Preface
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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.
-:
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,
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
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