removed, the loop filter is loaded again with another set of coefficients to perform carrier phase tracking. The coming section details design methods for the systems illustrated in Figure 8-5.
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Re{} .
x
-
sign{} . Freq. err. detector
Symbol Timing
Complex multiplier
x Baseband signal
zk
sign{} .
QPSK
Im{} .
x BPSK Phase error detector
Frequency offset detector
MF
+
+
exp(− j 2πθˆ) x NCO
z −1
Kp
+
Numerical controlled Oscillator
Loop filter +
+ z −1 acquire/ track
x Ki
Library of Coefficients
Figure 8- 5: Conventional Carrier Offset Acquisition and Tracking Loop for QPSK/BPSK.
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8.4.3
Loop Filter Design
The bandwidth of the loop filter (Figure 8-6) and the symbol rate (loop iteration rate) establish the performance of the carrier recovery feedback system in terms of acquisition time and the offset frequency capture range. A larger loop bandwidth has a shorter acquisition time, but provides noisier control signal, resulting in higher phase error variance. Usually the loop bandwidth is set between 5% to 10% of the symbol rate [9]. This has been shown to offer a maximum capture range of frequencies up to twice the loop bandwidth [10]. Thus for the 10 ksps system, the maximum frequency 'pull in" ⎛1⎞ range is 1000 Hz with BL = 0.05 * ⎜ ⎟ . ⎝T ⎠
θi (k ) = 2π∆fkT + θo + θ j (k ) θi (k )
+ -
θˆr (k )
θ err (k ) = θ i (k ) − θˆr (k ) x Loop filter Kp z
−1
+
+
+
z
x
−1
Ki
Figure 8- 6: Linear Model for the Carrier Recovery System in Figure 8-5. The loop filter in Figure 8-6 is implemented by a direct gain Kp path and an integrated path Ki to filter frequency discriminator error signals. The filter is a type-2 loop filter which has the property that its steady state error converges to zero for both phase and frequency offsets [11], and it can track a changing frequency rate proportional to the loop bandwidth given by,
θ i ( k ) = 2π∆fkT + θ o + θ n ( k )
(8-17)
where ∆f is the initial frequency offset, θ o is the initial carrier phase, and θ n (k ) is an AWGN. The system function can be shown to be Ki ) K p + Ki θˆr ( z ) = K p + Ki θ i ( z) z 2 − 2(1 − ) z + (1 − K p ) 2 K p + Ki (z −
(8-18)
The loop filter parameters Kp and Ki are designed so that they minimize the phase and offset errors while yielding minimum acquisition time. Both parameters are given by [12] Dr. Mohamed Khalid Nezami © 2003
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⎛ ⎜ 2BLT K p = 2ξ ⎜ ⎜ 1 ⎜ ξ + 4ξ ⎝ ⎛ ⎜ 2 BL T Ki = ⎜ 1 ⎜ ⎜ ς + 4ς ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
(8-19)
2
(8-20)
where BL is the noise loop bandwidth, ς is the second order damping factor, and T is the symbol rate (i.e., loop iteration rate). Given a set of desired receiver parameters, equations (8-17) through (8-20) can be used by computer-aided tools to optimize the carrier recovery loop for optimal timing acquisition and phase error variance. The error variance of the recovered carrier phase (8-18) is directly proportional to the loop bandwidth BL and is given by
σ θ2 =
BLT Es / N o
(8-21)
The required loop bandwidth in (8-21) is directly obtained using a desired minimum BER degradation according to (7-1). For instance, assume a QPSK system with BER of 10 −5 at E s / N o = 13dB ; for maximum BER deterioration less than 1 dB (from 7-1), the required loop bandwidth BL = 2000 Hz for the 10 ksps system (20% of the data rate). 8.4.4
Simulations Performance
Having established the loop parameters in (8-19),(8-20), and(8-21), the effect of ς and B L T on the dynamic response of the loop can be examined. The input to the second– order loop filter is given in (8-17) and let θ o = 300 , ∆fT = 0.01 , and BLT = 0.05 . Figure 8-7 presents the response of the phase detector output θ err (k ) versus sample index (time). A noisy input is plotted for various damping factors ς , along with the system acquisition time (7-3). Notice that the loop is able to drive the phase error θ err (k ) eventually to zero. As expected, the transient response of θ err (k ) becomes more oscillatory as ς gets smaller than 1. Traditionally, for type II loops, a damping factor of ς =0.707 is a considered as an optimal choice for carrier recovery. It is commonly suggested that the loop filter-damping ratio ς has little impact on acquisition time. Based on this, the acquisition time derived in (7-3) for a second-order loop can be approximated by Dr. Mohamed Khalid Nezami © 2003
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tacq. =
1.2 BLT
[symbols]
(8-22)
The impact of the loop bandwidth on the transient response for the second-order carrier recovery loop, as well as the analytical values of t acq . obtained using (8-22), are presented in Figure 8-8. 20 θ 0 = 30 0 BLT = 0.05
15
∆fT = 0.01
10
t acq. , Equ.4 − 3
Phase error-deg.
5 ξ =1
0 -5 ξ = 0. 5
ξ = 0.8
ξ = 0. 3
-10 -15 -20
20
40
60
80
100
120
140
160
180
200
Symbol-index
Figure 8- 7: Impact of Damping Factor ς on Carrier Tracking. 40 t acq. ( Equ.5 − xx )
35
θ 0 = 300 BL = 400,200,100,50 Hz ∆f = 800 Hz
30
ξ = 0.707
Phase error-deg.
25 BL = 50 Hz
20
15 BL = 100 Hz
10 BL = 200 Hz
5
0
BL = 400 Hz
-5
-10
20
40
60
80
100 120 Symbol-index
140
160
180
200
Figure 8- 8: Impact of Loop Bandwidth ( BL ) on Carrier Acquisition.
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The figure plots the phase error signal θ err (k ) versus sample index (time) for various values of BL ( BL =400, 200, and 100) for ∆f =800 Hz and ς of 0.707. From the observation that BL does not change the oscillatory nature of the system, one can conclude that BL only affects the tracking-speed of the loop. Further, there is acceptable agreement between t acq . using (8-22) and that obtained using simulations. For larger BL the tracking speed is increased, but the loop’s noise-suppression ability is reducedThis increases the phase error variance, and ultimately degrades the receiver BER performance. For small BL the noise is reduced, but the tracking speed is degraded (822). This tradeoff becomes a limiting factor in using this scheme for burst type systems. Also, the scheme is limited by the fact that it requires a decision on data {Iˆ, Qˆ } = {sign ( I ), sign (Q )} to generate an error signal. Further, as there is a tradeoff between the variance of the tracked phase, the acquisition timing and the loop bandwidth, the system may not work properly for short burst TDMA systems. Generally, conventional feedback systems have several disadvantages:
•
From a complexity point of view, the loop error signal for modulated signals requires two parallel error detectors. This can be computationally expensive and tedious to build, especially for mobile satellite receivers.
•
As the AFC frequency acquisition converges, and the phase tracking algorithm takes over, there is a sizable amount of noise contribution added from the AFC error signal to the loop filter.
•
Since acquisition (AFC) operation is limited to the CW portion of the preamble, if the receiver loses acquisition momentarily during message duration, the system has to drop the whole burst and wait until next burst to initiate re-acquisition.
•
Both frequency acquisition and phase tracking operation are heavily deteriorated by AM modulations or by signal fading caused by the satellite channel propagation effects.
•
The use of linearizing approximation in the phase detector ( sin(θ i − θ r ) ≈ θ i − θ r ), narrows the linear operating range of phase/frequency detection.
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8.5 Estimating and Tracking Carriers with Doppler Rate of Change
Algorithms derived above use second order type loops that can cope with carrier frequency offsets that are on the order of the loop bandwidth; however, some satellite signals have a high Doppler rate in addition to Doppler frequency and carrier phase offsets. . 8.5.1
Third Order Loops Feedback Doppler Rate of Change Estimator
One way to track carriers that have a Doppler rate is to use a third order loop digital cross-product AFC loop to combat high channel dynamics. This loop is implemented using a second order loop [7]. Using the phase trajectory computed by methods such as the arctangent, the baseband phase of the satellite signal that includes a Doppler frequency offset and a Doppler acceleration is given by
θˆk = θˆ0 + wˆ d t +
1 ˆ 2 1 ˆ && d t 3 + ........ w& d t + w 2! 3!
(8-23)
~ ⎛ Q(t ) ⎞ dθ k ⎟⎟ = θ (t ) + wt and wd (t ) = where θ (t ) = tan ⎜⎜ is the Doppler frequency and dt ⎝ I (t ) ⎠ ~ d 2θ k ~ is the Doppler rate (differential Doppler) and θ k is the baseband phase. w& d (t ) = dt
~
−1
First an error signal is formed by the cross product given by ek = Qk I k −1 − I k Qk −1
(8-24)
To estimate the differential Doppler frequency,
w&ˆ k +1 = w&ˆ k + γek where γ = 2 BL , and w& = 2π
(8-25)
d (∆f ) Another loop is also formed to estimate the Doppler dt
frequency,
wˆ k +1 = wˆ k + Tw&ˆ k + 2γek
(8-26)
Finally the carrier phase estimate updates are formed,
θˆk +1 = θˆk + Twˆ k
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8.5.2 Feedforward Doppler Rate of Change Estimator Another method to estimate the Doppler rate that is suitable for short burst TDMA is based on block-by-block estimation. Assuming that the received signal is given by
⎧ ⎛ α 2 ⎞⎫ x(k ) = c k exp⎨ j ⎜ ϕ + 2π∆fT (k − k 0 ) + (k − k 0 ) ⎟⎬ 2 ⎠⎭ ⎩ ⎝
(8-28)
where 0 ≤ k ≤ N − 1 , where N is the number of symbols for the observation interval, ck N −1 center of the observation interval, and α is the is the data modulations, k 0 = 2 ∆f Doppler rate being normalized by 1 / T 2 ( α = d Hz/sec). Removing the data using a dt preamble symbols, c k c ∗ = 1 , the signal in equation (8-28) is reduced to ⎧ ⎛ α 2 ⎞⎫ z (k ) = exp⎨ j ⎜ ϕ + 2π∆f (k − k 0 ) + (k − k 0 ) ⎟⎬ 2 ⎠⎭ ⎩ ⎝
(8-29)
by observing z(k) for N trials, each made of L samples. That is to obtain z (k ) = [z (0), z (1),......z ( N − 1)] , then samples within each segments are then summed together to produce P=N/L quantities of {y(k)} , which is computed by 1 L −1 (8-30) ∑ z (kL + m) L m =0 where 0 ≤ k ≤ P − 1 , and the Doppler rate is given by twice-differentiation of the phases. That is y (k ) =
ϕ (k ) = {θ (k ) − 2θ (k − 1) + θ (k − 2)}2π
(8-31)
α ⎛ 2⎞ where θ (k ) = arg{y (k )} and is equal to θ (k ) = ⎜ ϕ + 2π∆f (kL − k 0 ) + (kL − k 0 ) ⎟ , 2 ⎠ 2π ⎝ where 0 ≤ k ≤ P − 1 . Based on these measurements, the Doppler rate of change is estimated by 1 P −1 [Hz/sec] (8-32) α = T 2 2 ∑ w(k )ϕ (k ) L k =2 and the window function w(k ) in (8-32) is given by w(k ) =
30k (k − 1)( P − k )( P − k + 1) P ( P 2 − 1)( P 2 − 4)
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The Cramer-Rao for Doppler rate estimates is given by
CRB(α ) =
30 ⎛ 1 ⎞ ⎜ ⎟ 2 N ( N − 1)( N − 4) ⎝ SNR ⎠
(8-34)
2
8.5.3 Least Square Based Doppler Rate of Change Estimator Another algorithm for estimating the Doppler rate of change (acceleration) is based on the use of linear square estimation, assuming the received phase modulated baseband signal that has a frequency offset and a Doppler rate is given by
z (k ) = e jθ k e j 2π∆fkT e jφk e j 2πβ (kT )
2
(8-35)
where k = 0,....2 N , θ k is the carrier phase to be estimated and tracked, φ k is the data phase modulations, and β is the Doppler rate. This assumes that data is available (data aided estimation), so that the modulations φ k in (8-35) are removed. The carrier frequency offset and Doppler rate least squares based jointly(??) can be estimated by 2N 2 ⎧ j 2πkη jµk 2 ⎫ (a, ξ , γ ) = arg ⎨(min − z ( k ) ce e ⎬ ∑ ⎩ c ,η ,µ ) k =0 ⎭
(8-36)
Based on the LS estimates (8-36), the carrier frequency offset and carrier phase is given ξ 1 by ∆fˆ = , and θˆ = arg{a} , and the Doppler rate is estimated by MT M 1 γ ⎛ ⎞ (8-36) βˆ = 2 ⎜ ⎟ [Hz/sec] T ⎝M ⎠
where M is the dimension of the MPSK signal.
8.6 NDA Feedback Carrier Recovery Scheme
To alleviate the shortcomings of the previous conventional carrier recovery schemes, the NDA carrier recovery scheme for TDMA MPSK burst satellite receivers is proposed. The scheme has the following characteristics:
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•
It employs the phase and frequency arctangent function detector that does not apply the approximation of sin(θ i − θ r ) ≈ θ i − θ r , therefore extending the detector linear region (i.e., the ‘S-curve’ is made linear over a wider range of errors).
•
It incorporates some means of suppressing amplitude modulation and fading. Both carrier frequency and carrier phase error detectors use a normalized version of the inphase (I) and quadrature phase (Q) MF samples, which removes all amplitude distortions including fading.
•
It makes use of a single error detector for both frequency and carrier phase that has smaller implementation complexity.
•
The schemes acquire signals with large frequency offsets, yet use a narrow loop filter to yield optimal carrier phase tracking.
3
Error magnitude
2
imag{z M ( k )} * real{z M ( k )}
1
0
-1
imag{z M ( k )} -2
-3
1 arg{z M ( k )} M -0.6
-0.4
-0.2
0
F
0.2
0.4
0.6
d
Figure 8- 9: Comparison of the Proposed ATAN and Conventional Phase/frequency Detectors. Though the ATAN phase detector is not suitable for analog implementation, it can be efficiently implemented in software using digital processors. Figure 8-9 provides a comparison of the S-curve of the ATAN detector with those of other conventionally used phase/frequency error detectors. Figure 8-9 confirms the superiority of ATAN error detector (in terms of its wider linear range) over the conventional PLL/Costas detector [16]. The ATAN phase detector exhibits a linear segment saw-tooth transfer function as opposed to the ‘sine’ nonlinearity of a product detector used in the system shown in Figure 8-5. Another advantage is that the ATAN detector produces exact phase error Dr. Mohamed Khalid Nezami © 2003
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information and not an approximation. Since an explicit arctangent operation is used for the ATAN detector, the carrier recovery loop will be insensitive to amplitude variations in the I and Q components of the MF signal. Thus the carrier loop is decoupled from the AGC, which significantly improves the performance of the receiver operating in rapid fading environment. Further, the phase estimate is accurate during acquisition as well as during steady state operation, except for noise and interference.
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M-Power Non-linearity
Symbol Timing
z(k )
[z( k )]M
MF
1 arg{.} M
⎧4 , QPSK M =⎨ ⎩2 , BPSK x
mod(θ err ( k ),
Complex multiplier
Baseband signal
2π ) M
x Kp CNCO
z −1
z −1
+
+
+ z −1
Numerical controlled Oscillator
Loop filter x Ki
Library of Coefficients
Figure 8- 10: The Proposed NDA Satellite Burst Carrier Recovery System.
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8.6.1
Details of the Acquisition and Tracking Algorithm
Figure 8-10 shows the schematic of the proposed new carrier recovery scheme. The operation of this scheme can be explained with regard to the noiseless MF output given by z k = c k exp{ j ( 2πkT∆f + θ k )}
(8-37)
where ck are the transmitted MPSK symbols. For BPSK c k = a k + jbk , where {a k , bk } ∈ j
π
j (m)
π
{+1, -1}, and for QPSK, the symbols are given by c k = e e 2 , where m=0,1,2,3. In order to replace the polarity detector with the ATAN detector discussed above, the signal is first M-powered to remove the data modulations. That is,
[z k ]M = [c k ]M exp{ j( M 2πkT∆f
+ Mθ k )}
4
(8-38)
Since the term [c k ] in(8-38) is unity, the output signal of the nonlinearity defined by (838) is therefore a noisy CW signal containing a fundamental frequency that is equal to the symbol rate offset by an M-fold carrier offset and phase. Due to the imperfection of data modulation removal using M-power nonlinearity, an AWGN term representing phase residuals must be included in the relation of (8-38). To illustrate this, the simulation model created in Chapter 2 was used to generate 2500 symbols of shaped QPSK at a data rate of 10 kbps, and the output of the modulator was subjected to an intentional frequency offset of ∆f =-800 Hz for testing purpose. Figure 8-11 shows the PSD of the output of 1 the nonlinearity, which indicates the presence of a single tone at + 4∆f = 6800 Hz. To T extract the frequency offset ∆f , the signal(8-38) is fed into the ATAN phase/frequency error detector, which generates an error given by M
1 M arg{[z k ] } = 2πT∆f + θ ( k ) + θ j ( k ) M
(8-39)
2π phase ambiguity, the error M signal must be post processed to remove the phase ambiguity, as described in Chapter 7.
Since the use of the M-power nonlinearity produces
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delf=-800 Hz, 10000-4x800=6800 Hz 30 PSD ( z M ( kT ))
Es=10dB Roll-off=0.75
Power Spectrum Magnitude (dB)
20
1 + M ∆f T
10
0
-10
-20
-30
-40 0
0.5
1
1.5
2 Frequency
2.5
3
3.5
4 x 10
4
Figure 8- 11: Extracting Symbol Rate Signal Using M-power Nonlinearity. The error signal (8-39) is then fed into the loop filter designed to have a loop bandwidth ∆f of BL ≥ max . The CNCO is then loaded with the filtered phase error estimate which 2 eventually synthesizes a signal that is equal to the conjugate of the symbol rate frequency compensated by the carrier offset and the phase rotation in the incoming MF samples. That is, ⎛1 ± ∆f ⎝T
θ CNCO = θˆr = −2πkT ⎜
⎞ ⎟ − θ (k ) − θ j (k ) ⎠
(8-40)
The CNCO used to synthesize (8-40) must be able to cover the frequency range of f CNCO ± M∆f max , where ∆f max is the maximum carrier frequency offset experienced by the receiver. The loop then performs complex multiplication of the MF samples and the synthesized CNCO signal, resulting in the complex signal given by z d = c k exp{ j 2πkT (
1 ± ∆f ) + jθ i } exp{ jθ r } T
(8-41)
After a predetermined time, the CNCO signal will be matched to the incoming frequency offset and phase of the MF, resulting in zero phase and zero frequency error, and lock is declared by the receiver. 8.6.2
Performance Studies
Figure 8-12 shows an input of three consecutive bursts of 500 symbols of shaped QPSK and the error detector output after acquisition and tracking for the loop illustrated in Dr. Mohamed Khalid Nezami © 2003
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Figure 8-10. The MF symbols had an intentional frequency offset of ∆f =-1800 Hz. The bursts were fed into the proposed carrier recovery scheme of Figure 8-10, operating with Es BL =1000 Hz, ς = 0.707 , and = 10dB . Once the loop settled, the NCO was No 1 automatically set by the loop to the frequency of f = + ∆f = 8200 Hz to compensate T for the error offset in the incoming signal. This is confirmed by the PSD plot of the steady state output of the CNCO in Figure 8-13. The experiment was repeated for smaller positive frequency offsets of ∆f = +500 Hz. The waveforms and PSD plots are given in Figures 8-14 and 8-15. Though the loop acquires quickly in both experiments, the phase of the derived error is perturbed by the AWGN and never decays to zero. Note that the variance of this random phase error can be reduced by decreasing the loop bandwidth, but will result in narrower capture range. Figure 8-16 shows a comparison between acquisition performance of the proposed system for two different loop bandwidths in acquiring and tracking an offset of ∆f = +400 Hz. For BL = 1000Hz, the loop acquires in less than 50 symbols, which intuitively sounds excellent for a 500-symbol burst. However, the loop error signal is perturbed by AWGN, causing phase modulations (phase noise) in the CNCO signal, resulting in an unacceptable random symbol rotation. By lowering the loop bandwidth to BL = 200Hz, the final phase random walk is heavily damped; however, it took an unacceptable 350 symbols for the receiver to acquire and track the carrier frequency offset of 400Hz. To illustrate this problem further, the final phase variance was monitored using Monte-Carlo analysis with 20,000 transmitted QPSK symbols (40 bursts). Figure 8-17 plots the phase variance against Es/No for loop bandwidths of BL = 1000Hz, 500 Hz, 200Hz, and 5Hz. In most of these cases, the variance approaches an irreducible value for Es/No larger than 16dB. The final phase variance is reduced from σ θ = 5.73o at BL = 1000 Hz to less than σ θ = 0.005 o at BL = 5Hz.
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Es/No=10 dB delf=-1800 Hz zeta=0.707 Roll-off=0.75 QPSK Bl=1000 Hz Rb=10kbps
In 1 0 -1 -2
Burst-1 0
Burst-3
Burst-2 500
1000
1500
2000
2500
x2 0 -0.2 -0.4 Burst-1
-0.6 0
Burst-3
Burst-2 500
1000
1500
2000
2500
Symbol Index
Figure 8- 12: Carrier Acquisition and Tracking of Three Consecutive Bursts with -1.8 kHz Offset.
5
5
0
0
0
-5
-10
-15
-20
-25
Power Spectrum Magnitude (dB)
5
Power Spectrum Magnitude (dB)
Power Spectrum Magnitude (dB)
Frame 1: NCO, Fs=40khz Frame 2: NCO, Fs=40khz Frame 3: NCO, Fs=40khz 10 10 10
-5
-10
-15
-20
-25
-5
Es=10 dB delf=-1800 Hz zeta=0.707 Roll-off=0.75 QPSK Bl=1000 Hz Rb=10kbps
-10
-15
-20
-25
-30
-30
-30
-35
-35
-35
-40
-40 8000 10000 12000 Frequency
-40 8000 10000 12000 Frequency
8000 10000 12000 Frequency
Figure 8- 13: PSD of the Three Consecutive Steady State CNCO Signal with –1.8 kHz Offset.
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In Es/No=high delf=+500 Hz zeta=0.707 Roll-off=0.75 QPSK Bl=100 0Hz Rb=10kbps
1 0 -1
500
1000
1500
2000
2500
x2
0.1 0 -0.1
Burst-1
Burst-3
Burst-2
-0.2 500
1000
1500
2000
2500
Symbol Index
Figure 8- 14: Carrier Recovery for Three Consecutive Bursts with +500 Hz Offset.
Frame 1: NCO, Fs=40khz Frame 2: NCO, Fs=40khz Frame 3: NCO, Fs=40khz 10 10 10 Burst-3 Burst-1 Burst-2 5 5 5
-10
-15
-20
-25
0 Power Spectrum Magnitude (dB)
-5
0 Power Spectrum Magnitude (dB)
Power Spectrum Magnitude (dB)
0
-5
-10
-15
-20
-25
-5
-10
Es/No=high delf=+500 Hz zeta=0.707 Roll-off=0.75 QPSK Bl=1000 Hz Rb=10kbps
-15
-20
-25
-30
-30
-30
-35
-35
-35
-40
-40 8000 10000 12000 Frequency
-40 8000 10000 12000 Frequency
8000 10000 12000 Frequency
Figure 8- 15: PSD of the Steady-state CNCO for the Three Consecutive Bursts with +500 Hz Offset.
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Burst-3
Es/ No=∞ ∆f=+400 Hz zeta=0.90 Roll-off=0.75 QPSK Rb=10kbps
BL = 1000 Hz
BL = 200 Hz
Symbol Index
Figure 8- 16: Comparison of Frequency Acquisition with Two Different Loop Bandwidths.
-1
10
Carrier-offset=0 Hz zeta=0.707 Roll-off=0.75 QPSK Rb=10kbps
-2
Variance in [Rad 2]
10
BL = 1000 Hz
BL = 500 Hz
-3
10
BL = 200 Hz -4
10
BL=5Hz
BL = 50 Hz
-5
10
0
5
10
15
20
25
30
35
40
45
50
Es/No
Figure 8- 17: Variance of Phase Tracking Error for Several Loop Bandwidths. 8.6.3 Adaptive State Machine Based Carrier Recovery Scheme
To improve the performance of the proposed feedback scheme further, we incorporate an adaptive loop bandwidth scheme as shown in the state diagram of Figure 8-18. The loop Dr. Mohamed Khalid Nezami © 2003
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is designed as a state machine and, during the initial acquisition, the loop bandwidth is set high. As the receiver starts to acquire the signal, the loop gradually narrows the loop bandwidth and achieves the best of both situations: fast switching and optimal phase error variance. Acquisition State 1
± ∆f max
Start symbol timing And decoding
Carrier carrier is pulled in
Re-acquire
Re-acquire Re-acquire
Re-acquire
Acquisition State 2
State 5 Symbol Tracking
BL = 5Hz
Tracking State 4
BL = 1000 Hz
Tracking State 3
BL = 500 Hz
Acquisition State 4
BL = 50 Hz
BL = 100 Hz
Figure 8- 18: State transition diagram of the proposed adaptive loop bandwidth scheme. Figure 8-19 shows the time progress of the acquisition process for the 500 symbol QPSK burst, using a fixed loop bandwidth system of 1000 Hz and a 5 state machine loop with BL1 = 1000 Hz, 500 Hz, 100 Hz, 50 Hz, and 5 Hz. The proposed state machine based scheme acquires the incoming satellite signal with zero phase error in less than 100 symbols. The experiment was repeated using twenty consecutive bursts and the results shown in Figure 8-20 confirm the consistency in the steady state acquisition. Notice that the state machine based scheme has the same implementation complexity as that of the system in Figure 8-10, except for a library of coefficients which are to be loaded each time the state machine exceeds a threshold or a timer count.
Although the proposed scheme works well, it is only able to acquire and track signals with moderate offsets (up to 1800 Hz in this case). This makes it suitable for most geostationary mobile satellite receivers. However, for LEOs or for receivers deployed on board high-speed spacecraft, the carrier offset may be as high as the received data rate. In such cases, the proposed algorithm may not be able to cope with such large frequency offsets. In the next section, a new open loop scheme with wide frequency capture range and rapid acquisition is proposed for carrier recovery for such applications.
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0.25 0.2
BL = 1000 Hz
0.15 0.1 0.05
BL 0 = 1000 Hz BL1 = 500 Hz
BL1=1000Hz for 100 symbols BL2 =500Hz for 100 symbols BL2=100Hz for 50 symbols BL4=50Hz for 50 symbols BL5 =5Hz for 50 symbols delf=800 Hz
BL 2 = 100 Hz
0
BL 3 = 50 Hz
-0.05
BL 4 = 5 Hz
-0.1 -0.15
Burst-1
-0.2 0
50
100
150
200
250
300
350
400
450
500
Symbol Index
Figure 8- 19: Comparison Between the Adaptive and Fixed Loop Schemes.
BL1=1000Hz for 50 symbols BL2 =500Hz for 20 symbols BL4=100Hz for 20 symbols BL5 =50Hz for 20 symbols BL5 =5Hz for rest of frame Es=12dB delf=800 Hz
Burst-1,20
Burst-1,0
Symbol Index
Figure 8- 20: Carrier Acquisition and Tracking for Twenty Consecutive Satellite Bursts Using the Adaptive Scheme.
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8.7 DFT-aided Carrier Recovery Scheme
Stand-alone feedback loops have lost their popularity for use in systems with large frequency offset. One approach has been to use DFT-aided feedback loops2 [14], where the initial frequency offset is estimated through Discrete Fourier Transform (DFT) to within the pull-in range of the feedback loop. This method yields accurate offset estimation, but suffers from high computational complexity of DFT and feedback instabilities. Though feedforward acquisition loops can be used to circumvent this and accelerate acquisition, their capture ranges are limited to about 10% of the symbol rate (see Chapter 5) and thus unsuitable for the mobile LEO satellite receivers where the carrier offsets can be as large as the symbol rate. To alleviate these shortcomings, we propose the use of open loop DFT based schemes that have the following characteristics:
•
The estimator operates on the MPSK modulated data directly, eliminating the requirement of carrier CW preamble.
•
The system is an open loop implementation, which does not suffer from conventional feedback instabilities, hang-ups, and cycle slipping.
•
It uses a small portion of the alternating sequence available for symbol timing recovery and is not limited to using the CW preamble symbols.
•
The large capture range enables it to cope with large frequency offsets.
•
By implementing the coarse open loop prior to the matched filter, limitations imposed by the passband selectivity of the channel match filter is avoided.
8.7.1
Details of the Algorithm
Consider the noisy demodulated phase shift keyed (PSK) baseband signal in (8-37) redefined as
2 Although its is refereed to the algorithm as being a DFT-based algorithm, it is assumed that the DFT is obtained using FFT algorithm for efficiency of computations.
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z(t ) = e j ( 2π∆ft+θ (t )+φ ( t )) + n(t )
(8-42)
where n (t ) is the complex AWGN with variance of σ n2 = [N 0 / 2 E s ] with E s being the
energy per information symbol. Here ∆f is the frequency offset to be estimated, φ (t ) is a phase term representing the transmitted symbol, and θ (t ) is the phase term that includes carrier phase error and phase rotation introduced by the propagation channel. This signal can be processed by the proposed system shown in Figure 8-21, where the baseband signal is sampled at 1 / Ts and lowpass filtered to a bandwidth given by BW = 2∆f max + B s
(8-43)
where ∆f max is the maximum expected carrier frequency offset, and B s is the bandwidth of the received baseband signal. The filtered samples are decimated by a properly chosen factor (D), and then processed by a DFT routine working on a block size of LDFT samples. The DFT provides an initial offset estimate of ∆fˆ = ∆f , which is used by a DFT
complex numerical controlled oscillator (CNCO) to synthesize a signal of exp{− j 2π∆fˆDFT t} . This signal is used to correct the buffered baseband samples in the FIFO resulting in a smaller carrier frequency offset in the incoming received signal. That is ∆f DFT ≤ ∆f FF , max
(8-44)
where ∆f FF , max is the maximum capture range of the fine tuning loop. After channelmatched filtering, the corrected samples are presented to the M-power nonlinearity that removes PSK modulations (see Chapter 7). The signal is then fed to a FF frequency offset estimator that produces an accurate frequency offset estimation of ∆fˆ = ∆f FF ( ∆f FF << ∆f DFT ). The frequency estimates of estimates ∆f FF are further used by a second CNCO to synthesize a signal exp{− j 2π∆fˆ t} for final frequency offset correction as FF
shown in Figure 8-20.
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Differential detector Symbol Timing
Baseband signal
z−
*
(.)
T FIFO
x
(). M
Matched Filter ej
2π∆ f DFTt
x
FIFO
x ej
Non-linearity
LPF/DEC
Coarse Loop
1
CNCO
Fine tune Loop
CNCO
Figure 8- 21: Schematics of the Proposed DFT-aided Carrier Recovery Scheme.
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2π ∆f FFt
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8.7.2
Coarse Acquisition Loop (DFT-aided Open Loop)
Fourier series expansion of the noise-free demodulated PSK baseband signal s(t) during the alternating sequence preamble shown in Figure 22, indicates [17] a spectrum consisting of a number of discrete frequency components. However, due to receiver lowpass filtering, only two components are significant. The separation of the components equals twice the frequency offset and is unaffected by any phase offset or symbol timing errors. Thus s (t ) = A exp{ j 2π (1 / 2T − ∆f )t + θ (t ) + φ (t )}
(8-45)
+ exp{ j 2π (1 / 2T + ∆f )t + θ (t ) − φ (t )}
where A is the channel gain. To illustrate this, Figure 8-22 shows a strip of 10 kbps QPSK TDMA satellite burst containing 4000 samples of alternating preamble sequence followed by random data message, and subjected to an intentional frequency offset of ∆f = 1000 Hz. preamble
1 0 -1
Power Spectrum Magnitude (dB)
-2
20
4000
4500
5000
Lower Spectral component
Upper Spectral component
0 -20
5500
1 − ∆f 2T
1 + ∆f 2T
-40 4000
4500
5000 Frequency
5500
6000
6500
Figure 8- 22: QPSK Satellite Burst Showing the Spectral Contents of the Burst While Being Subjected to 1000 Hz Carrier Frequency Offset. Taking the PSD of a small slice that contains preamble samples reveals that the signal has 1 two components located at f u ,l = ± ∆f , viz., f l = 4000 Hz and f u = 6000 Hz. The 2T open loop estimator computes the DFT S(k) using Fast Fourier transform (FFT) over a pre-determined number of bins, and locates both f l and f u to produce coarse estimates. That is,
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S (k ) =
N DFT −1
∑ s(n) exp{− j 2πkn / N } DFT
(8-46)
n =0
where k is the bin number and NDFT is the DFT bin resolution. A peak search is employed on Pk = S (k ) to locate both f l = 1 / 2T − ∆f and f u = 1 / 2T + ∆f to produce the coarse estimates. That is, k l , max fˆl = N DFT T
(8-47)
k u ,max fˆu = N DFT T
(8-48)
and
The final coarse frequency estimates is obtained by taking the difference of (8-47) and (848), yielding ∆fˆDFT =
1 1 ( k u , max − k l , max ) N DFT T 2
(8-49)
Likewise, considering peak search over only the positive bins, the offset can also be located using only the upper spectral ( f u ) component that moves up and down with the 1 Hz, hence the coarse estimate is given frequency offset within the region from 0 Hz to T by ∆fˆDFT =
1 1 − k u , max 2T N DFT T
(8-50)
The resolution of the estimates depends on the DFT bin size and is chosen based on the maximum capture range of the second fine tuning loop. Figures 8-23 and 8-24 present one estimate for a MF input subjected to a frequency offset equal to ∆f = 4100 Hz for 1 two different bin sizes dfˆDFT ( df DFT = ). When the frequency offset lies exactly on TN DFT a bin, both fˆ and fˆ can be accurately located using the proposed search algorithm, as l
u
shown in Figure 8-23 for the case of dfˆDFT = 100 Hz. However, if the bin resolution is decreased to dfˆDFT = 200 Hz, the actual frequency offset of 4100 Hz is no longer an integer multiple of the bin resolution, as shown in Figure 8-24. This causes the frequency offset not to coincide with a bin number. Therefore, the bin corresponding to fˆl and fˆu Dr. Mohamed Khalid Nezami © 2003
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will no longer be a single bin (spike), rather there will be a multiple of components as the energy of the frequency offset gets distributed among several adjacent bins. In this case, the peak search algorithm proposed to locate the frequency offset would have to use either fˆu1 or fˆu 2 as shown in Figure 8-24, instead of the frequency component of fˆu which lies in the region of fˆ ≤ fˆ ≤ fˆ . u1
u
u2
∆f=4100 Hz ∆f DFT=100Hz
Hz
Figure 8- 23: DFT Coarse Estimates Using Bin Size of 100 Hz.
∆f=4100 Hz ∆f DFT=200Hz
fu
f u1
fu2
Hz
Figure 8- 24: DFT Coarse Estimates Using Bin Size of 200 Hz.
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Since the coarse loop is used only to derive an initial guess of the carrier frequency offset and not an exact value, and to keep the complexity of the receiver low, there is no attempt at this point to improve the picking process between fˆu1 or fˆu 2 . Note that either component can pull the carrier offset to within the capture range of the fine loop. Nevertheless, an improved approach is presented next that performs interpolation between fˆu1 and fˆu 2 and more accurately locates the bin corresponding to the actual fˆu . 8.7.3
DFT Peak Search Refinement
The DFT bin quantization error problem discussed above can be circumvented by interpolation. To illustrate this, consider Figure 8-25 which shows two scenarios for the possible location of either fˆu or fˆl . By comparing the energy level of the two immediate neighboring bins ( fˆ and fˆ ) or ( fˆ and fˆ ) around the quantized location of fˆ or u1
u2
l1
l2
u
fˆl , a more precise actual location of fˆu or fˆl can be estimated using interpolation. The first scenario is when Pu1 ≥ Pu 2 where P is the power of the DFT bin, then the spectral line
f u corresponding to the actual frequency offset must lie in the region of 1 1 ; hence, the more precise location of f u can be obtained ≤ f u ≤ (k u ) ( k u1 ) N DFT T N DFT T by fˆu =
where d =
1 N DFT T
(ku − d )
(8-51)
Pu1 Pu1 + Pu
The second scenario takes place when Pu 2 ≥ Pu1 , in which case the location of f u can be obtained as ( k u )
1 N DFT T
≤ f u ≤ (k u 2 )
1 N DFT T
, leading to the more precise location of
f u at fˆu =
where d =
1 (ku + d ) TN DFT
(8-52)
Pu 2 Pu 2 + Pu
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actual offset estimated offset
pu pu 2
Scenario 1: 1 pu1
threshold
Pu 2 ≥ Pu1 1 1 ku ≤ f u ≤ ku 2 N DFT T N DFT T
k k u1
ku
ku 2
1 N DFT T
Scenario 2:
actual offset estimated offset
pu
Pu1 ≥ Pu 2 1 1 k u1 ≤ f u ≤ ku N DFT T N DFT T
pu1 pu 2
threshold
k k u1 k u
ku 2
1 N DFT T
Figure 8- 25: Peak Search Refinement Method. If the refinement method is not used in the coarse algorithm discussed above, the coarse loop bin resolution can be set to produce a coarse estimate that is within the range of the fine tune FF loop. The bin resolution of the coarse algorithm is given by df DFT =
1 TN DFT
(8-53)
which means that the coarse acquisition loop has a capture range given by −
1 1 ± df DFT ≤ ∆f DFT ≤ ± df DFT 2T 2T
(8-54)
For the 10 kbps system, assuming a unity decimation factor, the coarse loop has a total 1 Hz. If using the acquisition range of − 5000 ≤ ∆f DFT ≤ 5000 Hz at a resolution of N DFT T
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1
≤ ∆f FF . For N DFT T instance, in order to bring the frequency offset associated with the 10 kbps system from 1 5000 kHz to 1250 Hz, the DFT coarse loop must use ≤ 1250 Hz , or N DFT T N DFT ≥ 8 bins.
proposed QPSK FF algorithm as the fine loop, this implies that
8.7.4
DFT Based Fine Acquisition Loop (FF Loop)
To achieve the stringent accuracy requirements of the fine frequency tracking, an additional differential detector estimator has to be employed. The output of the differential digital detector can be shown to be
z(kT ) z ∗ ((k − 1)T ) = exp{ j(2π∆f DFT kT − 2π∆f DFT ((k − 1)T ) + θ (kT ) − θ ((k − 1)T ) + φ (kT ) − φ ((k − 1)T )) + n ′(kT )}
(8-55)
'
where n ( kT ) is AWGN. Assuming negligible carrier phase variation from symbol to j (φ ( kT ) −φ (( k −1)T ))
symbol, terms involving θ cancel out and the differential phase term e in(8-55) can be isolated using an M-power nonlinearity, where M is the number of phase states in the PSK. The nonlinearity output is
y k = e jM ( 2π∆f FF kT ) + n ' ( kT )
(8-56)
which is a noisy sine-wave with frequency equal to M-fold carrier offset. This offset is then estimated using any of the estimation algorithms detailed in Chapter 7. Using the maximum likelihood estimation method detailed in Chapter 6, the final carrier frequency offset is given by
⎫ ⎧ L Im[( y k )]⎪ ∑ ⎪ 1 ⎪ ⎪ tan −1 ⎨ k L=1 ∆fˆ = ⎬ 2πMT ⎪ ∑ Re[( y k )]⎪ ⎪⎭ ⎪⎩ k =1
(8-57)
where L is the observation length in symbols. The maximum capture range of the second 1 . Thus for 10,000 symbol/sec satellite QPSK burst, the offset loop is ∆f FF ≤ ± 2 MT estimation capture range is − 1250 ≤ ∆fˆ ≤ +1250 Hz. 8.7.5
Simulations Performance
To validate the overall algorithm performance, simulation experiments were conducted using a 10,000 symbols/s QPSK satellite burst subjected to a sweep of intentionally Dr. Mohamed Khalid Nezami © 2003
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introduced carrier offsets of − 6000 ≤ ∆f ≤= +6000 Hz. The burst consisted of the alternating sequence preamble followed by random data message. The experiment used a Roll-off of 0.35, DFT bin size of 500 Hz (20 bins), and E s / N 0 = 10 dB. The performance of the coarse loop in terms of the DFT estimate versus carrier offset is shown in Figure 8-26 (the decimation factor was unity). The figure indicates a capture range of − 5000 ≤ ∆f ≤= +5000 Hz. The staircase behavior of the estimated coarse estimate is due to the quantization caused by the peak search algorithm in selecting ∆fˆDFT = fˆu 2 . As a result, the estimate is always off by one bin (800 Hz). This mismatch in estimation is of no concern because the coarse estimate is only used to pull in the total frequency offset to within the capture range of 1250 Hz for the subsequent precise FF fine tune loop. During the coarse estimation of signals having low E s / N 0 , extra measures have to be used to avoid false look, caused by the presence of spurs and noise spikes. Figure 8-27 depicts a typical situation where the estimate often yields false lock by locating a spur or a harmonic that is noise related. This problem can be alleviated using the following methods: •
Use of windowing techniques to taper-off nearby spurs and harmonics.
•
Use of overlapped averaging, where the coarse observation window (bins) is “time hopped” or moved by half the window width each time an estimate is made, then averaging all the overlapped estimates. The use of overlapping has the advantage of recycling calculated bins.
•
Use of smoothing filters similar to those developed in Chapter 6. 6000 Es/No=20 dB
DFT coarse estimates Hz
4000
coarse DFT loop
2000
0
-2000 actual offset
-4000
-6000
-6000
-4000
-2000
0
2000
4000
Actual carrier frequency offset Hz
Figure 8- 26: DFT Coarse Loop Offset Acquisition Performance.
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6000
Coarse estimates Hz
4000
2000
DFT loop
0
-2000
-4000
-6000 -6000
-4000
-2000
0
2000
4000
Actual carrier offset Hz
Figure 8- 27: Coarse Loop Acquisition Performance at Low SNR (3 dB). Figure 8-28 shows the operation of the coarse and fine loops working independently over several consecutive bursts having a carrier offset sweep of − 6000 ≤ ∆f ≤= +6000 Hz. The experiment confirms that the fine loop estimation algorithm is very accurate, but has a narrow capture range of − 1250 ≤ ∆f ≤ +1250 Hz. On the other hand, the coarse loop is able to acquire offsets that are as high as − 5000 ≤ ∆f ≤ +5000 with bin resolution of 833 Hz (12 bins). Figure 8-29 shows the results of cascading the two loops to operate sequentially; the first coarse loop pulls the frequency offset from − 5000 ≤ ∆f ≤ +5000 Hz down to − 500 ≤ ∆f ≤ +500 Hz, while the second loop using 200 symbol per estimation accurately acquires and tracks the smaller offset to zero. 6000
Coarse and fine tune estimates
4000
2000
0
fine tune Loop
-2000 coarse Loop
-4000
-6000 -6000
actual introduced offset
-4000
-2000
0
2000
4000
6000
Introduced offset Hz
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Figure 8- 28: Separate Performances of DFT-aided Coarse Loop and FF Fine Tune Loop. 6000
Estimated frequency offset (Hz)
4000
Eb/No=12 dB Res_freq=500 Hz DFT_bin=80 bins QPSK, Roll-off=0.35 L=200
2000
Fine loop estimates
0
-2000
Coarse loop estimates
-4000
-6000 -6000
-4000
-2000
0
2000
4000
6000
Introduced frequency offset (Hz)
Figure 8- 29: Overall Performance of the Proposed Scheme with E s / N 0 = 12 dB, L = 200 Symbols and a DFT Bin Resolution of 80 Bins. Figure 8-30 closely examines the overall performance for the range of 1000 ≤ ∆f ≤ 2400 for both loops. Figure 8-31 shows a comparison between the coarse estimation error and the fine loop error estimates of the coarse estimates. The error signal for both loops is illustrated in Figure 8-32. It can be further shown that the variance of the estimation error for the fine tune loop in Figure 8-32 is bounded by the Cramer-Rao bound given by
σ ∆2f ≤ 3 /(4π 2T 2 L( L2 − 1) E s / N o ) [Hz2]
(8-58)
The bound can also be represented for the coarse loop in terms of DFT bin size. That is 2 2 σ ∆2f ≤ 3N DFT /(4π 2 N DFT ( N DFT − 1) E s / N o ) [bin2]
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2400
Estimated frequency offset (Hz)
2200
2000 Fine loop estimates
1800
∆f DFT = 500 Hz
1600
1400
Coarse loop estimates
Actual frequency offset
1200
1200
1400
1600
1800
2000
2200
2400
Introduced frequency offset (Hz)
Figure 8- 30: A Closer Look at the Overall Performance of the Proposed Scheme with E s / N 0 = 12 dB, L = 200 Symbols and a DFT Bin Resolution of 80 Bins.
400
Eb/No=12 dB DFT-Freq-Res=500 Hz Bin-Res=80 bins Roll-Off=0.35 QPSK, L=200 symbols
Estimated Frequency (Hz)
300
200
100
Fine tune tracking estimates
0
Coarse loop residual error
-100
-200
-3000
-2800
-2600
-2400
-2200
-2000
Introduced frequency error (Hz)
Figure 8- 31: Comparison between the Coarse Estimation Error and the Fine Loop Error Estimates with E s / N 0 = 12 dB, L = 200 Symbols and a DFT Bin Resolution of 80 Bins.
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500 Coarse loop error (Hz)
Eb/No=12 dB Res_freq=500 Hz DFT_bin=80 bins
0
-500 -8000
-6000
-4000
-2000
0
4000
2000
6000
8000
Introduced frequency offset (Hz)
Fine loop error (Hz)
500 Eb/No=12 dB QPSK, Roll-off=0.35 L=200
0
-500 -8000
-6000
-4000
-2000
0
4000
2000
6000
8000
Introduced frequency offset (Hz)
Figure 8- 32: Estimate Error for Both Loops with E s / N 0 =12 dB. To further characterize the algorithm behavior versus E s / N 0 , the experiment was repeated for Es / N 0 = 6 dB. Figure 8-33 shows the estimation error for both loops. Notice that the coarse offset estimate error using the DFT coarse loop performance at E s / N 0 = 6 dB has maintained accurate estimation, as there are no errors that exceed the single bin resolution of 500 Hz as shown in Figure 8-34. On the contrary, the final estimation errors are caused by the FF algorithm (fine tune loop). This problem can be circumvented by using a longer FF observation interval. This can be further observed by inspecting the FF estimation performance shown in Figure 8-35. 6000
Estimated Frequency (Hz)
4000
Eb/No=6 dB DFT-Freq-Res=500 Hz Bin-Res=80 bins Roll-Off=0.35 QPSK, L=200 symbols
2000
0
-2000
-4000
-6000 -6000
-4000
-2000
0
2000
4000
6000
Introduced frequency error (Hz)
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Figure 8- 34: Overall Performance of the Proposed Scheme with E s / N 0 = 6 dB, L = 200 Symbols and a DFT Bin Resolution of 80 bins. 500
Estimated Frequency (Hz)
0
-500 -8000
-6000
-4000
-2000
0
4000
2000
6000
8000
500 Eb/No =6 dB DFT-Freq-Res=500 Hz Bin-Res=80 bins Roll-Off=0.35, L=200
0
-500 -8000
-6000
-4000
-2000
0
4000
2000
6000
8000
Introduced frequency error (Hz)
Figure 8- 33: Estimate Error for Both Loops with Es/No = 6 dB. 500
Eb/No=6 dB DFT-Freq-Res=500 Hz Bin-Res=80 bins Roll-Off=0.35 QPSK, L=200 symbols
Estimated Frequency (Hz)
400
300
200
100
0
-100
Fine tune tracking estimates
Coarse loop residual error
-200 -2400 -2200 -2000 -1800 -1600 -1400 -1200 -1000
-800
-600
Introduced frequency error (Hz)
Figure 8- 34: Comparison between the Coarse Estimation Error and the Fne Loop Error Estimates with Es/No = 6 dB, L=200 Symbols, and a DFT Bin Resolution of 80 bins. 8.8 NDA Extension of the DFT-aided Open Loop Algorithm
For transmission systems without alternating sequence preamble, it is possible to extend the use of the proposed DFT-aided algorithm to operate with random data symbols using M-power nonlinearity. The nonlinearity produces a CW signal at the symbol rate that can be explored by the proposed algorithm to obtain the initial carrier frequency offset as Dr. Mohamed Khalid Nezami © 2003
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shown in Figure 8-36. The idea is that by raising QPSK samples to the 4th-power, spectral 1 lines at f u = ± Hz and f l = 0 Hz are produced. Both spectral lines move up or down T with frequency offset of M ( ∆f ) . The NDA-DFT coarse carrier frequency offset estimation in Figure 8-36 obtains a number of bins corresponding to possible location of 1 . That is the frequency offset bin in the range of 0 ≥ ∆f ≥ MT
⎧ N DFT −1
⎫
⎩
⎭
{∆f 0 , ∆f1 , ∆f 2 ,......., ∆f N } = ⎨ ∑ z 4 (n) exp{− j 2πkn / N DFT }⎬ n =0
(8-60)
Then a peak search algorithm is used to sort {∆f 0 , ∆f 1 , ∆f 2 ,......., ∆f N } from the pool of bin(s) with the energy above a predetermined threshold. Finally one single component is 1 identified that corresponds to the location of the frequency offset f u = + M∆f . This T bin is then used to obtain the final frequency offset estimate given by ∆fˆ =
1 M
⎛1 ⎫⎞ ⎧ N DFT −1 ⎜ − 1 max ⎨ ∑ z 4 ( n ) exp{− j 2πkn / N DFT }⎬ ⎟ ⎟ ⎜ T TN DFT k ⎭⎠ ⎩ n =0 ⎝
(8-61)
⎫ ⎧ N DFT −1 where the bin max ⎨ ∑ z 4 ( n ) exp{− j 2πkn / N DFT }⎬ is the bin with maximum energy k ⎭ ⎩ n =0 corresponding to f u .
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Ts
Baseband Update rate = Ts signal z( kTs ) FIFO x
z
−1
M () .
Matched Filter
x
x
FIFO
j 2π ∆f DFTt
e
CNCO
Non-linearity Ts : Sampling rate
(). M
(.)*
T
e LPF
Differential detector
Symbol Timing
Coarse Loop
T : symbol rate
Update rate = T
Fine tune Loop
CNCO
Figure 8- 35: Schematics of the Proposed NDA Version of the DFT Carrier Recovery System.
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Figure 8-37 plots the PSD of the output of the NDA-DFT coarse algorithm for the 10 ksps QPSK signal subjected to an offset of ∆f =+500 Hz. There are two components present, the first located at f l = 0 + M∆f or 2000Hz, and the second located at 1 f u = + + M∆f or 12000 Hz. Either one of the components can be used to extract the T actual coarse frequency offset associated with the received baseband signal, using peak search detector in conjunction with the predetermined threshold. Figure 8-38 shows the acquisition of frequency offset of ∆f = −500 Hz, (i.e., received carrier is lower than the nominal receiver). The positive spectrum contains only one spectral peak at 1 f u = + + M∆f or 8000 Hz, the other spectral peak moved into the negative frequency T regions, or f l = 0 + M∆f =-2000 Hz. 1 T
4x500=2000
fl
30
fu
10000+4x500=12000
20 10
dB
0 -10 -20
PSD{z^4} delf=+500khz Es/N0=40 dB Roll-off=0.75
-30 -40 0
0.2
0.4
0.6
0.8
1 Hz
1.2
1.4
1.6
1.8
2 4
x 10
Figure 8- 36: NDA DFT Coarse Algorithm Performance for an Offset of +500 Hz.
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30
f u = 0 − M∆f = −2000 Hz
fu
10000-4x500=8000
20 10 0 dB
-10 -20
PSD{z^4} delf=-500khz Es/N0=40 dB Roll-off=0.75
-30 -40
0.2
0.4
0.6
0.8
1 Hz
1.2
1.4
1.6
1.8
2 4
x 10
Figure 8- 37: NDA DFT Coarse Algorithm Performance for an Offset of -500 Hz. Clearly the 4th power nonlinearity has the effect of multiplying any frequency offset in the received signal by a factor of four. This effect narrows the total useful range of estimation (i.e., one-fourth the range obtained using the earlier scheme). Further, the use of 4th power nonlinearity results in a loss of 6 dB in SNR. Both reasons make the NDA algorithm less attractive than the alternating sequence aided DFT scheme. As a final remark on the NDA-DFT scheme, notice that the coarse estimate in (8-61) is similar to the NDA symbol timing estimate in (2-50) where symbol timing estimates were given by
εˆ =
2π −j k⎤ ⎡ L −1 −T arg ⎢∑ z (k )e 4 ⎥ 2π ⎣ k =0 ⎦
(8-62)
Here z (k ) is the nonlinearity used and the coefficient 4 in the exponent is the ratio of samples to symbols. Notice that the inner term in (8-62) is simply the computation of the phase associated with the DFT bin corresponding to the symbol clock (i.e., 1/(4T)). The other bins are not needed since (8-62) is immune to frequency offsets, and only phase information of the particular bin is needed to sense symbol timing changes. This suggests that (8-62) and the NDA-DFT frequency offset estimation in(8-61) can be combined to reduce the receiver complexity, and eventually reduce hardware and software complexity.
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8.9 Dual-Chirp Tone Aided Carrier Acquisition Scheme and Tracking
Another method for acquiring and tracking both time offsets and frequency offsets is achieved by using a dual-chirped tone as shown in Figure 8-38. Here, the transmitted preamble signal of a frequency is generated according to a triangular voltage controlled oscillator as shown in Figure 8-38. This sweeping of the frequency preamble tone generates the time domain signal illustrated in Figure 8-39. To resolve timing and carrier offsets from a noisy offseted replica of this signal, the receiver generates the opposite of this frequency slope and multiplies the received signal by it. The result is a sine wave signal whose frequency depends on the frequency offset (Doppler ) and the timing offset. It is then possible to use the DFT to find both tones and then simultaneously estimate both frequency and timing offsets. Let the transmitted dual chirp signal be represented as s (t ) = A sin( 2π f (t ) t )
(8-63)
where the frequency chirp function f (t ) is given by Kt 0
(8-64)
and that the slope of the ramp chirp is given by K=
f max 2T
(8-65)
where f max is the maximum frequency reached at the end of the chirp as shown in the figure 8-39, and T is the period of either the decreasing or increasing chirp signal as shown in Figure 8-40.
f max
t 0
T
2T
Figure 8- 38: Frequency Slope of the Synchronization Preamble Signal. The received signal for both increasing and decreasing frequency is then given by
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rincr = Ae j 2π ( Kt + ∆f )(t +εT ) +θ inc ) + ninc (t )
(8-66)
and rdecr = Ae j 2π ( − Kt + 2 f max + ∆f )(t +εT ) +θ decr + ndecr (t )
(8-67)
To estimate the frequency and symbol timing offsets, the received then synthesizes a local replica of both signals according to the following sincr = Ae j 2π ( Kt ) t
(8-68)
sdec = Ae j 2π ( − Kt + 2 f max ) t
(8-69)
and
similar to that shown in Figure 8-39. Amplitude vs. Time 2 1.5
amplitude (volts)
1 0.5 0 -0.5 -1 -1.5 -2 -6
-4
-2
0 time (µsec)
2
4
6
Figure 8- 39: Chirp Tone Preamble Used for Carrier and Timing Recovery. The frequency offset is then estimated by multiplying (correlation) the received chirp with the reference complex signal chirp, followed by accomodating the DFT
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(implemented as FFT) in the results to locate the tone spectral line that is produced, as shown in Figure 8-41. That’s is x incr (t ) = rincr (t ) s incr (t ) = Ae 2πj ( f incr t + ∆fεT +θ i )
(8-70)
and x decr (t ) = rdecr (t ) s decr (t ) = Ae 2πj ( f decr t + ∆fεT + 2 f maxεT +θ decr )
(8-71)
where ∆fˆ is the frequency offset to be estimated, and εT is the symbol timing offset. By taking the DFT (8-70) and (8-71), the tone with the maximum energy that correspond to fˆincr = KεT + ∆f and fˆdecr = − KεT + ∆f is located. The frequency offset and timing offset are then estimated by ∆fˆ =
fˆincr + fˆdecr 2
(8-72)
and the sampling time is estimated by,
εˆT =
f incr − f decr ⎛ f max ⎞ ⎟ ⎜ ⎝ 2T ⎠
(8-73)
f = fˆinc FFT
e j2
s (t )
e
πft 2
fˆinc + fˆdec 2
− j 2π ft 2
FFT
∆fˆ
f = fˆdec
Figure 8- 40: Frequency Estimates Using Chirped Tone Preamble. 8.10
Practical DSP Implementation Issues
Common digital signal processors do not have the intrinsic capability to directly evaluate the complex mathematical functions that are needed to implement the proposed algorithms. Examples of such functions are x , tan −1 ( x) , x / y , and x . The alternative is to use mathematical approximation. The approximation results in performance Dr. Mohamed Khalid Nezami © 2003
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degradation, but it simplifies by allowing the use of low cost commercially-available digital signal processing chips. 8.10.1 Complex Magnitude Approximation
Magnitude calculation of a complex sample signal I + jQ = I 2 + Q 2 is difficult to implement on commercial digital signal processors (DSP) because of the computation involved in performing the square root. An approximation to this operation can be obtained as ⎧ I + 0.375Q I > Q I + jQ = ⎨ ⎩Q + 0.375I else
(8-74)
Figure 8-42 shows the closeness of the approximated and the exact floating magnitude function. 1.2
approximation
exact and approx. magnitude
1
0.8
0.6
exact
0.4
0.2
0 920
925
940 935 930 sample instance
945
950
Figure 8- 41: Approximation of Complex Sample Magnitude. 8.10.2 Complex Division Approximation
Division also presents a timely consuming and tedious operation for most DSP chips. 1 x One common approximation is to recast into the multiplication x( ) , and then use y y polynomial expansion of the function 1 / y . That is 1 = 2.5859 y 2 − 5.8182 y + 4.2424 + ... y
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where 0.5 ≤ y ≤ 1 . The division is then becomes, x = x{2.5859 y 2 − 5.8182 y + 4.2424} y
(8-76)
Equation (8-76) is simpler to implement as it avoids the use of binary shifting operation, normally performed when implementing division using DSPs. Figure 8-43 provides a comparison between (8-76) and its exact representation. Notice that the approximation of 1 by (8-76) is quite accurate for values of 0.5 ≤ y ≤ 1 . This condition is not limiting, as y the normalized magnitude of the MPSK complex vector and the values of I and Q channels can never exceed 1. 2.5
approx(1/y), exact(1/y)
exact
2 approximation
1.5
1 0.4
0.5
0.6
0.7 y
0.8
0.9
1
Figure 8- 42: Approximation for Sample Division Operation. 8.10.3 Complex ATAN Approximation
The tan-1(x) function is commonly implemented using table lookup. This leads to slow response, quantization error, and large memory use. Figure 8-44 shows a proposed approximation to the tan −1 ( x) function, which can easily be programmed in DSP chips [16]. The approximation is given by π ⎧ π ⎪ − 4 ratioI ( k ) + 4 0 0 ≤ θ ≤ 90 0 arg{x k } = ⎨ π π 90 0 ≤ θ ≤ 180 0 ⎪− ratioII ( k ) + 3 ⎩ 4 4
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[
]
M
Here x can be z k z k∗ −1 for the frequency offset detector in (8-6), (8-38), or (8-55). The terms ratioI (k) and ratioII (k) are Lagrange quadratic equations defined as [15]
ratioI ( k ) =
Re{z k z * k −1 } − Im{z k z * k −1 } Im{z k z * k −1 }] + Re{z k z * k −1 }
(8-78)
and
ratioII ( k ) =
Re{z k z * k −1 } + Im{z k z * k −1 } Im{z k z * k −1 }] − Re{z k z * k −1 }
(8-79)
The function in (8-79) computes the ATAN over the first two quadrants only. As depicted in Figure 8-45, the location of the angle with reference to the quadrants I and II can be found through the sign of Re{z k z * k −1 } , while the location of the angle in Quadrant III and IV can be found by checking the sign of Im{z ( k ) z * ( k − 1)} . This extends the operation of the ATAN function over all the four quadrants as shown in Figure 8-46.
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Received Complex Signal
Q = Im{z k }
complex sample
+j > 0, Quad I
Re{z k }
I II
r
θ
1 III
1
< 0, Quad II
I = Re{z k }
IV
<0
<0
Im{z k }
Im{z k }
Quad I
Quad III
-j
>0 Quad II
I
II
III
IV
Im{z k } + Re{z k } +
+
-
-
-
-
+
>0 Quad IV
θ = −θ
θ = −θ
I Execute ATAN Interpolating Function Q fetch next sample
Figure 8- 43: ATAN Approximation Algorithm. Dr. Mohamed Khalid Nezami © 2003
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To extend the approximation of (8-79) to quadrants III and IV, the sign of the imaginary part (i.e., sign {Q(k)}) can be used. It can then be used to initiate an inversion of the computed angle. Figure 8-46 compares the 4-quadrant approximation against the exact tan −1 ( x) representation. angle approximation 3.5 actual
3 approximate
2.5
{
}
sign[Re z (k ) z * (k − 1) ] = −1
2 Rad.
1.5
second Quadrant
first Quadrant
1
{
}
sign[Re z (k ) z * (k − 1) ] = +1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
Figure 8- 44: Algorithm Performance Compared with Exact tan −1 ( x) Representation in Half Quadrant. 4 Angle in [rad]
3
tan
−1
{z(k ) z (k − N )} *
2 ratioI, II
Im{z (k ) z * (k − N )}
1 0 -1 Re{z (k ) z * (k − N )}
-2 -3 -4
0
200
400
600
800
1000 1200 sample
1400
1600
1800
2000
Figure 8- 45: Approximated ATAN Function Performance in Full Quadrant. Dr. Mohamed Khalid Nezami © 2003
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In Figure 8-46, negligible error is noticed especially at the interpolation points of 0 0 , π 4 , π 2 , 3π 4 , and π . If accuracy needs to be improved, a higher order polynomial can be used to compute ratioI and ratioII in (8-79). Another attractive feature of this algorithm is that the amplitude variation in Q and I does not affect the angle estimation in (8-78) and (8-79). This is particularly useful in cases where the multi-path fading attenuation is present due to mobile terrain effects.
8.11
Illustrating Example: Carrier, symbol timing, and channel tracking
Illustrate a complete end- to- end algorithm decryption of acquiring and tracking carrier, symbol timing, and channel gain.
Figure E8-1 illustrates the channel, carrier, and symbol timing estimation and tracking algorithm.
8.11.1 Illustrating Example: Carrier phase racking
The final algorithm (shaded area in Figure E8-1) performed on the received signal before actual decoding of the bits is the fine tracking of symbol timing, carrier phase and the amplitude normalization (AGC) that is critical for maintaining a constant constellation reference level into the decoder. After the combined time, frequency and phase offset acquisition and corrections (detailed in the previous memos), the signal y 4(kT ) hypothetically now only contains the desired phase modulations and channel gain effects. However, up to now, we have assumed that all of the estimations and corrections done prior to this point were ideal and that neither residuals nor time variations are present. In reality, there will be a small residual error due to noise and inaccuracies of the acquisition algorithms. Furthermore, these inaccuracies and time varying parameters will tend to slowly drift from symbol to symbol, and thus have to be tracked using feedback loops as illustrated in Figure E8-1.
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y ( kTs ) = g k a k e z
y 2 ( kT s )
⎞ ⎛ ⎛ T ⎞ j ⎜⎜ 2π ⎜ k + ε ∆ f +θ k +ϕ k ⎝ N ⎠ ⎠ ⎝
− LT s
y 3( kT ) y1(Ts ) z
− LT s
1 gˆ k
z e
− LT s
z
INT − τˆ
− j 2 π ∆ fkT s
e
τˆ
gˆ
y 4 ( kT )
− LT
− jθˆk
θˆ
∆ fˆ j δθ + 2 π T δ f (k + δτ )+ ϕ k ) y 4 k = ak e (
decoder
dk
1 aˆ k error
NCO AGLF − j δθ + 2 π kT δ f k ) ∗ e ( k eθ = im dˆk y4k CLF
(
TLF
)
δτ
Figure E8- 1: Carrier Tracking Including AGC (i.e., channel gain tracking).
Let us represent that the input signal to the decoder by the following relation, y 4 k = a k e j (ϕ k +δθ k + 2πTδf (k +δτ )) + n k
(E8- 1)
where δτ represent the small fine symbol timing error offset or drift, δf is the fine frequency offset residual error, and δθ is the fine carrier phase offset residual error. The removal of the tracking error e j (δθ k + 2πTδf (k +δτ )) is illustrated in Figure E8-1. Here, if these residuals are small enough that we can start decoding data, decision-aided phase and symbol timing tracking algorithms can then be performed [1,2]. However, if data decisions (decoding) cannot be carried out, some of these residuals can be tracked incoherently, i.e., non-data aided. In either scheme, the feedback system now tracks separately the phasor e j (δθ k + 2πTδfk ) and e j (2πTδfδτ ) and then corrects the signal by multiplying (E8- 1) by the conjugate of the estimated phasor e j (δθ k + 2πTδf (k +δτ )) . As a result, the final decoder input signal is given by y 4 k = a k e jϕ k + n k
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The channel amplitude a k is also corrected using decision aided equalization (AGC). This yields the final desired normalized signal given by y 4 k = e jϕ k + n k
(E8- 3)
Next we detail derivation of the three error signals that are used to track carrier phase, symbol timing, and channel gain or AGC for the scheme shown in Figure 1 .Once carrier offsets have been corrected using the coarse estimation algorithm (detailed in previous memos), the residual and time varying phase error past the preamble can be generated based on taking the imaginary part of the cross-correlation between the decision complex symbol representation of dˆk and the complex samples that were used to make this decision, to cancel the modulation e jϕ k in (1). That is
(
eθ = im dˆk∗ck e j (δθ +2πTδf (k +δτ )) + nk dˆk∗
)
(E8- 4)
where δθ + 2πTδf (k + δτ ) are the relatively small cumulative residual errors accounting for the residual errors of the prior carrier and symbol timing offset estimation algorithms. After removing the symbol phase modulations from (4), the input signal to the decoder is now given by
(
eθ = im e jδθ + 2πTδf (k +δτ ) + nk'
)
(E8- 5)
Using the fact that e jx = cos( x) + j sin( x) , and small offset errors in (E8- 5) can vary only in the imaginary part, since cos( x ≈ 0) = 1 we find
(
)
im e j∆θ + nk' = sin (∆θ ) + σ n2
(E8- 6)
where σ n2 is the variance of the final phase error signal that can be minimized by the feedback loop filter. Using the small angle approximation in (E8- 6), the error signal is approximated as sin(∆θ ) ≈ ∆θ . Thus the final carrier phase error signal that is fed into the loop filter in Figure 1 is given by,
eθ = sin (∆θ )
(E8- 7)
This error estimate is the classic phase lock loop error signal. Using a type II loop filter in conjunction with a NCO, this error signal (E8- 7) can be used to track and removed carrier phase and small frequency residuals as shown in Figure E8-1. The design of this PLL is outlined as follow. First, it is well established that the transfer function that Dr. Mohamed Khalid Nezami © 2003
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relates the estimated to actual phase offset from a linear digitized feedback loop is given by
θˆ( z ) Kd F ( z) N ( z) = θ ( z) 1 + Kd F ( z) N ( z) where F ( z ) =
K p + K i z −1 1 − z −1
(E8- 8)
is the carrier loop filter, where K P is proportional coefficient,
K NCO describes the NCO 1 − z −1 transfer function, where K NCO is the NCO gain in radians/seconds/volts. The design procedure for a carrier-tracking loop with type II filter is as follows. First starting with a specified maximum tracking phase variance σ θ2 at the operating E s / N o , the loop bandwidth is found from the following relationship,
and K i is the integral coefficient.
σ θ2 =
The function N ( z ) =
BLT Es / N o
(E8- 9)
Then the loop natural frequency wn for a given damping factor 0.6 < ξ < 1 and BL is found from the relationship in (E8- 10), wn =
BL ξ + 1 4ξ
(E8- 10)
The natural frequency in (10) has been very often assumed as wn = 2πf n = 1.8BL .Using (E8- 10) and the known NCO and phase detector gains, the loop parameters can be obtained using (E8- 11) and (E8- 12). The integral loop coefficients is given by 2ζwn Twn2 + Ki = 2 K d K NCO K d K NCO
(E8- 11)
where T is the loop iteration time duration (i.e., symbol duration), and K d is the phase detector gain in radians/seconds. The proportional coefficients is given by, 2ζwn Twn2 Kp = − 2 K d K NCO K d K NCO
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The recursive loop phase error update equation can be found from the relation between the input phase error eθ in (E8-7) and the output of the loop filter F (z ) that feeds the NCO. That is
θˆk +1 = θˆk + xk
(E8-13)
xk = xk −1 + K p eθ ( k ) + K i eθ (k − 1)
(E8-14)
where
An important fact about the type II loop design using (E8-11) and (E8-12) is that it will tolerate a maximum frequency offset residual of ∆f max ≤ BL . This offset should be the maximum error that can take place using the previously employed frequency offset acquisition algorithms in Figure E8-1.
8.11.2 Illustrating Example: Channel Gain Tracking (AGC)
The tracking and corrections of the AGC gain a k in Figure E8-1 can be also derived using iterative loop strategy similar to that obtaining (E87). This tracking can be carried out using decision-aided or non-data aided techniques. The kth value of the AGC (channel gain) is given by aˆ k +1 = aˆ k + γe AGC
(E8-15)
where 0 < γ < 1 is the loop filter parameter , and e AGC is the AGC or channel real gain error. For NDA schemes, this error is given by
e AGC _ NDA (k ) = y k − Aref
(E8-16)
where Aref is reference level based on the modulation constellation average point given by the averaged symbol energies, that is Aref = E{ ai + jbi } . For CPFSK Aref is unity. For decision-aided channel gain tracking, the error signal used in (E8-15) given by
(
ˆ e AGC _ DD ( k ) = Re dˆ k∗ y k e jθ + nk dˆ k∗
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8.11.3 Illustrating Example: Symbol Timing Tracking
One of the most widely used symbol timing error detectors and tracking loop that is independent of the phase corrections (E8-7) and data decisions (NDA), but requires two samples per symbol, is the Early-Late gate timing detector. The kth normalized NDA symbol timing error signal is given by eτ _ NDA ( k ) = Re( y k [ y k +1 / 2 − y k −1 / 2 ])
(E8-18)
The decision-directed version of this algorithm that is dependent on carrier phase (E8-7) and data decision (thus a single sample per symbol) is given by
(
eτ _ DD (k ) = Re y k dˆk∗−1 − yk −1dˆk∗
)
(E8-19)
This error in either (E8-18) or (E8-19) is then used to implement a digital feedback locked loop by recursively deriving the symbol timing error as shown in Figure E8-1. That is
τˆk +1 = τˆk + γeτ
(E8-20)
Where the constant 0 < γ < 1 is related to the loop filter bandwidth and timing error detector gain by
γ =
4 BLT Aτ (1 + 2 BLT )
(E8-21)
The constant Aτ in (E8-21) represents the slope (or gain) of the S-curve of the timing detector. The loop iteration factor γ also could be derived based on a desired loop filter bandwidth in (E8-21), that is BLT =
8.12
γAτ 2(2 − γAτ )
(E8-22)
Reference
1. Wannasarnmaytha, S. Hara, and N. Morinaga, “Two-step Kalman filter based AFC for direct conversion type receiver in LEO satellite communications,” IEEE Trans. Commun., vol. 49, No. 1, Jan 2000. 2. Randy Katz, “CS 294-7: Mobile satellite systems,” Lecture notes from University of California, Berkley, 1996. Dr. Mohamed Khalid Nezami © 2003
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3. Yu T. Su and Ru-Chwen Wu, “Frequency acquisition and tracking for mobile LEO satellite communications,” Proceedings of GLOBECOM, 1997. 4. US-Department of Defense, Military Standard for UHF Communications, MILSTD-188-183A. 5. William Cowley, “Phase and frequency estimation for PSK packets: bounds and algorithms,” IEEE Trans. Commun., vol. 44, no. 1, pp. 26-28, Jan 1996. 6. Yu T. Su and Ru-Chwen Wu, “Frequency acquisition and tracking for mobile LEO satellite communications,” Proceedings of GLOBECOM, 1997. 7. Edward W. Chandler, “A frequency tracking technique for multiple access satellite communication networks,” TCC-1990. 8. M. Hebley, “The effect of diversity on a burst-mode carrier frequency estimator in the frequency-selective multipath channel,” IEEE Trans. Commun., vol. 46, No. 4, pp. 553-559, Apr 1998. 9. Mordechi Rennert and Ben Zion Bobrovsky, “Designing second order Costas loops and PLL's to track Doppler shift-analysis and optimization,” Proceedings of GLOBECOM, 1995. 10. U.L. Rohde, Digital PLL Frequency Synthesizers: Theory and Design, PrenticeHall, Englewood Cliffs, NJ, 1983. 11. M. Nezami, “CAD tools allow easy design of PLL frequency synthesizers,” Microwaves & RF Magazine, pp. 72-80, Apr 1995. 12. Y. Saito, I. Horikawa, and H. Yamamoto, “16 QAM carrier recovery PLL for channel transmission using FSK additional modulation,” IEEE Trans. Commun., vol. COM-30, No. 8, pp. 1918-1923, Aug 1982. 13. T. Almedia, M. Piedade, “DPLL design and realization on a DSP,” ICSPAT 1999. 14. Shah, J. Holmes, and S. Hinedi, “Comparison of four FFT-based frequency acquisition techniques for Costas loop BPSK signals demodulation,” IEEE Trans. Commun., vol. 43, No. 6, pp. 2157-2165, Jun 1995. 15. Boza Porat, A Course in Digital Signal Processing, John Wiley, NY, 1997. 16. J. Balodis, “Laboratory comparison of tanlock and phase lock receivers,” National Telemetry Conf., 1964.
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Chapter 9 Synchronization in Spread Spectrum Communication Systems As the wireless personal communications field grew over the last fifteen years, the method of communication known as spread spectrum has gained a great deal of prominence in personal communication systems such as cellular and wireless local area networks. Spread spectrum involves spreading the information signal over a bandwidth that is much larger than the minimum bandwidth necessary to transmit the information signal as shown in Figure 9-1.This is done to mitigate jamming, interference, multipath fading, and security, thus offering cellular system operators with larger capacity, such as the IS-95, CDMA2000, and ISM systems [1-13]. The spreading process is implemented either using direct spread (DS-SS) or via frequency hopping methods (FH-SS). This chapter introduces the theory and principles of spread spectrum synchronization for acquiring and tracking the pseudorandom noise (PN) code and carrier offsets. The chapter focuses on PN code acquisition and tracking because both carrier frequency and carrier frequency phase offsets are estimated and corrected using the synchronization algorithms that were used in narrow band signals and detailed in the past chapters.
9.1 Principle of DS-SS PN Code Acquisition and Tracking Unlike narrow band TDMA systems discussed in the past chapters, spread spectrum receivers must align the locally generated pseudo-noise (PN) code with the received PN code before fine carrier tracking can take place. The spread spectrum receiver knows the pseudo noise code sequence used to spread the signal at the transmitter in advance; however, the receiver has no priori knowledge of the starting time of the spreading chip upon signal arrival. The associated delay is represented in the phase of the received signal chips relative to the locally generated chip sequence. Since the signal undergoes delays and phase and frequency shifting while propagating through the channel, the chip delay varies in time and has to be aligned with the locally generated PN code. If the PN local code used to despread the received signal is misalignment by more than a half chip, the signal will be re-spread again and it will look like noise at the receiver. If the received chip timing phase is misaligned by less than a half chip (late or advanced) relative to the local replica, the signal will start to despread and to look like a narrow PSK modulation and the receiver will start picking signal energy presence. The SNR will be very low and will be proportional to the misalignment between the two codes. As the misalignment starts to get smaller (code starts to be acquired) due to the steering mechanism of the local variable delay controlled oscillator used to generate the local PN replica, the SNR starts to increase and energy will start being concentrated into the main narrow spike indicating successful PSK demodulation Thus the cross correlation function between both local PN code and the received signal will be maximized; hence the Dr. Mohamed Khalid Nezami © 2003
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PN code has been acquired and the tracking algorithm has to be initiated and continued throughout the transmission.
Un-spread signal
Spread signal
Figure 9- 1: Spectrum Comparison between Narrow Band BPSK and a Direct Sequence Spread Spectrum BPSK Signal.
The process of synchronizing the local and received PN signals is ordinarily accomplished in two steps as shown in Figure 9-2. First, a coarse PN code alignment (acquisition) is achieved so that the two signals (received and local PN code replica) are within small relative code timing offsets (less than a half chip period) [1,2,3], which is represented by the system enclosed in the upper dashed box in Figure 9-2. As soon as the energy of the cross correlator exceeds a predetermined threshold (VT), which indicates an offset alignment of less than a half chip, tracking will be initiated, which continually maintains the best possible alignment by means of a code tracking feedback loop [5,6,7,9]. PN acquisition can be classified into serial and parallel search methods [1,2,10-15]. For parallel search methods, fast PN code alignment can be achieved but more hardware is required as compared with the serial search method. In this chapter we focus only on the serial search method, since it is the one most commonly used and since these algorithms are also used in the parallel approach (i.e., serial search method is essentially a single branch of the parallel method).
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To RX
x
RF-RX PN(t −τ k )
Acquisition Loop swi tch
NCO
No
VCO
ACQUIRED
PN(t −τ k −
T
1 (.) T ∫0
2
(.)
BPF
Tc ± 2
NCO
≥ VT
yes
DLF
Tc ) 2
x
(.)2
BPF
+ I,Q PN( t −τ k +
LOCKED
Tc ) 2
x
+
+ -
2
(.)
BPF Tracking Loop
Figure 9- 2: Pseudonoise Code Synchronization Algorithm. For the PN acquisition algorithm, the lower limit on acquisition timing is the time required to cycle through the complete PN code sequence, while the upper limit acquisition time occurs when the sequence is totally missed and the search process has to be re-initiated. For instance, for a system with chip rate of 3M chips/sec using a 10-bit Gold PN code [3], yield a 1023 PN chip sequence, the best (lower limit) acquisition time is the time required to cycle through the 1023 chips, which is given by Tacqu = 1023 3x10 6 = 0.341sec . It can be assumed in general that the code acquisition time is a uniformly distributed random variable with its mean equal to half the PN sequence time length, or Tˆacqu = 1023 2 x 3x10 6 = 0.17 sec . The PN tracking is carried out by the delay locked loop (DLL) shown in the bottom dashed box in Figure 9-2. The process of PN code tracking is similar to the timing synchronization methods for narrow band MPSK modulations discussed in Chapter 4, namely the Gardner or the Early-Late timing synchronization algorithms. The DLL PN code tracking approach works as follows. The received signal is first correlated with two different reference signals, one advanced a fraction of a chip PN (t − τ k − ∇T ) , while another is delayed by the same fraction, PN (t − τ k + ∇T ) , where T is the chip period, τ k is the delay misalignment that has to be estimated and tracked, where T 2 ≤ ∇T ≤ T [1,2,3,4,10,11,12]. The correlation output is then passed through a bandpass filter (BPF) to remove the data modulations by being passed through a square law envelope detector. An error signal is then formed by the difference between the outputs of the quadrature envelope detectors of the complex cross correlation. The error signal is then filtered using a loop filter similar to those used in narrow MPSK carrier and symbol tracking loops discussed in Chapter 3 and Chapter 4. The filtered signal is then used to adjust the Numerical Code Oscillator (NCO) used Dr. Mohamed Khalid Nezami © 2003
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to clock the code generator and to closely null the error signal by aligning its code sequence with the sequence associated with the incoming signal. Once the timing chip offset error signal reaches minimum, a “Locked” status is declared. The lock indicator is obtained by the sum of both quadrature output branches of the envelope signal from the cross-correlation process, which is maxim and will exceed a predetermined threshold. Once the PN local code and the received signal have minimal timing offset, the signal is despread correctly and the power present at the BPF Is at its maximum. 9.2 PN-Code Acquisition Algorithm
Mathematically, the acquisition and tracking loop in Figure 9-2 can be derived using the illustrative system shown in Figure 9-3. Assuming that the normalized received modulated spread spectrum signal s (t ) in Figure 9-3 is given by s (t ) = c(t )d (t ) cos(2πf c t + +θ )
(9-1)
where c (t ) is the PN code used, d (t ) is the data transmitted, and cos(2πf c t + +θ ) is the carrier frequency signal with an associated arbitrary phase shift. The first stage of processing in the synchronization system is the quadrature down conversion that is performed by multiplying the signal in (9-1) with a local oscillator signal generated using local oscillator VCO. The inphase and quadrature signals of the NCO signal are programmed to a nominal carrier frequency fˆc having a frequency offset ∆f and phase offset θˆ , where ∆f = fˆ − f and ∆θ = θˆ − θ at baseband. The inphase and quadrature c
c
outputs of the quadrature down conversion is then given by
( (
)
)
y I (t ) = 2 cos 2π fˆ + ∆f c t + θˆ c(t )d (t ) and
( (
)
(9-2)
)
y Q (t ) = −2 sin 2π fˆ + ∆f c t + θˆ c(t )d (t )
(9-3)
Realizing that for BPSK, with binary symbols ‘1’ or binary ‘0’ (that is the phase shift of 0 degrees OR 180 degrees), the product c (t ) d (t ) = ±1 in Equation (9-3) is c (t ) d (t ) = ±1 Then after cross correlation and the integrate and dump filters (ID), the quadrature signals are given by I ( kT ) =
sin(π∆fTs ) ⎛ π∆fTs ⎞ R (τ ) cos⎜ + ∆θ ⎟ π∆fTs ⎝ 2 ⎠
(9-4)
Q ( kT ) =
sin(π∆fTs ) ⎛ π∆fTs ⎞ R (τ ) sin⎜ + ∆θ ⎟ π∆fTs ⎝ 2 ⎠
(9-5)
and
where Ts is the sampling period, ∆θ , and ∆f are the carrier offsets, and R(τ ) is the cross correlation outcome between c(t ) and cˆ(t − τ ) , where τ is the timing offset between the code speeding the signal and the local replica cˆ(t − τ ) . It can be seen that the correlation products in (9-4) and (9-5) which are used to derive the PN code timing error are a function of both carrier frequency and carrier phase offsets. Both signals are attenuated by a factor of sin x x due to the carrier frequency offset. However
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it is assumed most times that the offset is small enough, so that
sin(π∆fTs ) ≈ 1 . The output of the π∆fTs
envelope detector (squarer) is then computed by Y (kT ) = I 2 (kT ) + Q 2 (kT )
(9-6)
where T is the bit period (sampler rate in Figure 9-3). This signal is then averaged over a block interval of N B to minimize the noise variance and eliminate low frequency modulations due to multipath fading. That is NB
1 z= NB
∑ Y (kT )
(9-7)
k =1
The magnitude of (9-7) is an indication of the misalignment error between both local PN code c(kT ) and the received signal s (t ) . The PN code acquisition is then declared when the cross correlation value in (9-7) exceeds a threshold, z ≥ η , where η is based on the desired minimum SNR, timing capture range, and minimum tracking misalignment. ID
( )2 I (kT )
cos( 2πf ct + θˆ) s (t )
VCO
PN Generator
+
c(t − τ )
y (kT )
− sin( 2πf c t + θˆ) Q (kT )
( )2
ID
z Code Phase Adjustment Frequency Adjustment
z <η
1 NB
NB
∑ k =1
η
Figure 9- 3: Non-coherent Pseudo Noise Code Acquisition System.
Next we discuss methods by which we can evaluate the reliability of acquisition in (9-7), based on the probability of detection and of false alarm. 9.3 Probability of Acquisition Detection and Probability False Alarm
One common measure of receiver PN code acquisition performance is to form the probability of PN code detection and PN code probability of false detection (false alarm). Using the probability density function (PDF) of the envelope detector output in Figure 9-3, z and the threshold η [3,the probability of PN code detection when a valid signal is present is given by Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers ∞
PD = ∫ p s ( z )dz
(9-8)
η
Where p s ( z ) is the probability density function of z for the case of signal present.. The probability of false alarm is given by ∞
Pfa = ∫ p n ( z )dz
(9-9)
η
where p n ( z ) is the probability density function of z for the case of no signal present. One way to establish these measures when designing a receiver is to assume that the block length in (9-7) is relatively long, N B >> 1 and that R(τ ) ≈ 1 in (9-4) and (9-5), meaning that the loop is in tracking mode. Assume that the probability of detection when a valid signal is present is given by
PD = Q(α )
(9-10)
where Q( x ) is the complement cumulative Gaussian distribution, and α is defined in (9-12) . Using the same approximation technique, the probability of false alarm is given by Pfa = Q (β )
(9-11)
where
β − γ NB
α=
(9-12)
1 + 2γ
and
⎛ T
⎞
β = N B ⎜⎜ η − 1⎟⎟ ⎝ No ⎠
(9-13)
A2 2 where SNR value γ = T , A 2 2 is the power of the received carrier, and N o is the single-sided No noise spectral density.
The acquisition loop design is based on using a specific PD and Pfa (9-9) and (9-10) at a specific desired signal-to-noise ratio to obtain the values of α and β . Optimal threshold η is then found using the following equations,
η=
⎤ No ⎡ βγ ⎢1 + ⎥ T ⎣⎢ β − α 1 + 2γ ⎥⎦
(9-14)
And the block length is given by ⎡ β − α 1 + 2γ ⎤ NB = ⎢ ⎥ γ ⎥⎦ ⎣⎢
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As an example, with desired Pfa = 10 −4 , the value of α = −1.7 and with a desired probability of detection PD = 0.95 the value of α = −1.7 is calculated. Then using equations (9-12) through (9-15), the required signal to noise ratio has to be at least γ = −2dB and the minimum averaging length N B has to be at least N B =100 symbols. 9.4 PN-Code Tracking algorithm
Typically there are two structural options for code tracking: the delay locked loop (DDL) presented earlier in Figure 9-2 and Figure 9-3, which will be detailed in the next section [3]. In principle, the operation of PN tracking loops is similar to that of a classical Costas loop used for carrier phase tracking. Figure 9-4 illustrates the block diagram of the DLL scheme. The PN synchronization timing error signal is generated by taking the difference between the outputs of the two envelope detectors. The error signal is then filtered using the loop filter and subsequently used to adjust the code generator NCO to closely align the de-spreading code with the original spreading code.
on time PN Generator
early late
LF Tracking
e(τ )
(.)2
BPF
+ RF-RX
(.)
BPF
2
Figure 9- 4: Pseudo-noise Tracking System.
+1
δ=
T 2
e(τ )
-1 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
+1
δ =T
e(τ )
-1 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
+1
δ = 2T
e(τ )
-1 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
T time (normalized by T)
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Figure 9- 5: Ideal Error Detector for Half Chip Advance and Delay Offset.
In Figure 9-4 and 9-5, the PN code timing error detections obtained using the Early-Late gate timing loop discussed in Chapter 4. This is based on using two offset PN sequences, one is late (delayed) and one is early (advanced) both by fractions of a chip period T . These offset codes are separately multiplied with the input signal from the receiver as shown in Figure 9-4. The multiplied outputs are then passed through bandpass filter and envelope detector (square law detector). The difference between the resultant outputs e(τ ) is then used as a correction signal, which is subsequently filtered using a loop filter whose output is then used to adjust the reference code timing. The normalized error signal e(τ ) is given by
e(τ ) = Rc (τ + δ
) − Rc (τ − δ )
(9-16)
Where δ = Tc 2 is the delay used (lead lag value). The value of δ = Tc 2 is very often used. The value of Rc (τ + T 2 ) is the output of the crosscorrelation and is given by
Rc (τ − δ ) =
T
1 c(t )c(t + τ )dt Tc ∫0
(9-17)
The value of c (t ) is the pseudnoise spreading code. Figure 9-5 shows the error signal as a function of chip time offset. It is observed that the error signal is an even function and symmetrical, which is an ideal condition for loop steering operation. The process of cross-correlation to the error signal that is then used to control the delay of the locally generated PN sequence continues until the locally generated PN code correlation with the input signal is maximized, at which point the local PN code is accurately aligned with that used at the transmitter. In Figure 9-5 note that there is a tradeoff between the accuracy and range of the error detector that depends on the timing δ lag offset used in (9-16)
For the case of coherent BPSK PN-receivers, if the carrier is tracked prior, PN tracking needs only the inphase output of the Early-Late correlator. This error is given by ek =
1 (I E − I L )dˆ 2
(9-18)
where the I E and I L are obtained from the correlations over a full data symbol period using the scheme as shown in Figure 9-6, and dˆ is the modulated BPSK data polarity, where dˆ = sign{I } .
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Late-sample
Early -sample
On-time sample
RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
IL
IE t =0 t = +T t = −T Figure 9- 6: Pseudo-noise DLL Principle Showing the Loop While Sampling Too Fast.
The error signal in (9-18) can be modified for the case of quasi-coherent BPSK DSS receivers. Instead of feeding back the data decision, both I and Q samples are used to compute the error signal. That is ek =
(
)
(
1 2 1 I E − Q E2 − I L2 − Q L2 2 2
)
(9-19)
During fading, the error detector in (9-18) and (9-19) both degrade due to multipath presence. The impact of multipath can be very severe and often leads to a false lock. The additional signal paths propagating with a significant spread delay result in local maximums that may cause the DLL to false lock as shown in Figure 9-7. One way to improve the performance correlation of a timing detector in the presence of fading is by averaging the late and early correlations over multiple samples per symbol. That is ek =
1 2N
∑ ⎢⎣(I N
⎡
k =1
2 E ,k
)
− Q E2 , k −
(
)
1 2 ⎤ I L ,k − Q L2,k ⎥ 2 ⎦
(9-20)
where N is the over sampling factor and the subscript k is the sample index [1,2,3,4,12] . +1 Due to delayed path t = −Tc
t=−
Tc 2
τ t=+
Tc 2
t = +Tc
−1 Figure 9- 7: Error Signal in the Presence of Two Paths. Dr. Mohamed Khalid Nezami © 2003
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Based on (9-20), Figure 9-8 illustrates the timing error detector used to implement a DLL in conjunction with carrier tracking system. The carrier tracking loop operates on the “on time” version of the despread signal, right after the DLL has acquired the signal. Notice that the carrier error signal is also incorporated in the DLL correction since the carrier offset is related to the PN code by the phase shift, which is a large time offset .As a result, the carrier offset error being incorporated into the DLL helps to accelerate PN code tracking. Figure 9-9 illustrates the combination of PN code acquisition, PN code tracking, and Costas loop for carrier tracking, where all three synchronization loops are implemented digitally. The down conversion is the multiplication free digital down conversion scheme described in Chapter 1. Figure 9-10 shows a linear tracking model for the DLL loop. The loop bandwidth and the tracking error variances of the systems shown in Figure 9-8 and 9-9 are designed based on the model in Figure 9-10 using the analytical methods derived in Chapter 3 and Chapter 4. The loop dynamics that can be derived based on the closed loop transfer is given by Gc ( z ) =
z −1 z − 1 + K L K EL K NCO
(9-21)
where K L is the multiplication of loop gain factor, K NCO is the NCO gain, and K EL is the Early-Late gate timing discriminator gain derived from the S-curve as follows.
τ
KEL
+
KL
τˆ K NCO 1 − z −1
Figure 9- 8: Linearized Loop Model for the PN Tracking System.
The timing error detector gain K EL is derived from the linear region of the E-L detector, shown here in Figure 9-11, for a half chip offset δ = Tc / 2 . In order to express the interval in the number of symbols periods and not chip periods, a division by the symbol period Ts is required. Based on this and using Figure 9-11, the E-L gain is given by S (T / 4) − S (−T / 4) (9-22) (T / 2) / Ts where N = T / Ts is the number of chips per symbol. Substituting this into (9-22), the PN code timing error detector gain is given by K EL =
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(9-23)
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τ =0 τ =−
T 2
τ=
T 2
Linear region
τ =−
T 4
τ=
T 4
Figure 9- 9: Computations of the Early-Late Detector Gain Using the S-curve.
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Carrier Recovery
IE I OT
IL QE
s (kT )
ID ID ID
N B −1
∑ (I k =0
ID
2 E
N B −1
) ∑ (I
+ QE2 −
k =0
2 E
+ QE2
)
NCO QOT
QL
Early
ID ID
Late
On Time
DDL-LF PN Code Generator CR-LF
Figure 9- 10: Combined Pseudo-noise Code and Carrier Tracking System
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Loop Filter
VCXO
Demodulated data
dˆ
Narrow Band BPSK Demodulator
Nfs
∑
η
()
2
+
1,-1,1…
∑
PN Code Generator
( )2
Early PN
A/D
T
≥
s(kT )
τ = −T
corrections NCO
Late PN
2Nfs
On time PN
On time PN
τ =0 τ =T
τ = −T 2
∑
τ =T 2
( )2 τ =0
s(kT )
+ ∑
( )2
∑
()
∑
( )2
+
2
fs
Loop Filter
s(kT )
+
Figure 9- 11:Block Diagram of a Combined Carrier Recovery, PN code Acquisition, and Feedback Tracking System.
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9.5 Tau-Dither PN Tracking Loop
Another implementation of PN tracking is known as the Tau-Dither loop. As mentioned before, this is one way of reducing the hardware and software complexity of the DDL system [3] shown in Figure 9-12 by multiplexing the use of some of the common parts of the loop as shown in Figure 9-10. The loop uses only a single channel to perform the advance and delayed correlation. This arrangement generates the same code offsets as that generated using the DLL; however, only one multiplication branch is used at a time periodically alternating between the advanced and delayed versions of the generated PN code. The rate at which the branch is multiplexed is usually at half chip period and the multiplexing is usually achieved by means of gating the signal with a +/-1 sequence. This multiplexing process not only alternates the early and late versions of the local PN code, but also supplies the correct sense direction of the error. This is carried out using the following algorithm,
f (t ) = g (t )c(t − T 2) + [1 − g (t )]c(t + T 2)
(9-24)
depending on the gating function g (t ) , (9-24) is given by, ⎧c (t − T 2 ) , g (t ) = 1 f (t ) = ⎨ ⎩c (t + T 2 ) , g (t ) = 0
(9-25)
Depending on the value of g (t ) being one or zero, the loop error signal alternates between an error caused by either the advanced PN code or the retarded PN code.After performing cross correlation, the output of the envelope detector in Figure 9-12 is given by
e(τ ) = g (t ) Rc (τ + T 2) + (1 − g (t ) ) Rc (τ − T 2)
(9-26)
This error signal is then passed through the loop filter and gated by g (t ) to create the alternating early and late versions of the local PN code. The appropriate polarity associated with the PN code cross correlation with the incoming signal is carried out by multiplying the output of the envelope detector with the function 2 g (t ) − 1 as shown in Figure 9-12, which converts the gating sequence from 101…. to 1-11. The final error signal becomes, e(τ ) =
1 ( Rc (τ + T 2) − Rc (τ − T 2) ) 2
(9-27)
which is the same error signal obtained by the DLL shown in Figure 9-4.
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g (t )
late PN Generator
early f (t )
Tracking Loop BPF
RF-RX
LF
e(τ )
(.)2
s (t )
2 g (t ) − 1 Figure 9- 12: Tau-Dither Loop System
9.6 Feedforward PN Code Synchronization Algorithm
The Feedforward PN code acquisition and tracking scheme have also been used. Figure 9-13 illustrates one such system that utilizes the maximum likelihood estimation method to estimate and correct PN code timing offsets. Both the acquisition (coarse) and the tracking process (fine estimates) are using feedforward schemes. Assuming that the received signal in Figure 9-13 is down converted and then sampled at a rate of f s = 2 Rc , or two samples per chip, the digitized signal is then given by r ( k ) = s ( kTs + τ )e − j (2πkTs ∆f +θ )
(9-28)
The time delay τ is the unknown PN time code delay of the received signal, where, θ and ∆f is the carrier phase and frequency offset, s (k ) is the sample of the complex baseband signal given by s (t ) = 2 Eb d (t )c(t )
(9-29)
The estimation of both PN code delay and the carrier using feedforward is obtained as follows. First blocks of signal samples are stored in memory while a replica of it is passed to a coarse maximum likelihood (ML) estimator to estimate both PN code delay and frequency offset. Both estimates are then used to frequency correct and then despread the stored sample block. Also, the configuration of the coarse acquisition enclosed in the dashed box shown in Figure 9-13 is based on maximum likelihood estimation [3]. The received signal block stored in memory is multiplied by the local PN
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code, each with different delay, iTc , and then the power spectrum of each branch is estimated using FFT. That is X ik (l) =
lN + ( N −1)
∑ r (n)c(n − iT )e
2 − j ( 2πkf o nTs )
, l = 0,1,2,......
n = lN
(9-30)
The PN code delay and frequency offset are then estimated using a trial set that simultaneously maximizes the value of X ik . After taking the FFT of each branch, the branch power is accumulated over an observation interval, yielding Z ik (l) =
l=M
∑X l =0
ik
(l )
(9-31)
The largest value of Z ik in (9-31) is selected, which correspond to the branch that has the coarse estimates of iˆTc = Tc + τˆ and kˆf o = ∆fˆ . The number of FFT bins is selected by
Tb (9-32) Ts After being corrected by the coarse estimates, the despread output is then integrated and dumped at every bit period, iTb . As a result of the coarse rough estimation, the baseband signal still has a fine carrier phase and frequency offset that has to be estimated. The resulting despread signal, including these fine offsets, is given by N≤
sin (π∆f 2Tb ) − j ⎜⎜⎝ 2πTb ∆f 2 ⎜⎝ i − 2 ⎟⎠ +θ 2 ⎟⎟⎠ R (i ) = Eb di e π∆f 2Tb ⎛
⎛
1⎞
⎞
(9-33)
where d i is the transmitted data, and ∆f 2 = ∆f − ∆fˆ is the transmitter-receiver frequency offset .Both offsets are now passed to a feedforward phase and frequency estimation algorithms such as those detailed in Chapter 7. For MPSK data modulations, the frequency offset can be estimated by ∆fˆ2 =
(
1 arg ∑ R (i ) R ∗ (i − 1) 2πTb
)
M
(9-34)
and once the signal is corrected, the phase can also be estimated by
(
)
1 ˆ M arg ∑ R (i )e − j 2π∆f 2i 2πTb which is then used to correct the signal as shown in Figure 9-13.
θˆ2 =
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c(n) FFT
∑
FFT
∑
c(n − T )
LPF
A/D
Frequency synthesizer
memory
c( n − 2T )
X ik
Select largest branch
∑
FFT
Frequency offset estimates
PN code Timing estimates
∑
FFT c( n − iT )
e
− j 2π∆fˆkT
t = iTb + Tb
c ( n − τˆ)
correction
∫
Data decoding
∆fˆ2 θˆ2 Fine Frequency estimator
Figure 9- 13: DS-SS Carrier Frequency and PN Code Synchronization.
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9.7 Pilot-aided PN Code Synchronization Algorithm
Very often, a pilot channel with a common PN code is transmitted and shared between the multi-users for control and synchronization, such as with a IS-95 system [1]. This pilot channel is usually transmitted at higher power to mitigate fading and increase the probability of detecting and tracking the PN code, as is the case with the IS-95 cellular system. Figure 9-14 illustrates a pilot-aided PN synchronization system. At the output of the down converter, the received IF signal rIF (t ) is pilot modulated and given by
r (t ) = I (t ) cos(2πf IF t + θ ) + jQ(t ) sin(2πf IF t + θ )
(9-36)
Ignoring interference, the pilot quadrature modulations in (9-36) are given by I (t ) = a o PI (t )
(9-37)
Q (t ) = a o PQ (t )
(9-38)
and
where ao is the channel gain, and PI (t ) and PI (t ) are the pilot quadrature symbols. The down converted sampled IF signal is then given by r (kT ) = rIF (kT ) cos(2πf NCO kT ) + jrIF (kT ) sin( 2πf NCO kT )
(9-39)
sampling the IF signal at a rate that is 4-times, f s = 4 f IF , the down converted signal is given by 2πf k 2πf k r (kT ) = rIF (kT ) cos( IF ) + jrIF (kT ) sin( IF ) (9-40) 4 f IF 4 f IF After cross correlating (9-40) with the locally generated PN signal, the inphase signal is given by
y I (kT ) =
NN corr −1
∑ k =0
πk ⎤ ⎡ ⎢rIF (kT ) cos( 2 ) PI (k + n0 )⎥ + j ⎦ ⎣
NN corr −1
∑ k =0
πk ⎤ ⎡ ⎢rIF (kT ) cos( 2 ) PQ (k + n0 )⎥ ⎦ ⎣ (9-41)
and the quadrature branch is given by
y Q (kT ) =
NN corr −1
∑ k =0
πk ⎡ ⎤ ⎢⎣rIF (kT ) sin( 2 ) PI (k + n0 )⎥⎦ + j
NN corr −1
∑ k =0
πk ⎡ ⎤ ⎢⎣rIF (kT ) sin( 2 ) PQ (k + n0 )⎥⎦ (9-42)
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Where N is the number of samples per chip and Ncorr is the total symbols over which the correlation is performed. Simplifying both (9-41) and (9-42) results in the following quadrature signals, NN corr −1 NN corr −1 a0 a0 y I (kT ) = cos θ ∑ [PI (k ) PI (k + n0 )] + j sin θ ∑ PQ (k ) PQ (k + n0 ) 2 2 k =0 k =0 (9-43) and for the quadrature path,
[
]
NN corr −1 NN corr −1 a0 a0 y Q (kT ) = − sin θ ∑ [PI (k ) PI (k + n0 )] + j cos θ ∑ PQ (k ) PQ (k + n0 ) 2 2 k =0 k =0 (9-44)
[
]
where the term cos θ , and sin θ resulted from carrier offset. Taking the magnitude of each channel and then summing to generate the envelope of the correlation process, will cancel the carrier phase terms due to the fact that cos 2 θ + sin 2 θ = 1 . That is
⎛a y (kT ) = ⎜⎜ 0 ⎝ 2
2
⎞ ⎛ [PI (k ) PI (k + n0 )]⎟⎟ + ⎜⎜ a0 ∑ k =0 ⎠ ⎝ 2
NN corr −1
⎞ PQ (k ) PQ (k + n0 ) ⎟⎟ ⎠
NN corr −1
∑[ k =0
]
2
(9-45) Now by systematically changing the phase of the locally generated pilot signal (scanning) in the pilot-aided closed loop of Figure 9-14, the receiver will eventually adjust the shift n0 so that the local pilot replica is very close in phase to the actual transmitted pilot signal and thus code alignment is achieved. Since carrier phase and carrier frequency offsets are also present in both branches of the system in Figure 9-14, carrier phase and carrier frequency synchronization can also be achieved using the pilot symbols using the conventional loops discussed early in Chapters 4 and Chapter 5.
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
NN corr
∑
x
k =0
cos(2πf IF kT + θˆ)
PI (kT )
rIF (kT )
NCO
+
PILOT Generator
− sin( 2πf IF kT + θˆ)
PQ (kT ) NN corr
∑
x
k =0
Loop Filter
Figure 9- 14: Pilot-aided Pseudo- noise Code Synchronization System.
9.8 Decision-Directed PN Code Synchronization Algorithm
The non-coherent delay locked loops discussed previously are commonly used. However these loops suffer from increased tracking jitter due to the noise enhancement as a result of the use of the square law detector used to obtain the envelope from the crosscorrelation between the locally generated Early-Late PN code and the received signal. The Decision-directed coherent PN code synchronization scheme, however, circumvents the use of the square law detectors and, as a result, there is no noise enhancement and the scheme can superseded the DLL performance by 3dB in noise performance. Figure 9-15 shows one such system, where the data decisions are used to remove the modulations (as an alternative to squaring). The disadvantage of this algorithm however is that it requires accurate estimation of the received carrier phase. The system in Figure 5-19 works as follows. The complex spread signal is first crosscorrelated with an advanced and a retarded PN code replicas, given by c(t − τ + ∆T ) and c(t − τ − ∆T ) respectively. The output of the correlation is then integrated and dumped. Both early and late branches are then subtracted to form the chip timing error signal which contains data modulations and carrier phase error [11]. Thus, the timing error signal is given by
1 εi = T
( i +1)T
∫ r (t )[c(t − τ + ∆T ) − c(t − τ + ∆T )]dt
(9-46)
iT
This error signal is then reverse-modulated using the detected data dˆi and the carrier estimated rotations e − jθ . The error signal (9-46) contains the timing error measure, but ˆ
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
obtained without the square law detector, that causes of 3dB noise enhancement (?). This error signal is then used in conjunction with a feedback loop filter to control the epoch timing of the NCO that clocks the PN code generator.
ID/filter
c (t − τ − ∆T )
r(t )
+
ek
ID/filter
c(t − τ + ∆ T )
NCO
Re{ }
LP F
dˆ ∗ Carrier recovery
θˆ
Re ( {)∗ }
e− jθ
ˆ
dˆ
RX
Received data
Figure 9- 15: Decision-directed Coherent DLL.
9.9 Frequency-Hopped Speared Spectrum Synchronization
Frequency-hopped spread spectrum is the other major type of spread spectrum system in Use., Here the signal itself is not spread across the entire large bandwidth like the DS-SS methods in Figure 9-1 but instead, the wide bandwidth is divided into N sub-bands. The signal “hops” from one band to the next based on a pseudorandom sequence, so the center frequency of the signal changes from one hop to the next, as shown in Figure 9-16. The frequency-hopping scheme is implemented using direct digital synthesizers (DDS) which have the capability of switching very rapidly while producing accurate frequencies. Figure 9- 17 illustrates one such hopped frequency modulator implementation. Here the fast frequency hopping and generation is obtained using Direct-Digital Synthesizer (DDS). In this particular example, the frequency-hopping band is 22-44 Mhz (22Mhz bandwidth). This frequency hopping is then up converted to the final transmitter carrier frequency using a quadrature mixer. The hopping at lower frequency is able to use direct modulations using a quadrature DDS signal because it Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
voids the difficulties of building a DDS at very high frequencies. In this example, the final output of the transmitter is a FH-SS operating in the ISM band of 902-928 Mhz.
Frequency - Hz
Figure 9- 16: Spectrum of Frequency Hopped Spread Spectrum Signal.
Quadrature Up converter
F=902-928Mhz
Quadrature DDS & DAC
PA
Filter
Filter
0o
90o
Hopping: F=22-48
Mhz
Carrier Oscillator
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Figure 9- 17: Frequency Hopping Pseudo-noise Synchronization.
Figure 9-18 illustrates the receiver topology used for receiving and synchronizing the FHSS transmitter in Figure 9-17. To obtain an un-hopped signal (fast or slow), the received signal from the transmitter in Figure 9-17 (902-928 Mhz) is down converted first to an IF signal that uses a hopped local oscillator with frequency of 817-843 Mhz. The IF signal is then down converted again from IF to zero IF (baseband) using the quadrature DDS as a local oscillator. Both baseband quadrature channels are then digitized and then processed digitally. The synchronization parameters are then estimated numerically producing a correction signal that is fed to the DDS that de-hops the first local oscillator. F=902 -928M hz LPF
ADC
LNA
Preselection Filter
IF Filter
IR Filter
LPF
Filter
0 Quadrature Up converter
o
ADC
Baseband Digital Signal Processing
90 o
IF Oscillator
Quadrature DDS & DAC
Hopped LO
F=817 -843 M hz
0o
90 o
Carrier Oscillator
F=853.8M hz
Figure 9- 18 : Frequency Hopping Pseudo-noise Synchronization.
Commonly, initial FH synchronization is obtained using a preamble as that shown in Figure 9-19. The preamble here is designated purposely for FH frame and PN code acquisition purposes. Even though the modulation may be M-ary FH system, often the preamble is a binary modulation (dual tone) designed to ease the process of detecting the pilot tones. The frame structure in Figure 9-19 consists of a binary pilot tone that is a constant tone +Ftone, This tone is used for obtaining the initial acquisition, and then followed by a reversal of polarity all zeros that is series of -Ftone sequence, then a unique word that is used for indicating the end of the preamble and the start of the unique word that is used for ambiguity resolution and to estimate the start of the packet data field. The acquisition starts by the receiver first tuning to receiving the long sequence of all ones, that is all +Ftone that indicates carrier present by detecting energy levels above a predetermined threshold using a narrow tuned BPF or PLL with a center frequency of +Ftone. Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
After that, the reset of +Ftone is is then used to detect the transition from +Ftone to -Ftone. Once the transmission is detected, the receiver starts to correlate a local replica of the unique word with the incoming signal. Once the maximum correlation score is detected, acquisition of the frame is achieved. Pilot tone +Ftone
+Ftone - Ftone
Unique Word
Start of Message
Message
Figure 9- 19: Frame structure with preamble for synchronization of FH-SS signals. Similar to acquiring and tracking PN code in DS-SS systems. The process of acquiring and tracking in FH-SS is achieved by aligning a local hop generator with the received signal sequence in two stages,namely,FH acquisition and FH tracking. Coarse frequency synchronization (i.e., acquisition) is carried out by finding the degree of alignment between both received hopping signal and the generated local hopped sequence. Figure 920 shows one implementation of a FH-SS tracking system. The received signal here is correlated in a wideband mixer with the local hop sequence produced by the FH synthesizer driven by a PN code generator whose epoch time is controlled in accordance with the decision to continue the search operation. The higher mixing product terms are then filtered out because the IF filter is designed with a narrow bandwidth that only permits the IF and the maximum frequency offset . The signal from the IF filter is then followed by an envelope detector. The resultant envelope detector output is then accumulated (integrated) to produce a signal with a mean of zero when the received signal and the local PN code are not partially correlated (aligned). If the PN code generated locally and the received signal starts to align partially, the envelope signal will then rise and eventually exceed the threshold η , which is based on both signals being within an alignment offset of less than a half chip. The IF signal is down converted again to baseband and sampled separately, samples of the quadrature baseband signals that are processed by the subsequent demodulation algorithms result. The low pass filter after the quadrature down converter is designed, such that its bandwidth is given by BLPF ≥ 1 Th , where 1 Th is the hopping rate.
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LPF Received FH-SS signal
( )2
cos(2πf IF t )
IF Filter
Quadrature oscillator
+
sin( 2πf IF t )
Frequency synthesizer
LPF
PN generator
Epoch Control clock
( )2 z <η
z
1 NB
NB
∑ k =1
η
Figure 9- 20: Frequency Hopped Spread Spectrum FH-SS Spreading Sequence Acquisition System. Figure 9-21 and 9-22 illustrate the late and early timing offset tracking principle used to acquiring and tracking the PN code that enables the FH-SS in Figure 9-20 to de-spread and demodulate the data correctly. Here, the received frequency hopped signal and the locally generated frequency hoped local oscillator are shown in Figure 9-21. In Figure 921, the bandwidth of the IF filter is chosen to be less than twice the hop frequency spacing, so that all of the frequency components out of the mixer are filtered out, the output of the correlation process (output of mixer) will then be Gaussian noise. Once filtered by the LPF, it will be zero. On the other hand, if the received FH-SS signal and the synthesizer sequence are partially aligned by less than a hop interval as illustrated in Figure 9-22, the mixer output signal will then contain frequency components that are within the bandwidth of the IF filter. Assuming that the frequency error offset ∆f is less than the hop spacing, the correlation value is significant because there is a correlation peak. If the alignment is less than a half chip, the level will eventually exceed a predetermined energy threshold , thus indicating that PN acquisition has been achieved as shown in Figure 9-22.
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Th f 2 + ∆f
f 1 + ∆f
τ
f3+∆f
f4+∆f
f5 +f F5IF+FIF
f4 +fIF
f6 +fIF
Th − τ f 4 − f1 +
f4 − f2 +
f IF − ∆f
f IF − ∆f
f5 − f2 +
f6 − f3 +
f5 − f3 +
f IF − ∆f
f IF − ∆f
f IF − ∆f
f6 − f4 + f IF − ∆f
Figure 9- 21: Received Signal and Local PN Hopping Sequence Are Misaligned by More Than One Chip.
Th f1 +∆f
f2 +∆f
f3+∆f
τ
f4+∆f
f2 +fFIF5 +FIF
f1 +fIF
f3 +fIF
Th − τ fIF − ∆f
f1 − f 2 + f IF − ∆f
f2 − f3 + f IF − ∆f
f IF − ∆f
f IF − ∆f
f3 − f4 + f IF − ∆f
Cross correlation
threshold
τ Th = −1
τ Th = 0
τ Th = +1
Figure 9- 22: Received Signal and Local PN Hopping Sequence Are Misaligned by Less Than One Chip.
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9.10
References
1. Savo Glisic and Branka Vucetic, Spread Spectrum CDMA Systems for Wireless Communications, Artech House Mobile Communications Series, April 1997 2. R. C. Dixon, Spread spectrum Systems with Commercial Applications, 3ed, John Wiley & Sons, New York, 1994 3. J.K. Holmes, Coherent Spread Spectrum Systems, New York, Wiley, 1982. 4. J. Chuang, “The effects of multipath delay spread spectrum on timing recovery,” IEEE Trans. Vehicular Tech., vol. VT-35, No. 3, pp. 135-140, August 1987. 5. Gossink, J. Asenstorfer, and S. Cathcart, “A new digital code tracking estimator for direct sequence spread spectrum,” University of South Australia, 1999. 6. H. Olson, “Differential PSK detector ASIC design for direct sequence spread spectrum radio,” www.ele.kth.se/esd. 7. H. Olson, “Direct sequence spread spectrum digital radio performance analysis with simulation,” www.ele.kth.se/esd. 8. De Leon and B. Scaife, “Spread spectrum carrier with unknown Doppler Shift,” NASA publication, 1999, NAG-5-1491. 9. C. Chan, and C.D. Marsh, “software implementation of a PN spread spectrum, receiver to accommodate dynamics,” IEEE Trans. Commun., vol. COM-25, Aug 1977. 10. S. Su, and N. Yen, “Acquisition performance of PN synchronization loop for DSSS signals with Doppler shift,” IECE Trans. Fundamentals, vol., E80-A, No. 12, pp. 2372-2381, Dec 1997. 11. M. Sawahashi, and F. Adachi, “Decision-directed delay-locked PN tracking loops for DS-CDMA,” Proceedings of ICE-1999, pp. 82-87. 12. M. Shane, and A. Yamada, “Uplink timing detection for frequency hopping communications,” Proceedings of ICC, 1999, ISBN-0-7803-5538/99. 13. M. Sawahashi, and F. Adachi, “Decision-directed delay-locked PN tracking loops for DS-CDMA,” Proceedings of ICE-1999, pp. 82-87. 14. Gossink, J. Asenstorfer, and S. Cathcart, “A new digital code tracking estimator for direct sequence spread spectrum,” University of South Australia, 1999.
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15. Wern-Ho Sheen, and Gordon Stuber, “Effects of multipath fading on DelayLocked Loops for spread spectrum systems,” IEEE Trans. Commun., vol. 42, NO. 2/3/4, February/March/April, 1994.
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Chapter 10 Synchronization in Orthogonal Frequency Division Multiplexing (OFDM) Systems
This chapter introduces the theory and principles of OFDM modems and synchronization of both carrier frequency and symbol timing in the receiver. First we detail relevant signal processing in both modulator and demodulator. Next we examine the impact of synchronization errors on the performance of OFDM receivers. Finally several synchronization algorithms are illustrated and analyzed for their performance and complexity. These algorithms include both non-data aided algorithms based on correlation characteristics of the cyclic prefix, and algorithms that are based on the use of pilot symbols. The main focus here is to avoid some of the tedious mathematical details, yet offer design engineers and researchers enough detail to implement and experiment with these algorithms. 10.1
Introduction
The use of OFDM modulation technique was motivated by the need to combat frequency selective faded channels that limited the transmission rate through the wireless channel due to inter-symbol interference caused by scattered delay paths with negligible spread delays. Figure 10-1 shows the transmission of 50kbps BPSK signal in a frequency selective channel with a spread delay of 50 microseconds. Here the bandwidth of the information (100khz) is wider than the coherent bandwidth of the propagation channel, which is proportional to the inverse of the delay spread (6-79). The figure shows the frequency selective channel indicating that the coherent channel bandwidth is of the order of 20khz. Clearly the BPSK is being distorted by the notches within the 50khz bandwidth because of the spread delay of 50 microseconds [1]. The purpose of using OFDM to transmit higher data rates through frequency selective channels is to split the high data rate into a number of lower data rates, which are then modulated using different frequency carriers.
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Rb=50kbps, spread delay 50us, channel nulls at 1/50us=20khz 0
-10
Channel response
-20
-30
-40
PSD of 50kbps BPSK
-50
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2 x 10
Hz
5
Figure 10- 1: BPSK signals in Frequency selective channel The bandwidth efficiency of OFDM can be attributed to subcarriers being spaced at frequency intervals equal to the data rate, as shown in Figure 10-2. This produces orthogonal adjacent carriers that, despite being overlapped, are separable at the receiver so that no inter-carrier interference is taking place at the receiver. The complexity issue of multiple modulators and demodulators is circumvented through the use of the efficient a Fast Fourier Transforms (FFT) algorithm in creating both the modulated subcarriers and the demodulation of the subcarriers. The FFT processing is easily implemented using inexpensive digital signal processing chips [2] or field programmable gate arrays (FPGA). 1 T 1 1 = 5 sps T T 1 1 BW = ( N − 1) = 4 Hz T T R η = s = 5 / 4 bits/sec/H z BW Rs = N
N 4 = T T
Figure 10- 2: OFDM orthogonal carrier scheme.
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The OFDM subcarrier overlapping process as shown in Figure 10-2 also offers an increase in spectrum efficiency, since each one of the subcarriers is 50% overlapped with the adjacent subcarrier. To realize the overlapping technique, however, inter-carrier interaction between subcarriers must be minimized. This means that we require that subcarriers be orthogonal. To maintain this orthogonality, phase noise, frequency carrier offsets, and symbol timing offsets must be controlled, which means very stringent synchronization requirements. Figure 10-3 illustrates the spectral efficiency of OFDM over
conventional non-overlapping multicarrier FDMA technique-schemes [3]. For BPSK signals with a data rate of R and a single modulated carrier, the required transmission null-to-null bandwidth W is given by W = 2 R . Using three subcarrier OFDM systems (middle figure), the total required bandwidth to transmit R bps is W = 1.5R . With four subcarriers (bottom figure), the required bandwidth is W = 1.33R and so forth. So clearly four subcarrier OFDM modulations is 60% more efficient than a single FDMA modulation.
Figure 10- 3: Concept of OFDM versus conventional modulation technique.
10.2
OFDM Modulator
Figure 10-4 illustrates the block diagram of OFDM digital modulator that includes the. following :
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• •
Bits are encoded. Convert the serial high data rate binary data into symbols according to the number of bits/symbol selected.
•
Convert the serial symbol stream into parallel segments (S/P) according to the number of carriers (N).
• • • • • •
Convert each symbol to a complex phase modulations (BPSK, QPSK, MQAM…, etc.). Pass each complex symbol to the appropriate IFFT bin (subcarrier) Take the IFFT of the N streams. Add the cyclic prefix samples, and then convert the samples from parallel to serial (P/S). Convert the digital signal samples to analog. Up convert the modulated symbols to a carrier frequency (fc).
Bits
00110
PSK encoder
S/P
add cyclic prefix
N-IFFT
P/S
D/A & Filter
Filter Amplifier
exp(j2π fc t)
Figure 10- 4: OFDM demodulator.
Mathematically, the baseband data samples d [k ] at the output of the N-IFFT block for the transmitted signal are expressed by, N −1
s (t ) = ∑ d [k ]e j 2πkf scs t ,0 ≤ t ≤ Ts
(10-1)
k =0
where Ts is the symbol duration, and d [k ] are the corresponding complex symbols. With the subcarriers 1 , the baseband signal in (10-1) is represented by the following discrete system, being equal to f scs = NTs N −1
s (n) = ∑ d [k ]e
j 2πkf scs
Ts N
(10-2)
k =0
For orthogonal purposes, the subcariers must be placed at a uniform distance equal to the data rate used to modulate each one of the N-subcarriers, that is f scsTs = 1 . Substituting this in (10-2), the based band signal is simplified to N −1
s (n) = ∑ d [k ]e
j
2πkn N
(10-3)
k =0
N −1
Realizing that
∑ d [k ]e
j
2πkn N
is the Inverse Fourier Transform (IFFT) performed over the transmitted symbol
k =0
samples, that is IDFT {d [k ]} , clearly the heart of the OFDM modulator is simply a digital signal processing Dr. Mohamed Khalid Nezami © 2003
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implementation of IFFT using integrated circuits such as those available from Texas Instruments [2], or using a field programmable gate arrays (FPGA). 10.3
OFDM Demodulator After the signal is up converted to its assigned high frequency carrier and propagates through the transmission channel, the signal is received first by down converting modulated carrier frequency to an intermediate frequency, which is then sampled using an analog-to-digital (A/D) converter as shown in Figure 10-5. The OFDM symbol demodulation is then carried out using the following steps:
• • • • • • •
After samples are properly decimated and filtered, cyclic prefix samples are removed from the received samples. Convert the input serial stream into parallel (N) channels representing each symbol period. Take the FFT of each symbol. Demodulate the phase modulated transmitted symbols from each of the FFT bins. Using the detected phase and amplitude of each symbol, decode the data (if error control coding is used). Decode the symbols (if applicable) Convert the parallel decoded data to serial stream.
00110 P/S
decoder
remove S/P cyclic prefix
N-FFT
receive filter + A/D
down converter
Figure 10- 5: OFDM digital modulator.
Mathematically, the OFDM demodulation of the received signal s (n) in Equation (10-3), with prefect timing, and no frequency offsets, is given by,
dˆ[k ] = DFT [ s(n)]
(10-4)
Substituting the standard definition of the DFT into (10-4), the demodulated data samples are given by 1 dˆ (k ) = N N −1
with s (n) = ∑ d [k ]e
j
2πkn N
N −1
∑ s(n)e
−j
2πkn N
(10-5)
n =0
in equation (10-5), the demodulated data symbols are given by
k =0
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1 dˆ (k ) = N
2πkm j ⎞ − j 2πNkn ⎛ N −1 N ⎟ ⎜ d [ m ] e ∑ ⎟e ⎜∑ n=0 ⎝ m=0 ⎠ N −1
(10-6)
This is further reduced to 1 dˆ (k ) = N N −1
Realizing that
∑e
j
2πn ( m − k ) N
N −1
N −1
m=0
n=0
∑ d[m]∑ e
j
2πn ( m − k ) N
(10-7)
= Nδ [m − k ] in Equation (10-7), this yields the expected result of dˆ ( k ) = d [ k ] .
n=0
10.4
Guard Interval Insertion
In figure 10-4, after the samples of the symbol have been processed by the IFFT block, to avoid ISI and ICI guard period samples must be formed by a cyclic extension of the symbol period as shown in Figure 10-6. GI insertion is carried out by taking symbol samples from the end and appending them to the front of the symbol. This effectively creates a cyclic prefix by copying the last part of the OFDM symbol that is then prepended to the transmitted symbol. This makes the transmitted signal periodic and does not affect the orthogonality of the carriers. With the cyclic extension, the symbol period is now longer, but it represents the exact same frequency spectrum. The insertion of guard intervals serve an additional purpose. It serves as a way to obtain symbol timing and frequency offset synchronization, as will be detailed in the coming sections.
Copied samples
Copy the end tail and past to the beginning of the symbol
Guard TG
t=-∆
Ts=N/W Effective symbol period (Ts+TG) t=NT
t=0
Figure 10- 6: Guard interval using cyclic extension.
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Mathematically, the cyclic prefix illustrated in Figure 10-6 works as follows. Between consecutive OFDM symbols a guard period is inserted that contains a cyclic extension of the OFDM signal. The OFDM signal is extended over a period of ∆ , that causes the signal in (10-1) [23] to be given by, N −1
s (t ) = ∑ xk e j 2πf scs t , − ∆ < t < NTs
(10-8)
k =0
when the signal (10-8) passes through the propagation channel, modeled by a finite length impulses response h(t ) limited to the interval of [0, ∆ h ] , where the length of the cyclic prefix ∆ is chosen such that ∆ > ∆ h , the received OFDM symbol evaluated on the interval [0, NTs ] in (10-8) becomes N −1
r (t ) = s (t ) * h(t ) = ∑ H k xk e j 2πf scs t , 0 < t < NTs
(10-9)
k =0
∆h
where H k = ∫ h(τ )e − j 2πf kτ dτ is the Fourier transform of h(t) evaluated at the frequency f k . Note that within 0
this interval the received signal is similar to the original signal, except that H k x k modulates the kth subcarrier instead of x k . In this way the cyclic prefix preserves the orthogonality of the subcarriers. As a result, to undo the channel effect, a simple N-parallel one-tap equalizer can be used (i.e. channel estimator at each subcarrier). This is less complex than the high order number of taps needed to equalize a wideband modulated single carrier system [3,4,5]. The cyclic prefix does, however, introduce a loss in efficiency and thus a signal-to-noise reduction. However, the SNR loss is usually negligible when compared to its effect in mitigating interference and its use for synchronization. The SNR loss [11,13] is given by
⎛ T ⎞ SNRloss = 10 log⎜⎜1 − G ⎟⎟ ⎝ Ts ⎠
(10-10)
TG is the ratio of cyclic prefix period to the overall symbol period. Therefore, a longer cyclic prefix Ts would mean a higher SNR loss. Typically, the relative length of the cyclic prefix is small and on the order of TG ≤ 20% . Ts
where
10.5
OFDM System Parameter Design In conventional TDMA systems, the channel delay spread leads to inter-symbol interference. This takes place due to the delayed multipath signal overlapping with the successive symbols, which causes significant errors in high bit rate systems. Figure 10-7 illustrates the inter-symbol interference due to delay spreads on the received OFDM signal. It illustrates the effectiveness of the guard interval in minimizing the effect of multipath fading due to four delayed versions of the transmitted symbol. The direct path and the four additional scatters arriving with different delays are summed at the receiver input; however, as the figure illustrates, there will be no ISI as long as the cycle prefix period is designed such that it is greater than the longest delayed path. This ensures that all delayed paths are still within the DFT window for a single symbol. Dr. Mohamed Khalid Nezami © 2003
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Symbol Duration: with no ISI, only phase shift and magnitude attenuation
Direct path Multi-paths
Inter-symbol: interference is present here in the guard band
Figure 10- 7: Principal of combating fading using the guard interval in OFDM transmission.
The delay spreads that are encountered in most modern communication systems are well documented and widely known. For example with indoor communications, the delay spreads are on the order of 40nsec to 200nsec, since the reflected paths are caused by being bounced off objects with path lengths of 12 to 60 m (furniture, walls, and steel foundations). For outdoor communications, the delays are a little longer, on the order of 1usec to 20usec. These are caused by objects at distances of 300m - 6km (hills and high buildings). Table 10-1 lists spread delays experienced by the three major cellular systems [1,13] that utilize OFDM. System Pico cell Micro cell Macro cell
Cell size 100m 5km 20km
Max Delay Spread 300ns 15us 40us
Table 10- 1: Spread delay associated with various cellular systems. The spread delay experienced in a channel is directly responsible for the maximum data rate that can be transmitted through the frequency selective channel, since it controls how narrow the channel coherence bandwidth [3] is. The channel coherence bandwidth Bc is a statistical measure of the range of frequencies over which the channel passes all spectral components with approximatly the same gain and with linear phase. The approximate value of Bc for a wireless chanel is related to the reciprocal of the dominent spread delay as seen in Figure 10-1. A popular approximation of Bc corresponds to the bandwidth interval having a correlation of at least 0.5 [1] and is given by 1 (10-11) Bc = 5σ τ Dr. Mohamed Khalid Nezami © 2003
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where σ τ is the rms spread delay [1]. OFDM links are designed such that the high data rate impacted by frequency selective channel is divided to N channels, each of which has a symbol rate that is low enough that its subchannel bandwidth is less than the coherent bandwdth, resulting in a frequency flat faded channel. For MPSK OFDM modulations, the maximum data rate that can be transmitted in a frequency selective channel without being effected by frequency selectivity is given by 2 Rb Bc ≤ (10-12) log 2 M Using Equation (10-12) and the propagation delays listed in Table 10-1, the maximum BPSK data rate allowed before the channel becoming frequency selective is computed as follow: For Pico cell, with 100m cell size, with σ τ =300ns, the coherent Bc is 667khz, and R s , max = 330ksps . For Micro cell, with km cell size, with σ τ =15microsec, the coherent Bc is 66.7khz and R s , max = 33ksps . For Pico cell, with 100m cell size, with σ τ =40microsec, the coherent Bc is 25khz and R s , max = 12ksps . The following example illustrates the design of OFDM parameters based on the measured spread delay: •
Given that the desired transmit rate is 1.2Mbps using QPSK, the channel bandwidth is given by BW=800khz; the measured delay spread is τ rms = 40 microseconds. Using conventional OFDM modulations and based on the spread delay measured, this channel is frequency selective with nulls at 1 τ rms = 25khz , with a coherence bandwidth of 1 5τ rms = 5khz . Using OFDM to minimize the channel effect, the subcarrier spacing (i.e., data rate) has to be f scs ≤ 5khz , which will serve as the subcarrier frequency for the OFDM modulator. With f scs = 5khz the number of subcarriers to cover the total available bandwidth is N = 800 / 5 = 160 , so the subcarriers are spaced at f scs = 800 / 160 = 5khz . The symbol duration is then Ts = 1 / f scs = 200 µ sec , choosing a guard interval that is 25% of the symbol interval, or TG = 50µ sec . The total symbol time will then be Te = 250µ sec . The data rate transmitted using QPSK (2bits/symbol) on each subcarrier is R b _ sc = (1 / Te )(2bits ) = 8kbps ; to transmit 1.2MBPS, a total of 150 subcarriers are needed. The rest of the ten subcarriers can be used for pilot or as guard bands. The European digital broadcast system (DAB) [28,30] uses the parameters:
•
BW = 7 Mhz , Rs = 5.6Mbps , N=448 subcarriers, f scs = 15.625khz , TG = 16 µ sec , thus Te = 80µ sec , R b _ sc = 30kbps using QPSK modulations.
Ts = 64µ sec ,
In the USA, the wireless LAN system designated by IEEE802.11a [10] uses the parameters: •
BW = 20Mhz , N=48 data subcarriers and N=4 pilot subcarriers out of 64 subcarriers. f scs = 312.5khz , Ts = 3.2 µ sec , TG = 0.8µ sec , thus Te = 4µ sec Another system that has recently been advocated is the 4G cellular systems. These use the parameters:
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•
BW = 800khz , N=192 σ τ = 2 microseconds, Ts = 240µ sec , TG = 48.5µ sec , thus Te = Ts + TG = 288.5µ sec .
data
subcarriers,
f scs = 4.17khz ,
In summery, OFDM transmission has the following key advantages over conventional transmission systems: • •
Efficient use of the spectrum by overlapping intercarriers. By dividing the wideband frequency selective channel into narrow subchannels, the modulated carriers become more resistant to frequency selective fading than a single wideband modulated carrier.
•
By using cyclic prefix samples, the problem of ISI and multi-path is eliminated.
•
Using adequate channel coding (COFDM) and interleaving one can recover symbols lost due to the frequency selectivity of the channel.
•
Channel equalization becomes simpler than adaptive equalization techniques used with single carrier systems to overcome frequency selective fading channels.
•
The modem implementation is well suited for digital implementation since it uses computationally efficient FFT algorithms in both modulator and demodulator.
•
In conjunction with differential modulation, implementing a channel estimator is unnecessary.
•
Using a proper long guard interval at the start of each symbol, the impact of sample timing offsets is reduced.
•
The use of a cyclic prefix facilitates the use of non-data aided timing and carrier recovery algorithms. In terms of drawbacks, OFDM modulation schemes have the following disadvantages:
•
OFDM modulated signals have amplitudes with a very large dynamic range, namely the average-to-peak power ratio, which requires very linear RF power amplifiers and wide dynamic range analogto-digital converters.
•
Since orthogonality is a condition for extracting the overlapped sub-carriers with minimum inter-carrier interference (ICI), the scheme is very sensitive to carrier frequency offsets and phase noise, thus requiring accurate frequency carrier , timing recovery, and spectrally clean oscillators [25,26,27]. since frequency offset and sampling frequency errors are the main contributors to deterioration of OFDM transmission. The next sections will investigate this impact and will detail several synchronization algorithms used at the receiver. Dr. Mohamed Khalid Nezami © 2003
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10.6
Impact of Frequency Offset Synchronization Errors on OFDM Reception
Both carrier and symbol timing synchronization are an improtant part of sucessful OFDM receiver design. Figure 10-8 illustrates the impact of both carrier and timing errors on QPSK signal transmitted using OFDM. At first the received signal with frequency and timing offsets (left graph) is severly distorted and symbols are all over the constellation and unrcognizable for being QPSK. After sucessful frequency offset compensation (middle), the symbols are in the correct quadrant, however, due to the presense of timing erros, the symbols will experience phase rotations and attenuation in level. After timing error estimation and correction, the symbol phase rotations are removed and, as seen in the left part of Figure 10-8, are regonizable QPSK symbols that are ready to be detected [5,8,9].
recovered signal with carrier and timing offsets
OFDM recovered signal after carrier correction
OFDM signal after carrier and timing correction
Figure 10- 8: Impact of carrier frequency offset and timing offset synchronization errors on OFDM reception. Figure 10-9 illustrates a block diagram of a typical OFDM baseband receiver including synchronization. Here the quadrature digitized baseband samples (I’s and Q’s) at the output of a complex rotator are fed to a syhcronization algorithm block that extracts both carrier frequency offsets and sampling frequency offsets. The frequency offset estimates are passed to the FFT block and to the numerical controlled oscillator (NCO), where the frequency offset is removed. If the frequency offset is too large or if the estimates are out of range, the offset correections are fed to the radio frequency down converter in the analog section (dashed line) n, where it is used to offset the local oscillator used in the down converter. The sampling frequency errors are also fed to the A/D variable clock (VCXO) and the FFT block to correct its frequency. Symbol timing is estimated and passed to the FFT algorithm to mark the start of the FFT window.
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I (k ) RF receiver
A/D
FFT
Q(k ) o
90
0
demodulator
o
synchronization algorithms
NCO
Figure 10- 9: OFDM carrier and timing synchronization system. In the next sections, we derive the mathematical model for synchronization error impact on the OFDM received signal by evaluating the resulting distortion and the SNR loss that takes place as a result of these errors. Then numerus schemes for both carrier and timing synchronization schemes are derived and detailed. These algorithms are categorized into acquisition and into tracking. The acquisition or coarse synchronization algorithms are either based on characteristics of the cyclic prefix interval, or on pilot aided techniques [14,1518]. To illustrate the effect of frequency carrier and sampling frequency offsets mathematically, assume that N 1 is the number of sinusoidal carriers (DFT bins), and that the frequency spacing is , where Ts is the useful NTs symbol period (without the guard intervals). The complex envelope of the transmitted signal in equation (103) is again given by Xn =
K 2
∑z e
j 2πkf scs nTs
(10-13)
k
K k =− 2
substituting Nf scs =
1 in (10-13), the transmitted signal is given by Ts
Xn =
K 2
∑z e
j
2πkn N
(10-14)
k
K k =− 2
and K is the number of active channels. Defining zk as the complex data symbols modulating the kth channel, equation (10-14) given by, +∞
zk = ∑ zk + rN
(10-15)
−∞
where R[k ] is given by ⎧1 ,0 ≤ k ≤ N − 1 R[k ] = ⎨ , else ⎩0
(10-16)
substituting (10-16) into (10-15), the transmitted signal is then given by
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Xn =
1 N
N −1
∑ zk R[k ]e
j
2πkn N
(10-17)
k =0
At the receiver, the received signal including the frequency offset ∆f is given by 1 yn = N
N −1
∑z R k =0
k
N
[ k ]H k e
j
(
)
2π k +δ f n N
(10-18)
∆f is the f scs normalized frequency offset error. After taking the FFT of the received signal (10-18), the frequency representation of the received symbol at the receiver is given by, 2πkn 1 N −1 − j N Yl = ∑ yn e (10-19) N k =0
where H k is the transfer function of the propagation channel at the kth subcarrier, and that δ f =
Substituting the received signal samples in (10-18), this is reduced to, Yl =
1 N
N −1
N −1
∑ z k RN [ k ]H k ∑ e k =0
j
(
)
2π k +δ f n − 2πln N
(10-20)
n =0
which is further reduced to, Yl =
1 N
N −1
N −1
∑ z k RN [ k ] H k ∑ e k =0
j
[
]
2π k +δ f −l n N
(10-21)
n =0
If the frequency error δ f is an integer, we have
Yl = zk −δ f RN [k ]H k −δ f
(10-22)
Equation (10-22) indicates that the entire OFDM signal Yl in (10-22) has been shifted in frequency without any ICI components [20,22-24]. When δ f is a fractional number, or δ f ≤ 1 , the received signal in (10-21) has an additional term that is given by N −1
∑e n =0
j
[
]
2π k + δ f − l n N
=
sin (π (k + δ f ) − l )
⎛ π (k + δ f ) − l ⎞ ⎟⎟ sin⎜⎜ N ⎝ ⎠
e
j
[
2π k + δ f − l N
]
( N −1)
(10-23)
Substituting (10-23) into the received signal in (10-21), the demodulated signal becomes, Dr. Mohamed Khalid Nezami © 2003
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sin (πδ f ) j 2πδ f N( N −1) 1 1 + Y l = z l R N [ l ]H l e N N ⎛ πδ f ⎞ ⎟⎟ sin⎜⎜ ⎝ N ⎠ sin(πδ ) ⎛ π (k − l + δ f sin⎜⎜ N ⎝
) ⎞⎟ e
j
(
2π k − l + δ f N
)
( N −1)
N −1
∑z
k
R N [k ]H k (− 1)
k −l
.
k =0 k ≠l
(10-24)
⎟ ⎠
This indicates that the received signal (10-24) is the summation of Yl = X l + I l , where I l is the ICI part. The magnitude of the desired signal X l in (10-24) is amplitude attenuated by the channel gain H l and another sin (πδ f ) , which can be further simplified to, term due to the frequency error, that is given by ⎛ πδ f ⎞ ⎟⎟ sin⎜⎜ ⎝ N ⎠ sin (πδ f ) sin (πN∆fTs ) (10-25) = sin (π∆fTs ) ⎛ πδ f ⎞ ⎟⎟ sin ⎜⎜ ⎝ N ⎠ For example, when the carrier of IEEE802.11 a LAN terminal 5GHz has 10ppm crystal offset. This offset corresponds to a frequency offset of (5000x10 6 )(10 / 10 6 ) = 50kHz. j
(
2π k − l + δ f N
)
( N −1)
A third distortion term in (10-24) is due to frequency offset, represented by a phase rotated by e . This term introduces a constant phase shift that is common to all frequency bins, and can either be neglected if differential decoding is employedor can be included in the channel estimation and correction algorithm (i.e., estimated channel gain is complex gain). Figure 10-10 graphically illustrates ICI and amplitude distortion components associated with (10-24). Figure 10-11 shows carrier frequency offset impact on 16QAM symbol constellation, with frequency errors of δ f = 0 , δf = 1% , δ f = 10% , and δ f = 50% .
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∆f
amplitude reduction
ICI distortion
Frequency (Hz)
Figure 10- 10: The effects of a frequency offset, reduction in signal amplitude, and intercarrier interference.
10.7
Impact of Symbol Timing Synchronization on OFDM Reception
At the receiver the demodulation process starts at an arbitrary point in time with a shift of ∆τ relative to the start of the symbol. Then the subsequent sampling unit exhibits a different sampling period with a normalized ∆T error of δ t = [28]. In addition, the receiver starts recording N samples with a time delay of εTa , after Ts applying the FFT process to these N samples, the output of the FFT block is z n +ε (k ) for subcarrier k, which means that the demodulation window is shifted ε samples relative to the transmitter symbol period. This means that the observed N samples contain samples from neighboring symbols resulting in ISI. As a result of this error, the number of useful symbols decreases from N to N − ε . The resulting baseband signal is given by z n +ε ( k ) = z n ( k ) e
j 2 πk
ε N
−e
j 2πτ 0
1 s −1 NTs
∑ z n ( n + r )e
j 2 πk
( s−r ) N
r =0
+e
j 2πτ 0
1 s −1 NTs
∑ z n ( n + N + r )e
j 2 πk
( s −r ) N
r =0
(10-26) The two right hand side terms in (10-26) are ISI caused by samples that do not belong to the transmitted symbol beginning at nTs and by samples of the transmitted symbolwhich are missing in the demodulation window. In addition, the first useful signal part in (10-26) includes a phase shift rotation whose magnitude depends on the position of the subcarrier (k) in the spectrum (10-26). The impact of this rotation was illustrated in Figure 10-8.
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Frequency offset =0% of sub-carrier
Imaginary
Frequency offset =1% of sub-carrier
Real Frequency offset =10% of sub-carrier
Frequency offset =50% of sub-carrier
Figure 10- 11: Carrier frequency offset impact on 16QAM symbol constellation, δ f = 0 , δf = 1% ,
δ f = 10% , and δ f = 50% .
10.8
Impact of Sampling Time Synchronization Errors on OFDM Reception
As mentioned above, timing errors in OFDM receivers take place due to starting with wrong timing offset εTs and to an offset sampling timing (1 − δ t )Ts . As a result, the sampling time at index n of OFDM symbol m takes place at the time instant of t = εTs + ( n + mN )(1 − δ t )Ts , with no timing errors ( δ t = 0 ). Sample time strobe would be just t = εTs + ( n + mN )Ts . With the initial window offset removed and started at the start of symbol boundary ( ε = 0 ), sample timing would be t = ( n + mN )Ts , which is the optimal sampling time with n=0…..N-1 for each symbol. Including all three synchronization errors in the baseband signal for the subcarrier l (coarse time, sample time offset, and frequency offset), the baseband OFDM signal [14] is given by z m ( n) = ⎛
ε ⎞
⎛
ε ⎞
j 2πk ⎜ m (1−δ t ) + ⎟ jπ ((1−δ t )( k + δ f ) − l ) 1 j 2πδf ⎜⎝ m (1−δ t ) + N ⎟⎠ N −1 N⎠ e X m (k ) H k e ⎝ e ∑ N k =0 (10-27)
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N −1 N
[ sin [π ((1 − δ )(k + δ
]
sin π ((1 − δ t )(k + δ f ) − l ) t
f
) − l)/ N
]
10-16
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Equation (10-27) now is re-arranged into Equation (10-28) so that distortion terms due to synchronization errors are emphasized. These terms include distortions that are for k = l (current subchannel), and the ISI terms from the neighboring subcarriers k ≠ l , ICI terms. ⎛
zm ( n ) =
ε ⎞
⎛
ε ⎞
j 2πl ⎜ m (1−δ t ) + ⎟ jπ ((1−δ t )( l + δ f ) − l ) 1 j 2πδf ⎜⎝ m (1−δ t ) + N ⎟⎠ N⎠ e X m (l ) H l e ⎝ e N
N −1 N
+ ICI (l, δ t , δ f , ς )
[ sin [π ((1 − δ )(l + δ
]
sin π ((1 − δ t )(l + δ f ) − l ) t
f
) − l) / N
]
(10-28) This describes the impact of synchronization errors due to both timing and frequency offset inflected on the baseband samples X m (l) . By inspection of the equation (10-28), frequency offset errors δf causes continuous rotation of received signal constellation. This rotation is equal for all of the subcarrier positions sin[πδ f ] which is (see Figure 10-8). In addition, the signal amplitude is being attenuated by the term, sin[πδ f / N ] directly proportional to the frequency offset δf (see Figure 10-10). The frequency offset δf also contributes to the ICI distortion term. Sampling time error δ t also results in continuous phase rotations, which are a function of the position of the subcarrier location, causing outer subcarriers to have large rest phase shifts. In addition, the symbol timing errors δ t contribute to both amplitude distortion and phase rotation as shown in Equation (10-28). 10.9
SNR Degradation Due to Carrier Frequency Offsets ∆f , the SNR degradation f scs due to this frequency offset as a result of the power drop in the useful signal level [6] is given by
With the frequency offset being normalized to the subcarrier frequency offset δ f =
Dδf =
10 (πδ f 3 ln(10)
)
2
Es [dB] No
(10-29)
Note here that the degradation (10-29) increases with the square of the number of subcarriers since δ f = This degradation was extended to OFDM signals operating in fading channels [6] and is given by, 2
E ⎛ ⎞ ⎜ 1 + 0.5947 s sin 2 (πδf ) ⎟ No ⎟ [dB] Dδf ≤ 10 log10 ⎜ 2 ⎜ ⎟ [sin (δf ) (δf )] ⎜ ⎟ ⎝ ⎠
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Figure 10-12 illustrates the SNR degradation (10-29) and (10-30) as a function of frequency offset for both AWGN and fading channel. The figure shows that to obtain a ICI ratio of 20 dB (100 times), the frequency 5 offset δf has to be in the range of δf ≤ f scs . 100 1
10
without fade Es/No=20 with fade
SNR degradation-dB
0
10
Es/No=10 -1
10
Es/No=0
-2
10 0.01
0.015
0.02
0.025 0.03 0.035 Normalized frequency offset
0.04
0.045
0.05
Figure 10- 12: Degradation in SNR due to a frequency offset δf (normalized to the subcarrier spacing) in both AWGN and fading channels.
10.10
SNR Degradation Due to Sampling Frequency Offsets
The SNR degradation (in dB) due to sampling frequency clock errors in parts per million (PPM) [6] mathematically is given by ⎛ 1 Es ⎞ Dn ≤ 20 log10 ⎜⎜1 + nπ 10 −6 δf s ⎟⎟ [dB] (10-31) ⎝ 3 No ⎠
(
)
where δf s = ∆f s Ts is the sample clock offset in PPM at the nth subcarrier normalized to the OFDM symbol timing Ts . Figure 10-13 shows a plot of (10-31) versus the sample clock offsets δf s (in PPM) normalized by the sampling frequency for the subcarrier of n=256 .
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1
10
Es/No=20
SNR degradation-dB
0
10
Es/No=10
-1
10
Es/No=0
-2
10
0
50
100
150 200 250 300 350 400 Normalized sampling clock offset (PPM)
450
500
Figure 10- 13: SNR degradation versus the sample clock offsets δf s normalized by the sampling frequency for the subcarrier n=256 .
10.11
SNR Degradation due to Carrier Phase Noise Offsets
Carrier phase noise is caused by imperfection in the transmitter and receiver oscillators [29]. For OFDM, no distinction can be made between the phase rotations induced by timing error or carrier phase due to the channel or the oscillator phase noise. Phase noise for the carrier oscillator used in the receiver down converter are modeled by a Lorentzian model [29], where the single-sided noise density spectrum is given by Sd ( f ) =
2 πf φ ⎛ f ⎞ 1+ ⎜ ⎟ ⎜f ⎟ ⎝ φ⎠
(10-32)
2
where f φ is the 3dB linewidth of the oscillator and f is the frequency offset from the carrier frequency in Hz. Assuming that phase noise is given by φ (n) , the baseband OFDM signal affected by this phase noise is given by r (n) = x(n)e jφ ( n ) . After the DFT is performed at the receiver, the exponential term e jφ (n ) can be approximated by e φ ( n ) ≈ 1 + jφ ( n )
(10-33)
Since sin φ ( n ) < 1 rad .
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As a result, the received signal r (n) = x(n) + e jφ ( n ) (??). After performing the DFT on the received signal, N −1
with x(n) = ∑ s k e
j
2πkn N
, the demodulated signal becomes,
k =0
2πk ( r − k )n
j j N −1 N −1 N y (k ) = s k + ∑ s k ∑ φ (n)e N r =0 k =0 14444244443
(10-34)
ek
Thus the effect of phase noise on the received OFDM baseband signal in (10-34) has been an error term ek at each subcarrier that includes also a common phase error that is equal for all subcarriers. In Equation (10-34), when r = k , the error term ek is reduced to ek =
j N
N −1
N −1
r =0
k =0
∑ s k ∑ φ (n)
(10-35)
1 N −1 ∑ φ (n) is the average of the phase noise. This implies that the impact of phase noise results in a N k =0 constant rotation for all symbols at all sub-carriers and thus can be removed when removing (??) in the presence of the sampling time errors. For the case when r ≠ k in Equation (10-34), the error term ek is given by
The term
j ek = N
N −1
N −1
∑ s ∑ φ (n)e r =0 r ≠0
k
j
2πk ( r − k )n N
(10-36)
k =0
This corresponds to both symbol rotations and an inter-carrier interference term (ICI). The spectral components of the phase noise that contribute to the magnitude of this error are those from f scs up to the total phase noise bandwidth f φ . Because of its random nature, it cannot be corrected and thus will induce phase rotations that are like those induced by AWGN. Figure 10-14 shows this impact on the constellation of 16QAM OFDM modulations with two different values of f φ . The Figure on the left clearly shows that the phase noise present produces a total phase rotation, which title (??) the whole constellation (clockwise in this case).
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fφ f scs = 0.001%
Imaginary
fφ f scs = 0.01%
Real Figure 10- 14: Carrier frequency offset impact on 16QAM.
The figure on the left illustrates the case when the phase noise bandwidth is close to f scs . Here inter-carrier interference dominates over common phase errors. It can be seen that the constellation undergoes both rotation and smearing in the case where f φ << f scs (left constellation) [25,26]. The SNR degradation [6] due to phase noise is given by,
Dθ =
f ⎞E 11 ⎛ ⎜⎜ 4π φ ⎟⎟ s [dB] f scs ⎠ N o 6 ln(10) ⎝
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1
10
Es/No=20
SNR degradation-dB
0
10
Es/No=10
-1
10
Es/No=0
-2
10
1
2
3
4 5 6 7 Normalized phase noise BW
8
9
10 -3
x 10
Figure 10- 15: SNR degradation versus phase noise bandwidth f φ normalized to the subcarrier frequency f scs .
The degradation in Equation (10-37) also can be represented in terms of the phase noise variance for the case when σ θ2 << 1 rad and the phase noise variance is given by σ θ2 = 4πf φ [26]. That is
⎛ E ⎞ Dθ = 10 log10 ⎜⎜1 + σ θ2 s ⎟⎟ [dB] (10-38) No ⎠ ⎝ E As an example, for QPSK with b = 10dB and σ θ2 = 0.5 rad2, the SNR degradation due to phase noise is No equal to Dθ = 11 dB. By reducing this phase noise to σ θ2 = 0.1 rad2, the degradation is reduced to Dθ = 5 dB. In general, for a given phase noise 3dB bandwidth f φ , the rms phase noise variance σ θ2 is given by [6]:
σ θ2 =
f L fφ
1 + ( f L fφ )
4
[
]
1 ⎛ ( f L fφ )2 − 1 ⎞⎟ ⎜1 + 2 ⎝ ⎠
(10-39)
where f L is the loop bandwidth of the RF synthesizer generating the local oscillator signal in the receiver. Equation (10-39) shows that the RMS phase noise variance is a function of the ratio of f L f φ , that is the ratio of the carrier oscillator phase locked loop tracking bandwidth to the Lorentzian linewidth f φ . Clearly the narrower the loop bandwidth, the smaller the phase noise variance [6,29] will be, which leads to small degradation Dθ . Dr. Mohamed Khalid Nezami © 2003
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10.12
OFDM Synchronization Algorithms
Figure 10-16 shows the carrier frequency offset symbol timing and the sampling frequency feedback synchronization algorithms for OFDM system. The complex baseband OFDM signal, after having been rotated and frequency translated using the NCO, is fed to the prefix removing block and then fed to the FFT algorithm that converts the signal to frequency domain symbol-by-symbol. The frequency-offset corrections are fed to the NCO for compensation, while timing error frequency offsets are feedback both, the A/D clock at the cycle prefix remover, or the FFT window buffer.(??)
The principle of frequency and timing offset error detection is based on the fact that, since the guard interval includes part of the symbol at the end, it effectively makes the OFDM symbol cyclic in natureThe OFDM signal with carrier frequency and sampling frequency offsets causes a phase rotation between the replica signal and the original signal. The phase integration (accumulation of phase) over an interval of one symbol ( Ts ) will correspond to the frequency offset error. Cycle prefix remover
RF receiver
A/D
FFT
Demodulator
∆wt ∆wt
ej AFC
∆f s
Synchronization
Figure 10- 16: Analog OFDM carrier and timing synchronization system. The delay in the estimator in Figure 10-16 delays the OFDM signal by one effective symbol duration Ts . Correlation between the delayed version of the signal and the original signal is calculated in the positive and negative frequency region respectively. The tangent function then calculates φ1 and φ2 as the positive and negative frequency correlator as shown in Figure 10-17. Suppose the carrier frequency and the sampling (A/D) frequency each have a frequency offset. The output of the correlator vector has the following phase of φ1 (???) is the phase rotation of the positive frequency component correlation, while φ2 is the phase rotations of the negative frequency component correlator as shown in Figure 10-17. Assuming that the symbol timing error ∆Ts , which is the reciprocal of the sampling frequency f s , is small compared to the symbol duration, or ∆Ts ≤ Ts . Frequency error is estimated by [18]: Dr. Mohamed Khalid Nezami © 2003
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φ + φ2 ∆fˆ = 1 4πTs
(10-40)
and the sampling frequency offset is estimated by:
∆fˆs =
φ1 − φ 2
(10-41) 2Ts After estimation, error estimates are fedback to a loop filter, then the filtered corrections are used in conjunction with VCXO to correct the sampling frequency. A local oscillator is used to correct for the carrier frequency offsets.
L −1
rk
∑
−1
()
φ1
−1
()
φ2
k =0
Positive frequency Filter1
tan
Ts L −1
∑ k =0
Negative frequency Filter2
tan
Ts
Figure 10- 17 : Carrier frequency and sampling frequency offset estimator.
10.13
Synchronization Algorithm using Cyclic Prefix
Using the cyclic prefix, and considering that the first TG seconds of each symbol is identical to the last part of the symbol, can be exploited for both coarse timing and frequency synchronization. Ignoring the AWGN in the received signal y (t ) , this process is based on the following correlation process,
x(t ) =
TG
∫ y(t − τ ) y(t − τ − T )dτ s
(10-42)
0
where y (t ) is the received signal, and y (t − Ts ) is a one symbol delayed version of the received signal. The correlation function (10-42) produces several peaks from which both frequency offset and sampling clock timing are extracted. The phase of the correlation output is equal to the phase drift between samples that are Dr. Mohamed Khalid Nezami © 2003 10-24
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Ts seconds apart. Hence, the frequency offset can simply be found as the correlation phase divided by 2πTs . This method works well up to a maximum absolute frequency offset of half the subcarrier spacing ( ∆f max = f scs 2 ). The next section presents one practical implementation of this approach.
10.13.1
Maximum Likelihood Synchronization using Cyclic Prefix
The uncertainty in carrier frequency, which is due to a difference in the local oscillators in the transmitter and receiver, gives rise to a shift in the frequency domain that represents the received signal after the DFT block. Such behavior is modeled as a complex multiplicative distortion in the time domain as derived in (10-24) and equal to e
j
2 πk δf N
,
where δ f denotes the difference in the transmitter and receiver oscillators as a fraction of
the inter-carrier spacing ( δ f =
∆f ). Hence, the received signal including both frequency and timing offset f scs
is given by
r ( k ) = s( k − ε )e
j
2 πk δf N
+ n(k )
(10-43)
where ε is the integer-valued unknown arrival time of a symbol. The appearance of a cyclic prefix in (10-43) yields a correlation between some two shifted symbols, spaced by N samples apart. This produces a peak value at a shifted offset that corresponds to symbol timing ε . Once observed for several symbols a time position shift that corresponds directly to frequency drift (i.e., offset) will result.. Hence r(k) in (10-43) contains information about the timing offset ε and carrier frequency offset δ f . This is the crucial observation that offers the opportunity for simultaneous estimation of the timing and frequency offsets from r(k).
Assume that we observe 2N+L samples of the OFDM symbols at the output of the A/D. The starting position of the symbols within the observed block of samples is unknown because of the delay ε is unknown to the receiver. Define the index set of I = {ε ,...., ε + L − 1} and another N-delayed index of I ' = {ε + N ,...., ε + N + L − 1} to address the observed samples. The set I ' thus contains the index of the data samples that are copied into the cyclic prefix, and the set I contains the indexes of this prefix. With 2 N + L samples of the received signal r(t), or r = [r (1),....r (2 N + L)], a log-likelihood function for δ f and ε is formed using these observed samples, and is given by [19] Λ (ε , δ f ) = log f (r / θ , δ f
)
(10-44)
The maximum likelihood (ML) estimate of δ f and ε is the argument that maximizes the ML function Λ (ε , δ f ) , which is extended to [19],
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Λ (ε , δ f ) = log
ε + L −1
f (r ( k ), r ( k + N ) )
∏ε f (r(k ), r(k + N ) )
(10-45)
k=
where f ( x ) denotes the PDF of the variable x. Under the assumption that r is Gaussian, the function in (1045) is given by
Λ (ε , δ f ) = γ (ε ) cos(2πδ f + arg{γ (ε )}) −
σ s2 r (ε ) σ s2 + σ n2
(10-46)
where
γ (m ) =
m + L −1
∑ r (k )r
∗
(k + N )
(10-47)
k =m
and e(m ) =
1 m + L −1 2 2 r(k ) + r(k + N ) ∑ 2 k =m
A measure of the SNR ratio
(10-48)
σ s2 , is computed as the correlation coefficient given by the signal to noise ratio. σ n2
One of the drawbacks of this algorithm is that it assumes that the SNR is known at the receiver. It has been reported though that the algorithm deteriorates very little when the estimated SNR
σ s2 is not accurate. σ n2
The first term in Λ (ε , δ f ) of (10-46) is a weighted magnitude of the sum of the correlation between L consecutive pairs of samples. This contribution is weighted by a factor depending on the frequency offset. The term δ f (ε ) is an energy term, independent of the frequency offset δ f , and contributes negatively to the loglikelihood function. Notice that this contribution is also SNR-dependent. The maximization of Λ (ε , δ f ) over
the frequency offset ε is obtained when the cosine-term in (10-46) equals one, cos(2πδ f + arg{γ (ε )}) = 1 , or
when 2πδ f + arg{γ (ε )} = 2nπ , where n is an integer. This yields the ML estimates of the frequency offset,
δˆ f (ε ) =
−1 arg{γ (ε )} 2π
(10-49)
Substituting this into the ML function in (10-48) ,
(
)
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(10-50)
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To estimate the timing delay ε in (10-49), use the scheme shown in Figure 10-18 and the equation,
{ (
)}
εˆ = arg max γ (δ f ) −
σ s2 δ (ε ) σ s2 + σ n2 f
(10-52)
This estimate is then substituted into (10-49), and the frequency estimates is found by −1 arg{γ (εˆ )} 2π
δˆ f =
(10-53)
1 The estimate (10-49) is bounded to avoid the ambiguities by δˆ f ≤ . Figure 10-19 shows the estimates based 2 on (10-52) and (10-53) for a system with N = 1024 DFT subcarriers, L = 128, δ f = 0.25, and SNR=15dB.
Notice that the peak of the correlation is maximum at the symbol boundary and that the frequency offset is observed by taking the phase of the ML function that corresponds to that. Clearly the frequency offset estimate does not have to be at the maximum correlation values, as can be seen from the flat region of arg{γ (εˆ )} in the bottom of Figure 10-19. The frequency offset can be observed by inspecting the movement of the peak of the correlation in the top of Figure 10-19, in which case the frequency offset is positive. ε (m ) =
1 m + L −1 2 2 ∑ r (k ) + r (k + N ) 2 k=m L −1
∑ k =0
(.) 2
(.) 2
1 ⎛ σ s2 ⎞ ⎜ ⎟ 2 ⎜⎝ σ s2 + σ n2 ⎟⎠
rk
z −N ∗
(.)
γ (m ) =
∑ r (k )r ∗ (k + N )
Symbol timing
-
m + L −1
θεˆˆ
(.)
k=m
L −1
∑ k =0
Frequency offset arg{(.)}
−1 (.) 2π
δεˆˆ f
Figure 10- 18 : Block scheme of the ML estimator.
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8 6 4 2 0 -2
0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
4 2 0 -2 -4
Sample #
Figure 10- 19: The signals that generate the ML-estimates with N = 1024, L = 128, ε = 0.25, −1 The frequency offset is given by the angle of ( δˆ f = arg{γ (εˆ )} ) that is taken at the frame start position. As 2π shown in (??), this angle sits in a fairly flat region around (??), which indicates that the estimates will not be effected significantly by the timing offset. The estimator can resolve a frequency offset of less than half the inter-carrier spacing. It is assumed that the frequency offset will be smaller than this after the analog front end. The accuracy of the estimation is adequate. The algorithm implementation in Figure 10-18 can be further reduced by observing that the magnitude of the complex samples of y k can be reduced to,
rk = Re{rk } + Im{rk }
(10-54)
Substituting the approximation of (10-54) into (10-52), the timing estimates become,
εˆ = arg{max(Re{γ (ε )}+ Im{γ (ε )})}
10.13.2
(10-55)
Another Variant of Maximum Likelihood Synchronization Scheme Dr. Mohamed Khalid Nezami © 2003
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To circumvent the need to have SNR estimate available at the receiver ,Figure 10-20 shows one alternative algorithm that is also based on the principle detailed above. Here the sampled complex OFDM signal is split into two branches. In one of the branches, the sample of the signal is delayed by N samples, conjugated, and then multiplied by the non delayed path (correlation problem). The sample delay network can be implemented using dual port memory as a “first-in- first- out” (FIFO). The other branch of the signal is converted to its complex conjugate (i.e. the imaginary part changes sign), then the two branches are multiplied together in a complex multiplier. The frame synchronization timing (symbol boundary) is estimated by searching for the index sample (n), where absolute value of the correlation of is maximum.(??) That is, the estimates of samples are given by , 2 ⎧ ⎫ n ⎪ ⎪ ∗ nˆ = arg ⎨max ∑ y k y k − N ⎬ (10-56) n 1 = − + k n N g ⎪⎩ ⎪⎭ and the normalized frequency offset at the time instant nˆ in (10-56) is estimated by
∆fˆ =
⎛ n ⎞ 1 arg⎜ ∑ y k y k∗− N ⎟ ⎜ k = n − N +1 ⎟ 2πTs g ⎝ ⎠
(10-57)
Realizing that arg{x}max = π , this frequency estimate is bounded by ∆f ≤ y (n)
L −1
∑
max tan −1 (
( )2
n
k =0
z−N
()
∗
y ∗ (n − N )
{
f 1 , or ∆f ≤ scs . 2 NTs 2
Frequency detector
)}
nˆ
∆kˆ
Figure 10- 20 : Block diagram of the pre-FFT synchronization algorithm.
10.14
Pilot-Aided OFDM Synchronization Algorithms
The synchronization techniques based on the cyclic extension discussed above are particularly suited to perform blind synchronization in systems where no training symbols (preamble) are available, or to obtain an initial coarse estimates so that pilot aided (??) can be used. After the demodulation starts, both frequency offsets and timing estimates need to be continuously refined (tracked) using pilot aided techniques. Figure 10-21 shows a two dimensional view of OFDM pilot symbols in both time and frequency domain. The relative number of pilot symbols to the total OFDM symbols is given by 1 pilot _ eff = (10-58) M f Mt
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where the pilot symbols are transmitted over every M f subcarrier in every M t OFDM symbol.
Frequency
Mt
Mf
data pilot
Time
Figure 10- 21: 2D time-frequency pilot pattern for OFDM synchronization. The values of M f and M t in Figure 10-21 depends on the normalized maximum expected carrier offset δ f , cyclic prefix period TG , and the subcarriers spacing f scs . The special time distance in the 2D pilot pattern in Figure 10-21 is given by 1 (10-59) Mt ≤ 2(1 + f scs TG )δf and the special pilot subcarrier is given by Mf ≤
1 f scs TG
(10-60)
Because of the pilot presence, the spectrum efficiency is reduced. This reduction in efficiency can be represented as SNR loss due to pilot symbols, and given by
⎛ 1 D p = 10 log10 ⎜1 − ⎜ MM t f ⎝ And the total net data rate is reduced form
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(10-61)
N to Ts
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Rb =
1 N ⎛⎜ 1− ⎜ Ts ⎝ M t M f
⎞ ⎟ ⎟ ⎠
(10-62)
For example, assume N=1024, signal bandwidth is BW=5Mhz, TG = 10 µ sec , and that the system is suppose to cope with a maximum Doppler frequency offset of δ f =5%. Using these values in the relations of (10-59) and (10-62), the pilot symbols have to be sent every M t ≤ 9.534 symbols and using every M f ≤ 20.48 subcarrier. Assuming that M t = 4 and M f ≤ 10 , then SNR loss due to symbol pilot is D p = 0.11dB , and the data rate is reduced from 1024 / 40 µ sec , or 25.6Msps to 25.54 Msps .
Figure 10-22 illustrates the IEEE 802.11a WLAN that has a preamble that consists of 10 short pilot symbols (t1:t10), and 2 long pilot symbols (T1:T2) as shown in Figure 10-24 purposely designed for pilot aided synchronization (??). The data fields then follow the preamble.
Figure
10-
22:
802.11a OFDM training structure The short OFDM training symbol consists of 12 out of 52 subcarriers. The subcarriers used are those at every fourth bin, with indices of k= ±4, k= ±8, k= ±12, k= ±16, k= ±20, and k= ±24. These bins (subcarriers) are modulated by the sequence given by
A short OFDM training symbol consists of 12 subcarriers, which are modulated by the elements of the sequence S, given by: S [-26, 26] =
13 x {0, 0, 1+j, 0, 0, 0, -1-j, 0, 0, 0, 1+j, 0, 0, 0, -1-j, 0, 0, 0, -1-j, 0, 0, 0, 1+j, 0, 0, 0, 0, 0, 6 0, 0, -1-j, 0, 0, 0, -1-j, 0, 0, 0, 1+j, 0, 0, 0, 1+j, 0, 0, 0, 1+j, 0, 0, 0, 1+j, 0, 0}
The multiplication by a factor of
13 is in order to normalize the average power of the resulting OFDM symbol, which 6
utilizes 12 out of 52 subcarriers.
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The long training symbol preamble consists of 53 subcarriers plus a zero value at dc. These subcarriers are modulated using the following sequence
A long OFDM training symbol consists of 53 subcarriers (including a zero value at DC), which are modulated by the elements of the sequence L, given by: L [-26, 26] = {1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 0, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, 1, 1}
The timing estimate in (10-56) is a coarse estimate. This estimate can be refined and tracked using a data aided symbol timing tracking scheme as shown in Figure 10-24. The data modulations here are removed using the pilot symbols. The timing offset ∆nˆ can be accurately estimated as follows [31]: N ⎛ P ⎞ (10-63) arg⎜ ∑ z l∗, pi z l , pi −1 ⎟ 2πM f ⎝ i=2 ⎠ is the training pilot sequence such as that used above for the 802.11a.
∆nˆ =
where sl∗, pi
Freq uen cy correctio n Lo op
yn
Yl ,k
P− 1 N arg ( 2πM f k= 2
zl ,k
FFT
∑
z −1
)
∆nˆ
( )∗
P ilot Adjust windo w
Loop Filter
Figure 10- 23: Pilot-aided synchronization loop.
The underlying principle of frequency tracking algorithms using pilot symbols is reduced to a phase estimation problem by considering the phase shift between two subsequent subchannel samples using the algorithms discussed in Chapter 7. The influence of the symbol modulation is removed in this case by the multiplication with the conjugate complex value of the transmitted symbols, for exampleQPSK ODFM modulations. Using the NDA frequency offset estimator detailed in Chapter 7, the frequency offset using the fourth order nonlinearity is given by [27], n=+
1 ∆fˆ = 8πTs
N −1
1 ⎡ N 2 −1 4⎤ ∑N −1 2 ⎢ 4 − n(n + 1)( yn +1 yn ) ⎥ ⎣ ⎦
(10-64)
n=−
If pilot tones were available, the noise enhancing nonlinearity in (10-64) can be avoided, and pilot symbol aided estimation can be used. One way to achieve this is by modulating the pilot subcarriers with ‘all ones’ Dr. Mohamed Khalid Nezami © 2003 10-32
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(i.e., CW) symbols. The frequency estimates then can be obtained using another algorithm that was detailed in Chapter 7 given by [12,27]
∆fˆ =
M 1 arg ∑ R(k ) 8πTs ( M + 1) k =1
(10-65)
where the estimated autocorrelation function R (k ) is given by R (k ) =
10.15
N 1 arg ∑ y i y i∗− k N −k i = k +1
(10-68)
Dual Pilot Tone Synchronization Method
One simple pilot aided synchronization can be obtained using a dual tone system as shown in Figure 10-24. By using CW tones from adjacent subcarriers that have positive and negative indexes as shown in Figure 1029, both frequency and timing estimates can be obtained. Figure 10-25 shows the phase error associated with OFDM subcarriers for both sampling and carrier frequency. Carrier frequency phase errors are an even function, so to extract the frequency estimate from any adjacent carriers (one positive and one negative), multiplication between a carrier and the conjugate of its mirror image is carried out. Contrary to this, to obtain the frequency offset associated with the phase error associated with the sample clocks function (bottom), since it is an odd function, needs only multiplication .
Figure 10- 24: Carrier frequency error and sample clock errors adjacent OFDM bins.
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For the timing estimates in Figure 10-24, the same training pilot symbols are transmitted at symbol 1 and 2 in a frame. The frame synchronization is estimated by
nˆ = Im(Y2 (k )Y1∗ (k ) )
(10-69)
This error signal is filtered using a low pass filter, and then feedback to the frame position adjuster, before the samples are processed by the DFT block. After taking the DFT of the received samples, the output for the kth subcarrier is given by Y1 (k ) and for the second subcarrier is given by Y2 (k ) . Frequency offset can be estimated by
∆f =
[ [
⎛ Im Y2 (k )Y1∗ (k ) 1 tan −1 ⎜⎜ ∗ 2π ⎝ Re Y2 (k )Y1 (k )
]⎞⎟ ]⎟⎠
(10-70)
Y1 (k ) r (k)
Frame Position
Remove Guard interval
Im( )
N-FFT
Y2 (k)
Loop Filter
Figure 10- 25: Dual tone symbol synchronization.
10.16
Compensation of Carrier Frequency Offset
After estimation of frequency sampling and carrier frequency offsets, the correction process may involve feeding back these estimates to both the local oscillator in the frontend down converter and the clock of the A/D. It is of great interest to perform this correction in the digital domain. This can be achieved by realizing that the effect of carrier frequency can be easily compensated for by multiplying the samples of the received signal r ( k ) with the correction phase rotator factors of e following samples at the DFT algorithm:
j
2πkδ f N
. The effect of this rotation will produce the
2πδ f 4πδ f 2π ( N −1)δ f ⎤ ⎡ j j j N N N r = ⎢r (0), r (1)e , r (1)e ,......., r ( N − 1)e ⎥ ⎥⎦ ⎣⎢
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Instead of using feeding back analog correction signals to a variable A/D clock, the correction can be carried out in the digital computational domain. Since the influence of the sampling frequency offset can be eliminated by incorporating the frequency offset correction into the twiddle factor used to compute the FFT i (1+ δ ) algorithm in the receiver. This can be carried by modifying the DFT twiddle factor WMi → WM f . That is e
j
2πki N
→e
j
2πki
(1+ δ f )N
.
00110 P/S
N-DFT
decoder
N FIFO
N FIFO
X
S/P
receive filter + A/D
{r (0) : r ( N − 1)}
e
j
2πki N
→e
j
{e
2πki
(1+δ ) N
j
2πkδf N
}0≤ k ≤ N −1
Frequency Estimator
down converter
~ Fixed Crystal
Free running Oscillator
Frequency Estimator
Figure 10- 26 : Carrier Frequency offset and Sampling frequency offset digital correction. Likewise, for symbol timing error δ t , the timing offset of the receiver is then
Ts + δ t which translates into a phase factor given by e j 2πf (Ts + δ t ) in frequency domain (i.e., after taking the k DFT). Since in OFDM data is encoded on the frequency domain sub-carrier f k = where k is sub-carrier Ts j 2π
k (Ts + δ t ) Ts
for each carrier that has to be index and Ts is the FFT symbol period. There is then a phase factor e compensated for. This phase shift factor can be lumped in either the channel equalizeror in the DFT twiddle factor. Figure 10-30 shows the correction process in the digital domain.
10.17 Illustrative Example
Here we use the algorithms detailed earlier to illustrate a modem acquisition, tracking and data decoding for a wireless IEEE802.11a/g receiver. We illustrate the frame detection, carrier frequency offset (coarse and fine) acquisition, symbol timing (coarse and fine) offset, and then illustrate the pilot-subcarrier tracking. Dr. Mohamed Khalid Nezami © 2003
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1 2
3
4
5 6
7
8
9 10
Frame Detection, AGC, Diversity Coarse Frequency Offset Estimation, and Symbol Timing
GI2
T1
GI
T2
Channel, Fine Frequency Offset Estimation, Fine Symbol Timing
10 x 0.8 = 8 µ S
Signal
GI
Data 1
GI
Rate & Length
2 x 0.8 + 2 x 3.2 = 8 µS
0.8 + 3.2 = 4 µ S
Data 2
GI
Data N
User data detection, Carrier Frequency Tracking Sampling clock tracking 0.8 + 3.2 = 4 µ S
Figure : Illustration of the 802.11a preamble and payload transmission burst. Digital Down Converter
BPF
LPF
LPF
RF Section
PLL
ADC
DEC
LPF
DSP Section
INT
DEMOD
NCO
Figure : Illustration of a direct or near zero conversion RF receiver and digital modem.
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Frame detected
ADC
LPF
DEC
NCO
INT
Fine Timing
Coarse Timing
Coarse CFO
Fine CFO
FFT Channel Estimate
To RF AGC
AGC
Pilot sub-carriers
Data sub-carriers
Tracking algorithm Decoder “FEC”
data
Figure : Illustration of the modem DSP algorithms.
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0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2
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channel-es timate H
5 0
Channel Gain-dB
-5 -10 -15 -20 -25 -30 -30
-20
-10
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8
x 10
5
Coarse CFO
Estimated Frequency offset-Hz
6
Fine CFO
4 2 0 -2 -4 -6 -8 -8
-6
-4
-2
0
2
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8 x 10
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Actual Frequency offset-Hz SNR=12dB
10.18 Reference
1. Theodore S. Rappaport, Wireless Communications, Principle and Practice, Prentice Hall, NY, 2002, chapters 5. 2. www.ti.com/dspvillage. 3. Richard D.J. Van Nee, and Ramjee Prasad, Ofdm for Wireless Multimedia Communications, Artech House, January 2000, ISBN 0890065306. 4. J.A.C. Bingham, "Multicarrier modulation for data transmission: An idea whose time has come," IEEE Comm. Mag., pp. 5-14, May 1990. 5. P.H. Moose, "A technique for orthogonal frequency division multiplexing frequency offset correction," IEEE Trans., on Comm., pp. 2908-2914, Oct. 1994.
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6. T. Pollet, P. Spruyt, and M. Moeneclaey, The BER performance of OFDM systems using nonsynchronized sampling," Proc. of Globecom '94, pp. 253-257. 7. K.W. Kang, J. Ann, and H.S. Lee, "Decision-directed maximum-likelihood estimation of OFDM frame synchronization offset," Elect Lett., pp. 2153-2154, Dec. 1994. 8. J.S. Oh, Y.M. Chung, and S.U. Lee, "A carrier synchronization technique for OFDM on the frequency selective fading environment," Proc. of VTC '96, pp. 1574-1578. 9. F. Daffara and O. Adami, "A novel carrier recovery technique for orthogonal multicarrier systems," European Trans. on Telecomm., pp. 323-334, July-Aug. 1996. 10. IEEE standards, http://standards.ieee.org/reading/ieee/std/lanman/802.11a.pdf. 11. H. Nogami and T. Nagashima, "A frequency and timing period acquisition technique for OFDM systems," IEICE Trans. on Comm., pp. 1135-1146, Aug. 1996. 12. M. Luise and R. Reggiannini, "Carrier frequency acquisition and tracking for OFDM systems," IEEE Trans. on Comm., pp. 1590-1598, Nov. 1996. 13. M. Speth, F. Classen, and H. Meyr, "Frame synchronization OFDM systems in frequency selective fading channels," Proc. of VTC '97, pp. 1807-1881. 14. L. Hazy and M. El-Tanany, "Synchronization of OFDM systems over frequency selective fading channels," Proc. of VTC '97, pp. 2094-2098. 15. T. Schmidl and D. Cox, "Robust frequency and timing synchronization for OFDM," IEEE Trans. on Comm., pp. 1613-1621, Dec. 1997. 16. Y. Kim, D. Han, and K. Kim, “A new fast symbol timing recovery algorithm for OFDM systems”, IEEE Trans. On Consumer Electronics, vol. 44, No. 3, pp. 1134-1141, Aug 1998. 17. Chorng Shue, Y. Huang, and C. Huang, “Joint symbol, Frame, and Carrier synchronization for Eureka 147 DAB system”, Proceedings of ICC, 1997. 18. J. Beek, M. Sandell, M. Isaksson, and P. Borjesson, “Low-complex frame synchronization in OFDM systems” Lulea University of Technology, Sweden, 1997. 19. J.J. van de Beek, M. Sandell, and P. O. Börjesson, "ML estimation of timing and frequency offset in OFDM systems," IEEE Trans. of Sig. Proc., pp. 1800-1805, July 1997. 20. D. Lee and K. Cheun, "A new symbol timing recovery algorithm for OFDM systems," IEEE Trans. on Consum. Comm. Elect., pp. 767-775, Aug. 1997. 21. Jan-Jaap Van de Beek, “ML estimation of timing and frequency offset in OFDM systems”, Ph.D. Thesis at Lulea University, Sweden.
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22. Giovanni Santella, “A frequency and symbol synchronization system for OFDM signal: architecture and simulation results”, IEEE Trans. Vehic. Tech., vol. 49, No. 1, pp. 254-275, Jan 2000. 23. Hyoung-Keu Song, Y. You, J. Paik, and Y. Cho “Frequency-offset synchronization and channel estimation for OFDM-based transmission”, IEEE Commun. Lett., vol. 4, No. 3, pp. 95-97, Mar 2000. 24. M. Morelli, A. D’Andrea, and Mengali, “Frequency ambiguity resolution in OFDM systems” IEEE Commun. Lett., vol. 4, No. 4, pp. 134-136, Apr 2000. 25. John R. Pelliccio, Heinz Bachmann, and Bruce Myers, “ Phase noise effects on OFDM wireless LAN performance,”, applied microwave & wireless, 2001. 26. Ana Garcia Armada, “ Understand the effects of phase noise in OFDM,”, IEEE Trans. On Broadcasting, VOL. 47, NO. 2, June 2001. 27. Marco Luise, and Ruggero Regiannini, “ Carrier frequency acquisition and tracking for OFDM systems,”, IEEE Trans. Commun. VOL 44, NO. 11, Nov. 1996. 28. Yung-Liang Huang, Chorng-Ren Sheu, and Chia-Chi Huang, Joint synchronization in Eureka 147 DAB system based on abrupt phase change detection,”, IEEE Trans. Selected areas in Commun. VOL 17, NO. 10, Oct. 1999. 29. Thierry Pollet, Mark Van Bladel, and Marc Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and Weiner phase noise,”, IEEE Trans. Commun. VOL 43, NO. 2/3/4, February/March/April 1995. 30. Y. Harda, T. Kimura, K. Hayashi, A. Kisoda, S. Kageyama, S. Sakashita, and H. Mori, “An implementation of OFDM receiver for digital terrestrial television broadcasting and its technologies,”, International Broadcasting Convention, Sept 12-16, 1997. 31. Hanli Zou, Bruce McNair, and Babak Daneshard, “An integrated OFDM receiver for high-speed mobile data communications,”, IEEE 2002.
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Chapter 11 Techniques for Acquiring and Tracking Signals with Efficient Modulations Efficient modulations include minimum shift keying (MSK), gaussian minimum shift keying (GMSK), offset phase shift keying (OQPSK), and continuous phase modulations (CPM). All of which tend to occupy very narrow bandwidth, something that is highly desirable for crowded bandwidth limited systems such as satellite and cellular communications. For most of these signals, phase shaping reduces the adjacent channel interference but tends to complicate synchronization [1,2,3,4,5,6]. Another degree of spectral efficiency is added when using a powerful channel coding techniques such as turbo coding, Trellis coding, or the concatenation of convolution encoding with Reed-Solomon encoder. Using these channel encoders require the receiver to operate at very low SNR while preserving bandwidth. Previously, convolutional coding using forward error convolutional coders employing a Viterbi decoder were used. These offered coding gains of up to 7 dB using practical constrain length codes but at the expense of increasing the bandwidth. In 1993, a new channel-coding scheme was proposed using Turbo codes [7]. Unlike conventional convolutional encoding, turbo coding increases the coding gainwithout bandwidth expansion. Turbo encoding can achieve a performance of 0.7 dB from the Shannon limit using interleavers of several thousand bits deep. Trellis coded modulation has also gained popularity [27], with coding gains of 3dB for rate 2/3 8PSK and up to 5dB for rate ¾ 16PSK signals. This chapter is concerned with schemes for acquiring and tracking carrier and symbol timing of signals using high spectral efficiency modulations and high coding gain channel encoders such as turbo coded, Trellis coded, and CPM signals. The main challenge and difference of achieving synchronization with these signals from conventional linear modulated signals is two fold. First, the signal has to be acquired and tracked at very low SNR with typical Eb/No values of 2dB to 3dB, and ,second, complexity increases when synchronization is coupled with the decoder operation. First, we introduce techniques used for synchronization of turbo coded multilevel phase shift keying (MPSK), focusing only on parallel-concatenated turbo coding schemes. The synchronization schemes introduced here are also applicable with serially concatenate turbo coding. Then we discuss in detail acquiring and tracking both symbol timing and the carrier phase of signals with continuous phase modulation (CPM). Finally we introduce several variants of synchronization schemes used with Trellis coded MPSK signals.
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11. 1 Concatenated Turbo Coded Signals Most conventional wireless systems use convolutional (Viterbi) coding or a serial concatenation of Viterbi with Reed-Solomon coding. These techniques have been used in a wide range of voice and data applications for many years but the increased demand for bandwidth-hungry services and the growth in data services has motivated researchers to look for other effective ways of increasing capacity and improving transmission quality. Turbo coding is a new development in forward error correction that enables more information to be transmitted through an existing channel. Replacing Viterbi with Reed-Solomon modems with turbo coding can produce 50% more data transmission using the same bandwidth. The original results of turbo codes used a rate R=1/2 code, with constraint length of K=5 and data block size of 65,536 bits yielding a BER of 10E-5 at around 0.7dB, which is 0.7dB off the Shannon capacity [7,8]. With moderate interleaver lengths, however, a more moderate performance is obtained. For instance, turbo coded BPSK using a rate of R=1/2, 16-state and a block length=1024, produces a BER of 10 −5 at Eb N o =2.0dB. With a block length of 16384, the BER of 10 −5 is achieved at Eb N o =1.0dB [7]. For turbo coded QPSK signals, using a rate of R=2/3 and a block length of 188, a BER of 10 −5 is achievable at Eb N o =2.7dB [7]. For 8PSK modulation, using R=1/2 and a block length of 376, a BER 10 −5 is achieved at Eb N o =4.5dB. Figure 11-1 illustrates the required Eb N o for achieving a BER of 10 −5 using different turbo block lengths. The figure illustrates that for most practical block lengths of less than a few thousand-symbol interleaver, the required Eb N o is in the range of 1dB to 3 dB. Achieving reliable synchronization at these low SNR levels using conventional methods is a challenging task.
Performance at BER=10-5 (448, 16)
(Code word length, number of states)
Eb/No-dB
3 2 1
(1334, 16) (2048, 16)
(8192, 16) (16384, 16)
(229376, 16)
0
Block length , N
Figure 11- 1: Turbo Coding Eb N o Versus Block Length Performance for BER=10-5.
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11.1.1 Turbo Encoder The details of turbo coding are widely covered in many textbooks and journals, and thus beyond the scope of this book [8]. We will only cover the basic principle of parallel-concatenated turbo coding. Figure 11-2 below depicts a typical rate R=1/3 turbo encoder. The output consists of three parts. The first is a straight copy of the input bit (systematic bit), the second bit is coded using recursive systematic convolutional (RSC) encoder #1, while the third bit after being interleaved is coded using another identical RSC encoder #2. The three parallel streams of data are then multiplexed into one stream and sent to the modulator. To obtain a rate R=1/2, puncturing is used between the parity streams (second and third bits). An example of puncturing would be to send the even bits of the first RSC encoder, and the odd bits of the second encoder.
RSC Encoder #1
Data
Output
Modulator/ Transmitter
interleaver RSC Encoder #2
Turbo Encoder Figure 11- 2: Block Diagram of Turbo Encoder-modulator. 11.1.2 Turbo Decoder Turbo decoders operate in an iterative fashion as shown in Figure 11-3, with two decoders corresponding to the two constituent encoders. The first decoder makes an estimate of the probability for each data bit as to whether it is a 1 or a 0 by operating on the received data and parity bits (soft decision) produced by the first constituent encoder. This estimate is then sent to the second decoder along with the interleaved received data and the parity bits (soft decision from decoder #1) produced by the second constituent encoder. This process of having two passes through the decoding algorithm is considered to be one iteration and is repeated for a fixed number of iterations producing the BER curves shown in Figure 11-4. The accuracy of the final data decision gets more reliable as the number of iterations is increased at the expense of greater complexity and longer delay. deinterleaver APP APP
systematic data
decoder 1
Receiver/ Demodulator parity data
Interleaver
decoder 2 Hard decisions
demux
interleaver
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11-3
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Figure 11- 3: General Block Diagram for a Turbo Code Decoder-receiver.
100
Bit Error Rate
10-1 Iteration=1
10-2
10-3 Iteration=2 10-4 Iteration=6
Iteration=3
-5
10
0.0
0.5
1.5
1.0
2.0
2.5
Eb/No-dB
Figure 11- 4: Turbo Coding with Block Length of 65536, Rate ½..
11.2 Carrier and Symbol Timing Acquisition and Tracking of Turbo Coded Signals The acquisition of turbo-coded signals [7-20] can be achieved using conventional methods such as DFT, based on exploring the spectral characteristics of the CW, a dot pattern preamble, or pilot symbols to recover initial frequency and phase offsets. It will be shown later in this chapter that the DFT has processing gain that enables it to estimate initial synchronization offsets reliably even at Eb N o values below zero dB. For turbo-coded signals, acquiring the signal is a classical approach. However, the tracking of these synchronization parameters is a difficult task using either NDA or DD algorithms feedback or the feedforward loops that were discussed in the previous chapters.
11.2.1 Tentative Decision-aided Turbo Coded Signal Carrier Phase Tracking It is possible to use an externally generated hard decision with a MLE carrier and symbol timing tracking algorithms. But such a scheme is not reliable at low SNR values, due to frequent decision errors. Demodulators utilizing turbo decoder decisions for timing and carrier acquisition and tracking are shown in Figures 11-5 and 11-6 respectively. Here, instead of using an external hard decision to aid in the carrier-tracking loop, a turbo earlier data decision is used. To Dr. Mohamed Khalid Nezami © 2003
11-4
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
reduce the delay associated with interleavers, the tentative decision is extracted from the first turbo decoder during the decoding process, and not after a complete decoding cycle. Although the reliability of the tentative decision improves along with the iterative process, decisions with excessive delay cannot be used since the feedback loop may become unstable. The maximum likelihood decision aided (ML-DD) timing error can be derived by differentiating the log-likelihood function (see Chapter 6, Equation 6-8) with respect to timing error ε , assuming that the carrier phase θˆ and data decision aˆ n are available.
(
)
2 dl ε , aˆ , θˆ d ⎧ N −1 ˆ⎫ z (nT + εT )e − jθ ⎬ α − 2 Re ⎨∑ aˆ n∗ dε σ n ⎩ n = 0 dε ⎭
(11-1)
Where T is the symbol time and z (nT + εT ) is the interpolator output as shown in Figure 11-5, and aˆ n∗ is the tentative decision extracted after the first iteration from the turbo channel decoder. d The differentiation z (nT + εT ) can be approximated using the Gardner early-late timing error dε detector derived earlier in Chapter 4. Thus timing error en is given by ⎧ ⎛ T T ⎞ ⎛ ⎞⎫ en = Re ⎨aˆ n∗ z ⎜ nT + + εT ⎟ − aˆ n∗ z ⎜ nT − + εT ⎟⎬ 2 2 ⎠ ⎝ ⎠⎭ ⎩ ⎝
(11-2)
T T ⎧ ⎛ ⎞ ⎛ ⎞⎫ Re ⎨aˆ ∗n z ⎜ nT + + εT ⎟ − aˆ ∗n z ⎜ nT − + εT ⎟⎬ 2 2 ⎠ ⎝ ⎠⎭ ⎝ ⎩
NCO
IF
Carrier recovery
Interpolation
Loop Filter
T iming detector
T entative decision
delay
aˆ n∗
Iteration # (1) Iteration # (2)
Decoded bit
Iteration # (N)
T urbo Decoder
Figure 11- 5: Decision Aided Turbo Coded Symbol Timing Synchronization Carrier recovery is also derived in a similar fashion as shown in Figure 11-6. Defining an objective function that is directly proportional to the carrier phase offset, and assuming that the Dr. Mohamed Khalid Nezami © 2003
11-5
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
tentative decision aˆ and symbol timing error εˆ were obtained ., the carrier phase offset is then given by,
{
eθ = Im aˆ n∗ z n (εˆ )e − jθ
}
(11-3)
{
}
{
}
∗ ∗ The error signal in Equation (11-3) can be approximated by using Im z n aˆ n ≈ sin z n aˆ n , and
{
then applying the small angle assumption, sin z n aˆ
∗ n
} ≈ z aˆ n
∗ n,
which clearly leads to the well
known PLL error signal given by sin θ ≈ θˆ . The DSP based PLL implementation of a 2nd order PLL is then carried out by the following recursive function,
χ k = χ k −1 + (1 − γ )ek − ek −1
χk
where k is the loop iteration index, the signal the recursive relatiosnhip given by,
(11-4)
is then use to update the NCO by performing
θˆk +1 = θˆk + ρχ k −1
(11-5)
The loop parameters ρ and γ determine the dynamics and variance of the loop estimates and are related to the normalized bandwidth B L T .
NCO
IF
Timing recovery
Loop Filter
phase detector
delay
Tentative decision
Iteration # (1) Iteration # (2)
Decoded bit
Iteration # (N)
Turbo Decoder
Figure 11- 6: Decision Aided Turbo Coded Carrier Synchronization
11.2.2 Non-Data-Aided Turbo Coded Tracking
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11-6
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Turbo coded Non-data aided (NDA) carrier recovery and tracking has also been considered [11]. Non-data and non-decision aided algorithms operate on the principle of extracting a CW tone from the random received signal by passing the random received symbols through an M-order nonlinearity that cancels the data modulations (see Figure 7-16). This algorithm is one variant of the Viterbi-Viterbi phase estimator derived in (7-53) and given by,
θˆVV =
⎛ P 1 M ⎞ tan −1 ⎜ ∑ [zk ] ⎟ M ⎝ k =1 ⎠
(11-6)
The order of the non-linearity is M=2 for BPSK and MSK, and M=4 for QPSK and 4-ary CPM, P is the observation interval (block size). The phase estimates are then used to remove any nonambiguities and kept in the region of θˆ < ± π M . In [10, 11] it was shown that below SNR levels of Eb N o = 4dB, NDA algorithms are effected by numerous cycle slipping that made their phase estimate variances depart rapidly from the Cramer-Rao bound as shown in Figure 117.
10 0
Viterbi & Viterbi L=50 symbols
Tentative decisionaided
σ θ2
10 −1
d-7
10 −02
Cramer-Rao bound L=25 symbols
10 −3 0
1
2
3
4
5
6
7
8
9
10
Eb/No-dB
Figure 11- 7: Estimated Phase Variance for QPSK Rate ½ with a Window Length of 25 Symbols. Figure 11-7 shows that below Eb N o =5dB, the Vitervi&Viterbi are affected by so many cycle slips that their variance deviates from the CRB early , making them unsuitable for Tubo coded signals having to operate at Eb N o ranges of 1 to 3 dB. The tentative decision-aided algorithm, even as early as td=2 (traceback=2), has a viariance which remains close to the CRB down even at levels of Eb N o =2dB to 3 dB region. Increasing the tentative decision depth further improves the phase variance, as can be seen with a traceback of 7. Dr. Mohamed Khalid Nezami © 2003
11-7
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
So such carrier recovery schemes are considered unsuitable to transmissions using turbo codes which operate with typical SNR values of Eb N o = 2dB for binary MSK and PSK. This is expected since NDA algorithms utilize a non-linearity to remove the modulations, which enhances the AWG noise. Removing this variance because of noise requires a large number of observation intervals (of the order of L=1000 of symbols This may not be practical for short burst type systems.
11.2.3 Pilot-Aided Tracking Synchronization of Turbo coded signals in a rapidly changing fading environment can be very difficult. One way to cope with such channels is by inserting a pilot sequence into the packets. Periodically inserted symbols within the packet can help in the tracking of the carrier phase. Several methods have been reported in the literature that deals with this approach [9, 19].
Figure 11- 8 illustrates the most common techniques gathered from literature and past products for constructing a frame structure that supports data aided block type acquisition and tracking of data packets. A preamble in Figure 11-8 is used to implement data-aided algorithms to estimate the offsets of symbol timing, carrier frequency, and channel gains. Then pilot symbols are periodically inserted between data blocks to track the initial preamble data-aided acquired estimates. In addition to using pilot symbols for tracking, data-decision feedback algorithms can also be used to track these initial estimates as they vary slowly in time. Figure 11-9 illustrates another method that is not very common from a spectrum efficiency point of view. This method relies on transmitting a pilot tone to perform acquisition and tracking. These tones can be either at the edge of the band or in band as illustrated. M
M Preamble
Engage all data-aided feedforward block estimators.
D P D D
D P D D
D P D D
Engage only pilot aided channel estimator and decision-aided carrier (feedback) tracking.
Postamble
TBD
Figure 11- 8: Preamble-pilot Symbols Based Acquisition and Tracking Strategy. Dr. Mohamed Khalid Nezami © 2003
11-8
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Adjacent 25kHz subcarrieres with pilot carrier 0
8-ary CPFSK with Rs=9600sps, h=1/8
Power SPectral Density-dB
-2
-4
-6
-8
-10
-12 0
1
2 3 4 5 Frequency offset from center of RF channel-Hz
6 4
x 10
Figure 11- 9: Adjacent 25kHz Subcarrieres with Pilot Carrier Inserted Inband for Synchronization and Channel Estimation.
Pilot symbol aided demodulation and channel estimation has been detailed in several references. The pilot-tracking scheme is based on sending and receiving known symbols that are periodically inserted into the data prior to phase modulation, as shown in Figure 11-8. These symbols are then used for channel estimation (fading gains). The periods and how often the pilot symbols are inserted are a tradeoff between transmission efficiency, power loss, and accuracy of the final channel estimates. The period of pilot symbol insertion can be treated as a clock sampling, a slowly varying AM signal. As a result, the Nyquist sampling criteria method can be used to estimate how often the pilot symbol has to be inserted once the channel parameters are known, such as average fade duration and the Doppler frequency. Based on this, the pilot symbol insertion period is given by 1 ≥ 2 f DT M
(11-8)
where M is the distance (in symbols) between the pilot symbols, T is the symbol duration, and f D T = 10 −3 is the normalized Doppler frequency. As an example, for a 2400bps MSK with f D T = 10 −3 ,and using (11-8), the pilot symbols are required to be inserted every M = 500 symbols, for a rate of R=1/3 and a block length of 1024 bits. This indicates that the efficiency and power loss is negligible. To illustrate channel tracking using inserted pilot symbols as illustrated in Figure 11-10, consider that the received signal is given by, rk = s k + n k Dr. Mohamed Khalid Nezami © 2003
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where n k is AWGN, k is the pilot symbol index, and s k is the original transmitted phase modulated baseband signal Once this signal is received and converter to baseband, the receiver picks out the samples corresponding to the received signal portion that correspond to the pilot symbols at every kM symbols, and that is given by rkM = a kM s kM + nkM
(11-9)
where a kM is the fading gain experienced by the signal. The fading gain estimates aˆ kM are then obtained from the pilot symbols at time t = kMT , ∗ aˆ kM = s kM rkM
(11-10)
substituting the received signal (8-9) into (11-10), the channel gains are given by ∗ (a kM s kM + nkM ) aˆ kM = s kM which is further reduced to ∗ ∗ aˆ kM = s kM a kM s kM + s kM n kM
(11-11)
∗ Since s kM s kM = 1 , the channel gains are then given by
∗ aˆ kM = a kM + s kM n kM
(11-12)
∗ where s kM n kM is a noisy term. Since the fading gains are needed at every symbol and not just at periods of t = kMT , the fading gain estimates of the channel at data symbols are then obtained using interpolation as shown in Figure 11-10.
Interpolated estimates
a3kM aˆ 2 kM
aˆ kM
Fading envelope
D P D D
D P D D M
D P D D M
Figure 11- 10: Fading Channel Gain Estimation Using Pilot Symbols.
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11-10
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Figures 11-11 illustrate the algorithms that are suitable for use with reception of turbo coded packet transmission carrier and symbol tracking. Here the synchronization scheme can be implemented using either data-aided ornon-data aided loops. Figure 11-11 illustrates the dataaided acquisition synchronization scheme in conjunction with the pilot symbol-aided tracking implementation detailed above in conjunction with iterative Turbo data decoder. In this scheme the data ak in the preamble is used to obtain timing and carrier initial estimates, then, the pilot symbols are used for the final phase and channel tracking.
y (kTs ) = g k sk e z
− LTs
aˆkM
⎛ ⎛ T ⎞ ⎞ j ⎜⎜ 2π ⎜ k +ε ∆f +θ k ⎠ ⎝ ⎝ N ⎠
y3(kT )
y 2(kTs )
y1(kTs ) z
− LTs
1 gˆ k
z e
− LTs
− j 2π∆fkTs
y 4(kT )
z − LT
INT
− τˆ
εˆ
e
− jθˆk
θˆ
∆fˆ ak
ak
ak
ak
( ( )) y 4k = ∆ak sk e j δθ + 2πTδf k +δτ
rk
dˆ j
Remove pilots & INT-1
z − LT ⎛ 2 ∗⎞ ⎜⎜ σˆ 2 cˆk ⎝ n ⎠ Iterative Channel & noise variance estimator
Turbo decoder
λ(qi )
(q)
F (x)
yˆ
INT & re-insert pilots
(q) k
yp
xˆi( q ) ⎛ λ( q ) ⎞ soft decision, xˆi( q ) = tanh⎜⎜ i ⎝ 2 ⎠ Hard decision, xˆi( q ) = sign(λ(i q ) )
Figure 11- 11: Data-aided Block Signal Processing in Conjunction with Pilot symbols-aided
Iterative Turbo Channel Estimation and decoding and Decoding. Dr. Mohamed Khalid Nezami © 2003
11-11
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
Now we focus on detailing the algorithms involved in the turbo based acquisition and tracking schemes illustrated in 11-11. In the scheme shown in Figure 11-11, a feedforward AGC is first used to numerically scale all samples to avoid any dynamic range problems and correct attenuation caused by the slowly changing amplitude g k . . The fast fading gains a k are usually estimated and corrected using decision-aided tracking loop as will be illustrated later.. After AGC correction, the signal in Figure 11-11 is now given by y1( kTs ) = a k e − j (2π∆f ( kTs +τ )+θ k +φk ) + nk
(11-13)
where ak is the fast fading gain , φk is the transmitted data symbol, τ k is the symbol timing offset, ∆f is the frequency carrier offset, and θ k is the carrier phase offset. The received signal (11-13) now contains channel fading gains, symbol timing and carrier offsets. The easiest way to decouple the multivariable in (11-13) is to estimate the frequency offset ∆f first using differential detection methods that remove the slowly changing phase and timing between consecutive symbols. Realize that the frequency estimation and correction can be carried out without the symbol timing offset error information τ k and without knowledge of the carrier phase θ k . So the frequency offset correction is then obtained by multiplying the signal in (11-13) by a phasor that conjugates the frequency shift. That is given by, y 2(kTs ) = y1(kTs )e − j 2π∆fkTs = ak e − j (2π∆fτ +θ k +φk ) + nk ˆ
(11-14)
Even though the correction was done at the sample rate in (11-14), the estimation of ∆f itself does not have to be carried out at the sample rate since the maximum offset that ∆f can take on may be known beforehand. After frequency offset correction (11-14), the consequent algorithms of carrier phase acquisition, carrier tracking, and data decoding only need a single sample per symbol (N=1) to operate. As a consequence, sample decimation has to be performed prior to these two stages. However, since the clock used in the A/D in Figure 11-11 is a free running clock oscillator, the samples are asynchronous and thus the symbol timing offset ( τ ) in (11-14) has to be determined to pick the middle sample corresponding to each symbol. The estimation of the symbol timing offset is first obtained using the over sampled signal. This estimate is then used in conjunction with an interpolating filter to resample the baseband signal to the correct sampling instance. Once timing errors are removed, the signal (11-14) is decimated to yield a single sample per symbol ( N = T Ts ). This is necessary since the following carrier phase acquisition, and the subsequent decoding and tracking algorithms, operate on a symbol-by-symbol basis. As a result of timing offset correction and decimation by (N), the resulting baseband signal at the output of the combined decimator-interpolator filter is given by y3( kT ) = a k e − j (θ k +φk ) + nk Dr. Mohamed Khalid Nezami © 2003
(11-15) 11-12
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
The signal in (11-15) now contains carrier phase offset rotations, the fast channel gain a k , and the original desired symbol phase modulations ( φk ) . The next step is to estimate and remove carrier phase rotations. This is carried by the carrier phase estimation algorithm illlustrated in Figure 1111, which is based on feedforward phase offset estimators detailed in Chapter 8. The phase corrected samples are obtained by multiplying the signal (11-15) with a phase that is given by ˆ
y 4(kT ) = y3(kT )e − jθk
(11-16)
Substituting (11-15) into (11-16), the resulting baseband signal at the output of the carrier phase estimator-corrector is given by y 4( kT ) = a k e − j (φk ) + nk
(11-17)
The signal in (11-17) now contains the fading channel gain a k that has to be removed for proper soft decoding. Using the frequently inserted pilot symbols, these gains are removed using the algorithm detailed in Figure 11-10 As a result the baseband signal at the input of the decoder is given by
⎛ 1 y 4(kT )⎜⎜ ⎝ aˆ k
⎞ ⎟⎟ = e − j (φk ) + nk ⎠
(11-18)
Notice that the channel gain aˆ k in (11-18) can be either complex or real. The schemes in Figure 11-10 and Figure 11-11 assume that the channel gain is real and that the channel phase tracking is performed using the data decision aided feedback loop. However, in the iterative schemes shown in Figure 11-11 the channel gain estimates aˆ k are assumed to be complex since they account for both channel gain and channel phase rotations.
Up to now, we have assumed that all of the estimates and corrections are ideal and that no residuasl are present. Furthermore, we also assumed that the carrier and channel parameters are constant throughout the whole packet (past the preamble). In reality, there will be a small residual error due to noise and inaccuracies in these algorithms and the parameters will tend to slowly drift from symbol to symbol. Let us represent these residuals on the input signal to the decoder in Figure 11-11 by the following relation, y 4 k = (1 − δa k )e j (ϕ k +δθ k + 2πTδf (k +δτ )) + nk
(11-19)
where δak represents the error or drift in the channel estimation, δτ represents the fine symbol timing error offset or drift, δf is the fine frequency residual error, and δθ is the fine carrier phase error offset. The removal of e j (δθ k + 2πTδf (k +δτ )) is carried out using feedback decision-aided phase and symbol timing tracking algorithms as shown in all of the Figure 11-11. Dr. Mohamed Khalid Nezami © 2003
11-13
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
The feedback system now tracks separately the phasor e j (δθ k + 2πTδfk ) and e j (2πTδfδτ ) and then corrects the signal by multiplying (11-19) by the conjugate of the estimated phasor e j (δθ k + 2πTδf (k +δτ )) . As a result, the final decoder input signal in Figure 11-11 is given by y 4 k = (1 − δa k )e j (ϕ k ) + nk
(11-20)
The fine channel errors (1 − δa k ) are still present in (11-20. The estimate of the (1 − δaˆ k ) will be obtained by a decision-aided AGC loop that will be illustrated separately next.
11.2.4 Iterative Turbo Coded Channel Gain and Noise Variance Estimation Assuming that the final signal at the input of the decoder in Figure 11-11 is given by rk = ck y k + nk
(11-20)
where ck are the channel fading complex gains, and the variance of the AWGN is given N by σ n2 = 0 , and assuming that the symbol energy is given by E s = rEb , Eb is the bit energy, 2Es L is Turbo coding rate, i.e., the ratio of interleaver length of the block code length. and r = N With pilot symbols inserted every M data symbols as illustrated in Figure 11-10, the overall rE (M + 1) symbol energy is then given by E s = b . The channel complex gains will be assumed to M be
c(t ) = ( AR + X (t ) ) + jY (t ) where X (t ), Y (t ) are considered a zero mean complex numbers consisting of independent real and imaginary parts with variance σ R2 . The Rician factor is given by the ratio of specular to multipath (diffused) powers, that is ( A )2 KR = R 2 . 2σ R If the multipath gains X (t ), Y (t ) are minimal, the link is a pure Rician channel (line of sight) and K R = ∞ . On the other hand, if there are no line of sight components, AR = 0 , the link is dominated by Rayliegh multipath power, and K R = 0 . The channel fading rate is then dependent on the Doppler frequency ( f d ) a. This channel fading rate measure can be obtained from the channel autocorrelation function given by Rc (τ ) = σ R2 J o (2πf d T ) , Dr. Mohamed Khalid Nezami © 2003
11-14
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
where J o is the zero Bessel function, and T is the correlation lag time equal to a single symbol duration. The samples corresponding to the pilot symbol insertion index are shown in Figure 11-13.??? , where the pilot symbols at the ( k , i ) = ( k ,0) is given by rk , 0 = ck , 0 y k , 0 + nk , 0
where y k , 0 is the symbol transmitted during (k ,0) and the pilot symbol at (k , i ) = (k + 1,0) is rk +1, 0 = ck +1, 0 y k +1, 0 + nk +1, 0
where y k , 0 is the symbol transmitted during (k + 1,0) . Now we have available 2 pilot samples from two adjacent frames, frame K and frame k+1. We also know that the pilot symbols can be fixed to a single value (P). Assuming that we constantly send the same pilot symbol every L-1 symbol, then the channel gain at (k ,0) and (k + 1,0) is given by
and
cˆk ,0 = (ck ,0 yk , 0 + nk , 0 )P ∗
(11-21)
cˆk +1,0 = (ck +1,0 y k +1, 0 + nk +1, 0 )P ∗
(11-22)
since yk ,0 P ∗ = y k +1,0 P ∗ = 1 , the cˆk ,0 = ck , 0 + nk ,0 P ∗ and cˆk +1,0 = ck +1,0 + nk +1,0 P ∗ , which is the desired channel gain estimates using the previously known symbol pilot P. To obtain the channel estimate for data symbols between the pilot symbols, that is the channel gain cˆk ,i , where, 0 < i < L − 1 , we use linear interpolation between the obtained channel estimates at two pilot symbols, that is cˆk , 0 and cˆk +1, 0 . This gain is then given by,
cˆk ,i =
(L − i )cˆk ,0 + icˆk +1,0
Dr. Mohamed Khalid Nezami © 2003
(11-23)
L
11-15
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
L symbols kth Frame Preamble
(k+1)th Frame
D P D D
D P D D
Pk , 0
Pk +1, 0
rk , 0
rk +1, 0
cˆk ,0
cˆk +1, 0
{cˆk ,i } 0 < i ≤ L −1
D P D D Known pilot symbols
Available measured received signal sample Computed channel gains
interpolated channel gains
Figure 11- 12: Illustration of Channel Estimation Using Two Discrete Pilot Symbols.
For Turbo decoders, the channel estimates can be iteratively refined. First the channel estimation is performed prior to decoding, as illustrated above. The sequence {rk } is sent to a channel estimator algorithm that computes {ck } of the fading process and the noise variance σ n2 . The cˆ algorithm then outputs the sequence 2 k2 , which is then multiplied by the received sequence σn {rk } . The result is then passed to the turbo decoder after stripping off the symbol pilots since these are not needed by the decoder. The decoder then outputs data estimate dˆ of the data j
sequence The receiver then improves the estimates using the feedback process shown in Figure 11-3. Using the ) estimates of the code symbols {λ(q i } after each decoder iteration (q), the decoder log likelihood ratio (LLR) is first passed through a nonlinear function F ( x ) which either produces a hard decision estimate ) of the code symbols {xˆ i(q ) } using the function F ( x) = sign (λ(q i ) or a soft decision of the code symbols using the function
F ( x) = tanh(
) λ(q i
) . This is shown in Figure 11-14. After reinserting the pilot 2 symbols y p , the symbol sequence { yˆ i( q ) } is fed to the channel estimator to be used in a decision-aided cˆ type channel gain estimator and variance of the noise to produce the sequence 2 k2 . Now again, these σn estimates are fed into the decoder (after removing the pilot symbols) to produce another refined code cˆ word that is then fed back to the channel estimator for yet another more refined estimate of 2 k2 . This σn process goes on and on, until terminated [xx] intentionally or using a CRC. Dr. Mohamed Khalid Nezami © 2003
11-16
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
1 0.8
( )
) sign λ(q i
0.6 0.4
⎛ λ(q ) ⎞ tanh ⎜⎜ i ⎟⎟ ⎝ 2 ⎠
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
λ
(q ) i
Figure 11- 13: The Output of the Nonlinearity Used in the Feedback of the Iterative Channel
Estimate. 11.2.5 Extrinsic Information-Based Turbo Coded Signal Carrier and symbol Tracking Another method for synchronizing Turbo coded signal is based on the use of extrinsic information supplied by the decoder [12,16,17]. This is based on the fact that when there is carrier phase present the effective result is a reduction of the signal power at the input of the iterative MAP decoder used to implement the turbo decoder. Traditionally, the extrinsic information is only used as the a priori information for the next iteration. Here, however, we also use it to aid synchronization, as in data/decision-aided synchronization. In the data-aided case, the system needs to increase redundancy in the form of a long preamble or, in the decision-aided case, the decision information may be unreliable until synchronization is performed. Using extrinsic information avoids such problems. Intuitively, if the synchronization joins the iteration loops of the turbo decoding, the quality of the synchronization and the decoding will both be improved if aided by this increasingly reliable extrinsic information. In figure 11-15, the histogram of extrinsic information as a function of the decoder iteration number [xx] is used to illustrate its statistical character more clearly. The same phenomenon is also experienced when the signal is gradually tuned out of carrier phase offsets. This can be exploited as an error signal that controls a closed loop to track the incoming signal. Carrier phase offset presence also, in turn, reduces the power of the decoder extrinsic values Lle 21,k shown in Figure 11-16, where the notations mean, the kth bit of the lth frame. This allows the use of the estimated power of the decoder extrinsic values Lle 21,k as an error signal indicating the magnitude and direction of the carrier offset. This is used in conjunction with NCO to track and compensate the carrier phase offset as shown in Figure 11-16 .
Dr. Mohamed Khalid Nezami © 2003
11-17
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The scheme in Figure 11- 16 works iteratively and recursively as follows. First, initial carrier synchronization parameters are obtained using a CW or dot pattern preamble with the aid of a DFT feedforward loop. Next, the received sequence is corrected using the estimate obtained from the phase estimator made with the aid of the extrinsic information of the last iteration from the turbo decoder. The corrected sequence is then sent to the demodulator and the decoding blocks to produce the extrinsic information for use in the next iteration both for decoding and phase estimation, and so on until sufficient iterations have been carried out and the final decision can be made. Hence the number of iterations is the same for the phase estimation and the decoding, which are therefore combined.
Histogram
Last iteration
Iteration 5
Iteration 1
-1
1
0
2
3
Normalized Extrinsic Information
Figure 11- 14: Histograms of Extrinsic Information after Turbo decoding Iteration #1, 3 and 7.
To reduce the implementation complexity, we adopt the following simplified measure for the power of the decoder’s extrinsic values, 1 N −1 l (11-24) ∑ Le 21,k N k =0 is the extrinsic value of the kth bit of the l th frame computed by decoder 2 (see Figure M 1l =
where Lle 21,k
11-16) on the final decoder iteration. So the carrier phase offset can be compensated for by adjusting θˆ l to maximize M 1l in Equation (11-24). The following stochastic gradient algorithm is used to implement a phase lock loop. That is
{ (
)
}
θˆ l = θˆ l −1 + µ sign θˆ l − 2 − θˆ l −1 (M 1l − M 1l −1 )
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(
)
where 0 < µ < 1 is the update step size. The term sign θˆ l − 2 − θˆ l −1 is necessary to direct the carrier phase estimates in the right direction. The algorithm in Equation (11-825 was used with SNR=Eb/No=2dB for BPSK using µ =0.07, showing a convergence of the phase estimates in less than 20 symbols [17].
digital tuner
IF
ADC
DDC
Timing Estimation
εT
ˆ
e jθ
Interpolation
Extrinsic aided tracking k
Phase Estimation
L1e 21 fixed NCO
DFT -based acquisition
Acquisition Loop
DEINT
INT
DEC1
aˆ
DEC2
INT
DEMUX
DEINT
Turbo Decoder
Figure 11- 15: Carrier Tracking Schemes for Turbo Coded Waveforms For the symbol timing, a similar approach can be used for estimation and correction. Figure 1117 shows an example of symbol timing correction with turbo-coded signals [16]. This system is based on the principle of using two sets of samples separated by a symbol (T) that are fed independently to the turbo decoder, which performs the standard decoding procedure for one iteration. After that, the soft decision bits generated for both sets of samples are squared and then subtracted, thus forming a metric that measures the timing error. The error is then scaled using the scaling function shown in Figure 11-17. This scaling is weighted to be in the range of − 0.5 ≤ εT ≤ +0.5 , which is then fed to a Farrow interpolator filter for correction. The new corrected samples may be used for data detection or for improvement of the timing estimation in the next iteration. e (kT )
Scaling e (kT )
+
-
εT = +0.5 εT
0
( )2
()
2
εT = − 0.5
εT Farrow Interpolator Dr. Mohamed Khalid Nezami © 2003
z −1
Turbo Decoder
dˆ 11-19
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Figure 11- 16: Turbo Coded signal Soft Decision-aided Symbol Timing Synchronization.
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11.3 Continuous Phase Modulated Signals Continuous phase modulated signals (CPM) have a very efficient spectrum [3,4,6,21] and are able to work at low SNR due to the inherent coding gain obtained from the memory used in the modulation process. The mathematical description of CPM signal is given by
s (t , α) =
2E s cos(2πf c t + φ (t , α) + θ 0 ) Ts
(11-26)
where E s = A 2 2Ts is the symbol energy, and A is the amplitude of the carrier, α is the M-ary symbol alphabet to be transmitted, Ts is the symbol duration and f c is the carrier frequency, and θ 0 is the arbitrary carrier phase. The information carrying phase component in (11-26) is given by: +∞
φ (t , α) = 2πh ∑ α i q (t − iTs )
(11-27)
i = −∞
where h is the modulation index. The alphabets α for M-ary signals take the values of α ∈ {± 1,±3,... ± ( M − 1)} , and q(t ) is the value of the phase shaping function. For 4-ary CPM [21], the mapping of the sequence {α} from binary symbols to quaternary CPM symbols is done by the following [, {00}→ α =+3, {11}→ α =-3, {01}→ α =+1, {10}→ α =-1. The spectral characteristics of the full response quaternary CPM signals [ is shaped using the phase-shaping function given by,
⎧t / 2T , 0 ≤ t ≤ Ts ⎪ q(t ) = ⎨ 0, t ≤ 0 ⎪1 / 2 , t ≥ T s ⎩
(11-28)
Substituting (11-28) into (11-27), the shaped modulated phase signals in (11-26), over the period of nTs ≤ t ≤ ( n + 1)Ts is described by
where and
φ (t , α n ) = θ (t , α n ) + θ n
(11-29)
θ (t , α n ) = 2πhnα n q(t − nTs )
(11-30)
θ n = θ n −1 + πhn α n
(11-31)
Equation (11-31) is used to drive the trellis for the quaternary CPM signals, 4-ary CPM and MSK . Notice that Equation (11-31) computes the current symbol by accumulating the previous phase value adjusted by an amount of phase that is proportional to the current modulation index ( hn ) current alphabet α n to be transmitted. This means that the current symbol phase starts where the last waveform left off and ramps the phase smoothly by a rate that is proportional Dr. Mohamed Khalid Nezami © 2003
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(slope) to the current symbol and the current modulation index. Hence, the name “continuous phase modulation“ because there are no abrupt phase discontinuities at the symbol transitions that result in spectral re-growth. And thus the spectrum occupied by this signal is much narrower than that of linear modulations. The memory introduced into the phase modulation process can be exploited by a Viterbi decoder to improve the BER performance and yield a coding gain. Minimum shift keying (MSK) can be implemented using a single CPM signal with hn = hn −1 = 8 / 16 (i.e., h=1/2). The use of MSK preamble dot pattern {+1+1-1-1….} or {+11+1-1…} is constructed by observing the full-response CPM signal over two symbols nT ≤ t < (n _ + 1)T , with the MSK alphabet of α ∈ {± 1}, phase modulations can be represented in discrete time domain by
φ ( n, α n ) = θ n −1 + πhnα n
(11-32)
where α ∈ {± 1} and hn = 1 / 2 . As a result, the phase Trellis (phase trajectory in time) can be constructed using the phase states given by, ⎧ 2π ⎫ (11-33) θ n ∈ ⎨i ⎬ mod(2π ) i = 0,1,....i < P − 1 ⎩ P⎭ With the modulation index given by hn =
2k i , where k i and P are integers. For MSK with P
hn = 1 / 2 and P = 4 , the number of states in the trellis representing the phase trace (Trellis) is given by, ⎧ π⎫ θ n ∈ ⎨i ⎬
(11-34)
⎩ 2⎭
where i = 0,1,2,3 , or equivalently,
θ n ∈ {0, π 2 , 2π 2 , 3π 2}
(11-35)
This implies that the maximum phase change between any two trellis states is given by θ n − θ n −1 = π 2 . In general, with MSK at any point in the trellis, the next state is calculated from (11-32) by
θ n = θ n −1 ± π
1 2
(11-36)
Next we use (11-36) to construct CPM preamble that are purposely designed with characteristics that enable us to extract carrier and time offsets, such as periodical dot patterns and other periodical trellis patterns.
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11.3.1 Acquiring CPM Signals Using MSK Preamble Figure 11-18 shows one possible implementation of DFT based acquisition and decision aided tracking of CPM signals. Here the resultant digital IF is obtained after down converting the RF carrier. This IF is sampled and then translated using a digital tuner to baseband (see Figure 1-18). Since the A/D is running at a frequency that does not yield an integer number of samples per symbols, Farrow interpolator filters are used to resample the baseband samples to yield an integer ratio of samples to symbols. The same interpolator can also be used to keep track of the sampling instances by finely adjusting the interpolation instance 21,22,23,24,25. The resampled samples are then processed using spectral algorithms that estimate the initial large frequency and phase offsets, which are then fed forward to a numerical controlled oscillator (NCO) to remove carrier phase and frequency offsets. The signal is then fed into the decoder, which supplies a tentative decision that is used to remove modulations from a delayed sample of the input signal. This is then used as a carrier phase error signal that is filtered and then fed back to the NCO to track the received carrier. The decoder is also responsible for supplying the symbol timing error signal that is fed back to the interpolator to resample the baseband at an optimal timing instance.
cos(2π ( f c + ∆f )t + φ (t ) + θc (t ))
Timing Estimation
εT RF
IF
ADC ADC DDC
z k = ak e jθ
Baseband Interpolation DSP
zz
∆fˆ
fixed NCO LO
DFT-based acquisition
cos(2πf LO t + θ LO (t ) )
e
decoder Turbo Decoder Decoder −1
aˆ ∗k−1
NCO
θˆ ⎛ ⎞ ⎜ − j 2 π ∆ˆf +θˆ ⎟ ⎜ f s k ⎟⎠ ⎝
k
k
LF
Figure 11- 17: DFT-based Acquisition and Decision-aided MLE Tracking Principle.
To view the trellis of MSK preamble dot symbol sequence of 101010… or 11001100. (sentence fragment_ The dot Binary pattern is first binary symbol that is changed to “+/-1” MSK symbols. As an example, for the sequence of 11001100.., and assuming that we started the phase trellis at θ n −1 = 0 , the phase trellis is computed using (11-36),
1 2
θ n = θ n −1 + π α n = 0 + π 2 (+ 1) = π 2
(11-37)
The next symbol being the same, or “+1”, will produce θ n = π 2 + π 2 (+ 1) = π . The next two symbols, which are “-1” and another “-1”, both have the opposite polarity of the previous two
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symbols in the preamble and, as a result, the current phase state θ n = π will be used again to create a third trellis state given by
θ n = π + π 2 (− 1) = π 2
(11-38)
Finally, with the last symbol in the sequence being another “–1”, the trellis phase state again decrements by another π 2 to arrive back at the state θ n = π 2 + (− 1)π 2 = 0 . This is where the phase trace (Trellis) had started. With the sequence 1100 repeating for the whole duration of the preamble, the phase trellis again travels two states up and then comes down another two states, thus producing a triangular phase trellis path between θ 1 = 0o and θ 4 = 180o as shown in Figure 11-19(bottom graph). Since it took four symbols to make one trellis cycle, the waveform resulting from 1100…MSK preamble in [22] is basically a periodic wave having a frequency of one-fourth of the symbol rate or a repetition rate of 4 symbol (8 bits for quaternary CPM) duration. Substituting the phase symbols in (11-36), the modulations will yield a baseband real and imaginary signals that have unique periodicity as illustrated in Figure 11-19.
Figure 11-19 shows the time domain representation of the real and imaginary parts of the complex waveform MSK preamble, and its corresponding trellis phase representation. Notice that the real part of the preamble waveform is a sine wave with a frequency equal to one fourth of the symbol rate being sent. The imaginary part now is a full-rectified wave with a frequency of half of the symbol rate, which also peaks whenever the real part crosses zero, hence, keeping the preamble signal having a constant envelope.
Real Part
2
0
-2
0
10
20
30
40
0
10
20
30
40
0
2
4
6
8
Imag Part
2
60
70
80
90
100
50 60 samples
70
80
90
100
10 symbols
14
16
18
20
0
-2 200 Phase
50 samples
100
0
12
Figure 11- 18: MSK Preamble
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To view the preamble characteristics in the frequency domain for synchronization purposes, we utilize the Fourier transform and Fourier series representation of the preamble time sequence computed using Fast Fourier Transform (FFT) methods [25]. Figure 11-20 shows the Discrete Fourier Transform (DFT) of the preamble sequence of 1100 shown in Figure 11-19 for the rate of 9600 symbols/second. It can be seen that, as a result of the use of the 1100 alternating sequence, several strong tones are present. The dominant one is at DC (at the carrier itself), and the others are at f = ± n 4Ts , with n=0, 1,2,3,…. Fortunately, the spectral peaks at f = ±0 and f = ± 1 4Ts (on both sides of the carrier offset by one-fourth of the symbol rate) have most of the energy (see figure 11-20) . For synchronization purposes, therefore, only these components are used from which estimates of carrier and timing initial offsets are extracted. For instance, for a carrier offset of 20 degrees, Figure 11-21 shows the constellation resulting from sending this preamble when preceding a random CPM message. The tilt of the preamble distinct pattern (right graph) is now a direct indication of the misalignment between the local oscillator phase and that of the incoming received carrier. This is the principle of carrier acquisition using MSK preamble dot pattern at the start of CPM messages. In the coming section, we illustrate several algorithms to extract and track these offsets. 0.8
Amplitude
0.6
Rs/4
Rs/4
0.4 0.2 0 -8000
-6000
-4000
-2000
0
2000
4000
6000
8000
200
Phase
100
90o
0o 0
0o
-100 -200 -8000
-6000
-4000
-2000
0 Hz
2000
4000
6000
8000
Figure 11- 19: Preamble Spectrum without Frequency Offset for 28000SPS CPM Waveform.
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1
1
0.5
0.5
preamble 0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
preamble with 20 degrees shift
-1
-0.5
0
0.5
1
Figure 11- 20: Constellation of Constant Envelope Preamble before and after 20 degrees of Carrier Phase Shift.
11.3.2 Principle of DFT-Based MSK Preamble Acquisition The process of extracting initial synchronization parameters from an alternating MSK preamble sequence such as the binary sequence of 1100.. or 1010.. is carried out using the DFT spectrum estimation principle. For any periodic waveform, such as the tone created using the alternating sequence using the binary sequence given by x n =110011, The Discrete Fourier Transform (DFT) is given by N −1
DFT {x n } = ∑ x n exp( j n =0
2πnt ) Tp
(11-39)
where n=0,1,…N-1, and T p is the preamble repetition period. The DFT of the alternating preamble strip at the receiver is basically a summation of n-periodic sine waves with both positive and negative frequencies including a DC term at n=0. Since the total power of this series representation is concentrated mainly in the first three bins (n=0, ± 1), while the rest are filtered or considered noise, Equation (11-39) can be expanded to include only these three terms,
DFT {x n } = X −1 exp(− j
2πt 2πt ) + X 0 exp(0) + X +1 exp(+ j ) Tp Tp
(11-40)
where X −1 , X −0 , and X +1 are the Fourier amplitude coefficients of the sequence at DC and f = ± 1 T p . With carrier phase +/- θ , symbol timing error τ , and no frequency offset for now, the series representation of Equation (11-40) becomes, Dr. Mohamed Khalid Nezami © 2003
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⎡ 2π (t ± τ ) ⎤ ⎡ 2π (t ± τ ) ⎤ DFT {s (t )} = X −1 exp( − j ⎢ ± θ ⎥) ± θ ⎥ ) + X 0 exp( ± jθ ) + X +1 exp( + j ⎢ ⎢⎣ T p ⎥⎦ ⎢⎣ T p ⎥⎦ 41)
(11-
Once the preamble sequence containing an alternating sequence with phase and timing offset represented by Equation (11-41) is sampled and filtered, the samples are passed to the DFT algorithm (Figure 11-18) to obtain the spectrum shown in Figure 11-20 as it contains information on the carrier phase, frequency offset and symbol timing error. It is evident by inspection of Figure 11-20 and from the well-established properties of the DFT transform in (11-41) that carrier phase alone can be estimated from the center bin (tone). The carrier phase offset is estimated by
θˆ = arg{k 0 }
(11-42)
where arg{k 0 } is the angle or phase associated with the center bin (DC bin) after conversion to baseband . Symbol timing is extracted by finding the angle associated with either the lower tone (left) arg{k −1 } , or the upper tone (right) arg{k +1 } bins after removing the carrier phase offset that was estimated in Equation (1-42). That is angle{k +1 } − θˆ (11-43) T 2π where the multiplication by 4 used since the preamble period is one fourth of the actual period of the transmitted symbols (the count of 1100 bits). The symbol rate also is estimated by subtracting the frequency corresponding to either of the two side tones from the frequency of the center and then multiplied by 4. That is
τˆ
=4
Rˆ s = 4( fˆ±1 − fˆ0 )
(11-44)
where fˆ±1 is the estimated frequency that corresponds to the bins k ±1 and fˆ0 is the estimated frequency that corresponds to the bin k 0 . With the presence of a frequency offset ∆f in the received preamble signal defined in (1-40), all three spectral tones will be shifted in frequency by a magnitude and direction that are proportional to ∆f . As a result, the frequency-offset estimation can be obtained in several ways. One way is to estimate the frequency corresponding to the center tone, k 0 . That is
∆fˆ = k 0 f s N DFT
(11-41)
where f s is the sampling frequency of the samples processed by the DFT algorithm, and N DFT is the number of DFT bins. The same information can also be obtained from the location of
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either the shifted left or right tones and their displacement from their nominal location at frequencies of ± 1 4T . That is
∆fˆ = 1 4T − k ±1 f s N DFT
(11-42)
11.3.3 Interpreting the DFT Complex Preamble Spectrum The complex DFT algorithm in Equation (11-39) calculates bins from bin 0 to bin ( N DFT − 1) . As an example, with N DFT =1024, the bins are 0 through 1023 . The bins 0 to bin (N DFT 2 − 1) correspond to the positive frequencies, while the bins of N DFT 2 to bin N correspond to the negative frequencies. Assuming a sampling frequency of eight times the symbol rate, f s = 8Rs , for instance with a rate of Rs =28000 symbols/sec without frequency offset presence, the three preamble tones are expected at 0Hz, and +/- 7000 Hz as shown in Figure 11-20. After computing the 1024 complex DFT, the bins that correspond to these frequencies will be k −1 = 481 (left tone), k 0 = 513 (center tone), and k +1 = 545 (right tone). To compute the frequencies corresponding to these bins, the left tone is processed by,
k−1 = 481 − (1024 2 + 1) = −32
(11-43)
which corresponds to the frequency of
fˆL = −32 f s 1024 = −7000Hz
(11-44)
the center tone is at k 0 = 513 − (1024 2 + 1) = 0
(11-45)
and that corresponds to fˆc = 0 Hz : then the right hand side tone is
k+1 = 545 − (1024 2 + 1) = +32
(11-46)
which corresponds to fˆr = 32 f s 1024 = 7000Hz . 11.3.4 DFT Preamble SNR Performance Using
fs N DFT
as the DFT bin resolution in Hz, the signal-to-noise ratio of the DFT input signal
in term of the receiver’s
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E f /2 ⎛S⎞ = b − 10 log( s ) ⎜ ⎟ Rs ⎝ N ⎠ DFT N o
(11-47)
8R / 2 fs / 2 ) = −10 log( s ) = −6dB , then Rs Rs S E C as follow, = b − 6dB . This result can also be represented in term of No N No E C C = = b + 10 log(Rb ) (11-48) N o kT N o The SNR for the DFT obtained tones (not the input MSK signal) is given by
With a sampling rate of 8 samples/symbols − 10 log(
C ⎛ f ⎞ ⎛S⎞ = − 10 log⎜ s ⎟ ⎜ ⎟ ⎝ N ⎠ DFT kT ⎝ 1024 ⎠
(11-49)
As an example, for a symbol rate of 2400sps and
Eb =0dB, No
C = 0 + 10 log(2 x 2400 ) = 36.81dB , resulting in DFT SNR given by, kT
8 x 2400 ⎛S⎞ ) = 24.08dB = 36.81 − 10 log( ⎜ ⎟ 1024 ⎝ N ⎠ DFT and for 2400sps and
(11-50)
Eb ⎛S⎞ = 34.08dB =10dB, ⎜ ⎟ No ⎝ N ⎠ DFT
Similarly for 28000sps and
Eb C =0dB, = 0 + 10 log(2 x 2800 ) = 47.481dB No kT
8 x 28000 ⎛S⎞ = 47.48 − 10 log( ) = 24.08dB ⎜ ⎟ 1024 ⎝ N ⎠ DFT and for 28000sps and
(11-51)
Eb ⎛S⎞ = 34.08dB =10dB, ⎜ ⎟ No ⎝ N ⎠ DFT
Figure 11-22 shows the DFT spectrum of 2400sps MSK preamble using two different SNR E E values. The left figure shows the case for b =0dB and the left for the b =10 dB. Clearly the No No obtained DFT SNR for the center tone is close to the expected values computed by using
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Equation (11-50) and (11-51). Figure 11-23 shows the same results being obtained for the higher data rate of 28000sps. 5
5
0
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
-40
-45
-45
-50
-1000
0
-50
1000
-1000
0
1000
0
5000
Figure 11- 21: DFT for 2400SPS Preamble with Eb N o =0dB and 10dB. 5
5
0
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
-40
-45
-45
-50
-5000
0
-50
5000
-5000
Figure 11- 22: DFT for 28000SPS Preamble with Eb N o =0dB and Eb N o =10dB.
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11.4 Preamble Acquisition Estimates Accuracy and Estimates Lower Bounds The following analysis is based on CW preamble. However for the three MSK preamble tones, the analysis is valid for the center tone. For the side tones, the SNR will have a loss of 2-3 dB depending on the DFT bin resolution value. Defining the complex baseband CW preamble signal used for acquisition by
x ( n) = Ae
−
j 2πk o n N DFT
(11-52)
where A is the magnitude of the tone, with the number of complex samples made equal to the DFT bins N DFT and the center DFT bin of k 0 . If the preamble center tone were to coincide with a DFT bin, then the energy associated with the bin extracted using the DFT from the preamble samples of x(n) is given by
X (k ) =
N DFT −1
∑ n=0
⎡ j 2πk o n ⎤ − j 2πkn ⎢ Ae N DFT ⎥e N DFT = N DFT A ⎢⎣ ⎥⎦ −
(11-53)
j 2π ( k o − k )n N DFT
where X ( k ) is the DFT of x(n) , and e = 1 indicating that the tone is coincident exactly with a DFT bin. The SNR of the preamble tone corresponding to bin k 0 , is given by,
where N 0 = 2σ 2f
⎛ X 2 ( k = m) ⎞ ⎟ (11-54) SNR DFT = ⎜ 2 ⎜ ⎟ 2 σ f ⎝ ⎠ is the power of the noise in the DFT domain. Substituting the value of the bin
magnitude from Equation (11-54), the DFT SNR is then given by ⎛ ( N A) SNR DFT = ⎜⎜ DFT 2 ⎝ 2 N DFT σ n
2
⎞ ⎟ = N DFT SNR ⎟ ⎠
(11-55)
Where 2σ n2 = 2 Nσ 2f is the equivalency of the noise power in time-sampled domain to that obtained using the DFT frequency domain. A second case would be when the preamble tone does not exactly coincide with a DFT bin, but has a frequency due to carrier offset that is located somewhere between two bins. This shows a worst-case scenario because the tone is in the middle of two adjacent bins. Substituting the bin location, which is an offset of the current bin by a half bin, into Equation (11-53), the tone magnitude is given by
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⎡ j 2π (Nk0 +0.5 )n ⎤ − Nj 2πkn 1 − e jπ DFT X (k ) = ∑ ⎢ Ae ⎥e DFT = A 1 − e jπ N DFT n =0 ⎢ ⎥⎦ ⎣ N −1
(11-56)
As a result, the DFT SNR using Equation (11-56) is given by SNR DFT =
2 ⎡ ⎛ π N DFT ⎢1 − cos⎜⎜ ⎝ N DFT ⎣
⎞⎤ ⎟⎟⎥ ⎠⎦
(11-57)
SNR
Clearly, the SNR in the DFT acquisition is dependent on the location of the preamble tone with respect to two adjacent tones. This dependency is represented by the magnitude of the DFT frequency bin response and is given by
X k (∆f f s ) =
sin N DFT π (∆f / f s − k / N DFT ) sin π (∆f / f s − k / N DFT )
(11-58)
where ∆f is the frequency of the center tone that indicates the carrier frequency offset. Notice that Equation (11-58) is a sin(x)/x function with the first side lobe being 13 dB below the peak and that the first zero is located at f / f s = (k ± 1.5) / N . For two adjacent bins, the frequency response crossover at ∆f / f s = (k ± 0.5) / N , and the peak to crossover point is –4dB. Thus a signal with frequency offset (Doppler) half way between any two adjacent bins would suffer a loss of –4dB in the SNR. One way to circumvent this problem is by using zero padding. This will be detailed later in the chapter. Figure 11-24 shows Monte-Carlo simulations of a number of DFT estimates of the MSK preamble, highlighting the SNR loss associated with the two side bands. This is close to 4 dB as predicted by (11-58).
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
5
0
-5
dB
-10
-15
-20
-25 -8000
-6000
-4000
-2000
0
2000
4000
6000
8000
Frequency- Hz Figure 11- 23: Preamble Side Tone Variance with Eb N o =+2dB and Rs=28000SPS.
The accuracy of tone estimation is lower bounded by the Cramer-Rao bound [xx] that is given by,
σ 2f T 2 =
3
π N 2
3 DFT
SNR
[Hz2]
(11-59)
and for the tone phase estimates is given by
σ θ2 =
1 [rad2] 2 N DFT E s N o
(11-60)
The value of E s N o in (11-59) is related to the SNR by SNR = E s N o + 10 log(Rs ) − 10 log(B )
(11-61)
where (B) is the noise bandwidth. As an example, for B=25khz channels with E s N o = 8dB and symbol rate of Rs = 28000sps, the calculated SNR using Equation (11-61) is 8.5dB. The CRB 11
for the preamble center tone frequency estimate using 1024 DFT bins is σ 2f / T 2 = 4x10 - , or a standard deviation of σ f = 2Hz. The phase CRB is calculated using Equation (11-60) and is Dr. Mohamed Khalid Nezami © 2003
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σ θ = 1o . Notice that for either one of the MSK preamble side tones, the bound is made worse since both have lower energy than the center tone, which is close to 1 dB less SNR. 11.5 Preamble Tone Estimate Refinement Using DFT Bin Interpolation The acquisition peak search algorithm used to estimate all three MSK tones detailed above often runs into cases where the tone does not coincide with a bin number, and hence it’s tone spectral energy will leak into the adjacent bins. As a result, instead of a single spectral peak, there will be multiple spectral lines as illustrated in Figure 11-25. One way to reduce the loss of SNR is to pad the preamble acquisition samples captured with zeros and then perform the DFT. This will be equivalent to frequency domain interpolation, so now the actual carrier frequency offset tone that did not coincide with a DFT bin can be better estimated when the values between bins are now interpolated using the two neighboring bins. Another method for obtaining more accuracy is bin interpolation, as illustrated in Figure 11-25, can also be used in conjunction with the zero padding of time domain samples. Here, instead of taking the bin that corresponds to the maximum energy as an estimate of the tone, a method of interpolating between at least three adjacent bins above a certain threshold is carried out to increase the accuracy of the DFT estimates. The increase in accuracy comes at no additional computational cost because the bins have already been calculated prior. The interpolation technique is then utilized to refine the estimates that correspond to the actual synchronization parameters that are needed for the estimates in 11-41, 11-44, 11-43, 11-42 DFT bin magnitude
Actual offset Estimated offset
Pi +1
Pi threshold
Pi −1
ki −1 ki ki +1
DFT bin #
Figure 11- 24: Preamble Tone Energy Leakage and Peak Location Refinement. As illustrated in Figure 11-25, the interpolation technique is based on the use of the three largest adjacent bins k i −1 , k i and k i + 1 , as illustrated above, that are scattered near the tone location that are above a predetermined threshold as shown in Figure 11-25. Without the use of this interpolation, the estimates would be a half bin at worst . By observing that the left hand side
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bin k i − 1 has a magnitude less than the right hand side adjacent bin k i +1 , that is Pi +1 ≥ Pi −1 . Sentence fragment the difference in amplitude indicates that the actual frequency offset of the tone must be located in the region where k i N DFT T ≥ fˆ ≥ k i −1 N DFT T , where T is the DFT sample period. An adjustment is then applied to tilt the final frequency estimate to the left by the weight of this amplitude difference. That is Pi −1 1 fˆ = (k i − ) (11-62) Pi −1 + Pi N DFT T Likewise, if
Pi +1 ≥ Pi −1 , then the actual frequency of the tone must be in the region of
k i N DFT T ≤ fˆ ≤ k i +1 N DFT T .Hence the actual tone frequency estimate is tilted to the right by an adjustment factor that is proportional to the adjacent bin amplitude differences. That is fˆ =
Pi +1 1 (k i + ) TN DFT Pi +1 + Pi
(11-63)
Once the frequency bins corresponding to the three tones are more accurately estimated using the interpolation given in Equations (11-62) and (11-63), the same process is carried out to adjust their corresponding phase values. For the case with Pi −1 > Pi +1 , the interpolated phase is given by, ⎛ N DFT Pi −1 ⎞ ⎟π ⎟ N 1 P P − + DFT i − 1 i ⎠ ⎝
θˆint = arg{k i −1 } − ⎜⎜
(11-64)
and for the case of Pi −1 < Pi +1 , the interpolated phase value is given by
θˆint = arg{k i +1 } −
⎞ N DFT ⎛ Pi +1 ⎟π ⎜ 1 − ⎟ N DFT − 1 ⎜⎝ Pi +1 + Pi ⎠
(11-65)
11.6 Preamble Tone Estimate Refinement Using DFT Bin Interpolation We illustrate the performance improvement caused by padding the captured preamble samples with zeros before performing the DFT. Assuming that the preamble N-data samples used in (1152) are, x(0)…x(N-1) , by padding the N-samples by N samples with value of zero padded to the end of this preamble N samples, , the DFT of the zero padded 2N sequence results, thus X (k ) =
2 N −1
∑ x(n)e
−
j 2πkn 2N
n =0
Since x(n) is zero for N points, there is no need to include it in the sum above. This leads to
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X (k ' ) = ∑ x(n)e
−
jπkn N
(11-66)
n =0
Which is identical to an N-point DFT. Notice that the summation stops at N-1 and not over the 2N-1 period because the remaining signal samples are zeros. As a result, the bin frequency response is now given by
sin N DFT π (∆f / f s − k / 2 N DFT ) sin π (∆f / f s − k / 2 N DFT )
X k (∆f f s ) =
(11-67)
Here the adjacent DFT spectral bins now crossover at ∆f / f s = (k ± 0.5) / 2 N DFT , causing the adjacent peak tones to crossover at -1 dB instead of -4 dB for the case with no zero padding. Hence for frequency offsets laying in between two adjacent bins ( worst case scenario) the SNR reduction is only 1 dB. Figure 11-26 shows a comparison of the energy of the adjacent bins computed using Equation (11-67) and (11-58). This illustrates the advantages gained by including zero padding for SNR enhancement of detecting the preamble tones. The figure shows that without zero padding, the SNR exhibits the worst-case SNR loss of 4dB, as predicted by Equation (11-58). Zero padding does not change the amplitude and phase of the even numbered bins, but provides new interpolated bins at the odd numbers. By doing so, the DFT resolution did not increase since that depends on the number of data points. The peak bin location now can become easier to process as shown in Figure 11-27. Realize also that if the captured N DFT samples are only 50% of the preamble signal, and the rest is noise. then with 50% zero padding, the SNR loss is 3dB compared to 7dB if zero padding has not been used. no zero padding
zero padding
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
overlap at -4dB
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.9
1
1.1
1.2
0.9
(f/f ) *N s
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1
1.1 (f/f ) *N s
DFT
1.2
DFT
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Figure 11- 25: Effect of Zero Padding on DFT Bin Energies. 5 0
with zero padding no zero padding
-5 -10
dB
-15 -20 -25 -30 -35 -250
-200
-150
-100
-50
0
50
100
150
200
250
Hz
Figure 11- 26: Effect of Zero Padding and Windowing on Rs=2400sps Preamble Tones.
11.7 Preamble Probability of Detection and Probability of False Alarm Now we turn to statistical methods that are used to evaluate the performance of acquiring the preamble synchronization offsets using tone frequency and phase estimation methods using the DFT function. One of the problems in detecting the preamble tone x(n) in the presence of AWGN q ( n) is formulated as follows. Given the set of real and imaginary data points obtained from the DFT of the preamble samples x(n) , a decision has to be made between two statistical hypotheses [24]. H o : x ( n) = q ( n) H 1 : x ( n) = A sin( 2πf p n + θ (n)) + q ( n)
(11-68) (11-69)
The first hypothesis assumes that x(n) has a CW preamble tone signal, consisting of only AWGN, while the second hypothesis assumes the signal contains a CW tone and AWGN added together. The aim of the acquisition algorithm is to find the probability of false detection by evaluating Pfa = p ( D / H o ) and the preamble tone detection probability Pd = p ( D / H 1 ) , where D is the decision of hypothesis H 1 . This is possible by comparing the bin magnitudes to a
threshold r1 such that, H o : X (k ) ≤ r1 for noise only and H 1 : X (m) > r1 for the tone presence. The probability of detecting the tone from the noisy I and Q samples of the DFT with the envelope of the signal being above a fixed threshold r1 is given by ∞
Pd =
∫ p(r )dr
(11-70)
r1
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where r = I 2 + Q 2 Thus p (r ) can be easily shown to be either Rayleigh distributed function when r samples are noise only and no signal is present ( H o ) or Rician distributed when signal is present in the r samples ( H 1 ). Assuming Rayleigh distribution with variance of σ 2 for the rms values of the independent spectral lines (bins) of noise the PDF when no tone is present, but only noise, is given by −r 2
p q (r ) =
r N DFT σ
2
e
2 N DFT σ 2
(11-71)
Using (11-71), the probability of a false alarm is given by ∞
Pfas =
∫p
q
(r )dr = e
−
r 12 2 N DFT σ 2
(11-72)
r1
Solving for the normalized threshold r1 yields the value required for making a decision, that is related to (11-72) and is given by, −
r1 2 N DFT σ 2
= ln( Pfas )
(11-73)
for tone presence, the PDF is Rician and is given by
pcw (r ) =
−
r N DFT σ
2
e
r2 + Xk
2
2σ 2 N DFT
⎛ r Xk ⎞ ⎟ Io ⎜ ⎜ N DFT σ 2 ⎟ ⎝ ⎠
(11-74)
where A is the amplitude of the tone, σ 2 is the variance of the noise which, if obtained from the 2 = Nσ 2 , and I o (.) is the modified Bessel function noise samples of the DFT, will become σ DFT
of zero order. Using (11-74) the probability of detection is then computed by r1
Pdet = 1 − ∫ p cw (r )dr
(11-75)
0
The probability of false alarm is usually specified, so the detection algorithm optimally derives the threshold r1 using Equation (11-73) that satisfy Pd and Pfas defined in Equation (11-59) and (11-75) respectively. Equation (11-72) and (11-74) are plotted in Figure 11-28 and Figure 11- 29 respectively. Indicated on figure 11-28 is the threshold and the areas corresponding to Pd and Pfas for SNR of 0dB, 5dB, and 10dB.
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As an example, with an IF signal that is digitized by f s = 250 Msps , with N DFT = 64 bins, and assuming that specification calls for Pfas = 2.56 x10 −9 and Pdet = 90% ,Equation (11-73) is used to compute the threshold from the given Pfas , that is
Pfas = 2.56 x10
−9
=e
−
r1 2σ 2
,
from which the normalized optimal threshold r1 σ DFT is given by r1 σ DFT = −6.29 . Using this result in Equation (11-72) and Equation (11-75) for p (r ) in conjunction with the specification of Pdet = 90% , the required SNRDFT ≥ 14.5dB as also illustrated in Figure 11-28. Using Equation (11-55) and assuming that the preamble tone that has to be acquired coincides with a DFT bin and that 100% of the samples are data (no zero padding and no noise samples), the SNR = SNRDFT − 10 log( N DFT ) = −3.6dB for Pdet = 90% , which is very good. However, for the worst case, where the captured samples used to evaluate the DFT are only 50% actual preamble tone and the remaining 50% of the samples are noise., the SNR required to achieve Pdet = 90% is given by
SNR = SNRDFT − 10 log( N DFT ) − 10 log(1 − cosπ N DFT ) − 3dB = −0.6dB , which is 3dB deterioration in SNR acquisition. 0.7 No signal, only noise
0.6 0.5
SNR=0dB
p(r)
SNR=5dB
0.4
SNR=10dB
0.3 0.2 0.1 0
0
1
2
3
4
5
r1
r
σ
σ
6
7
8
9
10
Figure 11- 27: The PDF of Permeable Tone Present Versus Only AWG Noise Present for SNR of 0dB, 5dB, and 10dB.
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1 0.8
Pd
0.6 0.4 0.2 0
6
8
10
12 14 DFT-SNR
16
18
Figure 11- 28: Probability of Detection of the Tone in AWGN , Showing a Detection Probability of 90% for a CW tone in SNR=15dB.
11.8 Tracking of Continuous Phase Modulated (CPM) Signals After successful acquisition of the MSK preamble using the prior algorithms, carrier and symbol tracking have to be initiated to maintain tracking and correcting these offsets in order to start performing data decoding [21,22,23,24,25]. Figure 11-30 illustrates a block diagram of the combined acquisition, tracking, and demodulation of the CPM signal. Assuming that the received signal is given by A cos (2π (∆f + f c )t + φ (t ) + θ c (t ) )
(11-76)
where φ (t ) is the CPM phase modulations and θ c (t ) is the varying carrier phase to be tracked. This signal is first down converted to an intermediate frequency (IF) that is low enough to be directly sampled using an A/D converter as shown in Figure 11-30. The resulting digitized IF is then digitally down converted again from the digital IF to baseband using the digital down converter (DDC) in conjunction with a numerically controlled oscillator (NCO). The acquisition algorithm discussed above then adjusts the down converted signal to zero IF by multiplying it Dr. Mohamed Khalid Nezami © 2003
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with e − j (2π∆f kT +θˆ ) , where 2π∆f DFT kT + θˆ is the carrier phase and frequency offset phase estimated using the DFT-based acquisition algorithm described earlier. These estimates are then refined again using the carrier phase tracking adjustment obtained by using the maximum likelihood estimate of phase error estimates obtained by a data decision-aided algorithm as shown in Figure 11-30. DFT
The data decoder used in Figure 11-30 is the maximum likelihood sequence estimation (MLSE) algorithm using the Viterbi decoding algorithm. Since it takes on the order of 20 to 50 tracebacks for a typical Viterbi-based data decoder algorithm to make a bit decision, the data decision obtained after this long delay in the data detector cannot be used in implementing a stable decision-aided carrier-tracking loop. Instead, an alternative method, based on the use of a decision that is made with a shorter trace-back (0, 1 or 2) is used, and it has been verified by several sources that this has sufficient decision accuracy for decision-aided carrier and timing tracking purposes. A single trace-back is sufficient to implement the carrier phase tracking loops shown in Figure 11-30, where the carrier phase error is derived as follow. First, the tracking phase error in Figure 11-30 is based on the use of a buffered symbol sample given by z k −1 = a k −1 e − j (θ k −1 )
(11-77)
where a k is the transmitted symbol (complex value), . Using (11-77) and its corresponding decoder decision complex value aˆ k −1 , the decision directed carrier phase error is given by,
eθ ,k −1 = Im(aˆ k∗−1 z k −1 )
(11-78)
where aˆk∗ −1 is the conjugate of the tentative decoder output symbol decision illustrated in Figure 11-30 (with traceback length of one). Substituting (11-77) in (11-78) ,
(
eθ ,k −1 = Im a k −1 e − j (θ k −1 ) aˆ k∗−1
)
(11-79)
With ak −1aˆk∗ −1 = 1 , and removing the memory indices notation, the phase error signal (11-79) is then reduced to, eθ , k ≈ sin (θ k )
(11-80)
which is the very common phase error signal resulting from linear phase locked loops. The small angle approximation now can be used, thus sin (θ k ) ≈ θ k . This error signal is then used in conjunction with a first order loop filter, shown in Figure 11-31, which filters this signal and adjusts the NCO with the carrier phase error. The design of the loop parameters used in Figure 11-30 is detailed next.
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Re {
AGC-LF
RF
RF section
IF
A/D DDC
AGC loop Farrow filter
X
Tracking Buffer
X
}
ak −1
M LE
X
Im {
Carrier Acquisition Acquisition Buffer
DFT
NCO
z
−1
Carrier tracking loop CR-LF Symbol timing tracking loop
zk −1 Phase error Timing error
ST-LF
Figure 11- 29: CPM DFT-based Acquisition and Decision-aided MLE Tracking Principle.
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}
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11.8.1 CPM Tracking Loop Parameter Design One way to obtain the design parameters for the MLE-based tracking loop filter (LF) shown in Figure 11-30 is to use the linearized carrier recovery feedback loops as shown in Figure 11-31. The loop filter used here has its steady state error converge to zero for both phase and small frequency offset and it can track a changing frequency rate proportionally to the loop bandwidth. The digital loop filter implementation has a direct gain Kp path and an integrated path Ki as shown in Figure 8-6. The system function that describes the ratio of carrier phase input to the tracked estimated phase output in Figure 11-31 is given by, Kd F ( z) N ( z) θˆ( z ) = θ ( z) 1 + Kd F ( z) N ( z)
(11-81)
The value of K d is the phase detector gain (volts/radians). The loop filter function is given by
F (z) =
K p + K i z −1
(11-82)
1− z −1
and the discrete time transfer function of the NCO is given by
N ( z ) = 1− zo−1 K
(11-83)
where K o is the NCO gain (Hz/volts). By equating (11-81) to the classical second order canonical transfer function (3-25); the loop filter parameters Kp and Ki are obtained based on the desired natural loop frequency wn as a function of the loop bandwidth BL and damping ratio ξ is given by, wn =
BL ξ + 1 4ξ
(11-84)
Both proportional and integral constants, as a function of the natural loop frequency and damping ratio, are given by, 2ζwn Twn2 Ki = + (11-85) 2K d Ko K d Ko and 2ζwn Twn2 − (11-86) 2Kd Ko Kd Ko where T is the symbol rate (i.e., loop iteration rate). With the residual error frequency resulting from the DFT acquisition loop (Figure 11-30), the tracking loop acquisition time is approximated by (3-18), Kp =
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(
t acq ≈ 4 ∆f
2
BL3
)
(11-87)
After the loop reduces this offset to a negligible value, the loop phase lock time is also approximated by (3-19), t lock = 1.3 B L
(11-89)
After the loop locks, the phase error variance of the loop is estimated by (3-21), BT σ θ2 = L [rad2] (11-90) Es N o where E s N o is the symbol energy to noise spectral density. The effective loop signal-to- noise ratio, SNRL , is related to the loop input signal bandwidth and SNR by, SNR L = SNRi + 10 log10 ( Bi 2 BL )
(11-91)
where SNRi and Bi are the loop input signal SNR and noise bandwidth. Equations (11-81) through (11-91) are used to design the carrier tracking loop parameters in Figure 11-31. The loop parameters are then fine-tuned using computer simulations. A narrow loop bandwidth in (11-90) is usually desirable for noise immunity and low phase variance. However, it cannot be arbitrarily narrow for a reasonable acquisition time (11-87). For Bi =25 kHz channel bandwidth, 1/T=Rs=28000sps, specifying a tracking phase error of σ θ = 2 o based on worst BER degradation using (5-24), the required loop bandwidth is then calculated using (11-90). This results in BLT = 0.0086 , or 241 Hz for a loop that has an update rate equal to the symbol rate used of 28000 Hz. With DFT acquisition using 1024 points, as discussed previously in (11-28), the maximum residual carrier error from the acquisition loop is related to the DFT bin resolution error, which is assumed as a half bin residual error, ∆f DFT = (28000)(8)(1 / 1024) / 2 = 109.3Hz . This is well within the pulling range of the tracking loop since its loop bandwidth is 241Hz.
Phase detecto r model
θ (z )
+
NCO model
Loop Filter
F ( z)
Kd
N ( z)
-
z Dr. Mohamed Khalid Nezami © 2003
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Figure 11- 30: Linear Model for the MLE Carrier Tracking. 11.9 Acquiring and Tracking Trellis-Coded MPSK Signals Trellis coded signals are also another class of bandwidth efficient class of waveforms. In particular, the 8PSK and 16PSK are often used as bandwidth efficient modulation schemes [26,27]. It has also become more popular when combined with Trellis coding (TC-8PSK and TC-16PSK). In this section, we present two examples of carrier and symbol timing synchronization used with TC-8PSK signals. Figure 11-32 shows one example of 8PSK trelliscoded signal. Here a pair of information bits u k 1 and u k 2 are passed to the encoder, where the most significant bit u k 1 is unchanged, while the least significant bit u k 2 is encoded using a rate ½ convolutional encoder. This results in the coded bits c k 2 and c k 3 . The coded three bits ck1 , c k 2 , and c k 3 are then mapped to an 8-PSK signal constellation as shown in Figure 11-33. Notice that the most significant coded bit ck1 controls the selection of either upper or lower half plane as shown in Figure 11-32. This fact is used at the demodulator to estimate the most significant bit, while a Viterbi decoder is used to estimate the least significant information bit. The Trellis coding produces memory into the signal that is also used by the Viterbi decoder to improve the BER performance. Figure 11-34 shows a simulation performance of R=2/3 TC-8PSK BER performance comparison with uncoded 8PSK signals. Clearly the trellis coding has resulted in a 5.5dB of coding gain at BER of 10-2.
0_11 ck 1
uk1
0_10
0_01 ck 1 , c k 2 , c k 3
1_00
0_00
ck 2 uk 2
Convolutional Encoder
ck 3
1_10
1_01 1_11 8PSK modulator
Figure 11- 31: Rate 2/3 Trellis Encoder/modulator.
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1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Figure 11- 32: TC-8PSK Constellation Diagram at Eb/N0=15dB.
10
-1
BER
10
2/3-TC8PSK
0
10
-2
BER of uncoded 8PSK
BER of TC8PSK K=7, R=2/3, 10
-3
0
1
2
3
4
5 6 E b /No -dB
7
8
9
10
Figure 11- 33: Bit Error Rate Performance of Uncoded 8PSK Signals Versus TC-8PSK Signal. Figure 11-35 illustrates one scheme for both symbol timing and carrier tracking. The decision aided carrier phase error is generated by
{ }
eθ (k ) = Im s k dˆ k∗
(11-92)
where dˆ k = aˆ k + jbˆk is the decision derived at an earlier time than the final trellis decoded data symbol. The real and imaginary parts of the symbol decision are given by [26], Dr. Mohamed Khalid Nezami © 2003
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1 ⎡ (1 − z k )⎤⎥ aˆ k = sign(Re{s k })⎢ z k + 2 ⎣ ⎦
(11-93)
1 ⎡ (1 + z k )⎤⎥ bˆk = sign(Im{s k })⎢− z k + 2 ⎣ ⎦
(11-94)
and
where z k = sign( Re{s k } − Im{s k } ) is the sample corresponding to the symbol being decoded. If a first-order carrier loop is used, the DPLL equation is given by
θˆk +1 = θˆk − γ θ eθ (k )
(11-95)
where γ θ is the iterative loop coefficient less than unity. The timing error signal is given by the real part of the multiplication of the lead-lag difference and the conjugated decision. That is
{
eτ (k ) = Re (s k +1 / 2 − s k −1 / 2 )dˆ k∗
}
(11-96)
Likewise, if a first-order timing loop is used, the DPLL equation is given by
τ k +1 = τ k − γ τ eτ ( k )
(11-97)
where γ τ is the iterative loop coefficient of less than unity. Both timing and carrier phase tracking loops are designed using the DPLL feedback principles for CPM signals.
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1/2T Early-Late
sampling/ interpolator
On-time ˆ
e − jθ k Received IF signal
sk
Make complex
Acquisition loop
NCO
decisions
Loop Filter
dˆk
( )∗
Im(
) Trellis decoder
sampling/ interpolator
Received data
dˆk∗ On-time 1/2T Early-Late
Make complex
derivative
NCO
Re(
)
Loop Filter
Figure 11- 34: 8PSK Carrier and Symbol Timing Detector. Dr. Mohamed Khalid Nezami © 2003
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Another approach used to track the TC-8PSK signals is shown in Figure 11-36. This algorithm is based on the tentative decision carrier tracking principle detailed earlier [27]. Here the compromise has to be made between confidences in the tentative decision made with the delay (trace back) versus the delay introduced into the loop, which highly impacts the stability of the loop. In most cases the delay or a trace back of one is sufficient [18]. The Viterbi decoder in Figure 11-29 takes the correlation values between the input samples and a set of eight constellation points. The carrier phase error is then updated using the following relationship,
θˆk +1 = θˆk − γ θ eθ (k − D) and the carrier error is computed by,
{
eθ ( k − D ) = Im c k∗− D z ( k − D )e − jθ ( k − D )
}
where z (k − D)e − jθ ( k − D ) is the input sample corresponding to the symbol at t=D associated with the surviving path in the add-compare select block of the Viterbi decoder. The tentative decision delay t=D, is much less than the full traceback for normal decisions. Typical values for the tentative delay are D=0, D=1, while the full traceback depth is usually five times the constraint length of the Viterbi encoder for a constraint length of K=7, as illustrated in Figure 11-34. Figure 11-37 illustrates a non-data aided (NDA) TC-8PSK synchronization and demodulation system based on the NDA system detailed in Figure 8-10. Here the 8PSK modulations are removed using the modulated samples to the power eight. This system is not appropriate for low SNR due to the noise enhancement using the 8-times multiplier. Nevertheless, in some applications that are not power limited this algorithm may become a viable choice.
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( zi + jz q )e − j (2πkT∆f +θ ) ˆ ˆ
AFC
λ0
∆fˆ IF
Timing recovery
Decoded data
Add compare select
Branch metric generator
Decoding
λ8 NCO Phase synchronization
θˆ Loop Filter
Most likelihood state for the tentative decision at t=D.
dˆk −D
Most likelihood state for the decision with full traceback
dˆk −tb
Figure 11- 35: Demodulator and the Trellis Decoder for R=2/3 8PSK. Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
LPF
1 N
N −1
∑
kT
k =0
kNT
zk
DFT Acquisition
NCO
Loop filter
Kp
θˆk + 2π∆fˆkT
8 tan
−1
x
()
Ki
+
x
+ z−
kNT LPF
Trellis Decoder
Received data
1
+
1 N
N −1
∑
kT
z−
k
k =0
1
NCO
Symbol Timing Estimation
Figure 11- 36: NDA TC-8PSK Synchronization System.
Dr. Mohamed Khalid Nezami © 2003
11-53
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
11.10 References:
1.
Terrance J. Hill, “An enhanced constant envelope, interoperable shaped offset QPSK (SOQPSK) waveform for improved spectral efficiency,” Milcom 2001.
2.
Frank Amoroso, “Fractional Out-of-band power formulas for BPSK, QPSK and MSK,.” Microwave Journal, June 2000.
3.
Stephen G. Wilson and Richard C. Gaus, "Power Spectra of Multi-h Phase Codes," IEEE Trans. on Comm., Vol. COM-29, No. 3, pp. 250-256, March 1981.
4.
J. B. Anderson, T. Aulin, and C. E. Sundberg, Digital Phase Modulation, Plenum, Press, NY 1986.
5.
Terrance J. Hill, “A Non-Proprietary, Constant Envelope, Variant of Shaped Offset QPSK for Improved Spectral Containment and Detection Efficiency,” MILCOM 2000.
6. Sidney Graser, “ Techniques for Improving Power and Bandwidth Efficiency of UHF MILSTACOM Waveforms,,” Milcom 2001.
7.
Christian Schlegel, and Lance Perez, “On error bounds and Turbo-Codes,”, IEEE comm. Letters, VOL 3, NO. 7, July 1999.
8.
http://www.ee.virginia.edu/CCSP/turbo_codes/tcodes-bib/tcodes-bib.html
9.
Achilleas Anastasopoulos, and Keith M. Chugg, “Adaptive Iterative Detection for Phase Tracking in Turbo-Coded Systems,” IEEE Trans. Commun., vol 49 No. 12, pp. 2135 – 2144, Dec. 2001.
10. Lijune Lu, and Stephen G. Wilson, “Synchronization of Turbo Coded Modulation Systems at Low SNR,” ICC 98, PP: 428 -432 vol.1Volume: 1 , 1998. 11. Tyczka, P.; Wilson, S.G. , “On the Performance of Turbo Coded Modulation Systems with DD and NDA Phase Synchronization,” Proceedings International Symposium on Information Theory, 2000. PP: 223, 2000. 12. Eric Hall, and Stephen G. Wilson, “Turbo codes for noncoherent channels,” xxxxxxxxx 13. John Gass, Jr., Peter Curry, and Christopher Langfor, “An application of turbo coded modulation to tactical communications,,” Milcom 99. incomplete reference? 14. Charlotte Langlais, Maryline Helard, and Marc Lanoiselee, “ Synchronization in the carrier recovery of satellite link using Turbo-Codes with the help of tentative decision,” IEE. Incomplete reference 15. Catherine Morlet,“ Synchronization algorithms for multimedia satellite communications payloads,” IEEE 2001. 16. Wangrok Oh, and Kyungwhoon Cheun, “Joint decoding and carrier phase recovery algorithm for Turbo codes,”, IEEE comm.. letters VOL 5, No. 9, Sept. 2001.
17. Li Zhang, and A. G. Burr, “Joint phase recovery with turbo decoding and its improvement for BPSK over AWGN channel,,” IEEE 2001. incomplete reference. 18. Bartosz Mielczarek, “Joint adaptive turbo decoding and synchronization on Rayleigh fading channels,” IEEE 2001.
Dr. Mohamed Khalid Nezami © 2003
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RF Architectures and Digital Signal Processing Aspects of Digital Wireless Transceivers
19. Shigeo Nakajima, and Eiichi Sato “ Carrier recovery circuit using periodically inserted deterministic symbols for a turbo-coded binary PSK signals, “IEICE Tran. Comm. VOL. E84-B, NO. 5 May 2001. 20. Vincenzo Lottici and Marco Luise , "Iterative Carrier Phase Synchronization for Coherent Detection of TurboCoded Modulations," IEEE 2002.
21. Bruce Wahlen, Sidney Graser, Calvin Mai, and Tracy Burrr, “ Continuous Phase Modulation Waveform Simulations,” Space and Naval Warfare Systems Center, Dan Diego, 2000 22. M. K. Nezami, “Techniques for Acquiring and Tracking MIL-STD-181B Signals,” Milcom 2002. 23. M. Morelli, Umberto Mengali, Giorgio M. Vitetta: Joint Phase and Timing Recovery with CPM Signals. ICC (1) 1997: 1-5 24. G. Dillard, and B. Summers, “ Mean-level detection in the frequency domain,,” IEE Proc. Radar, sonar Navig., Vol. 143, No. 5, October, 1996. 25. ________“Comparison of various perdiodogram and frequency estimation,,”IEEE Trans. Aerospace and electronics systems, Vol. 35, No. 3, July 1999.
26. Riccardo De Gaudenzi, Vieri Vanghi, “ All-Digital carrier phase and clock timing recovery for 8PSK,”IEEE 1991. 27. Gottfried Ungerboeck, “Channel coding with multilevel phase signals,” IEEE trans. On information theory, Vol. IT28, No. 1, Jan. 1982.
Dr. Mohamed Khalid Nezami © 2003
11-55
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Author Biography: Dr. Nezami teaches part time at USA universities, and is a Sr. Principal Engineer with the Raytheon Company in St. Petersburg Florida, where he mainly works on Digital ground radios and satellite transponder design. He has a Ph.D. from Florida Atlantic University (2001) and both M.S. and B.S. in Electrical Engineering from the University of Colorado (1988 and 1991). His experience uniquely combines DSP, RF, and digital communication systems. His experience spans over a period of more than fifteen years in designing RF circuits, systems, and digital signal processing algorithms for radios. His emphasis is in the design of physical layer and analysis, modem signal processing, RF system design, RF circuit designs (DC-6GHz), and power amplifier linearization signal processing & RF system design and testing. He is also a USA ham radio operator with the sign call KI4CUA. He can be reached at
[email protected].
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