Multirate and Wavelet Signal Processing, Volume 8

Preface

The field af multirate and wavelet signal processing &ads applications in speech and image compression, the digital audio and digital video industries, adaptive signal processing, and in many other applications. The utilization of muleirate techniquw is becoming an. ind-irspensable tool of the electrical engineering profession, This point can be illustrated in three ways, First, if a performace specification is controlling the design of a particular system, that is, the performance specification exewds the eurrent state-of-art , then by converting the syskm to a multirate system, the overall system specification can be met with slower camponents. Secondly; if the d o l l ~ cost r specification is controllng the design of a particular systern, that is, the design of a competitive commercial system where bottom line cost is most important;, then by converding the system to a multirate system, the overall system cost will be reduced through the utilization of slower, cheaper devices. Thirdly, if power czansumption, is eoxlCrofling the design of a particular system, that is, the design of a hand-held system powered by a couple AA batteries, or possibly a satellite system, then by converting the system to a muleirate system will reduce power consumption through the utilization of devices with slower switching speed, and zls a result, lower power dissipation. Wavelet transforms are closely related to fitter banks. As such, a background in filter banks will make it eaier for the reader to understand, design, and implement wavelet transforms. Many of the most important applications, such as video compression, and many challenging research problems are in the area of multidimensional multirate. As such, multidimensional multirate is integrated throughout the book, The focus of this book is to present a sound theoretical foundation by emphasizing the general principfes of multirate. This book is setf-contained for readers who haw some prior exposure to linear algebra (at the level of Horn and Johnson's Matrix Analysjs) and multidimensional signal processing (at the level of Lirn's Two-DimensionalSignal and Image Processing or Dudgeon md Mersereau's MuItr"dimmsiond Digitd Sign& Processing).

xii

Preface

Moreover, this text will bring the reader to a point where helshe can read, understand, and apprecii%tiethe vast muftirate literature. The organization of this book is as fallows. The first two chapters are devoted t o basic multirde ideas including decirnators, expanders, polyphae notation, etc. This presentation is first given for one-dimensional signals in Chapter 1 and then generalized to multidimensional signals in Chapter 2. The next two chapters deal with filter banks, Chapter 3 presents the theory of Pilter b a n k for both one-dimensional and multidimensional signals. Chapter 4 deals with lattice structures, an, eEeiextt inzpEementation strategy for filter banks. Chapter 5 highlights an important applieaeion of mu1tir;tt;e -- the impEernentation of wavelets. I would also like to talre this opportunity to thank Professor Charles Chui for his enthusiasm about this projwt and for ineluding this text in his distinguished wavelet series. The hlXowing people have provided very useful feedback during the writing of this book. They include: Bill Cowan, Tom Folttz, Jerry Gerace, Ying Huang, You Jang, Matt Kabrisky, Mark Qxiey, b b e r d Parks, Juan. Vmquez, and Dan Zahirniak. Fairborn, Ohio February 9, 1997

Bsuee VV. Suter

Chapter 1

Mult irate Signal Processing

Thics chapter provides the b a i c concepts used in the study of multirate and wavelet signal processing. Some of the earliest contributions to the study of the fundamentals of multirate were due to Schafer and Rabiner[40], Meyer and Burrus[32], Oetken et a1.[37],and Crochiere and Rabiner[lO]. The idea of polyphme representation ie a key concept throughout the development of this book. This nontrivial idea war first articulated by Bellanger et a1.[3]. Much more recently, Evangalistaflq carefully examined another important idea - digitd comb filters, Many of the concepts developed in. this chapter are also discussed in the other multirate texts by Crochiere and Rabiner(ll1, Fliege[lS], Strang and Nguyen [46] and Vaidyanat han [49]. Section 1 2 presents a framework b r mtlltirate and it introduces two important representations for discrete signah. Section f -3 introduces the basic: building blocks. Section 1.4:provides ways to interchange the basic building blocks. Section 1.5 presents a filter bank example. 3.2

Foundations of md$ira;l;e

First we will. examine some samplixlg considerations and then present some basic transforms for andyzing signah,

haultirate is the study of time-varying systems. As such, the sampling rate will change a t various points in time in an impIementation. This will require us t o w r y the gain (magnitude) of filters in series with the timevarying building blocks so that the raulting gain is consistent with what one would expect if the sampling internal after the time-varying block had

2

Chapter 1 Muleirate Signal Processing

been the original sampled frequexlcy. Towards this end, let us axlalyze a train of impulses, Theorem 1.2.1.1. xzD=_m6(tkT) = $ x z = - m e ~ p jftnmt ( ). Proof: Let us expand

c

Ego=_, 6 ( t - kT) in a Fourier series. So,

00

G(t - kT) =

5

a(m)exp

where,

or equivalently,

Let

T ==:

t

- k T , Then,

We recognize this as a sum of integrals with adjoining limits artd simplify ta 1

Hence,

m

Let F denote the Fourier transform. So that if z ( t ) is a signal, then

Let us examine the Fourier trarlvform of an impulse train. Theorem 1.2.1.2.

7[

-kT)

~ ~ m : _ 6 (ot o

=

+ Cz==_,a(f - .).

rn

Praofi R o m the previous theorem, 00

6 ( t - k T ) = lI'

2

exp

Therefore,

+ [C;"=-,6 ( t - kT)] =

1 c@ FC,=-,6(fm

Now, performing the Fourier transform of a sum of the product of the input z(t) and Dirac delta functions, which can be expressed as the convolution of the corresponding functions, produces

=

J-",x(f -f')+C:"=-,s

= =

+CZ"=-,J-",x(f- f ' ) 6 +XE"=_,X

At this point, we will interpret these results for linear time-varying systems, If the sampling intewal is incremed by an; integer fwtor of M ,where M > 1, then the magnitude of the Fourier transform will need t o be deerewed by a factor of to reconstruct the original system, that is

&

Similarly; if %hesampling interval is d e c r e ~ e dby an integer factor of E, where i;> I, then the magnitude of the Fourier transform will need to be increllsed by a factor of .L t s reconstruct the original sign&, that is

For completeness, the definitian, of z-transfarms and the discrete Etsurier transforms will, be pwented. Then, we wilt present two sampled signal representations: the modulation representation and the polyphase representation. The theory of mutt irate and wavelet signal processing utilizes both af these representations

Chapter 1 Muitirate Sign& Pracessing

Befinition 1.2.2,1, The z-tf?amfown of a sequence z(n) is defined by

An important property of x- transforms is the following scaling theorem. Theorem 1.2.2.1.

Xlf the x-transform ofz eksts aad a is a sedar, then

Proof: By defix~iticsn,

or equivalently, QO

n=-oa

Hence,

Xf the z-transform converges far aU. z of the form z = exgljw) for real w, then the z-transform can be represented as the sum af harmonicdly related

which ics sometimes edled the discrete-time Fourier draasform.

1.2.2.2

Discrete Fomfer tramform

Defini-Lian3L.2.2.2. The di~ewrteF~uGert r ~ n s f o mof a periodic sequence s(n) sf length .lV b given by. IV-l

X(k)=

C z(n)exp

gt=@

and the correapondiag inver~ediscrete F0aTiier tmnsfim is @ven by

5

f .2 Foundations of mrxltirate

(q),

= exp the (principal) M t h root of unity. Then, the Let discrete Fourier transform matrk, denoted WN, is an N x N matrk, defined by [WNIk,% = WE= exp 1.2.23

ModuXa&iomrepresentation

DefiniGion 1.2.2.3. Given a, sequence s(n) and a positive integer M , then

the components of the modulation representation of the r-transformo f z ( n ) are defined X(ZW$), k = 0,1,.. . ,M 1.

-

If M == 2, then the eomponen%aof the modulation representation arts X(z) and X(--z). The t e m modulation representation can be most eilvily visualized in the time domain. Using the scaling theorem of x-transfarms, we obtain

~ - ~ [ x ( z w= ~ ) ]

f t is interesting to note that the components of the modulation represen-

tation can be combined pairwise t o form a r e d signal, that is

Definition 1.2.2.4. Given a sequence z(n)a d a positive integer M , then the Type-f polgphase contponents of as(%) are defined ask(n)= z ( N n + k ) , k = O 9 l? ' . ' , M - I .

If M ;;:2, then so(n)would be the even-numbered ~amplesand zl(n) would be the odd-numbered sarriples, Now, let us investigate the z-transform of the Type-$ poXypXlwe components, that is

or equivalently,

then X ( z ) becomes

The success af this decomposition rests on its usage of the well-known Division Theorem for Integers, whidl says any integer p can be represented it5

p;z=Mn+k wfiere, the quotient n and the remainder It;: are unique integers.

Definition 1.2.2.6. Give11 a sequence z ( n ) and a positive integer M , its Type-lsr polgphase componenfs of z ( n ) are given by a&(%) = z ( M n + M 1 - k ) , k = = 0 ,...,M - 1"

-

The ttransforrn of z(n) written in the Type-I1 polyphme representation is given by

or equivalently,

Let

then

It is also important to natc that Type-Xf pdyphase components are identical to Type-X polyphme components - only the indexing i s performed in reverse arder.

1.3 Bmie building Mock8

Figure 1.1. L-fold apmdler.

This section defines and analyzes the two types of bwic building blocks used in multirate signal processing. One type of building block deals with changes in ampl ling rate of the input sign& and the other type of building block deds with changes in filter length, We will first discuss two types sf building blocks which change the sampling rate of the input signal -decilna~tors t o reduce it and ezpanders to increme it. Secondly, another type of building block is discussed which changes the length of a given filter - comb filters to increme its length. In this way, a comb filter version of a given filter is andogoas to m expander acting on an input signd.

Definition 1.3.1.1, Let L be a yoYifive hteger, An L-fold ezpander (&a known as an upsampler) applied to an input signal z(n)inserts L - 1zeros between adjacent ssrupfes of the i n p d sign&. For a samp2ed sign& a(%), the outgue of dhe expander is givm by

An L-fold expander is depicted pictoridly in Figure 1.1.

Let us consider the following example to gain some intuition into the behavior of expanders. If L == 2, then the inputs and outputts of the expander me given by:

The operation. of expansion is invertible, or in other wards, it is possible to recover i ( n ) from samples of yB(n). We will now analyze the behavior of the &-fold =pander by taking a %transform of it, that i~

Chapter 1 Maltirage Sigad Proce~sing

8

By the definition of the expander, gE(n) = 0 if n is nat Therefore,

YB(Z)=

ia

multiple of L,

~E(%)z-*.

Let n == KL. Then, CIQ

-

By the definitian of the L-fold expander, zlk) yls(kl;). Hence,

aie equivalently,

YE(.) = x(st).

In the Iiterature, the ~ o t a t i o n

is oecasiondly used* If the region of convergence of Y E f z ) includes the points of the form z = exp(jccr), where w is real? then WE: can replace z with. e x p l j w ) to yield ra?(expfjw)> = X(exp(PLw)).

By suppressing the exponentid and, a such, changing the notation so that denotes &(exp(jw)), then vpe will write

&(w)

Therefore, YE(w) = X(Lw) means that a given frequency wo in X(w) is transformed to a new frequency in YE(@). A8 a r e ~ u l af t this transformation L 1 unmnted image spectra of the input signd s p e ~ t r appeared a after each of the rzrigiad spectra a t the output of the expander, This suggests the use of a lowpass filter with cutoff frequency f immediately following the expmder ZMillustrated in Figure 1.2. This Iowparrss filter i9 called an anti-imaging filter. RecaU from the previous section, if the sampling intervztl. is decrewed by a factor of L, then the product, of the p i n e of the expander and the anti-im+ag filter must equal L. Sinee the expander has unity gain, the anti-imaging filter must

-

Chapter 1 Mul tirate Signal Processing

Figure f .4.Spectrum of @ w a d e dsignai.

Figure 1.5. Fitter to recover input signal,

1.3 Basic building block8

Figure 1.6, M-fold decirn8tor.

The operation of decimation is not invertible, or in other wordpl, it is not possible to recover z(n) from yo(n). We will now analyze the behavior of the M-fold decimator by taking a z-transform af it, that i~

or equivalently,

At this point we can not make the substitution k: = Mn and proceed with a frequency domain expression for YD(z).Why? Because s(n)is not zero for noninteger multiples of Mn. So define an intermediate sequence that is zero for noninteger multiples of Mn, that is XI(%)

--

a;(%),

0,

where n is a multipfe of otherwise.

Therefore, at the decimated sample values yo(%) Hence,

=:

M

% ( M a )=

XI

f Mn).

DD

Y'(z) =

zl(Mn)z-". lt=-X3

Led b = Mn. Then,

Now, we need to express X1(1)in terms of X(z). By the definition of sl(n),we can write 21 (a)= C M (a)%(%)

Chapter 1 Mu1f;ir;st;eSign& Proces~ing

12

where, clle(njis a sampling fuwtion and is defined by C M ( ~= )

1, whenever n is a multiple of M

Q, otherwise.

So, for example, if 1M = 3 then cM(n)becomes

Mathematically, vve say that the sampling function cM(n)is a sequence of period M ,which can be repres-ented by the Fourier serim expansion N-'

cM (n)=

C C(k)exp M I@=@

where, the Fourier series coemcient C ( k )is defined by

on the interval (0, M )with samples at n = 0, ...,M a(n)= So,

- 1. a(%)is defined by

1, w h e n n = 0 0, otherwise.

CP) = 1 far all k. Thus,

Since zl(n)= cM(n)z(n), then the z-transform of zt ( n ) is given by

Substituting the equation for e M ( a )into the laad equation yields

11.3 Basic building I;rIor=ks or equivalently,

But Yo(z) = XI (z'/~). Hence,

or equivalently,

Y,(,) = -

M

M-1

CX

k=O

where W M = exp(+) is the (principle) Mth root of unity. The &th power af z means that the original spectrum is stretched by a factor of M. In the literature, the notation

is occasionally used. If the region of convergence of YD(z)includes the points of the form a = expfj w ) where w is real, then we can replace z with exp(ju)to yield

By s~ppressingthe expanenti& and, as such, cha~ghgthe natation so that YD(w)denoteg YD[exp(jw)J,then we will write

-

Hence, a frequency wa in X(w) is transformed to M new frequencies (wo 2nlc)M, ic = 0, ...,M 1, in YD(w).Now, consider the response y (R) of B decimator to an input sequence z(n), where every other sample is zero. This is simply an example af an intermediate qzxence with decimation by 2. So,

-

YD(Z)= x(z=j2). But using the analysis of the decimator for an arbitrary input r ( n ) , we can say that

Chapter 1 Muldirade Signal Processing

Figure 1.7. Spectrum of input ~ i g n d .

In the first c a c , e v e q other input sample wzs zero valued. Hence, the support of X ( w ) is limited t o 5 w < 5. Under these conditions, decimatio~lby 2 will not create aliasing, that is x ( - z ' / ~=) X ( Z ' / ~ ) Therefore, the analysis is consistent for a signal bandlimited to - f w < f. For this example, if the spectrum of the bandlimited input signal is given by Figure 1.7 then the corresponding spectrum of the output, of the decirnator is given by Figure 1.8, This bandlimited signal could be realized by the use of a lowpass filter with cut-off frequency & immediately proceeding the decimator as illustrated in Figure 1.9. This lowpass filter is known as an anti-aliosing filter. Recall from Section 1.2, if the sampfixlg interval is increased by a factor of M , then the product of the gains of the decirnator and the anti-aliwing filter m s t equal &. Since the decimator has a gain of the anti-aliasing filter must have unity gain. It will be shown later in this chapter that these operators, M-fold deeimators and &-fold expanders, are conlntutative proGded that L and M arc relatively prime.

-:

<

&,

Deftmitian 1.3.3.1. Given the impulse respunse h(n) of a filter, one c a ~ l build a comb filter g ( n ) by inserting L - 1 zeros be tween the coeEcien ts of

Figure 1.8. Spectrum of decimated signat.

Figure 1.9. Decimator with m t i - d i ~ i n filter. g

1.3 1S2ilsr"ebuilding blockg

17

and tldditions required. The following example demonstrates the lower costs achievable with an lFXR filter implementation. Example 1.3.3.1. Suppose we wish Is design a linear phme FIR filter with

specifications:

= 0.001 peak stopband ripple: 62 = 0.001 passband edge: wp = 0 . 0 1 ~

peak passband ripple:

stopband edge:

A f ,is given by:

c~,

-- 0 . 0 2 ~ .

The normalized transition bandwidth,

Then, we can estimate the filter order by:

So.

The number sf multiplications required is @yen by

and the number of additions required is given by

Let us first

= % = 0.0005 peak stopband ripple: b2, = b2 = 0.001 passband edge: up, = (0.01n)15 = 0 . 1 5 ~ stopband edge: w,, = (8,02~)15== 0 . 3 ~ . peak passband ripple:

Again consider an equiripple design. The normalized transition bandwidth, (A f,,) is given by:

Chapter 1 Multirate Signal Processing

28

Then, we can estimate the filter order by:

The number of mubiplications required far G ( z ) is given by

and the number of additions required is gima by

(# adds)G = NG = 46. Let us now consider the filter I ( s ) . Then, its specifications become peak passband ripple: 6%,=

2 = 0.0005

peak stopband ripple: b2, r= Sz = 0.001

passbaztd edge: wp,

="

stopband edge: w,, =

0.01T

- 0 . 0 2 ~= 0 . 1 1 3 ~ .

Considering ark equiripple design, the rrormafized transition bandwidth, (A f ), is given by:

[Afll

- w,, - wP, - 0.103?r - 2% - 2n =. 0.0515.

Then, we em estimate the Alter order for I ( z ) by:

Z'he number of multiplications required far I ( z ) is given by

Figure f .If. Conversion from studio work to CD mwtering.

and the number of aidditions required is given by

Hence,

(# multiplies)lFIR = (# r n ~ l t i p l i e s + ) ~(# m ~ l t i p l i e s )=~ 113

(#

= (# adds)G + (# adds)l = 58.

So, using the IFIR design technique,

cskn s i e i k a n t l y deerease the computatianal complexity over canventional equiripfle filter design. 2.4

Interchanging bslsic building blacks

This section. first presents tihe conditions for interchanging decirnators a ~ l d exprtnders, Secondly, we consider the noble identities, an appraxh for interchanging filters with. the basic building blocks for decimation, and expansion.

31.4.1

fnterchangiurg deelmators and expanderrs

Sampling rate conversion is important h r many signal processing applications. For examphe, conversion between the three standards for digital audio, which are 48KHz for studio work, 44.1KHz for CD mwtering (both digital tape and cornpnct discs), and 32 KHz for broadcasting digital audio. Ta convert from tbe studio work frequency to the CD mastering frequency, one could usc: the arrangement in Fip;ure 1.11. Thus, sampling rake conversion will require the cmcade of an &-foldexpander and an M-fold decirnator, separated by a filter. Thus, by emcading an &-fold expander and an M-fold decirnator, one obtains a ra$ionat sampling rate change with rate

&/Me Under what conditions can the decinaator and the expander be interchanged? First consider the configuration in which the expander proceeds the decimator as depicted in Figure L12, Writing the wsaciated elemental equations yields

Figure 1.12. L-fold expader proceeding M-fold decimakor,

Figure 1.13. M-fold deeimator proceeding L-fold expmder,

Hence,

Second, consider the configuration in which the decimator proceeds the expander as depicted in Figure 1.13. Writing the aysociat;ed elemental equations yields

and

Therefore, an L-fold expander can be interchanged with an M-fold decintdor if and only if Yl(x) = G ( x ) .By inspection, this vvill only occur

-

), k = 0, ...,M 1, generates the M t h roots of unity. The following theorem provides the coxrditions on l; and M for this t a occur. )Ik: =t 0, ..-,A4 - 1) equiblfs the set The set (exp(M th roots of unit& if and only.if l; and M itre rdativejy prime, Ghat

Theorem 1.4.1.1. of the is

gcd(L, M )+= 1.

), k = 0, ...,M - I, generatea dl the MLh roots of unity. Assume, for contradiction purposes, that gcd(k, M ) =: chi that d > I. Then, there e;x;ist relatively prime . But, L = dL1 and M = dMl. Hence, exp() wiU generat;e a t most IM; mots of unity, Since A/I; < M , a eoatradictlon occurs. Catse 2: Assume 2; and M are refatiwly prime. Assume, for cor~tradiction purposes, that expf ), k = 0, ...,M - 1,generates MI< M roots of unity, where M = %MI for some relatively prime positive integers el )), Since k can take on aUI and J1. Then, exp(1, then el[ L. Since ell L values from 0 t o M - I an I and el M, then gcd(L, M ) = el. Therefore, a contradiction occurs. Proof: Case 1: Assume exp(-

H e ~ ~ can e , &-fold expander and an M-fald decimalor commute provided M are relatively prime, Let us illustrate this result with a couple 2 and M =. 3, then gcd(2,3) = 1. Thus, exp() , So, Eor k -- @,I,2 we have t be following t e m s

I/ and

exp(-j0)

=:

I; exp

) becomes exp(-

). So, for k = 0,1,2 vve have the

following terms

Therefore, as expected, the two expressions are equivalent. Now, as a, counterexample suppose f; = 2 and M = 4, then gcd(2,4) = ) becomes exp(- j x k ) , So, for Ic .- 0,1,2,3 we have the following terms

) becomes exp(-?).

So, for k = 0,1,2,3 we have the

following terms exp(- j0) = 1;exp

= -j ; exp(- jn) = -1; exp

Chapter 1 Muidirate Sign& Processing

Figure 1.14. A more complicated example,

Figure 1.16. Equivdent version of Figure 1.14,

Thus, as expected, these two expressions are not equivalent. Now, let us consider a skghtly more complicated example, which is depicted in Figure 1.14, Find an expression for y (lz) in terms of %(a). Applying Theorem 1,4.1,1, we can interchange the first two blocks, since the expansion ratio of 5 is relatively prime to the decimation ratio of 2. This will yield Figure 1.15, Now, consider the two middle blocks afs depicted in Figare 1.16, Then,

or equivalently, *

T ( z )=

4

CS

(t)

h=O

So, we can eliminate the two middle blocks without effecting the output;, Hence, Figure 1.17 shows the simplified version of Figure 1.14.

Figure 1.16. Middle Blocks.

Figure 1.17. Simplified version of Figure 1.24.

Figure 1.18. Expwder falowed by inherpolation filter.

By inspection,

w(n> = ~ ( 2 % )

(f) for n. even for n odd,

Therefore, ~ ( n )for

O

tn

even

for n odd.

In, mast applications involving interpolation, an interpolation filter follows an expander as in Figure 1.18, The fitter must handle data at an increwed sampling rate of I;. We would like t a find ways to reduce the speed at which the filter oper&es. One way to do this is to interchange the expander and the filter in an equivalent system ars in Figure 1.19.

Figure 1.19. Filter proceeding expander,

Figure: 1.20. Decimation filter followed by decimat~r.

Figure 1.21. Decimator proceeding filter.

Let us di~cussthe conditions under which, they me equivalent, Similarly, in. many applications involving decimators, a decimitt;ion 6lter precedes the deeimator sls in Figure 1.20. Analogously, we mufd like to allovv W ( z ) t a operrtte at the slower rate of the decimator. 1s there any equivalent system to allow this intercl~antge? What must G ( z ) be for the system depicted in Figure f .21? If H ( z ) is a comb filter, then can the buading blocks be interchanged?

.

Interchanging Alters and expanders

Consider the con6guration depicted in Figure 1.22, Writing the elemental, equations, we obtain V ( z )= x ( z h ) and

Y ( z )= G ( z h ) ~ ( z ) . Wence,

Y ( z )= G ( ~ ' ) X ( Z ~ ) * But, this equation could also be interpreted zw Figure 1.23. These two equivalent block diagrams in Figure 1.24 present iz systeraakic approach for

Figure 1X.Z2.Expander followed by comb filter,

Figure 1.23.. Filter proceeding L-faId e v a d e r .

Figure 1.24. Noble identity for expaders.

interchanging filters with expanders a d tagether they will be referred to zw the noble identity for expanders. Please note that the noble identiey far expanders does mot, in general, apply if G ( z )is irrational.

1.4.2.2

Inter

ging filters and deeimators

Conisider the configuration depicted in Figure 1.25. Writing the elemental equations, V ( L )= G ( z M ) ~ ( z ) and

Hence,

Chapter I MuIthtte Signal Processr'ng

Figuire 1.26, DecimaBoir pmoeeding filter.

Figure 1.27. Noble idmtity far deciaaatsrs.

But this equation could be interpreted aa Figure 1.26. These two equivalent block diqrams, depicted in Figure 1.27, present a, systematic approach for ixzterchanging filters with decirnatars and together they will be referred to m the noble ideneiGy for decirnatczrs. Piewe note that the noble identity for decinrators does not, in general, apply if G ( z )is irrational,

1.5

A filtar bank: example

Consider the following example depicted in Figure 1.28 to gain sorne intuition into the importance of polyphase and decimators and expanders for multirate filter banks. As part of thk example, let us examine the inputs, autputs, and intermediate results for this 8im;pXe filter bank. For this exaraple, let us assume that the first nonzero input sample to the filter bank will be designaed s(O). Then,

1.6 Problems

Figure 1.28, A simple filter Bank,

Thus, the upper declrnator pases only even-numbered samples, while the lower decimalor passes only odd-numbered samples. Therefore, the outputs of the decimators correspond to the polyphwe components of the filter bank,

1.6

Problems

1. Let an input signal z(n) be periodic with period N , that is, z(n) = rc(n + MI. Let; y (n) be an M-fold decimated version af $(a), that is, y (n)=. z ( M n ) , Shovv that y(n) is periodic with some period 8, that is, ~ ( n=r) ~ (+nP). What is the smallest value of P in terms af M &nd N? 2, In the systems depicted ixr Figures 1-12and 1.13, derive the sequence domairk equations for their outputs in terms of their input, that is, yl( n ) and yz(n) in t e m s of re(n). Then, prove that these two expressions yield the same result if and only if .L and M are relatively prime,

3. Prove the following:

4. Let the input 4%) be defined by s(n) =; &,. Apply this input to a decirnator and show that the d e c b a t o r could be interpreted as a periodic time-invariant systerr~,

5. Some authors have proposed a Type-Iff polyphme representation using the fdlouring definition: Given a sequence z(n), then its Qpe-IXI polyphase components are defined ;zs

Chapter d Maltirate Sign& Procesahg How are the Type-f and Type-I1 polyphme components related Z;a Type-111 polyphase companents"!

6. fn example 1.3.3.1, we compared the complekly of a conventional equiripph filter design with an XFXR desip with a stretching h t o r of 15, Repeat the analysis far a stretching f a ~ t o of r 40.

Chapter 2 Multidimensional Mult irate Signal Processing 2,1 Introduction This chapter extends the basic concepts of rnultirate signal proce~singto multidimensional multirate signal processing. Examples of multidimensional signals include images in two dimensions and video in three dimensions, Good reference texts for bwkground materid on. multidimensional signal promwing are Dudgeon and Mermresu[lfjj and Lim[26]. The important concept of sampling is related to the mathematics of lattices; see for example Cassels[41. The engineering analysis of sampling begari in the classic paper by Petersen and Middleton[38] and later extended to include decimation and expansion considerations by Mersereau and Speake[31] and Dubois[ltij. The multidimensional z-trclrrsform is carefully described by Viscito and Allebrzeh[55) The idea of the Smith form was first articulated by Smithpq, Many recent papers have dealt with applications of the Smith form to the multidimensional DFT (Guessoum and MersereaupO]), to the multidimensional DCT (Giindgzhan e t a1.[21]), and for the development of multirate CAD toolo (Evansf28I). Some of the concept8 dewloped in this chapter are also discussed in the text by Vaidyanathan[49]. Section 2.2 presents a framework for the study of multidimensional m l tira;te signd processing and it introduces two important representations far multidimensional signals. Section. 2.3 presents the multibirnensional building bloehs. Section 2.4 provides ways to interchange the mltidimensiorral building blocks.

A multidimnsional signal is a signal of more than one variable. This section systematically presents concepts that act as s kamework for our study

363

Chapter 2 Muftidimensionaf Muftirate Signal Processing

of the application of multirate s p t e m ~to multidimensional signals, These canceF>tsinclude sampling of a multidimensional signal, which involvw tbe mathematical concept of a sampling lattice, and introduce multidimensional sampled signals by way of multidimensional ;r-tramforms and multidirnensiclnal Fourier transforms.. Now, we will discuss the mathematics needed t o describe the s~mglingconsiderations of multidimensianal multirate. 2.3;.I Sampling lattices

Sampling a multidirnemional signal is more complicated than a one-dimensiorral signd because of the many ways t o choose the sampling geometry* Sampling points could be arranged on a rectangular grid in a straightforward manner, but many times there are more efZicient ways to sample mu1tidimensional signals, Some applications are mare suitalble for nonrectangular sampling than rectangular samplng. F"or example, the testing of a phared-array antenna rquires the measurement of the electric field on a plane in the near field of the antenna. Since phased-array antennm are typically designed with an hexagand arrangement of efennends, the resulting processing is mare wcurato if hexagonal sampling is used for the memurements. One reman to consider nonrectangular sampling is Lo minimize the number of points rtwded to charwterbe an nil-dimensional hypervolume. Moreover, if the functions of interest are bandlimited aver a circular region, then Figure 2.1 shows that significant savings are possible if hexagonal sampling is used instead of rwtangular sampling. In order to precisely describe both rectangular and nsnrectangulw sampling, we nwd a convenient way to describe an arbitrary sampling geometry. To do this activity9 we must appeal to the laxlguage af linear akgebra in order t o present the mathematical theory of sampling lattices. 2.2.1.1 Linear independence and sampling lattices

The idea of a vector in an integer-valued k-dimensional space is a generalization of vectors in a plane by ixl6eger-tralrred Cartesian coordinate. This leads to the foliowing definition. Definition 2.2.1.I. The set of a11 le-dirnensio~a1integer vectors wjfl be cdIed the fgndamental lattice and it will be denoted N.That is,

N = {r = [rl,rz,.. . ,rkjT

rI is an integer).

The set of all k-dimensional real vectors will be denoted 8. Mow, let us review a few definitions related to sums of vectors.

2.2 Multidimczngianal frame work

Signal Gtim-snsionaliQ

Figure 2.1. Percent swings: hexagon& versus rectangulw sampling.

Deftnition 2.2.1.2. A linear combination of N vectors {rl,. . .rN) E hi is an expression of the form N

where %, 1: = 1, . .. ,N , are integer8 and arc called cocfticients. The set of vectors {rl,. . .r ~ L4) aid t o be linearly independent if

Cniri =

* ni = 0 for all ii.

i= 2

If the set of vectors {rl,. .. rN), is linearly independent, then the totality of vectors of the forr~

is called a N-dimensional lattice and it will be denoted by R. IB other words, the space R spaaned by the set of vectors {rl ,. . . ~ J )J is the space consisting of all h e a r combinations of the vectors, that is,

The set; 08 vectors (rl,...rN) Is a h s i s for R if dhey are linearly indepersdent and the g p x e is spanned by {rl,. ..rlv) is egud to R. Then, we say that R has dimension N. To better understand the definition of the lattice, consider the following illustration in two dimensions, Let r1

==

TI 1

,

r2

"

T12

and

n, ==

c22

T21

a1 7%

Then,

or equivalently., 72.1r1

+ 7L212 =

nlrll

t nzr12

nl Tzl + rb2T22

This can be rewritten as a matrk-vector product, i.e,

where the matrix R is cdallfed the sampli~gmatris, In general, let ri be the ith column of the matrix IR, that is

then the sampling miaLrix R is said to generate the lattice R. As such, the lattice R is dso given by

R=LAT(R) = { ~ E NmI= R n for n E N 1.

If R is the identity matrix, then each ri is a unit vector painting in the ith direction, and the resulting lattice, 'R, is the fundamental lattice N. Let us present some examples of sampling lattices usirlg bfzk dots to represent the lattice points and white circles to represent points in N that are not in LAT(R). Example 2.2.1.1. Consider rectangular sampling defined by the sampling 2 0 rnatraX R = . It i s depicted in Figure 2 2 . 0 3

Figure 2.2. Lat tiee structure using rectangular sampling matrix.

Figure 2.3. Lattice structure using hexagand ;tlmplingmatrix,

Chapter 2 Multi&rnensr"o~a1Mulgirate Signal Processing

Figure 2.4, Latt;ice structure with quincunx smgfing matrix.

Example 2.2..1 ..2. Consider .. hexagonal sampling defined by the sampling 1 1 . It is depicted in Figure 2.3, matrjac R .=: 2 -2 Example 2,2.i ,3. Consider quincunx sampling defined by t he sampliag 1 1 matrix R -It is depicted in Figure 2.4, -1 1

.

Far a given sampling matrix R, the corresponding Fourier domain sam- ~the lattice it generates is called the reciprocal pling matrix is ~ T R and lattice.

The m a t r k that generates a, la~tticeis not unique, AS we wiH gee later in this subsection, the follawing matrices generate the same lattice.

The theory underlying the nonuniqueness af these sampling litLtiGes is based on unimadular matrices. Thus, in order to dlkcus8 the nonuniqueness of sampling lattices, we must first briegy discuss unimoduiar matrices.

2.2 Multidimensional framew~rk

35

DeBxritian 2.2.1.3. An integer-valued matrix A is a unimortulizr matrix if

The following theorems provide properties of these mizlriees. Theorem 2.2.1.1. If A is an integer-valued unimoduiar matrix, then A-'

eksts and is an integer- vafued unimodular matrix. Praof: Let A be an integer-valued unimodular matrix. Moreover, since det A f 0, then A-' e f i ~ t and s AA-' = I. Since det A det A"-' = 1, then = 1. Since A is unirnodular, then

where the (i,j ) elernelit of the adjugate(A1 is equd to the cofactor of the ( j ,i ) element of (A). Since A irs uriirnodufar, then A-' = f adjugate (A).

Since A is an integer-wlued matrix, then ad,jugatefG) ia an integer-valued matrix. Therefore, A-I is an integer-valued unimodular matrix. II With this background on integer-vaIued unimodular matrices, we are ready to discuss the isaue of nonurriqueness of sampling 1;jtetices. Theorem 2.2.1.2. Given a nonsingular maLrix M and an iateger-vdued

unimodular matrix V, then LATWV)

-- LATW),

Proof: If x E LAT(M), then there exists an n E N,such that x = Mn =

.

MVV-'n = MVm, where m = V-'n Since V is an integer-valued unimodular matrix, then V-I is also an integer-valued unimodular matrix by Theorem 2.2.1.1. Since V-' is integer-valued, then m = V-'n i s integervalued. Therefore, x E LAT(MV). Therefore, LAT(M) E LAT(MV). Obviously; LAT(MV) C LACP(M). Since N and MV generate the sarare la.ttice, then ZAT(MV) =: LAT(M). 1 FOF notation pufposes, let 3(M) =f det; M the absolute value of the determinant of the sampling matrix: ha. Theoxem 2.2.1.3. Given a nonsinguliar matrix R?1; and an integer-dued unirnodular matrix Y. Then, JfM) = J f M V ) = d(VM),

36

Chapttsr 2 Mul ti&mensiond MuItirate Signal Procesging

Prooh Sirice J(M) =I det M

1,

then

J(MV) = J(M)J(V) = J(V)J (M) = J (VM). Since V is an intqer-vrtlued unimodulax: matrix, then J(V) = 1. Herlce,

a

Therefore, J(M) is unique and independent af the chaiee of basis veetors. Moreover, J(M) can be interpreted g e o m e t r i d y a the k-dimensiond volume of the parallelepiped defined by M. Sometimes J(M) is called the sampling density. Now, consider the falltowing sampling matsices

R=

1

0

2 -4

and S =

1 1 2 --2

where J(R) = J(S) = 4. Sirice RE = S for unimodular matrix

then R and S generate the same sampling lattice. In this case, the sampling lattice is known aa an hexagonal saxnpEing laktice,

2.2,1,3 Unit c e b and fundamental p~dlelteplpeds Definition 2,3.1.4. Given an integer-vrtlued matrix R, a unit cell i ~ c f a d e s

one lattice point from &AT@) and J @ ) - l adjacent points in N that are not in &AT@).

If these unit cells are periodicdly replicated 0x1 LAT(R), then the entire spwe is tiled witfi no overlap. Thus, the unit cell is a footprint that characterizes tire samplirkg lattice. Definition 2.2.1,6, Given an irmteger-mlued matrix R, the fundawkentrzl parallelqipetf of 1Iattice L A T O , denobd FPD(a), is the unit cell fiat ineludes the origin mid i~ bounded by d Iat tice poini;s one positive unit away. Formdy, the f u a d a m e ~ t dpard1dep;tped is givm by

where E is the set crf aJf k-dimension& real wetors.

2.2 Multidimensional framework

31

Consider the fallowing example as an illustration of the concept of fun2 0 damental parallelepipeds. Assume the sampling matrix R = 0 3 Then, fundamental ppardlelepiped is given in Figure 2.5. By inspection, there are J(R) = 6 points in the fundomental parallelepiped. Theorem 2.2.3 -5%. Given an i n t e g e f i v d ~ e dunimod~larmatrix U and an

integer-vdued diagonitl rzlaitrk A, t f i e ~

Prosh By the definition of the fundamental psrauelepiped,

FPD(UA) = {y Let s = Ax. Then, x = A-'g.

EE y

= UAx for all x E

Hence,

Since iJ i s an integer-valued unirnodular matrix, then Uz is an integervalued vector if and only if a k an integer vector, Therehre,

FPD(UA) = U{y E 6' y = z for all A-'z E [ O , ~ ) ~ ) .

Chapter 2 Multidimc?nsionaiMuitirate Sigad Processing

38

But

2;

= AX. Eenee, FPD(f5A) = W(y E @ Y = Ax for all x E [O, I ) ~ )

or equivalently,

FPD(UA) = U FPDCA). 1

Sometimes im the literature, authors will refer to the Symmetric parallelepiped, denoted SPD(R). It is defined by

SPD(R) = {y E E y = Rx for all x E [-I., I.)~).

DefiniCisn 2.2.1.6. Let V1 and Vz be Ic x Ic integer matrices. LAT("V2) iu cdled ia sabladbice of LAT(ITl) if LATrJ2) C LATF1), that is, every point of LAT(V2)is d g a id p ~ j n of t L A T p lf .

Let V1and Vz be integer matrices, where LAT(V2)C LAT(VI). Then, for every m E H , there exist n E such that

or equivalently,

n =V ; ' V ~ ~ . Since n and m are integer vectors and since V l n = Vzm, then v;'v~ must be an integer-valued matrix. Let L = V;'V~. Then, VIL = Vz. Since det Y rL = (det Vr)(det L) , then det V2 = (det V t ) (det L)

so that,

J(Vz) = J(V1) (det L) .

Hence, J(Vz) is an integer multiple of 3(V1). For example, if Vz =

2 0

4 0

0 4

, then V z is a sublattice of V1.AS such, J(Y2) = I6 0 2 and J(V1) = 4, then J(V2)iu, as expected, an integer multiple of J(V1), i.e. J(Vz) = 4J(V1). An important special case results when the lattice LAT(V1) coincides with the fundamental lattice N,i.e., LAT(V1)= LAT(I). In addition, p = represent8 the number of cells of FPD(V1) which can fit into FPD(V2). The lattice point in each one of these cells can m d V1 =

2.2 IMirltidI'mensr'ur~afframework

39

be thought of as a shift vector a, which if added t o each vector of LAT(V2) will generate an equivalence class of points called a coset. The union of all cosets is LAT(V1). Thus, the concept of a coset will provide a natural way to partition LAT(V1) into subsets, which is a necessary step for the generation of multidimensional polyghttse components. Befinition, 2.2.1.T. Let Vl and V2 be integer-vdued nzatrr'cm such that

Let a E LAT(Vl)fl FPD(V2). Define the coset C(V1,Vz,a) to be

If LAT(V1)= hi, then by convention, V1 is not explicitly identified, that is, the coset is simply

Given integer-valued matrices V1 and Vz such that LAT(V2)CLAT(V1), then denote the set of shifit vectors by

Similar to the convention for cosets, the convention for shift vectors when LAT(V1) = .A/' is not to explicitly identify V1, that is, the set of shift vectors is simpXy

4 0

and V1 = 0 4 then the cosets are uniquely defined by the following set of shift -vectors Returning to the example above, if TJZ =

Let us briefly review some elexnentary operatioas that be applied to integer matrices, These oper;;tlioxrs will be essential when we subsequently discuss the Smith form decomposition, a method for diagonalizing the sampling matrh.

Clxapeer 2 Multidimensional Muitirate Sign& Procrtsging

40

Etementary ruur (or column) operatima on integer matrices are important because they permit the patterning of integer matrices i&o simpler forms, such as triangular and diagonal forms. DeBnitian 2.1&.1,8. Any elementary row operatioxl on m integer-vdued m a t r k P i s d d e ~ e dto be any of the fo&owing:

Type-1: Interchaagtt two rows. Type-2: Multiply a row by a nonzerointeger constant c, Type-3: Add an i ~ t e p multiple r of a row to anather row, These aperatiotons can be represented by gremultiplying E) with an appropriaLLe square matrix called an. elementary m&rix, To illurJtrate these elementary operations, consider. the followhg examples, (By convellt;ion, the rows and columns are numbered starting with zero rather than one.) The first emmgle is a Type-l elemenlazy m a t r k that interchanges row 0 and raw 3, which h a the form

The second example is a Type-2 elementary matrix that multiplies elements in row 1 by c $ if:? which has the form

The third example is a Type-3 elementary matrix that replaces row 3 with row 3 fa * r w Q), which has the farm

+

Afl three types of elementary polynomial matrices w e integer-valued unimodular matrices.

41

2.2 Muft;idimensioxlal framework 2.lt.X.6

S d t h form decomposidfon

The Smith Form Decomposition p r o ~ d e sa method for dbgonalizing the sampling matrix, When matrices are diagonal, most one-dimensional results esn be edended automatically by performing operations in each dimension separrately. However, in the nondiagond cwe, these extensions are nontrivid and require complicated notations and m a t r k operations, Theorem 2.2.1.9, Every infeger-valued matrix corre~p ond i ~ Smith g form deeompo~itio~ aa

R ewl be expre~sedin its

where U and V are iateger-dued unimodular matn"ces md the Smitfr farm A is given by

where r is the rank of R and A;[ Xi+1,

i = 0,. .. ,T

- 2.

Proof: Assume the zeroth column of R contsbins a nonzero element, which may be broughL to the fQ,O) position by elementary operdions. This dement iar the ged of the zersth column. If the new (0,O) element does not divide all the elements in the zeroth row, then it may be replxed by the gcd of the elements of the zeroth rmv (the effect wilt be that it will contain fewer prime factors than belare). This process is repe&cd until ran elerneat in. the (0,O) position is obtained which divides every element of the zeroth row arid column, By elementary row and column operations, all the elements of the zeroth row and column, other than the (0, Of element, may be made zero, Denote the new submatrix formed by deleting the zeroth row and zeroth column by C. Suppose that the submatrix G contains an element C i j which is not divisible by ma. Add colu~lnj to column 0, Column O then consists of the elernents coo, el,j , . ,e,-1, j . Repeating the process, we replace by a proper divisor of itself using elementary operations. Then, we must finally reach the stage where the element in the (0,O) position divides every element of the matrix, and all other elements of the zeroth row and column are zero. The entire process is repeated with the submatrix obtained by deleting the zeroth row and column. Eventually a stage is reached when the m a t r k has the form

..

42

Chapter 2 Multidimensional MulGirate Signal Processling

where D = diag (A@, . . . , and &[&+li,i-- 0,. . . ,r - 2, But E must be the zero matrk, since otfierwiiie R woufd have rank larger than r. m

Note that although the two unimodular matrices U and '\r are not unique, the diagonal matrix II is uniquely determined by 33,. Example 2,2.1.4. To illustritte the Smith form decomposition, consider the

sampling. If we divide the (1,0) element, which corresponds to hexa,~~onal 2, by the (0,O) element, 1, ure obtain

2== 2 (I)+ 0 * quotient remainder Therefore, if we apply a Type-3 row operation, which is defined by

to R, we will reduce the (1,O) element to zero. Therefore,

Dansform the (@,I)element t o zero by a Type-3 column operation, which 1, -1 is defined by . Then, we obtain 0 1

Finally, the (1,l)element is forced t o be positive by a Type-2 row operation, 1 Q which is defined by . The~l,we obtain Q -1

Thus,

2.2 Multidjmerr~ionaiframework

Le%E be the product of efementary raw operations, i.e.

Let E" be the product of elementary column operations, i.e,

since only one elementary column operation was performed. Therefore,

Then, the Smith form decomposition is given. by

where,

and

Theorem 2.2.1.6, Let and TC" be unimodular matrices and let A be a &agoad matrix. H the Srnith form decomposition of smpfing matrix R is

given b;y R

==

UAVl then

ProoE Since LAT(AV) = LAT(II) by Theorem 2.2.1.2, then FPD(AV) = FPD(A). Therefore, FPD(UAV) = FPD(UA). Since FPD(UA) = U FPD(A) by Theorem 2.2.1.4, then FPD(R) = U FPD(A). rn 2.2.2

Mul$idimensionill sampled sign&

Unfortunately, some important sampling structures can not be represented as a, lat;lice, Fbr example, consider an important sampling structure h r High Definition Television(HDTV) called line quincunz, where two samples are placed one vertically above the other in place of every sample in

44

Chapter 2 MuItidimerlsiorrd Multirate Sig~aXPraces~ing

the sampling grid, But, line quincunx can be represented as the union af twa shieed lattices using the rnuftidimensiond ~ t r a n ~ ~ f o r m VVe i , will firat define some underlying vector mathematics and, then we wiu present the multidimensional ;=-transform and the multidimensianal discrete 1Faurit;r transform, Then we will present two nnuftidimen~iodsign.& representations - the modulation representation. ~ n the d pollyphmt: representation. The theory of multirate and vavelet signal processing is considerably simplified by the use of these representations. 2,2.2, J

Vector

In order to gencrdize the definitions tihat we have grown accustomed to seeing in one-dimension, we wilX provide the definition of a vector raised t o a vector power and, subsequently, the definition of a vector ra&ed ta it m a t r k power. T

Given eomplttx-vdued vectorr = ..., 7 ~ - 1] and integer-valued vector s = [ so, .=. , s ~ - 1 T . Then, the vector r r&sed to the vector s power is a sealtar and it is deaned to be

Definition 2.2.2.1.

Then, building on this definition, we will define a vector raised to a matrix power.

.

Definition 2.2.2.2. Given a complex-valued vectorr = [ ro, . . , ~ j v - 1 ] a d an i~teger-vduedma&rix&= Id0, ..* L N " 4 ,where hi is the i tlx column of L, the^, the vector r r&~edto the matrix L pawer is a row v e e t ~ and r if;is defined to be

.

Definition 2.2.2.3. defined by

The b-dimensional z- transform of z(no,...,lab - 1 ) is

T

2.2 Mtlltidimeasiand framework

where, z = [zO,..., z ~ - i~~ a ]complex-valued ~ vector, n = (no,...,n k - l l T is aa integer-vdued vecCor, and

Let L be an integer-dued noasingufar m;ttrix, &en by Elefiaitjon 2.2.22, zL i~ given by where, Li irs $he ith colrrrnn of L, that is,

Theorem 2.2.2.1. Let L he an integer-valued matrix where Zi, z' = 0,. . . k1, are the columns of Z1, Then,

Proof: Using the Definition 2.2.2.2, vve can write

where, Li is the ith column of L. Substituting the definition of a vector raised to a vector power yields

where Lmli is the .nth comporlent of Lie a wctor power gives a scalar. Hence, k-l

/k-X

By definition, a vector raised

',

to

46

Chapter 2 Muitidimensiond Multirate Sip& Processing

Sirrce the product of terms with the same base equals the base to the sum of the exponents, the last equatkn becomes

I

rn

If the z-transform converges for all a;, of the form z, =; exp(jwm), = OO?. .. k - f then the n-transform car1 be represented as the sum of

harmonicdly refated sinusoids, i.e.

which is the multidimensional generalization of the discrete-lime Fourier transform, In order to quickly distinguish vectors in Fourier space, the vector Fourier variitble 2 i s denoted by an underlined omega rather than a Bold omega. Theorem 13.2.2.2, Let I; be an i~lteger-valuedmatrix. Then, exp ( j w j L

= exp ( j ~ ~ g ) .

Proof: Using the Definitiozl 2.2.2.2, we find that

where, Li is the ith column af L. Since the exponents of each term are simply inner products between g and a coXumn of L, their order can be interchmged, that is,

The multidimensiond discrete hur-ier transform is an e x x t Fourier regresentation for periodicasjly sampled arrays, Therefore, it takes the form of a periodically sampled Fourier trax~sftzlrm,As irz the one-dimensionai case,

the nnultidimensiond discrete Fourier transform can be interpreted ES a Fourier series representation for one period af a periodic sequence. 111 this formulatim, we wiH have to address t m types of geriodicities --orre due to the sampling lat;tice and one due to the s i p & (tbat is, defined on lattice points) to be Fourier trazuformed. Let V denote the sampling mat^, L e e , hexagonal, quincunx, rectangular, etc. Let N denote the periodicity matrix, which characterizes the periodicity of the lattice points on which the signd to be Rurier trmsformed is defined. Assume LAT(M) is a sublattice of LAT(V). Then, we define equivalence eIwses betvveen periodic replicw of the data by

[n] = { m E LAT(V) n - m e LAT(N) ). Therefore, if parailelogribms are drawn between the elements of LAT(PJ), then any tm vectors that occupy the same relati= pasition are in the same equivalence cfms, Many properties of the periodicity matrix, N,follow by a n d 0 0 from the corresponding facts far sampling matrices. For example, the density of the periodiciQ matrix is uniquely defined by , denoted J(N); but for a given periodic sequence the periodicity matrix N is not uxlique, since it, can be multiplied by any unirnodular miztriur and still describe the same periodic signd. In additioa, the columns of fiJ indicate the vectors along which it is periodically replicated, RefOniLion 2.2.2.4. A multidhcn.siond sequence a(n) is pesio&c FviGh pen'od N,that i ~for ? all n?r E JI/, z(n) = ~ ( nNr).

+

Let XM represent one period of z(n). Tllea,

where V defines the unddying sarrrpling lattice, Moreover, since z(m) is periodic with period N, X(g) crzn also be written as

X(w) =

x

nEZr-3

But,

z (n)exp

- j g ~ n exp[] jwTv~r].

Therefore, exp[- j w T v ~ r= ] 1,

which is equivalent to the condition,

where m is a vector of integers. Upon further examination of observe t bat uT = 2?rmT(VN)-I,

2,we

.n ."

or equivalently, Therefore, w =2 n ( ~ ~ ) - ~ m . -

The matrix ~ ? I ( v N )serves - ~ as a Fourier domain sampling matrix. Substituting this equation inta the equation for Xfg) yields

or equivalently,

Let us further examine the inner product which occurs in the argument of the exponeatiaf,

Suppose the multidimenaiand sequence X(mf is periodic with period P, that is, X(m) = X(m Pq) for m, q E N.Also, let IF reprevent one period of X(m). Then, by analogy with the one-dimensional discrete Fourier ) the follovvlng form for some constarlt a, transform, assume that ~ ( ahas

+

Invoking the periodicity of X(m),that is, X(m) = X(m z(n) t o become

+ Pq),will cause

ar equivalently,

But,

Therefore,

2 z(n) = X(m)exp[jnT( 2 ? r ~ - ~ ) m ] . a mEZp

e ~ ~ [ j n ~ ( 2 ? r ~=-1~for) ~allqq] E N.

Since xl and q are integer-valued vectors, then

or equivalently,

P = bIT. Therefore, X(m)is periodic with period NT,that is, X(m) = X(m + NTq). Hence,

+

Now let us determine the constant er: by substituting the equation for z(n) into the equation for X(m), Hence,

which is as expected, since J(N) =;I det N I is the number of samples in one period for LATIN), Therefare, the multidimensiond discrete Fourier transform pair are given by:

It should be noted that these equations reduce to the usuaf discrete Fourier trantzform. pair in the one-dimensional cme and t o the familitzr rectangular muldidimensional discrete Fourier transform when N is a diagonal matrk. As an illustration of this theoretical development, sometimes it is of interest to input data from an arbitrary lattice and output it an iz rectangular lattice, so that it could be conveniently displayed an a eompzoter display. Assume that V is degned by

For hexagonal input: a = 2, b = 1, c = 2, Moreover, for quincunx input: a == 2, b = 1, c = 1. In addition, for rectangular input: b == 0, Select ht periodicity madrk so that VN is a diagonal m a t r k so that the resulting Fourier analysis will1 be on a rectangular grid, Now, let ur~pick N to be

then this N m a t r k is a good choice for a periodicity mittrk. Therefore, the DFT becomes

where,

2.2 Muftidimensional framework This suggests the following algorithm: (I) Compute &HI-point FFTs, one for each row in the

51

7&1 direction.

(2) Apply. a. phauc shift to each paint of the resulting data.

Since we are mrking with a sampling grid with samples a t integer-valued locations, it is important that we perform the phase shift for integer multiples of But :nzml is real-valued. Therefore, we will need to quantize :nzml to integer values through the use of the round function. ( 3 ) Compute &&-point FFTs, one fbr each column in the 7 ~ 2direction.

R.

2.2.2.4

The S d t b foran and the DFT

First;, let us begin with the mul$idirnensionaI discrete Fourier trrtrrsfornn, that is, F(k) = f (n) exp

C

DE~M

Replace M with its Smith form. decornpositian, that is, M = TJAV. Then, using Theorem 2.2.1.2, LA'I"(UAV)=: LAT(UA). Novv, using elementary linear dgebraic operations, let us simplify the exyorrent of the exponentid

= exp [- j 2 ? r ( ( ~ h ~ ) - ' n ) ~ k = exp [-j2nkT(UhV)-In

Then, the mzrltidirnensiond DF'T becomes

where

m = U-'n and pT= k T v - ' .

52

Chapter 2 Mult;idimensi'onalMuf tirate Signal Proeegsing

Therefore, the Smith form permits the use of a rectangulm DFT, when the initial data lies on. a nonrectangular sampling grid. The initial data must have pardlelepiped spatial support which becomes rectangular after being mapped by U-I. Moreover, the larger the values of the elements of U-I , the more the spatial suppart will be skewed. Thus, the aill;ar&hrrrfor the Smith form tversion of the multidimensiollal DFT is given by the following: (1) Shufee the input data samples by U-I. (2) Perform a septtrabk ~kultidimensionalDFT with len@'fis equal to the diagonal elements of A, (3) Shufee the output data samples by VT. Let us examine more carefu;tfy the mapping between the spwe of the initial data samples and the space defined by A. Assume that the data is defined on a quincunx grid with the hllowing SmitEl form decomposition.

Then, II characterizes the intermediate space for the Snrrith Farm decompasition and the mapping from M to A can be visualized by Figures 2.6 and 2,7. Si~iilarly,the inverse DFT is given by

where, det M

I=I det U

Defiinitrion 2.3.2.15. Let M be the sampfing matrix, Then, given a, multidimtm~ionalsequence ~ ( n )the , eompoxreats of the muZdidimensionaI modulat;ion representalion of the multidimensional n-transformof z f n) are defined

To more easily interpret this equation, assume z takes on the miue of .xp(jg). Then,

Figure 2.6, Ilnput data.

Figure 2.7. After shamng input data.

But z, = exp(jo,) for

m,

.- 0 , . .. ,k

- 1. Therefore,

Example 2.2.2.3. This example illustrates the idea of the multidimensional modulation representation of X ( z ) with respect to a rectangular 2 0 . In this case, I P Vwill I ~ contain four va.1sampling matrix; M = 0 2

X~?'(Z)

2.2.2.G

=

X(zo,zl)

for h a =

X(z0, -21)

for

X(-zo,zl)

for h z =

X(-al-xr)

far h3 =

hl = [ 0 1

2'

T

T

Polyphase represenkation

Definition 2.2.2.8. Given a multidimensional sequence s(n) and a nonsingular matrix R, then its Tgpe-I multidimensional polyphase components are defined as z.(n) = z (Rn a), where n E and a E N(R).

+

Now, let us investigate the multidimensional z-transform of the Type-I muEtidimexlsioxla1 polypfime components, that is,

or equivalently,

Far notatioriaf purposes, let

then, X ( z j becomes

55

2.2 MuXtidim ension& framework

where zea corresponds to shifts of the multidimensional signal by a vector amount a, Novv consider the Eollawing example, which illustrates the idea of a Type-I multidimensional polyphase decomposition of the multidimensional filter Xfs) . Example 2.2.2.2. What is the Type-I multidimensional polyphase decom-

position of the multidimensiorkal filter X (1;) with respect to a rectangular 2 0 ? In this case, there are J(R) = ldet R I = sampling mabtrk R = a 2 4 values of the shift vectors are given by [0, O I T , [I,O I T , (0,1 Let s = [za, rl] T and a = [(a), ,(a),lT then, zma = z~"'~z;(a'z. ThusI the quantity ia given by for a = 10, 0jT

( z,z' ;' =

r,'z;'

for a = [l,OIT

zcaz;'

for a = 10,

z

for a = (I,ljT,

I'

Since for arbitrary R = Hence, the quantity

llT

for a = [o,o

zo

=Cl

Yo

o1

x i R ) (sR)is

] ~

,zR is given by zR = ( ~ , 2 ~ zz:' Y , z? ) . defined by

So, the polypkitse decomposition of Xfz;) is giiven by

X(z) =

+

xiR) (zi,2:) + z;lxLR) -1

z1

x,

(R)

2

(Zi, zf )

2

(Z~,L~)+Z~=Z;~X~~~(Z~Z,I~).

Definition 2.2.2.7. Given a multidimensional vector z(n)and a nonsingu1ar matrix Et, its Type-11nzultidimewionad palyphme components are given by z,(n) = g(Rn - a), where a E hi(R) and n E N.

Chapter 2 A/lulfidirneusioud Mul tirate Signal Processi~g

56

Now, led us inwstigate the multidinrensionat ~~transfornr of the Type-I1 multidimensionalt polyphasse components, that is,

then

where za advances the multidimensional signal by a vector amount a. T ~ u s , the set of dl integer xctora can be partitioned into J(R) equivalence classes using either Type-I or Type-I1 muttidimensiona! polypbitse decomposition. Moreover, the success of these decompositions rests on the following theorem. Theorem 2.2.2.3. (Division Theorem. for Integer Vectors) Let R be a k x k nonsingular integer matrix, let; p and n he m integer veetors, and let a be s sbr'ft vector of R, the^, w can uniquely express p w

Praof: Write g

+EM

P'PP+P~; where, pp and pr, are unique vectors with

Since R is an integer matr2, then pr; ia an integer vector, Moreover, since p and E)L are integer vectors, then p~ is an integer vector. Moreover, since p~ E FPD(R), it follows that p~ E M(R). By letting p~ = a and p ~ = Rn, we obtain

p=Rn+a. I

The remainder, a, can be expressed

%s

a ZE p mod R or simply ((a)) =:

2-3 Multidimensional building blocks

Figure 2.8. Mult idimensiond ezcpantler,

2.3

Mdtfdimemional building blocks

Thits section defines and analyzes two types of multidimensional building blocks used in multidimensional rnultirate signal processing. One type of building block deals with changes ia the sampling rate of the input signal, and the other type of building; block deals with changes in filter length, Analogous t o the one-dimensional c a e , there are two types of multidirnensionaf building blocks, which change the sampling rate of the input, signal ---- decimators to reduce it and expanders t s increwe it. But in the multidimensional cme, deeirnators iznd expanders aRect not only the sampling rate but also the geometry of the sampling lattice, Then, a multidimensional building block is discussed which cha~rgesthe length of a given iilter - multidimeni~ionalcomb filters to increase its Imgtfr. fn this way, a multidimensional comb filter version, of a given filter is analogous to ~nultidimen~io~al expander acting on an input signal. 2.3.1

Multidlmemiond expanders

Definition 2.3.1.1. Let x(n) be a k-dimmsiond signd m d let L be an integer-vdued matrix. Then, $file muftidimensional procesg of L-fold expruldillg mtqps a signat on I\( to another signal that;is nonzero only rct points on $he subIattice LATP), The output of the R-dimensional expander is related to the input ~ i g n dz(n) by

where XI is a nonsingular k x k integer matrix. Since LAT(L) denotes all vectors of the form Lm, where m E N9then the condition L-'n E N is equivalent to n 6 LAT(L). The matrix L is knowxk the expansion matrix. The multidimensional expander is depicted pictorially in Figure 2.8, Now, led us anitlyze the multidimensianal expander using the definition of the multidimensional 2-transform,

Chapter 2 MuliCidimensional Muidirate Signal P r o c e s ~ i ~ g

Figure 2.9, Cascade of multidimensional expmders.

Then, applying the definition of the nrultidimension.Ellexpander will require that Y ( z ) is zero for the lattice points defined by L-'n $! N.So, let n. = Lm.Then, Y(z) = y(~m)e-Lm.

By the definition of the rnultidimensiond expander, z(m) = y(L@

m E hl* . Henee,

Y(z)=

for d l

~(rn)z-~~, EN

or equivalently, using Theorem 2.2.2.1, the equation becomes

Hence,

Y ( s ) = x(zL).

In order to investigate the behavior of the multidimensionizl expander in the Fourier domdn, replace rt with exp(jg) to yield

Utilizing Theorem 2.2.2.2 this equation becomes

By suppressing the exponential and, as such, changing the notation so that Y (2) denotes Y (exp(jo)), then

Now consider the carrcade of two expanders fcrl and L2,which can be depicted grapkicdly by Figure 2.9. Substituting ][I F= LILz into the equation yields for Y (x)

Y(o)= x ( ( L , L ~ ) ~ s ) , or simply,

Y ( o )=

x(~TLT~),

2.3 Multidimensional building Mocks

Figure 2.11. Smith form cascade of expanders.

which can be represented by Figure 2.10. This cwcade of tvvo multidimensional expanders can be verified by redizing that

~(= o~(LTo) ) and

S ( 0 ) = x(L:*).

Therefore,

Vu) = X ( L f ( L T w ) ) = X((LIL~)~A) = X(LTr??). So, for example, if L is repfaced by itis Smith form decomposition, that is, -L=: ULnlLVLt then we can graphically depict it in Figure 2.11..

L

Definition 2.3.2.1. Let z(n) bc a k-dimensional signal and let M be a k x k integer-vdued matrix. Tlxe multidimensianal M-fold decimalor sitmples input z (n) by mapping points on the sublat tice LAT(M) to .A/' according $0

yln) = s(Mn) and dises~rdingsamples ofaln) not on LATfM).M irj. ca,Ifed the decimatioa matrix.

The multidimensionaf decimator is depicted pictoridly in Figure 2.12. Let us analyze the muXticiimenaiorra1 decirnator using the definition of the

Figure 2.12. Multidlixeewianal deeinn8tor.

multidimensional z-transform, i. e.

Substituting the definition of the muXtidimensiona1decimator., we obtain

Note that zfn) is not zero for noninteger multiples of Mn, So, define an intermediate mapping of points that i s zero f'or aoninteger multiples of Mn, that is, z(n), where M-'n EN

ZI(~) =

0,

otherwise

.

So, y (xr) = 4 M n ) = zl(Mn). Hence,

Let m = Mn.Then, by the definition of zl(m),if M-'n $N,then sl(m)= 0. Therefare, Y ( z )= al(rn)a-M-'my

C

m€N

ar equivalently,

Y(.) =

C zl(m) (8M-'

)--me

=EN

Next, we need t o express XI (z) in terms of X(e). By the definition of sl(n), we can write the foliowing:

2.3 Multidimensional building blocks

61

where CM(n)is a scalar-valued sampling function associated with the sampling matrix M, that is,

G M ( ~=)

1, whenever M-'n E

0, otherwise

.

Since CM(n)is periodic in LAT(M) with spatial variable n, i.e.

+

C M ( ~=) C M ( ~Mm), it can be expressed as a complex Fourier series

Applying the definition of the multidimensional rtransform to zl(n)yields

Substituting the equation for GM(n) yields

or equivalently,

Then, performing a rnultidimensiond z-transform yields

Since Y ( B ) = X I (zM-I), then

Y (z) becomes

If z-" equals e x y ( - j g n ) , then sM-' is equivalent to e ~ ~ ( - j ~ - ~ ~ ) . Therehre,

Chapter 2 Multl"dimensiondt1Multirate Sign& Proces~ing

Figure 2.13. Cwcade af mtlltidimensiond dedmstars.

Figure 2.14. Interpreting the cmcade of multidimertsiond decinzators.

Utilizing Theorem 2.2.2.2, Y(exrpba]) becomes

By suppressing the exponential and, za such, changing the notation so that Y (o) denotes Y (expb*]), then

Now consider the emcade of two decirnators MI and M2, which can be depicted grqhically in Figure 2.13. Substituting M = M5M2i ~ t the o definition of the multidimensional ddecimator yields

which can be repre~entedin Figure 2.14. This cascade of multidimensional deeimators can be veriGeb by realizing that

Therefore, y(n) = z(M1M2n).

So, for examplie, if M is r e p k e d by its Smith hrrn decomposition, that is, M = UMAMVnn, then we can graphicdly depict it as Figure 2.15. Let ucr brieAy examine unimodular deeimators with decimation matrix Y. Consider multidimensional decimator,

2.3 Multidimensio11d building block8

Figure 2.15, Smith form cascade of decimatms.

Figure 2.16, Unimodular decimator,

Since V is a, unirnodular, then J(V) = 1. Therefore,

Therefore, there is no diasing and, as such, unimodvtar decimation can be viewed as just a rearrangement of samples. This inte~pretationof unirnodular decimation is depicted in Figure 2.16.

Definition 2.3.3.1. Given the impulse response h(n) o f a mull-idimensioaal filter, one can build a muttidimensional comb filter g(n) by mapping the filter h(n) on to another one that is nonzero only at points on the subfat tice LATCL), i.e.

As mentioned earlier, &-In E N is equivalent to n E LAT(L). The impulse response of a multidimensional comb filter can be represented by

where 6,,L, is the multidimensional Kronecker delta function. Taking the multihensional x-transform of gfai), we obtain

Chapter 2 Malt;idimensiond Maltirate Signal Proce~~ing

64

or equivalently,

If LAT(L) is not rectangular, that is, L is not a diagonal malrix, then the components of ~ ( z ' ) are no longer separable. To illustrate F

this point, let L =

I0

%I

90

YI

, then ~ ( a l =) H(z,'~r?,

z,Z1zp) and

H(z) = H(ro,zl). For example, we will consider both rectangular Sampling and hexagonal sampling, For rectangular sampf ng, let

then ~ ( s =~~ () r i z yz:z,:) = H(zi z:) and H ( z ) = B(ro,zl). This indicates the separability of components of ~ ( zfor~ a )rectangular sampling. Far hexagond sampling, let

) and H(z) = H ( z & zl). , then ~ ( z ' ) = x ( z ~ z ~ , z ~ z=; ~H(z~z:,z~z;~) T h k indiea-tes the lack of separability of components of the m-tlllidimensiortal comb filter H(zL)for non-rectangular sampling.

2.4

Interchanging fsdlding blocks

This section defines w q s to interchange multidimensional Building blockis. First, the conditions for irtterchanging muldidimensiond dceimators and expanders Tor sampling rate conversion are presented. Secondly, we consider the multidimensional noble identities, an approach for interchanging multidimensional building blocks with multidimensiond decimation and interpolatiox~filters. 2,4.1.

Irrderchangimg deciurralors and expanders

Multidimensional sarnplirxg rate conversion is irnportaxxt for many signal processing applications, because many times it is necessary to interfact: irnagc or video data between systems which use different sampling lai;t;ices. Examples include the conversion Between European and American television systems and the conversion between high definition television (HDTV)

2.4 I~Cercha~ging building blocks

Figure 2.17. Decimatoz preceding expander.

Figure 2.18. Substituting Smith form dewmpositrioxls.

signals and conventional tefevision. signals, Thus, sampling rate conver~ion will require the cascade of a multidimensional I;-foid exprlnder and a multidimensional.M-fold decirnator, separated by a filler. Under what conditions can the decimdor and the expander be interchanged? We will see that we will need to a s s u m that (a) M and L commute, that is, L M = ML, and (b) M and L are coprime. We wiH see that through the utilization sf the Smith h r m that coprimeness in muftidimensions can be xhieved in each dimension independently: Conr~iderthe configuration in whicEl the decinzator precedes the exy a d e r , which is depicted in Figure 2.17, Let us assume the following Smith farm decompositions for M and L:

M == iJMAMVlyr and XI = fJLALVL where UM,VM,UL, VL are unimodular matrices and AM,AL are diagonal matrices. Therefore, the resulting configuraLion is depicted in Figure 2J.8, or equivalently in Figure 2.19, Without loss of generdity, we can ilssurvze that

Hence, we obtain the configuration depicted in Figure 2.20, Since AM and

Figure 2.19. Substituting cmcade definitions.

Chapter 2 Multidimensional Multirate S i g ~ a Procesgiag l

Figure 2.20. Simplifying the cascade.

Figure 2.2 1, Interchan[;;inglambda matrices,

AL are diagonal matrices, then interchanging AM and .ALcan be? achiemd in each dimension individually, provided that the associated decimation and expansion ratios in each dimension are relatively prime. In this way, the Smith form decomposition. has llelped us transform a multidimensional problcrn into a series of one-dimensional problems. Therehre, we obtain the conliguration depicted in Figure 2.21, Since unirnodular deeimators me equivalent to inverse uxxirnodrrlsr expanders (as depicted in Figure 2-22), then for a ~ i yunirnariulw matrh T, Figure 2.23 results. Now choose a unimadulatr m a t r k UA such that

or equivalently,

UA = &UMh;'. Choose a matrix VB such. that

or equivalently,

Figure 2.22. Unimodular deeimatars and unimodular expanders,

2.4 In terehangirjlg building block8

Figure 2.23, Rmulting figure.

Substituting Ui' into the last equation yields

Since AM and AL are relatively prime in each. dimension, we can interchange A 2 and AL. Therefore,

vB= U

~ A ~ A ~ ~ U ~ L - ~ M .

Since VM = VL,then VLvkl ; :1. Therefore,

VB = u ~ A ~ v ~ v & ' A ~ ' u G ' L - ~ M , or equivalently,

vB= L M - I L - ~ M .

If L and M are assumed to commute, then L-'M = ML-'. Therefore,

VB = 1. Hence, we ltave shown that the nrultidimensionaf L-fold expar~derand the multidimensional M-fold decimator can commute provided (a)3L ztr~dM commute and (b) L artd M are coprime, Exampla 2.4,li.T. Let us consider the following example. Assume a multidimensional decirnator is defined by the decimation matrix D is given

and a multidimensional expander is defined by the expan~iorlm a t r k E is

Clan these multidimensiond Building blocbs be inderchlzngedl First, we will find the Smith form decomposition of rnaLtrices D a ~ l dE, that is,

Chapter 2 MaltidimensionaJ Malgirate Sign& Proee~sing

Figure 2.24. Multidimeasionial;expander followed by after,

Figure 2-25. Multidimensionaf dedmator preeded by filter.

Since the diagonal entries in Ax> and are relatively prime, then r f r , and ljlE me coprime, which in turn implies that 33 and E are coprime. Secondly; we need to check to see whether D and E commute, that is,

and

Since DE = ED, then I)m d E commute, Since D and E are coprime and since D and E commute, then the multidimensional decimator (defined by the decimation matrix D) and the multidimensional expander (defined by the expansion matrix E)can be interchanged.

In moat izppkcations involving multidimensional interpolation, an interpolation filter foUows an expander as in Figure 2.24. Similarly, in many app1icatiorra involving multidimensional becimators, a decimation filter precedes the decimator as in Figure 2.25. If H ( z ) and K ( B )are multidimensional comb filters, then can the multidimensional building blocks be interchanged?

Figure 2.26. MuILidimensional, expmder with comb filter.

Figure 2.27. Filter preceding mnltidimensiond expmder .

Consider the configuration depicted in Figure 2.26. Writing the elerrlentd equ;ztions, vve obtain V ( z )= x(sL)

But, this equation could dm be interpreted as Figure 2.21, where

and T ( z ) =: G ( z ) X ( e ) .

These two equivalent block diagrams present a systematic approwh for interchanging multidimensional filters with multidimensional expanders, and together they will be referred to cw the Nable identity for multidimensional expanders,

Consider the configuration depicted in Figure 2.28, Writing the! elemental equations, V ( e )= c(sM)X(Z)

Chapter 2 Mull;r.dimensiond Muftirate Sign& Proces~ing

Figure 2.28. Comb fitter preceding malt idirnensional decimator.

Figure 2.29. Mufdidimensiond decimator preceding filter.

Hence, using the multidimensional z-transform identities msociated with x raised to a matrix power, we obtain the following

But this equatioxt. could be interpreted Y (z)

a r ~Figure

2.29, where

=: G(z-)T(z)

These two equivalent block diagrams present, a systematic approxh for interchanging multidimensianal filters with multidimensional decimatorts, and together they will be referred to as the Noble identity far multidimensional decimadors, 2.5

Problem

1, Consider the following sampling matrix:

Sketch the lattice LAT(T). Clearly indicate the fundmental parallelepiped FPD(T) and highlight the J(T) points in N which belong to FPDIT).

fat

n

72

Ghapter 2 Multidimensio~~d Multirate S i p & Processing 7, Let us generalhe the notion cvf multidimensional expanders and dlecimatarg. Let N be a nonsingular mat^. In the definition of the multidimensional expander, replace JV'with LAT(N), where LAT(L) C LATIN), Similarly, in the definition of ttte mullidirnensional decimator, replace A/' with LAT (N), where LAT(M) C LAT(N). Interpret the resulting operators and provide examples to illustrate the use of eaeh of them,

Chapter 3

Mult irate Filter Banks

This chapter introduces the notion of maltirate filter banks. Using eonventiond filter design techniques, a high-order filter is required to obtain a faat roll-off capability, Filter bank8 are bwed on an alternative approach to reaiizing a high-order filter consistjng of tire cascade of lower-order (analysis) filters, that are designed with aliwing, and (synthesis) filters, that ape designed to cancel the alim-componerrts of the analysis filters. Working independently, Smith and BarnwellL44j and Mintzer[35j reported the e ~ s t e n c eof a two-channel filter bank, which permitted perfect reconstruction of the input signal. Subsequently, Smith and Barnwell[45] developed the aliars-componentm a t r k formulation for analyzing M-channel filter banks. Meanwhile, Vetterlil53) and Vaidyanathan(481 independently discovered the polyphwe formulation of analyzing M-channel filter banks. The theoretical underpinnings of filter banks were carefully examined by Vaidyanathan and Mitra[5q, who showed the connection between pseudocirculant matrices and aliars-free filter banks, and by Vaidyanatthan[48j, who showed a connection between paraunitary matficetl and perfect reconfstruction. These developments in multirate signal processing were recently complemented by research involving the joint consideration of quantization effects and filter bank design(KovaEevie(25l; Westerink et a1.[56]). Subsequently, Koilpillai and Vaidyanathan [24] developed perfect reconstructing cosine-modu1at;cd filter banks, whicb have the advantage that d l filter8 are derived from a single prototype filter. Xn an effort to eliminate blocking effects in low-bit rate image compression, Malvar[29,30] independently developed cosine modulated filter banks and he called them lapped arthogonal tmnsfoms (LOTS). Subsequently, using the very general conditions for windows, which were developed by Coifman and Meyer(S], Suter and Oxley[47], and independently Auucher, Weiss, and Wickerhauser[l] developed the corttinuous generalization of LBTs to permit the conytruction o_f

74

Chapter 3 Muftirate Filter Banks

diEerent l a c 4 b ~ e ins diRerent time intervalis. Yves Meyer cdled the eontinuous generalization of LOTS by the name Malvar wavelet[33), since they complement elwsical wavelet theory. Recently; Xia and Suter have gerreralized the theory of Malvar wavelets to include two-dimensional nonseparable Malvar wavelets[58] and Malvar wavelets on hexagons[59]. Vetterli[52] was the first person to write a paper on multidimensional multirate filter banks. His paper dealt with two-channel filter banks with quincunx decimation, Using an analogy with one dimensiand systems, Viscito and Allebach[55] formalized the theory of multidimensional multirate operations for arbitrary multidimensional lattices. Karlsson and Vetterli[23] and Chen and Vaidyanathan[G] formulated polyphase representations of mullidimensionril signals. Some of the concepts developed in this chapter are also discussed in the texts by Fliege[lS], by Strang and Nguyen[46], and by Vaidyanathan[49]. Section. 3.2 introduces quadrature mirror filter banks, Then, Section 3.3 presents the theoretical foundations; of multirate filter banks, Section 3.4 presents filter banks for spectrd analysis. Section 3.5 introduces multidimensional quadrature mirror filter banks. 3.2

Qua&ature mirror Alter b a d s

This section define8 and andyzea quadrature mirror filter f&MF) banks, The role of $ME" banks in source coding is examined first, This is fallowed by a, discussion of two important QMF bank formulations -- aliacomponent and polyphase. Then, a multirate source coding design example is presented that illustrates many of the QMF bank concepts of this section. Finally, a brief discussion of quantization and their egects on filter banks is presented.

3.2.1

Sawee coding and QMF b a d s

Consider the two-channel QMF bank that is depicted in Figure 3.1, The ana'fysis bank together wiLh the decirnaLors decompose the input signal z(n) into tvvo subband signals, y@(n)and yl(n). This is follwed by expanders and the synthesis bank, which produce an output sigxlal that reconstructs the input signal, For the QMF bank, the number of samples per unit time for the sum of both subband signals equals the number of samples per unit time for the input signal, But, the power of the subband signals is usually much lower than. the origin4 signitf.. After decoding, the input signal is reconstructed, Before we can design a system bitsed on these ideas, we will need t o become acquainted with the approaches that are used to analyze filter banks.

3.2 Quadrature mirror filter banks

Figure 3.1, Two-chansel gllter bmk,

3.2.2

Filter b a d formulations

We will discuss two difFercnt approaches to the analysis of filter banks: (li) diw-component formulation, and (2) polyphwe formulation. Then, we will show that the two formulations are equivalent,

3.2.2.1

rlliaa-campanent farmulation of ftXter ba&s

Consider two-channel filter banks. Let &fw) and El(w) represent the frequency response sf & ( z ) and L;l; ( z ) , respectively. For a QMF bank, the magnitude responsest 1 Ho(w) ] and I HI(@) are mirror-images of each other with respect to frequency $,which is one quarter of the sampling frequency 27r. For M-channels ( M > 21, t k h structure should not be called a QMF bank, because the traditional tw~-cftannclmeaning does not hold, However, QMF hm been used by niany ather authors, so we will retain the same nomenclature. An M-channel QMF hank, that is depicted in Figure 3.2, partitions the signd spectra, into M barrds of equal bandwidth and later recornbitres these frequency bands, The input is denoted z(n) and the output is denoted y(n). It consists of M decinlators (each with a decimation ratio of M ) ; M expanders (eadl with an expansion ratio of M); M a~lalysisfilters (denoted by Hk(z)?k = 0,.. . ,M - 1); and M synthesis filters (denoted FL(z), k = 09". , M - 1). The basic philoaoyhy behind the design of QMF banks is to permit a l i ~ i n gin the filters of the andgsis bank and then. choose the filters of the synthesis bank so that the alia-components in the filters of the analysis bank are cancelled, Now, we will proceed with the andysis af the quadrature mirror filter bank, by exaxnining ihtage-by-stage. First, we will consider the avralysis tank, which is depicted in Figure 3.3.

Chapter 3 Muitirate Filter Banks

Figure 3.2. M-Gfiwnel filter bank.

Figure 3.3, Andyais bank.

3.2 Quadrature mirror filter banks

Figure 3.4. Bank of decinaators,

The elemental equation for this stage is given by

for each k = 0, I,. .. M -- 1. Then, we will consider the bank of decirrtators, which is depicted in Figure 3.4. The elernentat equation, for this stage is given by

The following stage?consists of a bank of expanders, which is depicted ixr Figure 3.5. The elemenitaf equation for this stage is given by

Uk( z ) = v k ( z M ) . Lwtly, we will consider, the synthesis bank, which is depicted in Figure 3.6, The elernentd equation for this stage is @ven by

Combining the equation for X k ( z ) with the equation for Vk(z) yields

Chapter 3 Multirate F2ter Barrks

Figure 3.5. Bank of expanders,

pthests bank

Figure 3.6. Synthesis bank,

3.2 Quadrature mirror filter banks Conlbining this equation with the equation for U k ( z )yields

Finally, combining this equatioxl with the equation for Y ( z )yields Y ( z )=

desired terms

terrrrs due to ajliming

A few observations can be made. First, the desired term can be interpreted as the input signal weighted by the mean of the product of the analysis and synthesis lifters, Secondly, if the coefficients of the aliasing term can be set to zero then a linear time invariant (LTI) system can be constructed out of linear time varying (LTV) components - decirnators and expanders. In this ease, when aliasing is calcelled the distortion function is given: by

Unless T ( z )is an allpass filter?that is, ) T(w) = c j f O f o r a ~ ~ w , w e s . y Y ( x ) suffers from amplitude distortion. Similarly, unless T ( I ) has linear phase, that is, aarp;I"(u) = a bw for constcants a and B, wc say Y ( x ) suEers from phase di~torbios~. This leads us to the definition af a perfect reconstruction system.

+

Definition 3.2.2.2, Let z ( n ) and y(n) be the input and ouCgud, rmpect i v e l ~to the filter bank. Then, a perfect reconstruction system is a gygtem free from &wing; ampfitude distortion, and phase distortion, As such, y (n)iu a scaled and delayed vervion o f z(n), that is,

Let us recxamiac the output of the filter bank Y(x),which

or equivalently,

wa63

given

Chapter 3 Multirate Filter Banks

80 where

m i t i n g the s p t e m of equations A,( z ) , n = By. .. ,M yields I

A(z)= --H(r)f M

-f

(

in rniitrijc form

(z),

where,

= [Aa(z),

(z),

. .. ,A M - 1

(z)IT9

the synthesis bank is given by

and the so-cdrled alias-cornpaned (AC) matrix H(x) is given by

In this formulation, aXiasing can be eliminated if and only if the gain for each of tbe aliasing terms equitfs zero, that it^, .Alc(2) = 0 for JG == 1,. .. ,M - 1, Moreover, to illsure peqect recolastaction Ao(z)must be a delay9 %hatis, AO(x)= z - ~ @ . The solution of this system of equations for yynthesis filters f(z), may have prxtical diaeulties. It requires the inversion of the alias-component matrix H ( r ) . Even if successful, that is, if H(z) is nonsingular, there is no guarantee that the resulting filters f(r) are stable, that is, the poles o f f (2) are inside the unit circle. The approach given in the next sectiolz presents a. difirent technique in which dl of the above CtiBculties are remowd. Let us provide an example to illustrate the use of the alias-component m;tt;rix. Assume the analysis bank is cftnract;erized by the folowing filters:

&(+) = l and H 1 ( z )= + - I . Find the analysis filters Fa(+) and F~(z) that satisfy the alias cancellatian property, So, the alim-componmt forrnul&ion of the filter bank is characterized by

3.2 Quadrature mirror filter banh Writing this m a t r k equation in terrns of its terms yields

To insure alias cancellation, Aa(z)= a ( r ) and A l ( r ) = 0. Hence,

Let us construct the alias-component matrix H ( x ) , i.e.,

Solving for H-' ( z )yields

Substituting H-'(z) and A(z) into this equation yields

Solving for f ( r ) yields

For perfect reconstruction, let a ( r ) = Z - ~ O , where rno 2 1, then

8?

a,.

-42

z

& 8 u 7 .k Z

-s

83 u crr !2w !=,

-g .a -5Q 2 E =J

-4.a

$ 4s a,

!ij

29

4

B 4

55 2i *%

cd

it'

&

D

c;l rcc

"3

gz *2

8 43 g 4-(

23

2.9

3

=J,

3.2 Quadrature mirror filter banks or equimlently in m a t r k notation,

where the synthesis bank is given by

and the analysi~bank is given by

Wrltiug H k ( z ) ,k

=z

0,. . .M

- 1, in terms of Tme-I polyphase yields

M-X

z - " ~ k , , ( r ~ ) for k = 0,. ..M

f f k ( z )=

- 1.

m=@

This systenl of equations can be rewritten in m a t r k notation as

where the delay vector (called a delay chain) e~ ( z ) is given by

eM( r )= [I, z-',

... ,z - ( ~ - ' ) ] *

and the polyphwe-component m&rk for the analysis b n k E(z) is given

Then, writing Fk(r),k = 0,. .. M

- 1 in terms of Type-I1 polyphase yields

M-1

z - ( ~ - ' - ~ ) & ~ ~ ( for z ~k) = O ? . .

Fk(z) =

. M - 1.

m=O

This system of equations can be written in mzttrh notation as

where the paraconjugation of e~ (t),denoted ZM ( z ) ?is given by

Chapter 3 Maltirate Filter Bank8

84

and the polyphase-component matrix for the synthesis bank R(r) is given

Substituting these equations for h(r) and fT(r)irlto the equation for Y ( z ) yields

Let P(x>= R(x)lE(z). Then,

ur equivalently, V(z) = T(z)X(z)

where the distortian function T(z) is given by

These equations suggest the filter bank in Figure 3.7.

3.2.2.3

Refatioraship between formulatiolns

Aa seen by the previous subsections, the study of filter banks can be gerformed using either afiw-component or polyphme matrices. Now let us examine the relationship between these twa formulations. The AC m a t r k XX(z) is given by

3.2 Quadrature mirror filter bank

Figure 3.7, Poliyphase-component filter bank,

RecaU that h(z) is given tu

Hence,

H~(Z) = [h(z),h(zWM),...,~(zw$-')].

ReeaU that h(z) b related to the palypkae-component matrix: E(z) by

Therefore, H T ( z ) can be written as

E ( z ~( z)) , .~..,~E ( Z( W ~ ;-~)~)~~(ZW;-')

86 But by definition ~ M ( Z W ;=) 11, ,-I

w ~ L. ,Z, - ( ~ - ~w)-M( ~ - l ~ i k : l ~ 9

or equidently,

where A(+) = diag[l, z-',

... ,2 - f M - ' ) I . Note that

is just a single column of the W s matrix, where WM is the M x M discrete Fourier transform matrix. So,

Since ~

f= (Wg)T, i then

With this equation, any results obtained in terms of the polyphase formulation can be applied to the alias-component formulation and vice versa.

3.2.3

A mdtirate sornrce-coding d e s i e example

1 s t subsection to a nrultirate soureecoding design problem. Let z(n) be an arbitrary sequence and let zl(n) be its first divided difference, i.e, VV;e will now apply the results of the

Consider two sequences yo(n) and

yl

(n) that are defined by

~ o ( n=) ~ ( 2 %and ) yl(n)= z1(2n).

3.2 Q u h t ure mirror fif ter banb

Figure 3.8. Filter bank equivdent circuit,

Can we recover z ( n ) from yo(%) and yl(n)? It is important to note that the even-numbered samples are already available. Can the odd-numbered samples Be recovered? To salve this problem, let us first recast it as a two-channel QMF problem, wbem the aniltysiv filters ares Bo(x) = I. and HI ( z ) = 1 - z - I , the %-transformdomain expression for the first divided difirence, So the filter bank is depicted in Figure 3.8, where the goal is to determine Fo(r) and F 1 ( z ) S U C ~that we have perfect reconstruction. Let us first detemine the polyphase matrix E(z). From the last subseclioa, vve found that H(Z)= W ? A ( Z ) E ~ ( Z ~ ) . Recall that w2Wf = 21z and h ( z ) i ( z ) = 12. So, mlving for ET(z2)we obtain 1 E ~ ( Z ~=) X(Z)(~WZ)H(L) and substituting the appropride errtries into the matrices results in

or simply

E(x) = Now, choose R ( r ) = E-'(z) to ins a(% - 1). Then,

R(z) =

reconstruction, i.e, y(n) = 1 f

0 -I,

88

Chapter 3 MuI1;I'sateFdter Banks

Since the entries of R(z) are not functions of t,then R(zZ)= R(z). Therefore,

Compute the synGhesis filters using and substituting entries into %hematrices results in

or equivafently, Hence, Fo(z) = I + Z-I. and Fl(r) = -1. Assume z(n) is a slowly varying sequence, that is, adjacent samples differ by a smdl amount. If each sample of s(n) requires 16 bits for its representation, then the first divided difirence, beirtg very small, requires only 8 bits Tor its representation. Now instead of storing (or trzlnsmitting) all samples af z(a),we dore for transmit) yo(n) using 16 bits per sample a ~ d yr(n) using 8 bits per sample. Thus, we have reduced the bats rate from 16 bits per sample do an average of 12 Bits per sample, Nate, bowever, ttzat if the polypbwe matrices are utilized directly, that is,

then the filter bank can be operated at half the rate. The regulling filter bank is depicted in Figure 3.9. The regular structure of this realization foreshadows the discussion of lattice structures in the next chapter.

If the fllter bank is used flor compression, then there will be information loss between the analysis bank and the synthesis bank. This lass is due t a coding errors that age the result of quantization effects, This subsection will show that if the quantizer xloise is correlated with the input signal as in a Lloyd-Mw quantizer, then a sfiight modification to a filter bank can result in the cancellation of all signal-dependent errors, For completeness, we will briefly review Zeibnitz's rule and conLinuous randon1 variables, since they wit1 be used in the derivation of the LloydM w q-ztantizer,

3.2 Quadrature mirror Alter Banks

Figure 3.9. Eseient realization of dmign prabbm.

Zleibnitzk rule describes haw we can diRerentiate a function which is af the form I; ,(:! g(z, t)dz. Theorem 3.2.4.X. Let

where vl m d vz may depend orr the parameter t and a 5 t 5 b. Then,

<

for a 5 t b, if g ( s , t ) and $f are continuous in both z and t in some region of the st; plane induding y a vz, a t b and if y and vz are continuous and have continuoulr;.derivatives far a t b.

< <

< < < <

Proof: Let pzlt)

P(t) ==

g(z, t)&*

Let t E (a,b) and At $ O such that t -+At; E (a,b) , then

+

-

= ~ ( tat) p(t)

Chapter 3 Multirade Filter Banks or equivalently,

How, grouping the second and fourth terms yields

NOWscale this equatiorl by At yields

Applying the mean-value theorem far integrals Lo the last two terms yields

where v,(t

52 is between

vz(d) and

v ~ (-t-t A t ) a d tI is hetvvee~~ vz(t) and --.* 0, the equation becomes

+ At). Then, taking the limit as At

3,2.4.2

Continuaus raindam variables

Suppose X is a continuous random variable or1 the interval with a probab is given bility density function, f. Then, the probability that a 5 X

<

P(a<X
where, I( is edled the probabilie density function. The ezpeeted value of X on the interval [a,b], denoted E [ X ] ,is given by

3.2 Quadrature mirror filter b a n h

where a

< X < b. Moreover, for axry positive integer k,

where a 5 X

< b. The variance of X , denoted

Q ,;

is giverl by

Substituting the definition of the expected value yields

or equivalently,

Simplifying this equation using the definition of E [ X ]and E [ X 2 )yields

or simply, G$

= i!z[x2] -[E[x]]~.

Fbr completeness, the following descriptioxl of Lloyd-Max quarktizers [Max, 19601 is included. These quantizers minimize the expected value of the distortion (or error between the input and the output of the quantizer) of a signal by a quantizer when the number of levels of the quantizer is fixed, Let u be a randonr variable with a csntinuous probability density function p ( ~ ) It. i~ desired to find the set of increaving transition or decision levels {t,J rn = 1,. . . L ) with tl and t~ a9 the mi~kimumand maximum values of ?I,respectively. If u lies in the interval [t,, t,+%), then it is mapped to rm, the mth reconstruction Level. The Lloyd-Ma quantizer minimizes the: mean square error for a given number of quantization levels. Therefore, for an L-level quantizer, the meark square error

is to be minimizd. The necessary conditions for this minimization are obtained by differentiating E with respect to t, and r, and setting the r a u k equal to zero. Now let us consider It yields

E.

By inspmtion, this partial derivative is only nonzero for k = rn. &nee,

Setting

= 0 yields

Thus, the recsmtruction; levels of the Lloyd-Ma quantizer tie at the cen.t;er of mms of the pmbability densiw betwertn trmsition levels, Using the definition of expected value of a, continuous randons vz~riabfe,we can reinterpret

where IP is a rmdom variable and t, 5 EP Now let us consider It yields

g.

< t,+l.

93

3.2 Qrxdrafure mirror filter bank8

By inspection, this partial derivative is only Elonzero for k = m - 1 and k -- rn. Hence,

Then, using Leibnitz's rule, we obtain

Setting

rm 4-

0 requires us to set to zero the coefficient of p(t,).

Thus,

or equivalently,

Hence,

2t,(rrn

- ~ m - I )= F& -

~ % - ~ r

or equivalently, C,

rr=

Tm

4- Tm-t

2 ' Thus, for the Lloyd-Max quantizer, the transition levels lie halfway between the reconstruction levels, We can reinterpret 1, as

where X is a uniformly distributed random variable and

T,-1

X

< rm*

As we just observed, the Lloyd-Max quantizer is an example of a quantizer, whose quantization noise i s correlated with the input signal. The most obvious model for such a quantizer is given by Figure 3.10, or equivalently,

where the input signal t(n)and the additive noise q ( r ~are ) correlated. We will show that we could equivale~ltlyuse the gain plas additive noise model, wltich can be expressed as

Chapter 3 Mu1tirate Filter Bank8

Figure 3-10. First qumtiaer model.

Figure 3.11. Second quantizer model,

or equivalently in. block diagram form in Figure 3.11, where a is "cbe gain. filctor (0 < a 5 I) and r ( n ) Is the d i t i v e noise term. By making the proper choice of the gain a, we will be able to agsurne that r ( n ) is additive uncorrelated noise. For these quantizers, assume that the expected value of the quantization noise q(n) is zero, that is,

Let us briefly examine soxae interrelationships between the two models, R a m the first model,

or equivalently,

Els(n)l = E[t(n)l+Eiq(n)l* Since E [ Q ( ~= L 0? ) ~then

E[s(n)l= E[t(n)l* from the secorid model,

or equivalently,

E[.(")] = .Bit(.)]

+ E[.(n)].

Equati~lgthe two equations for E[s(n)]yields

e

,-"%

"-"-, w k .--@d

Fz

u CS

PsS

&--a

f2

/--.

-+ u I-.

Q

r?

e . -----, ,+-.

k

v r*3*

w *S

r"."41

r;3

I-.

u

Ft

@""-.

&2 u U

Ezr -4-r

I

Q

P-.

b

Nu

I

T4

il

u

i

b

N Cr"

t;;

$;,

!z

cll

--'I

?I

sf

I . 4

i? 8

3.2 Quadrature mirror filter b a ~ k ~

Figure 3.12. M-channel filter bank with quantizeas.

Figure 3.13. Additive: noiw eqnident circuit,

3.2A.5

Filter b

with qum$izers

The filter bank with quantizers will consist of the insertion of the quantizer made1 in each channel between the analysis bank and the synthesis bank, Thia ia depicted by the block diagram in Figure 3.12* The basic philosophy behind the design of QMF banks with quantiliser~ is to permit aliwing in the filters of the an&Iysisbank and coding gain in the quantizers, and then to choose the filters of the synthesis bank so that the dias-components in the filters of the andysb bank and the coding g d n of the quantizers are cancelled. The andysis of the QMF bank can be performed easily. First, realize that the eRect of ak is simply a scale factor for h t h the desired terms and the terms due Lo a-liming, Second, tke egect of the additive noise source .rk(n)can be described by the equivalent circuit per cllannel as depicted in Figure 3.13. Thus, the filter bank with quantizers yields

Now choose new synthesis filters G k ( z )to be defined iry

Chapter 3 Maltirate Filter Banks

98

Then, the equation for V(x) becontea

-.-

desired terms

I

terms due to aliasing

terms due to quantization Therefore, the errors due t o quantization are random signal-independent errors, which could be eliminated by post processi~zgthe output of the filter bank with a filter that is designed to remove additive uneorrelsted rioise. In addition, the analysis that was presented in. the previous subsections is still applicable for all signal-dependent terms,

In the previous section, we showed that the polyphme matrix P ( 4 characterizes the behavior of a QMF bank. Wfiat are the properties of PCx) that correspond to alias-free ar perfect reconstruction filter banks? This section will address these questions. 3.3.1

Alias-free Alter bmka

First, a background section on pnreudocirculant matrices is presented. This is foXlawed by the theory of diw-free filter banks. 3 3.X. I. Pseudocirculant matrices

Definition 3.3.1.1. A matrix P(z) is sa,id to be pseudociwulant if it is of the following form

Two important special eases of the pseudocircufant property are given below. f 1) If z = 1, then P(z) is circu1ant, (2) If z

=.

-1,

then P ( z ) is skew circulant.

For example, a pseudocirculant matrix P(s) of dimension three is given by

3.3.1.2

Elimina&ingalias distortion

Consider the filter bank depicted in Figure 3.14. Now, we will proceed with the a~nalysisaf the filter bank, by examining it stage-by-stage, First, we will con~ider,the bank of decimators, which is depicted in Figure 3. f 5. The elemental equatiorl for this stage is given by

Next, we will consider the P(z) block, which is depicted in Figure 3.16. The elemental equation for this stage is given by

Lastly, we consider the expander block as giwn in Figure 3.17. The elemental equation for this stage is given by

Ghapde-r 3 Muftirate Filter B m k ~

Figure 3.14, Polyphwe reprmentatiorz of M-cfiiiwlnel%lterbank.

Combining the equation for C,(r) and B.(r) yields

Combining this equation with the one for Y ( z )yields

or equivalently;

where

M-l

V,(z) =

C z-"z-(~-'-')P

#?a(z").

Since x ( ~ w & k) ,$0, represents the alias terms, then the resulting equation for Y ( z ) is free from aliasing if and only if the coefficients of x(zw&), k $ 0, are zero, that is, M-2 R=O

w&'"v,(z)= 0 for k jk 0,

3.3 Foundations of filter banks

Figure 3.13. B a k of dmirrza-k-or.

Fiere 3.16. P(x) Mock.

Figure 3-17, Bank of expanders.

ar equivalently in matsix notation,

where * is a nonzero term. Premultiply both sides of this equation by W M . Then, using the fact that WM = MI1 this equation becomes

Wg

Since all the entries in the first colu~nnof W M are equal to one, then & ( z ) = &(.)

Therefore, let

= V,-,(r).

3.3 Foundations of fillter banks and the alias ttems become

M-I

M-I

C W;'"

n==O

=

Ce

j2~rFn ~

p

I-exp(j2nk) = O for k $0. (= ~ )

n=O

Thus,

U(.) =: V(z)X(x) and the M-channel filter bank is free from sliming. Now, let us examine & ( z ) and & ( z ) more carefully. By definition, ( z ) is given by

Perform a change of variables by letting q = M

- 1 -- s, Then,

Similarly, by definition & ( z ) is given by

or equivalently,

Perfarm a change of variables by letlbg q = M

- s. Then,

Because of the requirement that h ( z ) = Vl(z), the polyphase-components should be the same. So, Vo(z)is simply a pseudocirculant shift of & ( z ) . Similarly, the nth column is obtained from the (n 1)st. Thus, P ( r ) must be pseudocircuEant for the filter bank to be alias-free,

+

Example 3.3.1.1. Consider the following example that illustrates these results. Consider the following P(r)matrix

(4 .'&@I(2)

pi2

(z)

pie(z) f'i~(r) &(z) &a(%)

P2l(Z) Pz2(r)

Thus,

Since then

Bence, we earr write

Let P2(2)= Poz(r),Pl(r)= q2(+), and Po(z)= q2(z). Then, P(z) becomes

s ( ~ )4(+)S(+) Po(z) PI(%) r-'Pl(r) z-'P,(z)PO(z) z-'&(z)

3.3 Foundations of filtc3r ;banks

105

First, a background section on paraunitary matrices is presented. This is followed by the theory of perfect reconstruction filter banks.

Definitian 3.3.2.1. A matrix P(z) is stkid ta be parraupzitary ifibnd only if

where c is a pa~itivecanstant. If c = I, thee PCs) k n o m as a namaiized puraunitraq matrix.

Two important speeid eases of the paraunitary property are given belaw when P ( z ) is a constant matrix.

(1) Xf VV is: a matrix of complex-valued constants, then W is unitary if and only if WaW = I. (2) If U is a mat^ of ~al-valuedconstants, then U is odhogonal if and only if UTU = I. Let us consider the following t hearem, which provides a fundamental result for products of paraconjugated matrices. Thearenn 3.3.2.1. Let A(x),B(x), and C(x) be maGrices suelr that

then ~ ( t ) = ~ ( z ) ~ ( z ) . Proof: Let A(z) be I x n, C(L)Be Ix rn and B j z ) be m x n so thal C ( z ) B ( r )is defined. Then, e ( z ) is m x I and g(z) -- is n x m SO g ( z ) c ( z ) is also defined and it has the same dimension as ~ ( z ) Let [A(z)lij= a i j ( z ) , = r i l i ( ~a) , d [B(z)]kj = B h j ( + ) * Then, the ( i , j ) element of A(%)is given by

[c(J)lit

ij

= a;i(l/z*). Then,

.

3.3 Foundations of filter b a n k

107

First, consider the distortion function, which corresponds to an aliasfree filter bank, i.t?,, (see Secti~rr3.2.2)

or equivalently,

For simplicity, represent P(z) in terms of its zeroth row, that is, &,a(.)

=

PO,,-k(z), for k = O7... 1~ Z - ' ~ @ , ~ - ~ + ~ ( Z for ) , k = n + I.,. . .,M - I..

Perform a change of variables in the first summation by letting k = n - s and in the second summation by letting k = n- s+M. Then, the distortion f~~nedion beearnes

or equivalently,

Thus, the M-channel filter bank is free from a l i ~ i n gif a d only if the P ( x ) is pseudocirculant, Moreover, the distortion function, corresponding to this alias-free filter bank, is given by

where S(z) is the polyphase representation of the zeroth row of P(z),that

or equivalently,

where B ~ ( z ) the , delay chain, is given by

e M ( r )= [ I , z-',

... ,z - ( ~ - ' ) ] * .

Then, the kth rovv fk > 0) is a riglzt circularly shifted version of S(xf, or,

The corresponding system of equations ia given by

Performing a change of variables by replacing z with fallowing equation

ZW;'

yields the

or equivalently,

where e M ( z w i k )= ( z ~ & ~ ) -. .' ,~(2. tions can be described by

. This set of equa-

3.3 Foundations of filler banks where

A(z) = diag[l, r -I,

... ,z - ( ~ - ' ) ] .

Since t h i ~equation holds for all values of k for k = 0,. .. ,M write it as A ( . ) ~ M Q ( .= ) P(~~)~(.)wM where W M is the M x M DFT matrix and

- I, we can

Q ( z ) = diag [S(z), . . . ,S ( z W-(M-ll)]* ,

Thus, any prseudocirculant matrix can be diagonalized, 3.3.2.3

AmpUtude distort$an-free Alter bank

Since A(r) and WM are paraunitary matrices, then Q(a) is a paraunitary matrix if and only if the peeudocirculank P(xf is paraunitary Moreover, Q ( z ) is a diagonal matrix with element8 ~ ( r ~ i 'Therefore, ) . Q ( t ) is a paraunitary mat;rix if and only if S ( z ) is allpws, that is the magnitude of S(z) is constant, or equivalently, Q(z) is a paraunitary matrix if a ~ d only if T ( z )is allpass, since T(z) is simply a delayed version of S(a). Note that T(x) is allpass if and only if the filter bank is amplitude dislorliunfree with m o n o t o n i a l - r i g phase, hence, TCx) is dlpms if and only if the pseudocirculant P(z) k paraunitmy or equivnlently, the filter bank is amplitude distortion-&ee with monotonically-varying phme if and only if the pseudocirculant m a t r k P(x) is paraunitary, 3'3.2.4

Perfect recomtrmcdioa. mt er b a d 8

Since P(.~)A(z)wM = & ( ~ ) W M $ ( Z ) then det(p(zM)) = det($(z)). However, M-l

Therefore,

a s(zw~"$

M-1

det(p(zM)) =

I%=@ OP

eqtuivalexttly in. the Fourier domain,

Chapter 3 Muldiratc? Filter Banks h s u m e S ( z ) is allpw8, then,

for some nonzero d and a monotonically varying phave rp

n

M-l

det(P(exp ijwMj)) = d M

exp

k=O

or equivalently;

Assume S ( I ) hau linear phase, then, 'p(w) = gw

+f

fer constants g and f . Hence,

Recall t h a t

cE~=;' k = M ( 7 - g .Therefore,

or equivalently,

where a = Mg and b = M f - z g ( M

- 1).Hence,

+

det(P(exp ijwM]))= d M exp [ j ( a w b)] or simply, det(P(exp b w M ] ) )= c exp(- j w m )

111

3.3 Fouadstions of filter banks

where c = dM exp(jb) and rn = -a. In the z-transform domain, this equation becomes det(p(zM)) = C Z - ~ . This equation implies that PoSr(z)must have a single norizero term. Let po(t) be a vector of the polyphase terms in the zeroth row of P(z),that is?

Since each po(z) can have only one noIiaero component, then successive circular shifts of p@(z) will result in M different P ( r ) matrices, each of which corresponds to a different perfect reconstruction system. Now, we wilt enumerate them using the notation p,(k)(z) to denote the fact that k circular shifts have been applied to po(t). Now, aYvume the initial unshifted fa) (2) is given by vector po

which represents tlre yseudocilieulaxlt matrix

Apply one circular shift to pr'(z) to yield

pt'( z ) ,that is,

The corresponding pseudocirculant m a t r k

Therefore, after k circular shifts of

pr'

( z ) , we obtain

which represents

To illustrate this result, a two-channel filter bank has perfect reeoxrstructiori property if and orrly if

Chapter 3 Maltirate Filter Banks

112 3.3.3

Filter bank formulations in retrospect

Xn this chapter, we introduced two fitter bank formulaliana: alia-component (AC) and polyphme formulations. of tlie The st;rategy of the AC formulation is to farce the ~oefticient~ Aimed terms to zero by imposing conditions on Ak(x),k =z 0 . . . ,M - 1, the components of the gairk vector. Attributes of Filter Bank

Characterizadian of A(z)

Perfect Reconstructioxl

AEI(z) = zWmoand AL(x)= 0; k f O

As we saw in Section 3.2.2, this approwh can lead to dificulties. Xt requires the inversion of the alias-cornyonerrt xxratrk and, even if this is successful, there is no guarantee that the resulting analysis filters are stable. The strategy of the polyplime formulation is to =same the desired result and then to impose conditlarks oxr the polyphase ma-lrix P ( z ) in order to obtain the correct behavior of the filter bank. Properties of Pix) Attributes of Filter Bank Alia-Ree, Anrplitude Distortion-fiee, and Monotonically-Varying P h a c

Paraunitary, Pseudoeircufank

Perkct Reconstruction

Paraunitstry, Pseudocireulant Linear P hme

We will exarmine two types of filter bar~ksfar spectral analysis, rtamelly the DFT filter bank and the cosine-modulated filter baxik. Bath of the filter bariks dkeussed in this sectiorh have the advantage that all their filters are derived from a single prototype filter. In addition, we will see th& the cosixie-modulated filter bank car1 achieve perfect reconstruction.

3,4,1 DFT Alter bank Traditionally, the OFT haa been used to analyze the spectrum of a given Bignal and an eacient, imglernentat;ian using FFT makes DFT a popular choice. Let h(n) be a towpass FIR prototype filter of length N" Then, the bmk of andysis filtenl, Hk(x), k = 0,. . . M 1, can be defined as

-

Assume that the filter length N = m M , where M is the number of channels in the filter bmb md m is an arbitrary integer constant. Using the Integer Division Tbeorenn, let n = M p q, then Hk(a)becomes

+

Since w$~'+')= W of Ek(z) is given by

wZ,

~ ~ =P W then ~ the polyphase representation

or equivalently, M-I

H ~ ( z=))3 Gs(zM) W$

2-9

q=O

where,

Now expressing the analysis filter bank form yields

tlk(t),k

= 0, ...,M

- 1, in matrix

h(z)= w M

where WM is a M x M DFT matrh, Let

where the delay chain eM(2) = 1, z , , . ., ] decomposition of h(z) is defined to be

then, by inspection,

E ( z ~=) W M ~ ( Z ~ ) -

21'

. Since the polyphwe

Chapter 3 Mulkirate Filter Banks

If-$

Assume that, the polyphme-comporrent matrix of the synthesis baizk R(xf = E(z). Then, P(z)= R(z)E(z)= %(z)E(z)* or equivalently,

p(4 = ~ ( ~ ) w E w M ~ ( . ) . Since W g W M= M I M , then

or equivalenll~r,Pfz) is a diagonal m a t r k whose elements are given by

Since P ( x ) is a diagoxlal matrix, it can becomc psendocirculant if all the diagonal elenlents are equal, that is,

Orrce P(z)is pseudocireulant, then, by definition, the filter bank is aliasfree, But, if all the diagonal elerrierlts are equal, that; is,

then this corresponds to a square window for %heDFT corngutillion, which will cause artifact8 in Lire form of large main and side lobes. To reduce these artifwts we could use overlapping windows, but dltk would force us t o give up the aliw-free property. A way around this difficulty is to use cosine-rrtodulated filter bax~ks-

This cstudy of eosine-alodctlated filter banks will be broken into two parts. First, we will define reversal matrices and then we will, examine properties of cosine-xnodulated matrices. Then, we will analyze cosine-modulated filter banks,

3.4.2.1

Reversal matrices

Definition 3.4.2.1. Reversal nlatrices are matrices with zeroes in all ea tries

except on the anticIiagonal. Reversal matrices of dimensio~im are denoted

ISy Jm*

3,4 Filter banlrrs for spectral m a l y ~ i ~ For example, J3 is given by:

3.4.2.2

Gosinc?-moddatedmatrices

In order to andyze cosine-modulated filter banks, we will need to provide a couple of theorems. Theorem 3.4.2.1.

l)&i7(~ -mM

For each k , q = O ,... , N - 1, l e t Then, C L , ( ~ M ~ + = ~)

ckvq

+ 3) + (-I)"]*

Proot: By definition C

~ , ( Z M ~ is + ~given )

Recall that for two angles a and Hence,

= 2cos[(2k +

Ck,g.

by

P, cos(a + P ) = cosa cos @ - sin a sin P.

+ + $1 +(-1)"] + 2 sin[(2k + l)p?r] x sinl(2k + l ) & ( q - m M + $) + (-I)"]. Since sinL(2k + l)plr] = O and cos[(2k + l ) p ]= (- 1)P, then C L , ( ~ M ~ += ~)

Theorem 3.4.2.2.

2 C O S [ ( ~ ~l ) p r ]C O S [ ( ~l)&;i(;i(4 ~ -mM f

ctTef= 2

~+ 2 ~ 1( -I)("-')~

JM is an M x M reversal matrix and for every ~ by.

~ k

,

~

5I

T-4

3C4

;ri

+ -5 c'"

w

*g

8I 4

42

II

-

3

gi

*-

, , t:

a

b 4-z

Q

Ed

ti!

4.a

2

h B a

&

.El

a

* 3 8 '12

F

kt*

-

I

3.. 11.1

u

hm.""d

@-%

En

r f i

E:

-

&IN

@r,

+

kt$

u

/-%

+

4

ISI?

CIJ

re*

a

& . & &,"".,a

w3

I

CJ

co~[(-l)~$] = $ and sin[(-I)"] IAZIL,~ become

Since

= (-I)'+$

therl ( A o ] and ~,~

s r equivalently,

Recall that cos(a-p) = cos a cos p+sin a sin /3 and sin(&-p) = sin a cos Pbecame cos a sin p. Then, ( A Q ] and ~.~

+

Since cos[(2k I)$] = O and sin[(Zk [Al]h,p become

and

+ I):]

= &c..[(zk + l)&(p -(-l)kJZsin[(2k

= (-I)', then [A@]k5p

+ ?)I

+ 1)&(g + f ) ] .

Let G and S denote Type-IV discrete cosine transforms (DCT) and Q p e - I V discrete sine tran~forms(DST) matrices, respectively. Then, the elements of C and S are given. by

and

= G s i n [(2X

+ 11-t21M %=

Let A be an M x M xnadrk defined by

[A]Lp = ( - 1 ) ~ 6 k * ~ ; k , q = 09 * * M -- 1. a

Then,

laslk,

= (-Ilk [SIkq.

Therefore, [ A o ] ~and , ~ [ A I ] ~can , ~ be written as

and

[-&I,,

= Jii;i [C - ASI,, * Consider the equations for A F A ~A , ~ AA~T ,A ~and ,

A~AI=

and

aTa,

M

ATA~.

+ ~ $ (C1- AS) ~ cTc - C ~ A +S S ~ A -~SC~ A ~ A S ) ,

(C

= M (C -

AS)^ (C - AS)

k,k = (-llk (-1)' = 1. Therefore, A ~ = AInr In addition, since the Type-IV DCT and DST matrices are orthogonal, then CTC = IM and ST$= IM.IIence,

119

3.4 Fijter banks for spec trd andysis

Let J M be the M x M reversal matrix. Then, ASJM = C. Moreover, since 52, = IM,then CTAS = J M and STnTC= Jnn Therefore,

which proves the special cave of m = 1. We must now extend our results for arbitrary integer > 1, by expressing the interrelationships between {A@> Al) and {A0,Al). For m even, A;

= (-1)T~x

A;

= (-1)Y-lAo

Utilizing these relationships gives the follawing: For rn even, A ~ A ; = ~ M ( I M - JM)

nbTa; = o

For rn odd,

A;TA~

= o

n;Ta;

= ~ M ( IJM). ~

A ~ ~ A=;

AiTA;

+ 2M(Ins + JM)

= O

a;Tab = o A ; ~ A ; = 2M (IM- J M ) . Combining the odd and even results yield8

Chapter 3 Muidirate Filter BanErs

120 3.4.2.3

Cosine-mad&Led filter banks

Let h(n) be a low-pass FIR prototype filter of length N . Then, the bank of analysis filters, Hk(z), k = 0,. .. M - 1, can be defined as

where and h(n)is a linear phase, low-pass FIR prototype filter. Assume that the glter length PJ = m2M, where M is the number of channels in the lilter bank and m is an arbitrary integer constant. Using the Integer Division Theorem, let n. = ( 2 M ) p q, then Hk(z)becomes

+

Since G , ( ~ M = ~ (+ ~~ ))P c ~then , ~ ,the polyphase representation of is given by

Hk( r )

where, n-l

Now, expressing the analysis filter bank H k ( z ) , k = 0, ...,M - 1, in matrix h r m yields G ~ ( - z ~ ~ )

Z-'G~(-Z~~) h(r) = C'

9

z-(zM-l)~zM-l( - z Z M ) where C' is a M x 2M matrix defined by

Let go(-.)

= diag [Go(-z), GI(-z),...,GM-I (-z)]

where the delay chain eM( z )= 1, I-', decomposition of h(x) i s defined lo be

... , z - ( ~ - ' ) ] I" . Since the polyphase

then, by inspection,

Atraume that the polyphase-component matrix of the synthesis bank R(z) = @ ( I ) . Then, P ( z )= g ( z ) ~ ( z )

+

Since c ' ~ c='2M12M 2 ~ ( - l ) ( ~ - I )

JM

0 -JM

0

, then

122

Chapter 3 Mul6l"rat.eFilter Bank8

Assume H(r) has linear phase. Hence, the polyphase-components of H(z) are related by G L ( z )= z-'m-1'G2M-1-k(%). Since and

2 g , ( - ~ 2 ) = diag [ ~ M - l ( - r ~ )GM-2(-z , ), ... r GL)(-z')

%hen r"C

go(-+') = z-~("-') JM diag

and GI(-z2) = z - ~ ( ~ JM - ~ diag ) [ G Q ( - Z ~ ) , G ~ ( - Z .Z. .) ,GM-l(-z , "C

or equivalently, and

2

)

go( - z 2 ) = ~ - ' ( ~ - l J) Mlpl (-Z2)

-gl(-zZ) =

J ~ ~ ~ ( - z ~ ) . Substituting these equations for go(-z2) and g1(-2') into the equation for P (z) yields

Since go (-rZ) and g1(-r2) are both diagonal matrices, then mutes with (-zZ), d.e.

( - z 2 ) com-

Therefore, the second term of the equation for P ( r ) equals zero. Hence, or equivalently, P ( z ) is a diagonal matrix whose elements are given by

Since P(z) is. a diagonal matrix, it can became pseudocirculat If all the diagonal elements art: equal, i,e., IP(z)lO,O = [p(z)ll,l = * = p(z)lM-l* M - 1 * Once P f a ) is pseudocirculant, then, by definition, the filter bank is dimfree. Moreover, if all the diagonal elements of pseudocirculant P ( z ) are equal t a a constant timers a deEw then the cosine-modulated filter bank satisfies the perfect reeoxlstruetion property.

,5=J

,--,

I

p w ti.

Y Y k-' "5'

F a Q M

%a.

+- -'* F tzl

G2

hS,

.+-% '

3 R"

Proof: Obserw that z ~ i , i i :h a compact support and by construcGion satisfies ~ ~ j , k l l & . (< ~ ) m for each j E % and k E N. Therefore, { ajVk[ j E Z, E NJ c L ~ ( R ) .

Next we demonstrate that { 1 j E Z, k E N)is an orthonormal set. By the nonoverlapping prop rty of the positive support intervals, it i s clear that (U j,h, % i , l ) = 0 if li 2 for d l b,l E N,"f"herefore, we need to consider two cases: (I) i =: j and (11) Case 1: We will show (uj,k,u j , l ) = 6 ~ , To ~ . see this, observe that

- >

P

From the definitions of even and odd extensions one sees that

Note that C = 6 k t I by choice. For A, let a: = aj -0 and for B, let z = aj + a to yield

BY Property (c), wj(z

hence, A z

== aj+l

- C) = w j - l ( s + 0)and so property (d) implies

+ B = 0. Similarly, in integral D,let a: = aj+l - CF and in E, let 4.o ta get

Properties (c) and (dl) imply

k,rT

D -1- E: .- 0. Consequently,

-

Case II: We will ~ h o w( u ~ ~ = ~ O~forz iL=~j ~ l~for) every j E Z* E N then the ewe i = j 3. wi11 follow ewily.

+

3.4 Filter banlrs for spectral analysis

125

By construction, %j-l,k and uj,i are possibly nonzero only on (aj - E j ,

aj

+ ~ j )hence ,

From the definitions of the extensions

In the first integral, let s = ai -cr and in the second integral, let s = aj +o, t hen

Proper@ fc) inrpller~

Therefore, ( ~ j - ~ ,ukj J, ) = 0. Cases I and II demonstrate that { ujtk[j E Z, k E N) is an orthonormal set. Lastly, we prove that .( %j,k ( j E 2,k E N) is a basis for L2(R). To do this, we will show that given s E L2(R)there exists a set of scalars .( aj,kl j E 21, k E N) such that

in the LZ(R)sense. Let s E L2(R)and define sj(z) = s ( z ) ~ j ( zfor ) each j E 2.Since sj has positive support on ( a jW E j , aj+l +ej+l) We fold sj(z)on (aj - Ej7aj)and (aj+I,aj+l Ej+l) into the interval [aj,aj+l]by defining

+

kj(~)

Now hj is supported on [aj,aj+l]. Consequently, there exist real numbers for k E N such that

aj,k

Chapter 3 Mtlltl'rate Filter Banks

226

where convergence and equality is in the L 2 ( I j )sense, and ajsr,is given by the i~lnerproduct rule

Applying the furretion rules at both exldpoirrts yields

n

where h j h a an odd extension about z = aj and an evctn ext;ensicsn about z = a j + l . That is, the use of the extension rules for fjtk(z)applied t o h j ( z ) yields the following )c

Multiplying

^hj (z) by wi(z) and summing over j

produces

To coxnplete the proof, we wiU show

If z is a point where s(z) is defined and z fixed J E 25 then

If

2:

E (aJ - E

J , ~ J

+ , then EJ)

E [aJ

+

E J ,a

+ E $ + ~ ] for some

~+i

+

[(n:

- f ~ zf'g) - (a)fsj(z)fm+

[(z- r ~ ) g ) x - r+~ ( Z ) I - T S ] ( z ) I - f m

=

(r)gy(z)~m z3[3 w

Chapter 3 Maltirate cFilter Banks

Figure 3.19. M-channel muldidimensionaf.Glter bank,

and polyphase, Then, in a manner andogous t o the one-dimensional case, vve will closely examine the polyphme matrix, whkh characterizes the behavior of the multidimen~ionalQMF banks. For simplicity irr Clrapter 2, we were able to refer to shift vectors as a without having to msume an ordering of them. In multidimen~iorlalfilter banks, we will. need ta asociate an ordering t o the shift vectors, so they T vvill be denoted ai, where a0 = [Q, 0,. .. ,O] by convention. 3 .5,1 Mufrtidimensional Alter b a d formulationis

We wilt discuss two digerent approaches t o Lhe analysis of multidimensional. filter banks: the abas-component formulation and the polyphwe formulation. Then, we vvill show that the two formulations me equivalent.

The input is denoted s(n) lund the output is denoted y(n). It consists of J(M) multidimensional decimators; J(M) multidimensional expanders; the J(M) multidimensional analysis filters denoted Hk(s), k = 0, ... , J(M)-1; and the J(M) multidimensional synthevis filters denoted Fk(a), k = 0,. . . , J f M)- 1, Figure 3.19 depi&s an M-channel multidimensional filter bank. The basic philosophy behind the design of MD-QMF banks is t o permit aliasing in the multidimensiond filters of the analysis bank and then choose the multidimensional filters of the ayn"cfiesis bank so that the alias-

Figure 3.20. Multidimensionill mafysis bank.

compments in the mullidimensionaI, filters of the analyois bank are cancelled. Now, we wilt proceed with the analysis of the nrultidimensiorld yuadratare mirror filter bank, by examining it stage-by-stage. First, we will consider the analysis bank, which is depicted by Figure 3.20. The elemental equation for this stage is given by

Then, we will consider the bank of deeimatom, which is depicted t?y Figure 3.21. The elctmentrtl equation for this stage is given by

The following stage consists of a bank of expanders, which is depicted by Figure 3.22. The elemental equation for this s t q e is given by

Lastly, we will consider the synthesis bank, which is depicted by Figure 3.23. The elemental equation for this stage is given by

Chapter 3 Multirate Filter Bank8

Figure 3.21. Multidimensiond bmk af decimators.

Figure 3.22. Multidimeasional;b m k of expmder;~.

Figure 3.23. Multidimensional;synthwis bmk.

Combining the equations for X k ( z ) , V k (z), and Uk(I)yields

Finally, combining t hi8 equation with the equation for Y (z) yields

desired terms

terms due to aliasing

where,

arld fies a t the origin. A few observations cm be made. First, the desired term can be interpreted as the multidimensional input signal weighted

by the mean of the product of the multidimensional analysis and multidimeasion& synthesis filters. Secondly, if the coefficients of the aliasing term can be set to zero then a linear time invariant (LTf) system can be constructed out of linear time varying (LTV) canrponents - multidimensional decirnators and multidirnensionat expanders. In this c a e , when diwing is cancelled the rnultidimnsianal distortion function is given by

= e $0i for all g, we say V ( z ) Unless: T ( s ) is allpass, i,e. suEers from amplitude disto ly, unless T ( z )hw linear pphase, i.e. #(g)= arg g(T(exp(jg))= a f bg for constant a and b, we say Y ( z ) suffers kern phme distortion. If the s y ~ t e mis free from dirtsing, amplitude distortion, and phase distortion, then T(z) is a pure delay, i.e. T(z) = CZ-"? c 0. In such a system, yfn)is a sealed and delayed version of z(n),i.e. y (n)= cz(n - no),and the resulting system is called a perfect reconstr2~clionsystern, Let us reexamine the output of the filter bank Y(z), which was given BY JIM)-l 1 Y ( z )= r n A i ( e ) X z exp [- j(2?r~-~)a~

+

C

i=O

where

Writing the system of equations A ~ ( z ) i, = 0,. ..,J(M) yields 1. A(z) = -H(z)f(z),

J(M)

where the vector of gain terms A(z) is defined by

[A(%)],= A k ( g ) , the synthesis bank f (z) icr defined by

and the alias-component matrix H(z) is defined by

- 1, in matrix form

In this formuladion, aliasing can be eliminated if and only if the gain for eaeh of the aliasing terms equals zero, that is, A*(%)= 0 for i = 1,.. . , J(M) 1, M o r e m r , to insure perfect recuwfwction Aofz) must be a delay, i.e. A@(%) = z-rnO for some integer rector m. The solution of this system of equ&ions for f ( ~ may ) have practical difficulties. It requires the inversioll of the alias-component matrix H(z). Even if successful, that is, Hfz) is nonsingular, there is no guara~lteethat the resulting filters f(z) are stable. The approach given in $he next section presents a, CfiRerenL technique in which all of the above diEculties vanislli,

Now, we consider the polyphme representation formu1a;tion of multidimensional filter banks. Towards this end, we will expand the desired result in terms of pofyphanse, Then, we wifl determine the conditions t o be placed on this result su aij to achieve perfect reconstruction or simply slim canceUation, Now, recalf that the desired result is given by

or equivderrtly in matrix notation 1

Y ( z )= -fT(g)h(z)~(z),

JIM)

where the synthesis bank is defined by

[f(z)lr:= Fk(z), and the analysis bank is defined try

[h(e)l, = HL(E)* Writing

Hk(s), k = 0, . .., J (M) - 1, in terms of Type-I polyphase yields

for aj E N(M) and k = 0,. .., JfM) - 1. This system of equations can be rewritten in matrix notation at4

h(z) = E ( ~ (zM ) )eM (z),

Chapter 3 Multirate Filter Banks

I34

where the delay ehain eM(z) is defined by

and the J(Mf x JIM) polyphase-component nlatrix for the multidimerl-

sional analysis bank E(~)(z~) is defined by

E N(M) and le = 0,. .., J(M) - 1 . Then, writing Fk(a),k = 0,. .. , 3(Mf - I in terms of Type-II polyphwe yields

aj

for aj E N(M) and k = 0,. .. , J(M) - 1. This systenl of equations can be written in miltr-irtrnotation as

where the paraconjugation of eM(z),denoted gM (z),is given by

and the J(M) x J(M) polyphase-component matrix for the synthesis bank R ( ~ ) ( zis~defined ) by

E hr(M) and k = 0,. . .,J(M) - 1. Substituting these equations for h(z) and fT(z) into the equation for Y (z)yields

aj

Let ~

((aM) ~ =R 1 ( ~ ) ( ~ ~(zM), ) E (Then, ~ )

or equivalently, Ylzf =. T(x)X(z), where the multidimensional distortion function T ( z )is given by

These equations suggest the filter bank depicted in Figure 3.24.

Figure 3.24. Multidimensional filter bank,

3.5.1.3

Retalions;lftip between formdaklions

As seen by the previous subaeetions, the study of filter banks can be performed using either alim-component or polyphae matrices. Now let U S examine the relationship betwen these two formulations. The AG m & ~ x H(z) is defined by

so,

Recsll that h(z) is defined by

[h(z)l, = Hm(z)* Let the ith row of HT(z)be designated by

Recall that h(a) is related to the polyphase-component matrix E ( a ) by

Therefore, HT(a)can be defined by

(%)Ik

Since [eM

=

then

Chapter 3 Mu1tirate Filter Bank8

136

... ,

Let A(z) = diag[l, z-a' ,

Then,

z-~J(M)-'].

H~(z)= E(~)(z~)A(z)w&

With this equation, any results obtained in terms of the polgphwe formulation can be ayplied to the alias-compoaent formulation and vice versa.

First, a background section on generalized pseudocirculant matrices is presented. This is followed by the theory of multidimexrsionstf aEiw-free filter banks.

Definition 3.6.2.1. Gives a sampling matrix M a ~ ad specific ordering of shift vectors ai, i = 0,. .. , J(M) 1, in N(M). Then, a generdized pseudocirculant matrix P(z) is defined by

-

where

+ aj -

a* and

f ( i ,j ) is the integer defined by ((ai

at(i,j))

+

= af(i,j).

Algorithmically speaking, P(z) can be determined by the following sequence af operzations: 1. Evatuate ai aj; i=13."., J(M)-1, j=O,. ..,J[M)-I* and 2. Define f (i,j ) = m, where ((& aj))M =

+

+

3. Solve for gfi,j ) using

4. Evaluate $he folowing equation

Therefore, given a sampling matrix M with k cosets, then all the relationships that define the generalized pseudocirculant matrix are of the form

z ~ ~ ' ~ ~ ~ P=~ +Pfci,jl*i(z) ~ ( z ) ; where i , j , f ( i , j ) E { O , 1,

,

k-

Thus, P(z) is a k x k matrix, since dim(N(M)) = k. For example, let us determine the generalized pseudocirculant matrix

with cosets given by

In addition, note that M-'=!j

1

+

1

Case I (aa R ) :

(I) Evaluate a~+ a1 =

0

0

3-

(2) Since a0 + a1 E N(M), then

1 0 ((a0

-

1

0

+

= ax. = ao

+ al = ale

Thus, f (1,O) = 1.

(3) Since a0

+ a1 = af(lpo) = al, then Mg(1,O) =

Hence, g(l,0) =

0

0

0

0

(4) Therefore, .oO$~o,o(~)

= 4,1(.),

or equivalently, P0,0(~ =)Pl,l(~).

+ az): (1)Evaluate a@+ a2 =

Case I1 (%

(2) Since ao +

0 8

-4-

1 1

N(M), then ((ao+

= 832. =

+ = a2.

138

(3) Since a 0

Chapter 3 XMulLirade Filter Banks

+ a2 =

Hence, g(2,O) =

af(2.0)

= az, then Mg(2,O) =

0

0 0

(4) Therefore,

= Pz,z(.),

PO!@ .):.(.

or equivalently,

Po,o(s)= Pz,2(z).

(2) Since a1

+

a1

# N(M), then

Thus, f (I, 1) = 2.

(3) Solve for gf 1,1):

(4) Therefore, .:~!~t,o(~)

= P z 9 1(z),

or equivalently,

=

~ 0 ~ 1 , 0 ( ~&,I(Z). )

Case IV (al+ az): (I) Evaluate al + % =:

(2) Since a1

0

1 0

4-

+ a2 # N(M), then

1 1

-

.-,.

2 1

Figure 3.25. Multidirrteasianaf filter bank,

Hence, g(2,2) =

4

-

0

"m".

1

(4) Therefore,

.oO.:pz,o(.)

= Pt,a(g),

or equivalently, aPZ,o(g)'

211,2(%).

With the conditions that were obtained by considering the above cases, the follawing generdized pseudocirculant m a t r k i s obtained,

Consider the follovving multidimensional filter bank depicted in Figure 3.25. Xn the polyphae representation of one-dimensional filter bank, we utilize delay elements at both ends of the filter bank, since there is no need to make a naneausal filter bank because of delays. On the other hand, in a muldidirnensionl filter baak, we will ~ h i f ethe inputs and shift them back a t the outputs, We observe that Y ( z )=:

where

Since X ( z e ~ ~ [ - j ( 2 ? r M - ~ ) k ~ I $] )0,, represents the alias terms, then the resulting equatiun for Y (z;) is free from Aiming if and only if J(M)-t

C

~ X ~ [ - ~ ( ~ E M - ~ ) ~ ~ ]=- O' ~for V I~#( 0, ZI)

or equivalently JfM)-x

e ~ ~ l j 2 ? r a T ~ - ' ~ ] ~=, (O zfor ) I

# 0.

m=O

But exylj2akT~-'k,] is simply the (1, m)th element of the complex conjugate transposed of the generalized discrete Fourier transform matrix, w&'.Hence,

Pre~nultiplyboth sides of this equation by w$'. Then, using the fact that H

L=

J(IJlIr)I) this equation becomes

Since all the entries in the first column of

%(E) = %(Z)

= a s *

wZ' :)re equal to one, then

= VJ(M)-l(~)

Therefore, let

V(z) = V,(z) for m = 0,. .. ,J(M) - 1.

Chapter 3 Multz'rilte Filter Banks

142

Hence,

for d l i = 0,. . . , SCM)

- 1, or equivalently,

-

for all i = 0,. . . , J(M) 1. Using the Multidimensional Division Theorem, we can express ai $- a* as

where g ( i ,j ) E N and at($,) E N(M). Then,

for all i = 0,. . . , J(M)

- 1. This leads t a

for all i, j = 0,. . . , J(M) - 1. Polyr~omialmatrices satisfying this relati011 are called gerleralized pseudocirculaxll matrkcs with respect t o M for a specific ordering of ais in N(M). 3.5.2.3

Perfect reeomtructioa QMF bank

Tlie multidimensional filter bank xhieves perfect reeonstruetian if and only if P(z) i s a generalized pseudocirculant matrix and all the elements in the first column are zero except the one P 3 , 0 ( ~ that ) is equal to a delay, that is, if j == j0 is the index of the nonzero entry, then

P,%@(Z) = 3.6

czM if j = jO

0

otherwise.

Problems

I, Consider a two-channel aliw-free filter bank. Using the alia-cornpone& formuhtion, solve for the most general form of synthesis filters. If we assume perfect reconutructio~,then how do these equations simplify?

3.6 Problems

143

2. Let z(n) be an arbitrary sequence and let sl(n)be the first divided diEerencc, that is,

and let zzfn) be the sum of the laat t m samples, that is,

+

z2(n)=I z ( n ) z(lt - I). Consider the two sequences yl(n) ctnd yz(n) tllstt are defined by yl ( n )= z1(2%) and yz(n)= zZ(2n).

Can we recover z f n) from yl (n) and

(a)?

3. Consider a special case of the Lloyd-Ma quarrtizer, where the probability density function is a constant over the &-transition XeveXs of the quantizes. Wfi;tfi is the probability density functicm? What is the corresponding mcta~isquare error? 4. Let A(z) and Bfx) be k x k pseudocirculztnt matrices. Prove that A(z) commutes with B(x), tha'c is,

5. Construct the generalized pseudocirculant matrix using the sampling matrix

and with eosets defined by a0

=

0 0

and a1 r=

Chapter 4

Lattice Structures

This chapter introduces the notion: of lattice structures for the redization of filter banks, The origin of lattice structures for continuous lossless systems was a classicrtl theory of LC circuit networks, since they do not generate or dissipate energy (see, for example, Belevitch(21). Results on discrete time systems and their factorization can be found in Vaidyanathan et aI.[51] and Doganata and Vaidyanat han [14]. Sonle of the concepts developed in this chapter are d s o discussed in the texts by Fliege[lSI1 by Strang axkd Nguyen [46], and by Vaidyar~at ha11[49]. Section 4.2 introduces multi-input multi-output (MZMO) linear system theory: Section 4.3 presents lattice structures for lassless systems. 4.2

Ramework for latit ice structwes

This section systematieallly presctlts concepts that a t as. a framework for our study of lattice structure^. These concepts include an introduction to multi-input multi-output systcrris, the Smith-MeMil1an.n form, and the McMillan degree of a system. 4.2.1

Multi-input mufti-output systems

Befinition 4,2.1.1. Let u(n) be an input vector of length r , that is,

and let Y(R) be an output vector of length p, that is,

Cliapter 4 La t tice Structures

146

men, the multi-input multi-ou tpu t system is described by

where,

and [H(z)lk, denotes the transfer function from the m th input to the k th out;put* The matrix H(z) i~ called the transfer matrLv of the system. We will use the term r-iaplut p-ozdQut s ~ s t e mand p x r sgstem, iater&imgeably.

4.2.1.3,

Lossliess system

An irnpsrtartt elms of MfMO systems are lossless systems. De6nit;ion 4.2.1.2. Let H(z) be a p x r 8ys"tem. H(z) is said to be lossless if (a) each entry [H(z)jkm is stable and (b) H(z) is unitary on the unit

circle, that is,

H" (exp(jw))~(exp(jw))= cI, for all w E [O, 2%) and some real constant c. If c = 1, then H(z) is known as a nomalized lossteas ~y~ttjna, Since all FIR systems are stable, refere~leesto FIR systems tend to use the words pa~azlnitaryand lsssless interchangeably.

4.2'1.2

Imp&@ response matrix

Let hkm(n)denote the impulse response of the transfer function H k m ( z ) . In addition, let h(n) reprevent the p x r matrix of impulse reuponses, where hrm(n) = [h(n))L, Then,

where H(z) and h(n) are p x 7 matrices. The matrix sequence h(n) is called the impulse response of the system. To illustrate this idea, let

4.2 fimework for lattice sdrtrcturcs Tken, this can be writterr as

The following theorem provides a11important property of the impulse response matrices of laasless systems.

c,=,p(n) r-".

Theorem 4.2.1.1. Given P ( r ) = then p B ( ~ ) p (=~p)B ( ~ ) p ( 0=) 0.

I f P ( r ) is paraunitary,

Proof: Apply the definiltion of paraunitarinegs to P(z) to yield

or equivalently,

Since P ( L )is yaraunitary, then $ ( z ) ~ ( r= ) cI. Therefore,

Since @ ( r ) ~ ( will z ) be nonzero only for coefficientsof r', then pH( N ) ~ ( o=) pH(N)p(o)= a* It

Matrk polynomials play an important role in MXMQ systems, r

matrix whose entrie~are poIynomids in z. The matrix earl be expregsed

If p(k) is not the zero matrix, then k is called the order of the poIynomial matriz. For example, a causal FIR system is a polynomial matrix in I-' k t h order k, $fiat is, k

148

Chapter 4 Lattice Structures

Definition 4.2.1.4. A unimodular polynamid matrix U(E) in variable x is a square polynomial matrix in s vvith a coastan t noxlzero determinant,

To illustrate unimodular polynomial matrices, consider the hlluvving examples: is unimodular, because det (Ul (2)) = 1.

and u2(z)

-

1+z"

22

2

is unirnodular, because det (Uz f x ) ) = 2.

The following theorems provide properties of these matrices, Theorem 4.2.1.2. E A is a u n h d u l a r polynomial matrix, then A-lexists

and is a u~liartodularpo!ynorr;iial matrix. Proof: Let A be a unimodular polynomial matrix, Since det .A $ 0, then A-I exists a11d AA-' = I. Since det A det Am' = 1 and I det A1 =c , then Idet A-' 1 = $. Since det AB = (det A)(det B), then (det A)(de = 1. Since A is uaimodular, then ldet A1 = c. Hence, ldet A-I II Therefore, A - 5 s a unimodular polynomial matrix,

If A and B are unimobufar polynomial matrices, then A13 is a uriim~dltlarpolyn~micitfmatrix.

Tbeorevn 4.2.2.3,

Since A and B are unimodular matrices, =I (det A)(det B),then fore, AB is a urtimoduEar polynomial matrix,

= d. Since det AB

4.2.1.5

m&of a poliynomiaf matrix:

Definition 4.2.1.5. The raak of a poIynomial matrix is defi~edas the dimerrsiort of the su bmatrix that; corregpond~to the large86 Banzero de terminanGal minor. CIearIfi if f(L) is a p x r polynomial matrix, then

rank (P(z))5 minip, T }.

4.2 framework for lattice structures

149

The theory of the Smith-McMlllm form is developed f-or causal Linear Time Invariant(LT1) systems. The theory of the Smith form for polynomial matrices is prevented first in order to provide the necessary background for the Smith-McMillan form. The analysis of the Smith form, that ipl developed in this ehapter for polynomid matrices, is analogous to the theory developed for integer matrices in Section 2.2. This is true because the set of polynomid m a t ~ c e s and the set of integer matrices belong to a common algebraic structure l damain, called the p ~ i ~ e i p aideat

4.2.2.1

Elementary operations

Elementary raw (or column) operations an polynomial nratrites are important because they permit the patternixlg of polynomial matrices into simpler forms, such z s triztngular and diagond forms. Definition 4.2.2.1. AR elementary row operation on a polynomial matrix P(z) is dearled to be any of the follawing:

Type-1: Interchange two rows. Type-2: Multiply s row Izy a nonzero eonstant c. Type-3: Add a polynomial multiple af a row

$0a ~ o t h e row, r

These operations can be represented by premultiplying P(z) with an appropriate square m a t r k ,called an elementary matrix, To illustrate these elementary operaions, consider the Eallowing examples. (By convention, the rows and columns me numbered starting with zero rather than one.) The first example is a Type-l elementary matrix: that interchanges row 0 and row 3, which hm the form

The second example irs a Type-2 elementary m a t r k th& multiplies elements in raw f by e f 0, which Itas the form

1 0 8 0

o c o o O

O

f

Q

0 0 0 1

150

Chapter 4 Lattice Structures

The third example is a Type-3 elernexltary m a t r k that repfaces row 3 with raw 3 (a(zf * rowQ), which has tbe form

-+

A11 three types of elementary polynomial ~katrieesare unimodular polyrromial matrices. Elerneatary column operations are defined in a similar w;zv by post multiplying P ( x ) with the qprapriate square matrix, These elementary operations calk be used to diagondize a, polynomial matrk. The key theorem whiclr enables us to obtain this diagondization is the Division Theorem for Polynomialu. It states that if N ( t ) and D(r) are polynomials in x, where the order of N ( x ) >_ order of Dfx), then there exists unique polynorniala Q ( z ) and R(z) such that

where the order of R ( z ) < order of D ( z ) . 4.2.2.2

Smith form &eeo;mposit;ion

Theorem 4.2.2,1, E w r ~ poIy11omid l matrix A(z) can be exprcissed in its eorrespondi~gSmith form decomposition as

where TT(a), V(z) are uniraoduk polynomial matrices and the Smith form S(s) is given by $3f z )

=: diag

(so( z ) ,.

..

8,-

(z), 0, Q1

... ,0)

where r is the r a ~ ~ ofA(z) k &ad s i ( z ) s~+I(x),i = O,.. . ,T

- 2-

Proof: Assume that the zeroth column of A(z) contains a narkzero element, which may be brought to the (0,O) position by elementary operationu. This element is the gcd of the zeroth column. If the new (0,O) element does not divide all the ele~lenttrin the zeroth row, then it may be replaced by the gcd of the elements of the zeroth row (the eRect will[ be that it will coxttajn fewer prime kctors than before). The process is repeated until an element in the fO,O) position is abt;tined which divides every element of the zlflrottl rovv anid eolurrtn. By elementary row and column operations, all the elenients

151

4.2 Framework for lattice structures

in the zeroth rovv artd column, other than the (0,0) element, rnily be made zero. Denote this new submatrh formed by deleting the zeroth row and zeroth column by C(z). Suppose that the submatrix of C(z) contains an element s,j (z) which is no%divkible by coo(z), Add column j to colurnn 0. Columxr O then consists of element8 coo, clj,... ,c,-1, j , Repeating the previous process, we replace coo by a proper divisor of itself using elerrlentary operations. Then, we must finally rellch the stage where the element in the (Q,O) position divides every element of the nnatrk,;md all other elements of the zersttz. row and column are zero, The entire process is repeat;ed with the submatrk obtrtined by deleting the zeroth row and column, Eventually a stage is rexhed when the matrix has the form 0 Elz)

-

where D(z)=diag (so( z ) ,. .. , ( z ) ) and si(z) isi+%(z), i = 0, . . . ,I. 2. But E(s) must be the zero matrix, since otherwise A(x) would have a. rank larger than. r . rn By cenverltion the polyrlomids s i b ) , i = 0,.. . ,r - I, are rnonie poIynomials, that is the highest power of the yolyrromial hm a coeBeient af unity. Note that although the two unirnodular polynomial matrices Ufz) and YCz) are, in generd, not unique, tlre diagonal matrix S(z) is uniquely determined by A(z). Example 4.2.2.1, Ta illustrate the Smith form decomposition, consider the fallowing example, Let

Afz) =.;

z+l 2z2

+3

X

2(2

+ lla

If we divide the (1,0) element, 2z2+3, by the (0,O) element, z+ 1, we obtaill 2z"

3= quotient

remainder

Tlxereforc, if vve apply a Type-3 row operatiorr, whit-rlt is defined by

to A(z), we will reduce the (1,0) element t o the constmt value of 5. Therefore,

152

Chapter 4 Lattice Structures

Reduce the (0,0)element to a cor~stantwith a Type-3 row operation, which 1 - 25 is defined by . Then, vve obtain a I

Transform the (0,1) element to zero by a Type-3 column ope ratio^^, which 1 --$ % ( I - 2 t ) is defined by 5 . Then, we obtain 0 1

Finally, the (1,0)element is forced to zero by a Type-3 row operation, which 1.0 is defined by . Then, we obtain -5 X

Thus,

Let E ( r ) be the product of elementary row operations, i . e .

Let F ( r ) be the product of elementary column operationu, i.e.

since only one elementary column operation was performed. Therefore,

4.2 &&mework for lattice structures Then, the Smith farm decomposition is ghea by

where, U(z) = E-I (z) =

9

1

V(Z)= F-' (z) =

-

0

Let H ( z ) be a p x T transfer matrix of rational functions representing a causal Linear Time Invariant system. Assume each eelement [H(z)lkm has been expressed as [H(z))~,= where the polynoaaid d ( x ) i s a lemt common multiple of the denominators of polynomials of H ( r ) . Define a p x T miltrk P(z) wiLh elements fi,(x) and let

P (z) = w (z)]t""(z)V(z) be its Smith form decomposition, where VV(z), V(z) are unimodular polynomial matrices and the Smith form r f z ) is given by

q-4= diag

( 7 @ ( ~ ) 7 *. *

97F-l(z),o?aY

*. *

90)

and % must sattlsfy %(z)I%+l(z),i = 0,. .. ,T-2, Then, the Smith-McMillan decomposition is given by

where the Smith-McMilliln form is given toy A(z) = diag (Ao ( z ) ,.

..,A-,'

(z), 0,0,.

.. ,0)

and

Cancelling common factors between y i ( z ) and d ( z ) yields

chapter 4 Lattice Structures

154

where CV;(Z) and &(z) are relatively prime polynomials. In view of the divisibiliw property of the Smith form of P(z), since r;(~)l%~(z)~ then . In addition, the %+I (z) = c ( z ) % ( z ) ,and, as a result, ai+1 (2) and (z) O , . * * , T -2. 4.2.3

MeMlibn degree of a system

DeAnition 4,2.3.l,. The McMillan degree, p, of a p x r causd sry;stem H(z) is the minimum number of delay units (z elements) required to implement it, that is P = det? (H(z)).

-'

If the system is noncausal, then the degree is undefined. If W ( x ) =.. z-IR, where R is im M x N matrix with rank p, then

where "IP is M x p and S is p x N , Therefore,

Hence, we can implement the system vvith p deltays. So, the system hw a McMillan degree p. As an example, consider

<

We can rewrite N(z) i b ~

Thus, the system can, be imglennerrted vvith a single del;ty, illustrating that the MeMiUan degree of the system is unit;y. The Smith-McMillan decomposition prwides insight; into the determination of the McMillan degree af an M x M fossless system, The following result is centrd to the design of lattice structures. Theorem 4,2.3,1. Let H(z) be an M x M causal Iogsless system. Then, det W ( x ) is stable aIlpss and

deg H(z) = deg (det H(z) 1.

4.2 Ramework for lattice structures

155

Proof: The stable allparcs property of dct H ( z ) fallows from the lossless property of H(z), that is a ( z ) ~ ( r=) c21. From the Smith-McMillan decornposit ion,

n-

M-L

det H ( z ) = a

ali ( z )

*=, Pi(.)

'

Since wi(z)and &(+) are relatively prime for every i and j, there are no urnaneelled h t o r s in the equation, Becwse of the causality of H(x), the degree of p j ( z ) (as a polynomial in z) is at least as large as the degree of a ( r ) (as a polynomial in z). Thus, the degree of the det H ( z ) is equal to zEd,i1degP i ( z ) , which is the degree of H ( r ) . II Example 4,2,3,%. The following example glustrates this important result, Consider the following system:

What i s the degree of the system? W will use tvwa metbods to determine this degree.

Method 1 (DefilIlit;liola of Smith-MeMil[larr,Form)

The Smith-MeMiXlan decomposition of R(z) is given by

50, the Smith-MeMillan form of H(z) is given by

or equivalently,

so, ~ o ( z = ) a&) = 1, Po(.) = r , and & ( z ) = 1. Therefore, pa = deg h f z ) = 1and p1 =. deg & ( z ) =s 0. Hence, by the definition of the SmidhMcMillm form, the degree of H(x) = pl = 1.

+

Chapter 4 Lattice Structures

156

Method 2 (Theorem an McMfllan Degree of a Lossless System)

First, we n e ~ ddo determine if H(x) is paraunitary

so, H[(z) is paraunitary and Theorem 4.2.3.1 applies. So compute the determinant of EIfz), that is det H(z) = (1

+- z - ' ) ~ - (1- z - ' ) ~ = 4z-I.

So, deg [ det H(1;)f =; I. Therelare, by the theorem on the McMillsan degree of a, lossfess system, deg H(z) .= I.

4 3 Latkice structures for lossless systems This section syslematicdfy presents lattice structures for lossless systems, Topics included are a fundamental degree-oxre building block and structures for losvless syster~s.

Bouseholder factorizations are a;n important technique for the factorization of unitary matrices. They play an i~nportantrole in the theory and implementation. af lossless systerns, because lossless faetasrizations always involve a urlitary rnaittrix, Definition 4.3.1.1. Let x be an M-dime~sionaicomplex-.valued vector, such that the ith component d x is given by

where Bi is real-valued and j =

Assume

x

+ 0 and define

where ea is a unit vector in the direction of the zeroth cornponeat of x, (Pt, is important to note that i f x is red-valued, then the definr'gion tafu becomes. u = x 4- sign(z0) l2eo.) The Householder transformation is defined by

4.3 Sdrucl;ares for ~ o s ~ ~syg e sterns s

Figure 4.1. Impfernentation af Householder transformation.

The irnple~~entation, of a HousehoXder tranfi3fosmation is depicted ixz Figure 4.1. It is impostant to note that

Moreover,

Renee, H is a unitary tranrsforrwratisn, In prwtiec, H never has to be exylieitlg cdculated, This is because H x earl be predicted a prin~i,But first vve need to compute uWx and uWu,

or equivalently,

158

Chapter 4 Lattice Structures

Since xHeo = I z a I exp(-jOo), e F x = zo I exp(jeo),and ef ea = 1, then

1;

= x H x . By combining results, we observe that

= 1,

Hence,

or equivalently,

+

Hx = x- ( x e x p ( j 6 ) Therefore,

Hx = - e ~ p t j e ~ ) 4.3.2

sl. hndamental, degree-one building black

As we saw in tbe previous section, Housebolder transformations are given

Definition 4.3.2.1. Let ffz) he a function and Ict v, be a wclor. Thee, the fundamentd d e p e - m e structure for Io~slesssystems, denoted V, ( x ) , is dearzed aa an extension of the Householder transformation by

Let us apply v, to V,fz), that is,

So v, is an eigenvector with eigenvalue -f (2). Let u be a vector orthogonal to v, that is, vEu = 0, then

4.3 Structures for Io~sle~s wrf~terna.

159

So, u is an eigenvector with eigenvafue one. Let uk,k = 0,. .. ,M - 1, be rz set of M mutually orghogonal vectors, where u ~ . - .= 1 vm. Defi~rean M x M matrix U = (uDI.. . , U ~ - ~ I then ~ , we have

Take the deternlinant of both sides of the equation deb (Vmf x)U) =; det

But the determinant of the product of matrices is equal to the product of the determinants of each individual miZt;rkUE. Hence, det V , f z ) det U ==det U det

Therefore, det V,(z) Since V,(z)

=

.

f( z ) .

is lossless, then

deg Vm()

= deg (det V f,

a))

== degf-f(.)2. Since V,(z) charwterizes a degree-one building block, then dee; V ,

Tlrerefarc,

(t)Z=

deg f -f ( x ) ) = 1.

Let us examine V,(z) a little closer, Let

Then,

1,

Chapter 4 Lattice Structure8

160 and

Q~=(I-P)~=I-~P+P~=I-P=Q. Thus, the projection operators P and $ satisfy the idempotent property, that is, P" P and Q 2 = Q. In addition,

Thus, P and Q satisfy the Hermitian symmetric property. Since P arld Q satisfy both the idempotent axkd symmetric properties, then P and Q are orthogonetl projections. III addition,

Consider the fuxzction

For Vm(z)to be lossless, then

or equivalently,

f (zIf*(,)

f

= 1'

This implies that poles (and zeros) of f ( r ) are cancelled by zeros (and poles) of fs(+)* Con~equently,f ( z )f *($) = 1 implies that poles (and zeros) must have zeros (and polea) in conjugate-reciprocd locations, This suggests the following form for f ( z ) , that is,

Figure 4.2. Structure for lossIess system.

where a, is s constant, Therefore, f (2) has a pole at a, and a zero at 2 .Hence, the fundamental degree-one solution for IIR lossless systems is given by

h r stability, all, the poles will be located ixlside the unit circle so When V,(z) is quantized, it remains unilary, since every occurrence of v,vg is normalized by vEv,. An interesting special case occurs when a, .= 0, Vnder thew conditions, V,(z) becomes a fundamental building black for FIR lossless systems, that; is,

Note that the pure delay term z-l is also aXlpws, since it can be interpreted as a pole at zero and a zero at infinity,

This section presents lattice structures for lossless systems. Let W N f z) be an causal losslesa system with deg HN(xf = N . Tk~err,we can fxtorize it (as

H N ( z ) = V N ( ~ ) V N - I (.Z *. ) VI (z)Ho where Ho is unitary and

The structure realizing the factorization of HN(x)is given in Figure 4.2. Sometirncs, it, is desirable to replace Ho with its Householder factorization,

Chapter 4 La6l;ic.e Structures

162

Example 4,3.3,1, Factorizakisn of M x 1 FIR Lorssfess System

Given

8,023

Hi(z) =

-0.016

-

0.5772-I 0.829~~ .~

Find the M x I factorizatioxl of HI(zf, Hl(x) can be eqtaidently written a

Since h(llHh(0)=O, choose vl = h(1). Then,

q1(Z)

= I - h(l)h(llH+ h(lfh(l)H h(l) "h(I) h(l)Hh(l)'

Then,

[I-

+

= = h(O) h(l)z-'

+

==

h(0)

2

+ h(l)z-I

- h(l)z-I + h(1)

h(0) -t- fifl).

Hence,

P;br constant vector, perform

a,

Householder transformation, that is,

where

Therefore,

HI Cz)

-

Vi (z)Uo,

As such, the filter HI(2) can be realized by two stages Vt ( r )and Ua. This filter is depicted in Figure 4.3.

4.3 Scructtlres for I o s ~ i e asy-s ~ terns

Figure 4.3. Two-stage filter.

Example 4.3.3.2. Factorization of M x M Lossless Sysdem Given some positive ixlkger k , then the system Hk(z) is given by

Find the M x M ktoriziltion of Hk(z). By inspection, Hk(z)is FIR, Ifi) Hk(z)paraunitary?

Hence, Hk(z) is stable and paraunitary Therefore, deg He f z ) = deg (det Hk(z)). $0,

+

deg HI:(z)= deg [(I z-k)2 - (1 - r-')'I

= deg [ 4 ~ -=~k.]

The impulse response matrix of Hk(z) is given by

Choose v k such that vfh(0) = 0. So, let vr: =

1

-1

. Then,

Chapter 4 Lattice Structures

Figure 4.4, K-stage filter.

Hence,

Then, V(z) = V , ( I )= * * - = V l i ( z ) will. be the degree-one building black. Then,

Therefore, the filter can be realized by k stages V ( r )and a constant scaling factor of 2. This filter is depicted in Figure 4-4,

4.4 Problems

1. Let G ( z ) be given by the foUming:

Express G ( r ) as a sum of impulse response matrices. Is G ( z ) unimodular?

2, Let B be given by the following:

where a is a real-vdued constant, Use a Householder transformation to make B an upper triangular matrix, 3. Let H(z) be a system transfer function that is given by the follovving:

where e is a real-valued eonatant, Is BIZ)paraunitary? Draw the cascaded structure to implement H ( z f , 4. Let E ( z ) be given by the following

Is H ( z ) lossless? What is the degree of H ( r ) ? Obtai~ka cascaded structure to implement H(xf, 5. Find the Smith-McMillan decomposition of the H ( r ) matrix, which wau defixled in the Iafst problem. Then, uaing this result, determine the degree of the system.

Chapter 5

Wavelet Signal Processing

5.1

Introduction

This chapter introduces the notion of fundamentals of wavelet signal processing. Morlet et a1.[36] was the first paper to discuvs the idea of a wavelet, although a t this paint in time it WM a rather empirical idea. Later, Daubechieu[l2] showed that it was possible to develop a theory and an associated algorithm for the generatio11 of compactly supported orthonorrnal wavelets, Meanwhile, MallatftZq 24 developed the notions af muleiresolution analysis and, recently, Shapirok42j has developed an eEcient scherne for encoding them. Results on biorthogonal wavelets can be found in Cohen e t a1.[8].

Many of the corlceptrs presented In this chapter are also discussed in other multirate and wavelet texts, including the engineering-oriented texts by Chan[5], Fliege[lSj, Strang and Nguyen[46], Vaidyanathan[49], Vetterli and KovaEevii.(54], Wornell[5?1 and the mathematicdly-oriented texts by Chui[7], Daubechiesll31, Holschncider[22], and Meyer[33, 341. Good reference texts for background material an vector spaces include Riesz and Sz.-Nagy[39] and Schilling and Lee[.ll]. Section 5,2 introduces the wavelet transform, Section 5.3 presents multiresolution analysis for both orthogonal and Ibiorlhogonal wavelets. 5.2

Wavelet tramform

The name wavelet comes from the require~lexkttitat a fiunction should integrate to zero, ulavi~gabove and below the axis. The diminutive eonnotation of wavelet suggests the function hw to be well localized. At this point, some people might ask, why not use trditional Fourier methods? Fourier bwis functions are localized in frequency, but not in time, So, Fourier aniiLfysi8 is the ideal tool for the efickexrt representation of very smooth,

168

Chaptw 5 Wavelet; S i g ~ d Procesging

stationary signah, Wavelet bwis furrctions are localized in time and frequency. So, wavelet analysis is an ideal tool for representing signa1a that contain discontinuities (in the signal or its derivatives) or for signals that are not stationary.

Definition 9.2.1.l. The mvelet tranufarm of g ( ~ with ) regpect to wavejet

$ ( t ) is defined by

where, a # 0 a~idb are called the scale and traas1al;ion parameters, regpeetitrely firtherrnore, the Fourier $randorm of m d e t Ift(l),denoted 9 ( f ) , must sati~fythe followi~gadmissibillity condition:

which &om that $(C) has to oscillate md decay, eo~tinuotrswavelet tran~formis given by

The inver~eof the

It is often useful to think of functions and their wavelet transforms rn occupying two donlains, Then, the falowing properties show the correspondence between operations perfornred in one domain with operations in the other. 5.2* ?L. 1,

Linearity property

Theorem 5.2.1,11. If the wvelet transforms ofg and h exist &ad a and P are scafar~,then

WT {ag + ph; a, b ) = aWT {g; a, b ) Proof: By definition,

By the linearity of irttegraGion,

+ PWT {h; a, b).

Chitpter 5 Wavelet Sigaal h c e s s i n g

170 Proof: By definition,

WT (f(e Let zt = t

f ( t - a ) +*(-

- a);a, b ) =

t

- tr ) dt. a

- a. Then,

or equivalently,

WT{f(*

5.2.1.4

- a);a,b) = WT{f;a, b - a).

DiRerentiiatioai theorem

Theorem 9.2.1.4, I f the wavelet tra~gform of f exidsts and iff "exists, then the w a d e t traasform of f "is given by

Proof: By definition,

t-b

WT( f ';a, b ) = ----

fr(t)$* (y-) fit*

But for every red t ,

f'(t) = lim

f(t

+ e ) - f ft)

€--+a

E

Then, using Lebesque's DominiitGed Convergence Theorem, we obtain

Using the Linearity Thwrem, we obtain,

WT{ f ';a, b ) = lim,,~

-&J?=

!

1-cr0 "

f(tl$*(y

Using the Shift Theorem, we obtain W T { f a, b WT{ f';a, b ) = lim €+a

Therefore,

f(t+e)$*(V)dt

+ E ) - WT{f; a, 6 ) E

d WT{ f ';a, b ) = -WT( f a, b ) . db

5e2.311.5

Gonvalatian theorem

Theorem 5.2.1.6. If the wavelet t r a n ~ f o mof g exists and if f then the wavelet. Grmsform of J .*. g is given by

WT { f * g; a, b ) = f

rl. g

exi8ts1

WT { g ; a, b),

b

where * d e ~ o t e eonvdu s tion with respect to the b wriablc. Proof: By definition,

or equivalently,

Using the Shift Theorem we obtain,

or equivalently,

W T { f * $;a,b ) = f

-k WT{g; a, b). D

The following subsection ilkustrates the role that wavelets nliLLurijllly play in radar signal processing. 5,222

-dm

signal processing itnd wav~li~ats

The problem consists of estimating the location and velocity of some target in a radar application. The estimation procedure can be described by the following. Suppose z(2) is a knovvn emitting signal. In the presence of a target, this signal s ( t ) will retur~lto the source ay the received signal h(t) with a delay T , due to the target" location and a Doppler effect distartion, due to the tmget's velocity, If z ( t ) is a n a m w - b a d signal, then the The characteristics af Doppler effect amounts ta a single frequency shift the target can be determined by maimizing the cross-correlation function, This estimator is cdled tbe Ramow-Band Anzbiguily Fanetdon, that is, foe

~ ( th(t ) - T ) exp (-j2?rfot) d t

Chapter 5 Wavelet Signal Processing

172

whi& is s b p l y the Short Time Fourier Transform (STFT) with shift T &out frequency fa. However, for wide-bmd signals, the Doppler frequency shift varies the signal spectrum, causing a stretching or comprcssian of the emitted signal, The resulting estimstar is c d e d the Wde-Baad Ambiguity Fanctiort, that

which is continuaus wavelet transfornt, (CW'P) about a point T at a ~ c d e given by a. Thus, the wavelet transform is an operation that de8emines the similarity or cross correlation between the emitted sign& z(t) axld the received signal, the wavelet h at scale a and shift T .

y)

5.3

Mdtiresolutirrrn analysis

A8 we have observed, the wavelet transform maps ai fanetian of one dimension into a two-dimensionaj picture. Using this idea, a mathernia.ticaf microstcope krrown as musltimsslutzlan anulg~iswill permit us to look a t the details that are added m we go from one scale to another. In this wag, m d tiresolution andysis provides a mathematical frearnevrrork t o conceptualize problems linked to the wavelet decomposition of signals..

Before proceeding with the definition of multiresoliution andysis, we will first need to discuss four tapic8 of great importance far the study of maltiresolution analysis: inner products, biarthagandity, Riesa basis, and direct sums.

5*3.1.1 Imer products Refinition li.3,I.l. The inner product on a vector space V is a mappiag that assigns a scdm t;o every pair of elements s, t E V. The inner produet I"s denoted (s, t;) and it hias the following prsperliea:

A vector space V with an inner product is e 4 e d an inner product space. The operation of performing the inner product is defined by the fallowing theorem.

Theorem 5,3.1.1. Let 7J be a finite-dirrtendd i'nner product slpace, Suppose {al,. . . ,u,) and {xl,... ,x,) are two basis sets for V. Let vectors s and t be given in terms of the basis set {uf,.. . ,u,) and {xf,.. . ,x,), respectively. That is,

Let the weighting matrix W be a defined by

Thexl, the inner product is @ven by

Proof: Consider the ixlx~erproduct af s and t by first substituting their respective series rnpansions, t h a t is,

Utilizing the Xinearitg property of the inner product yields

By definition, [W]kj= ( U ~ ~ XHence, ~).

or equivalently, rt

(s, t) =

C t; ( W S ) ~ *

j= f

Therefore, (s, t) = tH ws.

Chilpt er 5 Wavelet Signal Processi~g

IT4

Definition 5.3.1.2. Two lineark independent sets of vectors S = {ul, .. . , u,) and T = {xl,.. .,x,) form biorthogonal sets if

(uj9 x ~ =) O f i r all j and k where j # k, and ( u j , x j )f O for dl j , or equivdently; W is a diagond matrk. firtherrnore, S and T are called bio~thonornzalsets if

ar ey uivden t l VV ~ is an iden tidy matrix. Moreover, any set of biorthogonaI vectors c a be ~ made hiorthonormal. If S = T, then biorthagond sets become orthog~najfsets m d biorthonormajl sets become orthonormal sets,

The following theorem deals wit Er transformations that preserve all the informalion af the origirirtf signal. Theorem 5.3.1.2. Let { x r ) and {ak)be biorthonormd bases of an inner product space V. Then, y E V can be expressed as

Proaf; Since y E V and since ( x k ) and (uk) are complete biorthoaormaf sets of nonzero vectors in art irrner product s p x e V,then y. caan be expressed

1%

Taking the inner product of both sides of this equation with x, yields

h

Since U

L ~and

xm are biorthonammal vectors, then

5.3 Maltiresola Lion andygis Therefore, the right-hand side is nonzero only if k = m. Hence,

The following theorem shows the ixlterrelationship between paraunitary matrices and biarthogonality. Theorem 5.3.1.3. The filter H ( r ) satisfies the normalized paraunitaryprop-

if and only if its individual filters saGisfy the bforthonormal cmditiurr.

where,

HL(.L),HL(zWM), ... ,H ~ ( Z W ~ - ' ) Proof; (Case 1: Assume ~ ( z ) H ( z = ) JM.) Then, the ( j , k)th entry in the matrix I ? ( z ) ~ ( z ) i s given by

Since ~ ( L ) H ( z=) IM,thexi

(Case 2: Assume (

= bj,k.) If t h i ~equation is interpreted as the (j,k)th entry of a matrix, the11 this equation can be rewritten as

Then, the corresponding matrix equation is giver1 by

116

Chapter 5 Wavelet Signd Processing

5,3,1,3 Riesz basis DeBnieion 5.3.1.3. Let V be an inner product gpace. Let norm induced by the inner product, that is.,

The set ifk) c V h called a Riesz Basis if every element s E V of the space can be written as s = c c k fk k

for some chuice ofsealam { e k ) and i f positive consta~ltsA and B exist such that

Riesz basis are also known w a stable bwii3 or u ~ ~ c ~ ~ k d bmis. i t i o ~Iaf ~ the Reisz basii~is an orthogonal basis, then A = B = I. Consider two biortl~onormalsets S = {al,. .. ,u,) and T = {x,, . .. ,x,), that is, ( u j ,XL) = 6j.k.

Then, S and T are biorthonsrmd (Riesz) basis if there exist positi.ve constarlts A, B , @, L), such. that for all vectors y

Now, we will define the concept of a direct sunk Z L I I ~subsequentlyl , we will grove the projection tlreurern. RefiniLrisn 5.3,f ,4, Let S be a subspaice and let U and V be subspaces of' S , Tflen, S can be decomposed into s dirwct:sum of U and V, that is

i f the following conditions are satided:

2, S is generated by U U V.

We are now in a position to decompose a vector into its projectiorrs onto subspaces. To introduce this concept, we begin by identifying suitable subspaces, Deftrrridioxz 5.3.1.5. The orthogonal compbemead- o f a subset V of an inner product space V is denoted UI and is defined by

Thus, UL will always be a subspace of V and that subspace will be closed. Moreover, if U is a closed subupace, then ULi = U. (Projection Theorem) Let U be a closed subspace ofa complete inner produet space V. Tjie~l,V can be decomposed in to a direct sum of U and UA,that is,

Theorem 5.3.1.4.

.

Proof: Let Y = {yl,. . ,y,) and S = {sl,... , s , ) be an biorthonormal basis for U. Let x E V. Then, x~ and XI are defined by

and

= x -- Xg. Clearly, x -. xo-+xl, firtherrnore, xo E U since rro is a linear combination of the baais elements Y, Next, we must demonstrate that xA E U' and XI

tllat the decampasitiot.1of x: is unique, Let us consider the inrler product between x l and yli, E Y, that is,

or equivalently, (XI,YL) = ( x , Y ~ ) (xo~Y~).

Now replasing xo wit11 its series expansio~tyields

or equivalently,

178

Chapter 5 Wavelet SigaLZ-lProcessing

Since Y and S form a biorthonormal set,

(sj,y k) = 6j,k. Hence,

Thus, x1 is orthogonal to ewh basis vector in Y. Since each wctor in V is a linear combination of the bmis vectors, it fallows that xl is orthogonal to every vector in U. Her~ce,xl E UL. To show the decomposition of x i~ unique, let us stssurne that x is rxot unique and show that this results in a contradiction. Let's msurne x = xo+xl and x = z ~ + r ; ~Then, .

or equivalently,

Il(xo+x1) - (EO+ZX)112

= 0. After rearranging terms this equation becomes,

Then, using the Pythagorean Theorem,

Therehre,

==

xo and

= X I . Hexlce, the decsrnposition is unique.

r

Dsfinilion $.3.1,8, Multire~olu ti011 a n d y ~ i s(MRA) can be viemd as a sequence of approimations of a given Eunction f (t) a t &Rerent resolutions, The approximation of f ( t ) at a resolution 2j is defined as an orthogonal projection off ( t )on a su bspace V j . Now, we will provide a list of properties

that tbrtese subspaces fh"i11 need to satisfy. They are: 2.

viC Vj+l;

2. njVj= {0)and clasure of UjVj= L2; 3. f E Vj+l o f(2*) E Vj; 4. f E Vj 5. { f (*

f ( * - k) E Vj for all integer k;

- k)} is a Reisz basis in Vo.

A level of a muftiresolution analysis is one of the Visubspwes arid one level is coarser (finer) with respect to another whenever the index of the corresponding subspace is smaller (bigger). By properties 3, 4?and 5 there e ~ s t as Rekx bwis cp in Vj of the form

wl~ere,c iv a constant. Assume that the energy wsociated with ~ is, the energy associated with ~ j( t ),,that

ip(t)

equals

or equivalently;

For the left-hand side, perform a change of variables by letting u = 2 j t - k. The11,

Therefore,

c

2j/2*

Hexlee, the scaling function for spaee VJ hm the form

In an analogous fauhion, the wavelet function for the subsyace Wj, which iu lfte complement space of Vj, has the form

Since information about revolution 2j+' is given by the approsirnation in subepace V j and the details in subspsee W j ,Lhc arthogand prajectiurr on the orthogorlal corrlplerme~ltof space Vj in space Vj+1,that is

where, fj

E Vj and

Since V1 = Vo @ Wo and So, irr gexreral,

V2

(f3+1

- fj) E W j .

= Vt @ Wl then, Vz = Vo @ [Wo@ Wlj.

or simply;

Vj+l = v 0 @ 0 5 ~ 5 j W k where, Va is the subvpace corresponding to the coarsest scale. Since 9 E Vo E Q O C-k < jWkYthen? an ixiput signal f (s) can be expressed as and

180

Gzlpter

5 Wavelet Sign& Proces8ing

where,

Let us consider the following example to illustrate many of the ideas of multtresolutio~lanalysis. Assume the input signal f (s) is defined by 16, O < s < f 1 -8, ~ S Z 1< $

16,

f b )=

-20,

$ < Z < Z3

a sz < 1

0, otherwise .

For simplicity of this example, we will use the Haw wwelet. Its wavelet,

$(XI, and scaling function, p(z), are defined by cp(.I and

I, @ < % < I

=

0, otherwise 1,

O
-1, f S z < l 0, otherwise . Now, let us compute the caeBcienCs.

Hence,

Thus,

-

where,

= [s, do,o, dl.@,dl,llT and y is a vector of input signal values, and W4 is the discrete Hmr wavelet transfor~l, The discrete wavelet transllarm matrix ia analogous to the discrete Fourier transform rnatrk - the principal difference being tliat Fourier caeacients come from vdues a t rr points, while H a r wavelet coeReients come from n subintervals,

According do the definition af the muXLiresoiluLian analysis, tp E Vo C V1. As such, the s c a l i ~ ~funetiozt g must satisfy

or equivalently,

This equation is known by several digereat narnes: the refinement equation, the dilation equation, and the two-scale difference equation. The generalization of the refinement equatiorr is given by

The following theorem provides an important property of the sequence of filter (or mask) coefficients {ha). Theorem 5.3.1.5.

Ckho(k)= 4.

Chapter 5 Wavelet Sigr~alProcessing

182

Proof: IntegraLing both sides of the refinerneat equation yields

or equivalently,

For the integrd on tlte right-11snd-side, perform s change of variables by letting u ;== 2t -- k. Then,

Hence,

The scaling function is uniquely defined by the refinement equation and

Sirkc, Vl = V, @ Wo,then $I E V1. As such, the wavelet must satisfy

C h i (k)p(2t - k),

$ ( t ) = fi

k

Ttle generdb~at;ionof this equation is giver1 by

The wavelet is uniquely darted by this equation aid by Gfic normalization

The f u l h i n g theorem provides an importarkt property of the sequence of coefficients { hl }. Theorem 5.3.1.6.

C khl (k) = 0.

Proof: Since the wavelet

+ must satisfy

then by integrating both sides of the equation yields

or equivalently,

For tfre integral an the right-halid side, perforal a change of variables by letting u = 2t - k. Then,

Since

J-z+(t)dt = 0 andm:S

p(u)dv, = 1, then

C hl ( k ) = 0. 5.3.1.8

Filter-ba& implementat ion

The projectio~lof a signal f ( t )E

Vj+lis given by

where, aj4-1( n )=

(f vj+19n) 9

*

In addition, f ( t )E Vj@ W i , so f ( t )can also be written

where

EM

Chapter 5 Wavelet Signal Processing

Figure 5.1. An analysis stage of pyramidal filter bank,

Let us examine the coefficients a little closer,

and

These equrttions suggest a pyramidal filter bank implementation, where the analysis bank is defined in terms of stages, where aj+l (m)= ( f , qj+l,m) is decompo~edinto aj(k)= (f,~ j , and ~ ) b j ( k ) = ( f ,$j,k) . This result can be depicted pictoriiaXly in, Figure 3.1, Thus, we have defined the andysis side of the fitter bank. Now, let us examine the synthesis side, Since Vj+l= V j @ Wj9we can write any vector in Vj+lar a linear cornbiglation of the basis vectors in V jand W j . Since Vj C V j + l ,a shift of n in Vj+lcorresponds to a shift of 2 in V j and Wj. That is,

Let

(2k) = qo (k)and 91 (21) = po ( l ) . Then,

5.3 Muf tiregala tion analysis

Figure 5.2. A. synthesis stage of pyrmidd filter bank,

Let ma = k

+ 5 and ml = I + f. Then,

Then, the projection of signal f ( t )on the basis elements ~ ~ + ~ *is, given (t) by

This equation suggests a wranridal filter-bank implementation, where the synthesis bar~kiu defined in terms of stages, where aj (ma)= (f,P*j,) and bj (ml) = ( f ,@j,m,) are merged to form aj+l( a )= ( f ,P~+~,,). This result is depicted pictoridly in Figure 5.2. 5.3.J.41

Initial pr0jecl;ion coemcients

The accurxy. of the multiresolutian analysis depends on the computation of a N ( k ) ,i,e,,

anr(k) = ( f ,rpnr,k) = 2"/2

f ( t ) p(zNt

- k) dt.

Let us perform a change of variables by letting z~= 2" t. Then,

Chapter 5 Mravdet Sip& Processing

186

Since p ( ~hm ) compitct support, thia integral can be interpreted to have finite limits. In addition, f ( 2 - N u )may place constraints on the sampling of f (u),dependirzg of course on the smoothness of (u). 5.3.1 .lO

ReBtections on the wavelet tramform

By examining muftiresolutian andysis, w are ready to draw some interesting canclusians about wavelets.

1. Concept, of Scale. The discrete wavelet transform is usefixl for representing the finer variations irr the signal f ( t )a t various scales. Moreover, the function f ( t )can be represented as a linear combination of functions that represer~tthe variations a t different wales.

2. Localized Basis. If a signal has enerl;y at a particular gcale concentrated in an interval in the time domain, then the corresponding coeficient has a large value. Therefore, the wavelet baits provides localization irrformation in both the time dom&bin a~ well wnj the scale donrairr. 5.3.2

Image corrzpressihn and wavelets

In multiresolution analysis, there are wme dependencies between coefficients at different scales. In. addition, for naturd images, it is unlikely to find significant high-frequency energy if there is little low-frequency energy in the same spatial location. These obsewations motivated a wavelet-based image coding method, that is based on a data structure called a zera-tree. A, zero-tree provides a compact representation to indicate the positians of significant wavelet coescients. A wavelet coefficient c is imignzfieant with respect; to a threshold T , if el < T. The zero-tree is bwed on the hypothesis that if a wavelet coeficient a t a give&scale is insignificmt with respect t o a given threshold T,%lien,in most eases, all wavelet coefficients a t the same orientation in the same spatial location, a t finer sedes are ixtsipificant with respect to 12". An exigineering trade-off can be made bet w e n the choice of the threshold 2" and the number af bits af precision for untllresholded caeEcients. Thus, the zero-tree approach can isolate interesting nonzero details by eliminating large insignificant regions from consideration; arid, therefore, it make8 more bit8 available to encode the significant coeBcients. 53.3

A generalisation: biorthogoeal wavelets

Except for the Eaar vvavetet, all other orthogonal. wavelets are not symmetric. This is a, problem in applicrttians, such as imal;.e compression, since

thcsc unsymmetrical wavelets generitte notieeabfe phave distarbions, So can orthogonal wavelets be generdized to elirni~l&cthis problem? Yes, using biort hogonat wavelets, VVc? will develop this idea of biorthogond wavelets by first imagining two intertvPining multiresolution analyaes, that is,

Vj+l = V j $ W j and Tj+l= where,

-

vj~3 Gj

C_*__

V j n W j = O a l t d V j n W j =0. Iastecld sf ssuming the spaces V jand Wj are perpendicular, we will wsume that V j is perpendicular to j . Similarly, we will wvume that the are not perpendicular, but rather that qj is perpenspaces qiand Siij dicular to W j .Then, the wavelets lirj,&(t) = 2jj2$(2jt k) and &i,m(t) = 2'i2&(2't - m ) are biorthogonal wavelet bases. That is, the wavelet bases satisfy the followirlg biort honurniditg condition:

-

In addition, the scali~lgfunctions v j , ~ ( = t ) 2 j l z p ( 2 j t - k ) and Fj,,(t) = 2 j i 2 5 ( 2 j t - rn) satisfy the following orthonormdity condition:

5.3.4

Case study: wavelets in data cammu~catlarzs

Although this cwe study is focused on data communications, many of the i d e a expressed are also appropriate for other applications. Now, let us eonsider a modulated signal, derrotcd sft),satisfying the dyadic self-similarity property is given by

z ( t )= 2-rnH2(2rnt) for some real constant If and d l integers m, This is the elass of signals of interest far fiactul modulatdoln applications. These waveforms have the property that an arbitrary short duration time segment is su%eient to recover the exttire waveform and hence tlte embeded information. Similarly; a low bandwidth approximation to the waveform is suflicient t o recaver the undistorted waveforn:~and also the embedded inhrmation. Now, conlsidcr the wavelet representation of z jt), that is,

Chapter 5 Wavelet Sign& Processing

188 where,

le .,,z , Let us e x a ~ n i ~further

Perform

a, change

Substituting the defirlition of z ( t ) into,,,z

of w i a b l e s by letting u = Zmt. Then,

where /3 = 22H+1. Let us denote z,,

by q(n).Then,

TNe wish ta let q(n) correspond to a finite Iengkh message of length L that wc wish to sextd, but the right-hand side of this equation is an infinite sum. Let's use periodic extexisions of (P("IL) 111 5(t), that is,

Moreaver, since p = 22W41, the p a r a ~ r x eH controls the relative power distribution among the freque~reybands and hence the overall transmitted power, As8un~ethat ~ ( tis )transmitted llicross the chmnel and received ;rs a wdvefarm denoted ~ ( 2 ) . Then, the receiver extrwts the wavelet coescients rm,, of ~ ( t usirig ) the discrete wavelet transform. These coefficients take the form T ~ = p, - m /~2 y ( n mod L) zm,?%

+

where x,?, are the wavelet co~fXiciexitsof the channel noise. The duration-bandwidth characteristics of the cf-lax~ne"l@ect;which observation coeficients r,,, xxtay be access&. If the chax~rielis bandlimited t a 2 M a Hz for some integer Mu, then the receiver call not access coefficients a t scdes nz > MU.Similarly, the duration conatraint of the channel resufts in a minimum decoding rate of 2 M symbols/second ~ for some integer MA, whicfi precludes access to scales m < ML. Therefore, the coeEcients available a t the receiver correspond to the indices fib = M&.M&4- I, ... , M u

where L is the length of the message q(n).Therefore, the number of noisy measurements of the message at, the receivcr is given by

Perform a change of rariables by letting p = m

- ML yields

Thus, the receiver can select the rate/bandwidth ratio dynamically since the transmitter is fixed. 5.4

Problem

1. Let

where q k ( w ) is a set of orthonormal functions in the range a that is,

5 w < b,

Suppose we wish to approximate F l u ) with the finite mmmatiorl

<

where a .(: w b. We wish to make the approximation best in the least square sense, that is b

(F(w)- F,

( w )lzdw

must be minimhed. Show that the choice of f i k ) = f ( k ) , k = -my.. . ,nz, achieves the desired result.

190

Chapter 5 Witvded Sig~alProcegsir~g

2. Let us examine the Mexiearr hat fu~ietion,which irs defirled by

Is h(t) a wavelet? Justify your altsuver. 3. Given n-dimexlsiortal vectors s and t and arr n x n matrix W. Show that t " ~ ssatisfies the properties of an inner product, i . e . (s, t) = tHws.

4. Prove the following: Let yj,k(t) = 2j/2y(2jt norm, then gj9t( t )also has unit norm.

- k).

I f g ( t ) has unit

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[53] M Vetterli, Filter ba~kksallowing for perfect reconstruction, Signal Proce~sing10 ( 19861,219-244.

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Index A Alim free filter bank, 99 Alias-component QMF, 75 Ambiguity function narrow band, 171 wide-band, 172 Anti-ali~ingfilter, 14 Anti-imaging filter, 8

B Bmis local sine, f 27 Riesz, 176 Biort hogonal vectors, 174 Building block degree-one, 158

C Comb filter definition, 114 multidimensional, 63 Coset, 39

D Deeirnstor definition, 9 multidimensional, 59 noble identityt 26

Division theorern for integer vectors, 56 for integers, 6 for polynomials, 150 Downsampler, 9

Expander definition, 7 multidimensionat, 57 noble identity, 25

Filter anti-aliasing, 14 anti-imaging, 8 comb, 14 Interpolated FIR, I6 Filter bank aliizs free, 99 amplitude distortion-free, 109 cosine modulated, 114 DFT, 112 multidimensional, 128 perfect reconstruction, 105 hndarnental lattice, 30 hrrdamental parallelepiped, 36

H

DFT definition, 3 multidimensional, 46

wavelet, 181 Householder transform, 156

H a p

multidimensional, 69 Impulse response matrix, f 46 Inner product, 172 Interpolated FIR filter, 16

L Leibnitsk rule, 89 Linear space orthogonal complement, 177 Lossless system, 146

Malvar wavelets, 123 Matrix circulanl, 99 elementary operations, 149 generalized pseudocirculant, 136 impulse response, 146 normalized paraunilar~t,105 orthogonal, f 05 paraconjugated, 105 paraunitary; 105 polynomial, 147 pseudacirculaxlt, 98 reversd, 114 skew circulant, 99 urlimodular integer-valued, 35 unitary, 105 MeMiflan degree, 154 Multidimensional comb filter, 63 Multidimensiond decirnator, 59 Multidimensional DFT, 46 Multidimensional expander, 57 MuItiresolution andysirs, 178

N Noble identity fbr decimators, 26 for expanders, 25

Paracanjugation of matrix, 82 of vector, 82 Parzetlelepiped fundamental, 36 symmetric, 38 Polynomial matrix definition, 147 elementary operations on, 149 rank of, 148 unimodutar, 148 Polyphase multidimensional, 54 Type-I representatioxl, 5 Type-I1 representation, 6 Palypbase QMF, 82

Q QMF bank, 74 QMF formulations diw-component, 75 polyphiwe, 82

Sampling laLtice coset, 39 definition, 31 fundamental, 30 hexagonal 34 rionuniqucxlesr~,34 quincunx , 3 4 reciprocal, 34 rectangular , 32 tf

Index sublattice , 38 unit cell , 36 Sampling matrk, 32 Scaling function, 181 Signal representation modulation, 5 polyphase, 5 Smith form decomposition for integer-valued matrices, 41 for polynorniaf matrices, 158 Smith-McMillan decomposition, 153 Sublattice, 38 Symmetric paralldepiped, 38 Sysstem theory lossless, 146 MeMillm degree, 154 multi-input multi-output, 146 normalized lossless, 146

Unirnodular matrix integer-valued, 35 polgnomiat, 148 Unitmy matrices Householder transform, 156 Upsampler, 7

kctors bior t hogonat, 174 linear combination, 35, linear independence, 31 to m a t r k power, 44 to vector power, 44

w Wavelet trftnsfbrm adnniseibilie condition, 168 biortkagond wavelet, 187 convolution theorem, 271 definition, 168 differentiation theorem, 1TO B m r wavelet, 181 implementation, 183 initial eoeBciertt, 185 linearity property, 168 Malatar, 123 scaling function, 281 shi& theorem, 169 similarity theorem, 169

z transform

definition, 3 nrultidimensionaf, 44 Vect;ar space basis, 32 span, 31.