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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 )