WAVELETS AND NYQUIST FILTER DESIGN

by

Mingyu Liu

A thesis submitted to the Department of Electncal and Computer Engineering

in conformity with the requirements for the degree of Doctor of Philosophy

Queen's University Kingston, Ontario, Canada

March 1999

Copyright @ ikfingyu Liu, 1999

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To My Wife

Abstract

This dissertation is concemed with the design of wavelets and optimized Nyquist

filters. The use of wavelets as signaling waveforms in communications is investigated

first. The time-frequency properties for the autocorrelation functions of Daubechies

scaling Functions are analyzed and computed in terms of rms bandwidth and time

durntion products. The resuits are compared with raised cosine functions which are

cornrnonly used in communications.

For the Meyer-iike scaling hinctions and wavelets, which are bandlimi ted, t here

exist a set of them which have no intersymbol interference both before and after

matched filtering at the receiver. In this thesis, it is shonm that for time-limited

orthonormal scaling hinctions and wavelets, there is no such a scaling Fwiction except

the trivial Haar functions.

A new approach is deveioped for designing optimal FIR factorable Xyquist fil-

ten. The stopband energy is used as the criterion h c t i o n subject to a constraint

on the peak-sidelobe level and a cons traint ensuring fac torabili ty. The resul t ing

constrained quadratic optimization problem is solved by using the Goidfarb-Idnani

algorithm. The optimization problem at the stopband edge frequency is overcome

by dianging the starting frequency of the peak-sideiobe level

by simulations that there is a tradeoff between the stopband

constraint. It is shom

energy ratio and peak-

sidelobe level. By adjusting the peak-sidelobe level and its starting frequency, the

optimized Nyquist filters perfom better than some known results.

A new approach is proposed to designing smooth orthonormal wavelets from

FIR factorable half-band fiiters, a special kind of Nyquist fiiters. The tradeoff idea

between the stopband energy and peak-sidelobe level is employed to obtain the o p

tirnal half-band fdters. Bernstein polynomial expansions are used to incorpornte

smoot hness conditions into the resulting optimization problem. The optimized half-

band filter is then spectraiiy factorized by Bauer's method. The coefficients of the

minimum phase factor is then used to construct the scaling functions and wavelets

by using the interpolatory graphical display algorithm. It is shown that by adjust-

ing the peak-sidelobe level and the starting frequency for applying the constraint

on the peak-sidelobe lewl, the smoothness of scaling huictions and wavelets can

be improved. The calcdated Sobolev smoothness and simulations show that our

approach can construct smoother scaling h c t i o n s and wavelets t han previoi isly

known results .

Acknowledgment s

I would like to thank my s u p e ~ s o r , Dr. Frederick W. Fairman, to whom 1 am

greatly indebted for his expert guidance and for his kindness, generousity, encour-

agement and support over the years. 1 appreciate the confidence that he hns shown

in my abilities and the patience he has exhibited in teaching me lots of valuable

things.

I would also iike to thank my CO-supervisor, Dr. Christopher J. Zarowski, for his

guidance. His expert knowledge in wavelets and signal processing has been critical

to my work.

I gratefully thank my parents for their love and understanding. Mso, 1 tvould like

to thank my father-in-law and mother-in-law for their encouragement and support.

1 am gratehil to Zhongping Fang, Hongzhu Liang, and Marius Dan Secrieu for

t hei r help.

Summary of Notation

Abbreviat ions

BT

CWT

GI

ii f

ISI

b m

PCLS

PSL

QW

QPP

m

SER

STFT

WT

rms bandwidth and time duration

continuous wavelet t ransform

Goldfarb- Idnani

if and only if

intersymbol interference

hlui tiresolut ion analysis

peak-constrained leost-squares

peak-sidelo be level

quadrature rnirror filter

problem

root mem-square

stopband energy ratio

short- tirne Fourier transform

wavelet t ransform

Symbols

set of integers

set of nonnegative integers

set of real nurnbers

set of complex nwnbers

jpace of N-element column vectors with real nurnbers

space of N-element column vectors with complex numbers

Hilbert space of finite energy anaiog signals

Hilbert space of finite energy sequences

roll-off factor

Kronecker delta

stopband energy

Sobolev çmoothness of t$

passband edge frequency

stopband edge frequency

List of Tables

3.1 The rms duration and bandwidth for the two methods for Daubechies scding

functions ................................................................ 54

3.2 The BT products of Daubechies scaling functions for the two methods ..... 55

3.3 The rms bandwidth and single-sided 3dB bandwidth for the nutocorrelation

................................ hinctions of Daubechies scaling functions 56

3.4 The BT products of the raised cosine functions for different P ............. 60

1 Performance for N = 24, bt = 4. /3 = 0.1 and different 6 .................. 90

......... 4.2 Performance for PI = 24. M = -1. .3 = 0.1. A = 0.1 and different 6 -93

4.3 Performance for N = 24, M = 4, f l = 0.1 d = 0.006 and different A ........ 96

4.1 Performance for N = 16. M = 1. 0 = 0.1. A = 0.15 and different 6 ........ -99

4.5 Performance for N = 36. Ad = 4. 0 = 0.1. A = 0.06 and different 6 ....... 100

4.6 Performance for N = 4. hl = -40 = 0.1. A = 0.04 and different 6 ....... 101

4.7 Performance for N = 24. hl = 2. f l = 0 . l .A = 0.065 and different 6 ...... 104

4.8 Performance for N = 24. M = 6. P = 0.1. A = 0.1 and different 6 ........ -105

vi

........ 4.9 Performance for N = 24. hl = 8. /3 = 0.1. A = 0.1 and different 6, 106

4.10 Performance for N = 24. hl = 4. /3 = 0.05. A = 0.11 and different 6 ..... -110

4.11 Performance for N = 24. M = 4. /3 = 0.2. A = 0.07 and different 6 ....... 111

4.12 Performance for N = 24. M = 4. ,û = 0.3. A = 0.11 and different 6 ....... 112

4.13 Performance for N = 24. h.I = 4. ,û = 0.4. A = 0.04 and different 6 ....... 113

4.14 Performance for N = 24. M = 4. /3 = 0.5. A = 0.03 and different 6 ....... 114

5.1 Comparison of Srnoothness For Daubechies Scoling functions and our

................................................................. Design 159

5.2 Cornparison of Sobolev srnoothness between Cooklev's and our approach with-

out PSL constraint ..................................................... 160

0.3 Sobolev smoothness for different choices of 6! A and xs .................. 161

List of Figures

................. 2.1 The Daubechies scaling function and wavelet of N = 3.. .31

........................................ 2.2 Time-frequency plane for STFT. .38

......................................... 2.3 Time-frequency plane for CWT. 40

............................................... 3.1 Baseband channel model. .47

3.2 The BT products for raised cosine hctions and Daubechies autocorrelation

............................................................... functions. 61

R - 4.1 Typical Nyquist response hk (Shown for M = 4, N = 12). ............... ,û

1.2 Xyquist filter and its spectrum with N = 24, i t l = 1, 0 = 0.1 and

1.3 Nyquist filter and its spectrum with N = 24, Ad = 4, ,8 = 0.1 and

.............................................................. 6 = 0.059. 90

4.4 The relation between SER and PSL for N = 24, M = 4, IJ = 0.1 and

A=O.l. ................................................................ 92

4.5 The Nyquist filter, its spectnun and factorization for N = 48, hl = 4, ,O = 0.1

and 6 = 100. ........................................................... -97

S..

Vlll

4.6 The tradeoff relation between SER and PSL for M = 4, f l = 0.1 and different

N . (a) N = 16, (b) N = 24, (c) N = 36, (d) N = 48. ................... 102

4.7 The tradeoff relation on one scale for M = 4, f l = 0.1 and different N. '*' for

............. N = 16, '.' for N = 24, 'x'for N = 36, and '+'for N =48. 103

4.8 The tradeoff relation between SER and PSL for N = 24'0 = 0.1 and different

M. (a) hl = 2, (b) M = 4, (c) M = 6, (d) il.! = 8. ...................... 107

1.9 The tradeoff relation on one scale for N = 24, P = 0.1 and different M. '*'

for M = 2, '.' for M = 4, 'x' for h.1 = 6, and '+' for M = 8. ............ 108

4.10 The tradeoff relation between SER and PSL for N = 24, hf = 4 and different

0. (a) ,O = 0.05, (b) P = 0.1, (c) P = 0.2, (d) ,B = 0.3, ( e ) ,8 = O.-&? ( f )

f l = 0.3. .............................................................. . i l 5

4.11 The tradeoff relation on one scale for N = 24, hl = 4 and different S. *+' for

@=0.05, '.' for P=0.1, "-'for 0 =0.2, 'x' for ,a= 0.3, 'O' for p = 0.04. and

'*' for 0 = 0.5. ........................................................ -116

4.12 The impulse response and magnitude response of Nyquist filter for Example

1 in [103] where N = 30, M =6, P = 0.52' A = 0 and 6 = 100. ......... 118

4.13 The impulse response and magnitude response of Nyquist filter for Example

1 in [103] where N = 30, Ad = 6, P = 0.52, A = O and 6 = 0.000075. .... 119

4.14 The impulse response and magnitude response of Nyquist filter for Example

1 in [103] where N = 30, M = 6, ,O = 0.52, A = 0.03 and 6 = 0.000031. . 119

4.15 The impulse response and magnitude response of Nyquist filter for Example

2 in [IO31 where N = 20, M = 4, P = 0.52, A = 0 and 6 = 100. . . . . . . . . . 120

4.16 The impulse response and magnitude response of Nyquist filter for Exmple

2 in Il031 where N = 20, M = 6, ,O = 0.52, A = 0.04 and 6 = 0.00004. . . .120

5.1 Typical haif-band filter response hk (Shown for N = 11). . . . . . . . . . . . . . . . . 125

5.2(a) The impulse response and magnitude response of the half-band filter with

N = 17, L =7, rn=30, x, =0.5, LI = L2 = 200, 6 = 100, A = 0. ....... 145

5.2(b) The resulting scaling function and wavelet from the half-band filter in Fig.

5.2 (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.3(a) The impulse response and magnitude response of the half-band filter with

N = 17, L = 7, m = 30, x* =0.5, LI = L2 = 100, 6 = 0-055, A = 0.15 .... 147

5.3(b) The resulting scaling function and wavelet from the half-band filter in Fig.

5.3(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-18

5 4 4 The impulse response and magnitude response of the half-band fiter by

Cooklev's method for N = 17, L = 7, .m = 30, x, = 0.5. . . . . . . . , . . . . . . . . . 149

5.4(b) The s c d i g fimction and wavelet constructed from the lowpass filter derived

by spectrally factorizing the half-band filter in Fig. 5.4(a).. . . . . . . . . . . . . . -150

5.5(a) The impulse response and magnitude response of the hdf-band filter with

N = 17, L = 7, m =30, X, =0.6, LI = L2 = 100, 6 = 0.020, A = 0.2 ..... 152

5.5(b) The resulting scaling function and wavelet from the haif-band fiter in Fig.

5.5(a). ................................................................. 153

5.6(a) The impulse response and magnitude response of the hdf-band filter with

N = 17, L = 7, m = 30, X, =0.7, LI = L2 = 100, 6 = 0.01, A = 0.2 ...... 154

5.6(b) The resulting scaiing function and wavelet from the half-band filter in Fig.

5.6(a). ................................................................. 155

5.7(a) The impulse response and magnitude response of the half-band filter with

..... N = 17, L =7, m = 3 0 , xS =0.7, LI = Lz = 100, 6=0.005, A = 0.2 156

5.7(b) The resulting scaling hinction and wavelet from the half-band filter in Fig.

5.8 The magnitude responses of the half-band filter and its transmit filter for N =

............ 17, L = 7, LI = L2 = 100, x, = 0.5, A = 0.15 and 6 = 0.055. 162

Table of Contents

Abstract .................................................................... i

... Acknowledgements ........................................................ iii

Summary of Notation ..................................................... iv

List of 'Iàbles .............................................................. vi

... List of Figures ............................................................ viii . .

Table of Contents ......................................................... xi1

Chapter 1 Introduction ..................................................... 1

1.1 &lotivation .............................................................. 2

....................................................... 1.2 Literature Review 4

1.3 Thesis Cvntributivns .................................................... 9

1.4 Thesis Guide ........................................................... 11

Chapter 2 Background and Preliminaries ................................ 12

2.1 The Definition of Wavelets ............................................. 12

2.2 Construction of Wavelets .............................................. -14

2.2.1 Construction of Wavelets from MBA ............................. 15

.................... 2.2.2 Construction of Wavelets kom Discrete Fiiters 13

xii

............... 2.3 Wavelets and Scding Functions in the Frequency Domnin 18

........................................... 2.4 The Smoothness of Wavelets 21

................................................... 2.5 Illustrative Examples 27

....................... 2.5.1 Daubechies Scaling Functions and Wavelets 27

............................ 2.5.2 Meyer Scaling Functions and Wavelets 32

.............................................. 2.6 Time-Frequency Analysis -33

................................................. 2.6.1 Some Defhitions 33

............................ 2.6.2 Time-Frequency Analysis Using STFT 36

............................. 2.6.3 Using Wavelets as Window Functions 39

..................................................... 2.7 Chapter Summary 42

Chapter 3 Wavelets as Signaling Waveforms in Communications ...... 43

............... 3.1 Daubechies Scaling Functions as the Signaling Waveform 45

... 3.2 The RMS Duration and Bandwidth of the Autocorrelation Functions -48

..................................... 3.2.1 RMS Duration Computation 49

................................... 3.2.2 RMS Bandwidth Computation 51

............... 3.2.3 The BT Products of the Autocorrelation Functions 32

....................... 3.2.4 The RMS Bandwidth and 3dB Bandwidth -57

........ 3.3 The RMS Duration and Bandwidth of Raised Cosine Functions - 5 8

................................................. 3.4 Performance Analyses -61

............................ 3.5 ISI-Free Finite-Supported Scaling Functions 63

........................ 3.5.1 The Coefficients of the Dilation Equation -63

................................... 3.5.2 The ISI-Free Scaiing Functions 67

..................................................... 3.6 Chapter Summary 71

............................... Chapter 4 Optimal Nyquist Filter Design 72

......................................................... 4.1 Nyquist Filters 74

.................................................. 4.2 Problem Formulation. 76

................................... 4.2.1 The Function to Be Minimized 77

.................................. 4.2.2 The Constraints to Be Satisfied 78

.............................. 4.3 The Solution to the Optimization Problem 80

................................. 4.3.1 The Goldfarb-Idnani Algorithm - 81

................. 4.3.2 A Solution Using the Goldfarb-Idnani Algorithm 85

................................................ 4.4 PSL Constraint Interval 88

............................................ 4.4.1 Illustrative Examples 88

....................................... 4.4.2 A New Constraint Interval 9 1

................................................ 4.4.3 Adjustment of A .94

............................... 4.4.4 Sensitivity of Optimal Filter to A - 9 5

4.4.5 An Example for the Optimal Filters .............................. 96

.................................... 4.5 The Tradeoff between SER and PSL 97

4.5.1 The Tradeoff for Different Filter Length N ...................... -98

................................... 4.5.2 The Tradeoff for Different hl -103

.................................... 4.5.3 The Tradeoff for Different 0 -109

..................................... 4.6 Cornparisons with Known Results 117

.................................................. 4.7 Concluding Remarks 121

Chapter 5 Wavelet Construction Fkom HaKBand Filters ............ -122

................................................ 5.1 FIR Half-Band Filters 125

...................................... 5.2 Bernstein Polynomial Expansion 126

................................................. 5.3 Problem Formulation 131

........................ 5.3.1 The Objective Function to Be Minirnized 131

................................. 5.3.2 The Constraints to Be Satisfied 132

....................... 5.4 A Solution Using the Goldfarb-IdnaniAlgorithm 134

................................................ 5.4.1 Computing 70, R 135

........................................... 5.4.2 Matching Constraints 136

5.5 Spectral Factorization and Wavelet Construction ...................... -138

5.5.1 Bauer's Spectral Factorization .................................. 139

........................................... 5.5.2 Wavelet Construction 140

........................ 5.5.3 Smwthness of the Constmcted Wavelets 111

.................................................... 5.6 Simulation Results 113

5.6.1 Wavelet Construction with Fixed xs ............................. 144

.......................... 5.6.2 Wavelet Construction with Different x, 151

5.6.3 Sobolev Smoothness Cornparisons ............................... 158

5.6.4 Some Considerations for Applications ........................... 162

........................................................... 5.7 Conclusions 163

Chapter 6 Concluding Remarks ........................................ -164

............................................. 6.1 Surxunary of Presentation 165

6.2 Conclusions .......................................................... 166

....................................... 6.3 Suggestion for Future Research -168

Bibliograp hy ............................................................. 171

Appendiv A Matlab Routines for Supporting Prograrns ........... 193

Appendix B Matlab Routines for Main Programs ................. 202

Vita ...................................................................... -220

Chapter 1

INTRODUCTION

For the past ten years, wavelet theory and its applications have received considerable

attention. R e c d that a Fourier transform represents a signal as a superposition of

sinusoids with different Frequencies, and the Fourier coefficients measure the contri-

butior. of the sinusoids at these frequencies. Similarly, a wavelet transform represents

a signal as a sum of wavelets with different locations and scales. The wavelet coef-

ficients indicate the strength of the contribution of the wavelets at these locatiuns

and scales. Wavelet analysis is far more efficient than Fourier analysis whenever a

signal is dominated by transient behavior or discontinuities, as we wiil see in later

chapters of this thesis. In this chapter we wiil give an overview of this thesis - its

motivation, contributions, literature review and presentation out line.

1.1 Motivation

Wavelet constructions c m be traced back to the work by Alfied Hnor in 1910, [61],

where he used the dilations and translations of a simple piecewise constant function

to generate an orthonormal basis, now cailed Haar wavelets. However the truiy

pioneering efforts which spawned the fascinating field of wavelet theory were nut

reached until the work of Grossman and Motlet in 1984 (601. This work, along

with the works of Haar (611, Gabor [53], Allen and Rabiner [5] and Portnoff [99],

led to work by Daubechies [32], 1371 and Meyer [89] on orthonormal wavelet bases,

and Mallat [85], [86] on multiresolution analysis. Cohen [20], Chui and Wang [Id],

Strang [121], and Nguyen and Vaidyanathan [131] [94] and others made significant

contributions to the early development of wavelets as well. These theoret i d works

greatly darified the relation between wavelets in the continuous context! which is

familiar to mathematical analysts, and wavelets in the discrete context, as needed

in digital signal analysis. This helped to motivate a tremendous interdiscipiinary

effvrk tu apply wwavelet m e t h d i tu many fields as f d u w s ((81 Il371 [1?0/ (1231 [1:0;j

1. Computer vision and image compression (1201 [23] (841 (91

2. Adaptive filtering [a] [19] (461 (1271

3. Digital communications (discussed further below )

4. Fractals [14] [142]

5. Numerical analysis [124]

6. Time-frequency analysis and denoising [21] [41] [.LOI [91] [107] [125]

In this thesis we focus on wavelet applications in communications. In communi-

cation systems, the properties m a t called for are the ability to distinguish between

desired and undesired signal components and the Rexibili ty of adap t ively irnproving

some aspect of the signal representntion. Frequency resolution is a tradeoff for time

resolution in a seamless and methodical manner. In addition, orthogonality across

both scale and translation makes wavelets interesting to the cornmunicat ions corn-

munity. Systems typicaily d e r from performance degrndation due to intersymbol

interference (ISI), where the overlapping of adjacent symbols in a aven transmis-

sion causes compiicat ions at the receiver. The orthogonality of wnvelets rectifies

this problem. The flexibility of a wavelet in the time-freguency plane may lend to

more efficient applications to communications because important information often

appears through a simultaneous analysis of the signal's time and frequency p r o p

erties. In addition, the relation between wavelets and filter banks has also been

revealed. A wavelet can be constructed from quadrature rnirror fdters, which satisfy

some smoothness conditions. This gives a possible way to generate wavelets under

desired criteria.

The study presented in this thesis has b e n motivated by the facts just men-

tioned. More specifically, the goal of this thesis is to investigate the possibility of

using wavelets as the sîgnailing waveforms in communications systems, to design

optimal Nyquist filters and to constmct wavelets fiom Nyquist filters

1.2 Lit erat ure Review

The application of wavelets to communication systems has received a great deal of

attention in the last six years. The time-fiequency nature of the wavelet construc-

tions (wavelet , wavelet packets, M-band wavelets and multiwavelets) is appealing.

Previous applications of wavelets in communications include :

1. Waveform Design

Channel coding [128] [129]

modulation including fractal modulation (1.121 [143] [102!, continuous time

waveform representation of data bits [M] [55] [56] [82], multi-scale mod-

ulation and M-band wavelet modulation (701 [71], wavelet packet rnodr-

lation [140] [145] [79] [BI].

spread spectnun and covert communications [811 [65] [I5] [16] [l?] [70]

multiple users [27] [28] [I l l ] (1 121 [77] (1261 [a]

2. Interference Mit igation

transform domain excision [88] [72]

adaptive filtering [46] [47] [a] [49] [88] [43]

3. Other Applications

symbol synduvnization [29]

signai detection [44]

Channel identification (1271

In this thesis we focus on Nyquist-type wavefom design in digital cornrnunica-

tions and wavelet construction using optimized Nyquist fdters. We first consider

wing wavelets as signaling waveforms. Previous results were presented in [54] [82]

[?O] [SI] [MO] [77], where wavelets or wavelet packet hinctions are used as signal-

ing waveforms. The time-frequency flexibility of wavelets is the main advantage

over the Fourier basis. The time-frequency properties of wavelets were proposed

by many researchers [83] [13] [30] 1631 [91] [151]. The usual way to rneasure the

time-frequency localization of a function is to compute the rms bandwidth and time

duration (BT) product provided that these exist. Because of the wicertainty prin-

ciple the BT product of a h c t i o n cannot be made arbitrarily smail. Wavelets

can provide Aexi ble tirne-frequency localiza tions. The BT products of Daubechies

wavelets were previously computed in [91]. It is knom that the autcJccJrrelatim

h c t i o n of any orthonormal wavelet packet function is a Nyquist pulse. Therefore

when the wavelets are used as the signaling waveforms, there wiU be no intersyrn-

bol interference at the receiver. In this sense the tirne-frequency properties of the

autocorrelation hinctions should also be considered. In [?O], Jones shows thnt the

square roots of the commonly w d raised coaine hinctions in communications are a

special case of the Meyer scaling hc t ions , Le., wavelets were already used in com-

munications. It is worth knowing whether or not there are other wavelets which are

better than the square roots of r a i d cosine functions in the sense of BT product.

Irt [146] [38], a family of signaling waveforms are proposed with ISI-free motched

and unmatched filter properties. These waveforms are actuaily Meyer-like scaling

hnctions - they are bandlimited but not time-limited. This raises the question : 1s

there any time-limited scaling function which has such properties? This question is

answered in Chapter 3.

In the matter of wavelet construction, we first consider digital filter design be-

cause wnvelets can be generated from hnlf-band filters. In the pnst, most digital

filters were designed according to the rninimax and lenst-squares optimality criteria.

The history of FIR filter design is dorninated by the Parks-McClellan algorithm 1871.

[98]. This algorithm is based on the minimax optimaiity criterion. hloreover, the

altemation theorem provides a simple test for evdunting the optimali ty of minimax

solutions. The minimax criterion is appropriate for the possband in many appli-

cations. It is usually important to minimize the maximum amplitude distortion

for signals to be passed by a filter. The rninirnnx cntenon, however, is frequently

not appropriate in the stopband because i t minirnizes the maximum error without

regard to the error energy. Both the maximum stopband level and the stopband en-

ergy are crucial for many applications, especiaLly for those using narrow-band fiiters.

Narrow-band Nters are frequently used to separate the channels in communication

systems using frequency division rnultiplexing. Narrow fdter bandwidt h is required

when the number of channels is large. Major Instances of filter design using the

minimax criterion are reported in [96], [78], [go] and [104]. On the other hand,

the least-squares criterion is appropriate for the case where the input spectrum is

wideband and distributed approximately uniformly in frequency. In this case it is

important to miniMze the total energy of aliased signals. This is especially true

when the passband is narrow and the decimation ratio is large. As a result substan-

tial energy can be aliased into the narrow passband. Least-squares approximations

are frequently used for multirate signal processing applications. Their solutions are

easy to compute, and easy to justify in simple terms. The disadvantage of the

lest-squares critenon is that the resulting filters have large gains nt the edges of

t heir stopbands which are caused by the Gibbs phenornenon. Least-squares approx-

imations produce large errors near discontinuities in the desired response. Some

important instances of filter design using the least-squares criterion are reported in

[92], [132], [26] and [Il?].

Based on the idea of rninimax and least-squares criteria, Adams [Il-[3] generalizes

these two critena into the class of peak-constrained least-squares (PCLS) optimiza-

tion problerns, where the total squared error is minirnized while the peak error is

constrained. This gives a tradeoff between the total squared error and the peak er-

ror. The minimax and least-squares criteria are proved to be the two extreme cases

of the PCLS criterion. There are some difficulties to be solved in PCLS optimization

problems (21, for instance, the choice of the constrained optimization a igor i th , and

difficulties in imposing constraints at the stopband edge frequency, etc.

Nyquist filters are commody used in communication systems [101]. Wany a p

proaches have been proposed for designing the optimal Nyquist filter under different

criteria [92], [113], 11321, [78], [go] and [104]. These criteria are either minimax or

least-squares, i.e., there is no control over the tradeoff between the stopband en-

ergy and the peak-sidelobe level. We will use the PCLS idea to design our optimal

Nyquist filters and to provide a solution to a difficulty which was encountered in

Pl- [31 In [37] the relation between wavelets and digital fiters was revealed, i.e., wavelets

c m be constructed from certain digital filters. In [76] and [Il], necessary and suf-

ficient conditions for constructing orthonormal wavelet bases are presented. In fnct

n special kind of Nyquist füters, named half-band füters, cnn be used to genernte

wavelets under certain constraints 1261. Some known wavelets, e.g.. Daubechies

wavelets and Meyer wavelets, were constructed kom some special functions [37],

[32] and (891. In [26], Cooklev presents an approach to generating wavelets from

half-band filters. He uses Bernstein polynomial expansions to obtnin some con-

trol of the smoothness of wavelets. The optimization method he used is bved on

the leas t-squares cri terion. The resulting half-band filters have a relatively high

peak-sidelobe level. A major problem in this approach is that the constrained least-

squares method he proposed is not guaranteed to converge to the optimal solution.

In addition, the smoothness of wavelets obtained in [26] may not be satisfactory.

These drawbacks motiva t ed our approach to constmc t ing smw t her wovele t s using

our idea for designing optimal Nyquist filters.

1.3 Thesis Contributions

The contribution of this thesis is pnmarily in the area of methods for the design

of orthogonal wawlets and Nyquist filters. It is anticipated that this wiil form the

fondation for the development of such things as signahg waveforms for digital

communications that are better than those presently in use such as square-root

raiseci cosine pulses. The following is a synopsis of the significant contributions

presented in this thesis.

1. In Chapter 3, t ime-frequency analyses for the autoconelation functions of

Daubechies scaling h c t i o n s have been proposed. The mis bandwidth and

time duration (BT) products for the autocorrelation h c t i o n s of Daubechies

scaling functions have been derived and computed via both iteration algo-

nthms and numerical integration. These BT products are compared with the

comrnonly used raised cosine func t ions in cornmunicat ions systems.

2. In Chapter 3, tirne-limited, ISI-free and orthonormal scaling functions have

been derived and the solutions proved to be square pulses. This is a solution

of the counterpart problem b r those scaling functions with i n f i t e support in

the tirne domain, e.g., Xia scaling hinctions. In Xia [Id61 it has been shuwn

that there exists a set of Meyer-like scaling functions whidi are ISI-free both

before and after their matched filters.

3. In Chaptar 4, a new approach has been proposed for generating a set of fac-

torable Nyquist filters with tradeoff between the stopband energy and peak-

9

sidelobe level. The Goldfarb-idnani Aigorithm is used to rninimize the s t o p

band energy subject to two constraints, one on the peak-sidelobe level and one

to ensure factorabili ty.

4. In Chapter 4, a scheme has been propased to overcome certain difficulties

indicated in [3] which involve the use of the stopband edge kequency.

5. Examples are given in Chapter 4 which show that the constraint on the side-

lobe level provides Nyquist fiters which may give better performance than

p reviously known result S.

6. In Chapter 5, the idea of factorable Nyquist filter design with tradeoff between

the stopband energy and peak-sidelobe level is employed to design optimum

factorable hdf-band filters. These half-band flters are then used to generate

scaling functions and wavelets by BauerTs spectral factorbation and the in-

terpolatory graphical display algorithm (IGDA). With Bernstein polynomial

expansions, the smoothness of scaling functions and wavelets can be incorpo-

rated in the optimum half-band filter design. Sobolev smoothness is chosen to

ob jec t ively evalua t e the smoot hness of the result ing wavelet s.

7. In Chapter 5, it is shown in the sense of both theoretical calculation and

waveform plots that scaling functions and wavelets which are srnoother than

those in 1321 and [26] can be obtained by adjusting the peak-sidelobe level and

the stopband edge frequency of half-band filters. This gives a flexible approach

to generating a ïariety of smooth wavelets.

10

1.4 Thesis Guide

the next chapter, basic wavelet theoretic concepts and properties are introduced

well as a discussion of time-frequency analysis. This basic background mnterial

1 be used in later chapters. In Chapter 3, the time-Frequency properties for the

autocorrelation functions of Daubechies scaling functions are derived and computed

iteratively. These are compared with the corresponding results for raised cosine

functions. The t ime-limi ted, ISI-free and orthonormal scaling funct ions are consi d-

ered therein as well. In Chapter 4, factorable Nyquist flters with tradeoff between

the s topband energy and the peak-sidelobe level are derîved and the Goldfarb-Idnani

algori thrn is int roduced to solve the resulting constrained op tirnization problem. A

nurnber of simulations are presented there. In Chapter 5, we use the idea of Chapter

4 to design optimum factorable half-band fdters. These filters are used to generate

orthogonal scaling functions and wavelets. The smoothness of the result inp scaiing

Functions and wavelets are investigated by both theoretical calculation and simula-

tion. Some cornparisons are made. Finaily, Chapter 6 contnins the conclusions of

this thesis as well as directions for future research.

Chapter 2

Background and Preliminaries

Wawlet mathematical theory is reaching a mature stage. In this chapter, how-

ever, we oniy choose to int roduce wavelets, wavelet construction and t ime-frequency

analysis arnong ail aspects of wavelet theory. These properties are used in the later

chapters of this thesis. Wavelet constmction wiil be used in Chnpter 5, and the

time-frequency properties of some wavelets wiii be investigated in Chapter 3.

2.1 The Definition of Wavelets

We begin with some basic definitions for the wavelet theory.

Dej id ion 2.1 : For a fùnction f ( t ) , whidi satisfies

its Fourier transform is defined as

and the corresponding inverse Fourier transform is

1 +- f ( t ) = , Lm ~ ( w ) e " " & -

Sometimes we use f (w ) to denote the Fourier transfom of f ( t ) in this thesis.

Definition 2.2 [18] : We c d multiresolution analysis (MRA) a sequence of

approximation subspaces ( y } of L2 (R) such that the foliowing requirements

are sat isfied:

1. The V, are generated by a scaling function 4 E &(a), in the sense that, for

each fixed j, the family

spans the space 4 and satisfies the C2 stability condition for {ak } C t2 (Z)

which is independent of the choice of the coefficients ak and has C 2 c > 0.

2. The spaces are embedded, that is, V, c V,+,.

3. The orthogonal projectors Pj onto 5 satis& . lim Pj f = / and lim Pj f = O l'+al 34-a0

for aU / E &(a).

Definition 2.3 [13] : (4j ,k(t)}7 as defineci in (2.1), is cailed a Riesz basis of &('R)

if the h e a r span of #j ,k( t) , j , k E Z is dense in f *(a) and positive constants A and

B exist, with O < A 5 B < ao, such that

for dl doubly bi-infinite square-summable sequences {c jVr} , that is,

In different fields, wavelets may appear in vanous forms: from discrete-time,

subband coding, and filter banks, to continuous time, wavelet series, and wavelet

transforms. We use the foliowing definition as the starting point.

Definilion 2.4 [37] : The families of functions $a,6

a 1 4 2 t - b =I +(-Il Q

which are generated from one singie function 11 by the operation of dilations and

translations, are called wavelets.

The parameters a and b can be chosen to satisfy the different types of appli-

cations. For a, b E 7Z and a # O, it is possible to obtain the continuous wavelet

transform of a function. For suitable discrete a and b, we can get the wavelet senes

of a signal. Since a and b are considered as the scale dilations and time translations.

wavelets can be used to perform time-frequency analyses.

2.2 Construction of Wavelets

Wavelets can be generated from multiresolution anaiysis (MRA) t heory [37], [85],

and discrete Bters. There is a strong connection between wavelets and discrete

filters (especially fdter banks) [120]. In this section we will discuss some methods of

constructing wavelets.

2.2.1 Construction of Wavelets from MRA

From the definition of M M , one knows for a basis hinction @(t) that

so there exkt h, such that

where $ ( t ) is called a scaling hinction, and (2.3) is calied a dilation equation. Then

the wavelets can be generated as

where

By dilations and translations of $( t ) , a set of wavelets can be constructed as

thj,&) = 2 ~ ' ~ ~ 1 ( 2 J t - k), j , k E S. (2.5)

2.2.2 Construction of Wavelets from Discrete Filters

We will see below that for the construction of orthonormal bases of compactly siip-

ported wavelets it is more natural to start from the coefficients h,, than from the

function 4.

The following theorem gives conditions for the constmction of wavelets from the

discrete filters.

Theorem 2.1 [37] : Let h,, be a sequence. Let g,, = (- l)nh-n+l and +Q( 0,

(v) SUPEE R I E n /neinC I < P-I

then we have

+( t ) = 2112Cgn&2t n - n) .

It foUows from the Tneorem that 4 j , k ( t ) = 2 j i24 (2 j t - k ) define an hIRA and the

lLjqk ( t ) are the associated orthonormal wavelet basis.

In practice an iterative algonthm known as the cascade algorithm can be used

to construct a wavelet basis from discrete filters.

Let be the filter coefficients. The iterations s tart from the box function qo(t) as

below. There are two steps in each iteration-filtering and rescaling. The algorithm

is summarized below.

(i) q ~ ( t ) = 1 on [O, 11, and O elsewhere.

The algorithm works with Functions in continuous time. These h c t i o n s are

piecewise constant and the pieces become shorter (their length is 2-'). If q ( t ) con-

verges suitably to a limit #(t ) , then this limit function solves the dilation equation

(2.3).

Another approach to constructing a wavelet frorn a discrete filter h,, is the in-

terpolatory graphical display algorithm (IGDA) 1131. It is a numerical algorithm to

compute the scding functions and wavelets on the dyadic points, which are

where J E {O, 1,2,3, }. The steps are outlined as follows:

1. Compute the scaling function on the integers

2. Compute 4(t) on the dyadic points

3. Compute @(t ) on the dyadic points

The detailed algorithm can be found in 1131 and [l5l]. W e will use the IGDA in

Chapter 5 to construct wavelets from a discrete filter &

Many wavelets have a remarkable feature of compact support, Le., a wavelet

function is zero outside some interval. For instance, if h, = O for n c O and n > N ,

17

and so

then the support of #( t )

2.3 Wavelets and Scaling Functions in the Fre-

quency Domain

In this section, we summarize some important facts about wavelets and scaliny

hinctions in the frequency domain, which we wili employ in comection with the use

of wavelets in comrnunications and wavele t cons truc t ion.

We know that the scaling equation and wavelet equation can be given as

Fourier t ransforming the two equat ions yields

where

Recursive application of the process gives

If we normalize 4(t) such that Pm 4( t )d t = 1, which is usuaily the case, then

For orthonormal { d j q k ( t ) ) and {?,bjqk(t)}, we can obtain (see [151] and [32])

As a matter of fact, (2.11) is the Nyquist pulse citenon in communications. Xotice

that 1 @ ( w ) l2 is the Fourier transform of # ( t ) * #(- t ) . This implies that if an

orthonormal scaling function is used as the signaling waveform, then there will be

no intersymbol interference after matched filtering at the receiver. This is a necessary

condition for wavelets to be used in communications.

Substituting (2.6) in (2.11) gives

Then we have the condition on h,

where r = eju.

Similarly we have

Substituting (2.7) yields

From (2.6) and (2.8), we obtain

Then we have from (2.12)

for real k. So we obtain

In sumrnary, we list these important frequency properties for orthonormal scaling

functions and wavelets, which will be used in the next section:

The f is t property is in fact a condition in the frequency domain for half-band

filters. In Chapter 5 we will use optirnized half-band füters to construct wavelets.

The second property implies that Li(&") has at l e s t one zero at w = a and G(eJU)

has at least one zero at w = O. This implies that H(eJw) is a lowpass filter and G(eJW)

is a highpass filter. This leads to a practicaily important and usefui connection

between wavelets and filter b a h . This property will be used to construct wavelets

fiom filter banks in Chapter 5, where some additional zeros at w = n will be imposed

for scaling functions to increase the smoothness of the resulting scaling hinctions.

2.4 The Smoothness of Wavelets

The smoothness of @ has mostly a cosmetic influence on the error introduced by

thresholding or quantizing the wavelet coefficients. When reconstmcting a signal

fÎom its wavelet coefficients

an error c added to a coefficient c f , %,, > will add the wavelet component c $ ~ , ~

to the reconstructed signal. If 11 is smooth, then E $ ~ , ~ is a smooth error. For image

coding applications, a smooth error is often less visible than an irreguiar error, even

thoiigh they have the same energy. Better quality images are obtained with wnvelets

which are cont inuously ciiflecent iable than with the discontinuous Haar wavelet .

The smoothness of a h c t i o n II, can be measured by the number of times it is

differentiable. To distinguish the smoothness of hinctions that are n - 1 times, but

not n times, continuously differentiable, we m u t extend the notion of differentia-

bility to non-integers. This can be done in the Fourier domain. Recall that the

Fourier transform of the derivative $'(t) is i w i I ( w ) . Parseval's Theorem states that

ly E C2(R) if

This suggests replacing the usual pointwise definition of the derivative by a definition

based on the Fourier transform. We Say that 11 E L2(R) is differentiable in the sense

of Sobolev if

This inequality implies that 1 i I ( w ) 1 must have a sufficiently fast decay when the

fiequency w goes to CG, typicdy faster than ( w 1 - * . The smuuthness uf IL. is

measured from the asymptotic decay of its Fourier transform. This definition is

generalized for any s > O to give the Sobolev smoothness.

The Sobolev smoothness of a function S(t) E C2 ('R) is defined [45] as

where 'Ha is the Sobolev space and

where s is real and s >_ O. A function is said to have s denvatives in the time domain

if its Sobolev smoothness is S. The following proposition gives the relation between

the Sobolev smoothness and the number of zeros of H ( d w ) at w = R.

Proposition 1 1831 : Suppose that H ( @ ) as giuen in the previous section hcis L

teros al w = n. Let us perjonn the faclorizalzon

I/ sup,,.~ 1 P ( P ) I= B then 11 and # have the Sobolev smoolhness of s for

This proposition proves that if B c 2L-' then so > O? which means thnt w and

4 are uniformly continuous. For any m > O, if B < 2L- l-m then sa > rn so q!~ and

4 are rn times continuously differentiable, i.e., $J and 4 have Sobolev srnoothness

of m. A priori, we are not guaranteed that increasing L wiu improve the wavelet

smoothness, since B might increase as well. However, for some important families

of wavelets, such as Daubechies wavelets, B increases more slowly than L, which

implies that wavelet smoothness increases with L. The following is a detailed analysis

of smoothness for H(;) and G(r) introduced in the previous section.

Since H ( t ) must have at least one zero at r = -1, as discussed before, we

suppose H(r ) has L zeros at z = -1 and that H ( z ) is FIR of degree N - 1. Then

where P ( z ) is a polynomial in t of degree N - 1 - L

We wodd like to see the effect of these L zeros at

i.e., the smoothness of $( t ) and + ( t ) . We have

with real coefficients.

t = -1 on the decay of O ( w ) ,

The sin^:)^ term contnbutes to the decay of @(w) provided that the second

term can be bounded. This form has been used to estimate the smoothness of 4(t).

One such estimate is as follows.

Let P(cjW) satisb

for some 1 > 1. Then h, defines a scaiîng function d( t ) that is rn-times continuously

different iable.

W W 1 H(&") = e-jWL12(cas - ) L ~ ( d " ) = (cos - - ) L ~ ( w ) = - 1 h,,e-jw". 2 fi n

and

This produces a smooth lowpass filter.

Since G(r) = -2 H (-2) (see Section 2.3), the highpass filter G ( ; ) has L zerus nt

z = 1. We can write

W G(etw) = (sin - ) L ~ l ( w ) .

2

The (sin :) term insures the vanishing of the derivatives of G(ejW) at w = O and

the associated moments, that is,

yielding

We need to investigate further the choice of L in (2.18). Since P ( z ) is a polyne

rnial in 2-' with real coefficients Q(r) = P(z)P(:- ') is a symmetric polynomial:

Therefore,

So 1 P(@) I 2 is some polynomial f ( O ) , in (sin2 :) of degree N - 1 - L:

where x = sin2 :.

Therefore from (2.12) we know

This equation has a solution of the form

where R(x) is an odd polynomial such that

For different choices of R ( x ) and L, we wiii obtain different wavelet solutions,

that is, we can get wavelets of m y high smoothness, of course at the expense of long

filter lengths.

For Daubechies wavelets, R(x) = 0,

This equation can be employed to compute the coefficients of Daubechies scaling

functions and wavelets in the next section. The smoothess of wavelets will be used

in Chapter 5 to evaluate our wavelets constmcted from the optimized half-band

filters.

2.5 Illustrat ive Examples

We will mainly consider two kinds of orthonormal wavelets in this thesis. The

first is compactly supported, i.e., the wavelet function is zero outside some interval.

Daubechies wavelets are a typical example of this. The second is bnndlimited, but

not time-limited. A typical exmple is the Meyer wavelet.

2.5.1 Daubechies Scaling Funct ions and Wavelet s

In this section we give the coefficient derivations for Daubechies scaling functions

and wavelets. The derivation results here are realized in the Matlab routine wc0f.m

in Appendix A. These coefficients are necessnry for the BT product curnputatiuns

for the autocorrelation functions of Daubechies scaling functions and wavelets. We

begin with Equation (2.20) in the previous section as follows.

where

We can dso obtain

where

In detail we can derive ck

so the coefficients for zk are

and then we can get c;s as above in (2.22).

Theorem 2.2 (the Theorem 7.17 in (131) : Let m, a ~ - 1 E R with a ~ - 1 # O

such that

Then there exis ts a polynornial

with real coefficients and exact degree L - 1 that satisfies

I B ( 4 12= 44, ' - e-" CI -

For our case we have

and

Hence

Then the desired polynoznial is

Therefore we c m obtain the coefficients for the scding function and wavelet by

the following equations

and

In order to show what a scaling function and wavelet look like? we fist give an

example with compact support computed from the IGDA. If the discrete filter is

given for N = 3 by

we obtain the Daubechies scaling hinction and wavelet with support length of 3,

which are illustrateci in Figure 2.1.

The Daubechies xaiîng funcnon

Figure 2.1 The Daubediies scaling function and wavelet of N = 3.

Aimost all the wavelets and scaling hinctions in R with compact support lack

symmetry or antisymrnetry. Daubediies [37] proved that the Haar basis is the oniy

orthonormal basis of compactly supported wavelets for which the associated scaling

h c t i o n 6 is syrnrnetrical, where the Haar scaling function and wavelet are &en ty

O g < l 7

else where

o < t < 1/2

1 / 2 g < 1

elsewhere

Note that the Haar scaiing fuction is called the NRZ (non-return-tezero) pulse

by comrnunicntions engineers, and the Haar wavelet is called a Manchester pulse

2.5.2 Meyer Scaling Functions and Wavelets

The Meyer wavelets are orthonormal wavelets defined over the entire set R, i.e..

they are not supported on a finite interval. Their properties are summarized in [32].

The Fourier transform of the Meyer scaling hinction is given by

where the real-valued h c t i o n v (x) satisfies

and the symmetry condition

on the interval [O, 11. This might be c d e d the classical definition of the Meyer scaling

function. However, by contrast, the Jones definition [70, p. 641 is

For /3 = 1/3 we obtain the special case in (2.26).

Note that the set {#(t - k) 1 k E S) is orthonormal because @ ( w ) satisfies

(2.11), Le., Soa, d(t)qû(t - k)dt = bk, and thus this establishes the orthonomality of

the Meyer wavelets.

2.6 Time-Fkequency Analysis

Time-frequency analysis will be used in Chapter 3 to investigate the possibility of

using Daubechies scaling functions in communications. To present the features uf

a signal simultaneously in time and fiequency, we need to transform a signal in

the tirne domain to one in the tirne-frequency plane. In factt we can decompose

a signal to illuminate two important properties: locahzation in t h e of transient

phenomena and the presence of specific frequency components, where the Fourier

transform does not work well. Ln other words, what is really needed in such a signal

processing method is to determine the time localizations (intervals) that yield the

spectral information on any desirable range of fiequencies. The tirne-frequency plane

is a 2-D space usefd for idealizing these two properties of transient signals.

One method of tirne-frequency analysis is to try to generalize standard Fourier

analysis. The idea is to extend the concept of the energy density spectrum 1 S(w) (*

to that of a two-dimensional function P(t , w ) which indicntes the intensity per unit

frequency and per unit time [63]. Such a function would describe how the spectrum

is evolving in tirne and is called a time-frequency distribution.

Another approach is to break up a signal into short time intervals and analyze

each interval by Fourier transformation. Usuaily a window is h s t applied to the

signal. The result is the short-time Fourier transform (STFT). For different windows

different transforms are obtained.

In this section, we concentrate our discussion on the latter methud. Winduw

functions, because of their localization properties, are used to obtoin the localization

in time and frequency. Different seiections of the window functions correspond to

different methods of transfomat ion. The wavelet funct ions and wavelet packet

functions are suitable for such transformations since they are local in time and

frequency as we discussed before.

2.6.1 Some Definitions

De/inilzon 2.5 [151] : A h c t i o n w ( t ) E &(R) is a window function if it satisfies

where w (w) = F{w( t ) } is the Fourier transform of w(t ).

Definition 2.6 [13] : The average time ta and rms duration A, of a window

function w are defined as

and

respectively; and the width of the window fiinction w is defined by 2Aw.

Alternatively, the average frequency wu and rms bandwidth are dehed re-

spectively as

tu and Aw are sometimes c d e d the center and radius of the window Function!

respect ively.

In k t , the rms duration and rms bandwidth cannot be very smail simultane-

ously. The uncertainty principle prevents us from making the product of & and

maller than a fixed constant. The principle is presented below.

Theorem 2.3 [13] : If w ( t ) E &(R) is a window function, then

Equality is attained iff

w ( t ) = c P g , ( t - b ) ,

where c # O, a > O, a, b E R and g,(t) = .&e-c2/49 is a Gaussian Function.

35

One of the applications for the product of &, and A, is in radar and sonar

design ([IO, pp. 981, (93, pp. 29211, where the product is a useful measurement of

the quality of estimating the tirne delay and Doppler frequency.

We wiil consider two types of windows and their corresponding transforms used

as the tools of time-frequency analysis:

1. the short-time Fourier transform (STFT) .

2. wavelet transform (WT).

2.6.2 Time-Requency Analysis Using STFT

The short-time Fourier transform (STFT), was initiaily introduced by Gabor (1946)

[53] who used Gaussian hinctions as the windows. In STFT we move a window "ver

the time function and extract the hequency content in the interval.

Definition 2.7 [13] : Given a window h c t i o n w(t) (Definition S.3), the short-

tirne Fourier transform of a signal f (t) E &(a) is defined as the Fourier transform

of the signal within the extent or spread of that window. which is

The STFT is cailed a Gabor transform if the window w ( t ) is Gaussian. Let

be the basis functions of this transform. We have

So (S f )(w , 6) gives local information of f in the time-window

where ta and A, are defined in (2.30) and (2.31).

The basic hinctions gb,,(t) are generated by modulation and translation of the

window huiction w(t) , where w and b are modulation and translation parameters,

respectively. The window h c t i o n w(t) is also cded a prototype function. As 6

increases, the prototype h c t i o n simply translates in t ime, while keeping the sprend

of the window constant.

Let j and gb,w be the Fourier transforms of the functions f and gb,,, respectively.

By Parseval's theorem for the Fourier trmsform, we have

Therefore, sirnilarly, (Sj)(b,w) also gives hcal spectral information of f in the

fiequency-window :

[ w . + w - A w , w , + ~ + A ~ ] , (2.39)

where wa and & are d e h e d in (2.32) and (2.33).

As the modulation parameter w increases, the transfonn simply translates in

kequency, retaining a constant bandwidth.

In summary, we have a time-fiequency window

with width 2A,, height 2A, and constant window area

The time-frequency plane with information c e b for the STFT is illustrated in

Figure 2.2.

O time(b)

Fig. 2.2 Time-frequency plane for STFT.

We can see from the figure that each eiement A, and hw of the information

cell A,Aw is constant for any frequency w and time shift 6. Any trade-off between

time and frequency resolution must be accepted for the whoie (w, 6) plane. That

is just the difficulty with the STFT-the fixed-duration window w( t ) leads to a

fked-frequency resolution and thus d o w s only a h e d time-frequency resolution.

On the other hand, we wiU see next that for the wavelet transform the basis

hinctions are formed by dilations and translations of a prototype funetion @(t).

These basis h c t i o n s are short-duration, high-frequency and long-duration, low-

frequency hc t ions . They are much better suited for representing short bursts of

high-fiequency signals or long-duration , slowly varying signals.

2.6.3 Using Wavelets as Window Functions

The continuous wawlet transform (CWT) for a function / E L2('R) is d e h e d by

where a,b E R with a # 0.

Let 6 be the Fourier transform of @. Suppose that S, is n window b c t i o n . In

fact, many wavelets are window hinctions because of their property of localization.

Let ta and A* be the average time and rrns duration of @, and wa and A, be the

average frequency and rms bandwidth of 4, respectively.

The basis h c t i o n s of the CWT are

So the CWT giws local information of an analog signal / with a tirne-window

Sirnilarly, we can obtain the Fourier transform of the basis hinction

The average frequency of i lap( t ) [13] is

and the rms bandwidth squared of i l>.~,(t) is

* ? = a ' $ 0

The conesponding frequency-window is

Therefore the time-frequency window is

for a > O. The time-fiequency plane with information ceUs for CWT is iliustrated

in Figure

frequency (w )

O tirne@)

Fig. 2.3 Tïe-frequency plane for C WT.

When a is large, the basis fwiction becomes a stretched version of the prototype

wavelet, that is, a low-frequency hinction. When a is srnail, the basis function

is a contracted version of the wavelet function, that is, a narrow duration, high-

frequency function. For the differerit scaling parameter a, the wavelet function $ ( t )

dilates or contracts in time, leading to the corresponding contraction or dilation in

the fiequency domain. Therefore, the wavelet transform provides a flexible time-

frequency resolution.

In practice, we often sample the parameters (a, 6 ) to get a set of wavelet functions

in discrete parameters. The sampling lattice is

so that the basis functions for wavelet series are

@jVk ( t ) := af2@(afgt - kbO)

where j, k E 2.

As discussed before, suppose that $j,k are complete and orthonormal, which is

usually the case as a basis. The corresponding time-window becornes

The frequency-window is

Therefore we obtain the time-frequency window as

We can see that the area of the window is the same as the original (prototype)

wavelet. But the time and hequency resolutions depend on the factors ai. When

the time resolution increases by a factor of ai, the conesponding frequency resolution

decreases by a factor of aii . This is the result of the uncertainty principle.

2.7 Chapter Summary

In this chapter we have presented some basic aspects of wavelet theory. We in-

troduced the defini tions and propert ies of wavelets, and t ime-frequency analysis.

Although many other aspects of wavelet theory are also important? e .g . wavelet

transforms, we only present those properties which are needed for the derivations

in the Iater chapters of the thesis. In the next chapter we will investigate the

time-frequency properties of wavelets used as signahg waveforms in cornmunica-

t ion systerns.

Chapter 3

Wavelet s as S ignaling Waveforms

in Communications

During the past several yenrs efforts have been made to use scaling functions and

wavelets in communications. One of the main applications has been to use scaling

h c t ions or wavelets as the signaling waveforms ([79], [70], [54], [146] , [77] and [I~o]) .

Two kinds of scaling functions or wavelets are usudy considered. The first kind

of scaling Function or wavelet is time-limiteci (has finite support) and is therefore

not bandlimited in the frequency domain ([79], [SI, [77] and [l4Oj). An example of

this is the Daubechies sc&g functions and wavelets. The second kind of scaling

function or wavelet, often referred to as a Meyer-like scaling h c t i o n or wavelet.

[7O], is bandlimited in the frequency domain and is therefore not tirne-Iimi ted ([?O],

(1461 and [114]).

For any one of the two kinds of scaling huictions and wavelets, if it is orthogonal.

then its autocorrelation function is a Nyquist-type pulse and therefore is ISI-free (has

no intersymbol interference) after the matched filter ([70], (1481 and [114]). Hence

orthogonal functions of these kinds can be used as signaling waveforms to avoid ISI.

Jones shows in [70, pp. 651 that the square root raised cosine function, commonly

used in communications, is a special case of a Meyer scaling h c t i o n . Besides this,

some Meyer-Li ke scaling functions have additional properties of possible interest in

ISI-free signal design.

In this chapter we consider the possibility of using the orthonormal scaling h c -

tions and wavelets in communications systerns. First we wili see that there is no ISI

in the receiver if an orthonormal scaling function is used as the pulse-shaping filter

since the autocorrelation function of the orthonormal scaling function satisfies the

Xyquist pulse criterion. Then we use the rms bandwidth and time duration (BT)

product as the criterion to investigate the time-fiequency properties of Daubechies

scaling functions, which are of finite support in the time domain. In this chapter

we make a cornparison between the autocorrelation h c t i o n s of Daubechies scnling

functions and the cornmonly used raised cosine fimctions. We wili see chat in some

cases the autocorrelation h c t i o n s of Daubechies scaling functions have smaiier BT

products than r a i d cosine functiom.

In [146] Xia presents a family of pulse-shaping filters that are ISI-free before

and after matched Eltering but otherwise are sùnilar to the Meyer scaling h c t i o n s

given in [70]. A generalization of the item in [116] appears in [38]. Notice that the

BI-fke scaling functions in [146] or [114] are not tirne-limited. This motivates a

question: Are there such hinctions of the first kind, which are ISI-Free before and

after the matched filter? If yes, this may have some additional benefits because

these functions are of h i t e support and thus don't need to be truncated to be

implemented as in the case of scaling hinctions of the second End. The truncated

functions are less desi rable as they have no orthonormal property.

In this chapter we show that the only ISI-kee, orthonormol scaling functions

with support on an interval are the rectangular functions of unit duration, i.e., Haar

functions.

3.1 Daubechies Scaling Funetions as the Signaling

Waveform

A signal x(t) is said to be a Nyquist pulse, i.e.,

if and only if its Fourier transform X ( w ) = rm x(t)e-jWtdt satisfies

where T is the symbol interval. The proof is in Proakis [101], or Ziemer and Tranter

While orthonormal scaling functions are not Nyquist pulses, their autocorrelation

functions are. Define the autocorrelation function of the scaiing function $ ( t ) to be

For an orthonormal scaling func tion, we have

where bk is the Kronecker delta sequence . Hence for the symbol interval T = 1, we

immediately conclude that q(r) satisfies Nyquist pulse-shaping critenon

We consider Daubechies scaiing functions (321 as the pulse-shaping filter, Le..

where only a finite nurnber of the hk's are nonzero. For the orthonormal set of

scaling hinctions { 4 ( t - k) 1 k E Z}, we let dktoc(t) = d ( t ) * Q ( - t ) , which is

the autocorrelation function of t$(t). Therefore $+otd(t) is a Nyquist-type pulse.

Consequently there is no ISI when d ( t ) is used as the pulse-shaping filter and 4 ( - t )

is used as the matched filter.

Assume that the impulse response of the Channel is c(t) and that the noise is

n(t) . The baseband mode1 is illustrated in Fig. 3.1.

Fig. 3.1 Baseband channel model.

The overall impulse response can be written as

Let 9 ( w ) denote the Fourier transform of &). If @(w) is bandlimited, 4(t) and

&,tal(t) have infinite support in time. On the other hand, if + ( t ) has h i t e support in

time, h(t) is finite if c ( t ) is also finite. However the bandwidth of 9 ( w ) is unlimited.

Since the decay of the tails of h(t) depends on the channel bandwidth, it may be

useful to have a transmitting pulse which is as smwth as possible and has srnall BT

product [Gd].

For scaling functions to be used as the pulseshaping filter, they need to have o

certain smoothness and s m d BT pmduct. The srnoothness of scaling functions is

definecl in Chapter 2, and the smoothness of scaling functions and wavelets wiU be

investigated in Chapter 5. In this chapter we only consider the BT products.

3.2 The RMS Duration and Bandwidth of the

Aut ocorrelat ion Funct ions

We wiil calculate and compare the rms bandwidth and rms time duration products

of &,cor(t) for the cases of raised cosine hinctions and the autocorrelation fimctions

of scaling hinctions. Note that we compare the autocorrelation functions of the

scaling h c t i o n s with the raised cosine functions. Livingston and Tung 1821 present a

comparison between raised cosine functions and wavelets in communication systerns.

They consider a raised cosine function as the signaling waveform, but in practice

it is the square root of the raised cosine fimction that is used as the signaling

waveform. The raised cosine function itself is the convolution of the wweform

filter and the matched filter. Although the autocorrelation functions and the raised

cosine func tions are both Xyquist-type pulses, they have different time-frequency

properties. Daubechies scaling functions are time-limi ted and thus not bandlimi ted,

and the raised cosine functions are not tirne-limited but are bandlirnited. If we

want to analyze the performance of the scaiing hinctionst we need to compare them

with the square roots of raised cosine functions. To avoid having to deal with the

fact that the tirne and kequency supports for these pulses are different, we use the

BT product as a measure for the comparison. Notice that in [91] the BT products

of Daubechies wavelets are calcdated. tiere we consider the BT products for the

autocorrelation h c t i o n s of the Daubechies scaling functions.

For those hinctions satisfying the tw+scde equation as (XI), Viliemoes in [135]

presents an approach to computing their rms duration and rrns bandwidth. Zarowski

[149] gives a detaiied review of 11351. The foilowing resdts of the BT product

computations for the two-scale equation are extracted from [l.L9].

3.2.1 RMS Duration Computation

A general tw-scale equation has the fom

N- l

u(z) = 2 C C ~ U ( ~ Z - k), &=O

where the elements of the two-scale sequence {ck} may be complex-vaiued, i.e..

Ck € Ca

The nth energy moment of u(x) in tirne is dehed as

for n = 0,1,2, *.

D e h e the t h e moments

and their discrete-time equidents

We c m see that

We shall obtain a recursion for the sequence (3.4) which will then give (3.3).

Define the operators

where R, may be expressed in matrix form as

Then we can obtain

so we have

This may be rearranged to yield

where I is an identity matrix of suitable order. The existence of a solution to (3.7)

is forrnally established in (1351. A normalization condition is as f o h s :

Let E =II u(x) Il2= 1 u(x) I2 dx, and then from the preceding results we

may determine the rms duration of u(x) as follows.

where

- u=

1 x 1 u(x) 12dx = x 1 u(x) l 2 dx.

II ~ ( 4 II2 Then we have

3.2.2 RMS Bandwidth Comput ation

The nth energy moment of u(x) in frequency is dehed as

for n = 0 ,1 ,2 , O . Given n E {O, 1,2, a } , define

It t u s out that if ( l + ~ ~ ) ~ / ~ û ( w ) E C 2 ( R ) , the absolute convergence of the integral

in (3.8) is guaranteed. We can see that

From [149] we have

We must find the eigenvaiues of A (in matrix fom) that are an integer power of two,

and the corresponding eigenvectors. We also need to normalize t hese eigenvectors.

We give the result directly as

From Parseval's Theorem we know that

and thus the nns bandwidth of u(x) may be obtained from

3.2.3 The BT Products of the Autocorrelation F'unctions

Frum the twvscale equacion for the scaling function, which is

(N is aiways even) we may write its autocorrelation huiction as

= r i P k d W - *) nP(2t + YT - l ) ] dt 1 LO1

which is also a tw*scale equation. hrthermore by letting r = k - 1, we obtain

where

Then we use the method in the last two sections to calculate the rms duration and

bandwidth of the autocorrelation hct ion .

It is easy to show that the autocorrelation huictions of the scaling hinctions still

satisfy the twwscale equation (3.2). Note that for Daubechies scaling functions with

N = 3 and N = 5 , their rms bandwidth cannot be obtained because the matrix

A has no eigenvdues of 112 and 1/1. This means that we cannot get a solution

fiom (3.9). We also use numerical integration to calculate the rms duration and

bandwidth to cab the results. All the resuits are listed in Table 3.1 and Table

3.2. The matlab program is giwn in Appendix B.

Table 3.1 The rms duration and bandwidth for the two methods for Daubechies

sc&g functions.

A@

Numerical

1.6850

1.6868

1.6921

1.6967

N

7

9

11

13

Error

(%) 0.21

0.02

0.00

0.00

A d

Numerical

0.3815

0.3995

0.4155

0.4298

A d Villemoes

0.3858

0.4019

0.4168

0.4305

Error

(%) 1.13

0.60

0.31

0.16

AQw Viliemoes

1.6769

1.6865

1.6921

1.6967

Table 3.2 The BT products of Daubechies scaling hct ions for the two methods.

Villemoes Numerical (%)

N 1 Agu (Villernoes) 1 3 dB Bandwidth (rads/sec)

Table 3.3 The rms bandwidth and single-sided 3dB bandwidth for the

autocorrelation hinctions of Daubechies scaling functîons.

3.2.4 The M S Bandwidth and the 3dB Bandwidth

The rms bandwidth is used to measure the frequency uncertninty of a signal, and

the 3dB bandwidth of a signal is used to measure the frequency i n t e d where most

energy of the signal is. In communications the bandwidth needeà to transmit a

signal is sometimes definecl to be its 3dB bandwidth. We compute the single-sided

3dB bandwidth of the autocorrelation functions of Daubechies scaling hc t ions , and

compare it with the corresponding rms bandwidth. The results are given in Table

3.3.

It can be seen that as N increases, both the rms bandwidth and the single-

sided 3dB bandwidth increase. As shown in [115], the single-sided 3dB bnndwidth

approaches n as N goes to infinity. As N goes to infinity, the frequency responses

of the autocorrelation functions of the Daubechies scaling func tions approach the

Shannon (or Li ttlewood-Paley) scaling function which is given in (831 bv

( O , elsewhere

The rms bandwidth and the single-sided 3dB bandwidth for the Shannon scaling

function are and T, respectively. They are given in the case of N = w in

Table 3.3.

3.3 The RMS Duration and Bandwidth of the

Raised Cosine Funct ions

The raised cosine pulse and its spectrum are given by [101, ~~.546]

Xrc(u) =

2nT Y 0 S b 15 (1 -P)*/T

T T { ~ +cos [& (1 w 1 -?)]) Y (1 -&/T 51 'd I i (1 +@)KIT . (3.11) O 7 I w 12 (1 f ) n l T

where 1;1 is called the rolloff factor, and it takes values in the range O 5 B 5 1. We

We can obtain (1 Xrc(w) 112=11 xrc(t) Il2 from Parsevai's Theorem and we can aiso

caiculate that

obtain its average t h e t,, average frequency w,,, rms time duration

bandwidth Aurc as follows.

trc = II xr&) l Il2 r t l z T c ( t ) 1 2 d t = o . --

2 1- sin2 ( f i ) cos2 (npt ) = Ta dt. n2(i - f) O ( 1 - 4 p t 2 ) 2

Then we obtain the BT pcoduct of the raised cosine function

Notice that the BT product A,, is independent of T. Using the rectangle d e

for numerical integration to calculate A:;,, &and then A:=, we obtain the BT

products for different values of P (P > O), and the results are given in Table 3.4.

The greatest lower bound on the BT product is 0.5131, which is achieved for /3 = 1

and is 1.03 times the Heisenberg limit (AtAu > 112). The 3dB bandwidth of the r a i d casine hc t ions can be obtained from (3.1 1)

by letting

Then we have the single-side 3dB bandwidth

For T = 1, as ,O decreases from 1 to O, W ~ B increases fiom 0.728~ to T.

B Arc 0.51 0.6312

P Arc 0.76 0.5431

Table 3.4 The BT products of the raised cosîne functions for different 0.

60

3.4 Performance Analyses

Based on the results descnbed above we compare the rms duration and bandwidth

of the autocorrelation Functions of the Daubechies scaling hct ions and the raised

cosine functions. First we plot the results obtained in Fig. 3.2.

BT product of raised cogne function 5 , I I I I 1 I I r I t

BT product of Daubeches autoconelahon funchons 1 I I I I I I 1 1

Fig. 3.2 : The BT products for the raised cosine functions and the Daubechies

autocorrelation hctions.

It can be seen that as increases, the BT product of the raised cosine function

decreases. For the autocorrelation function of a Daubechies scaling function, as

N increases, the BT product increases. In communications, a pulse-shaping filter

should h m enough side lobe attenuation to yield good performance. If we consider

4OdB as the desired attenuation, the Daubechies scaling functions would be satisfied

if N 2 21. It can be seen hom Table 3.2 and Table 3.3 that the autocorrelation

function of the Daubechies scaling function of N = 21 hss the same BT product

as the rnised cosine h c t i o n with p = 0.2979. We can see that if ,û < 0.2979, the

raiseci cosine functions have lnrger BT products t han the autocorrelation func t ion

of the Daubechies scaling function of N = 21, and more bandwidth is needed. Fur

the raised cosine f~nct ions~ it c m be seen from Fig. 3.2 that when ,fl approaches

zero, Le., when the single-sided 3dB bandwidth approaches R for 7' = 1, the BT

product increases very fast. For the autocorrelation hinctions of the Daubechies

scaling funct ions, when N increases, t hei r single-sided 3dB bandwidt h increases and

it approaches n as shown in Shen and Strang [115], and their BT products increase

as well. However, the increasing rate of the BT product for the Daubechies case is

slower than that of the raised cosine function.

3.5 ISI-Free Finite-Support ed Scaling Funct ions

In [146] Xia presents a family of pulse-shaping tilters that are ISI-free before and ofter

matched filtering, i.e., the filters and their autocorrelation functions both satisfy the

Nyquist pulse-shaping cr i terion. Notice that the ISI-free scaling h c t i o n s in [146]

are not time-limited. In this section we will investigate if there are such functions

with finite support, which are ISI-fke before and after the matched fdter. Our

proof shows that the only ISI-free, orthonormal scaling h c t i o n s with support on

an interval are the rectangular h c t i o n s of unit duration, Le., Haar functions.

3.5.1 The Coefficients of the Dilation Equation

R e d from [120, pp. 22, 1821 that if # ( t ) is a scaiing function with support on the

interval [O, N), has unit area, and has { $ ( t - i) : i E Z} forming an orthonormal

set. t hen

where LV is restricted to be odd and

Now in order for the scaling h c t i o n to be BI-free, Q(t) m~ist satisfy the Nyquist

pulse shaping cri terion [101, p. 543)

&) = C for sorne j E [O, N - 11

W) = 0 k # j k E [O, N - l]

where c is an arbitrary nonzero constant.

To obtain the ISI-free function d ( t ) , we need first to get the coefficients pc in the

dilation equation (3.12). This is done as follows.

Theorem 1 If condiliolis (3.12)-(3.15) are salàsfied lhen ezadly two of the pks a n

equal to one with ail olhw pks being zero. In addilion, the index of one of the Iwo

nonzero pks is even and the zndez of the other n o n z m pk is odd, vith one of these

indices bezng j .

Proof: The set of equations obtained by repeating the dilation equation, (3.12),

at the successive values of time

can be expressed in matrùr form as

where N is odd and

with

and

M N =

/or j even : p k = O k even k # j

Thus approximately haif the pks are determined by imposing the ISI-free condi-

- - Po 0 0 0 ... * . . O O

f i p, p,, O * * * * * * O O

.

P N - 1 P N - 2 "' " * " * " Pl Po

O P N PN-, ". P2 0 O O P N P N - ~ * h p4

.

O O O O O * ' * p ~ P N - I 1 -

tion on the dilation equat ion for integer values of time. The remainder of the pks are

Irnposing the constraint (3.15) for d ( t ) to be ISI-free on (3.16) yields

determined by imposing the unit area and orthonormality conditions, (3.13, 3.14).

We are going to complete the proof by using induction.

Consider the case when N = 1. Then (3.13, 3.14) yields po = pl = 1 and the

theorem holds for N = 1.

Next assuming the theorem holds for N = L, we have (3.13, 3.14) satisfied or

L

L-2m

s3(rn) = x pkpk+& = o1 for d integers m E (3.2 2) k=O

We show now that the statement in the theorem holds for N = L + 2. In this

case we c m write the constraints (3.13, 3.14) as

Now the ISI-free condition, (LIS), is satisfied by j E [O, L + 11. Suppose j is odd. From (3.18) and (3.19) we have that pj = 1' j < L + 1 and

p~ 12 = O since L + 2 is odd. (3.26) berornes pop^ = O. If p~ + 1 = O. then (3.23-

3.25) become the case for N = L and thus the statement in the theorem holds.

Altematively, if pL+, # O, then = O. We can use the following idea. Notice that if

(3.23-3.26) are satisfied for the given sequence of elements bk : k = Cl1 2,4, - O , L + 11, then (3.23-3.26) are stiil satisfied if we exchange two even indexed elements in the

sequence. Thw we can exchange with p ~ + ~ everywhere in the set of constraining

equations (3.23-3.25) so that (3.23-3.25) are again the constraining equations for the

case N = L, (3.20-3.22)) and the statement in the theorem holds.

For the case where j is even, from (3.18) and (3.19) we have two cases p ~ + i = O

or p ~ + l = 1 since L + 1 is even. If p ~ + l = 0, (3.26) becomes p ~ p ~ + z = O . Using the same method as in the last step we can prove that the stntement is true. If

p ~ + l = 1 then po = O and (3.26) becornes plp~+2 = O. For the case where PL+^ = 0,

(3.25) becornes the case for N = L and hence the statement is tme. For the case

where PL.+* # O, pl = O h m (3.26). Then we c m obtain from (3.23-3.25) that

P L + ( = pt+2 = 1 and aii others are zero, which proves that statement is true.

Thus we have shown that the statement in the theorem applies in general when

N = L + 2 and the proof of the theorem is complete.

3.5.2 The ISI-free Scaling hnctions

In the previous section we showed that al1 except two of the pks in the dilation

equation are zero. Li addition the two nonzero pks are each +1 with one index

being even and the other odd. Thus, denoting the two nonzero p& as pq and pr the

dilation equation (3.12) becomes

where O 5 q 5 N and O < r 5 N, and one of them is odd and the other is even. Now we want to determine # ( t ) to satisfy (3.27). This is done as follows.

Theorem 2 1' a funclion #( t ) has support on [O, N ) , N odd, and satisfies Ihe

follouing equnlzon

where n, rn are inlegers sat-ng

then the interual O f support is [2m, 2n + 1 ) . If in addition # ( t ) is conlinvous in the

interual of support Ihen d ( t ) is unzquely detennined to wXhin a constanl c as

Proof: We prove (3.29). The proof of (3.30) proceeds in a sirnilar fashion.

Let the interval of support for # ( t ) be denoted by [a, b) . Then the interval of

support for #(2t - 2m) is [y, y) md for #(2t - 2n - 1) is [y: y). Therefore since the support interval on eit her side of (3.28) must be equal we have

[a, 6 ) =

which implies that

so that

Next in order to show (3.29), we let t Lie in the Iower half of the interval of

support, Le.,

1 2 m < t < m + n + -

2'

Then it follows that

and we see from (3.31) that

I + ( 2 t - 2 n - 1 ) = O t < m + n + -

2 '

or 4(2m c E ) = 4(2m + 2 4 1 O < t < n - r n + , Now since 4( t ) is continuous on its support interval we see from (3.32) that p ( t )

must be constant in the interval [2m, 2n + 1 )