Oversampled Filter Banks and Predictive Subband Coders · Oversampled Filter Banks and Predictive...

Dissertation

Oversampled Filter Banks

and Predictive Subband Coders

ausgefuhrt zum Zwecke der Erlangung des akademischen Grades

eines Doktors der technischen Wissenschaften

unter Anleitung von

Ao. Univ.-Prof. Dr. Franz Hlawatsch

Institut fur Nachrichtentechnik und Hochfrequenztechnik

Technische Universitat Wien

und

Ao. Univ.-Prof. Dr. Hans G. Feichtinger

Institut fur Mathematik

Universitat Wien

durch

Dipl.-Ing. Helmut Bolcskei

Wien, im November 1997

ABSTRACT

This thesis is concerned with the theory and applications of oversampled �lter banks and

predictive subband coders� Filter banks are used in many modern data compression

schemes� So far� the interest of the signal processing community has mostly been

restricted to critically sampled �lter banks� Only recently� there has been an increased

interest in oversampled �lter banks�

We provide a new frame�theoretic treatment of oversampled �lter banks and we show

that oversampled �lter banks have noise reducing properties and yield more design free�

dom than critically sampled �lter banks� We introduce oversampled cosine modulated

�lter banks� which are practically attractive since they allow an e�cient DCT�DST

based implementation� Furthermore� a novel subband image coding scheme based on

the new class of linear phase even�stacked cosine modulated �lter banks is developed�

We demonstrate that the proposed subband image coder outperforms existing subband

coders based on nonlinear phase cosine modulated �lter banks from a perceptual point

of view� We provide a subspace�based noise analysis of oversampled �lter banks and

we show that there exists an important tradeo� between noise reduction and design

freedom in oversampled �lter banks�

Finally� we introduce new� highly e�cient methods for achieving quantization noise

reduction in oversampled �lter banks� The resulting oversampled predictive subband

coders are attractive for subband coding applications where the resolution of the quan�

tizers used in the subbands is low� In this case our techniques help to drastically improve

the e�ective resolution of the coder�

iii

KURZFASSUNG

Diese Arbeit besch�aftigt sich mit der Theorie und mit Anwendungen von

�uberabgetasteten Filterb�anken und pr�adiktiven Teilbandcodierern� Filterb�anke wer�

den in vielen modernen Datenkompressionssystemen eingesetzt� Bis jetzt wurden

haupts�achlich Filterb�anke mit kritischer Abtastung betrachtet� Seit kurzem besteht

jedoch vermehrtes Interesse an �uberabgetasteten Filterb�anken�

Wir f�uhren ein neuartiges auf der Theorie der Frames basierendes Konzept zur

Analyse �uberabgetasteter Filterb�anke ein� Wir zeigen� da� �uberabgetastete Filter�

b�anke ger�auschreduzierende Eigenschaften haben und mehr Entwurfsfreiheit als kri�

tisch abgetastete Filterb�anke bieten� Wir f�uhren �uberabgetastete cosinus�modulierte

Filterb�anke ein� welche praktisch interessant sind� weil sie e�ziente DCT�DST�

basierte Implementierungen erlauben� Weiters schlagen wir ein neuartiges teil�

bandbasiertes Bildcodierungssystem vor� welches auf der neuen Klasse der linear�

phasigen �even�stacked� cosinus�modulierten Filterb�anke beruht� Wir zeigen� da�

das resultierende Bildcodierungssystem bessere perzeptuelle Eigenschaften hat als

bestehende Bildcodierungssysteme� die auf der Klasse der nichtlinearphasigen odd�

stacked cosinus�modulierten Filterb�anke beruhen� Wir f�uhren eine Ger�auschanalyse

von �uberabgetasteten Filterb�anken durch�

Schlie�lich werden neuartige e�ziente Methoden zur Ger�auschreduktion in

�uberabgetasteten Filterb�anken vorgeschlagen� Die resultierenden �uberabgetasteten

pr�adiktiven Teilbandcodierer sind insbesondere f�ur Teilbandcodierungsanwendungen mit

Quantisierern geringer Au��osung interessant� In diesem Fall erm�oglichen die vorge�

schlagenen Methoden eine betr�achtliche Verbesserung der e�ektiven Au��osung des

Codierers�

v

Die Begutachtung dieser Arbeit erfolgte durch�

� Ao� Univ��Prof� Dr� F� Hlawatsch

Institut f�ur Nachrichtentechnik und Hochfrequenztechnik

Technische Universit�at Wien

�� Ao� Univ��Prof� Dr� H� G� Feichtinger

Institut f�ur Mathematik

Universit�at Wien

vii

To my parents and Karin

ix

Acknowledgments

I would like to thank the following persons who implicitly or explicitly contributed

to this thesis�

� Prof� F� Hlawatsch for his support during my research� He always had time for

discussions and helped to improve the presentation of the thesis�

� Prof� H� G� Feichtinger for many useful remarks concerning the mathematical as�

pects of my work� for the time he spent in countless discussions� and for initiating

my stay at Philips Research Laboratories Eindhoven�

� Prof� W� F� G� Mecklenbr�auker for his interest� support� and encouragement�

� Dr� A� J� E� M� Janssen for helpful discussions concerning both mathematical and

engineering aspects of my work� for arranging my stay at Philips Research Labo�

ratories Eindhoven� and for teaching me practical aspects of performing research�

� Dr� G� Kubin for useful discussions on predictive coding and subband coding�

� Dr� R� Heusdens for interesting discussions on �lter banks and on rate�distortion

properties of overcomplete expansions�

� Dr� W� Kozek for stimulating discussions on Gabor expansions and frame theory�

� Dipl� Ing� T� Stranz for producing many simulation results and �gures�

xi

Contents

� Introduction and Outline of the Thesis �

� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Outline � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� Oversampled Filter Banks and Frames �

�� Oversampled Filter Banks � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Oversampled Filter Banks and Redundant Signal Expansions � � � �

�� Frame Operator and Polyphase Matrices � � � � � � � � � � � � � � �

�� Frame Bounds � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Perfect Reconstruction and Frame Properties � � � � � � � � � � � � � � � �

�� Perfect Reconstruction � � � � � � � � � � � � � � � � � � � � � � � �

�� Completeness � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Frame Conditions � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Approximative Construction of the Synthesis Filter Bank � � � � � ��

�� Critical Sampling� Linear Independence� and Biorthogonality � � � ��

�� Oversampled Paraunitary Filter Banks � � � � � � � � � � � � � � � � � � � �

�� Equivalence of Oversampled Paraunitary Filter Banks and Tight

Frames � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Construction of Paraunitary Filter Banks � � � � � � � � � � � � � � �

�� Important Special Cases � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Diagonality in the Polyphase Domain � � � � � � � � � � � � � � � � ��

�� Diagonality in the Frequency Domain � � � � � � � � � � � � � � � � ��

�� Redundant Block Transforms � � � � � � � � � � � � � � � � � � � � ��

�� Oversampled FIR Filter Banks � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Smith Form Decomposition � � � � � � � � � � � � � � � � � � � � � ��

�� Analysis of Oversampled FIR Filter Banks � � � � � � � � � � � � � ��

�� Simulation Results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Oversampled DFT Filter Banks �

�� DFT Filter Banks and Weyl�Heisenberg Sets � � � � � � � � � � � � � � � � ��

�� Odd�Stacked DFT FBs � � � � � � � � � � � � � � � � � � � � � � � � ��

xiii

�� Even�Stacked DFT FBs � � � � � � � � � � � � � � � � � � � � � � � ��

�� Representations of the DFT FB Operator � � � � � � � � � � � � � � � � � � ��

�� Perfect Reconstruction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� PR condition in the time domain � � � � � � � � � � � � � � � � � � ��

�� PR condition in the frequency domain � � � � � � � � � � � � � � � ��

�� PR condition in the polyphase domain � � � � � � � � � � � � � � � �

�� PR condition in the dual polyphase domain � � � � � � � � � � � � �

�� Frame�Theoretic Properties � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Representations of the Frame Operator � � � � � � � � � � � � � � � ��

�� Time�Limited Prototype � � � � � � � � � � � � � � � � � � � � � � � �

�� Band�Limited Prototype � � � � � � � � � � � � � � � � � � � � � � � ��

�� Paraunitarity Conditions � � � � � � � � � � � � � � � � � � � � � � � ��

�� Integer Oversampling � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Design of Oversampled FIR Paraunitary DFT Filter Banks � � � � � � � � ��

�� Simulation Results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Oversampled Cosine Modulated Filter Banks �

�� Oversampled Odd�Stacked CMFBs � � � � � � � � � � � � � � � � � � � � � ��

�� Oversampled Even�Stacked CMFBs � � � � � � � � � � � � � � � � � � � � � ��

�� De�nition of Even�Stacked CMFBs � � � � � � � � � � � � � � � � � ��

�� Interpretation of the Subband Signals � � � � � � � � � � � � � � � � ��

�� Representation of CMFBs via DFT Filter Banks � � � � � � � � � � � � � � �

�� Odd�Stacked CMFBs � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Even�Stacked CMFBs � � � � � � � � � � � � � � � � � � � � � � � � ��

�� A Fundamental Decomposition � � � � � � � � � � � � � � � � � � � �

�� Relation with MDFT Filter Banks � � � � � � � � � � � � � � � � � ��

�� Representations of the CMFB Operators � � � � � � � � � � � � � � ��

�� Perfect Reconstruction Conditions � � � � � � � � � � � � � � � � � � � � � � ��

�� PR Conditions Using the CMFB Operators � � � � � � � � � � � � ��

�� PR Conditions in the Time Domain � � � � � � � � � � � � � � � � � ��

�� PR Conditions in the Frequency Domain � � � � � � � � � � � � � � ��

�� PR Conditions in the Polyphase Domain � � � � � � � � � � � � � � ��

�� PR Conditions in the Dual Polyphase Domain � � � � � � � � � � � �

�� Frame�Theoretic Analysis � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Relations between CMFBs and DFT FBs � � � � � � � � � � � � � � �

�� Paraunitarity Conditions � � � � � � � � � � � � � � � � � � � � � � � �

�� Time�Limited Prototype � � � � � � � � � � � � � � � � � � � � � � � �

�� Band�Limited Prototype � � � � � � � � � � � � � � � � � � � � � � � �

�� Integer Oversampling � � � � � � � � � � � � � � � � � � � � � � � � � �

xiv

�� Construction of Paraunitary Prototypes � � � � � � � � � � � � � � �

�� Design Methods � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Design via Constrained Optimization � � � � � � � � � � � � � � � � �

�� Linearized Design Method � � � � � � � � � � � � � � � � � � � � � � �

�� Lattice Design � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� E�cient Implementation � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Implementation of Even�Stacked CMFBs � � � � � � � � � � � � � �

�� Implementation of Odd�Stacked CMFBs � � � � � � � � � � � � � � �

�� General Oversampling � � � � � � � � � � � � � � � � � � � � � � � � �

�� Image Coding with Even�Stacked CMFBs � � � � � � � � � � � � � � � � � �

�� Subband Image Coder Based on Even�Stacked CMFBs � � � � � � �

�� Simulation Results � � � � � � � � � � � � � � � � � � � � � � � � � �

� Noise Analysis ��

�� Noise Analysis and Design Freedom � � � � � � � � � � � � � � � � � � � � � ��

�� A�D Conversion as a Frame Expansion � � � � � � � � � � � � � � � ��

�� Design Freedom in Oversampled A�D Conversion � � � � � � � � � ��

�� Noise Analysis in Oversampled A�D Conversion � � � � � � � � � � ��

�� Noise Analysis and Design Freedom � � � � � � � � � � � � � � � � � � � � � �

�� Noise Analysis for Oversampled Filter Banks � � � � � � � � � � � � �

�� Noise Reduction Versus Design Freedom in Filter Banks � � � � � �

Oversampled Predictive Subband Coders ��

�� Oversampled Predictive A�D Converters � � � � � � � � � � � � � � � � � �

�� Noise Predictive �Noise Shaping� Coders � � � � � � � � � � � � � �

�� Signal Predictive Coders � � � � � � � � � � � � � � � � � � � � � � � ��

�� Oversampled Predictive Subband Coders � � � � � � � � � � � � � � � � � � ��

�� Noise Predictive �Noise Shaping� Subband Coders � � � � � � � � � ��

�� Signal Predictive Subband Coders � � � � � � � � � � � � � � � � � � �

� Conclusion ��

A The Theory of Frames ��

A� Motivation and De�nition of Frames � � � � � � � � � � � � � � � � � � � � ��

A�� The Frame Operator � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

A�� The Dual Frame � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

A�� Signal Expansions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

A�� Exact Frames and Biorthogonality � � � � � � � � � � � � � � � � � � � � � � ��

A�� Frames and Bases � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

A�� Transformation of Frames � � � � � � � � � � � � � � � � � � � � � � � � � � ��

xv

A� Frames and Pseudoinverses � � � � � � � � � � � � � � � � � � � � � � � � � � �

A� Frames and the Gram Matrix � � � � � � � � � � � � � � � � � � � � � � � �

A�� Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

B Multi�Channel Levinson Algorithm � �

xvi

Chapter �

Introduction and Outline of the

Thesis

�� Introduction

Many modern data compression schemes are based on signal decompositions or �lter

banks �FBs� �� The aim of signal decompositions is to perform a

decorrelation� which means that the expansion coe�cients are uncorrelated or� in prac�

tice� nearly uncorrelated� Such a decorrelation is equivalent to a diagonalization of the

signal�s autocovariance matrix �� Exact diagonalization is achieved by the Karhunen�

Lo�eve transform �KLT�� However� the KLT is often impractical since the basis functions

depend on the signal� Furthermore� in general the KLT cannot be implemented e��

ciently� since there is no structure in the KLT basis functions�

A reasonable alternative to the KLT is given by transforms that come close to the

KLT in the sense that they perform an approximate diagonalization of the signal�s

autocovariance matrix� In general� so�called unstructured transforms �like the KLT

itself� � � will perform better in decorrelating a signal than structured transforms� A

structured transform is a transform with basis functions generated from a single or

several �prototypes� according to some speci�c rule� Structured transforms can be

implemented very e�ciently by exploiting the structure inherent in the transform� A

popular structured transform is the wavelet transform� where the basis functions are

obtained by time�shifts and time�scalings of a single mother wavelet �� Other

structured transforms are for example Gabor expansions ��

Wilson expansions �� and local cosine bases �� Gabor expansions

provide decompositions of a signal into time�shifted and complex modulated versions

of a prototype function� whereas Wilson expansions provide decompositions into time�

shifted and cosine �sine� modulated versions of a prototype function� Local cosine bases

are generalizations of Wilson bases in the sense that prototype functions with varying

� CHAPTER �� INTRODUCTION AND OUTLINE OF THE THESIS

widths are used�

An important class of structured signal expansions known as shift�invariant expan�

sions �� corresponds to uniform �lter banks �FBs� �� A uniform

FB provides a decomposition of a signal into time�shifted versions of a set of generating

functions �prototypes�� which are the impulse responses of the synthesis �lters� If we

impose a modulation structure on the generating functions� i�e�� the generating func�

tions are derived from a single prototype function by complex or real modulations� we

may end up with a Gabor or with a Wilson expansion� respectively� As a di�erence to

uniform FBs� tree�structured FBs provide wavelet expansions �� Well known signal

compression standards such as MUSICAM for audio �� JPEG for images �� and

MPEG for video �� are based on FBs� Another advantage of the use of FBs is that

quantization errors can be distributed in the transform domain �which is related to the

frequency domain� such that they match perceptual criterions �� For example� in

speech coding the masking e�ect between frequency bands can be exploited ��

Most of the schemes discussed above are based on orthogonal or biorthogonal signal

expansions or critically sampled FBs� In such decompositions the basis functions are

linearly independent and hence there is no redundancy in the transform coe�cients�

Redundant signal expansions �� and oversampled FBs �� did not

receive much attention so far�

In this thesis� we shall study redundant shift�invariant signal expansions or equiv�

alently oversampled uniform FBs� It is well known that the use of redundancy in

engineering systems improves robustness and numerical stability� Besides that� we will

also show that oversampled FBs o�er more design freedom than critically sampled FBs

and consequently better �lters can be designed in the oversampled case ��

The increased design freedom is due to the facts that for a given oversampled anal�

ysis FB the corresponding synthesis FB is not uniquely determined� and that perfect

reconstruction can be satis�ed with fewer side constraints�

We shall furthermore show that oversampled FBs o�er the possibility of noise re�

duction �� We introduce oversampled predictive subband coders and we

compare the rate�distortion performance of redundant expansions with that of orthog�

onal or critically sampled expansions� The noise reducing properties of oversampled

FBs are of particular importance in applications where the resolution of the quantizers

in the subbands is low� In this case� oversampling and predictive quantization im�

prove the e�ective resolution of the subband coder� The practical advantages of using

low�resolution quantizers at the cost of increased sample rate are indicated by the pop�

ular sigma�delta techniques �� Using low�resolution quantizers in the digital

domain increases circuit speed and reduces circuit complexity�

Since these advantages of oversampled FBs come at the expense of increased com�

putational cost� we shall devote a major part of this thesis to oversampled modulated

�� OUTLINE �

�complex or real� FBs �� This class of FBs allows e�cient FFT�based or

DCT�DST�based implementations and is therefore of great practical interest� Besides

that� we introduce the new class of �even�stacked� oversampled cosine modulated FBs

that allows both perfect reconstruction and linear phase �lters in all channels ��

Linear phase �lters are of particular importance in image coding since nonlinear phase

�lters lead to undesirable artifacts in the reconstructed image if the subband signals are

quantized �� We furthermore propose a complete subband image coding scheme

based on even�stacked cosine modulated FBs�

Further applications of redundant representations or oversampled FBs include�

� Denoising of signals ��

� Oversampled A�D conversion ��

� Singularity detection and processing ��

� Speech enhancement ��

� Image interpolation ��

� Multicarrier transmission ��

�� Outline

We shall next outline the organization of the thesis�

Oversampled �lter banks and frames �Chapter �� In this chapter we develop

the relation between oversampled FBs and frame expansions �� The use of

frame theory is motivated by the fact that oversampled FBs correspond to redundant

signal expansions for which frame theory is an appropriate mathematical framework�

We show that the polyphase matrix and the alias�component matrix provide matrix

representations of the frame operator �� This fundamental result allows an ef�

�cient frame�theoretic analysis of oversampled FIR and IIR FBs� We show that the

frame bounds of a FB can be obtained by an eigenanalysis of the polyphase matrices�

We formulate necessary and su�cient conditions for a FB to correspond to a frame ex�

pansion� We furthermore provide time� frequency� and polyphase domain parameteriza�

tions of all synthesis FBs providing perfect reconstruction �PR� for a given oversampled

analysis FB� Finally� simulation results demonstrating the bene�ts of oversampling are

presented�

Oversampled DFT �lter banks �Chapter �� Oversampled DFT FBs correspond

to Weyl�Heisenberg expansions �Gabor expansions� or� equivalently� discrete short�time

Fourier transforms� Although the connection between DFT FBs and short�time Fourier


transforms �or Gabor expansions� is well known� a frame�theoretic approach to over�

sampled DFT FBs leads to a number of new insights and results �� The results of

Chapter � are specialized to oversampled DFT FBs� The case of integer oversampling

is discussed in detail and a new procedure for the construction of oversampled parauni�

tary DFT FBs is introduced� Finally� we provide simulation results demonstrating the

increased design freedom in oversampled DFT FBs�

Oversampled cosine modulated �lter banks �Chapter �� Oversampled cosine

modulated FBs �CMFBs� allow an e�cient DCT�DST�based implementation� In prac�

tice� CMFBs are often preferred over DFT FBs since in a CMFB the subband signals are

real�valued if the input signal and the analysis prototype are real�valued� We introduce

two types of oversampled CMFBs �� thereby extending a classi�cation recently

proposed by R� Gopinath for critically sampled CMFBs� The �odd�stacked� CMFBs

extend the traditional CMFB type ��class B CMFBs� �� to the oversampled case�

�Even�stacked� CMFBs extend the �class A CMFBs� recently introduced for critical

sampling by Gopinath �� to the oversampled case� Even�stacked CMFBs are attrac�

tive since they allow both PR�paraunitarity and linear phase �lters in all channels� The

linear phase property is of particular interest for image coding applications� We propose

a new subband image coding scheme based on even�stacked CMFBs� It is demonstrated

that the proposed subband image coder outperforms existing subband coders based on

nonlinear phase CMFBs from a perceptual point of view� We furthermore show that

odd� and even�stacked CMFBs are closely related to odd� and even�stacked DFT FBs

with twice the oversampling factor� We provide perfect reconstruction and paraunitar�

ity conditions for oversampled CMFBs� and we discuss design methods and the e�cient

implementation of oversampled CMFBs ��

Noise analysis �Chapter �� In this chapter� we present a subspace�based noise

analysis for oversampled FBs �� We derive bounds on the variance of the recon�

struction error caused by noisy subband signals� and we discuss the dependence of the

error on the oversampling factor� A signal space interpretation of noise reduction is

given� and the minimum norm synthesis FB is shown to minimize the reconstruction

error in the case of white uncorrelated noise� We show that there is a fundamental

tradeo� between noise reduction and design freedom �� Speci�cally� in many

cases the minimum norm synthesis FB �which for white uncorrelated noise minimizes

the variance of the reconstruction error� does not have desirable properties such as good

frequency selectivity� On the other hand� other PR synthesis FBs which may have these

desirable properties lead to a larger error variance�

Oversampled predictive subband coders �Chapter �� This chapter introduces

two types of oversampled predictive subband coders� namely� oversampled noise shaping

and oversampled linear predictive subband coders �� Motivated by a new signal

�� OUTLINE �

space interpretation of noise shaping in oversampled A�D conversion� we establish a

framework for oversampled noise shaping and linear predictive subband coders� We

show that considerable noise reduction can be achieved by using predictive quantiza�

tion in oversampled FBs �� The proposed subband coders generalize oversampled

predictive A�D converters and exploit both interchannel �cross�channel� and intrachan�

nel redundancy� For a given oversampled FB and given noise and signal statistics� we

derive the optimal noise shaping system and the optimal linear prediction system� It

is shown that exploiting interchannel redundancy leads to signi�cant improvements in

coder performance�

The coder structures discussed in this chapter are speci�cally attractive for subband

coding applications where the resolution of the quantizers in the subbands is low� In

this case� oversampling combined with noise shaping or prediction improves the e�ective

coder resolution� Noise shaping and linear prediction in oversampled FBs can be viewed

as techniques to exploit �or remove� the redundancy introduced by oversampling� An�

other technique to remove this redundancy is entropy coding �e�g� Hu�man coding��

We present coding examples based on oversampled FBs using noise shaping or linear

prediction and subsequent entropy coding� and we perform an experimental error�rate

analysis that provides insights into the error�rate performance of oversampled FBs�

Conclusion �Chapter �� This chapter provides concluding remarks�

The theory of frames �Appendix A�� This appendix provides an introduction

to frame theory which provides a mathematical basis for Chapters �� We discuss the

basics of frame theory and we give detailed proofs of the most important theorems of

frame theory�

Multi�channel Levinson algorithm �Appendix B�� This appendix gives a de�

tailed derivation of the multi�channel Levinson algorithm that permits an e�cient cal�

culation of the optimum noise shaping and optimum prediction systems discussed in

Chapter ��

We conclude this outline by providing a summary of what we consider the most

important contributions of the thesis�

� Frame�theoretic study of oversampled FBs based on a matrix representation of

the underlying frame operator �Chapter ��

� Parameterization of all synthesis FBs providing perfect reconstruction for a given

oversampled analysis FB �Chapter ��

� Application of the theory of Weyl�Heisenberg frames to oversampled DFT FBs

�Chapter ��


� Analysis and design of two classes of oversampled cosine modulated �lter banks

�Chapter ��

� Fundamental relationship between cosine�modulated �lter banks and DFT FBs

with twice the oversampling factor �Chapter ��

� Image coding scheme based on the new class of even�stacked cosine modulated

�lter banks �Chapter ��

� Noise analysis of oversampled FBs �Chapter ��

� Tradeo� between noise reduction and design freedom in oversampled FBs �Chap�

ter ��

� Noise shaping subband coders using oversampled FBs �Chapter ��

� Oversampled signal predictive subband coders with cross�channel prediction

�Chapter ��

Chapter �

Oversampled Filter Banks and

Frames

Uniform �lter banks �FBs�� i�e�� lter banks with the same integer decimation factor

in each channel �� correspond to a class of discrete�time signal expansions

known as shift�invariant expansions �� The relation between discrete�time

signal expansions and maximally decimated �or critically sampled� FBs has been studied

in �� It has also been recognized that oversampled FBs ��

correspond to redundant signal expansions �� In ��

�� the use of the theory of frames �� for the study of oversampled FBs

has been proposed� In �� oversampled FIR FBs have been studied using the

theory of polynomial matrices �� A vector��lter framework for the study of

oversampled FIR FBs has been proposed in ��

In this chapter� we introduce a new frame�theoretic approach for the study of over�

sampled FIR and IIR FBs� Our approach is based on an extension of the Zibulski�Zeevi

method for the study of continuous�time Gabor frames �� Speci�cally� we show

that the polyphase matrices and the alias component matrices provide matrix repre�

sentations of the frame operator corresponding to a frame generated by a FB ��

Our approach is more general than those presented in �� since it allows to treat

both FIR and IIR FBs and it leads to several further original results that include

� a compact parameterization of all synthesis FBs providing perfect reconstruction

�PR� for a given oversampled analysis FB�

� methods for estimating the frame bounds� constructing paraunitary FBs from

non�paraunitary FBs� and approximately calculating PR synthesis FBs�

� a sensitivity analysis for oversampled FBs involving the frame bounds�

�For the sake of brevity� we shall use the term �lter bank �FB� instead of uniform �lter bank�

�

CHAPTER �� OVERSAMPLED FILTER BANKS AND FRAMES

We shall now summarize our main results and outline the organization of this chap�

ter� Section �� brie�y reviews oversampled FBs� introduces uniform �lter bank frames

�UFBFs�� and establishes the connection between frames and oversampled FBs� We

show that the polyphase matrices and the alias component matrices provide matrix

representations of the frame operator� We furthermore show that the frame bounds

can be obtained by an eigenanalysis of the polyphase matrices or the alias component

matrices �the importance of the frame bounds of a FB is discussed in Chapter �� We

furthermore outline an approach for the approximative calculation of the frame bounds

of a FB�

In Section �� we formulate necessary and su�cient conditions for an oversampled

FB to provide PR� We discuss the minimum norm synthesis FB and its approximative

construction� We formulate necessary and su�cient conditions for a FB to provide a

UFBF expansion�

In Section �� we show that oversampled paraunitary FBs correspond to tight UFBFs�

and we propose a new method for constructing paraunitary FBs from given nonpara�

unitary FBs�

Section �� considers important special cases where the calculation of the minimum

norm synthesis FB and the frame bounds is simpli�ed� Important examples of such

FBs are integer oversampled DFT FBs� �bandlimited� FBs� nondecimated FBs� and

redundant block transforms�

Oversampled FIR FBs are a special case of the theory presented in this Chapter�

In the FIR case the polyphase matrices are polynomial matrices� which has several

interesting consequences to be discussed in Section �� For the sake of completeness

we review some of the results on oversampled FIR FBs presented in �� We also

present original results and extensions of the results presented in ��

Finally� simulation results are presented in Section ��

�� Oversampled Filter Banks

In this section� we brie�y discuss oversampled FBs and the polyphase approach proposed

in �� for maximally decimated �critically sampled� FBs and in �� for

oversampled FBs� We furthermore introduce UFBFs� the type of frames corresponding

to uniform FBs�

We consider an N �channel FB �see Fig� � with subsampling by the integer factor M

in each channel� PR and zero delay�� so that x�n� ! x�n� where x�n� and x�n� denote the

input and reconstructed signal� respectively� The transfer functions of the analysis and

synthesis �lters are Hk�z� and Fk�z� �k ! �� N � �� with corresponding impulse

�We note that our theory can easily be extended to PR with nonzero delay�

�� OVERSAMPLED FILTER BANKS

� ��

v��m�

v��m�

vN��m��HN��z�

Analysis �lter bank

� M��

�

�F��z��M�

M� � F��z� �

�

�

M

�

� � FN��z� x�n�

Synthesis �lter bank

��

�

��

��H��z�x�n� M

�

� H��z� � M�

�

Fig� �� N�channel uniform �lter bank�

responses hk�n� and fk�n�� respectively�� The subband signals �see Fig� � are given by

vk�m� !�X

n��x�n� hk�mM � n� � k ! �� N � � ��

and the reconstructed signal is

x�n� !N��Xk��

�Xm��

vk�m� fk�n�mM � � ��

In a critically sampled �or maximally decimated� FB we have N ! M and thus

the subband signals vk�m� contain exactly as many samples �per unit of time� as the

input signal x�n�� In the oversampled case N � M � however� the subband signals are

redundant in that they contain more samples �per unit of time� than the input signal

x�n�� Oversampled FBs have noise reducing properties �see Chapters � and �� and o�er

more design freedom �see Section �� and Chapter �� than critically sampled FBs�

The noise reducing properties of oversampled FBs �redundant representations� ��

allow a coarser quantization of the subband signals� Furthermore� noise shaping and

linear prediction in oversampled FBs result in considerable additional noise reduction

�see Chapter �� The design freedom is increased since in the oversampled case fewer

side constraints arising from the PR property have to be satis�ed� Furthermore� for a

given oversampled analysis FB there exists a whole class of synthesis FBs providing PR

�see Section �� and Chapter ��

The polyphase decomposition of the analysis �lters Hk�z� reads ��

Hk�z� !M��Xn��

znEk�n�zM� � k ! �� N � � ��

where

Ek�n�z� !�X

m��hk�mM � n� z�m � k ! �� N � � n ! �� M �

�Here� for example� Hk�z� �P�

n�� hk�n� z�n denotes the ztransform of hk�n��

� CHAPTER �� OVERSAMPLED FILTER BANKS AND FRAMES

is the nth polyphase component of the kth analysis �lter Hk�z�� The N �M analysis

polyphase matrix is de�ned as

E�z� !

��

E��z� E��z� �� E��M��z�

E��z� E��z� �� E��M��z��

��

EN��z� EN��z� �� EN��M��z�

��

The synthesis �lters Fk�z� can be similarly decomposed�

Fk�z� !M��Xn��

z�nRk�n�zM � � k ! �� N � � ��

with the synthesis polyphase components

Rk�n�z� !�X

m��fk�mM " n� z�m � k ! �� N � � n ! �� M � �

The M �N synthesis polyphase matrix is de�ned as

R�z� !

��

R��z� R��z� �� RN��z�

R��z� R��z� �� RN��z��

��

R��M��z� R��M��z� �� RN��M��z�

��

�� Oversampled Filter Banks and Redundant Signal Expan�

sions

FB analysis and synthesis can be interpreted as a signal expansion �� The

subband signals in �� can be written as the inner products�

vk�m� ! hx� hk�mi with hk�m�n� ! h�k�mM � n� � k ! �� N � � ��

where � stands for complex conjugation� Furthermore� with �� and the PR property�

we have

x�n� ! x�n� !N��Xk��

�Xm��

hx� hk�mi fk�m�n� with fk�m�n� ! fk�n�mM � �

This shows that the FB corresponds to an expansion of the input signal x�n� into the

function set ffk�m�n�g with k ! �� N � and �� m � �� In general the set

ffk�m�n�g is not orthogonal� so that the expansion coe�cients� i�e�� the subband signal

samples vk�m� ! hx� hk�mi� are obtained by projecting the signal x�n� onto a �dual�

��x� y�P�

n�� x�n� y��n� denotes the inner product of the signals x�n� and y�n��

�� OVERSAMPLED FILTER BANKS

set of functions fhk�m�n�g� Critically sampled FBs provide orthogonal or biorthogonal

signal expansions �� see Appendix A� Secs� A�� and A�� whereas oversampled FBs

correspond to redundant �overcomplete� expansions ��

Following the discussion in Appendix A we shall now establish the relation between

FBs and frames in� l��ZZ�� The set fgk�n�g �k � K� de�ned in Appendix A is now given

by the set of analysis functions� fhk�m�n�g� In a FB� the linear operator T �cf� Section

A�� referred to as analysis operator in the following maps the input signal x�n� � l��ZZ�

into the subband signal space �l��ZZ��N � We have

T � x� vk�m� ! hx� hk�mi or equivalently �Tx�k�m ! vk�m� ! hx� hk�mi� ��

We shall next motivate the requirements on hk�m�n� and� equivalently� the operator T

from a FB point of view�

� The FB should satisfy the PR property� This means that for a given analysis FB

a synthesis FB providing PR exists� Note that this requirement is equivalent to

the requirement that the operator T has a left inverse� i�e�� it is invertible on its

range� This means that the signal x�n� can be perfectly reconstructed from the

expansion coe�cients hx� hk�mi�

� We would like to have �nite�energy subband signals if the input signal x�n� has ��

nite energy� In order to meet this requirement the operator T has to be continuous

and hence bounded�

� Another important requirement is numerical stability of the FB in the sense that

small perturbations of the subband signals �caused e�g� by quantization or some

other modi�cation� result in small perturbations of the output signal� Therefore�

the left�inverse T�� has to be continuous and hence bounded�

Combining all three requirements we get the frame condition �cf� Section A��

Akxk� �N��Xk��

�Xm��

jhx� hk�mij� � Bkxk� � x�n� � l��ZZ� ��

with the frame bounds A � � and B ��

De�nition �� A function set fhk�m�n�g de�ned as in �� satisfying ��

will be called uniform �lter bank frame �UFBF��

�Here l��ZZ� denotes the space of squaresummable sequences��Note that here we have an indexing that is dierent from that in Appendix A� namely the index

k �k � K� from Appendix A becomes the double index �k�m�� where k � �� N � � and ��

m �� Here k identi es the subband �channel� while m denotes the time index��The range of a linear operator T � X � Y is Ran�T �� fy j y � Txwith x � Xg�


Note that according to �� the UFBF functions hk�m�n� are generated by uniformly

time�shifting the N analysis �lter impulse responses h�k��n�� i�e�� hk�m�n� ! h�k�mM�n��In the mathematical literature such function sets are known as shift�invariant function

sets ��

For analysis �lters hk�n� such that fhk�m�n�g is a UFBF for l��ZZ�� the dual frame

�cf� Section A�� is given by ��

fk�m�n� ! �S��hk�m��n� � ��

Here� S�� is the inverse of the frame operator de�ned as

�Sx��n� ! �T �Tx��n� !N��Xk��

�Xm��

hx� hk�mihk�m�n� � ��

where the adjoint operator T � is de�ned by �T �v��n� !PN��

k��

P�m�� vk�m�hk�m�n��

Always assuming that fhk�m�n�g is a frame� the frame operator is a linear� positive

de�nite� self�adjoint operator mapping l��ZZ� onto l��ZZ� �� Using the frame operator�

Eq� �� can be written as

Akxk� � hSx� xi � Bkxk��

Finally� the operator �T �cf� Section A�� is given by

�T � x � hx� fk�mi or equivalently � �Tx�k�m ! hx� fk�mi ��

with its adjoint �T�de�ned as

� �T�v��n� !

N��Xk��

�Xm��

vk�m�fk�m�n��

If the analysis set fhk�m�n�g is a frame� then the synthesis set ffk�m�n�g de�ned by

�� is also a frame �the �dual� frame�� with frame bounds A� ! �B and B� ! �A�

We note that the dual frame ffk�m�n�g is a special synthesis set that provides PR� The

following theorem states that if fhk�m�n�g is a UFBF� then the dual frame ffk�m�n�g is

again a UFBF� i�e�� it is generated by uniformly time�shifting a dual set of functions

fk�n��

Theorem �� If fhk�m�n�g is a UFBF with parameters M and N � then the

dual frame ffk�m�n�g is again a UFBF with the same parameters M and N �

i�e��

fk�m�n� ! fk�n�mM �� k ! �� N � �

where

fk�n� ! �S��hyk��n� with hyk�n� ! h�k��n��

�� OVERSAMPLED FILTER BANKS �

Proof� Introducing the unitary time�shift operator Am as �Am x��n� ! x�n � mM ��

we can write

hk�m�n� ! �Amhyk��n��

Using AmAm� ! Amm� and A�m!A�m �with A�

m denoting the adjoint of Am�� it is

easily shown that both the frame operator S and its inverse S�� commute with the

time�shift operator Am� i�e�� AmS ! SAm and AmS�� ! S

��Am� We then obtain

fk�m�n� ! �S��hk�m��n� ! �S��Am hyk��n� ! �AmS��hyk��n� ! �Am fk��n� ! fk�n�mM ��

which concludes the proof� �

A frame is called snug if B��A� ! B�A and tight if B��A� ! B�A ! �see

De�nition A�� For a tight frame we have S ! A I �where I is the identity operator

on l��ZZ�� and hence there is simply fk�n� !�Ah�k��n�� According to Theorem A�� the

signal x�n� can be expressed as

x�n� !N��Xk��

�Xm��

hx� fk�mihk�m�n� !N��Xk��

�Xm��

hx� hk�mi fk�m�n��

This can equivalently be written as

N��Xk��

�Xm��

hk�m�n� f�k�m�n

�� !N��Xk��

�Xm��

fk�m�n� h�k�m�n

�� ! ��n� n��

�� Frame Operator and Polyphase Matrices

We shall now further establish the connection between FBs and UFBFs by showing

that the analysis operator T � the synthesis operator �T�� and the frame operator S

can be expressed in terms of the polyphase matrices of the corresponding FB� This

result extends a similar result on continuous�time Weyl�Heisenberg frames �� and

can therefore be seen as a generalization of the Zibulski�Zeevi representation for Weyl�

Heisenberg frame operators to shift�invariant systems�

We shall �rst show how the frame operator S can be expressed in terms of the

polyphase matrices�

Theorem �� Let y�n� ! �Sx��n� and x�n� ! �S��y��n�� where S is the

frame operator corresponding to a UFBF� Then� the polyphase components

Yn�z� !P�

m�� y�mM " n� z�m of Y �z� and the polyphase components

Xn��z� !P�

m�� x�mM " n�� z�m of X�z� are related as�

Yn�z� !M��Xn��

Sn�n��z�Xn��z� with Sn�n��z� !N��Xk��

#Ek�n�z�Ek�n��z� ��

Xn��z� !M��Xn��

S��n��n�z�Yn�z� with S��n��n�z� !N��Xk��

Rk�n��z� #Rk�n�z� ��

Here� for example� �Ek�n�z� � E�k�n��z��


or equivalently� using the polyphase vectors�

x�z� ! �X��z� X��z� �� XM��z��T and y�z� ! �Y��z� Y��z� �� YM��z��T �

y�z� ! S�z�x�z� with S�z� ! #E�z�E�z� � ��

x�z� ! S��z�y�z� with S��z� ! R�z� #R�z� � ��

Proof� With �� the signal y�n� ! �Sx��n� can be written as

y�n� !N��Xk��

�Xm��

�Xr��

x�r� hk�mM � r�

�h�k�mM � n��

The nth polyphase component of y�n� is given by

Yn�z� !�X

l��

N��Xk��

�Xm��

�Xr��

x�r� hk�mM � r� h�k�mM � n� lM �

�z�l

!�X

l��

N��Xk��

�Xm��

M��Xr��

�Xr��

x�r� " r�M � hk�mM � r� � r�M � h�k�l�M � n� z�mzl

�

!M��Xn��

N��Xk��

�Xm��

hk�mM � n��z�m�X

l��h�k�lM � n�zl

�Xr��

x�n� " r�M �z�r�

!M��Xn��

Xn��z�N��Xk��

Ek�n��z� #Ek�n�z��

which establishes Eq� �� Eq� �� can be shown in an analogous manner� �

Thus� the frame operator S is expressed in the polyphase domain by the M �M

UFBF matrix S�z� ! #E�z�E�z� de�ned in terms of the analysis polyphase matrix

E�z�� Similarly� the inverse frame operator S�� is expressed in the polyphase domain

by the M � M inverse UFBF matrix S��z� ! R�z� #R�z� de�ned in terms of the

synthesis polyphase matrix R�z��

Specializing to the unit circle �z ! ej�� we now show that the polyphase matrices

E�ej�� and R�ej�� provide matrix representations �� of the frame operator S and

the inverse frame operator S�� respectively� Much of our subsequent discussion of FBs

will be based on these matrix representations�

Corollary �� The positive de�nite M �M matrices

S�ej�� ! EH�ej��E�ej�� and S��ej�� ! R�ej��RH�ej��

are the matrix representations of the frame operator S and the inverse

frame operator S�� respectively� with respect to the basis fen��n��g of l��ZZ�Here� �E�z� � E

H��z�� with the superscript H denoting the conjugate transpose is the paracon

jugate of E�z��


given by�� en��n�� !

P�m�� n��n�mM � ej��

�M n��n� �n ! �� M � �

� � � � ��

Proof� Using Xn�ej�� ! hx� en��i� it follows from �� that

hSx� en��i !M��Xn��

Sn�n��ej�� hx� en��i �

This show that S�ej�� ! EH�ej��E�ej�� is the matrix representation of S with

respect to the basis en��n�� The positive de�niteness of S�ej�� follows from the pos�

itive de�niteness of S� In a similar manner� it follows from �� that S��ej�� !

R�ej��RH�ej�� is the matrix representation of S��

According to �� the analysis operator T maps the input signal x�n� into the

subband signals vk�m�� Transforming �� into the z�transform domain yields

v�z� ! E�z�x�z� �

where v�z� !P�

m�� v�m� z�m with v�m� ! �v��m� v��m� �� vN��m��T and x�z� !

�X��z� X��z� �� XM��z��T � Thus� the analysis polyphase matrix E�z� provides a

polyphase domain representation of the analysis operator T � Comparing S ! T�T

with S�z� ! #E�z�E�z�� it is furthermore clear that the adjoint analysis operator T � is

represented by the paraconjugate #E�z��

In a similar manner� transforming �� into the z�transform domain yields

x�z� ! R�z�v�z��

where x�z� ! � X��z� X��z� �� XM��z��T with Xn�z� !P�

m�� x�mM " n� z�m� This

shows that the synthesis operator �T�is represented in the polyphase domain by the

synthesis polyphase matrix R�z��

An important consequence of Corollary �� is the identity of the eigenvalues of the

frame operator with the eigenvalues of its matrix representation� the UFBF matrix�

Corollary �� Let n�� with n ! �� M � denote the eigen�

values of the UFBF matrix S�ej�� ! EH�ej��E�ej�� de�ned by the

eigenequation

S�ej��vn�� ! n��vn�� n ! �� M � � � � � � �

Any eigenvalue n�� is simultaneously an eigenvalue of the frame opera�

tor S� Conversely� any eigenvalue of S is simultaneously an eigenvalue of

S�ej��

��This basis induces the polyphase representation on the unit circle��x� en��

� Xn�e

j�� P�m�� x�mM � n� e�j��m� Equivalently� this is the Zak transform of x�n� ��


Proof� Using �� it can easily be shown that S!Z��S�ej��Z� where Z denotes

the polyphase transform operator� i�e�� the operator mapping a signal x�n� onto its

polyphase transform with z ! ej�� Xn�ej�� Since Z is an isometric isomorphism

�� it follows that S and S�ej�� are unitarily equivalent �� Therefore� S and

S�ej�� have the same eigenvalues ��

It follows that the eigenanalysis of the frame operator S �a matrix of in�nite size� is

equivalent to that of the UFBF matrix S�ej�� an M �M matrix indexed by a real�

valued parameter � � �� Similarly� the eigenvalues of the inverse frame operator

S�� are equal to those of the inverse UFBF matrix S��ej�� ! R�ej��RH�ej��

which will be denoted �n�� in the following� Since S�ej�� and S��ej�� are positive

de�nite matrices� their eigenvalues are positive� This result is important� since it allows

to reduce the eigenanalysis of the operator S� which is a matrix of in�nite size� to the

eigenanalysis of the M � M matrix S�ej�� However� the matrix S�ej�� depends

on the parameter � � �� This means that one has to perform an eigenanalysis of

S�ej�� for � � �� which of course is not possible in practice� One can resort to the

pragmatic approach of performing an eigenanalysis of S�ej��lL � for l ! �� L � �

However� this discretization approach will in general not give the exact eigenvalues�

In Section �� we shall discuss situations where the frame bounds can be calculated

without performing an explicit eigenanalysis�

�� Frame Bounds

Important numerical properties of the UFBF fhk�m�n�g� and thus also of the associated

FB� are determined by its frame bounds A and B or� equivalently� A� ! �B and

B� ! �A �� With �� the subband signals vk�m� ! hx� hk�mi of a FB providing a

UFBF expansion satisfy �see De�nition A��

Akxk� �N��Xk��

�Xm��

jvk�m�j� � Bkxk� � x�n� � l��ZZ� ��

with � � A � B � �� This double inequality generalizes the well�known energy

conservation equationPN��

k��

P�m�� jvk�m�j� ! kxk� in orthogonal FBs �� which is

reobtained for A ! B ! � For A ! B we havePN��

k��

P�m�� jvk�m�j� ! A kxk�� i�e��

energy conservation up to a constant factor A� Note that �� shows that the subband

signals vk�m� are in �l��ZZ��N if the input signal x�n� is in l��ZZ�� Since the frame bounds

describe important numerical properties of a FB �see Section �� their calculation is

of interest� The next theorem states that the frame bounds follow from the eigenvalues

of the UFBF matrix�

Theorem �� The �tightest possible� frame bounds A and B of a FB

providing a UFBF expansion are given by the essential in�mum and supre�


mum� respectively� of the eigenvalues n�� of the UFBF matrix S�ej�� !

EH�ej��E�ej��

A ! ess inf�� n��M��

n�ej�� B ! ess sup

�� n��M��n�e

j��

Proof� It is well known �� that the �tightest possible� frame bounds A and B

are the essential in�mum and the essential supremum� respectively� of the eigenvalues of

the frame operator S �see Theorem A�� Hence� Theorem �� follows using Corollary

��

Similarly� we have

A� ! ess inf�� n��M��

�n�ej�� B� ! ess sup

�� n��M��n�e

j��

where �n�� are the eigenvalues of the inverse UFBF matrix S��ej�� !

R�ej��RH�ej�� Note that in practice the calculation of the frame bounds is ac�

complished by sampling the matrix S�ej�� on the unit circle and performing an eige�

nanalysis of S�ej��lL � for l ! �� L� � As already mentioned in Subsection ��

this approach will not give the exact frame bounds�

An interesting consequence of Theorem �� is the following corollary� which has been

formulated for the FIR case in ��

Corollary �� Let fhk�m�n�g be a UFBF for l��ZZ� with frame bounds A

and B� Then

A �

M

N��Xk��

khkk� � B� ��

In particular� in the case of a tight frame �where A ! B� we have

M

N��Xk��

khkk� ! A �

Proof� The trace of the UFBF matrix satis�es

Tr fS�ej��g !M��Xn��

N��Xk��

jEk�n�ej��j�

and

Tr fS�ej��g !M��Xn��

n��

so thatM��Xn��

n�� !M��Xn��

N��Xk��

jEk�n�ej��j��

CHAPTER �� OVERSAMPLED FILTER BANKS AND FRAMES

From Theorem �� we conclude that MA � PM��n�� n�� MB� and� with ��

MA � PM��n��

PN��k�� jEk�n�e

j��j� � MB� Integrating both sides of this inequality

with respect to the frequency parameter � and usingPM��

n��

R �� jEk�n�e

j��j� d� ! khkk��we obtain ��

If we normalize the hk�n� such that khkk� ! for k ! �� N�� then �� yields

the following inequality relating the frame bounds with the oversampling factor NM�

A � N

M� B�

For a tight UFBF �corresponding to a paraunitary FB� see Section �� it follows that

the frame bounds are equal to the oversampling factor�

A ! B !N

M�

Thus� in the latter case the frame bounds provide a measure of the frame redundancy�

In Sec� �� we shall show that the reconstruction error variance in a FB resulting from

perturbations of the subband signals can be bounded in terms of the frame bounds�

which therefore serve as an important measure of the FB�s noise sensitivity�

�� Perfect Reconstruction and Frame Properties

In this section� we derive a PR condition for oversampled FBs� We show that� for a given

oversampled analysis FB� the synthesis FB providing PR is not uniquely determined�

and we parameterize all PR synthesis FBs for a given oversampled analysis FB� We

furthermore provide conditions for a FB to implement a UFBF expansion in l��ZZ�� The

approximative construction of the minimum norm synthesis FB is discussed� and the

reconstruction error resulting from this approximation is shown to depend on the frame

bounds� Finally� we consider the special case of critical sampling� Critically sampled

FBs have been studied in great detail in the literature �� Based on

our frame�theoretic approach we provide a simple proof for the well�known fact that

critically sampled FBs provide decompositions into Riesz bases or equivalently exact

frames�

�� Perfect Reconstruction

In many applications� PR is considered a desirable property of FBs� PR is always

satis�ed by FBs corresponding to UFBF expansions� However� in the following we

consider FBs that do not necessarily correspond to UFBFs� and we will derive PR

�� PERFECT RECONSTRUCTION AND FRAME PROPERTIES

conditions in the polyphase� frequency� and time domains for arbitrary oversampling

factors�

PR condition in the polyphase domain� Transforming the FB input�output

relation x�n� !PN��

k��

P�m�� hx� hk�mi fk�m�n� into the polyphase domain yields

Xn�z� !M��Xn��

N��Xk��

Rk�n�z�Ek�n��z�

�Xn��z� � n ! �� M � �

or more compactly

x�z� ! R�z�E�z�x�z��

This gives the following result�

Theorem �� An oversampled FB satis�es the PR condition x�n� ! x�n�

if and only if

R�z�E�z� ! IM � ��

where IM is the M � M identity matrix� For E�z� given� R�z� is not

uniquely determined� any solution of �� can be written as �assuming

rank fE�z�g ! M a�e��

R�z� ! R�z� " U�z�hIN �E�z� R�z�

i� ��

where R�z� is the para�pseudo�inverse of E�z�� which is a particular solution

of �� de�ned as R�z� !

h#E�z�E�z�

i�� #E�z� � ��

and U�z� is an M � N matrix with arbitrary elements �U�z��k�l satisfying

j�U�ej��k�lj ��

Proof and discussion� For critical sampling �N ! M�� E�z� and R�z� are square

�M�M� matrices and thus �� has the unique solutionR�z� ! E��z� ��

In the oversampled case �N � M�� the matrices E�z� and R�z� are rectangular �N�M

and M � N � respectively�� and thus the solution of �� is not uniquely determined�

in fact� any left�inverse of E�z� is a valid solution �such a left�inverse exists if E�z�

has full rank a�e�� It can now be shown ��p� �� that any left�inverse of E�z�

can be written in the form �� R�z� ! R�z� " U�z�hIN �E�z� R�z�

i� where

R�z� is any particular left�inverse� which can be chosen as in �� Indeed� using R�z�E�z� ! IM it is easily seen that R�z�E�z� ! IM � In the special case of critical

sampling �N ! M� we duly have R�z� ! R�z� ! E��z�� We note that� on the

unit circle� the para�pseudo�inverse in �� becomes the conventional pseudo�inverse�

R�ej�� !hEH�ej��E�ej��

i��EH�ej��

��a�e� stands for almost everywhere�

�� CHAPTER �� OVERSAMPLED FILTER BANKS AND FRAMES

In the oversampled case� the nonuniqueness of the synthesis FB for given analysis

FB entails a freedom of design that does not exist in the case of critical sampling�

Expression �� is a parameterization of R�z� in terms of the MN entries �U�z��k�l

that can be chosen arbitrarily�

The particular synthesis polyphase matrix given by the para�pseudo�inverse R�z� !h#E�z�E�z�

i�� #E�z� can be given an important interpretation� For given analysis �lter

impulse responses hk�n�� consider the particular synthesis �lter impulse responses fk�n�

provided by frame theory via �� i�e�� fk�n� ! �S��hyk��n�� or in other words� ffk�m�n�gis the UFBF that is dual to fhk�m�n�g� From hyk�n� ! �Sfk��n�� it follows with ��

by setting x�n� ! fk�n� in �� for k ! �� N � � that #E�z� ! #E�z�E�z�R�z��

This implies R�z� !h#E�z�E�z�

i�� #E�z� ! R�z�� Thus� the para�pseudo�inverse of E�z�

corresponds to the particular PR synthesis FB provided by frame theory� It is shown in

�� that this frame�theoretic solution minimizesPN��

k�� kfkk� among the class of all left�

inverses� In �� it has been previously observed for the FIR case that the para�pseudo�

inverse corresponds to the frame�theoretic solution� We shall hereafter restrict our

attention mainly to this minimum norm synthesis FB � i�e�� to the particular synthesis

polyphase matrix R�z� !h#E�z�E�z�

i�� #E�z�� which will henceforth be denoted simply

by R�z�� The signi�cance of the para�pseudo�inverse is highlighted by the fact �to be

proved in Subsection �� that in the case of white uncorrelated noise added to the

subband signals the para�pseudo�inverse minimizes the resulting reconstruction error�

PR condition in the frequency domain� The FB output signal in the frequency

domain �z�domain� reads

X�z� !

M

M��Xm��

N��Xk��

Fk�z�Hk�z WmM �

�X�z Wm

M ��

with WM ! e�j��M � The PR condition can therefore be written as

M

N��Xk��

Fk�z�Hk�z WmM � ! ��m��

Using

X�z W lM� !

M

M��Xm��

N��Xk��

Fk�z WlM �Hk�z W

mM �

�X�z Wm

M �

with l ! �� M � � the PR condition can be rewritten in matrix form as

F�z�H�z� ! IM � ��

where F�z� with �F�z��k�l !�pMFl�z W

kM� is theM �N synthesis alias component �AC�

matrix and H�z� with �H�z��l�k !�pMHl�z W

kM� is the N � M analysis AC matrix� In

the oversampled case N � M � F�z� is not uniquely determined for given H�z�� any

solution of �� can be written as

F�z� ! F�z� "V�z��IN �H�z� F�z��

�� PERFECT RECONSTRUCTION AND FRAME PROPERTIES �

where V�z� is an M � N matrix with arbitrary elements �V�z��k�l satisfying

j�V�ej��k�lj � � and F�z� is the para�pseudo�inverse of H�z�� which is a partic�

ular solution of �� given by

F�z� ! � #H�z�H�z�� #H�z��

The parameterization �� can equivalently be written as

Fk�z� ! Fk�z� " Vk�z��

M

M��Xi��

Fk�zWiM �

N��Xl��

Hl�zWiM �Vl�z��

where Fk�z� denotes the synthesis �lters corresponding to the dual frame and the Vl�z�

are related to V�z� as �V�z��k�l ! Vl�zWkM �� The analysis operator T and the synthesis

operator �T�are represented in the AC domain by H�z� and F�z�� respectively� The

frame operator S and the inverse frame operator S�� are represented by #H�z�H�z�

and F�z�#�F�z�� respectively� The basis underlying this matrix representation �with z !

ej�� is given by tk��n�� !

P�m�� n� � m� ej��

kM �n� with n ! �� M � and

� � � � �

The AC matrices can be obtained from the polyphase matrices as

H�z� ! E�zM�D�z�W and F�z� ! WHD��z�R�zM��

where D�z� ! diagfzngM��n�� and W denotes the M � M DFT matrix with elements

�W�m�n ! �pMWmn

M �

PR condition in the time domain� The input�output relation in the time domain

reads

x�n� !N��Xk��

�Xm��

hx� hk�mi fk�m�n��

which leads to the following time�domain formulation of the PR condition�

N��Xk��

�Xm��

fk�n�mM �hk�mM � r� ! ��r � n��

Rewriting this equation in matrix form� it follows that the FB satis�es the PR property

if and only if ��

BA ! I��

Here the analysis matrix

A !

��

� � � � � � � � � � � �

H� H� �� HL�� H� H� �� HL��

� H� H� �� HL��

��

��I� denotes the identity matrix of in nite size�


is an in�nite�size block T�oplitz matrix consisting of the sequence of N � M matrices

Hl �l ! �� L� � de�ned as

Hl !

��

hy��lM � hy��lM " � �� hy��lM "M � �

hy��lM � hy��lM " � �� hy��lM "M � ��

��

hyN��lM � hyN��lM " � �� hyN��lM "M � �

��

where hy�n� ! h��n�� The number L of matrices Hl is given by�� L !lLhM

m� where Lh

denotes the length of the longest analysis �lter� Similarly� the synthesis matrix

B !

��

� � �

� � � FT� �

� � � FT� FT

�

� � �� FT

� FT�

FTL��

�� FT�

� � �

FTL��

��

� FTL��

� � ��

��

is an in�nite�size block T�oplitz matrix consisting of the sequence of N � M matrices

Fl �l ! �� L� � � de�ned as

Fl !

��

f��lM � f��lM " � �� f��lM "M � �

f��lM � f��lM " � �� f��lM "M � ��

��

fN��lM � fN��lM " � �� fN��lM "M � �

��

The number L� of matrices Fl is L� !

lLfM

m� where Lf denotes the length of the longest

synthesis �lter�

In the oversampled case N � M � for given analysis matrix A� the synthesis matrix

B is not uniquely determined� any solution of �� can be written as

B ! B"P�I� �A B��

where P is an arbitrary matrix of in�nite size and B is the pseudo�inverse of A� which

is a particular solution of �� given by

B ! �AHA��AH �

��dae denotes the smallest integer exceeding a�

�� PERFECT RECONSTRUCTION AND FRAME PROPERTIES ��

The parameterization �� can equivalently be written as

fk�n� ! fk�n� " pk�n� �N��Xl��

�Xm��

h fk� hl�mi pl�m�n� � ��

where pk�m�n� ! pk�n �mM � with pk�n� �k ! �� N � � being arbitrary functions

and fk�n� denote the synthesis �lters corresponding to the dual frame�

The analysis operator T and the synthesis operator �T�are represented by the matrices

A and �B� respectively� Consequently� the frame operator S is represented by the matrix

AHA and the inverse frame operator is represented by the matrix B �BH�

Unitary equivalence of di�erent representations� We �nally note that the

polyphase domain� frequency domain� and time�domain representations of the operators

T � �T�� S and S�� are unitarily equivalent� Let us� for example� consider the various

representations of the frame operator S� Using �� we get

#H�z�H�z� !WH #D�z�#E�zM �E�zM�D�z�W ! #L�z�#E�zM �E�zM�L�z��

where L�z� ! D�z�W� It can be shown that #L�z�L�z� ! L�z�#L�z� ! IM � which

implies that the AC domain and the polyphase domain representations of the frame

operator� #H�z�H�z� and #E�zM �E�zM�� are unitarily equivalent� In a similar manner it

can be shown that the time domain and the frequency domain representations� AHA

and #H�z�H�z�� are unitarily equivalent� As a consequence of this unitary equivalence�

it follows immediately that the eigenvalues of the di�erent representations and hence

the frame bounds are equal� For the sake of simplicity� we shall henceforth work with

the polyphase matrices�

�� Completeness

The next theorem states a condition for the completeness of the analysis set fhk�m�n�g�Note that the completeness of fhk�m�n�g is a necessary condition for PR�

Theorem �� The set fhk�m�n�g with hk�m�n� ! h�k�mM � n� is complete

in l��ZZ� if and only if the analysis polyphase matrix E�ej�� has full rank�

i�e�� rankfE�ej��g ! M � a�e� on � � ��

Proof� Assuming completeness of fhk�m�n�g� it follows that hSx� xi !PN��k��

P�m�� jhx� hk�mij� � � for x�n� ! �� and hence the eigenvalues of S �simul�

taneously the eigenvalues n�� of S�ej�� see Corollary �� satisfy n�� a�e� for

� � � � and n ! �� M � � This shows that S�ej�� has full rank a�e�� Since

rankfS�ej��g ! rankfE�ej��g� we get rankfE�ej��g ! M a�e� The converse state�

ment is shown by reversing this line of reasoning� �


It is intuitively obvious that FBs cannot satisfy the PR property in the undersampled

case N � M � Indeed� for N � M the set fhk�m�n�g is incomplete in l��ZZ�� and hence

it is not possible to expand every signal x�n� � l��ZZ� into the functions hk�m�n�� This

can be seen as follows� Since rankfS�ej��g ! rankfE�ej��g� which for N � M is

maximally N � it follows using Theorem �� that fhk�m�n�g is incomplete in l��ZZ��

�� Frame Conditions

We shall next derive conditions for a FB to provide a UFBF expansion in l��ZZ�� FBs

providing UFBF expansions are always PR FBs� Besides the PR property� the frame

property guarantees a certain degree of numerical stability �see the discussion on frame

bounds in Sections �� and A��

Lemma �� The analysis set fhk�m�n�g has an upper frame bound B ��

i�e��

N��Xk��

�Xm��

jvk�m�j� !N��Xk��

�Xm��

jhx� hk�mij� � Bkxk� � x�n� � l��ZZ� �

if and only if the polyphase components Ek�n�ej�� are all bounded a�e�� i�e��

jEk�n�ej��j � K �� a�e� on �� for k ! �� N�� n ! �� M��

Proof� Let jEk�n�ej��j � K � � a�e�� It follows that the entries of the UFBF

matrix S�ej�� ! EH�ej��E�ej�� are bounded a�e�� which implies that the n��

are bounded a�e�� Using Theorem �� we conclude that B � �� We next prove the

converse� Let B ! ess sup �� n��M�� n�� It follows thatPM��

n�� n�� is

bounded a�e�� With �� this implies that the Ek�n�ej�� are bounded a�e��

Theorem �� An oversampled FB with BIBO stable�� analysis �lters

hk�n� provides a UFBF expansion in l��ZZ�� i�e�� the analysis set fhk�m�n�gis a UFBF for l��ZZ�� if and only if the analysis polyphase matrix E�z� has

full rank on the unit circle�� i�e��

rankfE�ej��g ! M for � � � � �

Proof� From hk�n� � l��ZZ� it follows that the Ek�n�ej�� are bounded� and hence we

conclude from Lemma �� that an upper frame bound B �� exists� It remains to show

that a full rank E�ej�� is necessary and su�cient for the existence of a lower frame

��BIBO stability means that hk�n� � l��ZZ�� i�e��P�

n�� jhk�n�j �� for k � �� N � ��We emphasize that E�z� is here required to have full rank everywhere on the unit circle� In

contrast� the completeness condition in Theorem �� merely required E�z� to have full rank a�e� on the

unit circle�


bound A� If E�ej�� has full rank on �� then S�ej�� ! EH�ej��E�ej�� has full

rank on �� which means that n�� for � � � � and n ! �� M � � From

hk�n� � l��ZZ� it follows that the n�� are continuous functions of �� and therefore

we can conclude that�� A ! ess inf �� n��M�� n�� We next prove that�

conversely� a full rank E�ej�� is necessary for the existence of A � �� Suppose that

E�ej�� does not have full rank on �� It follows that S�ej�� does not have full

rank on �� This implies that there is at least one eigenvalue with n�� ! � on

a measurable set with positive measure� Hence� using Theorem �� we conclude that

A ! ��

Alternatively� it can be shown that a FB corresponds to a UFBF for l��ZZ� if E�ej��

has full rank for � � � � and the Ek�n�ej�� are continuous and bounded functions

of �� Yet another condition is phrased in terms of the eigenvalues of the UFBF matrix

S�ej��

Corollary �� An oversampled FB provides a UFBF expansion in l��ZZ�

if and only if the eigenvalues n�� of the UFBF matrix S�ej�� !

EH�ej��E�ej�� satisfy

ess inf �� n��M�� n�� and ess sup �� n��M�� n��

Proof� It is known �� that fhk�m�n�g is a frame if and only if ess inf � �

and ess sup �� where fg is the set of all eigenvalues of the frame operator S� Due

to Corollary �� the eigenvalues of S equal the eigenvalues n�� of S�ej�� which

completes the proof� �

Using the fact that FIR �lters are inherently BIBO stable� and thus one of the con�

ditions of Theorem �� is always satis�ed here� it follows as an immediate consequence

of Theorem �� that an oversampled FB with FIR analysis �lters provides a UFBF ex�

pansion in l��ZZ� if and only if the analysis polyphase matrix E�z� has full rank on the

unit circle� i�e�� rankfE�ej��g ! M for � � � � � This condition has been reported

previously in �� We emphasize that Theorem �� is more general since it holds for

both FIR and IIR FBs� In the following� a FB providing a UFBF expansion in l��ZZ�

will be called a frame FB �FFB�� Throughout this text we will restrict our attention to

BIBO stable �lters hk�n��

��For a continuous function the essential in mum is the in mum �this is however not relevant to this

proof��


�� Approximative Construction of the Synthesis Filter

Bank

In the case of an FFB� it follows from �� that the calculation of the minimum norm

synthesis FB requires the inversion of the matrix #E�z�E�z�� which is a cumbersome

task in general� By analogy to the approximation of dual frames described in �� an

approximative calculation of the synthesis FB can be based on a series expansion of

S��z� !h#E�z�E�z�

i�� Indeed� applying the Neumann expansion �see Theorem A� ��

to the matrix �#E�z�E�z�� we obtain

h#E�z�E�z�

i��!

�

A "B

�IM �

IM � �

A "B#E�z�E�z�

��

!�

A "B

�Xi��

IM � �

A"B#E�z�E�z�

�i�

and consequently the minimum norm synthesis FB is expressed as

R�z� !�

A"B

�Xi��

IM � �

A"B#E�z�E�z�

�i �#E�z� � ��

The norm convergence of this series expansion follows from frame theory �� see The�

orem A�� using the correspondence between S and #E�z�E�z��

By truncating the expansion �� the synthesis FB can be approximated with

arbitrary accuracy� Estimates of the resulting reconstruction error are available ��

We shall here restrict our attention to the �rst order approximation of R�z� obtained

by retaining only the i ! � term in ��

R ��z� !�

A "B#E�z� �

which corresponds to an approximation of the minimum norm synthesis �lters fk�n� as

f ��k �n� !

�

A"Bh�k��n��

The reconstruction error resulting from this approximation can be bounded in terms of

the frame bounds A and B� With x ��n� denoting the signal reconstructed using the

above ��rst order synthesis FB�� we have from Theorem A� the error bound

k x �� xk � B�A�

A�B " kxk� ��

We can see that the reconstruction error is small if the tightest possible frame bounds

A and B satisfy B�A � The underlying UFBF is thus called snug� In this case

the synthesis impulse responses f ��k �n� are a good approximation to the true minimum

norm� PR impulse responses fk�n�� in the sense that the resulting reconstruction error


k x �� xk is small� In the tight case where B�A ! � the reconstruction error becomes

zero� and indeed the approximation is here exact�

f ��k �n� ! fk�n� !

Ah�k��n��

Besides the trivial �rst order approximation discussed above� the series expansion

�� also allows the iterative calculation of the synthesis �lters fk�n�� This iterative

calculation is outlined in Corollary A�� and can be reformulated in terms of polyphase

matrices using the correspondence between S and #E�z�E�z��

�� Critical Sampling Linear Independence and Biorthog�

onality

It is well known that critically sampled FBs provide decompositions into Riesz bases

or equivalently exact frames �� see Section A�� In �nite�dimensional spaces

exact frames have linearly independent frame functions� In in�nite�dimensional spaces

exactness of a frame is re�ected in the fact that the expansion coe�cients �subband

signals� are uniquely determined� By abuse of notation we shall sometimes use the

notion of linear independence for frames in in�nite�dimensional spaces meaning that

the expansion coe�cients are uniquely determined� Consequently� we shall use the

notion of linear dependence for frames in in�nite�dimensional spaces meaning that the

frame leads to nonunique expansion coe�cients� Equivalently� one can say that there

is redundancy in the expansion coe�cients�

Exact frames are distinguished by the linear independence of the frame functions�

If fhk�m�n�g is an exact UFBF� then the dual frame ffk�m�n�g is exact as well� and

the expansion coe�cients ck�m in the expansion x�n� !PN��

k��

P�m�� ck�m fk�m�n� are

uniquely de�ned for any given x�n�� Furthermore� an exact frame is minimal in the

sense that the removal of an arbitrary frame function from the set fhk�m�n�g leaves an

incomplete set �� In �� it has been shown that critically sampled FBs may

provide decompositions into Riesz bases or equivalently exact frames� Based on the

framework developed so far� we shall now give an alternative proof for the equivalence

between critically sampled FBs and exact frames�

Theorem �� An FFB provides an exact UFBF expansion if

and only if it is critically sampled�

Proof� Let fhk�m�n�g be a critically sampled UFBF �N ! M�� Assume that the

hk�m�n� are linearly dependent� Then we can �nd coe�cients� bk�m which are not all

��The coe�cients bk�m satisfyPM��

k��

P�m�� jbk�mj

� ��


zero such thatM��Xk��

�Xm��

bk�m hk�m�n� ! � � ��

Applying the polyphase decomposition to both sides of �� yields

EH�ej��b�ej�� ! � a�e��

where b�ej�� ! �B��ej�� B��e

j�� BM��ej��T with Bk�ej�� !P�

m�� bk�m e�j��m�� Since fhk�m�n�g is a frame� it follows from Theorem �� that the

analysis polyphase matrix E�ej�� has full rank a�e�� This implies that the only solution

of �� is the trivial solution b�ej�� ! � a�e�� But the bk�m were shown above to be

not all zero� which is a contradiction� We conclude that the hk�m�n� must be linearly

independent and therefore the set fhk�m�n�g is an exact frame for l��ZZ��

It remains to show that� conversely� exact UFBFs are critically sampled� Let N �M

�note that the case N � M is impossible� since the set fhk�m�n�g would be incomplete

in l��ZZ�� and assume that fhk�m�n�g is an exact frame� Due to the linear independence

of the functions hk�m�n� it follows thatPN��

k��

P�m�� bk�m hk�m�n� ! � only if the bk�m

are all zero� Equivalently� in the polyphase domain� EH�ej��b�ej�� ! � a�e� only

if b�ej�� ! � a�e�� This is true only if the rank of the N �M matrix E�ej�� is N

which� combined with N �M � implies N ! M � i�e�� critical sampling� �

As a consequence of Theorem �� see Corollary A�� in critically sampled FBs

the frame functions hk�m�n� and their duals fk�m�n� satisfy the biorthogonality relation

��

hhk�m� fk��m�i ! �k�k� �m�m� � ��

It follows furthermore that an oversampled UFBF cannot be exact� hence the corre�

sponding FB cannot be biorthogonal�

�� Oversampled Paraunitary Filter Banks and

Tight Frames

In this section� we show that oversampled paraunitary FBs provide tight UFBF expan�

sions in l��ZZ�� and we discuss a frame�theoretic method for constructing paraunitary

FBs�

�� Equivalence of Oversampled Paraunitary Filter Banks

and Tight Frames

The analysis UFBF fhk�m�n�g is tight if the tightest possible frame bounds satisfy A !

B� From frame theory we know that this implies S!AI �� With �� this implies

�� OVERSAMPLED PARAUNITARY FILTER BANKS �

that the frame�theoretic �i�e�� minimum norm� solution for the PR synthesis FB is

fk�n� !

Ah�k��n� �

or equivalently R�z� ! �A#E�z�� This is precisely the relation between the synthesis and

analysis �lters in a paraunitary FB �� In fact we can formulate the following theorem�

Theorem �� An oversampled FB provides a tight UFBF expansion in

l��ZZ� if and only if it is paraunitary � i�e��

S�z� ! #E�z�E�z� � A IM �

The frame bound is A ! Sn�n�z� !N��Pk��

#Ek�n�z�Ek�n�z��

Proof� From S�z� ! A IM � it follows with �� that y�z� ! Ax�z� which implies

y�n� ! Ax�n�� Hence� comparing with y�n� ! �Sx��n�� we conclude that S ! AI� i�e��

fhk�m�n�g is a tight UFBF with frame bound A� The converse statement is proven

by reversing this line of reasoning� Combining Sn�n�z� !PN��

k��#Ek�n�z�Ek�n�z� and

S�z� ! A IM � it follows that A ! Sn�n�z� !PN��

k��#Ek�n�z�Ek�n�z��

The equivalence of tight Weyl�Heisenberg frames �a subclass of UFBFs� and parau�

nitary DFT FBs �see Chapter �� has been observed in �� and independently in ��

For the FIR case the equivalence between tight frames in l��ZZ� and paraunitary FBs

has been reported previously in �� Note that Theorem �� holds also in the IIR

case� Furthermore Theorem �� also holds in the critically sampled case� where the

underlying frame is not only tight but also orthogonal� The equivalence of paraunitary

FBs and tight frames in the critically sampled case has been known for several years

��

Paraunitary FBs are often referred to as orthogonal FBs� However� the name �or�

thogonal� is justi�ed only in the critical case� In the oversampled case� paraunitary

FBs correspond to UFBFs that are tight but not orthogonal�

�� Construction of Paraunitary Filter Banks

From Corollary A�� we know that the function set obtained by applying the inverse

positive de�nite operator square root S�� to each of the frame functions fhk�m�n�gyields a tight frame� Since S corresponds to #E�z�E�z�� the inverse square root S��

corresponds to an inverse square root of the matrix #E�z�E�z��

Theorem �� Consider an FFB with polyphase matrices E�z� and R�z��

and let P�z� be an invertible� parahermitian�� M �M matrix such that

�A matrix P�z� is said to be parahermitian if �P�z� � P�z� ��


P��z� ! #E�z�E�z�� Then the FFB with analysis polyphase matrix

E p��z� ! E�z�P��z�

is paraunitary with frame bound A ! � i�e�� S p��z� ! #E p��z�E p��z� �IM � The corresponding synthesis polyphase matrix is given by R p��z� !#E p��z� ! P��z� #E�z�� If� in the critically sampled case� the original FFB

is moreover biorthogonal� then the FFB with analysis polyphase matrix

E p��z� is orthogonal�

Proof� We have S p��z� ! #E p��z�E p��z� ! #P��z� #E�z�E�z�P��z�� Inserting#E�z�E�z� ! P��z� in the right hand side and using #P�z� ! P�z�� it follows that#E p��z�E p��z� ! IM � which shows that the FB with polyphase matrix E p��z� is parau�

nitary with frame bound A ! �

From the theory of frames we know �� that applying the procedure described

above to an exact frame �corresponding to a biorthogonal FFB�� an orthogonal function

set �corresponding to an orthogonal FFB� is obtained� �

The matrix P��z� can be calculated by performing a factorization of P��z� !

R�z� #R�z�� A detailed study of this factorization problem for both polynomial and

rational matrices is given in � �� Another method to calculate the matrix P��z� is

to perform a series expansion similar to �� Using the correspondence between the

operator S and the matrix #E�z�E�z� we have ��

P��z� !

s�

A"B

�Xi��

��i�$

��i �i$��

IM � �

A "B#E�z�E�z�

�i�

�� Important Special Cases

In this section� we discuss FFBs whose frame operator becomes a simple multiplication

operator in the polyphase domain or in the frequency domain� i�e�� the polyphase repre�

sentation or the Fourier transform �diagonalizes� the frame operator� This class of FBs

comprises integer oversampled DFT FBs� nondecimated FBs� and bandlimited FBs� We

shall see that the calculation of the synthesis FB� of the frame bounds� and of parau�

nitary FBs is drastically simpli�ed in these cases� Furthermore� we apply our results

to redundant block transforms� which can be viewed as a special class of oversampled

FBs�

�� Diagonality in the Polyphase Domain

According to Theorem �� the frame operator is represented in the polyphase domain

by the UFBF matrix� Consequently� an FFB is �diagonal in the polyphase domain� if

�� IMPORTANT SPECIAL CASES �

the UFBF matrix is a diagonal matrix�

S�z� ! #E�z�E�z� ! diagfSn�n�z�gM��n�� with Sn�n�z� !

N��Xk��

#Ek�n�z�Ek�n�z� �

It follows from �� that the polyphase matrix of the minimum norm synthesis FB is

given by

R�z� ! diag

�

Sn�n�z�

�M��

n��

#E�z� � Rk�n�z� !#Ek�n�z�

Sn�n�z��

�note that S��z� ! diagf�Sn�n�z�gM��n�� in the diagonal case�� We can see that the

calculation of the synthesis FB� which in general requires the inversion of a matrix�

reduces to simple divisions in the polyphase domain�

Using the fact that the eigenvalues of the diagonal matrix S�ej�� !

EH�ej��E�ej�� are given by

n�� ! Sn�n�ej�� !

N��Xk��

jEk�n�ej��j��

it follows from Corollary �� that the FB is an FFB if and only if

ess inf�� n��M��

Sn�n�ej�� and ess sup

�� n��M��Sn�n�e

j��

and according to Theorem �� the frame bounds are given by


Sn�n�ej�� B ! ess sup


j��

In particular� the FB is paraunitary with frame bound A if and only if

Sn�n�z� !N��Xk��

#Ek�n�z�Ek�n�z� � A for n ! �� M � �

The construction of paraunitary FFBs from nonparaunitary FFBs �see Theorem ��

simpli�es as well� Consider an FFB with polyphase components Ek�n�z� and Rk�n�z��

and de�ne Pn�z� by P �n�z� ! Sn�n�z� with #Pn�z� ! Pn�z�� Then� the FB with analysis

polyphase components

E p�k�n�z� !

Ek�n�z�

Pn�z��

is paraunitary with A ! � i�e�� #E p��z�E p��z� � IM � Thus� the matrix factorization#E�z�E�z� ! P��z� reduces to the factorization of polynomials in z �in the FIR case� or

rational functions in z �in the IIR case��

Integer oversampled DFT FBs � �� are another important example

of FBs that are diagonal in the polyphase domain� This type of FBs will be discussed

in detail in Subsec� ��


�� Diagonality in the Frequency Domain

A FB is �diagonal in the frequency domain� if its frame operator is a simple multipli�

cation operator in the frequency domain �z�transform domain�� With y�n� ! �Sx��n��

this means

Y �z� ! Gh�z�X�z� with Gh�z� !

M

N��Xk��

Hk�z� #Hk�z� � ��

where X�z�� Y �z�� and Hk�z� denote the z�transforms of x�n�� y�n�� and hk�n�� respec�

tively� The eigenvalues of the underlying frame operator� which here corresponds to the

scalar function Gh�ej�� are

�� ! Gh�ej�� !

M

N��Xk��

jHk�ej��j��

Two important classes of FBs that are diagonal in the frequency domain are nondeci�

mated FBs and bandlimited FBs� Nondecimated FBs are FBs with no decimation in the

subbands�� i�e�� M ! �� Nondecimated FIR FBs have been studied previously in ��

Bandlimited FBs are FBs whose analysis �lters have bandwidth � �M � Bandlimited

FBs are clearly IIR FBs�

With �� it follows that the z�transforms of the minimum norm synthesis �lters

are obtained as

Fk�z� !#Hk�z�

Gh�z��

Furthermore� �� implies that the frame condition can be reformulated as

ess inf��

Gh�ej�� and ess sup

�� Gh�e

j��

With Gh�ej�� ! �

M

PN��k�� jHk�e

j��j�� the lower bound means that the set of analysis

�lters has to �cover� the entire frequency interval �� This condition is satis�ed if and

only if the analysis �lters have no zeros in common on the unit circle� The upper bound

is automatically satis�ed for BIBO stable hk�n�� i�e�� hk�n� � l��ZZ� �k ! �� N��

The frame bounds are given by

A ! ess inf��

Gh�ej�� B ! ess sup

�� Gh�e

j��

Paraunitarity with frame bound A implies

Gh�z� !

M

N��Xk��

Hk�z� #Hk�z� � A�

�We note that nondecimated FBs are also trivially diagonal in the polyphase domain�

�� IMPORTANT SPECIAL CASES ��

which means that the analysis �lters Hk�z� are power complementary �� Paraunitary

FBs with frame bound A ! can be constructed by solving the factorization P ��z� !

Gh�z� with #P �z� ! P �z�� the paraunitary analysis �lters are then given by

H p�k �z� !

Hk�z�

P �z��

�� Redundant Block Transforms

Block transforms �� can be viewed as a special case of FBs �� In this section� we

specialize our frame�theoretic results to redundant �oversampled� block transforms�

In a block transform� the input signal x�n� is cut into nonoverlapping blocks x�m� !

�x�mM � x�mM " � �� x�mM "M � ��T of M consecutive samples� which are linearly

transformed into blocks v�m� ! �v��m� v��m� �� vN��m��T of N transform coe�cients

vk�m�� i�e��

v�m� ! Ex�m�

with the N �M analysis polyphase matrix E which is given by

E !

��

h�� h�� h��M � ��

h�� h�� h��M � ��

��

��

hN�� hN�� hN��M � ��

��

Note that the polyphase matrix E does not depend on z� For N � M the transform

coe�cients are redundant� The basis functions hk�n� of the block transform are the im�

pulse responses of the analysis FB� and the transform coe�cients vk�m� are the subband

signals �� Note that the analysis �lter impulse responses hk�n� are supported within

the interval n ! �M " ��M " �� A classical choice for E in the nonredundant

�critically sampled� case is a unitary matrix satisfying EHE ! IM � This choice leads to

an orthonormal block transform or equivalently a paraunitary FB � ��

The synthesis relation is

x�m� ! Rv�m��

with the M � N synthesis polyphase matrix R� The PR property x�m� ! x�m� is

satis�ed if and only if RE ! IM � which presupposes rankfEg ! M �cf� Theorem ��

In the redundant case N � M � the synthesis matrix R is not uniquely determined� it

may be any left�inverse of E� Hence� the general form of R is �see ��

R ! R" U �IN �E R� � ��

where U is an arbitrary M � N matrix and R ! �EHE��EH is the pseudo�inverse of

E which corresponds to the synthesis basis vectors with minimum norm�


The block transform provides a UFBF expansion in l��ZZ� if and only if E has rank

M � The frame condition rankfEg ! M is seen to be no more restrictive than the PR

condition� this is explained by the �nite dimension of the spaces of signal vectors and

transform coe�cient vectors� The frame bounds A and B of the block transform are the

minimum and maximum� respectively of the eigenvalues of S ! EHE� The frame�PR

condition rankfEg ! M impliesPN��

k�� jhk�n�j � � for n ! �M " ��M " �� since

otherwise E would contain a zero column� This shows that the impulse responses hk�n�

must not all be zero for the same index n� i�e�� the set of analysis impulse responses

must �cover� the entire time interval n ! �M " ��M " ��

�� Oversampled FIR Filter Banks

Oversampled FIR FBs are a special case of the theory presented in the previous sections�

In the FIR case the polyphase matrices are polynomial matrices in z� which has several

interesting consequences to be discussed in this section� A detailed study of oversampled

FIR FBs has been provided in �� For the sake of completeness� we shall review

some of the results presented in �� We also present original results and extensions

of the results in ��

�� Smith Form Decomposition

The Smith form decomposition allows to decompose polynomial matrices into simpler

forms such as triangular and diagonal forms �� This �diagonalization� will prove

useful in what follows�

The N �M analysis polyphase matrix E�z� in a PR FB can be decomposed into its

Smith form according to ��

E�z� ! A�z�D�z�B�z��

where A�z� and B�z� are unimodular �� matrix polynomials of size N �N and M �M �

respectively� The N �M matrix D�z�� called the Smith form of E�z�� is given by

D�z� !

��

d��z� � ��

� d��z� ��

��

��

� � �� dM��z�

� � ��

��

��

� � ��

��

�

��A unimodular matrix is a square polynomial matrix in z with constant nonzero determinant�

�� OVERSAMPLED FIR FILTER BANKS ��

where we used the fact that E�z� has normal rank�� M � �Since we are considering

PR FBs� the set fhk�m�n�g has to be complete in l��ZZ�� hence� by Theorem �� the

normal rank of E�z� is M �� We note that a matrix polynomial is unimodular if and

only if it can be represented as the product of a �nite number of elementary matrices

�� Premultiplying a matrix with an elementary matrix provides elementary row

and column operations� such as interchanging two rows �columns�� multiplying a row

�column� with a nonzero constant� or adding a polynomial multiple of a row �column�

to another row �column�� Note furthermore that the matrices A�z� and B�z� are not

uniquely determined� A�z� and B�z� can be chosen such that the polynomials di�z� are

monic�� and that di�z� is divisible by di��z�� Furthermore the elements of D�z� can

be expressed in terms of the elements of E�z� �� For a detailed discussion of matrix

polynomials see ��

�� Analysis of Oversampled FIR Filter Banks

As an immediate consequence of Theorem �� we obtain the following corollary which

is due to Cvetkovi%c and Vetterli ��

Corollary �� An oversampled FIR FB provides a UFBF decompo�

sition in l��ZZ� if and only if the polynomials di�z� on the diagonal of its

Smith form have no zeros on the unit circle�

Proof� Noting that the polyphase matrix E�z� and its Smith form D�z� have the

same rank in the entire z plane �� it follows from Theorem �� that the FB provides

a UFBF decomposition in l��ZZ� if and only if D�z� is of full rank on the unit circle�

D�z� has full rank on the unit circle if and only if none of the polynomials di�z� has a

zero on the unit circle� �

Theorem �� For a given oversampled FIR analysis FB with analysis

polyphase matrix E�z�� the minimum norm synthesis FB is FIR if and only

if #E�z�E�z� is unimodular�

We shall next present a proof which is much shorter than the one originally presented

in ��

Proof� From R�z� ! �#E�z�E�z�� #E�z� it is obvious that the unimodularity of

the matrix #E�z�E�z� is su�cient for the synthesis FB to be FIR if the analysis FB

is FIR� We shall next show the necessity� Assume that both E�z� and R�z� are

��The normal rank of a matrix E�z� is the maximum possible rank in the entire zplane��A polynomial is said to be monic if the coe�cient of the highest power ocurring in the polynomial

is equal to ��


polynomial matrices� Then #E�z�E�z� and R�z� #R�z� are polynomial matrices� Now�

since R�z� #R�z� ! �#E�z�E�z�� we can conclude that both #E�z�E�z� and its inverse

�#E�z�E�z�� have to be polynomial matrices� It thus follows �� that the determinant

of #E�z�E�z� satis�es �z��! det �#E�z�E�z�� ! c z�K with c �CI and K � ZZ� Now� since

�z� ! #�z�� it follows that �z� has to be a constant independent of z �i�e� K ! ��

which concludes the proof� �

The above theorem is a generalization of a result for critically sampled FBs ��

which states that FIR analysis FB and FIR synthesis FB with PR is possible if and

only if the determinant of E�z� is a pure delay�

The following theorem �� gives a complete parameterization of oversampled FBs

with FIR analysis and FIR minimum norm synthesis �lters�

Theorem �� Consider an oversampled FIR FB with analysis

polyphase matrix E�z�� Then #E�z�E�z� is unimodular if and only if E�z�

has the form

E�z� ! H�A��z�D��z�B��z��

where the matrices H��A��z��D��z� and B��z� have the following form�

� H� is a constant N �M matrix satisfying HH� H� ! d IM �

� A��z� and B��z� are unimodular M �M matrices�

� D��z� is a diagonal M �M matrix of polynomials� with monomials on

the diagonal�

The proof of this result is given in �� Recall that any unimodular matrix can be

factored into a product of a �nite number of elementary matrices� Since A��z� and

B��z� in Theorem �� are unimodular and any unimodular matrix can be expressed as

the product of a �nite number of elementary matrices �� states that E�z�

can be expressed as the product of a �nite number of matrices of a special form�

The following theorem states an important result on the factorization of paraunitary

oversampled FBs ��

Theorem �� An oversampled FIR FB with analysis polyphase

matrix

E�z� ! VM�z�VM��z� ��V��z�H� ��

with

Vi�z� ! IN � vi vHi " z��vi v

Hi �

where vi is a unit norm vector of size N � and H� is a constant N �M

matrix satisfying HH� H� ! A IM � is paraunitary with frame bound A�

�� SIMULATION RESULTS ��

Theorem �� can be viewed as an extension of a result formulated for the critically

sampled case by P� P� Vaidyanathan in �� Note that� due to �� the N �M matrix

E�z� is parameterized in terms of M constant vectors vi of size N � and a constant

N �M matrix H��

We shall next present a factorization of the analysis polyphase matrix that guarantees

FIR �lters in the analysis and the minimum norm synthesis FB� This factorization

extends a result reported for the critically sampled case in ��

Theorem �� Let Am with m ! �� J � be constant nonsingular

M �M matrices and AJ a constant N �M matrix such that AHJ AJ is

nonsingular� For an FIR analysis FB with polyphase matrix

E�z� ! AJ ��z�AJ��z� ��A��z�A��

where

��z� !

�� IM��

� z��

��

the minimum norm synthesis FB is FIR� it is given by

R�z� ! A�� z�A��

� ��z� ��A��J��

��z� �AHJ AJ�

��AHJ � ��

where

��z� !

�� IM��

� z

��

Proof� Using det��z� ! z�� it is readily seen that det #E�z�E�z� is a nonzero con�

stant� Hence� using Theorem �� it follows that the minimum norm synthesis FB

is FIR� With R�z� !h#E�z�E�z�

i�� #E�z�� it is easily veri�ed that the minimum norm

synthesis FB is given by ��

This demonstrates that oversampled FIR FBs of arbitrarily high order can be con�

structed by structuring E�z� according to ��

So far� we have considered an FIR minimum norm synthesis FB� Apart from the min�

imum norm synthesis FB� the other PR synthesis FBs as given by the parameterization

�� are of interest as well� If the minimum norm synthesis FB of an FIR analysis FB

is FIR� then it follows from �� that other PR synthesis FBs that are FIR as well

can be obtained by choosing the matrix U�z� as a polynomial matrix�

�� Simulation Results

We now present simulation results demonstrating the importance of snug frames �B�A � and the bene�ts of oversampling�

Snug frame� We consider a DFT FB �see Chapter �� with N ! �� channels and

a ��tap lowpass analysis prototype �lter h�n� as depicted in Fig� ��a�� Figs� ��b��


50 100 150−0.5

0

0.5

1

(a)

Analysis Prototype

50 100 150

−2

−1.5

−1

−0.5

0

0.5

1

(b)

Oversampling by 2

50 100 150−0.5

0

0.5

1

(c)

Oversampling by 4

50 100 150−0.5

0

0.5

1

(d)

Oversampling by 8

Fig� �� Comparison of analysis and minimum norm synthesis prototype �lters for

various frame bound ratios and oversampling factors� a Analysis prototype h�n��

b�d minimum norm synthesis prototype f �n� for b oversampling by � resulting in

B�A ! �� c oversampling by � B�A ! �� and d oversampling by

B�A ! ��

��d� show the minimum norm synthesis prototypes for oversampling factors �� and

� respectively� The frame bound ratio B�A was obtained as �� and ��

respectively� Thus� more oversampling is seen to result in better frame bounds� It is

furthermore seen that for better frame bounds �i�e�� more oversampling�� the analysis

and synthesis prototypes are increasingly similar� Approximating the minimum norm

synthesis FB using the �rst order approximation in �� i�e�� essentially using the

analysis FB as synthesis FB� resulted in the following upper bounds b ! B�A��A�B�

on

the normalized reconstruction error k�x��xkkxk �see �� b ! �� for oversampling

by �� b ! �� for oversampling by �� and b ! �� for oversampling by � Thus�

the reconstruction error can be expected to be negligible if the oversampling factor is

su�ciently large�

Frame bound ratio� For the DFT FB analysis prototype in Fig� ��a�� Tab� shows

the frame bound ratio B�A as a function of the oversampling factor N�M �note that

N�M ! means critical sampling�� We can see that B�A� �i�e�� the frame is more

snug� for increasing oversampling factor� for N�M the FB is nearly paraunitary�

We caution� however� that choosing the prototype such that it does not �match� the

time�frequency grid determined by the parameters N and M it is not guaranteed that

the frame bound ratio will improve for increasing oversampling factor � �� Furthermore�

we note that paraunitary FBs �corresponding to tight frames� i�e�� A ! B� can of course

�� SIMULATION RESULTS �

N M N�M B�A

��

��

��

��

��

��

��

��

��

��

��

Table �� Frame bound ratio B�A as a function of the oversampling factor N�M �

also be constructed in the case of critical sampling� However� in the oversampled case

the �lters of a paraunitary FB tend to have improved frequency selectivity� This is due

to the fact that in the design of an oversampled PR FB� there are fewer side constraints

to be satis�ed than in the case of critical sampling� The simulation results described

above were obtained by performing all calculations within the framework of cyclic DFT

FBs �cyclic Weyl�Heisenberg frames� �� with period �� The dual windows and the

frame bounds we obtained are hence approximations to the true �i�e�� noncyclic� dual

windows and frame bounds�

Design freedom� As already discussed in Subsection �� oversampled FBs o�er

more design freedom than critically sampled FBs� In the next simulation example

we shall concentrate on the design freedom which is due to the nonuniqueness of the

synthesis FB for a given oversampled analysis FB� Consider a two�channel FB with no

decimation in the subbands� i�e�� N ! ��M ! and thus oversampling by �� The analy�

sis �lters are the Haar �lters H��z� !�p��"z�� and H��z� !

�p��z�� Since there

is no decimation� the analysis polyphase matrix is given by E�z� ! �H��z� H��z��T �

From #E�z�E�z� ! #H��z�H��z� " #H��z�H��z� ! � it follows that the FB is paraunitary�

it implements a tight frame decomposition with frame bound A ! B ! � �see Subsec�

tion �� The minimum norm synthesis �lters are therefore given by F��z� !��#H��z�

and F��z� !��#H��z�� For the computation of all synthesis FBs providing PR� one can


0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

(a)M

agni

tude

Frequency response

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

(b)

Mag

nitu

de

Frequency response

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

(c)

Mag

nitu

de

Frequency response

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

(d)

Mag

nitu

de

Frequency response

Fig� �� Optimization of PR synthesis FB for Haar analysis FB oversampled by ��

a F��z�� b F��z�� c optimized F��z�� b optimized F��z��

use any of the three equivalent parameterizations provided in Subsection �� Here� it

is convenient to use the frequency�domain parameterization�� which in our case

reduces to

F��z� ! F��z� " V��z�� F��z��H��z�V��z� "H��z�V��z��

F��z� ! F��z� " V��z�� F��z��H��z�V��z� "H��z�V��z��

Any choice of V��z� and V��z� will lead to PR� The length of the synthesis �lters is

determined by the length of the �lters V��z� and V��z�� Assuming that both V��z� and

V��z� are FIR �lters of length Lv� it is easily seen that the �lters F��z� and F��z� have

length Lv " �� The parameterization in terms of the �lters V��z� and V��z� can now be

used to perform an optimization over the class of all synthesis �lter pairs fF��z�� F��z�gproviding PR�

In the next example� we choose the �lters V��z�� V��z� such that F�� !

�PLf��

n�� f��n�� is minimal for Lf ! �� i�e�� Lv ! �� Note that this corresponds to

an unconstrained optimization over the class of all synthesis �lters with length Lf pro�

viding PR� Figs� ��a� and �b� show the transfer functions of the minimum norm synthesis

�lters� In Figs� ��c� and �d� the transfer functions of the optimized PR synthesis �lters

are depicted�

��Note that since M � � the frequencydomain parameterization �� is equal to the polyphase

domain parameterization ��

Chapter �

Oversampled DFT Filter Banks

The main advantages of oversampled FBs are increased design freedom �see Chapters

� and �� and noise reducing properties �see Chapters � and �� These advantages

come at the expense of increased computational cost caused by the need to process a

larger number of subband signal samples per unit of time� Therefore� oversampled FBs

allowing e�cient implementations are of particular interest� Oversampled DFT FBs

�also known as complex modulated FBs� �� and oversampled cosine

modulated FBs �see Chapter �� allow e�cient FFT� or DCT�based

implementations� It is well known that there are no critically sampled FIR DFT FBs

with good frequency localization � �� In the oversampled case� however� FIR DFT

FBs can be designed to have good frequency selectivity ��

In this chapter� we discuss oversampled DFT FBs and their relation to Weyl�

Heisenberg �WH� frames �� in l��ZZ�� In particular� we shall describe the

construction of FIR� paraunitary� oversampled DFT FBs with good frequency localiza�

tion� We apply the theory of WH frames � �� to FIR and IIR� oversampled DFT

FBs � �� Although the connection between DFT FBs and short�time

Fourier transforms or Gabor expansions �� is well established

�� a frame�theoretic approach to DFT FBs has been proposed only recently

�� We will not comment on the e�cient implementation of oversampled

DFT FBs� since this is discussed in great detail in ��

The organization of this chapter is as follows� In Section �� we discuss oversampled

DFT FBs and their relation to WH expansions� In Section �� we provide time� fre�

quency� and polyphase domain representations of the DFT FB operator describing the

DFT FB�s input�output relation� Based on these representations� Section �� provides

PR conditions in the time� frequency� and polyphase domain for arbitrary oversampling

factors� We furthermore give time� frequency� and polyphase domain parameterizations

of DFT synthesis FBs providing PR for a given oversampled DFT analysis FB� In Sec�

tion �� we apply the theory of WH frames to oversampled DFT FBs� We provide

�

�� CHAPTER �� OVERSAMPLED DFT FILTER BANKS

time� frequency� and polyphase domain representations of the WH frame operator� The

important special case of integer oversampling is discussed in detail� In Section ��

we discuss a simple method for the design of oversampled FIR paraunitary DFT FBs�

Finally� Section �� provides simulation results demonstrating the increased design free�

dom of oversampled DFT FBs�

�� DFT Filter Banks and WeylHeisenberg Sets

In the following we shall discuss two fundamental classes of DFT FBs� namely odd�

stacked and even�stacked DFT FBs ��

�� Odd�Stacked DFT FBs

The analysis and synthesis �lters of an odd�stacked DFT FB with N channels and

decimation factor M are derived from a single analysis prototype h�n� � H�z� and a

single synthesis prototype �lter f �n� � F �z�� respectively� through frequency�shifts by

integer multiples of �N � i�e��

hDFT�ok �n� ! h�n�W� k��nN � fDFT�ok �n� ! f �n�W

� k��nN � k ! �� N��

with WN ! e�j��N � or equivalently

HDFT�ok �z� ! H�zW

k��N � � FDFT�o

k �z� ! F �zWk��N � � k ! �� N��

This is illustrated in Fig� ��a��

The polyphase decomposition of the analysis prototype H�z� is given by

H�z� !M��Xn��

znEn�zM � with En�z� !

�Xm��

h�mM � n� z�m�

Note that furthermore

Ek�n�z� ! W k��nN En

�zW

M k��N

��

so that the analysis polyphase matrix E�z� is fully determined by En�z� �n !

�� M �� Similarly� the polyphase decomposition of the synthesis prototype F �z�

reads

F �z� !M��Xn��

z�nRn�zM � with Rn�z� !

�Xm��

f �mM " n� z�m�

and there is Rk�n�z� ! W� k��nN Rn

�zW

M k��N

��

�The superscripts DFT�o and DFT�e indicate that the respective quantity belongs to an odd and

evenstacked DFT FB� respectively�

�� DFT FILTER BANKS AND WEYLHEISENBERG SETS ��

12N

-32N-1

2-

(a)

HN-2 HN-1 H0 H1

0 12N

32N

12

(b)

012-

HN-1

2N

- 1N

H0

H1

2N

1N

12

-

HN-2 H2

Figure �� Transfer functions of the channel �lters in a an N�channel odd�stacked

DFT FB and b an N�channel even�stacked DFT FB�

The input�output relation of an odd�stacked DFT FB is

x�n� !N��Xk��

�Xm��

hx� hDFT�ok�m i fDFT�ok�m �n� � ��

where the analysis and synthesis functions are the Weyl�Heisenberg WH sets ��

generated by h��n� and f �n�� respectively�

hDFT�ok�m �n� ! h�k�mM � n� ! h��mM � n�W� k�� n�mM�N ��

fDFT�ok�m �n� ! fk�n�mM � ! f �n�mM �W� k�� n�mM�N ��

with k ! �� N�� m �� They are derived from h��n� and f �n� throughtime�shifts by integer multiples of M and frequency�shifts by integer multiples of �N �

Note that hk�m�n� and fk�m�n� are obtained from h�k��n� and fk�n� by by time�shifting

h�k��n� and fk�n�� respectively� and are thus consistent with the UFBF framework in

Subsection ��

�� Even�Stacked DFT FBs

The distinction between even�stacked and odd�stacked DFT FBs has been introduced

in �� For odd�stacked DFT FBs �considered so far�� the subbands are centered about

frequencies �k ! k��N

�k ! �� N�� in particular� the subband corresponding

to frequency index k ! � is centered about �� ! ��N

� For even�stacked DFT FBs� the

subbands are centered about frequencies �k ! kN

�k ! �� N�� in particular� the

subband corresponding to frequency index k ! � is centered about �� ! �� Fig� � shows

the transfer functions of the channel �lters in an odd�stacked DFT FB in comparison

to the transfer functions in an even�stacked DFT FB�


The impulse responses and transfer functions of the analysis and synthesis �lters in

an even�stacked DFT FB with N channels and decimation factor M are given by

hDFT�ek �n� ! h�n�W�knN � fDFT�ek �n� ! f �n�W�kn

N � k ! �� N�

and

HDFT�ek �z� ! H�zW k

N� � FDFT�ek �z� ! F �zW k

N � � k ! �� N� �

respectively� and the corresponding polyphase components are

Ek�n�z� ! W knN En�z W

MkN � ��

Rk�n�z� ! W�knN Rn�z W

MkN ��

The input�output relation of an even�stacked DFT FB is �� where the analysis and

synthesis functions are given by

hDFT�ek�m �n� ! h�k�mM � n� ! h��mM � n�W�k n�mM�N ��

fDFT�ek�m �n� ! fk�n�mM � ! f �n�mM �W�k n�mM�N ��

with k ! �� N�� m �� This is again consistent with the UFBF format

de�ned in Subsection ��

�� Representations of the DFT FB Operator

The reconstructed signal of an odd� or even�stacked DFT FB can be written as

x�n� ! �S h�f�DFTx��n��

where

�S h�f�DFTx��n� !

N��Xk��

�Xm��

hx� hDFTk�m i fDFTk�m �n��

Here hDFTk�m �n� ! hDFT�ok�m �n�� fDFTk�m �n� ! fDFT�ok�m �n� in the odd�stacked case and hDFTk�m �n� !

hDFT�ek�m �n�� fDFTk�m �n� ! fDFT�ek�m �n� in the even�stacked case� We shall next provide time�

frequency� and polyphase domain representations of the operator S h�f�DFT � These repre�

sentations are the basis for the PR conditions given in Section ��

Time domain� In the time domain� the DFT FB operator S h�f�DFT can be expressed

as


�Xl��

dl x�n� lN �

N

�Xm��

f �n�mM � h��n "mM " lN �

��

�� REPRESENTATIONS OF THE DFT FB OPERATOR ��

with dl ! ��l in the odd�stacked case and dl ! in the even�stacked case�

Frequency domain� The frequency �z�transform� version �S h�f�

DFT ! ZS h�f�DFTZ

��

�with Z denoting the z�transform operator� of the DFT FB operator S h�f�DFT can be

expressed as

�S h�f�

DFTX��z� !

M��Xl��

X�zW lM �

M

N��Xk��

FDFTk �z�HDFT

k �zW lM �

��

with FDFTk �z� ! FDFT�o

k �z� ! F �z Wk��N �� HDFT

k �z� ! HDFT�ok �z� ! H�zW

k��N � in

the odd�stacked case and FDFTk �z� ! FDFT�e

k �z� ! F �z W kN �� H

DFTk �z� ! HDFT�e

k �z� !

H�z W kN� in the even�stacked case�

Polyphase domain� The polyphase components�

Y h�f�n �z� !

�Xm��

�S h�f�DFTx��n "mM � z�m

of �S h�f�DFTx��n� are related to the polyphase components Xn�z� !

P�m�� x�n"mM � z�m

of x�n� as

Y h�f�n �z� !

M

Q

Q��Xl��

dlXn�lN�z�P��Xi��

Rn�zWivP �En�lN�zW iv

P ��

where NM

! PQ

with� gcd�P�Q� ! � v ! � and dl ! in the even�stacked

case and v ! Q�

and dl ! ��l in the odd�stacked case� Introducing the

polyphase vectors yn�z� ! �d�Yn�z� d�Yn�N�z� �� dQ��Yn� Q��N�z��T and xn�z� !

�d�Xn�z� d�Xn�N�z� �� dQ��Xn� Q��N�z��T � �� can be compactly written as ��

yn�z� ! S h�f�n �z�xn�z� ��

with the Q � Q matrices

S h�f�n �z� ! Rn�z�En�z� ��

for n ! �� M�� The Q � P matrices Rn�z� and the P � Q matrices En�z� are

de�ned as

�Rn�z��k�l !

sM

QRn�kN�zW lv

P �� k ! �� Q� � l ! �� P�

�En�z��k�l !

sM

QEn�lN�zW kv

P �� k ! �� P � � l ! �� Q��

�Usually� the polyphase components are de ned only for n � �� M�� However� in view of

the quasiperiodicity relation Xn�lM �z� � zlXn�z�� where Xn�z� �P�

m�� x�n � mM � z�m� this

de nition can be extended to arbitrary n � ZZ��gcd�P�Q� denotes the greatest common divisor of P and Q�


where v ! � in the even�stacked case and v ! Q�in the odd�stacked case� Note that

the polyphase representation �� is not formulated in terms of the N � M analysis

polyphase matrix E�z� and the M � N synthesis polyphase matrix R�z� introduced

in Section �� but rather in terms of the smaller matrices En�z� and Rn�z�� This

simpli�cation is possible because of the modulation structure inherent in DFT FBs�

Dual polyphase domain� Alternatively� in the case of DFT FBs we can de�ne

dual polyphase components as X �n�z� !

P�m�� x�mN " n� z�m �n ! �� N�� and

similarly for Y h�f��n � E �

n�z�� and R�n�z�� Note that the number of polyphase components

is now N instead of M � �For N ! M � i�e�� critical sampling� the dual polyphase

decomposition coincides with the polyphase decomposition�� In the dual polyphase

domain� the DFT FB operator is represented as

Y h�f��n �z� !

N

Q

Q��Xl��

X �n�zW

lQ�

P��Xi��

R�n�iM��z�E �n�iM��zW l

Q� ��

where � ! � in the odd�stacked case and � ! in the even�stacked case� The dual

polyphase matrix representation of S h�f�� i�e�� formulated in terms of the dual

polyphase decomposition� can be obtained from �� in a straightforward manner�

�� Perfect Reconstruction

From �� we can see that a DFT FB satis�es the PR property x�n� ! x�n� if and only

if S h�f�DFT ! I� In this case the output signal of the DFT FB can be written as

x�n� !N��Xk��

�Xm��

hx� hDFTk�m i fDFTk�m �n� � ��

where hDFTk�m �n� and fDFTk�m �n� are de�ned as in Section �� Hence� a DFT FB with PR

provides an expansion of the input signal x�n� into the WH set fDFTk�m �n�� This expansion

is known as the �discrete�time� Gabor expansion �� Thus� PR DFT FBs

and Gabor expansions are mathematically equivalent �� Based

on the representations of the DFT FB operator S h�f�DFT given in Section �� we shall

next formulate PR conditions for oversampled DFT FBs in various domains� We shall

furthermore show that the PR condition is independent of the stacking type�

�� PR condition in the time domain

From �� it follows that a DFT FB is PR if and only if ��

N�X

m��f �n�mM � h��n "mM " lN � ! ��l��

�� PERFECT RECONSTRUCTION ��

Note that the PR condition is independent of the stacking type� Therefore� an even�

stacked DFT FB is PR if and only if the corresponding odd�stacked DFT FB with the

same number of channels� the same decimation factor� and the same prototypes is PR�

It can be seen that if prototypes h�n�� f �n� satisfy the PR property �� in the critical

case� the scaled prototypespPh�n��

pPf �n� will satisfy this property in the case of

integer oversampling with oversampling factor P � For a given DFT analysis FB� the

minimum norm synthesis FB f fk�n�g is always a DFT FB �� i�e�� fk�n� ! f �n�W�knN in

the even�stacked case and fk�n� ! f �n�W� k��nN in the odd�stacked case� this follows

immediately from the fact that the dual frame of a WH frame is again a WH frame ��

�cf� Section �� However� in the oversampled case there are synthesis FBs providing

PR for a given DFT analysis FB that do not have DFT structure� Let us reconsider

the time�domain parameterization �� in terms of the �lters pk�n�� In �� it has been

shown in the context of WH frames that

pk�n� ! p�n�W� k��nN ��

with ! � in the even�stacked case and ! �� in the odd�stacked case is su�cient

for the synthesis FB to be a DFT FB� Inserting �� in �� we get

f �n� ! f �n� " p�n�� N�X

l�� f �n� lN �

�Xm��

h�mM � n" lN � p�n�mM �

��

which is a parameterization of the synthesis prototype f �n� in terms of the single �lter

p�n� that may be chosen arbitrarily� Note that the minimum norm prototype f �n� is

reobtained for p�n� � �� The parameterization �� is valid for both odd�stacked and

even�stacked DFT FBs�

�� PR condition in the frequency domain

From �� it follows that the PR condition in the frequency domain reads ��

M

N��Xk��

F�z W k

N

�H�zW k

NWlM

�! ��l� � ��

The parameterization �� becomes

Fk�z� ! F �z W k�N � " Vk�z��

M

M��Xi��

F �z W iMW

k�N �

N��Xl��

H�z W iMW

l�N �Vl�z��

where ! � in the even�stacked case and ! �� in the odd�stacked case� A su�cient

condition for the synthesis FB to have DFT structure is �cf� ��

Vk�z� ! V �z W k�N ��

� CHAPTER �� OVERSAMPLED DFT FILTER BANKS

which yields

F �z� ! F �z� " V �z��

M

M��Xi��

F �z W iM�

N��Xl��

H�zW iMW

lN �V �z W l

N ��

This parameterization of the synthesis prototype F �z� in terms of the single �lter V �z�

that may be chosen arbitrarily� Again this parameterization is valid for both the odd�

stacked and the even�stacked case�

�� PR condition in the polyphase domain

With �� the PR condition in the polyphase domain is ��

M

Q

P��Xi��

Rn�zWivP �En�lN�zW iv

P � ! ��l��

where v ! � in the even�stacked case and v ! Q�in the odd�stacked case� Rewriting

�� in terms of the matrices En�z� and Rn�z� from Section �� we obtain

Rn�z�En�z� ! IQ � n ! �� M��

In the oversampled case N � M � Rn�z� is not uniquely determined for given En�z��

any solution of �� can be written as

Rn�z� ! Rn�z� "Un�z� �IP �En�z� Rn�z�� n ! �� M � ��

Here� Rn�z� is the para�pseudo�inverse of En�z�� which is a particular solution of ��

de�ned as Rn�z� ! �#En�z�En�z��

�� #En�z�� n ! �� M �

and Un�z� is a Q � P matrix with elements �Un�z��k�l ! Un�kN�zW lvP �� where v !

� in the even�stacked case and v ! Q�

in the odd�stacked case� Note that for each

value of n with n ! �� M � we can have a di�erent Un�z�� The matrices

Un�z� are fully determined by the M polyphase components Un�z� satisfying the quasi�

periodicity relation UnlM�z� ! zlUn�z�� Eq� �� is a polyphase�domain version of

�� and �� Note that the parameterization �� automatically implies that the

corresponding synthesis FB has DFT structure�

�� PR condition in the dual polyphase domain

Finally� it follows from �� that S h�f�DFT ! I if and only if the following PR condition

in the dual polyphase domain is satis�ed�

P��Xi��

R�n�iM�z�E �n�iM�zW l

Q� !Q

N��l� � n ! �� N��

A matrix formulation similar to �� and a parameterization of all PR synthesis FBs

with DFT structure similar to �� can easily be obtained�

�� FRAMETHEORETIC PROPERTIES �

�� FrameTheoretic Properties

A WH set fhDFTk�m �n�g that is a frame �cf� �� is called a WH frame� WH frames are

an important special case of UFBFs� The dual frame fk�m�n� ! �S��hDFTk�m ��n� can be

shown to be again a WH frame �� i�e��

fk�m�n� ! fDFT�ok�m �n� ! f �n�mM �W� k�� n�mM�N

in the odd�stacked case and

fk�m�n� ! fDFT�ek�m �n� ! f �n�mM �W�k n�mM�N

in the even�stacked case� The synthesis prototype is in both cases given by

f �n� ! f �n� ! �S��DFT�ehy��n� with hy�n� ! h��n� � ��

Here� S��DFT�e is the inverse of the WH frame operator in the even�stacked case� which

is given by SDFT�e ! S h�hy�DFT�e where S

h�f�DFT�e denotes the DFT FB operator in the even�

stacked case �see �� Among all synthesis prototypes satisfying PR� �� de�nes

the synthesis prototype with minimum energy �norm� �� In the theory of

Gabor expansions �WH frames� the synthesis prototype given in �� is also known

as Wexler�Raz dual ��

We shall next show that an odd�stacked DFT FB provides a frame decomposition in

l��ZZ� if and only if the corresponding even�stacked DFT FB �with the same synthesis

prototype� provides a frame decomposition in l��ZZ�� We shall use the following

Lemma �� The function set ff �n�mM � ej��k��N

ng with k ! �� N�� m � � and � f�� g is a WH frame in l��ZZ� if and only if

the set ff �n�mM � ej��k��N

n�mM�g with k ! �� N�� m ��is a WH frame in l��ZZ��

Proof� Straightforward manipulations reveal that the Walnut representations

�cf� �� of the frame operators corresponding to the sets ff �n�mM � ej��k��N

ngand ff �n � mM � ej��

�k��N

n�mM�g with k ! �� N � � �� m � � and

� f�� g are equal� Therefore� the corresponding frame operators are equal� which

in turn proves the lemma� �

We are now able to formulate

Theorem �� An odd�stacked DFT FB provides a frame decomposition in

l��ZZ� if and only if the corresponding even�stacked DFT FB with the same

synthesis prototype� the same number of channels and the same decimation

factor provides a frame decomposition in l��ZZ��


Proof� Let us �rst consider the modi�ed synthesis functions fDFT�o�

k�m �n� ! f �n �mM �ej��

�k��N

n� These can be obtained by applying a unitary transformation U �fre�

quency shift� to the fDFT�e�

k�m �n��

fDFT�o�

k�m �n� ! �UfDFT�e�

k�m ��n� ! fDFT�e�

k�m �n� ej�Nn�

where fDFT�e�

k�m �n� ! f �n � mM �ej��kNn� Note that this transformation is the same for

all functions fDFT�o�

k�m �n� with k ! �� N � � �� m � �� Since kU �xk ! kxk�it follows from Theorem A�� that if ffDFT�e�k�m �n�g �k ! �� N � � �� m � ��

is a frame for l��ZZ� then fDFT�o�

k�m �n� is also a frame for l��ZZ�� Moreover� it follows from

kUk ! that fDFT�o�

k�m �n� has the same frame bounds as fDFT�e�

k�m �n�� Noting that the

fDFT�e�

k�m �n� are obtained from the fDFT�o�

k�m �n� by means of the unitary transformation

�U��x��n� ! x�n� e�j�Nn�

the same arguments as above can be used to show the converse� The result in Theo�

rem �� then follows by employing Lemma ��

�� Representations of the Frame Operator

We shall next provide time� frequency� and polyphase domain representations of the

WH frame operator for odd� and even�stacked DFT FBs� These representations follow

easily from the representations of the DFT FB operator S h�f� given in Section �� by

replacing f �n�� F �z� and Rn�z� by h��n�� #H�z� and #En�z�� respectively�

Time domain� The time domain representation of the WH frame operator� also

known as Walnut representation �� reads

�Sx��n� !�X

l��dlx�n� lN �

N

�Xm��

h��n "mM � h��n "mM " lN �

��

where dl ! ��l in the odd�stacked case and dl ! in the even�stacked case� The

inverse frame operator can be represented in a similar manner by replacing h�n� with

f ��n� in ��

Frequency domain� In the frequency domain� the WH frame operator can be

expressed as

� �SX��z� !M��Xl��

X�z W l

M

�

M

N��Xk��

#HDFTk �z�HDFT

k

�z W l

M

��

where �S ! ZSZ�� is the frequency domain representation of S and HDFT

k �z� !

HDFT�ek �z� in the even�stacked case and HDFT

k �z� ! HDFT�ok �z� in the odd�stacked case�

�� FRAMETHEORETIC PROPERTIES �

The inverse frame operator can be represented in a similar manner by replacingHDFTk �z�

with #FDFTk �z� in ��

Polyphase domain� The polyphase components Yn�z� !P�

m��Sx��n"mM � z�m

are related to the polyphase components Xn�z� !P�

m�� x�mM " n� z�m as

Yn�z� !M

Q

Q��Xl��

dlXn�lN�z�P��Xi��

#En�zWivP �En�lN�zW iv

P ��

where v ! � and dl ! in the even�stacked case and v ! Q�and dl ! ��l in the odd�

stacked case� The polyphase vectors yn�z� ! �d�Yn�z� d�Yn�N�z� �� dQ��Yn� Q��N�z��T

and xn�z� ! �d�Xn�z� d�Xn�N�z� �� dQ��Xn� Q��N�z��T are related as ��

yn�z� ! Sn�z�xn�z� with Sn�z� ! #En�z�En�z� � ��

xn�z� ! S��n �z�yn�z� with S��n �z� ! Rn�z� #Rn�z�

for n ! �� M�� where En�z� and Rn�z� have been de�ned in Section �� This

representation of S in terms of M matrices Sn�z� of size Q � Q is known in WH

frame theory as the Zibulski�Zeevi representation of the WH frame operator ��

In particular� the inversion of the frame operator&which� in the general UFBF case�

requires the inversion of the M �M UFBF matrix S�z�&here reduces to the inversion

of M matrices of size Q � Q� Note that in the case of integer oversampling �Q ! ��

i�e�� N ! PM � the matrices Sn�z� �n ! �� M � � reduce even further to scalars

and hence the inversion of the frame operator is accomplished by simple divisions in the

polyphase domain� Integer oversampled DFT FBs are discussed in detail in Subsection

��

Dual polyphase domain� In the dual polyphase domain� the WH frame operator

is represented as

Y �n�z� !

�Xm��

�Sx��mN " n� z�m !N

Q

Q��Xl��

X �n�zW

lQ�

P��Xi��

#E �n�iM��z�E �

n�iM��zW lQ��

��

where � ! � in the odd�stacked case and � ! in the even�stacked case�

�� Time�Limited Prototype

If h�n� has �nite length � N � it follows from �� that the corresponding WH frame

operator S is a multiplication operator in the time domain� i�e�

�Sx��n� ! bh�n� x�n� � �S��x��n� !x�n�

bh�n��

where the factor bh�n� is a periodized version of jh��n�j��

bh�n� ! N�X

m��jh��n "mM �j��


For bh�n� � A the DFT FB is paraunitary with frame bound A� Note that the parau�

nitarity condition is independent of the stacking type�

�� Band�Limited Prototype

If h�n� is band�limited in a frequency interval of length � �M � clearly all the modulated

versions of h�n� are band�limited within an interval of length � �M � In this case

the FB is diagonal in the frequency domain �see Subsection �� and hence S is a

multiplication operator in the frequency domain� i�e��

� �SX��ej�� ! Gh��X�ej�� S��X��ej�� !

X�ej��

Gh��

Here� �S ! FSF�� with F denoting the Fourier transform operator� and the factor

Gh�� is a periodized version of jH�ej��j��

Gh��!

M

N��Xk��

��H�ej�� kN��

��

where �� ! � in the even�stacked case and �� !��N

in the odd�stacked case� For Gh�� A� the DFT FB is paraunitary with frame bound A� Note that the paraunitarity

condition is again independent of the stacking type�

�� Paraunitarity Conditions

Using the Walnut representation �� of the WH frame operator� the time�domain

condition for paraunitarity with frame bound A is obtained as

N�X

m��h��n "mM � h��n "mM " lN � ! A��l� �

In the frequency domain �see �� the paraunitarity condition reads

M

N��Xk��

#H�z W k

N

�H�z W k

N W lM

�! A��l� �

The paraunitarity condition in the polyphase domain reads �cf� ��

M

Q

P��Xi��

#En�z WivP �En�lN�zW iv

P � ! A��l��

with v ! � in the even�stacked case and v ! QP�

in the odd�stacked case� Using the

Zibulski�Zeevi representation �� we can see that a DFT FB is paraunitary with

frame bound A if and only if

Sn�z� ! A IQ for n ! �� M��

�� FRAMETHEORETIC PROPERTIES ��

In the dual polyphase domain �cf� �� the paraunitarity condition reads

N

Q

P��Xi��

#E �n�iM�z�E �

n�iM �z W lQ� ! A��l��

Note that these paraunitarity conditions formally equal the PR conditions in Section ��

with f �n�� F �z�� Rn�z�� and R�n�z� replaced by h��n�� #H�z�� #En�z�� and #E �n�z�� respec�

tively�

�� Integer Oversampling

Let us consider an integer oversampled DFT FB �N ! PM with P � IN�� Using

N ! PM in �� and �� the elements of the UFBF matrix S�z� ! #E�z�E�z� are

obtained as Sn�n��z� !PN��

k�� W kv� n��n�N

#En�zWkvP �En��zW

kvP �� where v ! � in the

even�stacked case and v ! �� in the odd�stacked case� Substituting k � lP " r �l !

�� M � � r ! �� P � �� we �nally obtain

Sn�n��z� ! MP��Xi��

#En�zWivP �En�zW

ivP � ��n��n� � n ! �� M�� n� ! �� M��

��

which shows that the UFBF matrix S�z� is diagonal with diagonal elements

Sn�n�z� ! MP��Xi��


ivP ��

Therefore� integer oversampled DFT FBs are diagonal in the polyphase domain �see

Section �� The Q � Q matrices Sn�z� here reduce to scalars since Q ! and

furthermore S�z� ! diagfSn�z�gM��n�� or equivalently Sn�n�z� ! Sn�z��

The eigenvalues of S�ej�� follow from the frequency responses En�ej�� of the anal�

ysis prototype�s polyphase components according to

n�� ! Sn�n�ej�� ! M

P��Xi��

��En

�ej��

i�vP

��

Hence� it follows that an integer oversampled DFT FB corresponds to a WH frame if

and only if

ess inf�� n��M��

Sn�n�ej�� ess sup


j��

and that the �tightest possible� frame bounds are given by


Sn�n�ej�� B ! ess sup


j��


Note that the frequency responses En�ej�� of the analysis prototype�s polyphase com�

ponents determine the frame bounds �numerical properties� of the FB� An integer over�

sampled DFT FB is paraunitary with frame bound A if and only if

Sn�n�z� � A for n ! �� M��

It follows from �� that the polyphase components of the minimum norm synthesis

prototype are given by

Rn�z� !#En�z�

Sn�n�z��

Thus� in the case of integer oversampling the synthesis prototype can be calculated in

the polyphase domain by simple divisions and the matrix inversion in �� is avoided�

According to Theorem �� and the discussion at the end of Section �� a paraunitary

FB can be constructed by factoring the matrix S�z� of an arbitrary FB corresponding to

a frame� For integer oversampled DFT FBs� this factorization reduces to a factorization

of polynomials �FIR case� or rational functions �IIR case� in z�� Let En�z� be the

analysis polyphase components of an integer oversampled DFT FB corresponding to a

WH frame in l��ZZ�� Furthermore� let Pn�z� be such that

P �n�z� ! Sn�n�z� ! M

P��Xi��


ivP � and #Pn�z� ! Pn�z� ��

with Pn�ej�� v ! � in the even�stacked case and v ! �� in the odd�stacked

case� Then using �� with �� and �� it follows that the DFT FB with analysis

polyphase components

E p�n �z� !

En�z�

Pn�z�

is paraunitary with frame bound A ! � i�e�� #E p��z�E p��z� ! IM �

In the special case of critical sampling �P ! �� we have

Sn�n�z� ! M #En��z�En��z� � n�� ! Sn�n�ej�� ! M jEn�e

j�� v��j� �

where � ! in the even�stacked case and � ! � in the odd�stacked case� In this case

the above relations simplify accordingly� In particular� �� reduces to

#En�z�En�z� !A

Mfor n ! �� M� �

or jEn�ej��j� � A�M � which means that the polyphase �lters En�z� are allpass �l�

ters� Thus� the design of a critically sampled paraunitary DFT FB reduces to �nd�

ing an arbitrary set of M allpass �lters� Furthermore� �� simpli�es to P �n�z� !

M #En��z�En��z��

In � �� it has been shown that for critical sampling� a DFT FB with FIR �lters in

both the analysis and the synthesis section is possible only if all the polyphase �lters are

�� DESIGN OF OVERSAMPLED FIR PARAUNITARY DFT FILTER BANKS ��

pure delays� This leads to �lters with poor frequency selectivity� In �� this e�ect has

been interpreted as a discrete�time equivalent of the Balian�Low theorem �� In the

oversampled case� this restriction is relaxed� More generally� a necessary and su�cient

condition for an oversampled FB to have FIR analysis and FIR minimum norm synthesis

�lters is det �#E�z�E�z�� ! C with C ! � �see Theorem �� For integer oversampling�#E�z�E�z� is a diagonal matrix and thus the condition reads

det�#E�z�E�z�� !M��Yn��

� #E�z�E�z��n�n !M��Yn��

M

P��Xi��


ivP �

�! C�

For example� in the even�stacked case for P ! �� the above condition can be satis�ed

by choosing the polyphase �lters En�z� such that the power symmetry conditions

#En�z�En�z� " #En��z�En��z� ! Cn� n ! �� M �

hold with some Cn� These polyphase �lters are not necessarily pure delays �� In

particular� a paraunitary FIR DFT FB with frame bound A can be constructed by

choosing polyphase �lters satisfying the power symmetry conditions �� cf� ��

#En�z�En�z� " #En��z�En��z� !A

Mfor n ! �� M��

It is well known that the above power symmetry condition can be satis�ed with non�

trivial FIR �lters ��

�� Design of Oversampled FIR Paraunitary DFT

Filter Banks

Oversampled FIR paraunitary DFT FBs constitute a practically important subclass of

oversampled DFT FBs� One possible method to design oversampled FIR paraunitary

DFT FBs �even�stacked or odd�stacked� is to use a constrained optimization approach�

where the side constraints are given by the paraunitarity condition� The cost function

can for example be chosen such that the prototype�s stopband energy is minimized� In

this section� we shall present an alternative method that allows an explicit construction

of paraunitary prototypes� This method does not� however� allow to incorporate the

frequency selectivity of the �lters explicitly� Nonetheless� we shall give some hints how

to obtain good frequency selectivity� Our method is furthermore restricted to prototypes

of length � N � where N denotes the number of channels� For longer prototypes one

can use the approach presented in Section �� In fact� the method presented here

can be considered as a special case of the method presented in Section �� For an

analysis prototype h�n� supported in an interval of length N � the WH frame operator


10 20 30−0.2

0

0.2

0.4

0.6

0.8

1

(a)10 20 30

−0.2

0

0.2

0.4

0.6

0.8

1

(b)

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(c)

Mag

nitu

de(d

B)

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(d)M

agni

tude

(dB

)

Figure �� Construction of paraunitary DFT FBs N ! ��M ! ��a Initial nonparaunitary prototype h�n�� b paraunitary prototype h p��n��

c H�ej�� d H p��ej��

is a multiplication operator �see �� Thus� the inverse square root S�� is also a

multiplication operator and we have

�S��x��n� !x�n�qbh�n�

with bh�n� given by �� For a given nonparaunitary prototype h�n� supported within

an interval of length N � the corresponding paraunitary prototype h p��n� is hence given

by

h p��n� ! �S��h��n� !h�n�qbh�n�

�

In general� the resulting paraunitary prototype h p��n� may not have good frequency

selectivity� However� choosing the initial prototype h�n� such that it has good frequency

selectivity and that

s�!

maxn

bh�n�

minn

bh�n� �

it is guaranteed that the resulting paraunitary prototype h p��n� will have reasonable

frequency selectivity as well� This is so� since for s the paraunitary prototype

satis�es h p��n� h�n�� Although these restrictions on the choice of the initial prototype

h�n� seem to be very stringent� we observed that in practice this goal can be achieved

�� SIMULATION RESULTS ��

20 40 60 80 100 120−0.2

0

0.2

0.4

0.6

0.8

1

(a)20 40 60 80 100 120

−0.2

0

0.2

0.4

0.6

0.8

1

(b)

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(d)M

agni

tude

(dB

)

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(c)

Mag

nitu

de(d

B)

Figure �� Construction of paraunitary DFT FBs N ! � �M ! ��a Initial nonparaunitary prototype h�n�� b paraunitary prototype h p��n��

c H�ej�� d H p��ej��

rather easily� Fig� � shows the construction of a paraunitary prototype of length ��

for N ! �� and M ! � �i�e�� oversampling by �� using an initial prototype h�n� with

s ! �� Fig� � shows the construction of a paraunitary prototype of length � for

N ! � and M ! � �i�e�� oversampling by �� using an initial prototype h�n� with

s ! ��


In this section we present two simulation results demonstrating the increased design

freedom of oversampled DFT FBs�

Example �� In the �rst example we consider a ��channel DFT FB �even�stacked

or odd�stacked� with oversampling by � The analysis prototype h�n� �see Fig� ��a�� is

supported within an interval of length N ! �� which implies that the frame operator

S is a multiplication operator �see Section �� Therefore according to �� the

minimum norm synthesis prototype is given by

f �n� !h��n�bh�n�

�

Fig� ��b� shows the minimum norm synthesis prototype f �n� corresponding to the anal�

ysis prototype in Fig� ��a�� Although the analysis prototype had reasonable frequency

� CHAPTER �� OVERSAMPLED DFT FILTER BANKS

selectivity� the corresponding minimum norm synthesis prototype f �n� has poor fre�

quency selectivity �the attenuation of the �rst sidelobe is only about �dB��

Since the FB is oversampled� the PR synthesis prototype is not unique� Using the

parameterization �� with an arbitrary window function p�n� that is also supported

within an interval of length N ! �� we get the following parameterization of synthesis

prototypes providing PR�

f �n� !h��n�

NP�

m�� jh��n "mM �j��N

�Xm��

h��n "mM � p�n �mM �

�" p�n��

��

Note that this is not the most general parameterization of synthesis prototypes providing

PR since the function p�n� has been assumed to be supported within an interval of length

N � whereas in general� p�n� can have arbitrary support� We designed a PR synthesis

prototype by performing an unconstrained optimization over all synthesis prototypes

parameterized by �� the parameter function p�n� was chosen such that the cost

function J !��R��

jF �ej��j� d� was minimized� The result is depicted in Fig� ��c�� One

can see that the attenuation of the �rst sidelobe is about ��dB� which is dB better

than for the minimum norm synthesis prototype� Finally� in order to demonstrate how

much design freedom there really is for oversampling factor � we chose the parameters

p�n� as random numbers and obtained the PR synthesis prototype depicted in Fig� ��d��

Hence� for oversampling factors as high as there is a large amount of design freedom�

Example �� The second example serves to demonstrate that if the oversampling

factor is not high enough there is not enough design freedom to substantially improve

the synthesis �lter quality� We simulated a DFT FB with N ! � channels and deci�

mation factor M ! � i�e�� oversampling by a factor of �� Figs� �a� and �b� show the

analysis prototype and the minimum norm synthesis prototype� respectively� Figs� �c�

and �d� show optimized synthesis prototypes� The synthesis prototype in Fig� �c� was

obtained by minimizing the energy of the synthesis �lter in the stopband region �here�

the passband was assumed to range from � to �N ! �� The prototype in Fig� �d�

was obtained by minimizing the �rst derivative of the prototype�s transfer function

at � ! �� PR was guaranteed by performing an unconstrained optimization over all

synthesis prototypes parameterized by �� One can observe that all synthesis pro�

totypes look very similar� This is so� because oversampling by � o�ers considerably less

design freedom than oversampling by as in the previous example� We observed that�

for most practical applications� oversampling factors of � or greater are necessary to

have a reasonable amount of design freedom�

�� SIMULATION RESULTS �

0 0.1 0.2 0.3 0.4

−60

−40

−20

0

(a)

Mag

nitu

de(d

B)

Frequency response

0 0.1 0.2 0.3 0.4

−60

−40

−20

0

(b)

Mag

nitu

de(d

B)

Frequency response

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(c)

Mag

nitu

de(d

B)

Frequency response

0 0.1 0.2 0.3 0.4

−60

−40

−20

0

(d)M

agni

tude

(dB

)

Frequency response

Fig� �� channel DFT �lter bank with oversampling factor � a analysis

prototype� b minimum norm PR synthesis prototype� c PR synthesis prototype

with optimum frequency selectivity� d �random� PR synthesis prototype�

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(d)

Mag

nitu

de(d

B)

Frequency response

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(c)

Mag

nitu

de(d

B)

Frequency response

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(b)

Mag

nitu

de(d

B)

Frequency response

0 0.1 0.2 0.3 0.4−70

−60

−50

−40

−30

−20

−10

(a)

Mag

nitu

de(d

B)

Frequency response

Fig� � ��channel DFT �lter bank with oversampling factor �� a analysis

prototype� b minimum norm PR synthesis prototype� c PR synthesis prototype

with �improved� frequency selectivity� d �smooth� PR synthesis prototype�

Chapter �

Oversampled Cosine Modulated

Filter Banks

Oversampled cosine modulated FBs �CMFBs� �� allow e�cient DCT�DST�

based implementations� As compared to DFT FBs� CMFBs are advantageous since their

subband signals are real�valued for real�valued input signal and analysis prototype�

It seems that so far only critically sampled CMFBs have been considered in the

literature �� This chapter introduces and

studies oversampled CMFBs with PR� The chapter is organized as follows� Sections ��

and �� introduce two types of oversampled CMFBs� thereby extending a classi�cation

of CMFBs recently proposed for critical sampling by Gopinath �� The �odd�stacked�

CMFBs introduced in Section �� extend the traditional CMFB type ��

�� class B CMFBs� �� to the oversampled case� The �even�stacked� CMFBs

introduced in Section �� extend the �class A CMFBs� recently introduced for critical

sampling by Gopinath �� to the oversampled case� This latter class contains CMFBs

previously proposed �for critical sampling� by Lin and Vaidyanathan �� and the

Wilson�type CMFBs introduced in �� Even�stacked CMFBs are attractive since they

allow both PR�paraunitarity and linear phase �lters in all channels� The linear phase

property is of particular interest for image coding applications ��

Section �� shows that odd� and even�stacked CMFBs are closely related to odd� and

even�stacked DFT FBs� respectively� whose oversampling factor is twice that of the

CMFB� We also show that even�stacked CMFBs are related to MDFT FBs ��

��

Section �� provides PR conditions for oversampled CMFBs� These conditions are

formulated in the time� frequency� and polyphase domains� They are shown to consist

of two parts� one of them being the PR condition in a DFT FB of the same stacking

type and with twice the oversampling factor� Furthermore� a uni�ed framework for

CMFBs with critical sampling or oversampling is developed�

�

�� CHAPTER �� OVERSAMPLED COSINE MODULATED FILTER BANKS

In Section �� we discuss frame�theoretic properties of oversampled CMFBs� We

formulate conditions on oversampled CMFBs to correspond to a frame decomposition

in l��ZZ� and we provide paraunitarity conditions� We show that the frame operator of

an oversampled CMFB is related to that of a DFT FB with the same stacking type and

with twice the oversampling factor� and that the frame bound ratio of a CMFB equals

that of the associated DFT FB�

Section �� discusses design methods for oversampled CMFBs and presents design

examples� We propose a design procedure based on constrained optimization� Fur�

thermore� an iterative design procedure recently introduced in �� for the design of

near�PR� odd�stacked� critically sampled CMFBs is adapted to the design of odd�stacked

and even�stacked PR CMFBs with arbitrary oversampling� The resulting algorithm is

found to have good convergence properties and allows for the design of prototypes with

up to �� taps� Next� the lattice design method introduced in � � for critically sam�

pled odd�stacked CMFBs is extended to odd� and even�stacked CMFBs with integer

oversampling� We �nally present a design example that demonstrates the increased

design freedom in oversampled CMFBs�

Section �� discusses the e�cient implementation of oversampled CMFBs� Finally�

in Section �� we propose a new subband image coding scheme based on even�stacked

CMFBs �� The structure of even�stacked CMFBs requires a modi�cation of the

quantization matrix and the zig�zag sequence used in the JPEG standard� We show

that our lossy subband image coding scheme using even�stacked CMFBs and entropy

coding outperforms odd�stacked CMFBs from a perceptual point of view while achieving

comparable rate�distortion performance as DCT�based JPEG schemes�

�� Oversampled OddStacked CMFBs

In spite of the advantages of oversampled FBs described in Chapter �� it appears that so

far only CMFBs with critical sampling have been considered� In this section� we extend

the conventional type of CMFBs �� termed �class B�

in �� to the oversampled case� We shall call this CMFB type �odd�stacked� due to

its close relation to odd�stacked DFT FBs �cf� Section ��

In the following� h�n� and f �n� denote the analysis and synthesis prototype� respec�

tively� which may be FIR or IIR �lters� We de�ne the analysis and synthesis �lters of

an odd�stacked CMFB with N channels and decimation factorM �note that the CMFB

is oversampled for N � M� as�

hCM�ok �n� !

p� h�n� cos

��k " ��

Nn" ok

��

�The superscripts CM�o and CM�e indicate that the respective quantity belongs to an odd and

evenstacked CMFB� respectively�

�� OVERSAMPLED ODDSTACKED CMFBS ��

fCM�ok �n� !

p� f �n� cos

��k " ��

Nn� ok

��

for k ! �� N�� Extending the phase de�nition given for critical sampling �N ! M�

by Gopinath and Burrus �� to the oversampled case� we de�ne the phases ok as

ok ! � �

�N

k "

�

�" r

�

�with � ZZ � r � f�� g �

The choice r ! corresponds to replacing the cos in �� and �� by �sin and sin�

respectively� The above phase expression contains the phases proposed in ��

� � � � as special cases� The transfer functions of the analysis and synthesis �lters

in an odd�stacked CMFB are

HCM�ok �z� !

p�

hH�zW

k��N

�ej�

ok "H

�zW

� k��N

�e�j�

ok

i� k ! �� N� ��

FCM�ok �z� !

p�

hF�zW

k��N

�e�j�

ok " F

�zW

� k��N

�ej�

ok

i� k ! �� N��

Note that the channel frequencies in an odd�stacked CMFB are �k ! k��N

� In partic�

ular� the channel with index k ! � is centered at frequency �� ! ��N

� Fig� compares

the transfer functions of an odd�stacked CMFB and those of an odd�stacked DFT FB

with twice the number of channels�

The input�output relation in an odd�stacked CMFB with N channels and decimation

factor M is

xCM�o�n� !N��Xk��

�Xm��

hx� hCM�ok�m i fCM�o

k�m �n� � ��

with the analysis and synthesis functions

hCM�ok�m �n� ! hCM�o�

k �mM � n�!p� h��mM � n� cos

��k " ��

N�mM � n� " ok

��

fCM�ok�m �n� ! fCM�o

k �n�mM �!p� f �n�mM � cos

��k " ��

N�n�mM�� ok

��

Note that the analysis and synthesis functions are time�shifted versions of h�k��n� andfk�n�� respectively� and thus consistent with the UFBF format de�ned in Section ��

An important disadvantage of odd�stacked CMFBs is that the channel �lters do not

have linear phase even if the prototypes have linear phase �� Linear phase �lters are

especially important in image coding applications� where nonlinear phase �lters lead to

undesirable artifacts in the reconstructed image �� Other important aspects of

odd�stacked CMFBs� such as their relation to DFT FBs� PR conditions� frame theoretic

properties� design� and implementation� will be considered in Sections ��'��


14N

-34N

-2N-14N

-12

-

(a)

HN H2N-2 H2N-1 H0 H1

0 14N

34N

2N-14N 1

2

HN-1

14N

-34N

-2N-14N

-12

-

(b)

HN-1 H1 H0 H0 H1

0 14N

34N

2N-14N 1

2

HN-1

Figure �� Transfer functions of the channel �lters in a a �N�channel odd�stacked

DFT FB and b an N�channel odd�stacked CMFB�

�� Oversampled EvenStacked Cosine Modulated

Filter Banks

In this section� we extend the �class A� CMFBs recently proposed for critical sampling

in �� to the oversampled case� We call this CMFB type �even�stacked� due to its

close relation to even�stacked DFT FBs �cf� Section �� An advantage of even�stacked

CMFBs over the odd�stacked CMFBs considered in the previous section is that they

allow both PR�paraunitarity and linear phase �lters in all channels� We shall see that

the CMFBs recently introduced �for critical sampling� by Lin and Vaidyanathan ��

and the recently proposed Wilson FBs �� are special cases of even�stacked CMFBs�

�� Denition of Even�Stacked CMFBs

The analysis FB in an even�stacked CMFB with �N channels and decimation factor

�M �note that the CMFB is oversampled for N � M� consists of two partial FBs

fhCM�ek �n�gk��N and f(hCM�e

k �n�gk��N�� derived from an analysis prototype h�n� as�

hCM�ek �n� !

��h�n� rM �� k ! �p�h�n� cos

�k�Nn" ek

�� k ! � �� N�

h�n� sM � ��n�sM � k ! N

(hCM�ek �n� !

p� h�n�M � sin

�k�

N�n�M� " ek

�� k ! � �� N� �

�Note that here we consider FBs with �N channels instead of N channels� The reason for doing so

will become clear later in this section�

�� OVERSAMPLED EVENSTACKED CMFBS ��

Similarly� the synthesis FB consists of two partial FBs ffCM�ek �n�gk��N and

f (fCM�ek �n�gk��N�� derived from a synthesis prototype f �n� as

fCM�ek �n� !

��f �n" rM �� k ! �p� f �n� cos

�k�Nn� ek

�� k ! � �� N�

f �n" sM � ��nsM � k ! N

(fCM�ek �n� ! �

p� f �n"M � sin

�k�

N�n"M�� ek

�� k ! � �� N� �

Extending the phase de�nition given for critical sampling in �� we de�ne the phases

as

ek ! � �

�Nk " r

�

�with � ZZ � r � f�� g �

furthermore� s � f�� g with s ! r for even and s ! � r for odd�

The transfer functions of the analysis and synthesis �lters in an even�stacked CMFB

are

HCM�ek �z� !

��z�rM H�z�� k ! ��p�

hH�zW k

�N � ej�ek "H�zW�k

�N � e�j�ek

i� k ! � �� N�

z�sMH��z�� k ! N

��

(HCM�ek �z� !

jp�z�M

hH�zW k

�N � ej�ek �H�zW�k

�N � e�j�ek

i� k ! � �� N� � ��

and

FCM�ek �z� !

��zrM F �z�� k ! ��p�

hF �zW k

�N � e�j�e

k " F �zW�k�N � ej�

ek

i� k ! � �� N�

zsMH��z�� k ! N

��

(FCM�ek �z� ! �

jp�zMhF �zW k

�N � e�j�e

k � F �zW�k�N � ej�

ek

i� k ! � �� N� ��

respectively� Note that an even�stacked CMFB has �N channels but there are only

N " di�erent channel frequencies �k ! k�N

�k ! �� N�� as depicted in Fig� ��b��

In particular� the k ! � channel is centered at frequency �� ! �� which is an important

di�erence from odd�stacked CMFBs where the k ! � channel is centered at �� !��N

�

The input�output relation in an even�stacked CMFB with �N channels and decima�

tion factor �M is

xCM�e�n� !�N��Xk��

�Xm��

hx� hCM�ek�m i fCM�e

k�m �n��

with the analysis and synthesis functions

hCM�ek�m �n� !

�� hCM�e�

k ��mM � n� � k ! �� N

(hCM�e�

k�N ��mM � n� � k ! N " � N " �� N � ��

fCM�ek�m �n� !

�� fCM�e

k �n� �mM � � k ! �� N

(fCM�ek�N �n� �mM � � k ! N " � N " �� N � �

��


(b)

0

0

HNHN-1

12

- N-12N

-

H1H2

22N

- 12N

-

H0

H1 H2

22N

12N

HN-1

N-12N

12

12-

N-12N

- 22N

- 12N

- 22N

12N

N-12N

12

HN-1 H2 H1 H1 H2 HN-1

(a)

0

HNHN+1

12

- N-12N

-

H2N-1H2N-2

22N

- 12N

-

H0

H1 H2

22N

12N

HN-1

N-12N

12

Figure �� Transfer functions of the channel �lters in a a �N�channel even�stacked

DFT FB and b a �N�channel even�stacked CMFB�

Note that the analysis and synthesis functions are again consistent with the UFBF

format described in Section �� For any choice of the parameters � ZZ and r �f�� g� all analysis �lters in an even�stacked CMFB have linear phase if the analysis

prototype h�n� satis�es the linear phase �symmetry� property h�"��l"�N�n� ! h�n�

for some l � ZZ� Similarly� all synthesis �lters have linear phase if f ��l"�N�n� !f �n� for some l � ZZ� This is an important advantage over odd�stacked CMFBs� For

the special case of critical sampling� the linear phase property of even�stacked �class A�

CMFBs has �rst been recognized by Gopinath �� In the oversampled case� the linear

phase property of even�stacked CMFBs has �rst been noted in ��

Two speci�c even�stacked CMFBs have been proposed previously� The �rst one�

corresponding to parameters ! � and r ! � �whence s ! �� is the CMFB recently

introduced �for critical sampling and the paraunitary case f �n� ! h��n�� by Lin and

Vaidyanathan �� A second special even�stacked CMFB is the Wilson�type CMFB

�corresponding to the discrete�time Wilson expansion �� recently introduced in ��

it is obtained with the parameters ! N and r ! � �and hence s ! � for N even and

s ! for N odd��

�� Interpretation of the Subband Signals

In this subsection� we discuss certain aspects of even�stacked CMFBs that are relevant

in a subband coding context� We shall essentially follow the discussion given in ��

�� OVERSAMPLED EVENSTACKED CMFBS ��

for critically sampled paraunitary Lin�Vaidyanathan FBs� As we shall see� the interpre�

tations and conclusions given in �� apply to even�stacked CMFBs in general� For the

sake of simplicity� we restrict our attention to critically sampled even�stacked CMFBs

with �N channels� however� our discussion can easily be extended to the oversampled

case�

In a conventional �N �channel FB with critical sampling� the total bandwidth �includ�

ing positive and negative frequencies� of the channel �lters is roughly ��N

� The output

of the channel �lters is decimated by �N �critical sampling�� Clearly� this decimation

causes only aliasing due to the non�ideal bandlimiting property of the channel �lters�

In a PR FB� this aliasing is cancelled by the synthesis �lters�

In a �N �channel even�stacked CMFB� each channel �lter has total bandwidth �N

�see Fig� ��b�� i�e� twice the bandwidth of a channel �lter in a conventional �N �

channel FB� Yet� the output signals of the channel �lters in the even�stacked CMFB

are decimated by �N � This decimation does� of course� lead to severe aliasing even

if the channel �lters are ideal bandpass �lters� In an even�stacked CMFB with PR�

even this aliasing is still cancelled by the synthesis �lters� However� the meaning of the

subband signals in an even�stacked CMFB is di�erent from that of the subband signals

in a conventional �N �channel CMFB� In a conventional �N �channel FB� the subband

signals represent the spectral content of the input signal in the corresponding frequency

band� In an even�stacked CMFB there are two �lters for each subband� namely� Hk�z�

and (Hk�z� �k ! � �� N � �� which cover the same frequency region� We shall show

in the following that the corresponding subband signals taken together retain the usual

meaning of a subband signal� This means that it is still possible to exploit the energy

distribution of the input signal in the usual way� in the sense that the number of bits

assigned to the subband quantizers can be chosen to be proportional to the spectral

content of the input signal in the corresponding frequency band�

Let yk�n� and (yk�n� denote the outputs of the �lters hCM�ek �n� and (hCM�e

k �n�� respectively�

and consider

Yk�z� " jzN (Yk�z� !hHCM�e

k �z� " jzN (HCM�ek �z�

iX�z� !

p� H�zW k

�N � ej�ek X�z� �

or equivalently

yk�n� " j(yk�n"N � ! Z��fp� H�zW k

�N � ej�ek X�z�g

with k ! � �� N � � where Z�� denotes the inverse z�transform operator� If the

input signal x�n� and the analysis prototype h�n� are real�valued� this implies

yk�n� ! RenZ�� np� H�zW k

�N � ej�ek X�z�

oo(yk�n "N � ! Im

nZ�� np� H�zW k

�N� ej�ek X�z�

oo

� CHAPTER �� OVERSAMPLED COSINE MODULATED FILTER BANKS

Re(.)

Im(.)

x[n] yk [n]

yk [n+N]

2 H zW2Nk e j k

e

Figure �� Interpretation of the subband signals yk�n� and (yk�n� for real�valued input

signal x�n� and real�valued analysis prototype h�n��

for k ! � �� N�� Thus� for real�valued x�n� and real�valued h�n� the output of the

�lterp�H�zW k

�N � ej�ek has real part yk�n� and imaginary part (yk�n "N � �see Fig� ��

We conclude that� except for delays and scale factors� the signals yk�n� and (yk�n"N � can

be interpreted as the real and imaginary parts of the �hypothetical� one�sided complex

subband signal obtained by �ltering the input signal x�n� byp�H�zW k

�N � ej�ek � which

from �� is seen to be equal to the analytic signal of yk�n�� Thus� decimating the

signals yk�n� and (yk�n� by �N is equivalent to decimating the analytic signal of yk�n�

by �N � Since the analytic signal of yk�n� has total bandwidth��N

� decimating it by �N

does not lead to aliasing except for �lter nonidealities� Therefore� it still makes sense to

quantize and encode the decimated signals according to the frequency�domain energy

distribution of the FB�s input signal� This analysis has originally been given in �� for

the special case of Lin�Vaidyanathan FBs� but we emphasize that our analysis is valid

for the whole class of even�stacked CMFBs�

There remains� however� a fundamental di�erence between even�stacked CMFBs and

conventional FBs� Since in a �N �channel even�stacked CMFB there are only N "

di�erent center frequencies� from a subband coding point of view the performance

of �N �channel even�stacked CMFBs is roughly speaking equal to the performace of

conventional N �channel FBs�

In Section �� we will propose a subband image coding scheme based on even�stacked

CMFBs� We will demonstrate that our proposed scheme achieves comparable rate�

distortion performance as DCT�based JPEG schemes� In particular� we shall also verify

that� from a subband coding point of view� �N �channel even�stacked CMFBs perform

equally well as N �channel odd�stacked CMFBs�

�� Representation of CMFBs via DFT Filter

Banks

In this section� we shall show that an oversampled or critically sampled� odd� or even�

stacked CMFB is related to a DFT FB of the same stacking type but with twice the

oversampling factor of the CMFB� In Sections �� and �� these relations will be seen

�� REPRESENTATION OF CMFBS VIA DFT FILTER BANKS �

to yield a uni�ed and simpli�ed framework for the analysis and design of odd� and

even�stacked CMFBs�

�� Odd�Stacked CMFBs

Let us �rst consider an odd�stacked DFT FB with �N channels and decimation factor

M �see Subsection �� The FB is critically sampled for �N !M and oversampled

for �N � M � The analysis and synthesis �lters are derived from prototypes h�n� and

f �n�� respectively� by modulation�

hDFT�ok �n� ! h�n�W� k��n�N and fDFT�ok �n� ! f �n�W

� k��n�N � k ! �� N� �

where W�N ! e�j��N � The corresponding transfer functions are �see Fig� �a��

HDFT�ok �z� ! H

�zW

k��N

�and FDFT�o

k �z� ! F�zW

k��N

�� k ! �� N� �

��

Let us now consider an odd�stacked CMFB� Comparing �� and �� it follows

that the analysis �lters of an odd�stacked CMFB with N channels and decimation factor

M can be expressed in terms of the analysis �lters of an odd�stacked DFT FB with �N

channels and decimation factor M as

HCM�ok �z� !

p�

hHDFT�o

k �z� ej�ok "HDFT�o

�N�k��z� e�j�o

k

i� k ! �� N� �

The synthesis �lters� transfer functions can be similarly expressed as

FCM�ok �z� !

p�

hFDFT�ok �z� e�j�

ok " FDFT�o

�N�k��z� ej�oki� k ! �� N� �

The �lters HDFT�ok �z��FDFT�o

k �z� and HDFT�o�N�k��z��F

DFT�o�N�k��z� correspond to channel fre�

quencies � ! k��N

and � ! �k��N

� respectively� Thus� the CMFB analysis �lters are

obtained by combining the �lters of a DFT FB corresponding to positive and negative

frequencies� Note that the oversampling factor of the DFT FB� �N�M � is twice that of

the CMFB� N�M �

The relation between odd�stacked CMFBs and DFT FBs can also be characterized

in a di�erent manner that will be important subsequently� Let us compare the input�

output relation �� in an odd�stacked CMFB with N channels and decimation factor

M to the input�output relation in an odd�stacked DFT FB with �N channels and

decimation factor M �again the oversampling factor of this DFT FB is twice that of

the CMFB�� According to �� this latter input�output relation is

xDFT�o�n� !�N��Xk��

�Xm��

hx� hDFT�ok�m i fDFT�ok�m �n� � ��


with analysis functions hDFT�ok�m �n� ! hDFT�o�

k �mM � n� ! h��mM � n�W� k�� n�mM��N

and synthesis functions fDFT�ok�m �n� ! fDFT�ok �n � mM � ! f �n � mM �W� k�� n�mM��N �

It follows that the CMFB analysis and synthesis functions �see �� and �� can be

expressed in terms of the DFT FB analysis and synthesis functions� respectively� as

hCM�ok�m �n� !

p�

hhDFT�ok�m �n� e�j�

ok " hDFT�o�N�k��m�n� e

j�ok

i��

fCM�ok�m �n� !

p�

hfDFT�ok�m �n� e�j�

ok " fDFT�o�N�k��m�n� e

j�ok

i� ��

Thus� the analysis and synthesis functions in a CMFB are obtained by combining pos�

itive and negative frequencies in the corresponding DFT FB�

�� Even�Stacked CMFBs

In an even�stacked DFT FB with �N channels and decimation factor M � the impulse

responses are �see Subsection ��

hDFT�ek �n� ! h�n�W�kn�N and fDFT�ek �n� ! f �n�W�kn

�N � k ! �� N�

and the transfer functions are �see Fig� ��a��

HDFT�ek �z� ! H�zW k

�N � and FDFT�ek �z� ! F �zW k

�N � � k ! �� N� �

��

Comparing �� and �� it follows that the analysis �lters of an even�stacked

CMFB with �N channels and decimation factor �M can be expressed in terms of the

analysis �lters of an even�stacked DFT FB with �N channels and decimation factor M

as

HCM�ek �z� !

��z�rM HDFT�e

� �z�� k ! ��p�

hHDFT�e

k �z� ej�ek "HDFT�e

�N�k �z� e�j�ek

i� k ! � �� N�

z�sM HDFT�eN �z� ! z�sM HDFT�e

� ��z�� k ! N

(HCM�ek �z� !

jp�z�M

hHDFT�e

k �z� ej�ek �HDFT�e

�N�k �z� e�j�ek

i� k ! � �� N� �

The synthesis �lters� transfer functions are similarly given by

FCM�ek �z� !

��zrM FDFT�e

� �z�� k ! ��p�

hFDFT�ek �z� e�j�

ek " FDFT�e

�N�k �z� ej�ek

i� k ! � �� N�

zsM FDFT�eN �z� ! zsM FDFT�e

� ��z�� k ! N

(FCM�ek �z� ! �

jp�zMhFDFT�ek �z� e�j�

ek � FDFT�e

�N�k �z� ej�eki� k ! � �� N� �

The �lters HDFT�ek �z�� FDFT�e

k �z� and HDFT�e�N�k �z�� F

DFT�e�N�k �z� correspond to channel fre�

quencies � ! k�N

and � ! � k�N

� respectively� Again� the oversampling factor of the DFT

FB is twice that of the CMFB�

�� REPRESENTATION OF CMFBS VIA DFT FILTER BANKS �

The relation between even�stacked CMFBs and DFT FBs can also be characterized

using the analysis and synthesis functions� Let us compare the input�output relation

�� in an even�stacked CMFB with �N channels and decimation factor �M to the

input�output relation in an even�stacked DFT FB with �N channels and decimation

factor M � According to Subsection �� this latter input�output relation reads

xDFT�e�n� !�N��Xk��

�Xm��

hx� hDFT�ek�m i fDFT�ek�m �n� ��

with the analysis functions hDFT�ek�m �n� ! hDFT�e�

k �mM � n� ! h��mM � n�W�k n�mM��N

and the synthesis functions fDFT�ek�m �n� ! fDFT�ek �n �mM � ! f �n�mM �W�k n�mM��N � It

follows �after simple manipulations� that the CMFB analysis and synthesis functions

�see �� and �� can be expressed in terms of the DFT FB analysis and synthesis

functions� respectively� according to

hCM�ek�m �n� !

��

�p�

hhDFT�ek��m �n� e�j�

ek " hDFT�e�N�k��m�n� e

j�ek

i� k ! � �� N�

jp�

hhDFT�ek�N��m��n� e

�j�ek�N � hDFT�eN�k��m��n� e

j�ek�N

i�

k ! N " � N " �� N�

hDFT�e��m�r�n� � k ! �

hDFT�eN��m�s�n� � k ! N�

��

and

fCM�ek�m �n� !

��

�p�

hfDFT�ek��m �n� e�j�

ek " fDFT�e�N�k��m�n� e

j�ek

i� k ! � �� N�

jp�

hfDFT�ek�N��m��n� e

�j�ek�N � fDFT�eN�k��m��n� ej�ek�N

i�

k ! N " � N " �� N�

fDFT�e��m�r�n� � k ! �

fDFT�eN��m�s�n� � k ! N�

��

�� A Fundamental Decomposition

In the previous subsection� we showed that the analysis �synthesis� functions corre�

sponding to a CMFB can be expressed in terms of the analysis �synthesis� functions

corresponding to a DFT FB of the same stacking type but with twice the oversampling

factor� This implies the following important decomposition�

Theorem �� The reconstructed signal of an odd� or even�stacked CMFB

can be decomposed as

x�n� !

�

h�S

h�f�DFTx��n� " �T

h�f�DFTx��n�

i� ��


where the operators S h�f�DFT and T

h�f�DFT are de�ned as


�N��Xk��

�Xm��

hx� hDFTk�m i fDFTk�m �n� � ��

�T h�f�DFTx��n� !

�N��Xk��

�Xm��

ej��kcm hx� hDFTk�m i fDFTk�m �n� � ��

Here� hDFTk�m �n� ! hDFT�ok�m �n�� fDFTk�m �n� ! fDFT�ok�m �n�� fDFTk�m �n� ! fDFT�o�N�k��m�n��

k ! ok� and cm ! for an odd�stacked CMFB and hDFTk�m �n� ! hDFT�ek�m �n��

fDFTk�m �n� ! fDFT�ek�m �n�� fDFTk�m �n� ! fDFT�e�N�k�m�n�� k ! ek� and cm ! ��m for

an even�stacked CMFB�

Proof� Let us �rst consider the odd�stacked case� Inserting �� and �� in ��

gives

xCM�o�n� !

�

N��Xk��

�Xm��

Dx� hDFT�ok�m e�j�

ok " hDFT�o�N�k��me

j�okE

hfDFT�ok�m �n� e�j�

ok " fDFT�o�N�k��m�n� e

j�ok

i!

�

N��Xk��

�Xm��

hhx� hDFT�ok�m i fDFT�ok�m �n� " hx� hDFT�o�N�k��mi fDFT�o�N�k��m�n�

" hx� hDFT�ok�m i fDFT�o�N�k��m�n� ej��o

k " hx� hDFT�o�N�k��mi fDFT�ok�m �n� e�j��ok

i!

�

�N��Xk��

�Xm��

hx� hDFT�ok�m i fDFT�ok�m �n� "�N��Xk��

�Xm��

ej��okhx� hDFT�ok�m i fDFT�o�N�k��m�n�

��

which is �� Note that in the last step we have used e�j��o�N�k�� ! ej��

ok �

We next consider the even�stacked case� Inserting �� and �� in �� we get

xCM�e�n� !

�

N��Xk��

�Xm��

hx� hDFT�ek��m e�j�ek " hDFT�e�N�k��me

j�eki

hfDFT�ek��m �n� e�j�

ek " fDFT�e�N�k��m�n� e

j�eki

"

�

�N��Xk�N�

�Xm��

hx� hDFT�ek�N��m��e�j�e

k�N � hDFT�eN�k��m��ej�e

k�N ihfDFT�ek�N��m��n� e

�j�ek�N � fDFT�eN�k��m��n� ej�ek�N

i"

�Xm��

hx� hDFT�e��m�ri fDFT�e��m�r�n� "�X

m��hx� hDFT�eN��m�si fDFT�eN��m�s�n�

!

�

N��Xk��

�Xm��

hx� hDFT�ek�m ifDFT�ek�m �n� " hx� hDFT�e�N�k�mi fDFT�e�N�k�m�n�

" ��mhx� hDFT�ek�m ifDFT�e�N�k�m�n� ej��ek " ��mhx� hDFT�e�N�k�mi fDFT�ek�m �n� e�j��

ek

i"

�Xm��

hhx� hDFT�e��m�ri fDFT�e��m�r�n� " hx� hDFT�eN��m�si fDFT�eN��m�s�n�

i�

�� REPRESENTATION OF CMFBS VIA DFT FILTER BANKS ��

Now using e�j��e�N�k ! ej��

ek � it follows that

xCM�e�n� !

�

�N��Xk��k ��N

�Xm��

hhx� hDFT�ek�m i fDFT�ek�m �n� " ��m ej��

ekhx� hDFT�ek�m i fDFT�e�N�k�m�n�

i

"�X

m��hx� hDFT�e��m�ri fDFT�e��m�r�n� "

�Xm��

hx� hDFT�eN��m�si fDFT�eN��m�s�n��

Finally� with ej��e� ! ��r� ej��eN ! ��r� and s ! r for even and s ! � r for

odd� we obtain

xCM�e�n� !

�

�N��Xk��

�Xm��

hx� hDFT�ek�m i fDFT�ek�m �n�

"

�

�N��Xk��

�Xm��

ej��ek��m hx� hDFT�ek�m i fDFT�e�N�k�m�n��

which is ��

We emphasize that the �rst component� �S h�f�DFTx��n�� is the output signal of a DFT

FB of the same stacking type with twice the oversampling factor �cf� �� and ��

The above decomposition will be the basis for our subsequent analysis of CMFBs� in

particular� for formulating PR and paraunitarity conditions in Sections �� and ��

�� Relation with MDFT Filter Banks

Modi�ed DFT FBs �MDFT FBs�� proposed for critical sampling in ��

were the �rst modulated FBs allowing for real processing and linear phase �lters in all

channels� We now demonstrate a close relation between MDFT FBs and even�stacked

CMFBs of the Wilson type� It can be shown that the reconstructed signal in a �possibly

oversampled� MDFT FB can be decomposed as

x�n� !

�

h�S

h�f�DFTx��n� " �T

h�f�DFTx

��n�i�

where �S h�f�DFTx��n� and �T

h�f�DFTx��n� are de�ned as in �� and �� for the even�

stacked case with parameters ! N and r ! � �the parameters of Wilson FBs�� In the

above input�output relation� T h�f�DFT acts on the conjugate of the input signal x�n�� For

real�valued x�n�� the input�output relation in an MDFT FB equals that in a Wilson FB�

For general �complex�valued� x�n�� it can be shown �similarly to the proof of Theorem

�� in Subsection �� that an MDFT FB satis�es the PR property if and only if �

S h�f�DFT ! �I and T

h�f�DFT ! O� for T

h�f�DFT ! O� x�n� ! �

��S

h�f�DFTx��n�� This equals the

input�output relation of an even�stacked DFT FB �up to the factor of �� and� as

will be explained in Subsection �� also the input�output relation in a PR Wilson

�Here� I and O denote the identity and zero operator� respectively� on l��ZZ��


FB� Thus� MDFT FBs with PR are equivalent to PR Wilson FBs� Furthermore� it

follows from the above decomposition that the PR and paraunitarity conditions to be

formulated in Subsections �� and �� apply to MDFT FBs as well�

�� Representations of the CMFB Operators

The �CMFB operators� S h�f�DFT and T

h�f�DFT from Subsection �� are fundamental for

the analysis and design of CMFBs with PR �see Section �� We shall now provide

representations of the operator T h�f�DFT in the time� frequency� and polyphase domains�

The corresponding representations of the operator S h�f�DFT are obtained from the repre�

sentations provided in Section �� by replacing N by �N � These representations will

then be used in Sections �� and �� for formulating PR and paraunitarity conditions�

respectively�

Time domain� In the time domain� the operator T h�f�DFT can be expressed as

�T h�f�DFTx��n� ! ��r �N

�Xl��

�Xm��

dl am x��n " �mM� �lN� �

�f �n�mM � h�n�mM " " �lN � ��

with dl ! ��l and am ! in the odd�stacked case and dl ! and am ! ��m in the

even�stacked case�

Frequency domain� The frequency �z�transform� version �T h�f�

DFT ! ZT h�f�DFTZ

��

�here Z is the z�transform operator� can be expressed as

�T h�f�

DFTX��z� !

��rM

M��Xl��

X�zW lM �

�N��Xk��

FDFTk �z�HDFT

k �zW lM �W

k��N � ��

with FDFTk �z� ! FDFT�o

k �z�� HDFTk �z� ! HDFT�o

k �z�� ! �� FDFTk �z� ! FDFT�o

�N�k��z�� and

� ! ��in the odd�stacked case and FDFT

k �z� ! FDFT�ek �z�� HDFT

k �z� ! HDFT�ek �z�� ! �

��

FDFTk �z� ! FDFT�e

�N�k �z�� and � ! � in the even�stacked case�

Polyphase domain� It can be shown that the polyphase components T h�f�n �z� !P�

m�� T h�f�DFTx��n"mM � z�m of �T

h�f�DFTx��n� are related to the polyphase components

Xn�z� !P�

m�� x�n "mM � z�m of x�n� as

T h�f�n �z� ! ��r M

Q

Q��Xl��

dlX�n��lN ��z�P��Xi��

hRn�zW

�iv�P �E�n��lN��zW iv

�P �

" Rn��zW�iv�P �E�n��lN��zW

iv�P �

i� ��

where En�z� !P�

m�� h�mM � n� z�m and Rn�z� !P�

m�� f �mM " n� z�m are the

polyphase components of the prototypes h�n� and f �n�� respectively� NM

! PQwith P and

�� PERFECT RECONSTRUCTION CONDITIONS ��

Q relatively prime� and �nally dl ! ��l� v ! Q�� and � ! � in the odd�stacked

case and dl ! � v ! �� and � ! in the even�stacked case�

Dual polyphase domain� The dual polyphase components T h�f��n �z� !P�

m�� T h�f�DFTx��n " m�N � z�m of �T

h�f�DFTx��n� are related to X �

n�z� !P�

m�� x�n "

m�N � z�m as

T h�f��n �z� ! ��r �N

Q

Q��Xl��

�P��Xi��

aiX��n��iM�zW l

Q�R�n�iM��z�E �

�n��iM ��zW lQ�

! ��r �NQ

Q��Xl��

P��Xi��

aiX��n��iM�zW l

Q�hR�n�iM��z�E �

�n��iM��zW lQ�

" aP zQR�n�iM�NQ��z�E

��n��iMNQ��zW

lQ�i� ��

where � ! � and ai ! in the odd�stacked case and � ! and ai ! ��i in the


The above time� frequency� and polyphase domain representations �specialized to

the even�stacked case with !N and r!�� can also be used for MDFT FBs if x�n� is

replaced by x��n� in the various representations of T h�f�DFT �we recall from Subsection ��

that in an MDFT FB T h�f�DFT acts on x��n��

�� Perfect Reconstruction Conditions

In this section� we will formulate PR conditions for odd� and even�stacked� oversam�

pled and critically sampled CMFBs� These PR conditions are formulated in the time�

frequency� and polyphase domains� they will constitute a basis for the design methods

to be discussed in Section �� For the special case of critical sampling �N ! M�� our

conditions simplify to those derived in ��

�� PR Conditions Using the CMFB Operators

Inserting the PR relation x�n� ! x�n� in the CMFB input�output relation �� or ��

yields

x�n� !N ��Xk��

�Xm��

hx� hCMk�mi fCMk�m �n��

where N � ! N � hCMk�m�n� ! hCM�ok�m �n�� and fCMk�m �n� ! fCM�o

k�m �n� for an odd�stacked CMFB

and N � ! �N � hCMk�m�n� ! hCM�ek�m �n�� and fCMk�m �n� ! fCM�e

k�m �n� for an even�stacked CMFB�

We shall next derive a fundamental PR condition in terms of the operators S h�f�DFT and

T h�f�DFT � This PR condition follows from the decomposition in Theorem ��


Theorem �� A CMFB �even�stacked or odd�stacked� oversampled or

critically sampled� satis�es the PR property x�n� ! x�n� if and only if

S h�f�DFT ! �I and T

h�f�DFT ! O� ��

Proof� Inserting �� into the decomposition �� immediately gives x�n� ! x�n�� so

that �� is su�cient for PR� We now show that �� is also necessary for PR� From

�� there is x�n� ! ��

h�S

h�f�DFTx��n�" �T

h�f�DFTx��n�

i� According to �� T

h�f�DFTx��n� is

a linear combination of time�reversed and shifted signal versions x��n"�mM��lN��whereas according to �� S

h�f�DFTx��n� is a linear combination of shifted �but not time�

reversed� signal versions x�n � �lN �� For x�n� ! x�n�� it is clear that all time�reversed

signal versions x��n"�mM � �lN �� due to T h�f�DFT� must be weighted by zero since

they cannot be canceled by the signal versions x�n � �lN � �due to S h�f�DFT� unless the

signal x�n� has special symmetry properties� Hence� there must be T h�f�DFT ! O� which

entails x�n� ! ��S

h�f�DFTx��n�� This �nally implies S

h�f�DFT ! �I� �

If the second PR condition� T h�f�DFT ! O� is satis�ed� the CMFB�s input�output relation

�� reduces to

x�n� !

��S

h�f�DFTx��n��

which is �up to a constant factor� the input�output relation of a DFT FB with �N

channels and decimation factor M � This DFT FB is odd�stacked �even�stacked� for an

odd�stacked �even�stacked� CMFB� Thus� we conclude that CMFBs with PR correspond

to PR DFT FBs of the same stacking type and with twice the oversampling factor� In

view of this correspondence� it is not surprising that the �rst PR condition� S h�f�DFT ! �I�

is �up to a constant factor� the PR condition for a DFT FB with �N channels and

decimation factor M �cf� Section �� this PR condition is the same for odd�stacked

and even�stacked CMFBs� In the special case of critical sampling� a similar relation has

previously been shown to exist between MDFT FBs and DFT FBs ��

Using the time� frequency� and polyphase domain representations of the CMFB oper�

ators S h�f�DFT and T

h�f�DFT �see Subsection �� we next reformulate the two PR conditions

�� in the various domains�

�� PR Conditions in the Time Domain

ReplacingN by �N in �� it immediately follows that the �rst PR condition� S h�f�DFT !

�I� is satis�ed if and only if

N�X

m��f �n�mM � h��n "mM " �lN � ! ��l� � ��

�� PERFECT RECONSTRUCTION CONDITIONS ��

With �� it follows after some manipulations that the second PR condition� T h�f�DFT !

O� holds if and only if

�Xi��

bi f �n� iPM � h�n " iPM " " �lN � ! � � l ! �� Q��

where NM

! PQwith P � Q relatively prime and bi ! ��iQ in the odd�stacked case and

bi ! ��iP in the even�stacked case� In the special case of integer oversampling� i�e��

Q ! or N ! PM with P � IN� the latter condition simpli�es to

�Xi��

bi f �n� iPM � h�n" iPM " � ! � � ��

where bi ! ��i in the odd�stacked case and bi ! ��iP in the even�stacked case�

Note that critical sampling� N ! M � is a special case with P ! � It can be seen that if

prototypes h�n�� f �n� satisfy the �rst PR property �� in the critical case� the scaled

prototypespPh�n��

pPf �n� will satisfy this property in the case of integer oversampling

with oversampling factor P �this has also been observed for the paraunitary case in

�� However� we caution that a similar rule does not hold for the second PR condition

�� Thus� prototypes providing PR in the critical case do not automatically provide

PR for integer oversampling�

�� PR Conditions in the Frequency Domain

Replacing N by �N in �� it follows that S h�f�DFT ! �I for either stacking type if and

only if�N��Xk��

F �zW k�N �H�zW k

�NWlM� ! �M ��l� � l ! �� M��

Similarly� it follows from �� that T h�f�DFT ! O if and only if

�N��Xk��

FDFTk �z�HDFT

k �zW lM �W k�

�N ! � � l ! �� M��

where FDFTk �z� ! FDFT�o

�N�k��z�� HDFTk �z� ! HDFT�o

k �z�� and � ! � in the odd�stacked

case and FDFTk �z� ! FDFT�e

�N�k �z�� HDFTk �z� ! HDFT�e

k �z�� and � ! ��in the even�stacked

case�

�� PR Conditions in the Polyphase Domain

From �� it follows that S h�f�DFT ! �I if and only if

Rn�z�En�z� ! � IQ � n ! �� M��


Alternatively� for Q odd we have S h�f�DFT ! �I if and only if

Al�n�z��!

P��Xi��

hRn�zW

i�P �En��lN�zW i

�P � "Rn��zW i�P �En��lN��zW i

�P �i!

�Q

M��l�

��

for l ! �� Q�� n ! �� M�� For Q even� S h�f�DFT ! �I if and only if

dl Al�n�z� " z�P dlQ��AlQ��n�z� !�Q

M��l� � l ! ��

Q

�� n ! �� M��

where dl ! ��l in the odd�stacked case and dl ! in the even�stacked case�

Similarly� for Q odd it follows with �� that T h�f�DFT ! O if and only if

Bl�n�z��!

P��Xi��

�Rn�zW�i�P �E�n��lN�� zW i

�P � ��

" Rn��zW�i�P �E�n��lN�� zW

i�P �� ! � ��

for l ! �� Q� � n ! �� M � � where again � ! � in the odd�stacked case

and � ! in the even�stacked case� For Q even� T h�f�DFT ! O if and only if

dlBl�n�z� " ��az�P dlQ��BlQ��n�z� ! � � l ! �� Q

�� n ! �� M�

with a ! � in the odd�stacked case and a ! in the even�stacked case� For integer

oversampling �Q ! �� and �� simplify respectively to

A��n�z� !P��Xi��

�Rn�zWi�P �En�zW

i�P � "Rn��zW i

�P �En��zW i�P �� !

�

M

and

B��n�z� !P��Xi��

hRn�zW

�i�P �E�n�� zW i

�P � " Rn��zW�i�P �E�n�� zW

i�P �i! � �

for n ! �� M � �

�� PR Conditions in the Dual Polyphase Domain

Finally� it follows from �� that S h�f�DFT ! �I if and only if

�P��Xi��

R�n�iM�z�E �n�iM�zW l

Q� !Q

N��l� � n ! �� N��

and �� implies that T h�f�DFT ! O if and only if

R�n�iM�z�E ��n��iM �zW l

Q� " aP zQR�n�iM�NQ�z�E

��n��iMNQ�zW

lQ� ! �

�� FRAMETHEORETIC ANALYSIS �

for l ! �� Q� � i ! �� P � � n ! �� N�� where aP ! in the odd�

stacked case and aP ! ��P in the even�stacked case� For integer oversampling� these

conditions simplify respectively to

�P��Xi��

R�n�iM�z�E �n�iM �z� !

N� n ! �� N� ��

R�n�iM�z�E ��n��iM�z� " aP z R

�n�iM�N�z�E

��n��iMN�z� ! � �

with i ! �� P � � n ! �� N � �

�� FrameTheoretic Analysis

In this section� we will apply frame theory to oversampled CMFBs� The frame operator

of a CMFB is given by

�SCMx��n� !N ��Xk��

�Xm��

hx� hCMk�mi hCMk�m�n� ��

with N � ! N and hCMk�m�n� ! hCM�ok�m �n� for an odd�stacked CMFB and N � ! �N and

hCMk�m�n� ! hCM�ek�m �n� for an even�stacked CMFB� Our frame�theoretic analysis of CMFBs

will be based on the following fundamental decomposition of the CMFB frame operator�

Theorem �� The frame operator of an odd� or even�stacked CMFB can

be decomposed as

SCM !

��SDFT " TDFT� � ��

Here� SDFT is the frame operator of �respectively� an odd� or even�stacked

DFT FB with �N channels and decimation factor M �

�SDFTx��n� !�N��Xk��

�Xm��

hx� hDFTk�m ihDFTk�m �n� �

and TDFT is given by

�TDFTx��n� !�N��Xk��

�Xm��

ej��kcm hx� hDFTk�m i hDFTk�m �n� �

with hDFTk�m �n�� k� and cm as de�ned in Theorem �� and hDFTk�m �n� !

hDFT�o�N�k��m�n� in the odd�stacked case and hDFTk�m �n� ! hDFT�e�N�k�m�n� in the even�

stacked case�

The proof of this theorem is similar to that of Theorem �� and will be omitted� We

note that this important decomposition has �rst been given for orthogonal continuous�

time Wilson expansions in �� It has been extended to oversampled Wilson frames in

��


We emphasize that SDFT is the frame operator of a DFT FB with the same stacking

type and twice the oversampling factor of the CMFB� Furthermore� the operators SDFT

and TDFT are obtained from the CMFB operators S h�f�DFT in �� and T

h�f�DFT in ��

respectively� by replacing f �n� with hy�n� ! h��n��

SDFT ! S h�hy�DFT and T DFT ! T

h�hy�DFT with hy�n� ! h��n� �

This means that time� frequency� and polyphase domain representations of SDFT and

TDFT can immediately be obtained from the corresponding representations of S h�f�DFT

and T h�f�DFT presented in Subsection �� by replacing f �n�� F �z�� Rn�z�� and R

�n�z� with

h��n�� #H�z�� #En�z�� and #E �n�z�� respectively�

�� Relations between CMFBs and DFT FBs

Based on the above decomposition we shall now show that� under a speci�c condition�

the minimum norm synthesis prototype in an odd� or even�stacked CMFB is equal �up to

a constant factor� to the minimum norm synthesis prototype in the corresponding even�

stacked DFT FB� The minimum norm synthesis FB is of great practical interest since it

minimizes the reconstruction error variance resulting from additive white uncorrelated

noise in the FB�s subbands �Chapter �� We shall furthermore show that under a

speci�c condition the CMFB inherits important numerical properties of the DFT FB�

and that a CMFB is paraunitary if the underlying DFT FB is paraunitary�

Theorem �� Let h�n� and f �n� denote the analysis and synthesis proto�

type� respectively� in an odd�stacked CMFB withN channels and decimation

factor M � or in an even�stacked CMFB with �N channels and decimation

factor �M � Let h�n� be such that fhDFT�ok�m �n�g �for an odd�stacked CMFB�

or fhDFT�ek�m �n�g �for an even�stacked CMFB� is a frame in l��ZZ�� i�e��

ADFTkxk� � hSDFT x� xi � BDFTkxk� � x�n� � l��ZZ� � ��

Furthermore� let h�n� be such that TDFT ! O� Then� the following holds�

�i� the CMFB analysis functions fhCMk�m�n�g constructed from h�n� are a frame

in l��ZZ� with frame bounds ACM ! ADFT�� and BCM ! BDFT�� i�e��

ADFT

�kxk� � hSCM x� xi � BDFT

�kxk� � x�n� � l��ZZ� � ��

�ii� for f �n� ! � �S��DFT�ehy��n� with hy�n� ! h��n�� the synthesis CMFB

ffCMk �n�g constructed from f �n� is the PR synthesis CMFB with minimum

norm �lters�

�� FRAMETHEORETIC ANALYSIS

Proof� Statement �i� follows easily from Theorem �� for TDFT ! O� �� implies

that the CMFB frame operator reduces to SCM ! ��SDFT� Thus� using SDFT ! �SCM

in �� we obtain ��

Statement �ii� will be proved for the odd�stacked case� the proof for the even�

stacked case is similar� For TDFT�o ! O� we have SCM�o ! ��SDFT�o and thus

S��CM�o ! �S��DFT�o� With �� the minimum norm PR synthesis �lters are given

by fCM�ok�m �n� !

�S��CM�oh

CM�ok�m

��n� ! �

�S��DFT�oh

CM�ok�m

��n�� With �� it follows that the

minimum norm PR synthesis �lters are obtained from the analysis �lters as

fCM�ok �n� ! �

�S��DFT�oh

CM�ok��

��n� ! �S��DFT�o

�p� hy�n� cos

��k " ��

Nn" ok

��

!p�hS��DFT�o

nhy�n�W� k��n

�N

oe�j �

ok " S��DFT�o

nhy�n�W k��n

�N

oej �

ok

i�

Now with S��DFT�ofhy�n�W� k��n�N g ! W

� k��n�N �S��DFT�eh

y��n�� this becomes further

fCM�ok �n� !

p�h�S��DFT�eh

y��n�W� k��n�N e�j �

ok " �S��DFT�eh

y��n�W k��n�N ej �

ok

i

!p� � �S��DFT�eh

y��n� cos

��k " ��

Nn� ok

��

Comparing with �� we see that f �n� ! � �S��DFT�ehy��n�� which completes the proof�

�

The following interpretations and conclusions are valid for TDFT ! O�

� Eq� �� implies SCM ! ��SDFT� which means that the CMFB frame operator

reduces to the frame operator of the corresponding DFT FB� Thus� for TDFT ! O

the design of a CMFB reduces to that of a DFT FB of the same stacking type and

with twice the oversampling factor� Since according to �� S��DFT�ehy��n� is the

minimum�norm synthesis prototype of the corresponding DFT FB �even�stacked

or odd�stacked� �� the minimum norm PR synthesis prototype in the CMFB�

f �n� ! � �S��DFT�ehy��n�� is equal �up to a constant factor� to the minimum norm

PR synthesis prototype in the corresponding DFT FB�

� The CMFB frame bounds ACM ! ADFT�� and BCM ! BDFT�� are trivially re�

lated to the frame bounds ADFT and BDFT of the corresponding DFT FB� Since

BCM�ACM ! BDFT�ADFT� the CMFB inherits important numerical properties

�noise immunity� of the corresponding DFT FB �see Chapter �� even though it

has just half the oversampling factor of the DFT FB� This is remarkable� since

usually a decrease of redundancy leads to a deterioration of the numerical prop�

erties of a frame�

� In particular� if the DFT FB is paraunitary �which means ADFT ! BDFT or� equiv�

alently� SDFT ! ADFTI �� then the corresponding CMFB is paraunitary

as well �ACM ! BCM� SCM !ADFT

�I��


All these results hinge on the condition TDFT ! O� As noted further above� TDFT !

O is equivalent to the second PR property in Theorem �� T h�f�DFT ! O� with f �n�

replaced by h��n�� This means� in particular� that all conditions for T h�f�DFT ! O

formulated in Subsection ��'�� in the time� frequency� and polyphase domains can

immediately be reformulated as conditions for TDFT ! O� For integer oversampling� a

simple su�cient condition for TDFT ! O is stated in the following theorem�

Theorem �� For odd�stacked CMFBs with arbitrary integer oversampling

factor P � and for even�stacked CMFBs with odd P � the symmetry property

h�� " ��l " �PM � n� ! h�n� �with some l � ZZ� ��

is su�cient for TDFT ! O�

Proof� Using �� with f �n� ! h��n� shows that TDFT ! O if and only if

�Xi��

bi h��n " iPM � h�n " iPM " � ! � �

where bi ! ��i for odd�stacked CMFBs and bi ! ��iP for even�stacked CMFBs�

Now� for odd�stacked CMFBs with arbitrary integer P and for even�stacked CMFBs

with odd P � we have bi ! ��i and thus we have to show that

A�n��!

�Xi��

��i h��n " iPM � h�n " iPM " � ! ��

Inserting �� h�n� ! h��"��l"�PM�n�� and subsequently substituting �l"�i !j� we obtain

A�n� !�X

i��i h�n" ��l " � i�PM " � h��n " ��l " � i�PM �

! ��X

j��j h��n " jPM � h�n" jPM " � �

which when compared to �� is seen to be �A�n�� Thus A�n� ! �A�n�� which implies

A�n� ! ��

It can be shown that if h�n� is the prototype of a critically sampled� paraunitary

CMFB with parameter and� more speci�cally� h�n� satis�es the symmetry property

�� with a certain l� then the scaled prototypepPh�n� induces a paraunitary CMFB

also in the case of integer oversampling �with oversampling factor P as restricted in

Theorem �� provided that in the oversampled case is replaced by � ! � ��l "

��P � �M �

Condition �� implies that h�n� has linear phase� Thus� in the case of integer

oversampling� PR �with linear phase �lters in the case of an even�stacked CMFB� is


achieved by choosing h�n� according to �� and using f �n� ! � �S��DFT�ehy��n�� In

particular� the CMFB will then be paraunitary if SDFT ! �I �cf� ��

In certain special cases� the expressions for SDFT and the calculation of the minimum�

norm synthesis prototype using S��DFT�e simplify considerably� This will be discussed in

Subsections ��

�� Paraunitarity Conditions

It is easily seen that a CMFB �odd�stacked or even�stacked� oversampled or critically

sampled� is paraunitary if and only if it satis�es PR and f �n� ! h��n�� This implies

that the two PR conditions of Theorem �� with f �n� replaced by h��n�� are necessaryand su�cient for paraunitarity� Thus� an odd� or even�stacked CMFB is paraunitary if

and only if

S h�hy�DFT ! SDFT ! �I and T

h�hy�DFT ! TDFT ! O with hy�n� ! h��n� �

��

Furthermore� all PR conditions formulated in Subsections ��'�� become parauni�

tarity conditions if f �n�� F �z�� Rn�z�� and R�n�z� are replaced by h��n�� #H�z�� #En�z��

and #E �n�z�� respectively�

We recall that a su�cient condition on h�n� for the second paraunitarity condition�

T h��h�DFT ! O� was provided in Theorem ��

�� Time�Limited Prototype

If h�n� has �nite length � �N �recall that the corresponding DFT FB has �N channels��

the frame operator SDFT is a multiplication operator in the time domain �see Subsection

�� i�e�

�SDFTx��n� ! � bh�n� x�n� � �S��DFTx��n� !x�n�

� bh�n��

with bh�n� as de�ned in Subsection ��

bh�n� ! N�X

m��jh��n "mM �j��

For TDFT ! �� the frame bounds are given �independently of the stacking type� by

ACM ! minn��M�� bh�n� and BCM ! maxn��M�� bh�n�� for bh�n� � the CMFB is

paraunitary with frame bound ACM ! �


�� Band�Limited Prototype

If h�n� is band�limited in a frequency interval of length � �M � SDFT is a multiplication

operator in the frequency domain �see Subsection �� i�e��

� �SDFTX��ej�� ! �Gh��X�ej�� S��DFTX��ej�� !

X�ej��

�Gh��

with Gh�� de�ned as

Gh��!

�M

�N��Xk��

��H�ej�� k�N��

��

where �� ! � in the even�stacked case and �� !��N

in the odd�stacked case� For TDFT !

O� the frame bounds are given by ACM ! inf�� Gh�� and BCM ! sup�� Gh��

they are independent of the stacking type since for the two stacking types Gh�� di�ers

merely by a frequency shift� For TDFT ! O and Gh�� the CMFB is paraunitary

with ACM ! �

�� Integer Oversampling

In the case of integer oversampling� N ! PM � SDFT is a multiplication operator in

the polyphase domain �see Subsec� �� i�e�� the polyphase components Yn�z� !P�m�� SDFTx��n"mM � z�m are given by

Yn�z� ! �Gn�z�Xn�z�

with

Gn�z� !M

�

P��Xi��

h#En�zW

iv�P �En�zW

iv�P � " #En��zW iv

�P �En��zW iv�P �

i� ��

where v ! �� in the odd�stacked case and v ! � in the even�stacked case� For TDFT !

O� the frame bounds of the CMFB are given by ACM ! inf�� n��M�� Gn�ej�� and

BCM ! sup�� n��M�� Gn�ej�� they are independent of the stacking type� The

CMFB is paraunitary with ACM ! if TDFT ! O and Gn�z� � �n ! �� M��

this condition turns out to be the same for both stacking types�

Furthermore� it follows from �� that in the polyphase domain TDFT is essentially

a multiplication operator� i�e�� the polyphase components Tn�z� !P�

m�� T DFTx��n"

mM � z�m are given by

Tn�z� ! �Dn�z�X�n��z�where

Dn�z� ! ��r M�

P��Xi��

h#En�zW

�iv�P �E�n��zW iv

�P �

" #En��zW�iv�P �E�n��zW iv

�P �i�

with v ! �� and � ! � in the odd�stacked case and v ! � and � ! in the



�� Construction of Paraunitary Prototypes

In this subsection we shall describe a method for the construction of paraunitary CMFBs

from nonparaunitary CMFBs� This method is based on the approach described in

Theorem ��

Theorem �� Let h�n� be the analysis prototype of a �non�paraunitary�

even�stacked or odd�stacked CMFB providing a frame expansion with

TDFT ! O� Then the CMFB with analysis prototype

h p��n� !p� �S

��DFT�eh

y��n�and synthesis prototype f �n� ! h p�

�

��n� ! p��S

��DFT�eh

y��n� is paraunitary

with frame bound ACM ! �

Proof� From Corollary A� � we know that applying S��CM to each of the frame func�

tions hCMk�m �n� yields a tight frame with A ! � We shall �rst provide the proof for the

odd�stacked case and then outline the di�erences to the even�stacked case� Let us de�ne

the time�frequency shift operatorW ok�m as �W o

k�mx��n� ! x�n�mM �W� k�� n�mM��N �

With SCM�o ! ��SDFT�o and �� we then get hCM�o

pk�m�n� !

p��S

��DFT�oh

CM�ok�m ��n� !

�S��DFT�oW

ok�mh

y��n�e�j�ok " �S

��DFT�oW

o�N�k��mh

y��n�ej�ok � Now� using S

��DFT�oW

ok�m !

Wok�mS

��DFT�e we get hCM�o

pk�m�n� ! ��S

��DFT�eh

y��n �mM � cos� k��

N�mM � n� " ok

�and

hence hyp�n� !p��S

��DFT�eh

y��n�� which completes the proof for the odd�stacked case�

In the even�stacked case� the proof is similar using S��DFT�eW

ek�m ! W

ek�mS

��DFT�e where

�W ek�mx��n� ! x�n�mM �W

�k n�mM��N � �

The above construction of hp�n� simpli�es in the special situations considered in

Subsections ��'�� If h�n� has length � �N and satis�es TDFT ! O� then

h p��n� !p� �S

��DFT�eh

y��n� ! h�n�qbh��n�

with bh�n� de�ned in Subsection �� If h�n� is band�limited with bandwidth � �M �

then the Fourier transform of h p��n� is given by

H p��ej�� !p� � �S

��DFTH�� !

H�ej��qGh��

with Gh�� de�ned in Subsection �� Finally� for integer oversampling the polyphase

components of h p��n�� E p�n �z� !

P�m�� h p��mM � n� z�m� are obtained from the

polyphase components En�z� of h�n� as

E p�n �z� !

En�z�

Pn�z��

where Pn�z� is a �positive square root� of Gn�z� �see �� in the sense that

Pn�z� #Pn�z� ! Gn�z� with Pn�ej�� and #Pn�z� ! Pn�z� �see Subsection ��


�� Design Methods

In this section� we shall discuss three design methods for paraunitary� odd� or even�

stacked CMFBs with oversampling� For a paraunitary CMFB� there is f �n� ! h��n�and hence the CMFB design reduces to the design of the analysis prototype h�n�� We

here assume h�n� to be real�valued�

�� Design via Constrained Optimization

Our �rst design method minimizes

Ch�!Z ��

�jH�ej��j�W �ej�� d��

where W �ej�� is a nonnegative weighting function� subject to the paraunitarity side

constraints TDFT ! O and SDFT ! �I which are quadratic in h�n��

For the sake of simplicity� let us assume integer oversampling with arbitrary �integer�

oversampling factor P in the odd�stacked case and odd P in the even�stacked case�

Then� the symmetry property �� is su�cient for T DFT ! O� and it remains to

satisfy SDFT ! �I� �With arbitrary oversampling� T DFT ! O would have to be included

as a further side constraint�� Assuming a real�valued FIR prototype h�n� of length

L ! " ��l " �PM " with some l �ZZ� the assumed symmetry of h�n� implies that

onlyK samples of h�n� have to be determined� where K ! L�for L even and K ! L�

�for

L odd� Collecting these samples in the vector h� the cost function �� can be written

as the quadratic form Ch ! hT Ph �� with superscript T denoting transposition�� Here

P is a K � K matrix whose elements depend on the weighting function W �ej��

Hence� our optimization problem reduces to the minimization of Ch ! hTPh under the

side constraint SDFT ! �I�

Substituting f �n� ! h��n� ! h��n� in the time�domain PR condition �� the

paraunitarity side constraint SDFT ! �I is formulated as

�Xm��

h�n"mM � h�n " �m" �lP �M � !

PM��l� for n � ZZ� ��

For an FIR prototype h�n�� this amounts to a �nite number of scalar equations since

h�n "mM � and h�n " �m " �lP �M � will overlap only for a �nite number of di�erent l

values� We note that the side constraint SDFT ! �I can equivalently be formulated in

the frequency� polyphase� and dual polyphase domains �cf� Subsections ��'��

Fig� � presents a design example that has been obtained using the constrained

optimization discussed above� with W �ej�� ! for � � �� and W �ej�� ! �

else� The length of the prototype is L ! �� Fig� ��a� shows an odd�stacked CMFB

with N ! �� M ! � while Figs� ��b��c� show the �lters Hk�z� �k ! �� N�

�� DESIGN METHODS �

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Figure �� Design of paraunitary CMFBs via constrained minimization of stopband

energy� a Odd�stacked CMFB with N ! �� and M ! �� b�c even�stacked CMFB

with �N ! �� and �M ! �� b �lters Hk�z� k ! �� N� c �lters (Hk�z�

k ! � �� N � �

and (Hk�z� �k ! � �� N � �� respectively� in an even�stacked CMFB with �N ! ��

�M ! �� In both cases the oversampling factor is P ! �� Both CMFBs use the same

prototype since the paraunitarity condition �side constraint� SDFT ! �I is independent

of the stacking type�

We conclude this subsection with design examples that demonstrate the impact of the

increased design freedom in oversampled FBs on the achievable �lter quality �stopband

attenuation�� Figs� � and � show odd�stacked and even�stacked CMFBs with critical

sampling and oversampling� respectively� These CMFBs were again obtained via con�

strained optimization as discussed above� The prototype length was L ! �� and the

weighting function was chosen as W �ej�� ! for �� and W �ej�� ! �

else� One can observe that in the oversampled case the stopband attenuation is about

��dB higher than in the critically sampled case� This can be attributed to the reduced

e�ective number of PR�paraunitarity side constraints that have to be satis�ed in the

oversampled case�

�� Linearized Design Method

The optimization method presented in this subsection is based on a similar method for

the design of odd�stacked CMFBs with critical sampling and near�PR introduced in

�� We shall extend the method discussed in �� to a constrained optimization of

paraunitary� oversampled� even�stacked or odd�stacked CMFBs�

The optimization method discussed in Subsection �� is very time�consuming be�

cause of the quadratic side constraints �� Adopting an iterative approach� we may

CHAPTER �� OVERSAMPLED COSINE MODULATED FILTER BANKS

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Figure �� Design of critically sampled paraunitary CMFBs� a Odd�stacked CMFBwith N ! � and M ! �� b even�stacked CMFB with �N ! �� and �M ! �� upper

plot� �lters Hk�z� k ! �� N� lower plot� �lters (Hk�z� k ! � �� N��

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Figure �� Design of oversampled paraunitary CMFBs� a Odd�stacked CMFB withN ! � and M ! �� b even�stacked CMFB with �N ! �� and �M ! � upper plot�

�lters Hk�z� k ! �� N� lower plot� �lters (Hk�z� k ! � �� N��

linearize �� according to

chk�hk��l�

�! MP

�Xi��

hk��n� iN � hk�n� �i " �l�N � ! ��l� � ��

where k denotes the iteration index �k ! � �� This is indeed linear in the current

impulse response hk�n�� The kth iteration of the constrained optimization now goes as

follows�

� Minimize Ch ! hTPh subject to the linear side constraint ��

hk ! arg minh

hTPh subject to chk�hk��l� ! ��l� �

Note that hk depends on hk�� via the side constraint�

�� DESIGN METHODS

�� If khk � hk��k � �� where � is a prescribed tolerance level� stop the iteration�

The result is hk�� Otherwise� set hk � �hk " � � ��hk�� with some �xed �

between � and � and go back to Step with k � k " �

This iteration is initialized using a linear�phase lowpass prototype h� that is designed

by means of a standard method such as the Remez exchange algorithm� It should be

noted that the iteration stops only when hk hk�� in which case �� is a good

approximation to the true paraunitarity condition �� and thus paraunitarity is ap�

proximately satis�ed� By choosing the tolerance level � su�ciently small� paraunitarity

can be achieved with arbitrary accuracy�

This algorithm can be expected to converge once that the prototype changes little in

each iteration� Otherwise� the approximation of the paraunitarity side constraint will be

poor and the algorithm may not converge� In our design experiments� we observed that

� ! �� as recommended in �� yields good convergence properties in most cases�

however� in some cases the algorithm did not converge at all� Especially for critically

sampled CMFBs we observed convergence problems�

We conclude this section with some design examples� Fig� ��a� shows a paraunitary

odd�stacked CMFB with N ! channels and decimation factor M ! � �oversampling

by �� In Figs� ��b� and �c� a paraunitary even�stacked CMFB with �N ! channels

and decimation factor �M ! � �oversampling by �� is depicted� Both CMFBs use the

same prototype �lter �length L ! �� since the paraunitarity condition SDFT ! �I

is independent of the stacking type �see Section �� The cost function was chosen

as Ch !��R�s

jH�ej��j� �� d� with �s ! �� Straightforward calculations show that

this cost function can easily be expressed as a quadratic form Ch ! hT Ph� In this

example� the tolerance value was � ! �� Thus� the resulting prototype is an excellent

approximation to a �true� paraunitary prototype� Observe that the prototype has about

��dB stopband attenuation�

Fig� ��a� shows a paraunitary odd�stacked CMFB with N ! � channels and deci�

mation factor M ! � �oversampling by �� In Figs� ��b� and �c� a paraunitary even�

stacked CMFB with �N ! �� channels and decimation factor �M ! � �oversampling by

�� is depicted� Again� both CMFBs use the same prototype �lter �length L ! �� The

cost function was chosen as Ch !��R�s

jH�ej��j� �� d� with �s ! �� The tolerance value

was � ! �� Note that the prototype �lter has about ��dB stopband attenuation�

�� Lattice Design

The lattice design method for CMFBs has been introduced in � � for critically sampled�

paraunitary� odd�stacked CMFBs� We shall next outline an extension of this method


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Figure �� Design of paraunitary CMFBs via linearized method L ! ��

a Odd�stacked CMFB with N ! and M ! �� b�c even�stacked CMFB with

�N ! and �M ! �� b �lters Hk�z� k ! �� N� c �lters (Hk�z�

k ! � �� N � �

to odd� and even�stacked� paraunitary CMFBs with integer oversampling� As before�

we assume that TDFT ! O�

For integer oversampling� the paraunitarity condition SDFT ! �I can be formulated

in the dual polyphase domain as the �power�complementarity condition� �cf� ��

with R�n�z� replaced by #E �n�z��

�P��Xi��

#E �n�iM�z�E �

n�iM�z� !

PM� n ! �� PM��

In the special case of critical sampling �P ! �� this simpli�es to

#E �n�z�E

�n�z� " #E �

n�M�z�E �n�M�z� !

M� n ! �� M��

This condition has �rst been formulated in � � for critically sampled odd�stacked

CMFBs� and a lattice structure has been proposed for implementing �lters satisfying

it�

An extension of this lattice design method to integer oversampling and arbitrary

stacking type is obtained by substituting i ! l"�j �with l ! �� and j ! �� P��in the general condition �� which gives

P��Xj��

h#E �n�j�M�z�E �

n�j�M�z� " #E �n�j�M�M�z�E �

n�j�M�M�z�i!

PM

for n ! �� PM � � This condition can be satis�ed by using the original lattice

method � � to design� for each n� P polyphase �lter pairs fE �n�j�M�z�� E �

n�j�M�M�z�g�j ! �� P�� that satisfy a power complementarity condition of the simple type

�� EFFICIENT IMPLEMENTATION

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

Mag

nitu

de R

espo

nse

(dB

)

(a)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−200

−150

−100

−50

0

(c)

Mag

nitu

de R

espo

nse

(dB

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−200

−150

−100

−50

0

Mag

nitu

de R

espo

nse

(dB

)

(b)

Figure �� Design of paraunitary CMFBs via linearized method L ! ��

a Odd�stacked CMFB with N ! � and M ! �� b�c even�stacked CMFB with

�N ! �� and �M ! �� b �lters Hk�z� k ! �� N� c �lters (Hk�z�

k ! � �� N � �

�� i�e��

#E �n�j�M�z�E �

n�j�M�z� " #E �n�j�M�M�z�E �

n�j�M�M�z� !

P �M�

Imposing the symmetry property �� on h�n� to achieve TDFT ! O results in a

symmetry relation of the polyphase components� which reduces the e�ective number of

polyphase �lters to be designed� For details on the impact of prototype symmetry on the

polyphase �lters the interested reader is referred to � �� The advantage of this lattice

design method is that paraunitarity is inherently satis�ed and need not be included as

a side constraint�

�� E�cient Implementation

This section proposes e�cient DCT�DST�based implementations of oversampled and

critically sampled CMFBs of either stacking type�

�� Implementation of Even�Stacked CMFBs

We �rst extend the DCT�based implementation of critically sampled Lin�Vaidyanathan

FBs �� to arbitrary even�stacked CMFBs with integer oversampling� Let us de�ne

the �N �dimensional analysis �lter vector

h�z��! �H��z� H��z� � � �HN�z� (H��z� (H��z� � � � (HN��z��

T �


z 1

Ce

S

z r

z s

M

2

M

2

M

2

2

2

2

2

x[n]

z 1

z 1

v0 [m]

v1 [m]

vN-1 [m]

vN [m]

v1 [m]

v2 [m]

vN-2 [m]

vN-1 [m]

z

z

z

E2N 1 z2P

E0 z2P

E1 z2P

Figure �� Polyphase structure for even�stacked CMFBs analysis part with integer

oversampling�

Inserting the dual polyphase decomposition

H�z� !�N��Xn��

E �n�z

�N � zn ��

in �� and �� we obtain after simple manipulations

h�z� ! D�z��Ce

S

�G�z�N � e�z� � ��

Here� D�z� is a �N � �N diagonal matrix de�ned as

D�z��! diag

nz�rM � � � �� z �

N� times

� z�sM� z�M z�M � � � z�M� �z �N� times

o�

Ce is an �N " �� N matrix de�ned as

Ce !

��

� � � c�� c�� c�� c��N��

��

��

cN�� cN�� cN�� cN��N��

�� with ck�l !

p� cos

�k�

Nl � ek

��

S is an �N�� N matrix de�ned as

S !

��

s�� s�� s��N��s�� s�� s��N��

��

��sN�� sN�� sN��N��

�� with sk�l ! �

p� sin

�k�

Nl � ek

��

�� EFFICIENT IMPLEMENTATION �

z 1

Ce

S

M

2

M

2

M

2

x[n]

z 1

z 1

v0 [m]

v1 [m]

vN [m]

v1 [m]

v2 [m]

vN-1 [m]

z

z

z

2

2

2

E2N 1 z2P

E0 z2P

E1 z2P

Figure � � Polyphase structure for Lin�Vaidyanathan CMFB analysis part with

integer oversampling�

G�z� is a �N � �N diagonal matrix de�ned as G�z� ! diag fE �n�z�g�N��n�� and e�z� !

� z � � � z�N��T � Straightforward manipulation of �� using noble identities ��

leads to an e�cient polyphase structure of the analysis part of the CMFB that is shown

in Fig� �� Here (vk�n� denotes the subband signals corresponding to the �lters (Hk�z�

with k ! � �� N � � This implementation requires �N polyphase �lters plus the

matrix multiplications by Ce and S� these matrix multiplications can be implemented

using fast DCT and DST algorithms �� For ! � and r ! s ! � �Lin�Vaidyanathan

CMFB�� further simpli�cations are possible that yield the implementation depicted in

Fig� � This extends the implementation in �� to the case of integer oversampling�

Finally� an analogous derivation leads to similarly e�cient implementations of the

synthesis FB�

�� Implementation of Odd�Stacked CMFBs

The e�cient implementation of critically sampled odd�stacked CMFBs has previously

been studied in �� These implementations will now be extended to integer

oversampling �N ! PM��

Inserting the dual polyphase decomposition �� in �� and proceeding sim�

ilarly as in Subsection �� it follows that the analysis �lter vector h�z� !

�H��z� H��z� � � � HN��z��T can be written as

h�z� ! CoG��z�N � e�z� �where Co is an N � �N matrix de�ned as �Co�k�l !

p� cos

� k��

Nl � ok

�� G�z� !


Co

M

M

M

v0 [m]

v1 [m]

vN-1 [m]

z

z

z

x[n]

E1 z2P

E0 z2P

E2N 1 z2P

Figure �� Polyphase structure for odd�stacked CMFBs analysis part with integer

oversampling�

diag fE �n�z�g�N��n�� and e�z� ! � z � � � z�N��T � The use of noble identities and other

manipulations yield the e�cient polyphase structure shown in Fig� � This implemen�

tation requires �N polyphase �lters plus the matrix multiplication by Co� this matrix

multiplication can be implemented using fast DCT algorithms �� We note that a

similarly e�cient implementation exists for the CMFB�s synthesis part�

�� General Oversampling

The particularly e�cient polyphase structures proposed in Subsections �� and ��

can be used only in the case of integer oversampling or critical sampling� For general

oversampling� CMFBs still allow a fairly e�cient DCT�DST�based implementation if

FIR prototypes are used� For example� the subband signals in an oversampled or

critically sampled� odd�stacked CMFB can be written as

vk�m� ! hx� hCM�ok�m i !

p�

L��Xn��

x�mM � n� h�n� cos

��k " ��

Nn" ok

��

where L is the length of the FIR analysis prototype h�n�� Hence� the subband signals

can be e�ciently calculated by applying a fast DCT algorithm to x�n� locally windowed

by h�n�� A DCT�based implementation exists also for the synthesis part� Further�

more� a similar DCT� and DST�based implementation can be developed for even�stacked

CMFBs with arbitrary oversampling� The blockwise operation of the overall algorithm

is described in ��

�� Image Coding with EvenStacked CMFBs

It is well known in subband image coding that linear phase �lters are of particular

importance� because nonlinear phase �lters lead to undesirable artifacts in the recon�

structed image �� Since the new class of even�stacked CMFBs allows for both linear

phase �lters in all channels and PR� it is a promising candidate for use in subband image

�� IMAGE CODING WITH EVENSTACKED CMFBS �

CMFBQuantizer

CMFB-based encoder

specificationTable

Entropyencoder

Tablespecification

Compressedimage dataSource

image data

even-stacked

�a�

image data

Compressedimage data

decoderEntropy

Dequantizer

specificationTable Table

specification

CMFB-based decoder

even-stackedCMFB

Reconstructed

�b�

Figure �� Subband image coding based on even�stacked CMFB� a encoder�

b corresponding decoder�

coding� The special structure of even�stacked CMFBs requires� however� a modi�cation

of some of the elements used in the image coding standard JPEG� In this section� we

shall describe these modi�cations and we shall demonstrate the perceptual performance

and the rate�distortion performance of even�stacked CMFBs as compared to �nonlinear

phase� odd�stacked CMFBs� In particular� we show that subband image coding schemes

based on even�stacked CMFBs and entropy coding achieve a comparable rate�distortion

performance as odd�stacked CMFBs and hence also the DCT�based JPEG standard�

�� Subband Image Coder Based on Even�Stacked CMFBs

Fig� �� shows the encoder and the corresponding decoder� The encoder consists of

a transform coding part �even�stacked CMFB�� a scalar quantizer� and an entropy

�Hu�man� coder� For the sake of simplicity� we restrict our attention to separable two�

dimensional� even�stacked CMFBs� Nevertheless� many of the results presented in this

section carry over to the nonseparable case� The transform coder �rst applies an even�

stacked CMFB to the rows of the image and then to the columns of the result� The

transform coe�cients �subband signals� are then quantized using a quantization matrix


�a� �b�

Figure �� a Typical amplitude distribution in an � DCT�coded block�

b typical amplitude distribution in a block of subband signals that result from

an even�stacked CMFB�

that has been obtained by modi�cation of the quantization matrix used in the JPEG

standard �� see below�� The quantized coe�cients are zig�zag scanned and entropy

coded using a Hu�man coder� The decoder reverses these steps�

The special structure of even�stacked CMFBs �see Section �� requires a modi�ca�

tion of the quantization matrix and the zig�zag sequence used in the JPEG standard�

We shall next describe these modi�cations� According to Fig� ��b�� an even�stacked

CMFB has two �lters for each subband� i�e�� the �lters Hk�z� and (Hk�z� cover the

same frequency band� In order to illustrate this di�erence from the DCT and the odd�

stacked case� typical amplitude distributions in a DCT�coded block and in a block of

subband signals resulting from an even�stacked CMFB are shown in Fig� ��a� and

��b�� respectively� Here� Fig� ��b� corresponds to an analysis �lter vector

h�z� ! �H��z� H��z� � � �HN��z� (H��z� (H��z� � � � (HN��z� HN�z��T � ��

The reason for this speci�c ordering �note the position of HN�z� within h�z�� will be

explained further below� From Fig� ��b�� we see that the speci�c �lter ordering in

�� yields � subblocks of equal size N � N � whose amplitude distributions resemble

that in one DCT�coded block�

Quantization matrix� The quantization matrix accounts for the fact that lower

frequencies are quantized more accurately than higher frequencies� The entries of the

quantization matrix correspond to quantization stepsizes �i�e�� a larger value corresponds

to a rougher quantization�� Based on the amplitude distribution in Fig� ��b�� we now

propose a quantization matrix for even�stacked CMFBs� Consider an �N " �� N "


��

��

��

��

��

��

��

��

��

��

�a��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�b�

Figure �� a Quantization matrix used in the DCT�based JPEG

standard blocksize� � b quantization matrix used in a subband image coder

based on a ��channel even�stacked CMFB�

� quantization matrix Qo for an �N " ��channel odd�stacked CMFB� �Qo can be

obtained by suitable linear interpolation or decimation of the entries in the � JPEG

quantization matrix�� Then� we de�ne the �N � �N quantization matrix for an even�

stacked CMFB with �N channels as

Qe !

��A B

C D

��

with the N � N submatrices �A�k�l ! �Qo�k�l� �B�k�l ! �Qo�k�l�� C�k�l ! �Qo�k��l� and

�D�k�l ! �Qo�k��l� where k� l ! �� N�� Fig� ��a� shows the quantization matrix

used in the JPEG standard� It is clearly seen that lower frequencies �located in the

upper left corner� are quantized more accurately than higher frequencies �located in

CHAPTER �� OVERSAMPLED COSINE MODULATED FILTER BANKS

(1,1)0,0C 0,0

(1,1)C

(0,1)0,0CC

0,1C

C(0,0)C

(1,0)0,1C

(0,0)0,0 0,1

(0,1)0,1

(1,0)

�a� �b�

Figure �� a Zig�zag sequence used in a DCT�coded block� b zig�zag sequence used

in a block of subband signals that result from even�stacked CMFBs�

the lower right corner�� Fig� ��b� shows a quantization matrix Qe for use with a

��channel even�stacked CMFB� The four � submatrices A�B�C�D can easily be

distinguished�

Zig�zag scanning� In the DCT�based JPEG standard� the quantized transform

coe�cients are zig�zag scanned before entropy encoding� This produces long runs of

zeros which can be encoded very e�ciently� For even�stacked CMFBs� the fact that the

�lters Hk�z� and (Hk�z� for given k � f� �� N�g cover the same frequency band

requires a modi�ed zig�zag sequence since standard zig�zag scanning would result in

signi�cantly shorter runs of zeros�

The speci�c �lter ordering in �� yields � subblocks of equal size N � N � whose

amplitude distributions resemble that in one DCT�coded block �cf� Fig�� This sug�

gests to de�ne the modi�ed zig�zag sequence by interleaving the zig�zag scanned values

of the four N � N subblocks �see Fig� ��b�� That is� the modi�ed zig�zag sequence

for even�stacked CMFBs is

s ! �C �� C

�� C

�� C

�� C

�� C

�� C

�� C

��

where the superscript i�j� �i� j ! �� and subscript k�l �k� l ! �� N�� in C i�j�k�l

identify the subblock and the frequency index� respectively�

An alternative ordering of the analysis �lters is

h�z� ! �H��z� H��z� (H��z� �� HN��z� (HN��z� HN�z��T � ��

�� IMAGE CODING WITH EVENSTACKED CMFBS

�a� �b�

Figure �� Subband image coding using the DCT� a original image �Bridge��

b �Bridge� coded at �� bpp PSNR��dB using a DCT of block size �

This ordering� like the one in �� requires a suitable modi�cation of the quantiza�

tion matrix� However� there is no need for a modi�ed zig�zag sequence� i�e�� a standard

JPEG�type zig�zag scanning can be performed on the resulting �N � �N block of sub�

band signals� We observed that both orderings yield similar performance� Note that

performing a standard zig�zag scanning �without the above interleaving or the alterna�

tive ordering of the �lters as in �� in an even�stacked CMFB will lead to a deteri�

oration of the rate�distortion properties of the coder� since the resulting runs of zeros

will be signi�cantly shorter than those obtained using the modi�ed zig�zag sequence�


We conclude this section with a coding experiment demonstrating the perceptual per�

formance and the rate�distortion performance of even�stacked CMFBs� Fig� ��a� shows

the image �Bridge�� The result of coding this image using the DCT�based JPEG stan�

dard at �� bpp �PSNR!�� dB� is depicted in Fig� ��b� and seen to exhibit

heavy blocking artifacts due to the very low bitrate� Fig� ��a� shows �Bridge� coded

at �� bpp �as before� PSNR!�� dB� using an odd�stacked CMFB with N !

channels and �lter length L ! �� Here� blocking artifacts are not visible but the image

appears blurred� this is essentially due to the nonlinear phase of the �lters� Finally�

Fig� ��b� shows �Bridge� coded at �� bpp �again� PSNR!�� dB� using an even�

stacked CMFB with �N ! � channels and the same prototype as in the odd�stacked

CMFB� Here� the details appear much sharper than in Fig� ��a�� Thus� for equal


�a� �b�

Figure �� Subband image coding using CMFBs� a �Bridge� coded at

�� bpp PSNR��dB using an odd�stacked CMFB �N ! M ! ��

b �Bridge� coded at �� bpp PSNR��dB using an

even�stacked CMFB ��N ! �M ! ��

PSNR� the even�stacked CMFB outperforms the odd�stacked CMFB and also the DCT

from a perceptual point of view�

Fig� �� demonstrates that even�stacked CMFBs achieve comparable rate�distortion

performance as the DCT�based JPEG standard and as odd�stacked CMFBs� Note�

however� that the rate�distortion properties of even�stacked and of odd�stacked CMFBs

will be comparable only if the even�stacked CMFB uses twice the number of channels as

the odd�stacked CMFB� This is so� since in an even�stacked CMFB with �N channels

there are e�ectively N di�erent channels �frequencies�� Furthermore� the bandwidth of

the �lters in a �N �channel even�stacked CMFB is equal to the bandwidth of the �lters in

an N �channel odd�stacked CMFB� Since the rate�distortion performance of the subband

coders investigated in this section depends critically on the bandwidth of the channel

�lters �the channels are coded mutually independent�� it is not surprising that the rate�

distortion performance of an N �channel odd�stacked CMFB will be similar to that of a

�N �channel even�stacked CMFB� We emphasize that� even though the rate�distortion

performance of even�stacked CMFBs is not better than that of odd�stacked CMFBs or

the DCT� even�stacked CMFBs have a clearly improved perceptual performance due to

their linear phase property�


DCT

odd−stacked CMFB

even−stacked CMFB

0 0.5 1 1.516

18

20

22

24

26

28

30

bpp

PS

NR

/ dB

Figure �� Rate�distortion characteristics of the DCT and odd�and even�stacked

CMFBs for image �Lena��

Chapter �

Noise Analysis and Design Freedom

in Oversampled Filter Banks

In this chapter we show that oversampled FBs have noise reducing properties� We

present a noise analysis and we derive bounds on the variance of the reconstruction error

caused by noisy subband signals� We discuss the dependence of the reconstruction error

on the oversampling factor and we show that the frame bound ratio B�A �see Sec� ��

serves as a condition number of the FB�

We furthermore show that there exists an important tradeo� between noise reduc�

tion and design freedom in oversampled FBs� We prove that the synthesis FB corre�

sponding to the para�pseudo�inverse of the analysis polyphase matrix minimizes the

reconstruction error variance resulting from additive white uncorrelated noise in the

FB�s subbands� We furthermore derive the synthesis FB minimizing the reconstruc�

tion error variance for arbitrary perturbation noise statistics� Using other synthesis

FBs �generalized inverses of the analysis FB� leads to an additional noise component

that often signi�cantly increases the reconstruction error� We present a design example

demonstrating the tradeo� between noise reduction and design freedom in oversampled

FBs�

Our results are based on a signal space framework that can be applied to redun�

dant representations in general� In Section �� we shall �rst apply this signal space

framework to oversampled analog�to�digital �A�D� conversions� since here the under�

lying spaces have a very simple structure� We interpret oversampled A�D conversions

as continuous�time frame expansions� Based on this viewpoint� we then establish the

tradeo� between noise reduction and design freedom arising in oversampled A�D conver�

sions� In Section �� we then proceed with an extension of this signal space framework

to oversampled FBs�

��

�� CHAPTER �� NOISE ANALYSIS

�� Noise Analysis and Design Freedom in Over

sampled A D Conversion

The conversion from continuous or analog signals to digital representations is usually

called analog�to�digital conversion �A�D conversion�� It is well known that a band�

limited continuous�time signal can be perfectly recovered from its sampled version if

the sampling rate is twice the bandwidth of the analog signal �� So� in theory there

is no need to oversample the analog signal� In practice� however� oversampling methods

have recently become very popular because they avoid many of the practical di�culties

encountered with conventional methods for A�D conversion� Conventional converters

have attributes that make their implementation in VLSI technology di�cult� The use of

analog �lters� the need for high precision analog circuits� and the vulnerability to noise

and interference render conventional converters inferior to oversampling converters� The

advantage of conventional methods is a relatively low sampling frequency� usually close

to the Nyquist rate� i�e�� twice the signal bandwidth� Oversampling converters can

use simple analog components with relatively high tolerance� but they require fast and

complex digital signal processing stages ��

In the following� we interpret critically sampled and oversampled A�D conversions as

frame expansions� Based on this interpretation� we develop a signal space framework

that will serve to establish an important tradeo� between noise reduction and design

freedom in oversampled A�D conversion�

�� A�D Conversion as a Frame Expansion

Let us consider a band�limited continuous�time signal x�t� with bandwidth �counting

positive frequencies only� B�� From the sampling theorem �� we know that this sig�

nal can be perfectly recovered from its samples taken uniformly at a rate Fs � �B��

The cases Fs ! �B� and Fs � �B� are referred to as critical sampling �sampling at

the Nyquist rate� and oversampling� respectively� We shall next provide an interpreta�

tion of critically sampled and oversampled A�D conversion as a continuous�time frame

expansion�

Frame expansion� The signal space under consideration is the Hilbert space H of

B��band�limited functions� Due to the sampling theorem any signal x�t� � H can be

represented as ��

x�t� !

K

�Xk��

x�kT � sinc��B��t� kT ��

where sinc�t� ! sin tt� T ! �

Fswith Fs � �B� and K ! Fs

�B�is the oversampling factor�

The samples x�kT � can be written as the inner product of the signal x�t� with the

�� NOISE ANALYSIS AND DESIGN FREEDOM ��

functions gk�t� ! �B� sinc��B��t� kT �� Indeed� using the fact that the signal x�t� is

B��band�limited we get

x�kT � !Z B�

�B�

X�f� ej��kTf df !DX�Gk

E! hx� gki�

where� Gk�f� ! rectB��f� e�j��kTf is the Fourier transform of gk�t�� We can thus rewrite

�� as

x�t� !

Fs

�Xk��

hx� gki gk�t� with gk�t� ! �B� sinc��B��t� kT ��

Therefore� the interpolation of an analog signal from its sample values x�kT � can be

considered as the reconstruction of x�t� from its expansion coe�cients x�kT � ! hx� gki�The overall process of A�D conversion can be interpreted as expanding the signal x�t�

into the function set fgk�t�g with k�ZZ� Since every x�t� � H can be represented as a

linear combination of the functions gk�t�� the set fgk�t�g is obviously complete in H�

We shall next calculate the frame operator of fgk�t�g� Using De�nition � and the

Parseval identity� the frame operator S can be formulated in the frequency domain as

� �SX��f� !�X

k��hX�GkiGk�f� ! Fs rectB��f�X�f��

Since � �SX��f� ! FsX�f� for x�t� � H� fgk�t�g is a tight frame for the space H of

B��band�limited functions� with frame bound A ! B ! Fs� Hence� the dual frame is

given by

#gk�t� !

Fsgk�t��

Using Theorem A�� we get

x�t� !�X

k��hx� gki #gk�t� !

K

�Xk��


which shows that the reconstruction formula �� corresponds to a reconstruction using

the dual frame� Moreover� it is easily checked that for critical sampling �T ! ��B�

�

the gk�t� are orthogonal to each other� i�e�� hgk� gli ! �B� ��l � k�� Thus� a critically

sampled A�D conversion can be interpreted as an orthogonal signal expansion� In the

oversampled case the set fgk�t�g is a tight frame for H�

Sampling and interpolation operators� The sampling operation and the recon�

struction of the analog signal x�t� from its samples x�kT � can be interpreted as the

application of linear operators to the signal x�t� and to its samples x�kT �� respectively�

�Here rectB��f� denotes a function that is � in the interval ��B�� B�� and zero elsewhere�

�� CHAPTER �� NOISE ANALYSIS

The analysis operator T �cf� Section A�� acts on the signal x�t� � H and provides the

samples x�kT ��

T � x � x�kT � or equivalently �Tx�k ! x�kT ��

The sampling operator T thus provides a mapping from the Hilbert space H of B��

band�limited functions to the space l��ZZ� of square�summable discrete�time signals�

The synthesis operator �T � performs a reconstruction of the signal x�t� from its samples

x�kT � via interpolation� i�e��

�T�� x�kT � � x�t� with

��T�x�kT �

��t� !

K

�Xk��


�� Design Freedom in Oversampled A�D Conversion

The reconstruction of the analog signal x�t� from its samples x�kT � can alternatively

be interpreted as the application of an ideal lowpass �lter with bandwidth B� to the

signal x�t�P�

k�� t�kT � !P�

k�� x�kT � ��t�kT �� In the case of critical sampling�

the ideal lowpass �lter of bandwidth B� is the only �lter that provides PR of the signal

x�t� �see Fig� ��a�� In the oversampled case� there is in general an in�nite number

of reconstruction �lters providing PR� The only requirement is that the reconstruc�

tion �lter�s transfer function is constant within the signal band �B� � f � B� �see

Figs� ��b��c�� Therefore� in the oversampled case one has more freedom in design�

ing the reconstruction �lter �� In practice� this additional freedom is exploited for

designing reconstruction �lters with desirable �lter characteristics like e�g� rollo� �see

Fig� ��c�� This situation parallels the fact that� for a given analysis FB� the synthesis

FB providing PR is uniquely determined in the critically sampled case� whereas it is

not uniquely determined in the oversampled case �see Subsec� ��

�� Noise Analysis in Oversampled A�D Conversion

This subsection presents a noise analysis for oversampled A�D conversion� We �rst

assume that the reconstruction �lter is the ideal lowpass �lter with bandwidth B��

Let us model the quantization error as discrete�time white noise with power spectral

density ��q � Then� the reconstruction error variance� i�e�� the variance of the error in the

reconstructed signal due to quantization� is given by

��c ! ��q�B� ! ��qFs

in the critical case and

��o ! ��q�B� ! ��qFsK

�� NOISE ANALYSIS AND DESIGN FREEDOM ��

�Fs�! �B�

Fs�! B�

f

�a�

�Fs�

Fs�

�B� B�

f

�b�

�Fs�

Fs�

�B� B�

f

�c�

Fig� �� Reconstruction of analog signal by lowpass �ltering�

a critical case� b�c oversampled case� b ideal lowpass �lter with minimal

bandwidth� c generalized synthesis �lter�

in the case of oversampling by a factor of K �see Fig� � �� We therefore have

��o��c

!

K�

Hence� the noise power that falls into the signal band is inversely proportional to the

oversampling factor� Therefore� each doubling of the oversampling factor decreases the

in�band noise power by � dB�

When the reconstruction is performed using not the ideal lowpss �lter with bandwidth

B� but a generalized inverse� i�e�� some other �lter providing PR� it is obvious from

Fig� ��c� that the noise power in the reconstructed signal will increase� This is due to

the fact that a generalized reconstruction �lter will pass some of the noise in the out�of�

band region B� � f � Fs�� The amount of additional noise in the reconstructed signal

is determined by the bandwidth and the transfer function of the reconstruction �lter�

In this sense� there exists a tradeo� between noise reduction and design freedom in

oversampled A�D conversion� Practically desirable �or realizable� reconstruction �lters

�i�e� �lters with rollo�� lead to an additional reconstruction error�

� CHAPTER �� NOISE ANALYSIS

�Fs�! �B�

Fs�! B�

power spectral density of noise

f

�a�

�Fs�

Fs�

�B� B�

f

power spectral density of noise

�b�

Fig� � � Adding quantization noise to the sampled signal�

a critical case� b oversampled case�

We shall next interpret this result from a frame�theoretic point of view� Recall that

the reconstruction of the signal x�t� from its coe�cients x�kT � ! hx� gki using the ideal

lowpass �lter with bandwidth B� corresponds to a reconstruction using the dual frame

#gk�t� !�Fsgk�t�� The range space R of the sampling operator T is the space of discrete�

time functions band�limited in the interval � � �� K� ��K

�� Since reconstruction using

the ideal lowpass �lter of bandwidth B�� equivalently in the digital domain� bandwidth��K

corresponds to an orthogonal projection onto R� it follows that reconstruction with

the dual frame involves an orthogonal projection onto R �� Thus� all noise in the

orthogonal complement R of the range space of T �corresponding to the out�of�band

region � ��K� �� will be discarded� It is hence seen that reconstruction with the dual

frame leads to the minimum reconstruction error if noise is added to the signal samples�

In the next section� we shall see that a similar tradeo� between noise reduction and

design freedom arises in oversampled FBs� which are another special case of redundant

signal expansions� In oversampled FBs� however� the analysis is less intuitive since the

signal spaces R and R do not correspond to frequency bands�

�� Noise Analysis and Design Freedom in Over

sampled Filter Banks

One of the major advantages of oversampled FBs is their increased design freedom as

compared to critically sampled FBs� For a given oversampled analysis FB� the synthesis

�� NOISE ANALYSIS AND DESIGN FREEDOM �

FB providing PR is not uniquely determined �see Subsec� �� This nonuniqueness

allows for the design of a broad class of PR synthesis FBs having desirable properties

that are not shared by the minimum norm synthesis FB�� Besides PR we can therefore

impose additional restrictions on the synthesis �lters and perform an optimization over

the class of synthesis FBs providing PR and satisfying our restrictions� Using the

parameterizations of all PR synthesis FBs given in Subsec� �� this can be done using

an unconstrained optimization procedure since PR need not be incorporated via a side

constraint�

This increased design freedom is similar to the increased design freedom in over�

sampled A�D conversion� However� as in oversampled A�D conversion there exists a

tradeo� between noise reduction and design freedom in oversampled FBs�

In this section we shall investigate the sensitivity of oversampled FBs to �quantiza�

tion� noise� We shall show that the frame bound ratio B�A �see Subsec� �� serves as

a condition number of the FB� We �nally perform a coding experiment that corroborates

the tradeo� between noise reduction and design freedom in oversampled FBs�

�� Noise Analysis for Oversampled Filter Banks

In this subsection� we investigate the sensitivity of oversampled FBs to �quantization�

noise qk�m� added to the subband signals vk�m� ! hx� hk�mi �k ! �� N�� The

input to the synthesis FB is now given by the noisy subband signals

v�k�m� ! vk�m� " qk�m� ! hx� hk�mi" qk�m��

Let us collect the noise signals qk�m� in the N �dimensional vector noise process q�m� !

�q��m� q��m� � � � qN��m��T that is assumed to be wide�sense stationary �WSS� and

zero�mean� The N �N power spectral matrix of q�m� is de�ned as ��

Sq�z� !�X

l��Cq�l� z

�l

with the autocorrelation matrix

Cq�l� ! Efq�m�qH �m� l�g�

where E denotes the expectation operator ��

Variance of reconstruction error� It is convenient to redraw the FB in the

polyphase domain as shown in Fig� � �� Here� x�z� ! �X��z� X��z� �� XM��z��T

�This is especially important in coding applications� where the subband signals are quantized and

therefore errors are introduced in the signal� In this case� the synthesis lters determine the perceptual

impact of the quantization error on the reconstructed signal� In the DCTbased JPEG standard�

for example� the discontinuities of the synthesis lters� impulse responses at their boundaries lead to

blocking artifacts�


and x�z� ! � X��z� X��z� �� XM��z��T with Xn�z� !P�

m�� x�mM " n� z�m and Xn�z� !

P�m�� x�mM " n� z�m� and the noise q�m� is represented by its z�transform

q�z� !P�

m�� q�m� z�m� Assuming a PR FB� we have �see Fig� �� x�z� ! x�z� "

R�z�q�z�� so that the reconstruction error vector e�z� is given by

e�z� ! x�z�� x�z� ! R�z�q�z� � ��

Here� R�z� denotes an arbitrary left�inverse of E�z�� The reconstruction error e�n� is

again WSS and zero�mean� with M �M power spectral matrix

Se�z� ! R�z�Sq�z� #R�z� ��

and variance ��

��e !

M

Z �

�TrnSe�e

j��od��

For uncorrelated white noise signals qk�m� with identical variances ��q ! Efjqk�m�j�g�k ! �� N � �� it follows that ��

Cq�l� ! ��q IN ��l� Sq�z� ! ��qIN �

Here� with �� and �� the reconstruction error variance becomes

��e !��qM

Z �

�TrnR�ej��RH�ej��

od��

Frame�theoretic analysis of noise sensitivity� We now consider the case of

uncorrelated white noise signals� Furthermore� we assume that the FB provides a

frame expansion and that the reconstruction is performed using the dual frame� i�e��

R�z� ! R�z�� We recall from Subsection �� that the �tightest possible� frame bounds

A� and B� of a FB providing a frame expansion are given by

A� ! ess inf�� n��M��

�n�ej�� B� ! ess sup

�� n��M��n�e

j��

where �n�� denotes the eigenvalues of the inverse UFBF matrix S��ej�� !

R�ej��RH�ej��

With TrnR�ej��RH�ej��

o!PM��

n�� n�� and �� it follows that MA� �TrnR�ej��RH�ej��

o�MB�� Inserting this in �� we obtain

A� � ��e��q

� B� � ��

i�e�� the reconstruction error variance ��e is bounded in terms of the frame bounds A��

B� and the subband noise variance ��q � Let us assume normalized analysis �lters� i�e��

khkk ! for k ! �� N � � Then� it follows from Corollary �� and A� ! �B�B� ! �

A

�� NOISE ANALYSIS AND DESIGN FREEDOM

E�z� R�z�

q�z�

x�z� x�z�

Figure �� Adding noise to the subband signals�

that A� � �K� B�� where K ! N

Mis the oversampling factor� Hence� for A� B� or

equivalently B��A� � �� implies that

��e��q

Kwith K !

N

M�

Therefore small perturbations in the subbands yield a small reconstruction error� which

moreover is inversely proportional to the oversampling factor K ! NM� This shows that

FBs providing snug frame expansions are desirable in practical applications� since they

guarantee that small errors in the subband signals will result in small errors in the

reconstructed signal� In applications involving modi�cations of the subband signals�

this property is desired as well�

For a paraunitary FB with khkk ! we have A� ! B� ! �K� and hence �� becomes

��e��q

!

Kwith K !

N

M� ��

Since in the critically sampled case ��c�! ��e jK�� ! ��q � Eq� �� can be rewritten as

��e��c

!

K�

Thus� in the paraunitary �tight� case the reconstruction error variance is inversely pro�

portional to the oversampling factor K� which means that more oversampling entails

more noise reduction� Such a ��K behavior� has been observed in Subsection ��

for oversampled A�D conversion� In fact� as already noted in Subsection �� an over�

sampled A�D conversion is a tight frame expansion� Since paraunitary FBs correspond

to tight frames� the �K behavior of the reconstruction error variance in paraunitary�

oversampled FBs does not come as a surprise�

A �K behavior of the reconstruction error variance has also been observed for tight

frames in �nite�dimensional spaces �� and for the reconstruction from a �nite

set of Weyl�Heisenberg �Gabor� or wavelet coe�cients �� Recently� under addi�

tional conditions� a �K� behavior has been demonstrated for frames of sinc functions

�� and for Weyl�Heisenberg frames �� In �� a nonlinear


E�z� R�z�

Sq�z�

E q��z� R q��z�

��qIN

Figure �� Adding correlated colored noise to the subband signals�

iterative method based on principles from set�theoretic estimation is used to achieve a

�K� behavior of the reconstruction error� In Chapter �� we shall propose oversampled

predictive subband coders which can do even better than �K��

Unfortunately� our assumption of uncorrelated� white noise signals is not realistic for

K � � For arbitrary �possibly correlated and nonwhite� noise signals with power

spectral matrix Sq�z�� a whitening system can be employed to transform the noise into

uncorrelated white noise� which in turn allows to use the results obtained above for the

uncorrelated white noise case� Assuming that the noise power spectral matrix Sq�z�

can be factored as

Sq�z� ! S��q �z�#S��q �z��

the system depicted in Fig� � is equivalent to a system with noise power spectral matrix

Sq�z� ! ��qIN �corresponding to uncorrelated white noise� if E�z� and R�z� are replaced

by

E q��z� !q��q S

��q �z�E�z� and R q��z� !

q��qR�z�S��q �z��

respectively �see Fig� �� The double inequality �� continues to hold if the frame

bounds in �� are replaced by the frame bounds of the FB fE q��z��R q��z�g with

inverse UFBF matrix given by

S q��

�z� ! R q��z� #R q��z� !

��qR�z�Sq�z� #R�z��

Similarly� the �K behavior of the reconstruction error variance continues to hold if

R q��z� is paraunitary� i�e�� R q��z� #R q��z� ! I�

�� Noise Reduction Versus Design Freedom in Filter Banks

In Subsection �� we have seen that oversampled FBs o�er increased design freedom

as compared to critically sampled FBs� However� as in oversampled A�D conversion

there is a tradeo� between noise reduction and design freedom� This tradeo� is the

subject of this subsection�

Recall from Subsection �� that the analysis FB operator T that assigns to each in�

put signal x�n� the vector signal v�m� comprising the subband signals vk�m� ! hx� hk�mi�


�l��ZZ��N

R

R

Fig� �� Range space of analysis �lter bank and its orthogonal complement�

the frame operator S� and the adjoint operator T � are represented by the matrices E�z��#E�z�E�z�� and #E�z�� respectively� From Theorem A�� it follows that the orthogonal pro�

jection operator on the range space of the analysis operator T � R ! Ran�T � � �l��ZZ��N �

is given by PR!TS��T �� Consequently� the matrix representation of PR is ob�

tained as PR�z� ! E�z�h#E�z�E�z�

i�� #E�z�� Similar arguments can be used to show

that P�z� ! IN � PR�z� is the matrix representation of the projection operator

PR� ! I � PR on the orthogonal complement space �� of R denoted as R� The

spaces R and R are depicted in Fig� ��

Let us consider an oversampled FB withR�z� chosen according to �� i�e�� R�z� ! R�z� " U�z�

hIN � E�z� R�z�

i� such that PR is guaranteed� Inserting �� in ��

we obtain the following decomposition of the reconstruction error�

e�z� ! eR�z� " e�z� �

where

eR�z� ! R�z�q�z� and e�z� ! U�z�P�z�q�z��

Since R�z�PR�z� ! R�z�� the error component eR�z� in �� can equivalently be

written as

eR�z� ! R�z�PR�z�q�z��

which shows that eR�z� is reconstructed from the subband noise component PR�z�q�z�

in R� Similarly� e�z� ! U�z�P�z�q�z� is reconstructed from the subband noise

component P�z�q�z� in R�

For subband noise signals qk�m� that are uncorrelated and white� it follows from the

orthogonality of the spaces R and R that the error components eR�n� and e�n� are

uncorrelated� Hence� their variances� denoted respectively ��R and �� can simply be

added to yield the overall reconstruction error variance ��

��e ! ��R " ��


This leads to the following result�

Theorem �� For white and uncorrelated subband noise signals in an

oversampled PR FB �R�z�E�z� ! IM�� the minimum norm synthesis FB

�corresponding to R�z�� the para�pseudo�inverse of E�z�� yields the min�

imum reconstruction error variance ��e�min ! ��R among all PR synthesis

FBs�

Proof� According to �� the variance component ��R is independent of the param�

eter matrix U�z�� and thus of the particular R�z� chosen� The variance component ��

on the other hand� depends on U�z�� it is an additional variance that will be zero if

and only if R�z� ! R�z�� Indeed� it follows from �� that R�z� ! R�z� if and only

if U�z�P�z� � �� in which case e�z� ! U�z�P�z�q�z� � o and thus also �� ! ��

�

Using R�z�� all noise components orthogonal to the range space R are suppressed�

whereas any other PR synthesis FB �possibly with desirable properties such as im�

proved frequency selectivity� leads to an additional error variance �� since also noise

components orthogonal to R are passed to the FB output� In this sense� there exists

a tradeo� between design freedom and noise reduction� At this point� we recall the

tradeo� between noise reduction and design freedom in oversampled A�D conversion

�see Subsection �� where the range space was the space of ��K

�band�limited func�

tions� Even though in the FB case the range space and its orthogonal complement

take a somewhat more complicated form� the same interpretations and conclusions as

in oversampled A�D conversion apply�

In the case of correlated colored noise signals� i�e�� arbitrary noise power spectral

matrix Sq�z�� it follows from the noise whitening approach used in Subsection ��

that the above results continue to hold if the matrices E�z� and R�z� are replaced by

E q��z� and R q��z�� respectively� In particular� for a given analysis FB with polyphase

matrix E�z� and given noise with power spectral matrix Sq�z�� the equivalent synthesis

FB �cf� Fig� � �� minimizing the reconstruction error variance has polyphase matrix R q��z� ! �#E q��z�E q��z�� #E q��z�� which yields

R�z� !q��q R

q��z�S��q �z� ! �#E�z�S��q �z�E�z�� #E�z�S��q �z��

We �nally note that the tradeo� between noise reduction and design freedom is

inherent in redundant representations in general� It is not restricted to redundant shift�

invariant signal expansions� such as oversampled A�D conversion and oversampled FBs�

Loosely speaking� the range space R&and thus also the �xed noise component ��R&

becomes �smaller� for increasing oversampling factor K ! N�M � This explains why

more oversampling tends to result in better noise reduction�


2 4 6 8 10 12 14 16 18 2020

25

30

35

40

45

50

55

60

65

70

Quantization stepsize

SN

R/d

B

Fig� �� SNR as a function of the normalized quantization stepsize in a

two�channel FB with analysis �lters being the Haar �lters� The solid

line shows the SNR using the minimum norm PR synthesis FB� whereas the

dashed�dotted line shows the SNR resulting from the alternative

PR synthesis FB depicted in Figs� �c and d�

We conclude this section with a simulation result illustrating the tradeo� between

noise reduction and design freedom in oversampled FBs� We coded a piece of music using

the two�channel Haar FB with oversampling by � previously considered in Section ��

The subband signals were quantized using a uniform quantizer in each of the channels�

The stepsizes of the quantizers were equal in both channels� The solid line in Fig� ��

shows the SNR in dB as a function of the quantization stepsize for the minimum norm

synthesis �lters F��z� and F��z� �see Figs� ��a� and �b�� The same signal was coded

using the alternative PR synthesis �lters depicted in Figs� ��c� and �d�� These synthesis

�lters result in an increased amount of noise in the reconstructed signal� Indeed� the

SNR of the subband coder using the alternative synthesis �lters �dashed�dotted line

in Fig� �� is seen to be consistently about � dB below the SNR obtained with the

minimum norm synthesis �lters�

Chapter �

Oversampled Predictive Subband

Coders

In this chapter we introduce oversampled predictive subband coders� These coders will

be classi�ed as noise predictive �noise shaping� subband coders and signal predictive

subband coders� Both types of subband coders can be viewed as extensions of over�

sampled predictive A�D converters �� to oversampled FBs� Oversampling in a

FB introduces redundancy in the subband signals and therefore allows to reduce quan�

tization noise� In Chapter �� we proved a �K�type behavior �K is the oversampling

factor� of the reconstruction error variance in oversampled paraunitary FBs� This be�

havior can further be improved by using predictive techniques such as noise shaping

�� see Subsec� �� and linear prediction �see Subsec� �� In this sense� oversam�

pled FBs allow to trade sample rate and consequently coding rate �in an information

theoretic sense� for quantizer accuracy� Oversampled predictive subband coders are

therefore well suited for subband coding applications where for technological or other

reasons quantizers with low accuracy �even single�bit� have to be used� In this case�

oversampling in the FB and noise shaping or linear prediction improve the e�ective

resolution of the subband coder� Using low resolution quantizers in the digital domain

increases circuit speed and allows for lower circuit complexity� One�bit codewords� for

example� eliminate the need for word�framing�

The subband coders proposed in this chapter exploit both the intrachannel and the

interchannel redundancy� Exploiting the interchannel redundancy is important since

the channel �lters� transfer functions are always overlapping and therefore the channel

signals are correlated�

We shall now outline the organization of this chapter� In Sec� �� we review oversam�

pled noise predictive and signal predictive A�D converters� This discussion serves as

a basis for Section �� which introduces oversampled predictive subband coders� Both

noise predictive and signal predictive subband coders are discussed� We present cod�

�

CHAPTER �� OVERSAMPLED PREDICTIVE SUBBAND CODERS

ing examples that shall help to develop a quantitative feeling for the tradeo� between

quantizer accuracy and oversampling factor�

�� Oversampled Predictive A D Converters

In this section we shall review predictive techniques for oversampled A�D conversion

�� Our presentation essentially follows �� Based on the interpretation of oversam�

pled A�D conversion as a frame expansion �see Subsec� �� we provide a new signal

space interpretation of the principle underlying noise shaping A�D converters ��

Oversampled A�D converters �� avoid many of the di�culties encountered with con�

ventional methods for A�D and D�A conversion and have therefore found widespread

application� In oversampling converters� simple and relatively high�tolerance analog

components can be used� However� they require fast and complex digital signal process�

ing stages� Usually� the analog input is converted into a simple digital code �single�bit

words in general� at a frequency much higher than the Nyquist rate� In essence� over�

sampled A�D converters allow to trade resolution in time �sampling rate� for resolution

in amplitude in such a way that imprecise analog circuits may be used� They take

advantage of the fact that �ne�line VLSI is better suited for fast digital circuits than for

precise analog circuits� Because the sampling rate of oversampled A�D converters has

to be several orders of magnitude higher than the Nyquist rate� oversampling methods

are best suited for relatively low�frequency signals� �We note� however� that several in�

teresting approaches for high�speed A�D conversion based on an array of slower speed

A�D converters have been proposed recently �� So far� oversampled A�D

converters have found use in applications such as digital audio� digital telephony� and

instrumentation� Future applications include video and radar systems�

In the following we shall discuss two types of oversampled A�D converters� namely

noise shaping converters also known as sigma�delta converters �see Subsec� �� and

signal predictive converters �see Subsec� ��

�� Noise Predictive �Noise Shaping� Coders

This subsection discusses oversampled noise predictive coders also known as sigma�delta

converters or noise shaping converters� The overall coder is modeled as a digital system

by moving the A�D converter�s sampling gate �i�e� the A�D converter without quan�

tizer� ahead of the coder� In our analysis we assume that the A�D converter�s sampling

gate is ideal� i�e�� it provides exact samples of the analog signal without introducing any

distortions�

Principle� Fig� �� shows a general noise shaping coder with X�z� denoting the

z�transform of the input signal �the sampled version of the analog signal x�t�� Xq�z�

�� OVERSAMPLED PREDICTIVE A�D CONVERTERS

�G�z�

QX�z�

Q�z�

Xq�z�

Fig� �� General noise�shaping coder� The box labeled Q denotes the quantizer�

denoting the z�transform of the output signal and the noise shaping �lter

G�z� ! �LX

k��

g�k�z�k�

Here L denotes the order of the noise�shaping coder and the �lter coe�cients are as�

sumed real�valued� g�k� � IR� Note that the �lter � G�z� !PL

k�� g�k�z�k has to be

strictly causal since it operates in a feedback loop� The quantizer is modeled as an

additive noise source� the quantization noise q�n� is assumed to be a zero mean� WSS

process�

Straightforward calculations reveal that q�n� �cf� Fig� �� is given by

q�n� !LX

k��

g�k� q�n� k�� Q�z� ! ��G�z��Q�z��

where Q�z� is the z�transform of the quantization noise sequence q�n�� The goal of

an optimum noise shaping system design is to choose the �lter coe�cients g�n� such

that q�n� optimally estimates or predicts the inband component of the quantization

noise sample q�n�� in the sense that the inband component of q�n� � q�n�� i�e�� the

component of q�n� � q�n� that lies in R� is minimal �note that the corresponding out�

of�band component� i�e�� the component in R� will be removed by the reconstruction

lowpass �lter in the decoder �cf� Fig� �� The signal x�n� � q�n� presented to the

quantizer thus is the signal sample x�n� minus the estimate q�n� of the next sample of

the inband noise component� The noise shaping system therefore performs a prediction

of the inband component of the next noise sample by forming a linear combination of

the L past noise samples q�n�� q�n�� q�n�L�� In this sense noise shaping coders

can be interpreted as noise predictive coders�

It is easily seen from Fig� �� that the coder�s output signal is given by

xq�n� ! x�n� " q�n�� q�n�

�� CHAPTER �� OVERSAMPLED PREDICTIVE SUBBAND CODERS

��s �s��

Fig� �� Spectrum of oversampled analog signal�

or equivalently

Xq�z� ! X�z� "Q�z�G�z��

Note that x�n� is not a�ected by the noise�shaping system G�z� whereas q�n� is formally

passed through G�z�� For given noise statistics� the �lter coe�cients g�k� are chosen

such that the inband noise power� i�e�� the noise that falls into the signal band� is

minimized �see Fig� � �b�� In this sense� noise�shaping coders improve the resolution

of the A�D converter but leave the A�D converter�s dynamic range unchanged �this is

fundamentally di�erent from the signal predictive coder discussed in Subsection ��

Since the inband noise power is reduced relative to the quantization noise power of the

A�D converter� it is possible to increase the quantization intervals and thereby reduce

the overall converter complexity�

In principle� the optimal noise shaping �lter G�z� would be the ideal highpass �lter

with passband �s � j�j � �� where �s !��K

denotes the bandwidth of the sampled

signal� However� the ideal highpass �lter is of course not realizable� furthermore� it

would lead to a noncausal �G�z� and can therefore not operate in a feedback loop�

Optimum Noise Shaping System� We shall next describe two approaches for ob�

taining the �optimum� noise shaping �lter G�z� �� The �rst approach minimizes the

reconstruction error variance ��e � i�e�� the variance of e�n� ! xq�n�� x�n� corresponding

to

E�z� ! Xq�z��X�z� ! Q�z�G�z��

by choosing the coe�cients g�k� such that ��e�g�k�

! � for k ! � �� L� In this case�

of course� the coe�cients of the noise shaping �lter depend on the statistics of the

quantization noise and on the oversampling factor� In the following we assume that the

correlation function of the quantization noise� Cq�k� ! Efq�n�q��n� k�g� is known� Thecorresponding power spectral density function is given by Sq�z� !

P�k��Cq�k�z

�k�

Assuming that the analog signal x�t� is band�limited to jf j � B�� i�e�� X�f� !R�� x�t� e�j��ft ! � for jf j � B�� and that the coder performs oversampling at a

rate Fs � �B�� the �lter G�z� has to be chosen such that the quantization noise in the

signal band ��s � � � �s with �s ! B�

Fs! �

�Kis attenuated as much as possible �see

Fig� �� The remaining out�of�band noise in the frequency region �s � j�j � �� will

be removed by a lowpass �lter in the decoder�

�� OVERSAMPLED PREDICTIVE A�D CONVERTERS �

The power spectral density function of the reconstruction error E�z� ! Q�z�G�z� is

given by Se�z� ! G�z� #G�z�Sq�z� �� The reconstruction error variance ��e is obtained

as

��e !Z ��

��Se�e

j�� d� !Z ��

��jG�ej��j� Sq�ej�� d��

Since in the decoder the output of the coder is lowpass �ltered with cuto� frequency �s�

the integration in �� can be restricted to the interval j�j � �s and we get the �nal

expression for the reconstruction error variance as

��e !Z �s

��sjG�ej��j� Sq�ej�� d��

With G�ej�� ! � PLk�� g�k�e

�j��k� it follows after straightforward manipulations

that

��e ! )� � �LX

k��

g�k�)k "LX

k��

g�k�LXl��

g�l�)k�l� ��

where

)k !Z �s

��sSq�e

j�� cos��k� d��

In particular� )� !R �s��s Sq�e

j�� d�� De�ning the L � L matrix � with elements

��m�n ! �m�n

��m ! � �� L� n ! � �� L�� the L � vector h with elements

�h�m ! �m��

�m ! � �� L� and the L � vector g containing the coe�cients of the

noise shaping �lter according to �g�m ! g�m�� the reconstruction error variance ��e can

be rewritten as

��e ! )�� gTh " gT�g��

The optimum noise shaping �lter G�z� of order L is obtained by setting ��e�g�k�

! � for

k ! � �� L or equivalently ��e�g

! �L� where �L denotes the L � zero vector� Using��ggTh ! h and �

�ggT�g ! ��g �� it follows that the coe�cients of the optimum

noise shaping �lter have to satisfy ��

�gopt ! h or equivalently gopt ! ��h� ��

The matrix � has T�oplitz structure �� and can therefore be inverted e�ciently

using the Levison�Durbin recursion �� From �� it is clear that the

coe�cients of the optimum noise shaping �lter gopt depend on the oversampling factor

K ! ��s

and on the noise statistics Cq�k�� Therefore� the resulting coder is not robust

with respect to changes in the noise statistics and the oversampling factor�

Optimum noise shaping system � Alternative approach� We shall next present

an alternative approach �� for the calculation of an �optimum� noise shaping �lter�

This approach is based on a power series expansion of the reconstruction error variance

��e � The L noise shaping �lter coe�cients g�n� are chosen such that the �rst L terms of

this series expansion vanish� independently of the noise statistics� Expanding the cosine


term cos��k� in )k !R �s��s Sq�e

j�� cos��k� d� as a power series with respect to ��

we obtain the following expression of )k as a power series in �s ! B�Fs�

)k)�

!�Xl��

��l��l�$

k�l ��s��l *l with *l !

)�

Z �s

��sSq�e

j��

��

�s

��l

d��

Consequently the L � vector h de�ned further above can be expanded as

h !�Xl��

h�l� *l ��ls �

where �h�l��m ! ��l �l��

m�l��l� and the L � L matrix � can be expanded as

� !�Xl��

��l�*l��ls with ��l��i�j !

��l��l�$

�i� j��l��l �i� j ! � �� L�

Inserting this power series expansions of h and of � into �� we get a power series

expansion of the normalized noise variance ��e��

as

��e)�

!�Xl��

+l*l ��ls � ��

with

+l ! ��l�� gTh�l� " gT��l�g�

where we have used *� ! � The power series expansion method now selects the �lter

coe�cients g�k� �k ! � �� L� such that +l ! � for l ! �� L� � The solution of

the resulting system of equations is given by ��

g�k� ! ��k��L

k

�k ! � �� L�

from which it follows that the corresponding noise shaping �lter is given by

G�z� ! �� z��L�

The lowest�order nonvanishing term in �� is given by +L*L��Ls � For large oversam�

pling factors K ! ��s

� i�e�� small �s� higher powers of �s can be neglected in the series

expansion �� which implies that the decay of the reconstruction error variance ��e is

essentially proportional to ��Ls !�

��K

��L� �We note that +L does not depend on the

oversampling factor K�� Approximating the inband noise power by the lowest�order

nonvanishing term +L*L��Ls � we get

��e )�+L*L��Ls ! )�+L*L

�K

��L� ��

Although the noise shaping �lterG�z� is independent of the quantization noise statistics�

the performance of the coder of course depends on the quantization noise statistics�

�� OVERSAMPLED PREDICTIVE A�D CONVERTERS ��

��

��

power spectral density of noiseG�ej��

�

Fig� �� Typical noise shaping �lter�

Assuming for simplicity white quantization noise with power spectral density ��q � we

get *L !��q��

�s�L�

and consequently

��e +L

��L��q�L"

K�L��

Therefore� in the white noise case the reconstruction error variance is inversely propor�

tional to K�L�� Setting L ! �� we reobtain the �K�behavior of the reconstruction

error variance in a K�times oversampled A�D converter without noise shaping �see Sub�

section �� Recently it has been shown � � using a vector quantization approach

that the reconstruction error in an L�th order noise shaping coder cannot decrease faster

than �K�L�� In order to achieve this performance� �consistent� reconstruction of the

quantized values has to be used ��

Frame�theoretic interpretation� We conclude this subsection with a frame�

theoretic interpretation of noise shaping which will turn out to be crucial for the de�

velopment of the results in Subsec� �� In a noise shaping coder� the quantization

noise is �shaped� in such a way that the noise in the signal band is reduced� i�e�� the

noise shaping coder shifts the noise to the high frequencies which are then discarded by

the lowpass reconstruction �lter� Fig� �� shows the transfer function of a typical noise

shaping �lter along with the power spectral density of the �white� quantization noise

and the signal spectrum�

Since the signal band � ��K

� � � ��K

corresponds to the range space R of the

sampling operator T �see Subsec� �� the operation of the noise shaping coder can

also be interpreted as moving the quantization noise to the orthogonal complement R

of the sampling operator�s range space� Reconstruction with the dual frame� i�e�� low�

pass �ltering� then performs an orthogonal projection onto the range space R which

discards all the components in the complement space R� In the light of these expla�

nations� it appears that the ideal highpass �lter with passband ��K

� j�j � �� is

the optimum noise shaping �lter since it projects the noise entirely onto R� hence�

after the reconstruction lowpass �lter �orthogonal projection onto R�� no noise would

be left� However� as already mentioned the ideal highpass �lter is not realizable and

would furthermore lead to a noncausal feedback loop system�


�G�z�

QX�z�

X�z�

E�z�Xq�z�

�G�z�

Encoder Decoder

Fig� �� Signal predictive coder� The box labeled Q denotes the quantizer�

In an oversampled noise shaping subband coder �to be discussed in Subsec� ��

the operation of the noise shaping system is similar� the only di�erence being that the

signal spaces R and R have a somewhat more complicated form�

�� Signal Predictive Coders

This subsection discusses oversampled signal predictive coders �� In contrast to

noise shaping coders� which perform a prediction of the inband quantization noise �see

Subsec� �� signal predictive A�D converters attempt to predict the next sample

of the signal to be quantized� As in the noise predictive case� the overall coder is

modeled as a digital system by moving the A�D converter�s sampling gate �which is

again assumed to be ideal� ahead of the coder�

Principle� Fig� �� shows a signal predictive coder with

G�z� ! �LX

k��

g�k�z�k�

where L denotes the order of the predictor and g�k� � IR� In the following� we assume

that the input signal x�n� is a band�limited WSS zero�mean stochastic process with

bandwidth �s �� and correlation function Cx�k�� and that the quantization noise q�n�

is a WSS zero�mean process with correlation function Cq�k�� Furthermore� q�n� and x�n�

are assumed to be uncorrelated� Note that these assumptions are partly di�erent from

the assumptions in the noise predictive case� since here also the input signal to the

coder is assumed to be a stochastic process�

The predictor uses the past L quantized �noisy� signal samples to calculate the signal

estimate

x�n� !LXl��

g�l� �x�n� l� " q�n� l��

�� OVERSAMPLED PREDICTIVE A�D CONVERTERS ��

The input to the quantizer is the prediction error

e�n� ! x�n�� x�n� ! x�n��LXl��

g�l��x�n� l� " q�n� l��

or equivalently

E�z� ! G�z�X�z�� G�z��Q�z��

It is readily veri�ed that the coder output is given by Xq�z� ! X�z� " Q�z�� Thus�

the overall reconstruction error Xq�z��X�z� ! Q�z� is equal to the quantization error

Q�z�� In this sense� a predictive coder does not give improved accuracy� However� by

choosing the prediction error �lter G�z� such that the prediction error e�n�� i�e�� the

input signal to the quantizer� is reduced� signal prediction provides a reduced dynamic

range over which the quantizer must operate� This allows to improve the resolution of

the quantizer for a �xed number of quantization intervals�

An oversampled signal predictive coder exploits two types of redundancies� which

shall be called natural redundancy and synthetic redundancy hereafter� The natural

redundancy �correlation� is the redundancy that is inherent in the input signal whenever

the input signal has a non�at power spectral density function� The synthetic redundancy

is introduced by oversampling the analog signal� i�e�� by expanding the input signal into a

redundant signal set which in our case consists of time�shifted versions of sinc functions

�see Subsec� �� It is therefore clear that for increasing oversampling factor the

prediction will become more accurate� since there is less variation in the signal�s sample

values or equivalently more synthetic redundancy�

The estimate x�n� !LPl��

g�l��x�n � l� " q�n� l�� of the signal sample x�n� is based on

quantized values of the past �noisy prediction�� In the case of high�resolution quan�

tization� the e�ect of quantization noise can be neglected and x�n� LPl��

g�l�x�n � l��

However� since we are interested in the speci�c case of low�resolution quantization� this

assumption will not be justi�ed in general�

Optimum prediction system� We shall next describe two approaches for the

calculation of an �optimum� prediction �lter �G�z� �� Both approaches are based

on the following expression for the power spectral density function of the prediction

error

Se�z� ! Sx�z�G�z� #G�z� " Sq�z��G�z�� #G�z��

Here we used the assumption that q�n� and x�n� are uncorrelated� The prediction error

variance ��e !R �� Se�e

j�� d� is consequently given by

��e ! )� � �LX

k��

g�k�)k "LX

k��

g�k�LXl��

g�l��)k�l " ,k�l��


where )k !R �s��s Sx�e

j�� cos��k� d� and ,k !R �� Sq�e

j�� cos��k� d�� Note

that the integration in )k ranges only over the interval j�j � �s� since x�n� is band�

limited with bandwidth �s� De�ning the L � L matrix � with elements ��m�n !�m�n�m�n

��m ! � �� L� n ! � �� L�� the L � vector h with elements �h�m !

�m��

�m ! � �� L�� and the L � vector g containing the coe�cients of the prediction

�lter as �g�m ! g�m�� the prediction error variance can be rewritten as

��e ! )�� gTh " gT�g��

This is formally equivalent to �� the only di�erence from the noise predictive case

being the de�nition of the matrix �� Note that � is still a T�oplitz matrix�

The following derivations closely parallel those presented in the noise predictive case

�see Subsec� �� The �rst approach minimizes the prediction error variance by

choosing the coe�cients g�k� such that ��e�g�k�

! � for k ! � �� L� The coe�cients of

the optimum prediction �lter �G�z� are given by

gopt ! ��h�

Due to the T�oplitz structure of �� the coe�cients gopt�k� can be calculated using the

Levinson�Durbin recursion ��

The second approach for the calculation of an �optimum� predictor is a power series

expansion method analogous to that in Subsec� �� It assumes that the quantization

noise can be neglected �� i�e�� ,k ! � for k � ZZ� As explained above� this is an unreal�

istic assumption in the case of low�resolution quantization� The results of Subsec� ��

i�e�� the derivations of the power series expansion method based optimum noise shaping

�lter� readily carry over to the signal predictive case by simply replacing Sq�ej�� in

Subsec� �� with Sx�ej�� It follows that the resulting optimum prediction error �lter

of order L is given by G�z� ! �� z��L� Even the approximation �� remains valid

as an approximation for the prediction error variance�

Remarks� An important special case of signal predictive coders is delta�modulation�

which is obtained by setting L ! and using a one�bit quantizer� The use of one�bit

words eliminates the need for word�framing� which makes the corresponding coders

attractive for numerous practical applications� A detailed analysis of delta�modulators

is provided in ��

We conclude this section with some remarks that will facilitate the transition from

oversampled predictive A�D converters to the oversampled predictive subband coders

discussed in Section �� In an oversampled predictive A�D converter� �synthetic� redun�

dancy is introduced by oversampling the analog signal� As explained in Subsec� ��

oversampled A�D conversion can be viewed as a redundant signal expansion� This re�

dundancy in the signal samples can then be exploited to perform a quantization noise

�� OVERSAMPLED PREDICTIVE SUBBAND CODERS ��

reduction by using either noise prediction or signal prediction� Noise shaping coders

exploit the redundancy by moving some of the quantization noise to a frequency band

which is then discarded by the reconstruction �lter� Signal predictive coders exploit

the redundancy by subtracting from the quantizer input an accurate estimate of the

current sample to be quantized�

�� Oversampled Predictive Subband Coders

In this section we shall introduce oversampled predictive subband coders� Such coders

are useful for subband coding applications where low�resolution quantizers �even single�

bit� have to be used� In this case� oversampling and noise shaping or prediction improve

the e�ective resolution of the subband coder considerably� The use of low�resolution

quantizers in subband coding applications o�ers several advantages such as increased

circuit speed and simple hardware architectures�

The coders discussed in this section are classi�ed as noise predictive �noise shaping�

subband coders �Subsec� �� and signal predictive subband coders �Subsec� ��

The proposed coders are capable of exploiting both the intrachannel and the interchan�

nel redundancy in the subband signals� Since the channel �lters in a practical subband

coder are nonideal� i�e�� their transfer functions are overlapping� it is evident that there

will be correlations between the di�erent channels� So far� signal predictive subband

coders have been restricted to intrachannel prediction and critical sampling ��

Only recently� it has been demonstrated for the special case of a two�channel Haar FB

with critical sampling that using information from the high�frequency band to perform

prediction in the low�frequency band leads to a subband coder that is asymptotically

rate�distortion optimal ��

We show that the calculation of the optimum �noise or signal� prediction system

can be reduced to the inversion of a matrix that has block T�oplitz structure� An

e�cient algorithm for this inversion is the multi�channel Levinson�Durbin recursion

�� reviewed in Appendix B� We provide simulation results demonstrating the

advantage of oversampled subband coders for low�resolution quantization applications�

These simulation results serve to develop a quantitative feeling for the tradeo� between

oversampling factor and quantizer accuracy� furthermore� they will provide insight into

the rate�distortion properties of oversampled predictive subband coders�

�� Noise Predictive �Noise Shaping� Subband Coders

In Subsec� �� we argued that oversampled A�D converters with noise shaping ex�

ploit the redundancy inherent in the signal samples by pushing the quantization noise

to the orthogonal complement of the sampling operator�s range space� The fact that

� CHAPTER �� OVERSAMPLED PREDICTIVE SUBBAND CODERS

oversampled FBs correspond to redundant signal expansions again suggests the appli�

cation of noise shaping� Classical noise feedback coding �single channel case� has found

widespread use� The noise shaping subband coders introduced here combine the advan�

tages of subband coding with those of noise or error feedback coding� The noise shaping

system in an oversampled FB aims at pushing the quantization noise to the orthogonal

complement R of the analysis �lter bank�s range space R�

�� Principle

In a FB the analysis operator is represented by the analysis polyphase matrix E�z�

�see Subsec� �� We propose a noise shaping system that is cradled between the

analysis FB E�z� and the synthesis FB R�z� and represented by the N � N transfer

matrixG�z� �see Fig� �� The quantizer is modeled as a WSS zero�mean vector�valued

noise source q�m� with autocorrelation matrix Cq�k� ! Efq�m�qH �m � k�g and power

spectral matrix Sq�z� !P�

k��Cq�k� z�k� The quantization noise q�z� is fed back

through the noise shaping system IN �G�z� to yield the quantization noise estimate

q�z� ! �IN � G�z��q�z�� which is then subtracted from the subband signal vector

v�z� ! E�z�x�z�� Assuming a PR FB� i�e�� R�z�E�z� ! IM � the reconstructed signal is

obtained as

xq�z� ! R�z� �E�z�x�z� "G�z�q�z�� ! x�z� "R�z�G�z�q�z��

It follows that the reconstruction error equals q�z� �ltered by G�z� and then by the

synthesis FB R�z��

e�z��! xq�z�� x�z� ! R�z�G�z�q�z� � ��

Therefore� the M � M power spectral density matrix of the reconstruction error is

given by

Se�z� ! R�z�G�z�Sq�z� #G�z� #R�z��

Consequently� the reconstruction error variance is

��e !

M

Z ��

��TrnR�ej��G�ej��Sq�e

j��GH�ej��RH�ej��od��

Without further constraints� the noise could be completely removed using the system

G�z� ! IN � E�z�R�z��

Indeed� inserting �� in �� it follows with R�z�E�z� ! IM that e�z� � o� In

the case of reconstruction with the dual frame� i�e�� with the para�pseudo�inverse R�z��

the ideal noise shaper would therefore be the orthogonal projection operator on R

given by P�z� ! IN � E�z� R�z� �see Subsec� �� Thus� the ideal noise shaper

�� OVERSAMPLED PREDICTIVE SUBBAND CODERS �

E�z� Qx�z� xq�z�

IN �G�z�

q�z�

R�z�

v�z�

Fig� �� Oversampled subband coder with noise shaping� The box labeled Q denotes the

quantizer�

projects the noise onto the orthogonal complement R of the analysis FB�s range space

R� The projected noise is then suppressed by the minimum norm synthesis FB R�z��

which performs an orthogonal projection onto the range space of the analysis FB �see

Subsec� �� This is similar to the case of oversampled A�D conversion where the

theoretically optimal noise shaping �lter was seen to be the ideal highpass �lter with

passband �s � j�j � �� Unfortunately� the ideal system in �� is inadmissible

since it is not causal and can therefore not operate in a feedback loop� Hence� we

hereafter constrain the noise shaping system to be a causal FIR system of the form

G�z� ! IN �LX

k��

Gk z�k � ��

resulting in a strictly causal feedback loop system

IN �G�z� !LX

k��

Gk z�k�

Here L denotes the order of the noise shaping system� The quantization noise estimate

now becomes q�n� !PL

l��Glq�n�l�� In the following we shall assume that the synthesis

FB is potentially an IIR FB� i�e��

R�z� !�X

k��Fkz

�k with �Fk�i�j ! fj�kM " i��

Furthermore� the synthesis �lters are assumed to be real�valued� which implies that the

matrices Fk are real�valued�

�� Optimal Noise Shaping System

We now derive the optimal noise shaping system� i�e�� the matrices Gk minimizing the

reconstruction error variance ��e in ��


Inserting �� and �� in �� it follows after straightforward manipulations

that

��e !

MTr

��

�Xk��

�Xk��

Cq�k� � k�FT

k�Fk

�LXl��

�Xk��

�Xk��

Cq�k� � k " l�GT

l FTk�Fk

�LXl��

�Xk��

�Xk��

Cq�k� � k � l�FT

k�FkGl

"LXl��

LXl��

�Xk��

�Xk��

Cq�k� � k " l� � l�GT

l�FTk�FkGl

��

We next derive the optimum noise shaping system for the case of white uncorrelated

quantization noise with equal noise variances in all channels� i�e�� Cq�k� ! ��qIN��k��

We shall then proceed to show that the optimum noise shaping system for arbitrary

quantization noise statistics can be obtained from the optimum noise shaping system

for white uncorrelated quantization noise by means of a simple transformation�

Specializing �� to Cq�k� ! ��qIN��k�� we get

��e !��qM

Tr

��

LXl��

h�lG

Tl " �T

l Gl

i"

LXm��

GTm

LXl��

�m�lGl

��

with the N � N matrices

�l !�X

s��FTs Fsl ��

which satisfy

�T�l ! �l�

Setting ��e�Gs

! � and using the matrix derivative rules �cf� �� Section ��

�

�GsTr fAGT

s g ! A

�

�GsTr fATGsg ! A

�

�GsTr fGsAg ! AT

�

�GsTr fAGsG

Ts g ! ATGs "AGs

for s ! � �� L� we obtain the linear system of equations��

�� L�� L��

��

��

�L�� L��

��

��

G�

G�

��

GL

�� !

��

��

��

��

�L

��


or� more compactly�

LXl��

�i�lGl ! �i with i ! � �� L� ��

This system of equations has block T�oplitz form and can therefore be solved e�ciently

using the multichannel Levinson recursion �� summarized in Appendix B �the ma�

trices GTl and Cx�k� in Appendix B have to be replaced by Gl and ��k� respectively��

Inserting �� into �� we get the minimum reconstruction error variance as

��e�min !��qM

Tr

��

LXl��

�Tl Gl�opt

��

where Gl�opt denotes the solution of �� or ��

Paraunitary FB� For a paraunitary FB with normalized� real�valued analysis �lters

�hk�n� � IR and khkk ! �� we have R�z� ! �K

#E�z� and consequently Fl ! ET�l� This

implies

�l !

K�

�Xs��

EsETs�l ��

and furthermore Tr�� ! MK� The analysis polyphase coe�cient matrices are given by

�Es�i�j ! hi�sM � j� � IR� If we restrict the analysis �lters to be causal and of �nite

length Lh ! JM �with some J � IN�� we get Es ! � for s � � and s � J and hence

�l !�K�

PJs��EsE

Ts�l� From this expression� it can be seen that �l ! �N for jlj � J � In

particular� in the nondecimated case M ! we have Es ! � for s � � and s � J �

and hence �l !�K�

PJ��s�� EsE

Ts�l� Consequently� �l ! �N for jlj � J�� The maximum

possible order of the noise shaping system is determined by the rank of the block matrix

in �� containing the matrices �l� With Tr�� ! MK� the minimum reconstruction

error variance in the paraunitary case is given by

��e�min !��qM

M

K� Tr

�LXl��

�Tl Gl�opt

��! ��e

��L��

� ��qM

Tr

�LXl��

�Tl Gl�opt

��

where ��e��L��

!��qK

is the reconstruction error variance obtained without noise shaping

�cf� ��

Colored noise case� We now consider the calculation of the optimum noise shaping

system in the general case of nonwhite� correlated quantization noise� Assuming that

the noise power spectral matrix can be factored according to Sq�z� ! S��q �z�#S��q �z�

and inserting this factorization into �� we obtain

��e !

M

Z ��

��TrnR�ej��G�ej��S��q �ej��S��

H

q �ej��GH�ej��RH�ej��od��

��


which can be rewritten as

��e !

M

Z ��

��TrnR�ej��G��ej��G�H �ej��RH�ej��

od��

with G��z� ! G�z�S��q �z�� Comparing with �� we see that ��e is minimized ifG��z�

is the optimum noise shaping system for white uncorrelated noise with equal variances

��q ! in all channels �denoted G�opt�z� in what follows�� This system can be calculated

as explained above� The optimum noise shaping system Gopt�z� for possibly nonwhite�

correlated quantization noise with power spectral matrix Sq�z� is then given by

Gopt�z� ! G�opt�z�S

��q �z��

Thus� the calculation of the optimum noise shaping system for arbitrary quantization

noise statistics has been reduced to the calculation of the optimum noise shaping system

for white uncorrelated quantization noise and a spectral factorization of the noise power

spectral density matrix�

�� An Example

We next provide a simple example in order to demonstrate the basic idea of noise shaping

in FBs and the importance of exploiting both the interchannel and the intrachannel

redundancy� Let us consider a simple paraunitary two�channel FB �i�e�� N ! �� with

M ! and� hence� oversampling factor K ! �� Fig� � shows the FB without noise

shaping� The analysis �lters are the Haar �lters H��z� ! �p�� " z�� and H��z� !

�p�� z�� and the synthesis �lters �corresponding to R�z� ! R�z�� are F��z� !

��#H��z� and F��z� ! �

�#H��z�� We furthermore assume that the noise added to the

subband signals is white and uncorrelated with variance ��q in each channel� i�e�� Sq�z� !

��q I��

We shall �rst calculate the reconstruction error variance ��e for the case where no

noise shaping is employed� Using F��z� #F��z� " F��z� #F��z� !��in �� we obtain

��e ! ��q

Z ��

��jF��e

j��j� d� "Z ��

��jF��e

j��j� d��!

��q��

which is consistent with the �K result ��

We next calculate the optimum �rst�order noise shaping system� The analysis

polyphase coe�cient matrices are given by E� ! �p�� T and E� ! �p

�� T �

With �see �� !��E�E

T� "E�E

T� � !

��I� and

�� !

�E�E

T� !

��

� �

��


X�z� Xq�z�

H��z�

H��z�

F��z�

F��z�

q��z�

q��z�

Fig� � � Two�channel FB with oversampling by factor ��

and �l ! � for l � � it follows from �� that the optimal noise shaping system of

order L ! is given by G�z� ! I� �G� z�� with

G��opt ! �� ! �� !

�

��

� �

��

The corresponding �minimum� error variance is obtained from �� as

��e ! ��q

�

�� Trf�T

�G��optg�!

��q��

Comparing �� with �� we can see that with the �rst�order noise shaping

coder the variance has been reduced by a factor of � as compared to the subband coder

without noise shaping� It is instructive to compare this result with the optimum noise

shaping system GD�z� of order L ! obtained under the constraint that G�z� is a

diagonal matrix� which means that the interchannel correlation �i�e�� the correlation

between the two channels� is not exploited� In this case� G� is replaced by the diagonal

matrix

GD� !

��

� �

��

Straightforward calculations reveal that

��e !��q�� " � " � " ��

Di�erentiating ��e with respect to the parameters and � and setting the derivatives

to zero we obtain ! �� ! ��

�and consequently

GD��opt !

�

��

� �

��

Note that the diagonal elements of GD��opt equal those of G��opt� The corresponding

reconstruction error variance is given by

��e !�

��q �


0 0.1 0.2 0.3 0.4 0.5−25

−20

−15

−10

−5

0

5Frequency response

Mag

nitu

de(d

B)

(a)0 0.1 0.2 0.3 0.4 0.5

−25

−20

−15

−10

−5

0

5Frequency response

Mag

nitu

de(d

B)

(b)

0 0.1 0.2 0.3 0.4 0.5−25

−20

−15

−10

−5

0

5Frequency response

Mag

nitu

de(d

B)

(c)0 0.1 0.2 0.3 0.4 0.5

−25

−20

−15

−10

−5

0

5Frequency response

Mag

nitu

de(d

B)

(d)

Fig� �� Noise shaping �lters and synthesis �lters in an oversampled two�channel FB�

a jG��opt�ej��j� b jG��opt�e

j��j� c jF��ej��j� d jF��e

j��j�

Thus� as expected� failing to exploit the interchannel redundancy leads to a larger

error variance which� however� is still smaller than the error variance ��e !��q�obtained

without noise shaping�

The transfer functions F��z�� F��z� of the minimum norm synthesis FB and the trans�

fer functions G��opt�z�� G��opt�z� of the noise shaping �lters in the diagonal of Gopt�z�

�the same as in the diagonal of GDopt�z�� are depicted in Fig� �� It can be seen that

the noise shaping system G��opt�z� ! � ��z�� operating in the lowpass channel�

attenuates the noise at low frequencies �note that subsequently F��z� attenuates high

frequencies�� whereas the noise shaping system G��opt�z� ! " ��z�� operating in the

highpass channel� attenuates the noise at high frequencies �note that subsequently F��z�

attenuates low frequencies�� Thus� the noise shaping system shifts part of the quanti�

zation noise to those frequencies that are subsequently discarded by the corresponding

synthesis �lters�


In the following we present simulation results demonstrating the performance of noise

shaping subband coders�

Simulation � �Noise reduction�� In the �rst simulation we calculate the nor�

malized reconstruction error variance � log��e�min

��q� with ��e�min given by �� for three

paraunitary� odd�stacked CMFBs �with normalized analysis �lters� i�e�� khkk ! �� The

FB has N ! � channels� the prototype length is Lh ! � and the decimation factors

were chosen as M ! � �� and �� respectively �i�e�� oversampling factors K ! ��


1 2 3 4 5 6 7 8 9 10-40

-35

-30

-25

-20

-15

-10

-5

0

System order L

Nor

mal

ized

err

or v

aria

nce

(dB

)

K=2

K=4

K=8

Fig� �� Normalized reconstruction error variance � log��e�min��q� as a function of

the noise shaping system�s order L�

and � respectively�� The quantization noise was assumed to be white and uncorrelated

with variance ��q in each channel� Fig� �� shows the normalized reconstruction error

variance � log��e�min

��q� as a function of the noise shaping system�s order L for various

oversampling factors� We emphasize that the curves in Fig� �� have been calculated

using �� and �� and have not been measured on an implemented system� Yet�

Fig� �� gives an indication of the performance to be expected from an implemented noise

shaping subband coder� Note that for increasing L the reconstruction error variance

decreases up to a certain point� after which it remains constant� Fig� �� furthermore

shows that the maximum order of the noise shaping system �i�e�� the order after which

the reconstruction error variance does not decrease any more� depends on the over�

sampling factor� We observed that the rank of the block matrix in �� decreases

with increasing oversampling factor� For noise shaping system order L ! �� we can see

from Fig� �� that doubling the oversampling factor results in a dB improvement in

the reconstruction error variance� This corresponds to an error�sample rate behavior of

�K�� For system order L ! � we even get a �dB improvement� which corresponds to

a �K��behavior�

Simulation � �Audio signal�� We next present simulation results that demonstrate

the performance of implemented noise shaping subband coders on an audio signal �tele�

phone ringing�� We used oversampled paraunitary odd�stacked CMFBs with N ! ��

channels and oversampling factorsK ! � �� The length of the prototypes

was consistently Lh ! �� Uniform quantizers with equal stepsizes in all subbands were

employed� The noise shaping system was calculated under the assumption of white un�


L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90


SN

R /

dB

�a�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 100−2

0

2

4

6

8

10


SN

R−

diffe

renc

e / d

B

�b�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60


bps

�c�

K=8, L=0

K=8, L=2

K=4, L=0

0 10 20 30 40 50 6010

20

30

40

50

60

70

80

90

bps

SN

R /

dB

�d�

Fig� �� Noise shaping subband coder with oversampling factor � simulation results

for audio signal� a SNR as a function of quantization stepsize for di�erent noise

shaping system orders L� b SNR di�erences with respect to L ! � from a� c bps

as a function of quantization stepsize for di�erent system orders L� d distortion�rate

characteristic in comparison to alternative subband coders�

correlated quantization noise with equal variance in all channels��

For a noise�shaping subband coder with oversampling factor K ! � Fig� ��a� shows

the SNR ! kxk�kxq�xk� in dB as a function of the quantization stepsize for di�erent orders

of the noise shaping system� The curve labeled L ! � corresponds to a subband coder

without noise shaping� In Fig� ��b� the di�erences of the curves in Fig� ��a� with

respect to the L ! � curve are depicted� One can observe that for �ne quantization a

�It is well known from A�D conversion that the white noise assumption is not justi ed in the

oversampled case �� However� estimating the quantization noise statistics is not possible in practice

since the quantizer is placed within a feedback loop �� We therefore chose to stick to the white

noise assumption� We note that the performance loss of implemented noise shaping subband coders

as compared to the expected performance �see Simulation �� is mostly due to the fact that the white

noise assumption is not justi ed in the oversampled case�


L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

110


SN

R /

dB

�a�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 100−5

0

5

10

15

20

25

30

35


SN

R−

diffe

renc

e / d

B

�b�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 10050

100

150

200

250

300

350

400

450

500


bps

�c�

K=64, L=0

K=64, L=2

K=32, L=0

K=32, L=2

0 50 100 150 200 250 300 350 400 450 50010

20

30

40

50

60

70

80

90

100

110

bps

SN

R /

dB

�d�

Fig� �� Noise shaping subband coder with oversampling factor �� simulation results

for audio signal� a SNR as a function of quantization stepsize for di�erent noise

shaping system orders L� b SNR di�erences with respect to L ! � from a� c bps

as a function of quantization stepsize for di�erent system orders L� d distortion�rate

characteristic in comparison to alternative subband coders�

noise shaping system of order L ! � leads to SNR improvements of up to ��dB� For

system order L ! �� the block matrix in �� turned out to be singular up to working

precision� Therefore� the maximum order of the noise shaping coder was L ! ��

In our simulations we furthermore observed that noise shaping destroys long runs of

zeros and therefore the e�ective number of bits needed to encode the quantizer outputs

increases� Fig� ��c� shows the number of bits per sample �bps� as a function of the

quantization stepsize for di�erent orders of the noise shaping system� It can be seen

that the number of bps increases with increasing noise shaping system order� Finally�

we entropy�coded the quantizer outputs using a Hu�man coder� and performed an em�

�The Human coder operates on the outputs of all the channels jointly� i�e�� all subband signals are

collected and then jointly Human coded�


K=64

K=32

K=16

K=8

K=4

K=2

K=1

0 10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

110


SN

R /

dB

Fig� �� SNR improvement as a function of the oversampling factor�

pirical distortion�rate analysis of the resulting noise shaping subband coder� Fig� ��d�

shows the SNR as a function of the number of bps used to encode the signal� The

distortion�rate characteristic of the coder with oversampling by a factor of �with and

without noise shaping� is compared to that of a coder with oversampling by a factor �

and no noise shaping� One can conclude that the distortion�rate performance of a noise

shaping coder with oversampling factor and noise shaping system order � is better

than that obtained with K ! and no noise shaping but poorer than the performance

of a coder with oversampling by a factor � and no noise shaping� We furthermore ob�

served that the noise shaping coder was not able to compete with a critically sampled

FB from a rate�distortion point of view�

Fig� �� shows simulation results for a noise shaping subband coder with oversampling

by a factor �� Here� we can see in Fig� ��b� that an SNR improvement �relative

to L ! �� of up to ��dB is possible� This improvement is achieved with a noise

shaping system of order L ! � and for �ne quantization� �For system order L ! ��

the block T�oplitz matrix in �� turned out to be singular�� The distortion�rate

curves of the coder �using Hu�man coding� are depicted in Fig� ��d�� One can observe

that a noise shaping coder with oversampling factor �� and noise shaping system order

L ! � achieves comparable performance �or even outperforms� a subbband coder with

oversampling factor �� and no noise shaping� Again� the noise shaping subband coder

was not able to compete with a critically sampled subband coder from a rate�distortion

point of view�

In Fig� �� we depicted the SNR as a function of the quantization stepsize for di�erent

oversampling factors� For each oversampling factor� we used the noise shaping system

order that led to the best results� For oversampling factors between � and �� we can

observe a �K��dependence of the SNR� i�e�� a �dB increase in the SNR for each doubling


L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80


SN

R /

dB

�a�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 100−2

0

2

4

6

8

10

12


SN

R−

diffe

renc

e / d

B

�b�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700


bps

�c�

K=8, L=0

K=8, L=2

K=4, L=0

0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

80

bps

SN

R /

dB

�d�

Fig� �� Noise shaping subband coder with oversampling factor � simulation results

for �rst�order Gauss�Markov signal� a SNR as a function of quantization stepsize for

di�erent noise shaping system orders L� b SNR di�erences with respect to L ! �

from a� c bps as a function of quantization stepsize for di�erent system orders L�

d distortion�rate characteristic in comparison to alternative subband coders�

of the oversampling factor� The SNR increase in going from oversampling factorK ! ��

toK ! �� is about dB and therefore corresponds to a �K��dependence� We conjecture

that for larger oversampling factors we can get even better performance� �Of course� ��

is the maximum oversampling factor for a ��channel FB� In order to investigate the

behavior of noise shaping coders for larger oversampling factors� we would have to use

FBs with a larger number of channels��

Simulation � �Gauss�Markov signal�� In this simulation the experiments of

Simulation � have been redone for a �rst�order Gauss�Markov signal �AR� process�

de�ned by x�n� ! ax�n� �"u�n� with correlation coe�cient a ! �� and white driving

noise u�n� with variance equal to � Figs� �� show the results� We used the same FBs

and quantizers as in Simulation �� The same general trends as for the audio signal in


L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

110


SN

R /

dB

�a�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 100−5

0

5

10

15

20

25

30

35

40


SN

R−

diffe

renc

e / d

B

�b�

L=0

L=1

L=2

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4x 10

4


bps

�c�

K=64, L=0

K=64, L=2

K=32, L=0

K=32, L=2

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

10

20

30

40

50

60

70

80

90

100

110

bps

SN

R /

dB

�d�

Fig� �� Noise shaping subband coder with oversampling factor �� simulation results

for �rst�order Gauss�Markov signal� a SNR as a function of quantization stepsize for

di�erent noise shaping system orders L� b SNR di�erences with respect to L ! �

from a� c bps as a function of quantization stepsize for di�erent system orders L�

d distortion�rate characteristic in comparison to alternative subband coders�

Simulation � can be observed� We therefore conjecture that the simulation results we

obtained for our two test signals give an indication of the performance of the proposed

coder in a real environment with rather arbitrary signals�

Simulation �Improving quantizer accuracy�� In order to demonstrate that

noise shaping in oversampled FBs is capable of improving the e�ective resolution of the

subband coder� we performed the following experiment� We coded an audio signal using

a paraunitary� ��channel� odd�stacked CMFB with critical sampling and quantizers

with �� quantization intervals in each subband� The resulting SNR was �� dB� Then�

we coded the same signal using a paraunitary� ��channel� odd�stacked CMFB with

oversampling factor and noise shaping system with order L ! �� the quantizers had

� quantization intervals� The resulting SNR was ��dB� Thus� we can see that the

same SNR could be achieved in the oversampled case with a quantization that had


K=64

K=32

K=16

K=8

K=4

K=2

K=1

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120


SN

R /

dB

Fig� �� SNR improvement as a function of the oversampling factor�

far lower resolution than that used in the critical case� For oversampling factor ��

quantizers with � levels� and a noise shaping system of order �� we obtained an SNR

of ��dB� In order to achieve an SNR of ��dB in the critically sampled case one

has to use quantizers with �� levels� Thus� in the latter case we were able to save ��

quantization levels at the cost of increased sample rate� Equivalently� we were able to

achieve the performance of a ��bit quantizer in a critically sampled FB using a ��bit

quantizer in an oversampled FB with noise shaping� Tab� � summarizes the simulation

results described above�

We conclude this subsection by drawing the reader�s attention to the fact that we

were able to get an error�sample rate performance of oversampled FBs that is capable of

behaving better than �K�� So far� the best noise reduction in frame expansions �other

than continuous�time frames of sinc functions� i�e�� oversampled A�D conversion� re�

ported in the literature has been �K� �� Furthermore� it is important

to note that our noise shaping approach can be generalized to arbitrary frame expan�

sions� Thus� we have developed a method that is capable of achieving noise reduction

with an error�sample rate behavior better than �K� in arbitrary frame expansions�

�� Signal Predictive Subband Coders

In Subsec� �� we argued that signal predictive oversampled A�D converters exploit the

redundancy �correlation� inherent in the signal samples to form an accurate estimate

of the next sample value to be quantized� This reduces the dynamic range of the

quantizer input and consequently results in higher resolution with the same number

of quantization intervals� Since more oversampling introduces more redundancy in

the sample values� it is obvious that the prediction accuracy increases for increasing


K L SNR�dB NQ

� ��

� ��

� ��

��

Table �� Improving the e�ective resolution of a subband coder by means of

oversampling and noise shaping� NQ denotes the number of quantization intervals

required�

oversampling factor� Extending this idea to subband coding� we shall next introduce

oversampled signal predictive FBs that make use of the redundancy introduced by the

oversampled analysis FB�

�� Principle

Fig� �� shows the structure of an oversampled signal predictive subband coder� A major

di�erence between a signal predictive A�D converter and a signal predictive subband

coder is the fact that in a subband coder we have to deal with a vector prediction

problem� Consequently� the prediction system is a multi�input multi�output �MIMO�

system� The prediction error system is an N �N MIMO system given by

G�z� ! IN �LXl��

Glz�l� ��

which results in a strictly causal feedback loop �prediction� system IN � G�z� !PLl��Glz

�l� The predictor uses the past L noisy subband signal vectors to estimate

the current subband signal vector�

v�m� !LXl��

Gl �v�m� l� " q�m� l��

This is a �noisy� vector prediction problem� For subband coding using high�resolution

quantizers� the e�ect of quantization noise can be neglected and hence v�m� PLl��Glv�m � l�� However� here we are primarily interested in low�resolution quan�

tization�

The prediction error e�m��! v�m�� v�m� forms the input to the quantizer� We have

e�z� ! v�z�� v�z� ! G�z�v�z�� IN �G�z�� q�z��

By choosing G�z� such that the dynamic range of the quantizer input vector e�m� !

v�m�� v�m� is reduced� it is possible to improve the e�ective quantizer resolution for a

�xed number of quantization intervals�


E�z� Qx�z� xq�z�

v�z�

v�z� e�z� a�z�

IN�G�z�

IN�G�z�

Encoder Decoder

R�z�

Fig� �� Oversampled signal predictive subband coder�

It is easily seen that the quantizer output is a�z� ! G�z��v�z� " q�z�� which in

turn implies that the coder output is xq�z� ! R�z� �v�z� " q�z�� Using a PR FB �i�e��

R�z�E�z� ! IM�� we have R�z�v�z� ! R�z�E�z�x�z� ! x�z� so that

xq�z� ! x�z� "R�z�q�z� �

This yields the following result that can be interpreted as an extension of the funda�

mental theorem of predictive quantization ��

Theorem �� For an oversampled signal predictive subband coder with

PR� i�e�� R�z�E�z� ! IM � the overall reconstruction error xq�z� � x�z� is

given by R�z�q�z��

Theorem �� allows to draw an important conclusion� Since the overall reconstruction

error is equal to the quantization noise �ltered by the synthesis FB� it follows from the

discussion in Chapter � �see Theorem �� that the para�pseudo�inverse R�z� minimizes

the reconstruction error in the case of white and uncorrelated quantization noise�

Just like an oversampled signal predictive A�D converter� an oversampled signal pre�

dictive subband coder has to deal with two types of redundancies� namely natural

redundancy and synthetic redundancy �see Subsec� �� Here� the synthetic redun�

dancy is introduced by the oversampled analysis FB� i�e�� by expanding the input signal

into a redundant set of functions that consists of time�shifted versions of the analysis

�lters� impulse responses� Obviously� for increasing oversampling factor� the prediction

becomes more accurate since there is more synthetic redundancy in the subband signals�

Note� furthermore� that the matrices Gk will in general not be diagonal� so that we are

performing both an intrachannel and an interchannel prediction�

The estimate v�m� !PL

k��Gk �v�m� k� " q�m� k�� of the subband signal vector

v�m� is based on quantized values of the past� We therefore have to deal with a noisy

vector prediction problem� In the case of high�resolution subband coding� i�e�� subband

coding with high�resolution quantizers in the subbands� the e�ect of quantization noise


can be neglected and v�m� PLk��Gkv�m � k�� However� here we are primarily in�

terested in the speci�c case of low�resolution quantization� Therefore� in general� the

high�resolution assumption will not be justi�ed�

The multi�channel system G�z� is said to be minimum phase or minimum delay if all

the roots of detG�z� ! � lie inside the unit circle in the z�plane �� This condition

ensures that the inverse �lter G��z� will be stable� which is needed for stability of

the feedback loop in the signal predictive subband coder �� In the noiseless case

�q�z� ! �� the optimal signal predictive subband coder has a minimum phase prediction

error system G�z� if the process v�m� is stationary and nondeterministic �� A proof

of this important fact for the general case of vector prediction can be found in ��

Although in our case we were not able to prove the minimum phase property of G�z��

we always observed stability of G��z� in our simulation examples�

�� Optimum Multi�Channel Prediction System

We now derive the optimum prediction system� In contrast to the case of noise shaping

subband coders� the input signal x�n� will here be modeled as a random process that

is assumed wide�sense stationary� zero�mean� real�valued� and uncorrelated with the

quantization noise process q�m�� For simplicity� the analysis and synthesis �lters are

assumed real�valued as well� It will be convenient to introduce the �FB input vector�

x�m� ! �x�mM � x�mM " � �� x�mM " M � ��T with M �M correlation matrices

Cx�l� ! Efx�m�xT �m � l�g and power spectral matrix Sx�z� !P�

l��Cx�l�z�l� Using

v�z� ! E�z�x�z�� the power spectral matrix of v�m� is given by ��

Sv�z� !�X

l��Cv�l�z

�l ! E�z�Sx�z�#E�z��

where

Cv�l� ! Efv�m�vT �m� l�g !�X

i��Ei

�Xj��

Cx�j�ETij�l ��

with CTv ��l� ! Cv�l�� With �� and using the fact that x�n� �and hence also v�m��

is uncorrelated with q�m�� it follows that the power spectral matrix of the prediction

error e�m� ! v�m�� v�m� is given by

Se�z� ! G�z�Sv�z� #G�z� " �IN �G�z��Sq�z��IN � #G�z��

Hence� the prediction error variance is obtained as

��e !

N

Z ��

��TrnG�ej��Sv�e

j��GH�ej��

"hIN �G�ej��

iSq�e

j��hIN �GH�ej��

iod� �

Inserting �� into �� and using CTv ��l� ! Cv�l� and CT

q ��l� ! Cq�l�� we obtain

further


�� !

NTr

�Cv��

LXl��

�CT

v �l�Gl "Cv�l�GTl

�

"LX

m��

Gm

LXl��

�Cv�l �m� "Cq�l �m��GTl

��

In order to calculate the matrices Gl minimizing ��e � we set� ��e�Gi

! � and use the

matrix derivative rules from Subsection �� This yields the following block T�oplitz

system of linear equations�

LXl��

�Cv�l � i� "Cq�l � i��GTl ! CT

v �i� for i ! � �� L � ��

or� equivalently��

�� L�� L��

��

�� L�� L��

��

��GT

�

GT��GT

L

�� !

��CT

v ��CT

v ��

CTv �L�

�� with �l ! Cv�l� "Cq�l� �

��

Using �� in �� the minimum prediction error variance is obtained as

��e�min !

NTr

�Cv��

LXl��

Cv�l�GTl�opt

��

where Gl�opt denotes the solution of �� or ��

In the noiseless case� �� reduces to

LXl��

Cv�l � i�GTl ! CT

v �i� for i ! � �� L � ��

which can be solved e�ciently using the multi�channel Levinson recursion �� An�

other important special case where this is possible is the noisy case with white �but

possibly correlated� quantization noise� i�e�� Cq�l� ! Cq��l�� Here� �� reduces to

�� with Cv�� replaced by Cv�� "Cq�� We �nally note that the above derivation

can easily be extended to incorporate correlations between v�m� and q�m��

�� An Example

As a simple example� let us reconsider the paraunitary two�channel Haar FB in Fig� �

�see Subsec� �� with analysis �lters H��z� !�p��" z�� and H��z� !

�p�� z��

and minimum norm synthesis �lters F��z� !��#H��z� and F��z� !

��#H��z�� The input

process is an AR� process� i�e�� a �rst order Gauss�Markov process de�ned by x�n� !

�In the noiseless case� the optimum prediction system can be derived more elegantly using the

orthogonality principle ��


ax�n � � " u�n� with correlation coe�cient a ! �� and white driving noise u�n� with

variance equal to � The autocorrelation function of the AR� process is given by

Cx�k� !��

��

�jkj�� Inserting E� ! �p

�� T and E� ! �p

�� T into �� it

follows after simple manipulations that

Cv�� !

��

� ��

�� and Cv�� !

��

��

��

We shall now calculate the optimum �rst�order prediction system for the case where

no quantization noise is added to the subband signals �noiseless prediction�� With

L ! � there is G�z� ! I��G�z�� From �� it follows with L ! that the optimum

coe�cient matrix G� is given by Cv��GT� ! CT

v �� which yields

G��opt ! Cv��C�Tv �� !

��

��

��

The corresponding minimum prediction error variance is obtained from �� as ��e !��TrfCv��Cv��G

T��optg ! ��

We shall next calculate the optimum prediction system for the noisy case� We assume

that the quantization noise is white and uncorrelated with variance ��q ! in each

channel� i�e�� Cq�k� ! I��k�� We shall again consider a �rst�order predictive subband

coder� i�e�� G�z� ! I� �G�z�� From �� it follows that the optimum matrix G� is

given by

G��opt ! Cv��Cv�� " I��T !

��

��

��

The corresponding prediction error variance is obtained from �� as ��e !��TrfCv��

Cv��GT��optg ! �� Let us compare this result with a prediction system that

does not exploit the inter�channel correlations� and whose matricesGl are thus diagonal�

Setting

GD� !

��

� �

��

it follows from �� that the prediction error variance is given by

��e !�

��

� "

�� "

�

�� "

�

��

From ��e��

! � and ��e�

! �� we obtain the optimum coe�cients as ! �� and

� ! �� and consequently

GD��opt !

��

� ��

��


Note that the diagonal elements of GD��opt equal those of G��opt� The corresponding

prediction error variance is obtained as ��e ! �� We therefore conclude that exploiting

the inter�channel correlation is important� Finally� without prediction �i�e�� L ! � or

e�m� ! v�m�� the variance at the input of the quantizer is ��e ! ��

Usually predictive coders are designed under the assumption of high�resolution quan�

tization� i�e�� the quantization noise is neglected� For coarse quantization� this approach

will obviously lead to a deterioration of coder performance� In the following simulation

we shall demonstrate the behavior of a predictive subband coder designed under the

high�resolution assumption� For the Haar FB oversampled by a factor �� we calculated

the prediction error variance resulting from a �rst�order high�resolution subband coder�

i�e��

G��opt ! Cv��Cv��T �

for white and uncorrelated quantization noise with variance ��q between �� and ��

This is shown by the solid curve in Fig� � � For comparison� the dashed�dotted line

shows the prediction error variance using a predictor that has been adjusted to the

�white and uncorrelated� quantization noise� i�e��

G��opt ! Cv��Cv�� " ��qI��T �

The signal x�n� was an AR� process with a ! �� and driving noise variance equal to

� One can see that for small quantization noise variance� i�e�� in the high�resolution

range� the two prediction error variances are very close� For rougher quantization or

equivalently for lower resolution� the performance of the misadjusted predictor �solid

line in Fig� � � deteriorates signi�cantly� This shows the importance of including the

quantization noise in the design of low�resolution predictive subband coders� In prac�

tice this is done by assuming a quantization noise model and designing the predictor

accordingly�


In the following we present simulation results demonstrating the performance of over�

sampled predictive subband coders� In all the simulations we used a paraunitary� odd�

stacked CMFB �with normalized analysis �lters� i�e�� khkk ! � with N ! � channels

and a prototype of length Lh ! �� Furthermore� all the results have been obtained by

averaging over � realizations �of length �� of the input stochastic process�

Simulation � �Synthetic redundancy�� With the �rst simulation example we

demonstrate that linear prediction is able to remove synthetic redundancy introduced

by the oversampled analysis FB� or equivalently� the predictor is able to exploit syn�

thetic redundancy for improving prediction accuracy� In this experiment� the input


misadjustedadjusted

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Quantization noise variance

Pre

dict

ion

erro

r va

rianc

e/dB

Fig� � � E�ect of neglecting the quantization noise in subband predictor design�

stochastic process was white noise �signal length �� and no quantization was in�

volved �noiseless prediction�� The white noise input contains no natural redundancy�

However� for K � the analysis FB introduces synthetic redundancy in the subband

signals� Therefore� all the prediction gain can be attributed to synthetic redundancy�

Thus� for a critically sampled FB there will be no prediction gain since the underlying

function set is orthogonal and hence nonredundant�

In Fig� ��a� we show the prediction error variance � log ��e�min calculated according

to �� as a function of the order L of the prediction system for di�erent oversam�

pling factors K� Since the energy of the input signal was chosen equal to � there

is a one�to�one correspondence between the prediction error variance and the overall

prediction gain� Note that for increasing L the prediction error variance decreases up

to a certain point� after which it remains constant� We emphasize that the curves in

Fig� ��a� have been calculated using �� and �� and have not been measured

on an implemented system� Fig� ��b� shows the corresponding measured prediction

error variance � log ��e � For prediction system order L � � �not shown�� the per�

formance of the implemented coder deteriorated signi�cantly� This is probably due to

the near�singularity of the block matrix in �� for L � �� which implies that the

prediction system coe�cient matrices obtained using either simple matrix inversion or

the multi�channel Levinson algorithm �see Appendix B� are incorrect due to numerical

errors� Note that in the critically sampled case �the curve labeled K ! �� there is in

fact no prediction gain�

Simulation � �Noisy prediction�� In this simulation example� we shall investigate

the e�ect of additive noise on the coder�s performance� The subband signals have been

adulterated by white uncorrelated noise with equal variance ��q in all channels� i�e��


K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�a�

K=1

K=2

K=4

K=8

1 2 3−20

−15

−10

−5

0

5

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�b�

Fig� �� Noiseless prediction � Prediction error variance � log ��e for a white noise

input signal as a function of the prediction system�s order L for various oversampling

factors K� a computed� b measured�

Cq�k� ! ��qIN��k�� In order to get optimum performance of the predictive subband

coder� the prediction system design took into account the noise statistics by using the

system of equations �� to determine the optimum prediction system�

Fig� �� shows the results for a white input signal and for white uncorrelated noise

with variance ��q ! �� and ��q ! � respectively� Fig� � shows the results of the

same experiment for an AR� signal with correlation coe�cient a ! �� We can see

that due to the noise added to the subband signals� the overall coder performance

deteriorates �compare to Simulation � Fig� �� However� one can observe that the

di�erence between the computed and the measured coder performance is not as large

as in the noiseless case �see Fig� �� This seems to be due to the fact that the additive

noise has a regularization e�ect on the block matrix to be inverted for calculating the

optimum prediction system coe�cient matrices �see ��

Simulation � �Rate�distortion properties�� In this simulation example we inves�

tigate rate�distortion and related properties of oversampled predictive subband coders�

The predictor design took into account the quantization noise� which for the sake of

simplicity was assumed to be white and uncorrelated with variance ��

��in each channel

�- denotes the actual quantization stepsize used�� We caution� however� that especially

in the oversampled case this noise model will not be accurate� As explained above� the

variance of the quantizer input decreases for increasing oversampling factor and for

increasing prediction system order L� Therefore� for a �xed number of quantization

intervals �which in this case was �� we can reduce the quantization stepsize� thereby

reducing the quantization error and consequently the overall reconstruction error �see

Theorem �� Fig� ��a� shows the SNR ! kxk�kxq�xk� for an AR� signal with correlation

coe�cient a ! �� as a function of the prediction system order L for various oversam�


K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−12

−10

−8

−6

−4

−2

0

2

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�a�

K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−12

−10

−8

−6

−4

−2

0

2

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�b�

K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�c�

K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�d�

Fig� �� Noisy prediction � Prediction error variance � log ��e for a white signal and

white� uncorrelated noise� a computed for noise variance ��q ! �� b measured for

noise variance ��q ! �� c computed for noise variance ��q ! � d measured for

noise variance ��q ! �

pling factors K� In Fig� ��b� the di�erences of the curves in Fig� ��a� with respect

to the K ! curve are depicted� One can observe that a predictive subband coder of

order L ! � and oversampling factor K ! � leads to SNR improvements of more than

��dB as compared to the critical case�

Since in a K�fold oversampled FB the sample rate is K times higher than in a

critically sampled FB� the e�ective number of bits per input signal sample �bps��even

after entropy coding�will be higher than in the critical case� This is so� since entropy

coding �Hu�man coding� is not able to remove all the redundancy introduced by the

oversampled analysis FB� Furthermore� prediction destroys long runs of zeros �i�e�� it

whitens the signal�� which implies that the e�ective number of bits needed to encode the

quantizer outputs increases as compared to the nonpredictive case� Indeed� Fig� ��c�

shows that the number of bps using a predictive subband coder increases consistently


K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−12

−10

−8

−6

−4

−2

0

2

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�a�

K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−12

−10

−8

−6

−4

−2

0

2

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�b�

K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−6

−5

−4

−3

−2

−1

0

1

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�c�

K=1

K=2

K=4

K=8

1 2 3 4 5 6 7 8 9 10−6

−5

−4

−3

−2

−1

0

1

System order

Pre

dict

ion

erro

r va

rianc

e / d

B

�d�

Fig� �� Noisy prediction � Prediction error variance � log ��e for an AR�� signal

with a ! �� and white� uncorrelated noise� a computed for noise variance ��q ! ��

b measured for noise variance ��q ! �� c computed for noise variance ��q ! � d

measured for noise variance ��q ! �

with increasing oversampling factor�

Finally� Fig� ��d� shows the distortion�rate characteristic of the signal predictive

subband coder under investigation� The same Hu�man coder as in the simulations

for the noise predictive case �see Subsec� �� has been used� The distortion�rate

characteristic of a critically sampled subband coder �without prediction� is compared

to the distortion�rate characteristics of coders oversampled by factors � and � �with

and without prediction�� One can conclude that the distortion�rate performance of a

predictive subband coder with oversampling factor � and prediction system order

�which in this case is the maximum possible prediction system order� is poorer than

that of a critically sampled subband coder without prediction� Thus� the proposed

oversampled signal predictive subband coders cannot compete with critically sampled

subband coders from a rate�distortion point of view�


K=16

K=8

K=4

K=2

K=1

1 2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90

100

System order

SN

R /

dB

�a�

K=16

K=8

K=4

K=2

K=1

1 2 3 4 5 6 7 8 9 10−10

0

10

20

30

40

50

60

System order

SN

R−

diffe

renc

e / d

B

�b�

K=16

K=8

K=4

K=2

K=1

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

System order

bps

�c�

K=1, L=0

K=2, L=8

K=4, L=8

K=2, L=0

K=4, L=0

0 5 10 15 20 2520

25

30

35

40

45

50

bps

SN

R /

dB

�d�

Fig� �� Signal predictive subband coder with �� quantization intervals and various

oversampling factors� simulation results for an AR�� signal with a ! �� a SNR as

a function of prediction system order L� b SNR di�erences with respect to K !

from a� c bps as a function of prediction system order� d distortion�rate

characteristic�

In the next simulation example� the above experiment has been redone for an input

signal according to a tenth�order AR model for speech with the AR coe�cients taken

from �� Fig� �� shows the results� The same general trends as for the AR� signal

can be observed� We therefore conclude that the simulation results we obtained for

our two test signals give an indication of the performance of the proposed coder in a

practical environment with rather arbitrary signals�

Simulation �Improving quantizer accuracy�� In the last simulation example

we demonstrate that linear prediction of the subband signals is a powerful means to

improve the e�ective resolution of a subband coder� We coded realizations of an AR�

process �length �� with correlation coe�cient a ! �� using a paraunitary� ��

channel� odd�stacked CMFB with oversampling factor and prediction system order


K=16

K=8

K=4

K=2

K=1

1 2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90

100

System order

SN

R /

dB

�a�

K=16

K=8

K=4

K=2

K=1

1 2 3 4 5 6 7 8 9 10−10

0

10

20

30

40

50

60

System order

SN

R−

diffe

renc

e / d

B

�b�

K=16

K=8

K=4

K=2

K=1

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

System order

bps

�c�

K=1, L=0

K=2, L=0

K=2, L=8

K=4, L=0

K=4, L=8

0 5 10 15 20 2520

25

30

35

40

45

50

bps

SN

R /

dB

�d�

Fig� �� Signal predictive subband coder with �� quantization intervals and various

oversampling factors� simulation results for an AR�� speech�like signal� a SNR as a

function of prediction system order L� b SNR di�erences with respect to K !

from a� c bps as a function of prediction system order� d distortion�rate

characteristic�

L ! �� In each of the channels we used a quantizer with � quantization intervals�

The resulting SNR was �� dB� In order to achieve the same SNR performance with

a critically sampled subband coder without prediction� we had to use quantizers with

� quantization intervals in each channel� the SNR was here obtained as ��dB�

Hence� with a predictive subband coder with oversampling factor � prediction system

order L ! � and ��bit quantizer in each of the channels we were able to achieve the

performance of �bit quantizers in a critically sampled subband coder� We thus saved

� bits of quantizer resolution in each of the channels�

For a predictive subband coder with oversampling factor �� quantizers with � quan�

tization intervals� and a prediction system of order � we obtained an SNR of ��dB�

In order to achieve the same SNR in the critically sampled case one has to use quantiz�

ers with �� quantization intervals� Tab� � summarizes the simulation results described


K L SNR�dB NQ

� ��

� ��

� ��

� � ��

Table �� Improving the e�ective resolution of a subband coder by means of

oversampling and prediction� NQ denotes the number of quantization intervals

required�

above� We can conclude that the same SNR as in the critically�sampled� non�predictive

case could be achieved in the oversampled case with a quantization that had far lower

resolution than that used in the critical case�

Chapter �

Conclusion

This dissertation has investigated theoretical and practical aspects of oversampled �lter

banks� We have shown that the theory of frames is a powerful vehicle for the analysis

and design of oversampled �lter banks �FBs�� A key result on which most of our theory

was based is the fact that the polyphase matrices provide matrix representations of

the frame operator� We demonstrated that the frame bounds characterize important

numerical properties of FBs� and that they can be obtained by an eigenanalysis of the

polyphase matrices� For a given oversampled analysis FB� we provided a compact and

useful parameterization of all synthesis FBs providing perfect reconstruction �PR�� and

we discussed the PR synthesis FB with minimum norm �i�e�� the particular synthesis

FB obtained from frame theory� and its approximative construction� We formulated

conditions for an oversampled FB to provide a frame decomposition� We also proposed

a new method for constructing paraunitary FBs from given nonparaunitary FBs�

We furthermore studied oversampled modulated FBs� Speci�cally� we introduced

two classes of oversampled cosine modulated �lter banks �CMFBs�� thus extending a

classi�cation of CMFBs recently proposed by Gopinath for critical sampling� The class

of oversampled even�stacked CMFBs is especially attractive as it allows both PR and

linear phase �lters in all channels� The CMFB recently introduced for critical sampling

by Lin and Vaidyanathan has been extended to the oversampled case and shown to be

a special case of even�stacked CMFBs�

We derived PR conditions for oversampled odd� and even�stacked CMFBs and demon�

strated that� concerning both PR and frame�theoretic properties� CMFBs are closely

related to DFT FBs of the same stacking type and with twice the oversampling factor�

In particular� it has been shown that any oversampled PR CMFB corresponds to an

oversampled PR DFT FB� and that under an additional condition the frame bound

ratio of a CMFB equals that of the corresponding DFT FB� We brie�y discussed the

design and e�cient DCT�DST�based implementation of oversampled CMFBs�

We have introduced two methods for noise reduction in oversampled FBs� These

��

�� CHAPTER � CONCLUSION

methods are based on predictive quantization� they can be viewed as extensions of

oversampled predictive A�D converters� We demonstrated that predictive quantization

in oversampled FBs yields considerable quantization noise reduction&in fact� signi��

cantly larger noise reduction than previous methods&at the cost of increased rate� The

combination of oversampled FBs with noise shaping or linear prediction improves the

e�ective resolution of subband coders and is thus well suited for applications where&for

technological or other reasons&quantizers with low resolution �even single bit� have to

be used� Using low resolution quantizers in the digital domain increases circuit speed

and allows for lower circuit complexity� One�bit codewords� for example� eliminate the

need for word�framing�

We furthermore addressed the long�standing question of rate�distortion performance

of oversampled subband coders� Our simulation results suggest that� from a rate�

distortion point of view� oversampled subband coders are inferior to critically sampled

subband coders� However� this issue might appear in a di�erent light if perceptual

distortion measures are adopted and the increased design freedom in oversampled FBs

is exploited to obtain �lters with improved perceptual properties�

Appendix A

The Theory of Frames

A�� Motivation and De�nition of Frames

This appendix provides an introduction to the theory of frames� which has been intro�

duced by R� J� Du�n and A� C� Schae�er �� It has been worked out by I� Daubechies

�� and several other authors �� We shall review the general theory of frames�

The special type of uniform �lter bank frames� which is the type of frames relevant to

�lter banks� is discussed in great detail in Chapter �� Throughout this appendix� we

shall make extensive use of basic linear operator theory� which is discussed for example

in �� Most of the material in this appendix is taken from �� An excellent

overview of frame theory and related topics can also be found in ��

In the following� let� fgk�n�g �k � K� K a denumerable set� be a set of functions taken

from a Hilbert space H� Note that this set need not be orthonormal or orthogonal� It is

convenient to de�ne a linear operator T which assigns to each signal x�n� the sequence

of inner products �Tx�k ! hx� gki� Throughout this appendix we assume that fgk�n�gis a Bessel sequence� i�e��

Pk jhx� gkij� � � for all x�n� � H�

De�nition A�� The linear operator T is de�ned as the operator that

maps the Hilbert space H into the space l��ZZ� of square�summable complex

sequences�

T � H � l��ZZ��

it assigns to each signal x�n� the sequence of inner products hx� gki�T � x� hx� gkik�K or equivalently �Tx�k ! hx� gki�

Note that kTxk� !P

k j�Tx�kj� !P

k jhx� gkij�� i�e�� the energy kT xk� of Tx can be

expressed as

kTxk� !Xk

jhx� gkij�� A��

�We restrict our attention to discretetime signals�

��

� APPENDIX A� THE THEORY OF FRAMES

Since fgk�n�g is a Bessel sequence� there exists a constant B � � such that ��

kTxk� � Bkxk� � x�n� � H� �A��

We shall next formulate the properties which the gk�n� and the operator T should

satisfy�

� The linear operator T is continuous and hence bounded�� Since we are considering

Bessel sequences fgk�n�g this requirement is automatically satis�ed �� From

�A�� it follows that the expansion coe�cients hx� gki have �nite energy if the

signal x�n� has �nite energy� i�e�� the expansion coe�cients are square�summable�

�� The signal x�n� can be reconstructed from the coe�cients hx� gki� In order to

satisfy this requirement� the operator T has to be left invertible�� which means

that T is invertible on its range� Ran�T �� Note that in general Ran�T � will be a

subspace of l��ZZ�� The left inverse of T will be denoted T��

�� The reconstruction of the signal x�n� from the sequence hx� gki should be numeri�

cally stable� If two sequences ck ! hx� gki and dk ! hy� gki are �close� in the sense

that kc � dk� !P

k jck � dkj� is small� then x�n� and y�n� should be �close� in

the sense that kx � yk� !P

n jx�n� � y�n�j� is small� Thus� for example� small

errors in transmitted analysis coe�cients or round�o� errors in the reconstruction

process would not be disastrous� We therefore require that the left inverse T ��

be continuous and hence bounded� Note that the mere continuity of T does not

say anything about the continuity of T��

Requirement implies �A�� i�e��

kTxk� � Bkxk�� A��

with a positive constant B �� With �A�� this can be rewritten as

Xk

jhx� gkij� � Bkxk�� A��

which implies that the expansion coe�cients hx� gki have �nite energy if the signal x�n�

has �nite energy� In requirement � we demand that the left inverse T�� is continuous�

This is the case if and only if the operator T is bounded below �� i�e��

kTxk� � Akxk��A linear operator L � X � Y where X and Y are normed linear spaces is said to be bounded if

there is a real number M � � such that kLxk � Mkxk for all x�n� � X � If the operator satis es

kLxk � mkxk for all x�n� � X it is said to be bounded below��A mapping L � X � Y is said to be left invertible if a mapping M � Y � X exists such that

ML � I �the identity mapping� on the space X ��Ran�T � denotes the range of the operator T � i�e�� Ran�T � � fy j y � Tx with x � Xg

A�� MOTIVATION AND DEFINITION OF FRAMES �

with A � �� With �A�� this condition can be rewritten asXk

jhx� gkij� � Akxk��

Combining the last two requirements� we are now able to give the following de�nition�

De�nition A�� A set of signals gk�n� � H �k � K� is called a frame for

the Hilbert space H if

Akxk� � Xk

jhx� gkij� � Bkxk� � x�n� � H �A��

with A�B � IR and � � A � B � �� Valid constants A and B are called

frame bounds�

Using �A�� in �A�� an equivalent formulation of �A�� is

Akxk� � kTxk� � Bkxk� � x�n� � H� �A��

This means that the energy of Tx is bounded above and below by the product of the

frame bounds and the signal�s energy� The existence of a lower frame bound A � �

guarantees a certain degree of numerical stability� Besides that it also guarantees the

completeness of the set fgk�n�gk�K� i�e�� the only signal that is orthogonal to all the

gk�n� �k�K� is the zero signal x�n� ! ��

We next de�ne the operator T which maps the coe�cient l��ZZ� space to the Hilbert

space H�

De�nition A�� The linear operator T is de�ned as

T � l��ZZ��H �A��

�Tck��n� !Xk

ck gk�n�� where ck � l��ZZ�� A� �

We shall next show that the operator T is the adjoint� T � of T � Consider an arbitrary

sequence ck � l��ZZ�� We have to prove that

hT x� cki ! hx�T cki�This can be seen by noting that

hTx� cki !Xk

hx� gkic�k

hx�Tcki !Dx�Xk

ckgkE!Xk

c�khx� gki�

So we showed that the adjoint operator of T is T�

T ! T �� A��

In what follows� we shall always write T � instead of T ��Let L � X � Y be a bounded linear operator where X and Y are Hilbert spaces� Furthermore

x � X and y � Y � The unique bounded linear operator L� which satis es�Lx� y

��x�L�y

for any

x � X and y � Y is the adjoint of L�

�� APPENDIX A� THE THEORY OF FRAMES

A�� The Frame Operator

De�nition A�� For a frame fgk�n�g the operator

S ! T �T �

�Sx��n� !Xk�K

hx� gki gk�n�

is said to be the frame operator�

We can write

�Sx��n� !Xn�S�n� n�� x�n��

where S�n� n�� the kernel of the frame operator S �i�e� its in�nite matrix representation��

is given by

S�n� n�� !Xk�K

gk�n� g�k�n

�� A��

We note that S maps H onto H �� and that

Xk

jhx� gkij� ! kTxk� ! hSx� xi� �A��

We are now able to formulate the frame condition with the frame operator S� Equation

�A�� can also be written as

Akxk� � hSx� xi � Bkxk�� A��

An equivalent formulation of �A�� is �

AI � S � BI� �A��

where I is the identity operator on H� From �A�� it also follows that

kSk � B� �A��

This is seen as follows�

kSk ! supx�H

kSxkkxk ! sup

x�H

vuuthSx�Sxikxk� !

vuuut supx�H

hS�x� xikxk�

!q�sup ! sup�

By Theorem A� �see below� this implies kSk � B� Here we have used S� ! S�

We shall now formulate an important theorem on the eigenvalues of the frame oper�

ator��This relation is to be understood in the sense of quadratic forms� i�e�� A

�x� x��Sx� x

� B

�x� x

for all x�n� � H�

A�� THE DUAL FRAME �

Theorem A�� Let inf �sup� denote the in�mum �supremum� of the

eigenvalues of the frame operator S� Then

A � inf and B � sup�

Proof� With ��

inf � hSx� xikxk� � sup

and �A�� the proof follows immediately� �

While the frame bounds A�B are not uniquely de�ned� this theorem shows that the

tightest possible frame bounds are given by inf and sup� respectively�

We shall now discuss the properties of S� The frame operator S is

� linear�

�� positive de�nite� i�e� hSx� xi � � for all x�n� � H�

�� self�adjoint� S� ! S�

�� invertible on H� i�e�� the inverse operator S�� exists and is bounded with

BI � S��

AI�

The linearity of S follows from the fact that S is obtained by cascading a linear operator

and its adjoint �see De�nition A�� To see that S is positive de�nite note that� with

�A�� and �A��

hSx� xi ! kT xk� � Akxk� � �

for all x�n� ! �� Hence� S must also be self�adjoint� The invertibility follows directly

from A I � S � B I with A � ��

Besides the frame operator S ! T �T � also the composite operator TT � is important�

This operator will be discussed in Section A��

A�� The Dual Frame

We have seen above that S is invertible on H with bounded inverse S�� satisfying

BI � S

��

AI� �A��


Theorem A�� If fgk�n�g is a frame with frame bounds A and B� then

the family f#gk�n�g given by

#gk�n� ! �S��gk��n� �A��

is a frame with bounds A� ! �Band B� ! �

A� The associated operator which

assigns to each x�n� � H the sequence hx� #gki��T � H � l��ZZ�

�T � x� hx� #gkik�K or equivalently � �Tx�k ! hx� #gki�

is given by�T ! TS�� ! T �T �T ��

Proof �� From �A�� it follows with B � � that S�� is positive de�nite and thus

self�adjoint� Hence we have hx� #gki ! hx�S��gki ! hS��x� gki for all x�n� � H� Thus�

using �A�� we obtain

Xk

jhx� #gkij� !Xk

jhS��x� gkij� ! hS�S��x��S��xi ! hx�S��xi ! hS��x� xi�

So we can conclude from �A�� that

Bkxk� �X

k

jhx� #gkij� �

Akxk�� A��

i�e�� the #gk�n� constitute a frame with frame bounds A� ! �B

and B� ! �A� It remains to

show that �T ! TS��

� �Tx�k ! hx� #gki ! hx�S��gki ! hS��x� gki ! �TS��x�k� �

We shall call f#gk�n�g the dual frame associated to the frame fgk�n�g� For the dual

frame too� it is convenient to introduce the dual frame operator�

De�nition A�� The frame operator associated to the dual frame�

�S ! �T� �T � � �Sx��n� !

Xk

hx� #gki#gk�n��

will be called the dual frame operator�

Theorem A�� The dual frame operator �S satis�es

�S ! S�� A� �

A�� THE DUAL FRAME ��

Proof� For any x�n� � H� we have

� �Sx��n� !Xk

hx� #gki#gk�n� !Xk

hx�S��gki�S��gk��n�

!

�S��X

k

hS��x� gkigk��n� ! �S��SS��x��n� ! �S��x��n��

Now there arises the question how the dual frame f#gk�n�g can be obtained from

fgk�n�g� Due to Theorem A�� #gk�n� ! �S��gk��n�� The following theorem presents a

series expansion of S��

Theorem A� �� Let fgk�n�g and f#gk�n�g be dual frames in H� and let A

and B be the frame bounds of fgk�n�g� We then have

S�� !

�

A "B

�Xl��

I � �

A"BS

�l�A��

or equivalently

#gk�n� !�

A"B

�Xl��

I � �

A"BS

�lgk�n�

�� A��

The Neumann series �A�� is guaranteed to converge uniformly �� This

convergence is governed by the frame bounds A�B according to

!!!!I � �

A"BS

!!!! � B � A

B " A� �

Proof �� Recall that #gk�n� ! �S��gk��n�� We have S�� ! �AB

hI � �I � �

ABS�i��

�

When the Neumann series expansion ofhI � �I � �

ABS�i��

converges uniformly� we

obtain

S�� !

�

A"B

�I �

I � �

A "BS

��!

�

A"B

�Xl��

I � �

A"BS

�l�

The Neumann series converges uniformly if kI � �AB

Sk � � From �A�� it follows

that

��B I � ��S � ��A I j" �A"B�I

�A�B� I � �A"B� I � �S � �B � A� I

�B � A

A"BI � I � �

ABS � B � A

B " AI�

�The Neumann series expansion of �I ��A�� where A is a continuous linear operator and � �CI�

is given by �I � �A�� P�

k�� kA

k �� Note that this series expansion converges absolutely only

if kAk � �

j�j ��


So we conclude that!!!!I � �

A "BS

!!!! � B � A

B " A� � since � � A � B� � �A��

The closer A to B� the better the convergence� Frames with A B are called snug

frames� A special �optimum� case is a frame with A ! B� which is known as tight frame

�see Section A��

We shall next consider the operator which provides the orthogonal projection of an

arbitrary sequence onto the range of the operator T �

Theorem A�� The operator

P � l��ZZ�� Ran�T � � l��ZZ�

de�ned as

P ! �TT � ! TS��T � ! T �T�

is the orthogonal projection operator on Ran�T � �which is a closed subspace

of l��ZZ��

Note that P is de�ned for all sequences ck � l��ZZ�� and

�P c�k ! � �TT �c�k !

��TXl

clgl�n�

�k

!DX

l

clgl� #gkE!Xl

clhgl� #gki

or equivalently

�P c�k ! �T �T�c�k !

�TXl

cl#gl�n�

�k

!DX

l

cl#gl� gkE!Xl

clh#gl� gki�

Proof �� We have to show that

� P is the identity operator I on Ran�T ��

�� P is the zero operator on �Ran�T ��

We �rst prove that P is the identity operator for c � Ran�T �� If c � Ran�T �� c ! Tx�

then P c ! �TT �c ! TS��T �c ! TS��T �T x ! TS��Sx ! Tx ! c�

The second part of the proof is based on the fact �shown in �� p� �� that c ��Ran�T �� if and only if T �c ! �� Assuming now that c � �Ran�T �� one obtains

P c ! �TT �c ! �

since T �c ! � for c � �Ran�T ��

A�� SIGNAL EXPANSIONS ��

A�� Signal Expansions

The following theorem can be considered the central result of the theory of frames� It

states that any signal x�n� � H can be expanded into a frame� The expansion coe�cients

can be chosen as the inner products of x�n� with the dual frame functions� The question

whether or not these coe�cients are unique will be addressed in Section A��

Theorem A�� Let fgk�n�g and f#gk�n�g be dual frames in H� Any signal

x�n� � H can be expressed as the unconditionally norm�convergent series

x�n� ! �T � �Tx��n� !Xk

hx� #gki gk�n� �A��

x�n� ! � �T�Tx��n� !

Xk

hx� gki #gk�n�� A��

Note that� equivalently�

T� �T ! �T

�T ! I� �A��

where I is the identity operator on H�

Proof� We have

�T � �T x��n� !Xk

hx� #gkigk�n� !Xk

hx�S��gkigk�n�

!Xk

hS��x� gkigk�n� ! �SS��x��n� ! x�n��

which proves T � �T ! I or� equivalently� �A�� The expansion �A�� can be proved in

a similar manner� �

Equations �A�� and �A�� are �completeness relations�� since they can also be written

as Xk

gk�n� #g�k�n

�� !Xk

#gk�n� g�k�n

�� ! I�n� n�� A��

where I�n� n�� is the kernel of the identity operator I on H�

The duality of the families fgk�n�g and f#gk�n�g is also expressed by the following

corollary�

Corollary A�� For any x�n�� y�n� � H we have

hx� yi ! hTx� �T yi !Xk

hx� gki h#gk� yi �A��

and hx� yi ! h �T x�T yi !Xk

hx� #gki hgk� yi� �A��


Proof� With �T�T ! I and T � �T ! I� we have

hx� yi ! hIx� yi ! h �T �Tx� yi ! hTx� �T yi !Xk

hx� gkih#gk� yi

hx� yi ! hIx� yi ! hT � �Tx� yi ! h �Tx�T yi !Xk

hx� #gkihgk� yi� �

De�nition A�� A frame fgk�n�g with frame bounds A ! B is called a

tight frame�

Note that� for a tight frame�

hSx� xi !Xk

jhx� gkij� ! Akxk�� A��

Corollary A�� If fgk�n�g is a tight frame in H� then

S ! AI �A��

or equivalently

�Sx��n� !Xk

hx� gki gk�n� ! Ax�n� � x�n� � H� �A��

Proof� Combining A I � S � B I and A ! B� we obtain S ! AI and furthermore

x�n� ! �Ix��n� !

A�Sx��n� !

A

Xk

hx� gki gk�n��

If fgk�n�g is a tight frame� then the dual frame f#gk�n�g is tight as well� This can be

seen by noting that S ! AI implies

S�� !

AI�

Furthermore we have Xk

jhx� #gkij� ! hS��x� xi !

Akxk��

We can see that tight frames provide an easy way of reconstruction� because #gk�n� !�Agk�n� and thus calculation of the dual frame is trivial� It is evident that every orthonor�

mal system is a tight frame with A ! � Note� however� that conversely a tight frame

�even with A ! � need not be an orthonormal or orthogonal system� An interesting

special case is considered in the next theorem�

Theorem A�� A tight frame with A ! and kgkk� ! for all k � K

is an orthonormal system�

A�� SIGNAL EXPANSIONS ��

Proof �� Combining

hSgl� gli ! Akglk� ! kglk�

and

hSgl� gli !Xk

jhgl� gkij� ! kglk� "Xk ��ljhgl� gkij��

we obtain

kglk� "Xk ��ljhgl� gkij� ! kglk��

Since kglk� ! for all l� it follows thatPk ��ljhgl� gkij� ! �� This implies that the functions

gk�n� are orthogonal to each other� which completes the proof� �

As mentioned in Section A�� the convergence speed of the expansion �A�� depends

on the quotient of the frame bounds B and A� Let us consider the extreme case of

retaining only the �rst term �corresponding to l ! �� in the series expansion �A��

i�e�� the dual frame functions #gk�n� are approximated by

#gk��n� !�

A"Bgk�n��

Note that this approximation of #gk�n� corresponds to an approximation of the recon�

structed signal x�n� !P

k hx� gki #gk�n� by

x��n� !�

A"B

Xk

hx� gki gk�n��

The following theorem states a result on the error incurred when using this crude

approximation�

Theorem A� �� The norm of the error signal

R�n� ! x�n�� x��n�

incurred when approximating x�n� by

x��n� !�

A "B

Xk

hx� gki gk�n�

is bounded as

kRk �BA�

AB"

kxk� �A��


Proof �� From Theorem A�� and Theorem A�� we have

x�n� !Xk

hx� gki #gk�n� !Xk

hx� gki �

A"B

�Xl��

I � �

A"BS

�lgk�n�

!Xk

hx� gki �

A "B

gk�n� "

�Xl��

I � �

A "BS

�lgk�n�

��

Hence we obtain

R�n� ! x�n��Xk

hx� gki �

A"Bgk�n�

!Xk

hx� gki �

A "B

�Xl��

I � �

A "BS

�lgk�n�

and furthermore

kRk !

!!!!!Xk

hx� gki �

A"B

�Xl��

I � �

A"BS

�lgk

!!!!!!

�

A"B

!!!!!�Xl��

I � �

A"BS

�lXk

hx� gkigk!!!!!

� �

A"B

!!!!!�Xl��

I � �

A"BS

�l!!!!!!!!X

k

hx� gkigk!!!

!�

A"B

!!!!!�Xl��

I � �

A"BS

�l!!!!! kSxk� �

A"B

�Xl��

!!!!I � �

A"BS

!!!!l kSxk� �

A"B

�Xl��

!!!!I � �

A"BS

!!!!l kSk kxk�In the proof of Theorem A�� we have shown that

!!!!I � �

A"BS

!!!! � B � A

B " A�

Hence

�

A"B

�Xl��

!!!!I � �

A"BS

!!!!l kSk kxk � �

A "B

�Xl��

B � A

B " A

�lkSk kxk

!�

A "B

B�ABA

� B�ABA

kSk kxk�

With kSk � B� we �nally obtain

kRk �BA�

AB"

kxk� �

A�� SIGNAL EXPANSIONS �

We can also see that the reconstruction error is small when B � A� Thus� for

snug frames x��n� gives a good approximation of x�n�� For tight frames� kRk ! ��

x�n� ! x��n�� and �A�� is valid�

We shall now formulate an iterative algorithm for the reconstruction of the signal x�n�

from its expansion coe�cients hx� gki �� This algorithm is known to converge slowly�

Better algorithms were recently proposed in ��

Corollary A�� Let fgk�n�g be a frame for the Hilbert space H with frame

bounds A and B� Then every signal x�n� � H can be reconstructed from

the coe�cients hx� gki� k � K by the recursion

xl�n� ! xl��n� "�

A"B�S�x� xl��n�� l � �A��

initialized by x��n� ! �� That is� x�n� ! liml�� xl�n� with the error bound

kx� xlk � B � A

B " A

�lkxk�

The information needed for the iterative reconstruction of the signal x�n� is �Sx��n� !Pkhx� gkigk�n�� This requires the knowledge of the expansion coe�cients hx� gki and the

frame functions gk�n�� We shall now give the proof of Corollary A��

Proof �� From �A�� we know that!!!!I � �

A"BS

!!!! � B � A

B " A� �

Using �A�� we can write

x�n�� xl�n� ! x�n�� xl��n��

A"B�S�x� xl��n�

! I � �

A"BS

��x� xl��

��n��

Iterating this recursion gives

x�n�� xl�n� !

� I � �

A"BS

�l�x� x��

��n��

Taking the norm yields

kx� xlk �!!!!! I � �

A "BS

�l!!!!! kx� x�k

�!!!!I � �

A"BS

!!!!l kx� x�k�With �A�� and x��n� ! � it follows that

kx� xlk � B � A

B " A

�lkxk� �A��

Since B�ABA

� � �A�� guarantees convergence of the sequence xl�n� towards x�n� as

l � ��


A�� Exact Frames and Biorthogonality

A set of functions fgk�n�g is complete in a Hilbert space H if hx� gki ! � for all k � K

and for x�n� � H implies x�n� ! �� i�e�� the only function in H which is orthogonal to

every gk�n� is x�n� ! �� Every x�n� � H can be expanded into a complete set fgk�n�g�Obviously� frames are complete sets of functions �cf� �A�� and �A�� On the other

hand� the coe�cients ck occurring in the expansion x�n� !P

k ckgk�n� need not be

unique� Frames yielding unique coe�cients are called exact� A frame fgk�n�g for a

separable Hilbert space H is exact if and only if it is a Riesz basis �� We shall call a

frame �inexact� if the removal of an arbitrary frame signal gm�n� leaves a set fgk�n�gk ��mthat is again a frame�

De�nition A�� Frames fgk�n�g which become incomplete sets when an

arbitrary function gm�n� is removed are called exact�

In Section A�� we shall show that the expansion of a signal into an exact frame is unique

�i�e�� the expansion coe�cients are unique�� In order to give a condition under which

a frame is exact� we need two lemmas� The �rst lemma states that among all possible

expansion coe�cient sequences ck satisfying x�n� !P

k ck#gk�n�� the frame coe�cients

ck ! hx� gki have minimum l� norm or� equivalently� that among all possible expansion

coe�cient sequences ck satisfying x�n� !P

k ckgk�n�� the frame coe�cients ck ! hx� #gkihave minimum l� norm�

Lemma A�� Given a frame fgk�n�g and given x�n� � H� let ak !

hx� #gki so that x�n� !P

k ak gk�n�� If it is possible to �nd other scalars ck

such that x�n� !P

k ck gk�n�� then we must have

Xk

jckj� !Xk

jakj� "Xk

jak � ckj�� A��

Note that this impliesP

k jckj� �P

k jakj�� i�e�� the coe�cients ck have larger l� norm�

This statement will be reconsidered from a di�erent point of view in Section A� �

Proof �� We have

ak ! hx� #gki ! hx�S��gki ! hS��x� gki ! h#x� gki

with #x�n� ! �S��x��n�� Therefore�

hx� #xi !DX

k

akgk� #xE!Xk

akhgk� #xi !Xk

aka�k !

Xk

jakj�

and

hx� #xi !DX

k

ckgk� #xE!Xk

ckhgk� #xi !Xk

cka�k�

A�� EXACT FRAMES AND BIORTHOGONALITY �

We conclude that Xk

jakj� !Xk

cka�k !

Xk

c�kak� �A��

Hence�

Xk

jakj� "Xk

jak � ckj� !Xk

jakj� "Xk

�ak � ck��a�k � c�k�

!Xk

jakj� "Xk

jakj� �Xk

akc�k �

Xk

a�kck "Xk

jckj��

Using �A�� we get

Xk

jakj� "Xk

jak � ckj� !Xk

jckj�� A��

Lemma A�� Let fgk�n�g be a frame� Then for each m we have

Xk ��m

jhgm� #gkij� !� jhgm� #gmij� � j� hgm� #gmij�

��

Proof �� There is obviously gm�n� !P

k ckgk�n�� where cm ! and ck ! � for k ! m�

so thatP

k jckj� ! � Furthermore let ak ! hgm� #gki� Eq� �A�� yields

!Xk

jckj� !Xk

jakj� "Xk

jak � ckj�

!Xk

jakj� " jam � cmj� "Xk ��m

jak � ckj�

!Xk

jhgm� #gkij� " jhgm� #gmi � j� " Xk ��m

jhgm� #gkij�

! �Xk ��m

jhgm� #gkij� " jhgm� #gmij� " j� hgm� #gmij�

and hence Xk ��m

jhgm� #gkij� !� jhgm� #gmij� � j� hgm� #gmij�

��

We are now able to formulate a condition for a frame to be exact�

Theorem A�� The removal of a function gm�n� from a frame fgk�n�gleaves either a frame or an incomplete set� In fact�

hgm� #gmi ! for arbitrary m � fgk�n�g is exact

hgm� #gmi ! for arbitrary m � fgk�n�g is inexact�


Proof �� We �rst show that hgm� #gmi ! implies that fgk�n�gk ��m is incomplete and

hence fgk�n�g is an exact frame� From Lemma A�� we have

Xk ��m

jhgm� #gkij� ! � jhgm� #gmij� � j� hgm� #gmij��

�

Suppose now that hgm� #gmi ! � ThenP

k ��m jhgm� #gkij� ! �� so hgm� #gki ! h#gm� gki ! �

for k ! m� That is� #gm�n� is orthogonal to gk�n� for every k ! m� But #gm�n� ! � since

h#gm� gmi ! � Therefore fgk�n�gk ��m is incomplete� because #gm�n� is orthogonal to every

function of the family fgk�n�gk ��m�We next show that hgm� #gmi ! implies that fgk�n�gk ��m is still a frame� We can

always write

gm�n� !Xk

hgm� #gkigk�n� ! hgm� #gmigm�n� "Xk ��m

hgm� #gkigk�n��

If hgm� #gmi ! � this can be written as

gm�n� !

� hgm� #gmiXk ��m

hgm� #gkigk�n��

and for x�n� � H we have

jhx� gmij� !

��

� hgm� #gmi

��Xk ��m

hgm� #gkihx� gki��

�

j� hgm� #gmij��Xk ��m

jhgm� #gkij��Xk ��m

jhx� gkij��

Therefore

Xk

jhx� gkij� ! jhx� gmij� "Xk ��m

jhx� gkij�

�

j� hgm� #gmij��Xk ��m

jhgm� #gkij��Xk ��m

jhx� gkij�� " X

k ��mjhx� gkij�

!Xk ��m

jhx� gkij�� "

j� hgm� #gmij�Xk ��m

jhgm� #gkij��

! CXk ��m

jhx� gkij�

or equivalently

C

Xk

jhx� gkij� �Xk ��m

jhx� gkij�

with

C ! "

j� hgm� #gmij�Xk ��m

jhgm� #gkij��

A�� EXACT FRAMES AND BIORTHOGONALITY ��

With �A�� it follows that

A

Ckxk� �

C

Xk

jhx� gkij� �Xk ��m

jhx� gkij� �Xk

jhx� gkij� � Bkxk��

so fgk�n�gk ��m is a frame with bounds AC� B�

To see that� conversely� an exact fgk�n�g implies that hgm� #gmi ! for arbitrarym� we

suppose that fgk�n�g is exact and hgm� #gmi ! � But the condition hgm� #gmi ! implies

that fgk�n�g is inexact� which is a contradiction� It remains to show that an inexact

fgk�n�g implies hgm� #gmi ! for allm� Suppose that fgk�n�g is inexact and hgm� #gmi ! �

The condition hgm� #gmi ! implies that fgk�n�g is exact� which is a contradiction� �

Corollary A� �� If fgk�n�g is an exact frame� then fgk�n�g and f#gk�n�gare biorthogonal� i�e��

hgm� #gki ! �mk !

�� if k ! m

�� if k ! m�

Conversely� if fgk�n�g and f#gk�n�g are biorthogonal� then fgk�n�g is exact�

Proof �� If fgk�n�g is exact� then by Theorem A� we must have hgm� #gmi ! for every

m� and hence by Lemma A�� we haveP

k ��m jhgm� #gkij� ! � and thus also hgm� #gki ! �

for all k ! m� as claimed� It remains to show that� conversely� the biorthogonality of

fgk�n�g and f#gk�n�g implies that the frame fgk�n�g is exact� For biorthogonal functions

gk�n� and #gk�n� we have hgm� #gki ! �mk and hence hgm� #gmi ! for allm� So by Theorem

A� we conclude that fgk�n�g is exact� �

Corollary A�� A frame fgk�n�g is exact if and only if the dual frame

f#gk�n�g is exact�

Proof� This follows immediately from Corollary A�� and the symmetry inherent in the

inner product hgm� #gki� �

The next theorem establishes bounds on the energy of gk�n��

Theorem A�� Let fgk�n�g be a frame with bounds A�B� Then

kgkk� � B�

Furthermore if fgk�n�g is exact� then

kgkk� � A�


Proof ��

� For m �xed we have

kgmk� ! jhgm� gmij� � jhgm� gmij� "Xk ��m

jhgm� gkij�

!Xk

jhgm� gkij� � Bkgmk��

so kgmk� � B�

�� If fgk�n�g is an exact frame� then by Corollary A�� fgk�n�g and f#gk�n�g are

biorthogonal� Therefore� for m �xed we have

Ak#gmk� �Xk

jh#gm� gkij� ! jh#gm� gmij� � k#gmk�kgmk��

so kgmk� � A� �

A�� Frames and Bases

We shall now give some results about the relation between frames and bases� The

central result is that an exact frame is a basis� i�e�� the expansion of a signal into an

exact frame is unique� We �rst give three de�nitions�

De�nition A� � A set of signals fgk�n�g in a Hilbert space H is a basis for

H if for every x�n� � H there exist unique scalars ck such that

x�n� !Xk

ck gk�n��

De�nition A�� A basis gk�n� in a Hilbert space H is bounded if � �

infkkgkk � supkkgkk �� for every k�

De�nition A�� A basis fgk�n�g in a Hilbert space H is unconditional

if the seriesPkck gk�n� converges unconditionally� that is� every permutation

of the series converges �to the same limit��

We are now able to state

Theorem A�� A set of signals fgk�n�g in a Hilbert space H is an

exact frame for H if and only if it is a bounded unconditional basis for H�

A�� FRAMES AND BASES ��

Proof �� Assume that fgk�n�g is an exact frame with bounds A�B� Then from

Theorem A�� we have

kgkk� � A and kgkk� � B�

We have thus shown that the set of signals fgk�n�g is bounded in the sense of De�nition

A�� By Theorem A�� we have

x�n� !Xk

ckgk�n� with ck ! hx� #gki

for any x�n� � H� We now have to prove that this representation is unique� Assume

that there is a representation of the form x�n� !Pkak gk�n� with ak ! ck� Then�

cl ! hx� #gli !DX

k

akgk� #glE!Xk

ak hgk� #gli ! al

where we have used the biorthogonality of fgk�n�g and f#gk�n�g� So it follows that

al ! cl and we conclude that the representation is unique� Thus fgk�n�g is a basis for

H� and since every permutation of a frame is also a frame� we conclude that the basis

is unconditional� We have shown that the exactness of fgk�n�g implies that fgk�n�g is a

bounded unconditional basis�

It remains to prove that the converse is also true� Assume that fgk�n�g is a bounded

unconditional basis for H� In Hilbert spaces� all bounded unconditional bases are equiv�

alent to orthonormal bases� in the sense that if fgk�n�g is a bounded unconditional

basis� then there exists an orthonormal basis fek�n�g and a topological isomorphism�

U � H � H such that gk�n� ! �Uek��n� for all k �� Given x�n� � H� we

therefore have

Xk

jhx� gkij� !Xk

jhx�Uekij� !Xk

jhU �x� ekij� ! kU �xk� �

But on the other hand

kxk ! kIxk !!!!U ��

U�x!!! � !!!U ��

!!! kU �xk �

which implies thatkxk!!!U ��

!!! � kU �xk

and hencekxk�!!!U ��

!!!� � kU �xk� !Xk

jhx� gkij��

A topological isomorphism is a continuous linear transformation of H onto H such that the inverse

transformation exists and is continuous�


Furthermore we have

Xk

jhx� gkij� ! kU �xk� � kU �k� kxk� ! kUk� kxk��

Thus� fgk�n�g is a frame with frame bounds

A !!!!U ��!!!� and B ! kUk� �

It is clearly exact since the removal of any vector from a basis leaves an incomplete

set� �

Theorem A�� Inexact frames are not bases�

Thus� inexact frames allow to expand any signal x�n� � H as x�n� !P

k ck gk�n� but the

expansion coe�cients ck are not unique� In fact� inexact frames are overcomplete in the

sense that there is redundancy in the expansion coe�cients�

Proof �� Assume that fgk�n�g is an inexact frame� Then fgk�n�gk ��m is a frame

for some m� By Theorem A�� we have gm�n� !Pkak gk�n� ! am gm�n� "

Pk ��m ak gk�n��

where ak ! hgm� #gki� But we also have gm�n� !Pkck gk�n�� where ck ! for k ! m and

ck ! � for k ! m� By Theorem A� we must have am ! hgm� #gmi ! � i�e�� am ! cm�

Thus we have two di�erent representations for gm�n� in terms of fgk�n�g� and hence

fgk�n�g is not a basis� �

Theorem A�� A frame is tight �with frame bound A� and exact if and

only if it is an orthogonal basis �with kgkk� ! A��

Proof� For an exact frame we have hgk� #gmi ! �km� If the frame is tight we have

#gk�n� !�Agk�n�� Thus we conclude that hgk� gmi ! A�km� which proves that fgk�n�g is

an orthogonal basis with kgkk� ! A� To see that the converse is also true� one has to

recall that every orthogonal basis with hgk� gmi ! A�km is a tight frame with frame

bound A ! B� Thus we have #gk�n� !�Agk�n� and we obtain the biorthogonality relation

hgk� #gmi ! �km which implies the exactness of the frame� �

In the proof of Theorem A�� we have shown that an exact frame fgk�n�g yields

unique expansion coe�cients� In �nite�dimensional spaces H� this property is equivalent

to the linear independence of the frame functions� For an inexact frame the expansion

coe�cients are not unique� In a �nite�dimensional space H� this means that the frame

functions are linearly dependent�

A�� TRANSFORMATION OF FRAMES ��

A� Transformation of Frames

We shall next characterize frame�preserving mappings�� Starting from a frame fgk�n�gfor a space H�� we want to �nd frames fhk�n�g for some other space H�� One possible

approach is to construct a set of functions hk�n� ! �Ugk��n�� where U is a bounded�

linear operator from H� into H� with RanfUg ! H�� The following theorem states a

necessary and su�cient condition on the operator U in order for fhk�n�g to be a frame�

Theorem A�� Let fgk�n�g be a frame forH� with bounds A�B� Let

U be a bounded� linear operator mappingH� onto H� �i�e�� with RanfUg !H�� Then fhk�n�g ! f�Ugk��n�g is a frame for H� if and only if the adjoint

operator U � is bounded below� i�e�� if there exists a positive constant � such

that the adjoint operator U � satis�es

kU �yk� � �kyk� � y�n� � H�� A��

Frame bounds for fhk�n�g are given by C ! �A and D ! BkUk��

Proof �� We �rst prove that �A�� is a su�cient condition� Given y�n� � H�� we

have �U �y��n� � H� and thus� by �A��

Xk

jhy�Ugkij� !Xk

jhU �y� gkij� � AkU �yk� � �Akyk� � y�n� � H��

So we have found a lower frame bound C ! �A for f�Ugk��n�g� The next step is to

derive an upper frame bound for f�Ugk��n�g� From �A�� we have

Xk

jhy�Ugkij� !Xk

jhU �y� gkij� � BkU �yk�

� BkU �k�kyk� ! BkUk�kyk� � y�n� � H��

where the last equation follows from the fact that kU �k ! kUk� We conclude that an

upper frame bound is D ! BkUk�� which completes the proof of the su�ciency�

We shall next prove that �A�� is a necessary condition� Assume that f�Ugk��n�g is

a frame for H� with lower frame bound C�

Ckyk� �Xk

jhy�Ugkij� � y�n� � H��

Again using �A�� it follows that

Ckyk� �Xk

jhy�Ugkij� !Xk

jhU �y� gkij� � BkU �yk�

The results of this section have mostly been taken from �� However� we also present extensions

of the results given there�


where B is the upper frame bound of fgk�n�g� and hence

kU �yk� � C

Bkyk��

which is �A�� with � ! CB� The constant � is greater than zero because B ��

Corollary A�� If fgk�n�g is a frame for H�� U maps H� into H�� and

fhk�n�g ! f�Ugk��n�g is a frame for H�� then the frame operator Sh of

fhk�n�g is

Sh ! USU ��

where S is the frame operator of fgk�n�g� Furthermore if U is an invertible

mapping of H� onto H�� the dual frame f#hk�n�g is given by

#hk�n� ! U��

#gk�n��

Proof� We �rst show that the frame operator of fhk�n�g can be expressed as Sh !

USU� �

�Shx��n� !Xk

hx� hki hk�n� !Xk

hx�Ugki �Ugk��n�

!Xk

hU �x� gki�Ugk��n� !

�UXk

hU �x� gki gk��n�

! �USU �x��n��

Furthermore we have to show that f#hk�n�g ! fU ��

gk�n�g for U invertible� Using the

previous result about the frame operator Sh� we obtain �note that the invertibility of

U implies that of U ��

#hk�n� ! �S��h hk��n� ! �S��h U gk��n� !��USU ��U gk

��n�

! �U ��

S��U��U gk��n� ! �U ��

#gk��n��

where S is the frame operator of fgk�n�g� �

An important special case is an operator U that is unitary� i�e�� U ��

! U � In that

case we have#hk�n� ! �U #gk��n��

i�e�� the dual frames f#gk�n�g and f#hk�n�g are related by the same mapping �namely� U�

as the frames gk�n� and hk�n��

We shall now discuss a frame�preserving mapping of particular importance� namely

U ! S�� which maps H� onto H� �i�e�� H� ! H�� Here S�� denotes the positive

de�nite operator square root of S��

A�� TRANSFORMATION OF FRAMES �

Corollary A�� Let fgk�n�g be a frame for H� with frame operator S�

Then f�S��gk��n�g is a tight frame for H with A ! � Moreover if fgk�n�gis exact� then f�S��gk��n�g is an orthonormal basis for H�

Proof �� We know from Theorem A�� that f�S��gk��n�g is a frame if and only if

kS��xk� � �kxk� with � � �� We shall �rst show that kS��xk� ! kS��xk� � ��

This follows from �Bkxk� � hS��x� xi ! hS��x�S��xi ! kS��xk�� where we have

used the fact that S�� is self�adjoint� We are now able to show that f�S��gk��n�gis tight with A ! � According to Corollary A�� we have to show that

x�n� !Xk

hx�S��gki �S��gk��n��

Indeed� the right�hand side of this equation is

Xk

hx�S��gki�S��gk��n� ! S��X

k

hS��x� gki gk�n�

! �S��SS��x��n� ! �Ix��n� ! x�n��

For the proof of the second statement� we have to consider the inner product

hS��gk�S��gmi ! hgk�S��gmi ! hgk� #gmi�

Since fgk�n�g is exact we have

hgk� #gmi ! �mk�

and hence we conclude that f�S��gk��n�g is an orthonormal basis for H� �

We shall next ask whether the orthogonal projection of a frame fgk�n�g for H into a

subspace H� � H yields a frame for H�� The following theorem gives an answer�

Theorem A�� Let H� � H be a subspace of H� Let fgk�n�g be

a frame for H and f#gk�n�g the dual frame� Let P denote the orthogonal

projection operator from H into H�� Then f�P gk��n�g and f�P #gk��n�g are

dual in H� �not necessarily dual frames�� Moreover� the frame bounds A

and B of fgk�n�g are also frame bounds for f�P gk��n�g�

Proof �� The orthogonal projection operator P satis�es

kP �yk ! kP yk� ! kyk� � y�n� � H��

We thus have the conditions of Theorem A�� with � ! � It follows that f�P gk��n�g is

a frame for H�� We can furthermore write x�n� in terms of the frame fgk�n�g�

x�n� !Xk

hx� #gki gk�n��


Furthermore we use the fact that y�n� ! �P y��n� for y�n� � H� to obtain

y�n� ! �P y��n� !Xk

hP y� #gki gk�n� !Xk

hy�P #gki gk�n�� y�n� � H�

and further

y�n� ! �P y��n� !

�PXk

hy�P #gki gk��n� !

Xk

hy�P #gki�P gk��n� � y�n� � H��

From this equation� it follows that f�P gk��n�g and f�P #gk��n�g are dual in H�� Note

that this does not necessarily mean that f�P gk��n�g and f�P #gk��n�g are dual frames

in H�� Pure duality only means that the corresponding dual sets will provide perfect

reconstruction�

To complete the proof� we have to show that the frame bounds of fgk�n�g are also

frame bounds of f�P gk��n�g� Since H� � H� it follows from �A�� that

Akyk� �Xk

jhy� gkij� � Bkyk� � y�n� � H��

Furthermore y�n� ! �P y��n� for all y�n� � H� and hence

Akyk� �Xk

jhP y� gkij� � Bkyk�� y�n� � H��

which implies that

Akyk� �Xk

jhy�P gkij� � Bkyk� � y�n� � H��

A�� Frames and Pseudoinverses

Consider an inexact frame fgk�n�g for the Hilbert space H� Due to Theorem A�� any

signal x�n� � H can be represented as x�n� !P

k hx� #gki gk�n�� where f#gk�n�g is the dual

frame� A consequence of the �overcompleteness� of an inexact frame fgk�n�g is that the�frame coe�cients� hx� #gki do not constitute the only sequence ck satisfying

Xk

ck gk�n� ! x�n�� A��

However� Lemma A� states that the coe�cients ak ! hx� #gki have minimum norm

among all possible sequences ck� We shall now consider this property from a di�erent

point of view�

Let us reformulate the problem� We consider the operator T � �De�nition A�� which

acts on a sequence ck � l��ZZ� as

�T �c��n� !Xk

ck gk�n��

A� � FRAMES AND THE GRAM MATRIX

Due to �A�� the coe�cients ck satisfy the linear equation

�T �c��n� ! x�n��

As stated further above� it follows from the inexactness of the frame fgk�n�g that this

equation has more than one solution ck� Hence� the equation T �c ! x is underdeter�

mined� Accordingly one may be interested in the solution ak with minimum norm�

i�e��

ak ! argminckkckk�

According to �� pp� �� this sequence is given by the pseudoinverse �T ��y !

T �T �T �� of the operator T ��

ak ! �T ��yx ! T �T �T ��x ! TS��x ! �Tx�

Thus� the operator �T ! TS�� considered in Theorem A�� assigning to each signal x�n�

the coe�cient sequence ak ! � �Tx�k ! hx� #gki� is the pseudoinverse of T � which assigns

to each signal x�n� � H the coe�cient sequence with minum norm�

�T ! TS�� ! �T ��y�

We �nally note that the relation between frames and pseudo�inverses has been elab�

orated in ��

A�� Frames and the Gram Matrix

The Gram matrix G of a set of signals fgk�n�g is a Hermitian� positive semide�nite

matrix whose elements are the inner products of the signals gk�n��

Gik ! hgi� gki�Given a frame fgk�n�g� consider the operator TT � mapping l��ZZ� into l��ZZ� according

to

�TT �c�k !

�TXl

clgl�n�

�k

!DX

l

clgl� gkE

!Xl

clhgl� gki !Xl

Glkcl !Xl

�GT �kl cl ! �GT c�k�

We thus have shown

Lemma A�� The operator

TT� � l��ZZ�� l��ZZ�

�TT �c�k !Xl

hgl� gki cl

is equal to the transposed Gram matrix of the frame fgk�n�g�TT

� ! GT �


The following theorem relates the eigenvalues and eigenfuntions of the frame operator

S to the eigenvalues and eigenvectors of the Gram matrix G�

Theorem A�� The eigenvalues of the frame operator S ! T�T are the

same as the nonzero eigenvalues of the Gram matrix G� The eigenfunctions

u�n� of the frame operator S are related to the eigenvectors v of the Gram

matrix G according to

�v�k ! hu� gki�� k � K�

where �v�k denotes the kth component of the eigenvector v�

Proof� Let � � and u�n� be an eigenvalue and eigenfunction� respectively� of the

frame operator S� From the eigenequation of S�

�Su��n� ! u�n��

we obtain

hSu� gki ! hu� gki� �A��

On the other hand�

hSu� gki !DX

l

hu� gli gl� gkE!Xl

hu� glihgl� gki

!Xl

hgk� gli�hu� gli !Xl

G�kl hu� gli�

Combining with �A�� yields

Xl

Gkl hu� gli� ! � hu� gki��

De�ning the vector v with components �v�k ! hu� gki�� and using the fact that � !

due to the self�adjointness of S� this can be written as

Gv ! v�

which shows that and v are an eigenvalue and eigenvector� respectively� of the Gram

matrix� �

The rank of the frame operator S �which equals the dimension of H� equals the

number of nonzero eigenvalues of the Gram matrix G�

A�� EXAMPLES �

�a�

x�

x�

g�

g�g�

g�

x�

x�

g�

g� g�

�b�

Fig� �� a Frame for IR�� b tight frame for IR��

A�� Examples

In order to illustrate the theory of frames discussed so far� we shall now give two

examples� For simplicity� we consider vectors x � IR� instead of signals x�n�� The

underlying Hilbert space H is two�dimensional �the plane IR��

Example �� The four vectors

g� !

"#

�

$A � g� !

"# �

$A � g� !

"# �

$A � g� !

"# ��

$A

depicted in Fig� ��a� constitute a frame for IR�� Note that this frame consists of an

orthonormal basis �g� and g�� augmented by two vectors g�� g� which are obviously

linearly dependent with respect to g�� g�� Thus� this frame is inexact�

The frame operator is a �� matrix given by �cf� �A��

S !�X

k��

gkgHk !

"# � ��

$A �

where ghH denotes the outer �dyadic� product of the vectors g and h� The tightest

frame bounds are obtained as the minimum and the maximum of the eigenvalues of the

matrix S� which are

A ! min ! �� B ! max !�

��

The inverse frame operator �matrix� is obtained by inverting the matrix S�

S�� !

��

"# �

� �

$A �


The dual frame f#gkg is obtained by applying the inverse frame operator to the vectors

gk�

#g� ! S��g� !

��

"#

�

$A � #g� ! S

��g� !

��

"# �

�

$A �

#g� ! S��g� !

��

"# ��

�

$A � #g� ! S

��g� !

��

"# ��

$A �

Any vector x � IR� can be reconstructed from the expansion coe�cients hx� #gki ! #gTk x

as

x !�X

k��

hx� #gki gk�

Example � �� We shall now consider a tight frame for IR�� The frame vectors are

g� !

"# �

$A � g� !

"# �p��

��

$A � g� !

"# p

��

��

$A

�see Fig� ��b�� Obviously� these vectors must be linearly dependent� and thus the

frame is again inexact� The frame operator is given by

S !�

�I��

where I� is the � � � identity matrix� The frame bounds are A ! B ! �� Since the

frame vectors all have length � the frame bound ��can be interpreted as a �redundancy

factor� �we have � vectors in a ��dimensional space�� The inverse frame operator is

given by

S�� !

�

�I��

and the dual frame vectors are #gk !��gk�

Appendix B

MultiChannel Levinson Algorithm

In this appendix we provide a detailed derivation of the multi�channel Levinson al�

gorithm� which is sometimes also referred to as Levinson�Wiggins�Robinson algorithm

�� The discussion essentially follows the presentation in ��

The multi�channel Levinson algorithm allows an e�cient recursive solution of the the

multi�channel Yule�Walker equations

��

Cx�� Cx�� Cx�L� �

Cx�� Cx�� Cx�L� ��

��

��

Cx��L� �� Cx��L� �� Cx��

��

��

GT�

GT��

GTL

�� !

��

CTx ��

CTx ��

CTx �L�

�� B��

where the N � N matrices Cx�k� ! Efx�n�xH �n�k�g are the correlation matrices of the

N � vector process x�n� to be predicted� The N � N predictor coe�cient matrices

are denoted by Gl �l ! � �� L��

In the following we shall use both forward and backward linear predictors� The L�th

order forward linear predictor is de�ned according to

xf �n� !LXl��

Gfl x�n� l��

The corresponding multi�channel Yule�Walker equations are

LXl��

Cx�l � k�GfT

l ! CTx �k�� k ! � �� L� �B��

which is equivalent to �B�� except that we have written Gfl instead of Gl� The forward

prediction error power matrix is de�ned as

�f �! Cx��

LXl��

Cx�l�GfT

l � �B��

�

� APPENDIX B� MULTICHANNEL LEVINSON ALGORITHM

The L�th order backward linear predictor is given by

xb�n� !LXl��

Gbl x�n " l��

The corresponding Yule�Walker equations read

LXl��

Cx�k � l�GbT

l ! Cx�k�� k ! � �� L �B��

and the backward prediction error power matrix is de�ned as

�b �! Cx��

LXl��

Cx��l�GbT

l � �B��

Incorporating the equations for the prediction error power matrices into the Yule�

Walker equations �B�� and using Cx��k� ! CTx �k� we get the following augmented

system of forward equations

��

Cx�� Cx�� Cx�L�

Cx�� Cx�� Cx�L� ��

��

��

Cx��L� Cx��L� �� Cx��

��

��

�INGfT

��

GfT

L

�� ! �

��

�f

�N��

�N

�� B��

Similarly� for the backward case we obtain

��

Cx�� Cx�� Cx�L�

Cx�� Cx�� Cx�L� ��

��

��

Cx��L� Cx��L� �� Cx��

��

��

GbT

L��

GbT

�

�IN

�� ! �

��

�N��

�N

�b

�� B��

Derivation of the Multi�Channel Levinson Algorithm� In the sequel we as�

sume that the solutions for the forward and backward predictors of order k � are

already available and that the solutions for the kth order predictors are required� The

�k � �st order forward predictor satis�es the equations

��

Cx�� Cx�� Cx�k � �

Cx�� Cx�� Cx�k � ��

��

��

Cx��k � �� Cx��k � �� Cx��

��

��

�INGfT

� �k � ��

GfT

k��k � �

�� ! �

��

�f �k � �

�N��

�N

��

�B� �

where Gfl �k � � denotes the lth predictor coe�cient matrix for the �k � �st order

forward prediction system and �f �k � � is the corresponding prediction error power

APPENDIX B� MULTICHANNEL LEVINSON ALGORITHM �

matrix� The �k � �st order backward predictor satis�es

��

Cx�� Cx�� Cx�k � �

Cx�� Cx�� Cx�k � ��

��

��

Cx��k � �� Cx��k � �� Cx��

��

��

GbT

k��k � ��

GbT

� �k � �

�IN

�� ! �

��

�N��

�N

�b�k � �

��

�B��

with Gbl �k � � denoting the lth predictor coe�cient matrix for the �k � �st order

backward prediction system and �b�k � � denoting the corresponding prediction error

power matrix�

Enlarging the dimension of the forward equations by one we obtain

��

Cx�� Cx�� Cx�k�

Cx�� Cx�� Cx�k � ��

��

��

Cx��k� Cx��k � �� Cx��

��

��

�INGfT

� �k � ��

GfT

k��k � �

�N

��! �

��

�f �k � ��

�N

�f �k�

��B��

with

�f �k� ! �k��Xl��

Cx�l � k�GfT

l �k � �� B��

where GfT

� �k � ��! �IN � Enlarging the system of backward equations by one

dimension we get

��


Cx�� Cx�� Cx�k � ��

��

��

Cx��k� Cx��k � �� Cx��

��

��

�N

GbT

k��k � ��

GbT

� �k � �

�IN

��! �

��

�b�k�

�N��

�N

�b�k � �

��

�B��

with

�b�k� ! �k��Xl��

Cx�k � l�GbT

l �k � ��

where GbT

� �k � ��! �IN �

Multiplying the backward equations �B�� by the N � N forward re�ection coe��

cient matrix KfT �k�� which still is to be determined� from the right and adding them to

APPENDIX B� MULTICHANNEL LEVINSON ALGORITHM

the forward equations �B�� yields

��


Cx�� Cx�� Cx�k � ��

��

��

Cx��k� Cx��k � �� Cx��

��

��

��

�INGfT

� �k � ��

GfT

k��k � �

�N

��"

��

�N

GbT

k��k � ��

GbT

� �k � �

�IN

��KfT �k�

��

!

�

��

�f �k � �

�N��

�N

�f �k�

��

��

�b�k�

�N��

�N

�b�k � �

��KfT �k�� B��

We shall now determine the forward re�ection coe�cient matrix KfT �k� in a way that

�B�� is equivalent to the augmented Yule�Walker equations for the kth order forward

predictor in �B�� We obtain the condition

�f �k� "�b�k � �KfT �k� ! �N

and consequently

KfT �k� ! ��b��

�k � ��f �k�� B��

Furthermore� a comparison of the �rst equation in �B�� with the �rst equation in �B��

results in

�f �k� ! �f �k � � "�b�k�KfT �k�� B��

Similarly� comparing the remaining equations in �B�� with the corresponding equations

in �B�� yields

Gfl �k� !

�� Gf

l �k � � "Kf �k�Gbk�l�k � � for l ! � �� k �

�Kf �k� for l ! k��B��

Let us next generate the Yule�Walker equations for the k�th order backward predictor

in a similar fashion� Multiplying the augmented forward equations �B�� by the N �N

backward re�ection coe�cient matrix KbT �k� from the right and adding them to the

augmented backward equations �B�� yields

��


Cx�� Cx�� Cx�k � ��

��

��

Cx��k� Cx��k � �� Cx��

��

��

��

�INGfT

� �k � ��

GfT

k��k � �

�N

��KbT �k� "

��

�N

GbT

k��k � ��

GbT

� �k � �

�IN

��

��

!

APPENDIX B� MULTICHANNEL LEVINSON ALGORITHM

�

��

�f �k � �

�N��

�N

�f �k�

��KbT �k��

��

�b�k�

�N��

�N

�b�k � �

�� B��

These equations should represent the Yule�Walker equations for the k�th order backward

predictor� Proceeding as above one can see that this is the case if

�b�k� ! �b�k � � "�f �k�KbT �k� �B� �

KbT �k� ! ��f��

�k � ��b�k�� B��

The corresponding predictor coe�cient matrices are consequently given by

Gbl �k� !

�� Gb

l �k � � "Kb�k�Gfk�l�k � � for l ! � �� k �

�Kb�k� for l ! k��B��

Some simpli�cations to these equations are possible� It is shown in �� that

�b�k� !�fT �k��

Hence� the multi�channel re�ection coe�cient matrices in �B�� and �B�� are given

by

KfT �k� ! ��b��

�k � ��f �k� �B��

KbT �k� ! ��f��

�k � ��fT �k�� B��

Furthermore� inserting �B�� and �B�� in �B�� and �B� �� respectively� the predic�

tion error power matrices can be expressed as

�f �k� ! �f �k � ��IN �KbT �k�KfT �k�� B��

�b�k� ! �b�k � ��IN �KfT �k�KbT �k�� B��

The Multi�Channel Levinson Algorithm� The multi�channel Levinson algo�

rithm is initialized by �nding the solution for the �rst�order linear predictor� According

to �B��B��B�� and �B�� the �rst�order predictor coe�cient matrices are given

by

GfT

� �� ! �KfT �� ! C��x ��Cx�� B��

GbT

� �� ! �KbT �� ! C��x ��Cx�� B��

Using �f �� ! Cx�� and �b�� ! Cx�� see �B�� and �B�� respectively� in �B��

and �B�� the �rst order prediction error power matrices are given by

�f �� ! Cx��IN �KbT ��KfT �� B��

�b�� ! Cx��IN �KfT ��KbT �� B��

Finally� we can summarize the multi�channel Levinson algorithm as follows�

� APPENDIX B� MULTICHANNEL LEVINSON ALGORITHM

� Initialization�

GfT

� �� ! �KfT �� ! C��x ��Cx�� B��

GbT

� �� ! �KbT �� ! C��x ��Cx�� B��

�f �� ! Cx��IN �KbT ��KfT �� B��

�b�� ! Cx��IN �KfT ��KbT �� B��

For k ! �� L�

� Re�ection coe�cient matrices see B�� B�� and B��

KfT �k� ! ��b��

�k � ��f �k� �B��

KbT �k� ! ��f��

�k � ��fT �k� �B��

�f �k� ! �k��Xl��

Cx�l � k�GfT

l �k � � �B��

� Predictor coe�cient matrices see B��B��

Gfl �k� !

�� Gf

l �k � � "Kf �k�Gbk�l�k � � for l ! � �� k �

�Kf �k� for l ! k

Gbl �k� !

�� Gb

l �k � � "Kb�k�Gfk�l�k � � for l ! � �� k �

�Kb�k� for l ! k�

� Prediction error power matrices see B��B��

�f �k� ! �f �k � ��IN �KbT �k�KfT �k�� B��

�b�k� ! �b�k � ��IN �KfT �k�KbT �k�� B��

� Finally� the solution to the multi�channel Yule�Walker equations is given by

Gl ! Gfl �L�� l ! � �� L �B��

� ! �f �L�� B��

Bibliography

�� M� Vetterli� �A theory of multirate lter banks� IEEE Trans� Acoust�� Speech� Signal

Processing� vol� �� pp� �� March ��

�� P� P� Vaidyanathan� �Theory and design ofM �channel maximally decimated quadraturemirror lters with arbitrary M � having perfect reconstruction property� IEEE Trans�

Acoust�� Speech� Signal Processing� vol� �� pp� �� April ��

�� P� P� Vaidyanathan� Multirate Systems and Filter Banks� Englewood Cli�s �NJ�� Pren�tice Hall� ��

�� M� Vetterli and J� Kova�cevi�c� Wavelets and Subband Coding� Englewood Cli�s �NJ��Prentice Hall� ��

�� A� N� Akansu and R� A� Haddad�Multiresolution Signal Decomposition� Academic Press��

� � G� Strang and T� Q� Nguyen� Wavelets and Filter Banks� Wellesley�Cambridge Press��

�� N� S� Jayant and P� Noll� Digital Coding of Waveforms� Englewood Cli�s �NJ�� PrenticeHall� ��

�� S� G� Mallat and Z� Zhang� �Matching pursuits and time�frequency dictionaries� IEEETrans� Signal Processing� vol� �� pp� �� Dec� ��

�� I� Daubechies� �The wavelet transform� time�frequency localization and signal analysis�IEEE Trans� Inf� Theory� vol� � � pp� � �� Sept� ��

�� I� Daubechies� Ten Lectures on Wavelets� SIAM� ��

�� H� G� Feichtinger and T� Strohmer� eds�� Gabor Analysis and Algorithms� Theory and

Applications� Boston �MA�� Birkh�auser� ��

�� M� J� Bastiaans� �Gabor�s expansion of a signal in gaussian elementary signals�Proc� IEEE� vol� �� pp� �� April ��

�� A� J� E� M� Janssen� �Gabor representation of generalized functions� J� Math� Anal�

Appl�� vol� �� pp� ��

�� J� Wexler and S� Raz� �Discrete Gabor expansions� Signal Processing� vol� �� pp� ��

�� M� Zibulski and Y� Y� Zeevi� �Oversampling in the Gabor scheme� IEEE Trans� Signal

Processing� vol� �� pp� � �� Aug� ��

� BIBLIOGRAPHY

�� A� J� E� M� Janssen� �Duality and biorthogonality for Weyl�Heisenberg frames� J�

Fourier Analysis and Applications� vol� �� no� �� pp� ��

�� I� Daubechies� H� J� Landau� and Z� Landau� �Gabor time�frequency lattices and theWexler�Raz identity� J� Fourier Analysis and Applications� vol� �� no� �� pp� ��

�� H� G� Feichtinger and K� Gr�ochenig� �Gabor wavelets and the Heisenberg group� Gaborexpansions and short time Fourier transform from the group theoretical point of view�in Wavelets � A Tutorial in Theory and Applications �C� Chui� ed�� pp� ��Boston� Academic Press� ��

�� I� Daubechies� S� Ja�ard� and J� L� Journ�e� �A simple Wilson orthonormal basis withexponential decay� SIAM J� Math� Anal�� vol� �� pp� ��

�� H� G� Feichtinger� K� Gr�ochenig� and D� Walnut� �Wilson bases and modulation spaces�Math� Nachrichten� vol� �� pp� ��

�� P� Auscher� �Remarks on the local Fourier bases� inWavelets� Mathematics and Appli�

cations �J� J� Benedetto and M� W� Frazier� eds�� pp� �� Boca Raton� FL� CRCPress� ��

�� H� B�olcskei� H� G� Feichtinger� K� Gr�ochenig� and F� Hlawatsch� �Discrete�time Wil�son expansions� in Proc� IEEE�SP Int� Sympos� Time�Frequency Time�Scale Analysis��Paris� France�� pp� �� June ��

�� H� B�olcskei� K� Gr�ochenig� F� Hlawatsch� and H� G� Feichtinger� �Oversampled Wilsonexpansions� IEEE Signal Processing Letters� vol� �� pp� �� April ��

�� R� R� Coifman and Y� Meyer� �Remarques sur l�analyse de fourier �a fen�etre� C� R�

Acad� Sci�� vol� �� no� �� pp� ��

�� A� Ron and Z� Shen� �Frames and stable bases for shift�invariant subspaces of L��Rd��

Canadian Journal of Mathematics� vol� �� no� �� pp� ��

�� A� J� E� M� Janssen� �Density theorems for lter banks� Tech� Rep� �� PhilipsResearch Laboratories� Eindhoven� The Netherlands� April ��

�� A� J� E� M� Janssen� �The duality condition for Weyl�Heisenberg frames� in Gabor

Analysis and Algorithms� Theory and Applications �H� G� Feichtinger and T� Strohmer�eds�� pp� �� Boston �MA�� Birkh�auser� ��

�� Y� F� Dehery� M� Lever� and P� Urcum� �A MUSICAM source codec for digital audiobroadcasting and storage� in Proc� ICASSP�� Toronto� Canada�� pp� � �� May��

�� G� K� Wallace� �The JPEG still picture compression standard� Communications of theACM� vol� �� pp� �� April ��

�� D� LeGall� �MPEG� A video compression standard for multimedia applications� Com�munications of the ACM� vol� �� pp� � �� April ��

�� R� N� J� Veldhuis� M� Breeuwer� and R� G� van der Wall� �Subband coding of digitalaudio signals� Philips J� Res�� vol� �� pp� ��

BIBLIOGRAPHY �

�� C� E� Heil and D� F� Walnut� �Continuous and discrete wavelet transforms� SIAM Rev��vol� �� pp� �� Dec� ��

�� H� G� Feichtinger� �Coherent non�orthogonal expansions reencouraged� in Proceedingsof the �th Annual International Conference of the IEEE Engineering in Medicine and

Biology �N� F� Sheppard� M� Eden� and G� Kantor� eds�� pp� � a��a� Nov� ��

�� H� G� Feichtinger and K� Gr�ochenig� �Theory and practice of irregular sampling� inWavelets� Mathematics and Applications �J� J� Benedetto and M� W� Frazier� eds��pp� �� Boca Raton� CRC Press� ��

�� Z� Cvetkovi�c and M� Vetterli� �Oversampled lter banks� IEEE Trans� Signal Process�

ing� to appear�

�� H� B�olcskei� F� Hlawatsch� and H� G� Feichtinger� �Frame�theoretic analysis of oversam�pled lter banks� IEEE Trans� Signal Processing� submitted Feb� ��

�� H� B�olcskei� F� Hlawatsch� and H� G� Feichtinger� �Frame�theoretic analysis and designof oversampled lter banks� in Proc� IEEE ISCAS�� vol� �� Atlanta �GA�� pp� �� May ��

�� H� B�olcskei and F� Hlawatsch� �Oversampled lter banks� Optimal noise shaping� designfreedom� and noise analysis� in Proc� IEEE ICASSP�� vol� �� Munich� Germany��pp� �� April ��

�� H� B�olcskei and F� Hlawatsch� �Oversampled cosine modulated lter banks with per�fect reconstruction� IEEE Trans� Circuits and Systems II� Special Issue on Multirate

Systems� Filter Banks� Wavelets� and Applications� to appear ��

�� H� B�olcskei and F� Hlawatsch� �Noise reduction in oversampled lter banks using pre�dictive quantization� IEEE Trans� Inf� Theory� �� to be submitted�

�� J� C� Candy and G� C� Temes� Oversampling Delta�Sigma Data Converters� IEEE Press��

�� R� M� Gray� �Oversampled sigma�delta modulation� IEEE Trans� Comm�� vol� ��pp� �� April ��

�� S� K� Tewksbury and R� W� Hallock� �Oversampled� linear predictive and noise�shapingcoder of order N�� IEEE Trans� Circuits and Systems� vol� �� pp� �� July��

�� H� B�olcskei and F� Hlawatsch� �Oversampled modulated lter banks� in Gabor Analysisand Algorithms� Theory and Applications �H� G� Feichtinger and T� Strohmer� eds��ch� �� pp� �� Boston� MA� Birkh�auser� ��

�� H� B�olcskei and F� Hlawatsch� �Oversampled cosine modulated lter banks with linearphase� in Proc� IEEE ISCAS�� Hong Kong�� pp� �� June ��

�� H� B�olcskei and F� Hlawatsch� �Oversampled Wilson�type cosine modulated lter bankswith linear phase� in Proc� �� th Asilomar Conf� Signals� Systems� Computers� �PacicGrove� CA�� pp� �� Nov� ��

�� J� L� Mannos and D� J� Sakrison� �The e�ect of a visual delity criterion on the encodingof images� IEEE Trans� Inf� Theory� vol� �� pp� �� July ��

� BIBLIOGRAPHY

�� H� B�olcskei� T� Stranz� F� Hlawatsch� and R� Sucher� �Subband image coding usingcosine modulated lter banks with perfect reconstruction and linear phase� in Proc�

IEEE ICIP�� vol� �� Santa Barbara� CA�� pp� �� Oct� ��

�� D� L� Donoho� �De�noising by soft�thresholding� IEEE Trans� Inf� Theory� vol� ��no� �� pp� ��

�� Z� Cvetkovi�c and M� Vetterli� �Discrete�time wavelet extrema representation� Designand consistent reconstruction� IEEE Trans� Signal Processing� vol� �� pp� �� March ��

�� J� S� Lim and A� V� Oppenheim� �Enhancement and bandwidth compression of noisyspeech� Proc� IEEE� vol� �� pp� �� Dec� ��

�� S� G� Chang� Z� Cvetkovi�c� and M� Vetterli� �Resolution enhancement of images usingwavelet transform extrema extrapolation� in Proc� IEEE ICASSP�� vol� �� pp� ��

�� Z� Cvetkovi�c� Overcomplete expansions for digital signal processing� PhD thesis� Uni�versity of California� Berkeley� CA� Dec� ��

�� M� Sandell� Design and Analysis of Estimators for Multicarrier Modulation and Ultra�

sonic Imaging� PhD thesis� Lulea University of Technology� Lulea� Sweden� ��

�� H� B�olcskei� F� Hlawatsch� and H� G� Feichtinger� �Oversampled FIR and IIR DFT lterbanks and Weyl�Heisenberg frames� in Proc� IEEE ICASSP�� vol� �� Atlanta� GA��pp� �� May ��

�� R� A� Gopinath� �Modulated lter banks and wavelets � A unied theory� in Proc�

IEEE ICASSP�� Atlanta� GA�� pp� �� May ��

�� T� Chen and P� P� Vaidyanathan� �Vector space framework for unication of one� andmultidimensional lter bank theory� IEEE Trans� Signal Processing� vol� �� pp� �� Aug� ��

�� H� B�olcskei� F� Hlawatsch� and H� G� Feichtinger� �Equivalence of DFT lter banks andGabor expansions� in Proc� SPIE Wavelet Applications in Signal and Image Processing

III� vol� �� Part I� �San Diego �CA�� pp� �� July ��

�� R� E� Crochiere and L� R� Rabiner� Multirate Digital Signal Processing� EnglewoodCli�s �NJ�� Prentice Hall� ��

� �� M� J� T� Smith and T� P� Barnwell� �A new lter bank theory for time�frequency repre�sentation� IEEE Trans� Acoust�� Speech� Signal Processing� vol� �� pp� �� March��

� �� Y� Lin and P� P� Vaidyanathan� �Application of DFT lter banks and cosine modulatedlter banks in ltering� in Proc� APCCAS�� Taipei� Taiwan�� pp� ��

� �� R� N� J� Veldhuis� �A lter�notation for analysis�synthesis systems and its relation toframes� Tech� Rep� �� Institute for Perception Research� The Netherlands� Sept��

� �� R� J� Du!n and A� C� Schae�er� �A class of nonharmonic Fourier series� Trans� Amer�Math� Soc�� vol� �� pp� ��

BIBLIOGRAPHY �

� �� Z� Cvetkovi�c� �Oversampled modulated lter banks and tight Gabor frames in l��Z��in Proc� IEEE ICASSP�� Detroit �MI�� pp� �� May ��

� �� M� Vetterli and Z� Cvetkovi�c� �Oversampled FIR lter banks and frames in l��Z�� inProc� IEEE ICASSP�� vol� �� pp� �� May ��

� � T� Kailath� Linear Systems� Englewood Cli�s �NJ�� Prentice Hall� ��

� �� I� Gohberg� P� Lancaster� and L� Rodman� Matrix Polynomials� Academic Press� ��

� �� M� Zibulski and Y� Y� Zeevi� �Analysis of multiwindow Gabor�type schemes by framemethods� Applied and Computational Harmonic Analysis� vol� �� pp� �� April��

� �� N� J� Munch� �Noise reduction in tight Weyl�Heisenberg frames� IEEE Trans� Inf�

Theory� vol� �� pp� �� March ��

�� M� Vetterli and C� Herley� �Wavelets and lter banks� Theory and design� IEEE Trans�

Signal Processing� vol� �� pp� �� Sept� ��

�� A� W� Naylor and G� R� Sell� Linear Operator Theory in Engineering and Science� NewYork� Springer� �nd ed��

�� A� J� E� M� Janssen� �The Zak transform� A signal transform for sampled time�continuous signals� Philips J� Research� vol� �� no� �� pp� ��

�� C� Heil� �A discrete Zak transform� Tech� Rep� MTR��W�� The MITRE Cor�poration� Dec� ��

�� L� Auslander� I� C� Gertner� and R� Tolimieri� �The discrete Zak transform applicationto time�frequency analysis and synthesis of nonstationary signals� IEEE Trans� Signal

Processing� vol� �� pp� �� April ��

�� H� B�olcskei and F� Hlawatsch� �Discrete Zak transforms� polyphase transforms� andapplications� IEEE Trans� Signal Processing� vol� �� pp� �� April ��

�� A� Ben�Israel and T� N� E� Greville� Generalized Inverses� Theory and Applications�Wiley� ��

�� S� Li and D� M� Healy� Jr�� A parametric class of discrete Gabor expansions� IEEETrans� Signal Processing� vol� �� pp� �� Feb� ��

�� D� G� Luenberger� Optimization by Vector Space Methods� New York� Wiley� ��

�� H� B�olcskei and F� Hlawatsch� �Frame�theoretic analysis and design of lter banks�Part II� Oversampled and critically sampled uniform lter banks� Tech� Rep� ��II�Department of Communications and Radio Frequency Engineering� Vienna Universityof Technology� Nov� ��

�� V� Belevitch� Classical Network Theory� Holden�Day� ��

�� D� C� Youla� �On the factorization of rational matrices� IRE Trans� Inf� Theory�pp� ��

�� P� Vary� �On the design of digital lter banks based on a modied principle ofpolyphase� AE�U� vol� �� no� �� pp� ��

� BIBLIOGRAPHY

�� M� Vetterli and D� LeGall� �Perfect reconstruction FIR lter banks� Some propertiesand factorizations� IEEE Trans� Acoust�� Speech� Signal Processing� vol� �� pp� �� July ��

�� F� Hlawatsch and H� B�olcskei� �Time�frequency analysis of frames� in Proc� IEEE�SPInt� Sympos� Time�Frequency Time�Scale Analysis� �Philadelphia� PA�� pp� �� Oct��

�� K� Swaminathan and P� P� Vaidyanathan� �Theory and design of uniform DFT� parallel�quadrature mirror lter banks� IEEE Trans� Circuits and Systems� vol� �� pp� �� Nov� ��

�� M� J� Bastiaans� �Gabor�s expansion and the Zak transform for continuous�time anddiscrete�time signals� Critical sampling and rational oversampling� Tech� Rep� ��E�� Technical University Eindhoven� Eindhoven� The Netherlands� Dec� ��

�� I� Daubechies� A� Grossmann� and Y� Meyer� �Painless nonorthogonal expansions� J�Math� Phys�� vol� �� pp� �� May ��

�� J� J� Benedetto and D� F� Walnut� �Gabor frames for L� and related spaces� inWavelets�

Mathematics and Applications �J� J� Benedetto and M� W� Frazier� eds�� pp� �� Boca Raton �FL�� CRC Press� ��

�� H� G� Feichtinger and K� Gr�ochenig� �Gabor frames and time�frequency analysis ofdistributions� Journal of Functional Analysis� vol� �� no� �� pp� � ��

�� H� G� Feichtinger� O� Christensen� and T� Strohmer� �A group�theoretical approach toGabor analysis� Optical Engineering� vol� �� pp� � ��

�� M� Poize� M� Renaudin� and P� Venier� �The Gabor transform as a modulated lter banksystem� inQuatorzi�eme Colloque GRETSI� �Juan�Les�Pins� France�� pp� �� Sept��

�� M� J� Narasimha and A� M� Peterson� �Design and application of uniform digital band�pass lter�banks� in Proc� IEEE ICASSP�� Oklahoma�� pp� �� April ��

�� A� J� E� M� Janssen� �On rationally oversampled Weyl�Heisenberg frames� Signal Pro�cessing� vol� �� pp� ��

�� D� F� Walnut� �Continuity properties of the Gabor frame operator� J� Math� Anal�

Appl�� vol� � �� pp� ��

�� H� S� Malvar� Signal Processing with Lapped Transforms� Artech House� ��

�� J� H� Rothweiler� �Polyphase quadrature lters� a new subband coding technique� inProc� IEEE ICASSP�� Boston� MA�� pp� �� April ��

�� P� L� Chu� �Quadrature mirror lter design for an arbitrary number of equal bandwidthchannels� IEEE Trans� Acoust�� Speech� Signal Processing� vol� �� no� �� pp� ��

�� R� D� Koilpillai and P� P� Vaidyanathan� �Cosine�modulated FIR lter banks satisfyingperfect reconstruction� IEEE Trans� Signal Processing� vol� �� pp� �� April ��

BIBLIOGRAPHY �

�� T� A� Ramstad and J� P� Tanem� �Cosine�modulated analysis�synthesis lter bank withcritical sampling and perfect reconstruction� in Proc� IEEE ICASSP�� Toronto�Canada�� pp� �� May ��

�� R� A� Gopinath and C� S� Burrus� �Some results in the theory modulated lter banksand modulated wavelet tight frames� Applied and Computational Harmonic Analysis�vol� �� pp� ��

�� T� Q� Nguyen and R� D� Koilpillai� �The theory and design of arbitrary�length cosine�modulated lter banks and wavelets� satisfying perfect reconstruction� IEEE Trans�

Signal Processing� vol� �� pp� �� March ��

�� Y��P� Lin and P� P� Vaidyanathan� �Linear phase cosine modulated maximally deci�mated lter banks with perfect reconstruction� IEEE Trans� Signal Processing� vol� ��pp� �� Nov� ��

�� N� J� Fliege� �Computational e!ciency of modied DFT polyphase lter banks� inProc� � th Asilomar Conf� Signals� Syst�� Computers� �Pacic Grove� CA�� pp� �� Nov� ��

�� N� J� Fliege� Multirate Digital Signal Processing� Wiley� ��

�� T� Karp and N� J� Fliege� �MDFT lter banks with perfect reconstruction� in Proc�

IEEE ISCAS�� Seattle� WA�� pp� �� May ��

�� T� Karp� J� Kliewer� A� Mertins� and N� J� Fliege� �Processing arbitrary length signalswith MDFT lter banks� in Proc� IEEE ICASSP�� Atlanta� GA�� pp� ��May ��

�� H� Xu� W� S� Lu� and A� Antoniou� �E!cient iterative design method for cosine�modulated QMF banks� IEEE Trans� Signal Processing� vol� �� pp� � �� July��

�� H� S� Malvar� �Lapped transforms for e!cient transform�subband coding� IEEE Trans�

Acoust�� Speech� Signal Processing� vol� �� pp� � �� June ��

�� J� Kliewer and A� Mertins� �Design of paraunitary oversampled cosine�modulated lterbanks� in Proc� IEEE ICASSP�� vol� �� Munich� Germany�� pp� �� April��

�� P� Yip and K� R� Rao� �Fast discrete transforms� in Handbook of Digital Signal Pro�

cessing �D� F� Elliott� ed�� Academic Press� ��

�� R� J� Marks� Introduction to Shannon Sampling Theory and Interpolation Theory�Springer Verlag� ��

�� L� E� Franks� Signal Theory� Englewood Cli�s �NJ�� Prentice Hall� ��

�� L� L� Scharf� Statistical Signal Processing� Reading �MA�� Addison Wesley� ��

�� R� A� Haddad and K� Park� �Modeling� analysis� and optimum design of quantized M �band lter banks� IEEE Trans� Signal Processing� vol� �� pp� �� Nov� ��

�� V� K� Goyal� �Quantized overcomplete expansions� Analysis� synthesis and algorithms�Master�s thesis� University of California at Berkeley� July ��

BIBLIOGRAPHY

�� N� T� Thao and M� Vetterli� �Deterministic analysis of oversampled A�D conversion anddecoding improvement based on consistent estimates� IEEE Trans� Signal Processing�vol� �� pp� �� March ��

�� N� T� Thao and M� Vetterli� �Lower bound on the mean squared error in oversampledquantization of periodic signals using vector quantization analysis� IEEE Trans� Inf�

Theory� vol� �� pp� � �� March ��

�� N� T� Thao� �Vector quantization analysis of sigma�delta modulation� IEEE Trans�

Signal Processing� vol� �� pp� �� April ��

�� Z� Cvetkovi�c and M� Vetterli� �Overcomplete expansions and robustness� in Proc� IEEETFTS�� Paris� France�� pp� �� June ��

�� I� Galton and H� T� Jensen� �Oversampling parallel delta�sigma modulation A�D con�version� IEEE Trans� Circuits and Systems II� vol� �� pp� �� Dec� ��

�� R� Khoini�Poorfard� L� B� Lim� and D� A� Johns� �Time�interleaved oversampling A�Dconverters� Theory and practice� IEEE Trans� Circuits and Systems II� vol� �� pp� �� Aug� ��

�� A� Papoulis� Probability� Random Variables� and Stochastic Processes� New York�McGraw�Hill� �rd ed��

�� A� Weinmann� Uncertain Models and Robust Control� Vienna� Austria� Springer� ��

�� U� Grenander and G� Szego� Toeplitz Forms and Their Applications� New York� ChelseaPublishing Company� ��

�� R� M� Gray� �Toeplitz and circulant matrices� Tech� Rep� ISL� �� Stanford Uni�versity ISL� April ��

�� C� W� Therrien� Discrete Random Signals and Statistical Signal Processing� EnglewoodCli�s �NJ�� Prentice Hall� ��

�� S� Hein and A� Zakhor� �Reconstruction of oversampled band�limited signals from sigma�delta encoded binary sequences� IEEE Trans� Signal Processing� vol� �� pp� ��April ��

�� S� L� Tan and T� R� Fischer� �Linear prediction of subband signals� IEEE Journal on

Selected Areas in Communications� vol� �� no� �� pp� ��

�� S� Rao and W� A� Pearlman� �Analysis of linear prediction� coding� and spectral esti�mation from subbands� IEEE Trans� Inf� Theory� vol� �� pp� �� July ��

�� P� W� Wong� �Rate distortion e!ciency of subband coding with crossband prediction�IEEE Trans� Inf� Theory� vol� �� pp� �� Jan� ��

�� S� M� Kay� Modern Spectral Estimation� Englewood Cli�s �NJ�� Prentice Hall� ��

�� R� A� Wiggins and E� A� Robinson� �Recursive solution to the multichannel lteringproblem� J� Geophys� Res�� vol� �� pp� �� April ��

�� A� Gersho and R� M� Gray� Vector Quantization and Signal Compression� Boston�Kluwer� ��

BIBLIOGRAPHY

�� R� M� Young� An Introduction to Nonharmonic Fourier Series� New York� AcademicPress� ��

�� C� Heil� �A basis theory primer� tech� rep�� Georgia Institute of Technology� ��

�� K� Gr�ochenig� �Acceleration of the frame algorithm� IEEE Trans� Signal Processing�vol� �� no� �� pp� ��

�� A� Aldroubi� �Portraits of frames� Proceedings of the American Mathematical Society�vol� �� no� � pp� � ��

�� H� G� Feichtinger� �Pseudo�inverse matrix methods for signal reconstruction from partialdata� in Proc� of SPIE Conf� Visual Comm� and Image Processing� �Boston�� pp� � �� Nov� ��

�� O� Christensen� Frame Decompositions in Hilbert Spaces� PhD thesis� Aarhus Univ��Aarhus� Denmark� and Univ� of Vienna �Vienna� Austria��

�� O� Christensen� �Frames and pseudo�inverses� J� Math� Anal� Appl�� vol� �� no� ��

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