Lecture notes on wavelet transforms

Compact Textbooks in Mathematics

Lokenath DebnathFirdous A. Shah

Lecture Notes on Wavelet Transforms



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Lokenath Debnath • Firdous A. Shah

Lecture Notes on WaveletTransforms

Lokenath DebnathSchool of Mathematical and Statistical

SciencesUniversity of Texas – Rio Grande ValleyEdinburg, TX, USA

Firdous A. ShahDepartment of MathematicsUniversity of KashmirAnantnag, Jammu and KashmirIndia

ISSN 2296-4568 ISSN 2296-455X (electronic)Compact Textbooks in MathematicsISBN 978-3-319-59432-3 ISBN 978-3-319-59433-0 (eBook)DOI 10.1007/978-3-319-59433-0

Library of Congress Control Number: 2017942944

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To my wife Sadhana with love, gratitude, and respectLokenath Debnath

To my beautiful niece AleezaFirdous A. Shah

Preface

“A teacher can never truly teach unless he is still learning himself. A lamp can never lightanother lamp unless it continues to burn its own flame. The teacher who has come to the endof this subject, who has no living traffic with his knowledge but merely repeats his lessonsto his students, can only load their minds; he cannot quicken them.”

Rabindranath TagoreNobel Prize Winner for Literature (1913)

The origin of wavelet analysis can be traced to the classic theory of harmonicanalysis and the seminal contributions of Joseph Fourier, Alfred Haar, and PaulLevy. Since the appearance of the pioneering work of Morlet and Grossman inthe 1980s wavelet methodology has been introduced to the literature as a regularalternative for analyzing irregular situations where the data/signal contains scalingproperties, discontinuities, sharp spikes, etc. These contributions were followedby the introduction of the general idea of multiresolution analysis by Mallat andMeyer and the notion of orthogonal wavelet bases by Daubechies in the late 1980s.Thus, we can say that the development and advancement of the theory of waveletscame through the efforts of mathematicians with a variety of backgrounds andspecialties, and of engineers and scientists with an eye for better solutions andmodels in their applications. Nowadays, there is no doubt that the introduction ofwavelet transform was one of the most important events in mathematics over thepast few decades. They have fascinated the scientific, engineering, and mathematicalcommunity with their versatile applicability and are now considered as a nucleus ofshared aspirations and ideas. The application areas for wavelets have been growingfor the last two decades at a very rapid rate. They have been applied in a number offields including signal processing, image processing, sampling theory, turbulence,approximation theory, geophysics, astrophysics, quantum mechanics, computergraphics, statistics, economics and finance, quality control, differential and integralequations, numerical analysis, neuroscience, medicine, neural networks, chemistry,nano-technology, and even in political time series. A consequence of this interest isthe appearance of several books, journals, and a large volume of research articles onthis subject. Currently, there are many books in the market, with more being writteneveryday, which treat the subject of wavelets from a wide range of perspectives andwith several areas of a large spectrum of possible applications. Workers in the fieldjudge some of these “excellent.” So, why bother to publish an additional one?

vii

viii Preface

The answer lies in the fact that there seems to be no textbook that provides asystematic introduction to the subject of wavelet transforms. While teaching courseson integral transforms and wavelet transforms, the authors have had difficultychoosing textbooks to accompany lectures on wavelet transforms at the seniorundergraduate and/or graduate levels. Many hours of study convinced us that thereis a need for lecture notes on wavelet transforms for mathematicians, scientists andengineers that provide both a systematic exposition of the basic ideas and resultsof wavelet analysis. The selection, arrangement, and presentation of the materialin these lecture notes have carefully been made based on our past and presentteaching, research, and professional experience. In particular, drafts of these lecturenotes have been used by us for regular teaching courses in wavelet transformsand their applications at the University of Texas–Pan American, USA, and theUniversity of Kashmir, India. These notes differ from many textbooks with similartitles due to major emphasis placed on numerous topics and systematic developmentof the underlying theory before presenting applications and the inclusion of manynew and modern topics such as fractional Fourier transforms, windowed canonicaltransforms, fractional wavelet transforms, fast wavelet transforms, spline wavelets,Daubechies wavelets, harmonic wavelets, and nonuniform wavelets. Therefore, ourprimary goal is to show how different types of wavelets can be constructed, illustratewhy they provide us with a particularly powerful tool in mathematical analysis, andindicate how they can be used in applications. Our secondary goal is to developrequired analytical knowledge and skills on the part of the reader, rather than focuson the importance of more abstract formulation with full mathematical rigor. Indeed,our major emphasis is to provide an accessible working knowledge of the analyticaland computational methods with proofs required in pure and applied mathematics,physics, and engineering.

This monograph is written from the ground level and up. The presentation isas simple as possible, but to paraphrase Einstein “it should not be simpler.” Wehave attempted to make the monograph as self-contained as possible. Mathematics,science, and engineering students need to gain a sound knowledge of mathematicaland computational skills by the systematic development of underlying theory withvaried applications and provision of carefully selected fully worked-out examplescombined with their extensions and refinements through addition of a large setof a wide variety of exercises at the end of each chapter. Numerous standardand challenging worked-out examples and exercises are included so that theystimulate research interest among senior undergraduates and graduate students.Another special feature of this book is to include sufficient modern topics which arevital prerequisites for subsequent advanced courses and research in mathematical,physical, and engineering sciences.

Now it is time to give some indications on the contents of the monograph.The book has 5 chapters, which are described briefly here to show how themonograph’s main ideas are developed. Wavelet transforms can be considered asa modern supplement to classical Fourier transforms, and for this reason we give

Preface ix

a more detailed presentation of Fourier transforms in Chapter 1. We start with themotivation of Fourier series and Fourier transforms in L1.R/ and L2.R/ followedby their basic properties. Several important results including the approximateidentity theorem, general Parseval’s relation, and Plancherel theorem are discussedin some detail. Discrete Fourier transform, fast Fourier transform, and fractionalFourier transform are also discussed briefly for the purpose of comparing themwith the continuous, discrete, and fractional wavelet transforms. Applications ofthe fractional Fourier transform in solving generalized nonhomogeneous differentialequations including the generalized wave and heat equations are also given. Specialattention is also given to the Heisenberg’s uncertainty principle.

Chapter 2 is devoted to a fairly detailed examination of the joint time-frequencyanalysis of signals. The main goal here is to set the foundation for the developmentof continuous and discrete wavelet transforms. We begin with the time-frequencylocalization of signals which leads us to the windowed Fourier transform. This isfollowed by the Gabor transform and its basic properties, including the inversionformula. Special attention is also given to the Zak transform and its basic properties.Based on the relationship between the Fourier transform and linear canonicaltransform, a hybrid windowed transform, namely the windowed linear canonicaltransform, has been introduced. Its basic properties and several results including theorthogonality relation and inversion formula are also discussed.

The heart of the wavelet theory is covered in Chapters 3 and 4 in a compre-hensive approach. We start Chapter 3 with the introduction of wavelets and wavelettransforms with examples. The basic ideas and properties of wavelet transformsare discussed with special attention given to the use of different wavelets forresolution and synthesis of signals. This is followed by the discrete version ofwavelet transform and the construction of orthonormal dyadic wavelet basis. Specialattention is given to fairly exact mathematical treatment of the fractional wavelettransform, and several important results including Parseval’s formula and inversiontheorem are proved. Chapter 4 contains an exposition of the general notion of amultiresolution analysis together with several examples. Special attention is givento properties of scaling functions and orthonormal wavelet bases. This is followedby a method of constructing orthonormal bases of wavelets from an MRA. In theend, the fast wavelet transform is briefly discussed.

Chapter 5 is devoted to several generalizations and extensions of orthonormalwavelet bases in L2.R/. To construct wavelets with greater degrees of smoothnessand having compact support, we construct wavelets that are smooth and piecewisepolynomials, usually known as spline wavelets. The well-known Franklin andBattle-Lemarié wavelets are the special cases of these wavelets. This is followedby Daubechies algorithm for the construction of compactly supported wavelets.Then, we discuss another intersecting class of orthonormal wavelets called harmonicwavelets. Finally, we present a novel and simple procedure for the construction ofnonuniform wavelets associated with nonuniform MRA. In this nonstandard setting,the associated translation set is no longer a discrete subgroup of R but a spectrum

x Preface

associated with a certain one-dimensional spectral pair, and the associated dilationis an even positive integer related to the given spectral pair.

In preparing the monograph, the authors have been encouraged by and havebenefited from the helpful comments and criticisms of a number of faculty andpostdoctoral and doctoral students of several universities in the USA and India.The authors express their grateful thanks to these individuals for their interest inthe book. The editor from Birkhäuser-Springer, Benjamin Levitt, deserves a specialvote of thanks for his cooperation and for the exemplary patience he displayed. Itgoes without saying, however, that all responsibility for errors, imperfections andresidual or outright mistakes, is shared jointly by both of us. However, we hope thatthese are both few and obvious and will cause minimum confusion.

Edinburg, TX, USA Lokenath DebnathAnantnag, Jammu and Kashmir, India Firdous A. Shah

Contents

1 The Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Fourier Transform in L1.R/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Fourier Transform in L2.R/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5 The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.6 The Fractional Fourier Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2 The Time-Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2 The Time-Frequency Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3 The Gabor Transforms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4 The Zak Transform.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.5 The Windowed Linear Canonical Transform . . . . . . . . . . . . . . . . . . . . . . . . . 792.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3 The Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2 The Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.3 The Discrete Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.4 Orthonormal Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.5 The Fractional Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4 Construction of Wavelets via MRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2 Multiresolution Analysis in L2.R/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.3 Construction of Mother Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4 The Fast Wavelet Transform.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xi

xii Contents

5 Elongations of MRA-Based Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 The Spline Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 The Daubechies Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.4 The Harmonic Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.5 The Nonuniform Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

1The Fourier Transforms

Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it maybe said to furnish an indispensable instrument in the treatment of nearly every reconditequestion in modern physics.

Lord Kelvin

Fourier was motivated by the study of heat diffusion, which is governed by a lineardifferential equation. However, the Fourier transform diagonalizes all linear time-invariantoperators, which are building blocks of signal processing. It is therefore not only the startingpoint of our exploration but the basis of all further developments.

Stéphane Mallat

1.1 Introduction

Historically, Joseph Fourier (1770–1830) first introduced the remarkable idea ofexpansion of a function in terms of trigonometric series without giving any attentionto rigorous mathematical analysis (See Fourier 1822). The integral formulas for thecoefficients of the Fourier expansion were already known to Leonhard Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solutionof heat (or Fourier) equation in terms of Fourier series so that the Fourier seriescan be used as a practical tool for determining the Fourier series solution of partialdifferential equations under prescribed boundary conditions. Thus, the Fourier seriesof a function f .t/ defined on the interval .�L;L/ is given by

f .t/ DX

n2Z

cn exp

�in�t

L

�; (1.1.1)

© Springer International Publishing AG 2017L. Debnath, F.A. Shah, Lecture Notes on Wavelet Transforms, Compact Textbooksin Mathematics, DOI 10.1007/978-3-319-59433-0_1

1

2 1 The Fourier Transforms

where the Fourier coefficients are

cn D1

2L

Z L

�Lf .t/ exp

��

in�t

L

�dt: (1.1.2)

In order to obtain a representation for a nonperiodic function defined for all real t, itseems desirable to take limit as L ! 1 that leads to the formulation of the famousFourier integral theorem:

f .t/ D1

2�

Z 1

�1

ei!td!Z 1

�1

e�i!tf .t/ dt: (1.1.3)

Mathematically, this is a continuous version of the completeness property of Fourierseries. Physically, this form (1.1.3) can be resolved into an infinite number of

harmonic components with continuously varying frequency� !2�

�and amplitude,

1

2�

Z 1

�1

e�i!tf .t/ dt; (1.1.4)

whereas the ordinary Fourier series represents a resolution of a given functioninto an infinite but discrete set of harmonic components. Thus, using the notationof an inner product, the Fourier transform of a continuous function f .t/ can beexpressed as

Of .!/ D˝f ; ei!t

˛D

Z 1

�1

e�i!tf .t/ dt: (1.1.5)

This transform decomposes a signal into orthogonal trigonometric basis functionsof different frequencies and phases, and it is often called the Fourier spectrum. Itis generally believed that the theory of Fourier series and Fourier transforms is oneof the most remarkable discoveries in mathematical sciences and has wide spreadapplications in mathematics, physics, and engineering.

This chapter deals with Fourier transforms in L1.R/ and L2.R/ and their basicproperties. Several important results including the approximate identity theorem,general Parseval relation, and Plancherel theorem are proved. Discrete Fourier trans-form (DFT), fast Fourier transform (FFT), and fractional Fourier transform (FrFT)are also discussed briefly for the purpose of comparing them with the continuous,discrete, and fractional wavelet transforms. Applications of the FrFT in solvinggeneralized nonhomogeneous differential equations including the generalized waveand heat equations are also given. Special attention is also given to the Heisenberguncertainty principle.

1.2 The Fourier Transform in L1.R/ 3

1.2 The Fourier Transform in L1.R/

We begin by introducing some notation that will be used throughout this work. Theset of natural numbers (positive integers) is denoted by N and the set of integersby Z. The fields of real and complex numbers are denoted by R and C, respectively.Elements of fields R and C are called scalars. We will deal with various spacesof functions defined on R. The simplest of these are the Lp D Lp.R/ spaces,1 � p < 1, the vector space of all complex-valued Lebesgue integrable functionsf defined on R with a norm

��f��

p D

�Z 1

�1

ˇf .t/

ˇpdt

�1=p

< 1: (1.2.1)

The number��f��

pis called the Lp-norm. These signal classes turn out to be Banach

spaces, since Cauchy sequences of signals in Lp converge to a limit signal also in Lp.Since we do not require any knowledge of the Banach space for an understandingof wavelets in this introductory book, the reader needs to know some elementaryproperties of the Lp-norms. The Lp spaces for the cases p D 1; p D 2; 0 < p < 1;

and 1 < p < 1 are different in structure, importance, and technique, and thesespaces play a very special role in many mathematical investigations. The casep D 2 is of special interest: L2 is a Hilbert space. That is, there is an inner productrelation on square-integrable functions, which extends the idea of the vector spacedot product to analog signals. When p D 1, the space L1.R/ is the collection ofmeasurable functions which are bounded, after we neglect a set of measure zero.we neglect a set of measure zero (See Debnath and Bhatta, 2015; Debnath andMikusinski, 1999).

In particular, L1.R/ is the space of all Lebesgue integrable functions defined onR with the L1-norm given by

��f��1

D

Z 1

�1

ˇf .t/

ˇdt < 1:

Suppose f is a Lebesgue integrable function on R. Since e�i!t is continuous andbounded, the product e�i!t f .t/ is locally integrable for any ! 2 R. Also,

ˇe�i!t

ˇ� 1

for all ! and t on R. Consider the inner product

˝f ; ei!t

˛D

Z 1

�1

f .t/ e�i!tdt; ! 2 R: (1.2.2)

Clearly,

ˇˇZ 1

�1

e�i!tf .t/ dt

ˇˇ �

Z 1

�1

ˇf .t/

ˇdt D

��f��1< 1: (1.2.3)

This means that integral (1.2.2) exists for all ! 2 R. Thus, we give the followingdefinition.


Definition 1.2.1 (The Fourier Transform in L1.R/). The Fourier transform ofany function f 2 L1.R/ is defined by

Of .!/ D F˚f .t/

D

Z 1

�1

e�i!tf .t/ dt: (1.2.4)

Remarks.

1. Physically, the Fourier integral (1.2.4) measures oscillations of f at the frequency! and Of .!/ is called the frequency spectrum of a signal or waveform f .t/.

2. The factor 1=2� may be bundled with the Fourier transform or 1=p2� can

appear in front of the transform and the inversion formula to provide a symmetricappearance. All these approaches are found in the literature.

3. The Fourier transform is, in fact a continuous version of the Fourier series.A Fourier series decomposes a signal on Œ��; �� into components that vibrateat integer frequencies. By contrast the Fourier transform decomposes a signaldefined on an infinite time interval into a !-frequency component, where ! canbe any real number.

4. Another form for the Fourier transform of f used frequently in probability theoryreplaces the kernel exp.�i!t/ by exp.i!t/. In this case if f is the probabilitydensity function of the random variable x, then

g.x/ D

Z 1

�1

f .t/ eixt dt;

is called the characteristic function of f .5. In general, the Fourier transform Of .!/ is a complex function of a real variable !.

From a physical point of view, the polar representation of the Fourier transformis often convenient. The Fourier transform Of .!/ can be expressed in the polarform

Of .!/ D R.!/C iX.!/ D A.!/ exp fi�.!/g ; (1.2.5)

where A.!/ DˇˇOf .!/

ˇˇ is called the amplitude spectrum of the signal f .t/, and

�.!/ D argnOf .!/

ois called the phase spectrum of f .t/.

Example 1.2.1. The Gaussian function is one of the most important functions inprobability theory and analysis of random analysis. It plays a central role in Gabortransform. The Gaussian function with unit amplitude is given by

f .t/ D e�a2 t2 ; a > 0: (1.2.6)


f(t)

t

f^(ω)

ω

1

0.5

0.8

0.6

0.4

0.2

000-2 2-4 4 0-2 2-4 4

Fig. 1.1 Graphs of f .t/ D e�a2 t2 and Of .!/ with a D 1

The Fourier transform of f is computed as

Of .!/ D

Z 1

�1

e�

i!tCa2 t2�dt D

Z 1

�1

e�a2�

tC i!2a2

�2� !2

4a2 dt

D e�!2=4a2Z 1

�1

e�a2y2 dy D

p�

ae�!2=4a2 ; (1.2.7)

in which the change of variable y D

�t C

i!

2a2

�is used. Even though

�i!

2a2

�

is a complex number, the above result is correct. The change of variable can bejustified by the method of complex analysis. The graphs of f .t/ and Of .!/ are drawnin Figure 1.1.

It is interesting to note that the Fourier transform of a Gaussian function is alsoa Gaussian function (see Figure 1.1). The parameter a can be used to control thewidth of the Gaussian pulse. It is evident from relations (1.2.6) and (1.2.7) that largevalues of a produce a narrow pulse but its spectrum spreads wider on !-axis.

In particular, when a2 D1

2and a D 1, we obtain the following results

.a/ Fne�t2=2

oD

p2� e�!2=2; (1.2.8)

.b/ Fne�t2

oD

p� e�!2=4: (1.2.9)

Example 1.2.2 (Characteristic Function). This function is defined by

f .t/ D

�1; �a < t < a0; otherwise.

(1.2.10)


In science and engineering, this function is often called a rectangular pulse or gatefunction. Its Fourier transform is

Of .!/ D F˚f .t/

D

�2

!

�sin.a!/: (1.2.11)

We have

Of .!/ D

Z 1

�1

f .t/ e�i!t dt D

Z a

�ae�i!t dt D

�2

!

�sin.a!/:

Note that the Fourier transform Of .!/ vibrates with a zero frequency and hence weshould expect that the larger values Of .!/ occur when ! is near zero (see Figure 1.2).Also, it should be noted that f .t/ 2 L1.R/, but its Fourier transform Of .!/ 62 L1.R/.

Example 1.2.3. Find the Fourier transform of

f .t/ D

�1 �

jtj

a

�H

�1 �

jtj

a

�

where H.t/ is the Heaviside unit step function defined by

H.t/ D

�1; t > 0;0; t < 0:

(1.2.12)

0.8

0.6

0.4

0.21

-a a 00

-5 5-10 10 15-15t ω

Fig. 1.2 Graphs of f .t/ and Of .!/ with a D 1


Or, more generally,

H.t � a/ D

�1; t > a;0; t < a;

where a is a fixed real number. So the Heaviside function H.t � a/ has a finitediscontinuity at t D a: Then, it can easily be verified that

Of .!/ D a �sin2

�a!

2

�

�a!

2

�2 : (1.2.13)

Example 1.2.4. Find the Fourier transform of f .t/ D e�ajtj; a > 0.

We have

F˚e�ajtj

D

Z 1

�1

e�ajtj�i!tdt

D

Z 0

�1

e.a�i!/tdt C

Z 1

0

e�.a�i!/tdt

D1

a � i!C

1

a C i!D

2a

a2 C !2:

We note that f .t/ D e�ajtj decreases rapidly at infinity and it is not differentiable att D 0 (Figure 1.3).

0.8

1

0.6

0.4

0.2

00

0.5

1

f(t) f^(ω)

0-2 2-4-6 4 60-2 2-4-6 4 6t ω

Fig. 1.3 Graphs of f .t/ D e�ajtj and Of .!/ with a D 1


Example 1.2.5. F˚f .t/

D F

na2 C t2

��1oD�

ae�aj!j; a > 0:

Example 1.2.5 can easily be verified and hence left to the reader.

Before we discuss the basic properties of Fourier transforms, we define thetranslation, modulation, and dilation operators respectively, by

Taf .t/ D f .t � a/ (Translation),

Mbf .t/ D eibt f .t/ (Modulation),

Dcf .t/ D1pjcj

f� t

c

�(Dilation),

where a; b; c 2 R and c ¤ 0. Each of these operators is a unitary operator fromL2.R/ onto itself. The following results can easily be verified:

Ta Mb f .t/ D eib.t�a/f .t � a/;

Mb Ta f .t/ D eibt f .t � a/;

Dc Ta f .t/ D1pjcj

f� t � a

c

�;

Ta Dc f .t/ D1pjcj

f� t � a

c

�;

Mb Dc f .t/ D1pjcj

exp

�i

b

ct

�f� t

c

�;

Dc Mb f .t/ D1pjcj

exp

�i

b

ct

�f� t

c

�:

Theorem 1.2.1. If f .t/; g.t/ 2 L1.R/ and ˛; ˇ are any two complex constants, then

(a) Linearity: F˚˛f .t/C ˇg.t/

D ˛F

˚f .t/

C ˇF

˚g.t/

:

(b) Shifting: F˚Taf .t/

D M�aOf .!/;

(c) Scaling: FnD 1

af .t/

oD DaOf .!/;

(d) Conjugation: FnD�1f .t/

oD Of .!/;

(e) Modulation: F˚Maf .t/

D TaOf .!/:

The proof follows readily from Definition 1.2.1 and is left as an exercise.


Theorem 1.2.2 (Continuity). If f .t/ 2 L1.R/, then Of .!/ is continuous on R.

We shall discuss the derivative of the Fourier transform. As we know thatsmoother the function f , the more rapidly Of will decay at infinity and conversely.Therefore, more rapidly f decays at infinity, the smoother Of will be. There are variousways to measure the smoothness of a given function f but here we will measure thesmoothness of f by counting the number of derivatives it has.

Theorem 1.2.3 (Differentiation Theorem). If both f .t/ and tf .t/ belong to L1.R/,

thend

d!Of .!/ exists and is given by

d

d!Of .!/ D .�i/F

˚tf .t/

: (1.2.14)

Proof. We have

dOf

d!D lim

h!0

1

h

hOf .! C h/� Of .!/

iD lim

h!0

�Z 1

�1

e�i!tf .t/

�e�iht � 1

h

�dt

�:

(1.2.15)

Note that

ˇˇ1h.e�iht � 1/

ˇˇ D

1

jhj

ˇˇe� iht

2

�e

iht2 � e� iht

2

�ˇˇ D 2

ˇˇˇˇ

sin

�ht

2

�

h

ˇˇˇˇ

� jtj:

Also,

limh!0

�e�iht � 1

h

�D �it:

Thus, result (1.2.15) becomes

dOf

d!D

Z 1

�1

e�i!tf .t/ limh!0

�e�iht � 1

h

�dt

D .�i/Z 1

�1

tf .t/ e�i!t dt D .�i/F˚tf .t/

:

This proves the theorem.


Corollary 1.2.1. If f 2 L1.R/ such that tnf .t/ is integrable for finite n 2 N, thenthe nth derivative of Of .!/ exists and is given by

dnOf

d!nD .�i/nF

˚tnf .t/

: (1.2.16)

Proof. This corollary follows from Theorem 1.2.3 combined with the mathematicalinduction principle. In particular, putting ! D 0 in (1.2.16) gives

"dnOf .!/

d!n

#

!D0

D .�i/nZ 1

�1

tnf .t/ dt D .�i/nmn; (1.2.17)

where mn represents the nth moment of f .t/. Thus, the moments m1;m2; : : : ;mn canbe calculated from (1.2.17)

Theorem 1.2.4 (The Riemann-Lebesgue Lemma). If f 2 L1.R/, then

limj!j!1

ˇˇOf .!/

ˇˇ D 0: (1.2.18)

Proof. Since e�i!t D �e�i!.tC �! /, we have

Of .!/ D �

Z 1

�1

e�i!.tC �! /f .t/ dt D �

Z 1

�1

e�i!xf�

x ��

!

�dx:

Thus,

Of .!/ D1

2

�Z 1

�1

e�i!tf .t/ dt �

Z 1

�1

e�i!tf�

t ��

!

�dt

�

D1

2

Z 1

�1

e�i!thf .t/ � f

�t �

�

!

�idt:

Clearly,

limj!j!1

ˇˇOf .!/

ˇˇ �

1

2lim

j!j!1

Z 1

�1

ˇˇf .t/ � f

�t �

�

!

�ˇˇ dt D 0:

This completes the proof.


Observe that the space C0.R/ of all continuous functions on R which decay atinfinity, that is, f .t/ ! 0 as jtj ! 1, is a normed space with respect to the normdefined by

��f�� D sup

t2R

ˇf .t/

ˇ: (1.2.19)

It follows from above theorems that the Fourier transform is a continuous linearoperator from L1.R/ into C0.R/. Theorem 1.2.4 gives a necessary condition for afunction f to have a Fourier transform. However, that belonging to C0.R/ is not asufficient condition for being the Fourier transform of an integrable function.

Theorem 1.2.5.

(a) If f .t/ is a continuously differentiable function, limjtj!1

f .t/ D 0 and both

f ; f 0 2 L1.R/, then

F˚f 0.t/

D i!F

˚f .t/

D .i!/Of .!/: (1.2.20)

(b) If f .t/ is continuously n-times differentiable, f ; f 0; : : : ; f .n/ 2 L1.R/ and

limjtj!1

f .r/.t/ D 0 for r D 1; 2; : : : ; n � 1;

then

F˚f .n/.t/

D .i!/n F

˚f .t/

D .i!/nOf .!/: (1.2.21)

Proof. We have, by definition,

F˚f 0.t/

D

Z 1

�1

e�i!tf 0.t/ dt;

which is, integrating by parts,

D e�i!tf .t/

�1�1

C .i!/Z 1

�1

e�i!tf .t/ dt

D .i!/Of .!/:

This proves part (a) of the theorem.


A repeated application of (1.2.16) to higher-order derivatives gives result(1.2.21).

We next calculate the Fourier transform of partial derivatives. If u.x; t/ is continu-

ously n times differentiable and@ru

@xr! 0 as jxj ! 1 for r D 1; 2; 3; : : : ; .n � 1/,

then, the Fourier transform of@nu

@xnwith respect to x is

F

�@nu

@xn

�D .ik/nF fu.x; t/g D .ik/n Ou.k; t/: (1.2.22)

It also follows from the Definition 1.2.1 that

F

�@u

@t

�D

d Ou

dt; F

�@2u

@t2

�D

d2 Ou

dt2; : : : ;F

�@nu

@tn

�D

dn Ou

dtn: (1.2.23)

Remark. This result has an important consequence: The smoothness of f is man-ifested in the rate of decay of its Fourier transform Of . We have already noted thatthe Fourier transform of a L1.R/ function must decay to zero at large frequenciesOf .!/ ! 0 as ! ! 1. If the nth derivative f n is also in L1.R/ and the derivativesvanish at infinity, then its Fourier transform F

f n.t/

�D .i!/nOf .!/ must go to zero

as ! ! 1. This requires that Of .!/ go to zero more rapidly than j!j�n. Thus, thesmoother f , the more rapid the decay of its Fourier transform. As a general rule ofthumb, local features of f such as smoothness are manifested by global features ofOf .!/, such as decay for large j!j. The symmetry principle implies that reverse isalso true: global features of f correspond to local features of Of .!/. This local-globalduality is one of the major themes of Fourier theory.

We now introduce the concept of convolution f �g of two functions f ; g 2 L1.R/.Recall that if f and g are integrable functions onR, then the convolution is defined by

f � g

�.t/ D

Z 1

�1

f .t � �/ g.�/ d�: (1.2.24)

The existence of the integral is justified by the following argument:

Z 1

�1

Z 1

�1

ˇf .t � �/ g.�/

ˇdt d� D

Z 1

�1

ˇg.�/

ˇd�Z 1

�1

ˇf .t/

ˇdt D

��g��1

��f��1:

It is clear that .f � g/.t/ 2 L1.R/ and in fact, we have


��f � g��1

��g��1

��f��1:

It is easy to verify that convolution is commutative, associative and distributive.That is;

f .t/ � g.t/ D g.t/ � f .t/;f .t/ �

g.t/ � h.t/

�Df .t/ � g.t/

�� h.t/;

f .t/ �g.t/C h.t/

�D f .t/ � g.t/C f .t/ � h.t/:

Proposition 1.2.1. If f 2 L1.R/ and g 2 L1.R/, then the convolution f � g iscontinuous on R.

Proposition 1.2.2 (Young’s Inequality). If the exponents p; q and s satisfy1=s D 1=p C 1=q � 1, then

��f � g��

s��f��

p

��g��

q: (1.2.25)

Theorem 1.2.6 (Convolution Theorem). If f ; g 2 L1.R/, then

F˚

f � g�.t/

D F˚f .t/

F˚g.t/

D Of .!/ Og.!/: (1.2.26)

Proof. Since f � g 2 L1.R/, we apply the definition of the Fourier transform toobtain

F˚

f � g�.t/

D

Z 1

�1

e�i!tdtZ 1

�1

f .t � �/ g.�/ d�

D

Z 1

�1

g.�/Z 1

�1

e�i!tf .t � �/ dt d�

D

Z 1

�1

e�i!� g.�/d�Z 1

�1

e�i!uf .u/ du; .t � � D u/

D Of .!/ Og.!/;

in which Fubini’s theorem was utilized.


Corollary 1.2.2. If f ; g; h 2 L1.R/ such that

h.x/ D

Z 1

�1

g.!/ ei!xd!; then (1.2.27)

f � h

�.x/ D

Z 1

�1

g.!/Of .!/ ei!xd!:

We now turn to the problem of inverting the Fourier transform. That is, we shallconsider the question: Given the Fourier transform of an integrable function f .t/,how do we obtain f .t/ back again from Of .!/? The reader, who is familiar with theelementary theory of Fourier series and integrals, would expect

f .t/ D1

2�

Z 1

�1

ei!t Of .!/ d!: (1.2.28)

Unfortunately, it is not necessary that if f 2 L1.R/, then its Fourier transform Of alsobelongs to L1.R/ (see Example 1.2.2), so that the Fourier integral

R1

�1Of .!/ ei!td!

may not exist as a Lebesgue integral. However, we can introduce a function K�.!/ inthe integrand and formulate general conditions on K�.!/ and its Fourier transformso that the following result holds:

lim�!1

Z �

��

Of .!/K�.!/ ei!td! D f .t/ (1.2.29)

for almost every t. This kernel K�.!/ is called a convergent factor or a summabilitykernel on R which can formally be defined as follows.

Definition 1.2.2 (Summability Kernel). A summability kernel on R is a family˚K�; � > 0

of continuous functions with the following properties:

(i)Z

R

K�.x/ dx D 1; for all � > 0;

(ii)Z

R

ˇK�.x/ dx

ˇ� M; for all � > 0 and for a constant M;

(iii) lim�!1

Z

jxj>ı

ˇK�.x/

ˇdx D 0; for all ı > 0:

The idea of a summability kernel helps to establish the so-called approximateidentity theorem.


Theorem 1.2.7 (Approximate Identity Theorem). If f 2 L1.R/ and fK� ,� > 0g 2 L1.R/ is a summability kernel, then

lim�!1

��f � K��

� f�� D 0: (1.2.30)

Proof. We have, by definition of the convolution (1.2.24),

f � K�

�.t/ D

Z 1

�1

f .t � u/K�.u/ du;

so that

ˇf � K�

�.t/ � f .t/

ˇD

ˇˇZ 1

�1

˚f .t � u/K�.u/ du � f .t/

ˇˇ

D

ˇˇZ 1

�1

˚f .t � u/� f .t/

K�.u/ du

ˇˇ ; by Definition 1.2.2(i);

�

Z 1

�1

ˇK�.u/

ˇˇf .t � u/ � f .t/

ˇdu:

Given " > 0, we can choose ı > 0 such that if 0 � juj < ı, thenˇf .t � u/

�f .t/ˇ<"

M, where

��K��1

� M. Consequently,

��f � K��.t/ � f .t/

�� D

Z

R

ˇf � K�.t/ � f .t/

ˇdt

�

Z

R

dtZ 1

�1

ˇK�.u/

ˇˇf .t � u/� f .t/

ˇdu

D

Z 1

�1

ˇK�.u/

ˇ�f .u/ du;

where

�f .u/ D

Z

R

ˇf .t � u/� f .t/

ˇdt � C:


Thus,

��f � K��.t/ � f .t/

��

Z

juj<ı

ˇK�.u/

ˇ�f .u/ du C

Z

juj>ı

ˇK�.u/

ˇ�f .u/ du

� "C CZ

juj>ı

ˇK�.u/

ˇdu D ";

since the integral on the right-hand side tends to zero for ı > 0 by Definition 1.2.2(iii). This completes the proof.

We have the following convolution theorem with respect to the frequencyvariable.

Theorem 1.2.8 (General Modulation). If F ff .t/g D Of .!/ and F fg.t/g D

Og.!/, where Of and Og belong to L1.R/, then

F˚f .t/ g.t/

D

1

2�

�Of � Og

�.!/: (1.2.31)

Proof. Using the inverse Fourier transform, we can rewrite the left-hand side of(1.2.31) as

F˚f .t/ g.t/

D

Z 1

�1

e�i!tf .t/ g.t/ dt

D1

2�

Z 1

�1

e�i!tg.t/ dtZ 1

�1

eixt Of .x/ dx

D1

2�

Z 1

�1

Of .x/ dxZ 1

�1

e�it.!�x/g.t/ dt

D1

2�

Z 1

�1

Of .x/ Og.! � x/ dx

D1

2�

�Of � Og

�.!/:



1.3 The Fourier Transform in L2.R/

In this section, we discuss the extension of the Fourier transform onto L2.R/. Itturns out that if f 2 L2.R/, then the Fourier transform Of of f is also in L2.R/ and��Of��2

Dp2��f��2; where

��f��2

D

�Z 1

�1

ˇf .t/

ˇ2dt

� 12

: (1.3.1)

The factorp2� involved in the above result can be avoided by defining the Fourier

transform as

Of .!/ D1

p2�

Z 1

�1

e�i!tf .t/ dt: (1.3.2)

Theorem 1.3.1. Suppose f is a continuous function on R vanishing outside abounded interval. Then, Of 2 L2.R/ and

��f��2

D��Of��2: (1.3.3)

Proof. We assume that f vanishes outside the interval Œ��; ��. We use the Parsevalformula for the orthonormal sequence of functions on Œ��; ��,

�n.t/ D1

p2�

eint; n D 0;˙1;˙2; : : : ;

to obtain

��f��22

DX

n2Z

ˇˇ 1p2�

Z 1

�1

e�intf .t/ dt

ˇˇ2

DX

n2Z

ˇˇOf .n/

ˇˇ2

:

Since this result also holds for g.t/ D e�ixtf .t/ instead of f .t/, and��f��22

D��g��22,

then

��f��22

DX

n2Z

ˇˇOf .n C x/

ˇˇ2

:

Integrating this result with respect to x from 0 to 1 gives


��f��22

DX

n2Z

Z 1

0

ˇˇOf .n C x/

ˇˇ2

dx DX

n2Z

Z nC1

n

ˇˇOf .y/

ˇˇ2

dy; .y D n C x/

D

Z 1

�1

ˇˇOf .y/

ˇˇ2

dy D��Of��2

2:

If f does not vanish outside Œ��; ��, then we take a positive number a for which thefunction g.t/ D f .at/ vanishes outside Œ��; ��. Then,

Og.!/ D1

aOf�!

a

�:

Thus, it turns out that

��f��22

D a��g��22

D aZ 1

�1

ˇˇ1a

Of�!

a

�ˇˇ2

d! D

Z 1

�1

ˇˇOf .!/

ˇˇ2

d! D��Of��2

2:


The space of all continuous functions on R with compact support is dense inL2.R/. Theorem 1.3.1 shows that the Fourier transform is a continuous mappingfrom that space into L2.R/. Since the mapping is linear, it has a unique extensionto a linear mapping from L2.R/ into itself. This extension will be called the Fouriertransform on L2.R/.

Definition 1.3.1 (Fourier Transform in L2.R/). If f 2 L2.R/ and ffng is asequence of continuous functions with compact support convergent to f in L2.R/,that is,

��f � fn�� ! 0 as n ! 1, then the Fourier transform of f is defined by

Of D limn!1

fn; (1.3.4)

where the limit is taken with respect to the norm in L2.R/.

Theorem 1.3.1 ensures that the limit exists and is independent of a particularsequence approximating f . It is important to note that the convergence in L2.R/does not imply pointwise convergence, and therefore the Fourier transform of asquare-integrable function is not defined at a point, unlike the Fourier transform ofan integrable function. We can assert that the Fourier transform Of of f 2 L2.R/ isdefined almost everywhere on R and Of 2 L2.R/. For this reason, we cannot say that


if f 2 L1.R/ \ L2.R/, the Fourier transform defined by (1.2.4) and the one definedby (1.3.4) are equal.

An immediate consequence of Definition 1.2.2 and Theorem 1.3.1 leads to thefollowing theorem.

Theorem 1.3.2 (Parseval’s Identity). If f 2 L2.R/, then

��f��2

D��Of��2: (1.3.5)

Theorem 1.3.3. If f 2 L2.R/, then

Of .!/ D limn!1

1p2�

Z n

�ne�i!tf .t/ dt; (1.3.6)

where the convergence is with respect to the L2-norm.

Proof. For n D 1; 2; 3; : : : ; we define

fn.t/ D

�f .t/; for jtj < n0; for jtj � n:

(1.3.7)

Clearly,��f � fn

��2

! 0 and, hence,��Of � Ofn

��2

! 0 as n ! 1.

Theorem 1.3.4 (Change of Roof). If f ; g 2 L2.R/, then

Df ; NOgE

D

Z 1

�1

f .t/ Og.t/ dt D

Z 1

�1

Of .t/ g.t/ dt DDOf ; NgE: (1.3.8)

Proof. We define both fn.t/ and gn.t/ by (1.3.7) for n D 1; 2; 3; : : : : Since

Ofm.t/ D1

p2�

Z 1

�1

e�ixtfm.x/ dx;


we obtainZ 1

�1

Ofm.t/ gn.t/ dt D1

p2�

Z 1

�1

gn.t/Z 1

�1

e�ixtfm.x/ dx dt:

The function exp.�ixt/ gn.t/ fm.x/ is integrable over R2, and hence, the Fubini

theorem can be used to rewrite the above integral in the form

Z 1

�1

Ofm.t/gn.t/ dt D1

p2�

Z 1

�1

fm.x/Z 1

�1

e�ixtgn.t/ dt dx

D

Z 1

�1

fm.x/ Ogn.x/ dx:

Since��g � gn

��2

! 0 and��Og � Ogn

��2

! 0, letting n ! 1 with the continuity of theinner product yields

Z 1

�1

Ofm.t/ gn.t/ dt D

Z 1

�1

fm.t/ Ogn.t/ dt:

Similarly, letting m ! 1 gives the desired result (1.3.8).

Lemma 1.3.1. If f 2 L2.R/ and g DNOf , then f D NOg.

Proof. In view of Theorems 1.3.2 and 1.3.3 and the assumption g DNOf , we find

Df ; NOgE

DDOf ; NgE

DDOf ; OfE

D��Of��2

2D��f��22: (1.3.9)

Also, we have

˝f ; NOg˛DDOf ; OfE

D��f��22: (1.3.10)

Finally, by Parseval’s relation,

��Og��22

D��g��22

D��Of��2

2D��f��22: (1.3.11)


Using (1.3.9)–(1.3.11) gives the following:

��f � NOg��2

2D˝f � NOg; f � NOg

˛D��f��22

�˝f ; NOg˛�˝f ; NOg˛C��Og��22

D 0:

This proves the result f D NOg.

Example 1.3.1 (The Haar Function). The Haar function is defined by

f .t/ D

8ˆ<

ˆ:

1; for 0 � t <1

2;

�1; for1

2� t < 1;

0; otherwise:

(1.3.12)

The Fourier transform of f .t/ is given as

Of .!/ D

Z 1

�1

e�i!tf .t/ dt D

"Z 12

0

e�i!t dt �

Z 1

12

e�i!t dt

#

D1

i!

�1 � 2e� i!

2 C e�i!�

De� i!

2

.i!/

�e

i!2 � 2C e� i!

2

�

D e� i!2

sin2�!4

�

!

4

: (1.3.13)

The graphs of f .t/ and Of .!/ are shown in Figure 1.4.

t10

-1

1

ω

f(t)f

^(ω)

Fig. 1.4 The graphs of f .t/ and Of .!/


f(t)f

^(ω)

0

02

4

ω√2-√

Fig. 1.5 Graphs of f .t/ and Of .!/

Example 1.3.2 (The Second Derivative of the Gaussian Function). If

f .t/ D1 � t2

�e�t2=2; (1.3.14)

then the Fourier transform of f .t/ can be computed as

Of .!/ D Fn1 � t2

�e�t2=2

o

D �F

�d2

dt2e�t2=2

�

D �.i!/2Fne�t2=2

o

D !2 e�t2=2:

Both f .t/ and Of .!/ are plotted in Figure 1.5.

Theorem 1.3.5 (Inversion Formula). If f 2 L2.R/, then

f .t/ D limn!1

1

2�

Z n

�nei!t Of .!/ d!; (1.3.15)

where the convergence is with respect to the L2.R/-norm.


Proof. If f 2 L2.R/ and g DNOf , then, by Lemma 1.3.1,

f .t/ D Og.t/ D limn!1

1

2�

Z n

�ne�i!t g.!/ d!

D limn!1

1

2�

Z n

�ne�i!t g.!/ d!

D limn!1

1

2�

Z n

�nei!t g.!/ d!

D limn!1

1

2�

Z n

�nei!t Of .!/ d!:

The formula (1.3.15) is called the inverse Fourier transform. If we use the factor�1=

p2��

in the definition of the Fourier transform, then the Fourier transform and

its inverse are symmetrical in form, that is,

Of .!/ D1

p2�

Z 1

�1

e�i!t f .t/ dt; f .t/ D1

p2�

Z 1

�1

ei!t Of .!/ d!: (1.3.16)

Theorem 1.3.6 (General Parseval’s Relation). If f ; g 2 L2.R/, then

˝f ; g˛D

Z 1

�1

f .t/ g.t/ dt D

Z 1

�1

Of .!/ Og.!/ d! D˝Of ; Og˛; (1.3.17)

where the symmetrical form (1.3.16) of the Fourier transform and its inverse is used.

Proof. It follows from (1.3.3) that

��f C g��22

D��Of C Og

��2

2:

Or, equivalently,

Z 1

�1

ˇf C g

ˇ2dt D

Z 1

�1

ˇˇOf C Og

ˇˇ2

d!;

Z 1

�1

f C g

�Nf C Ng

�dt D

Z 1

�1

�Of C Og

� �NOf C NOg

�d!:


Simplifying both sides gives

Z 1

�1

ˇfˇ2

dt C

Z 1

�1

f Ng C gNf

�dt C

Z 1

�1

ˇgˇ2

dt

D

Z 1

�1

ˇˇOfˇˇ2

d! C

Z 1

�1

�Of NOg C Og NOf

�d! C

Z 1

�1

jOgj2 d!:

Applying (1.3.3) to the above identity leads to

Z 1

�1

f Ng C gNf

�dt D

Z 1

�1

�Of NOg C Og NOf

�d!: (1.3.18)

Since g is an arbitrary element of L2.R/, we can replace g; Og by ig; iOg respectively,in (1.3.18) to obtain

Z 1

�1

f .ig/C .ig/Nf

�dt D

Z 1

�1

hOf�

iOg�

C .iOg/ NOfi

d!:

Or

�iZ 1

�1

f Ng dt C iZ 1

�1

g Nf dt D �iZ 1

�1

Of NOg d! C iZ 1

�1

Og NOf d!;

which is, multiplying by i,

Z 1

�1

f Ng dt �

Z 1

�1

g Nf dt D

Z 1

�1

Of NOg d! �

Z 1

�1

Og NOf d!: (1.3.19)

Adding (1.3.18) and (1.3.19) gives

Z 1

�1

f .t/ g.t/ dt D

Z 1

�1

Of .!/ Og.!/ d!:


The following theorem summaries the major results of this section and is usuallycalled Plancherel theorem.

1.4 The Discrete Fourier Transform 25

Theorem 1.3.7 (Plancherel’s Theorem). For every f 2 L2.R/, there exists Of 2

L2.R/ such that

(i) If f 2 L1.R/ \ L2.R/, then Of .!/ D1

p2�

Z 1

�1

e�i!tf .t/ dt;

(ii) Of .!/ D limn!1

1p2�

Z n

�ne�i!t f .t/ dt;

(iii) f .t/ D limn!1

1p2�

Z n

�nei!t Of .!/ d!,

(iv)˝f ; g˛D˝Of ; Og˛;

(v)��f��2

D��Of��2;

(vi) The mapping f ! Of is a Hilbert space isomorphism of L2.R/ onto L2.R/.

1.4 The Discrete Fourier Transform

The Fourier transform deals generally with continuous functions, i.e., functionswhich are defined at all values of the time t. However, for many applications,we require functions which are discrete in nature rather than continuous. Inmodern digital media audio, still images or video-continuous signals are sampledat discrete time intervals before being processed. Fourier analysis decomposes thesampled signal into its fundamental periodic constituents sines and cosines or, moreconveniently, complex exponentials. The crucial fact, upon which all modern signalprocessing is based, is that the sampled complex exponentials form an orthogonalbasis. To meet the needs of both the automated and experimental computations, thediscrete Fourier transform (DFT) has been introduced.

To motivate the idea behind the discrete Fourier transform, we take twoapproaches, one from the approximation point of view and other one from discretepoint of view.

Consider the function f with Fourier transform

Of .!/ D

Z 1

�1

f .t/ e�i!t dt: (1.4.1)

For some functions f , it is not always possible to evaluate the Fourier transform(1.4.1), and for such functions, one needs to truncate the range of integration to aninterval Œa; b� and approximate the integral for Of by a finite sum as

Of .!/ D

N�1X

kD0

ftk�

e�i!tk h:


Now, for sufficiently large a < 0 and b > 0

Z b

af .t/ e�i!t dt

is a good approximation to Of . Therefore, in order to approximate this integral, wesample the signal f at a finite number of sample points, say

t0 D a < t1 < t2 < � � � < tN�1 D b; a < 0; b > 0:

For simplicity the sample points are equally spaced and so

tk D a C kh; k D 0; 1; 2; : : : ;N

where h Db � a

Nindicates the sample rate. In signal processing applications, t

represents time instead of space and tk are the times at which we sample the signalf . This sample rate can be very high, e.g., every 10 � �20 milliseconds in currentspeech recognition systems.

Thus, the approximation g of Of is given by

g.!/ D

N�1X

kD0

ftk�

e�i!tk h

D e�i!aN�1X

kD0

ftk�

e�i!.b�a/k=Nh:

We now take the time duration Œa; b� into account by focusing attention on the points

(frequencies), !n D2�n

b � a; where n is an integer. Then, the approximation g of Of at

these points becomes

g.!n/ D e�ia!n

N�1X

kD0

ftk�

e�i2nk�=Nh:

By neglecting the term e�ia!n h in the R.H.S. of the above expression and focusingattention on the N-periodic function Of W Z ! C, we obtain

Of .n/ D

N�1X

kD0

ftk�

e�i2nk�=Nh; n 2 Z

D

N�1X

kD0

ftk�

w�nk; (1.4.2)

where w D e2� i=N : Equation (1.4.2) is known as discrete Fourier transform (DFT).


From a discrete perspective, one is dealing with the values of f at only a finitenumber of points

˚0; 1; 2; : : : ;N � 1

. Consider f as defined on the cyclic group of

integers modulo the positive integer N:

ZN DZN�Zmodulo N

�

whereN�

D˚kN W k 2 Z

and f W ZN ! C. This function f can be viewed as the

N-periodic function defined on Z by taking

fk C nN

�D f .k/; 8 n 2 Z; k D 0; 1; : : : ;N � 1:

But ZN is finite. Therefore, any function defined on it is integrable. Thus, L1ZN�

D

L2ZN�

D CN is the collection of all functions f W ZN ! C. One gets the discreteFourier transform for f W ZN ! C as

Of .n/ D

N�1X

kD0

fk�

e�i2nk�=N ; n 2 ZN :

This formula for the discrete Fourier transform is analogous to the formula for thenth Fourier coefficient with the sum over k taking the place of the integral over t.

In matrix notation, the above discussion can be summarized by the following.Here we replace ZN by the cyclic group of Nth root of unity. Therefore, f and its

discrete Fourier transform Of can be viewed as vectors.

f D

0B@

f .0/:::

f .N � 1/

1CA and Of D

0B@

Of .0/:::

Of .N � 1/

1CA

with

WN D

0BBBBBBBBBBBB@

1 1 1 � � � 1

1 e�1�2� i=N e�2�2� i=N � � � e�.N�1/�2� i=N

1 e�2�2� i=N e�2�2�2� i=N � � � e�2.N�1/�2� i=N

::::::

::::::

:::

1 e�.N�1/�2� i=N e�2.N�1/�2� i=N � � � e�.N�1/2�2� i=N

1CCCCCCCCCCCCA

:

Then, clearly Of D WNf . Here, WN is also called the Nth-order DFT matrix.


Example 1.4.1. Let f W Z4 ! C be defined by

f .n/ D 1; for all n D 0; 1; 2; 3:

Then,

Of .n/ D W4f D

0BBBBBB@

1 1 1 1

1 �i �1 i

1 �1 1 �1

1 i �1 �i

1CCCCCCA

0BBBBB@

1

1

1

1

1CCCCCA

D

0BBBBB@

4

0

0

0

1CCCCCA:

Here some properties are analogous to the corresponding properties for the Fouriertransform given in Theorem 1.2.1.

Theorem 1.4.1. The following properties hold for the discrete Fourier transform:

(a) Time shift: Of .n � j/ D Of .n/ e�2� ijn=N ;

(b) Frequency shift:�

f .n/ e2� ijn=N�^

D Of .n � j/;

(c) Modulation:

�f .n/ cos

2�nj

N

�^

D1

2

hOf .n � j/C Of .n C j/

i:

Proof. Let g.n/ D f .n � j/. Then, by using the fact that f is N-periodic, we obtain

Og.n/ D

N�1X

kD0

g.k/ e�2n� ik=N

D

N�1X

kD0

f .k � j/ e�2n� ik=N

D

N�1�jX

mD�j

f .m/ e�2n� i.mCj/=N

D

�1X

mD�j

f .m/ e�2n� i.mCj/=N C

N�1�jX

mD0


D

N�1X

mDN�j

f .m/ e�2n� i.mCj/=N C

N�1�jX

mD0


D

N�1X

mD0

f .m/ e�2n� im=N

!e�2n� ij=N

D Of .n/ e�2n� ij=N :

This proves part (a).


(b) Let g.n/ D f .n/ e�2n� ij=N . Then

Og.n/ D

N�1X

kD0

g.k/ e�2n� ik=N

D

N�1X

kD0

f .k/ e2� ijk=N � e�2n� ik=N

D

N�1X

kD0

f .k/ e�2� ik.n�j/=N D Of .n � j/:

Hence�

f .n/ e2� ijn=N�^

D Of .n � j/:

(c) Since

f .n/ cos2�in

ND1

2

hf .n/ e2� ijn=N C f .n/ e�2� ijn=N

i

the desired result is obtained by using part (b).

Our next task is to compute the inverse of the discrete Fourier transform. Wehave already computed the inverse Fourier transform, since Theorem 1.3.5 tellshow to recover the function f from its Fourier transform. The inverse discreteFourier transform is analogous and it allows us to recover the original discretesignal f from its discrete Fourier transform Of .

Theorem 1.4.2 (Inversion Formula). Let f W ZN ! C be such that

Of .n/ D

N�1X

kD0

f .k/ e�2� ikn=N D

N�1X

kD0

f .k/w�kn

where w D e2� i=N. Then

f .n/ D1

N

N�1X

kD0

Of .k/ e2� ikn=N D1

N

N�1X

kD0

Of .k/wkn: (1.4.3)

Proof. In order to establish the result (1.4.3), we must show that

N�1X

kD0

w.n�j/k D

�1 if n D j0 if n ¤ j

(1.4.4)


Since w D e2� i=N ;wkn � w�kj D wk.n�j/: Therefore, in order to sum this expressionover k D 0; 1; 2; : : : ;N � 1, we use the following elementary observation that

1C x C x2 C � � � C xN�1 D

8<

:N if x D 1

1 � xN

1 � xif x ¤ 1

Set x D wn�j and note that xN D 1 because wN D 1. Also, we note that wn�j ¤ 1

unless n D j for 0 � n; j � 1. Thus

N�1X

kD0

w.n�j/k D

�1 if n D j0 if n ¤ j

Hence

1

N

N�1X

kD0


N

N�1X

kD0

Of .k/wkn

D1

N

N�1X

kD0

0

@N�1X

jD0

f .j/w�kj

1

Awkn

D1

N

N�1X

jD0

f .j/

N�1X

kD0

w.n�j/k

!:

Therefore, by using (1.4.4), we get

1

N

N�1X

kD0


Nf .n/ � N D f .n/:

Example 1.4.2. Consider the discrete sinusoid f .n/ D cos.!n/:

Note that f .n/ is periodic if and only if ! is a rational multiple of 2�: ! D 2�p=q,for some p; q 2 Z. If p D 1 and q D N, then ! D 2�=N, and f .n/ isperiodic on Œ0;N � 1� with fundamental period N. Therefore, the functions ofthe form cos.2�kn=N/ are also periodic on Œ0;N � 1�, but since cos.2�kn=N/ D

cos2�.N �k/n=N

�, they are different only for k D 1; 2; : : : ; bN=2c: Similar results

hold for g.n/ D sin.!n/, except that sin.2�kn=N/ D � sin2�.N � k/n=N

�.

1.5 The Fast Fourier Transform 31

Note that for k D 0; 1; : : : ; bN=2c; we have

cos

�2�nk

N

�D1

2exp

�2�ink

N

�C1

2exp

�2�in.N � k/

N

�; (1.4.5)

sin

�2�nk

N

�D1

2iexp

�2�ink

N

��1

2iexp

�2�in.N � k/

N

�: (1.4.6)

Equations (1.4.5) and (1.4.6) thereby imply that

Of .k/ D Of .N � k/ D N=2 with Of .m/ D 0; for 0 � m � N � 1;m ¤ kI and

Og.k/ D �Og.N � k/ D .�iN/=2 with Og.m/ D 0; for 0 � m � N � 1;m ¤ k:

The factor of N in the expressions for Of .k/ and Og.k/ ensures that the inverse DFTrelation (1.4.3) holds for f .n/ and g.n/, respectively.

For more about DFT, the reader is referred to Sundararajan (2001), Butz (2006),and Wong (2010).

1.5 The Fast Fourier Transform

Although the ability of the discrete Fourier transform to provide information aboutthe frequency components of a signal is extremely valuable, the huge computationaleffort involved meant that until the 1960’s, it was rarely used in practical appli-cations. Two important advances have changed the situation completely. The firstwas the development of the digital computer with its ability to perform numericalcalculations rapidly and accurately. The second was the discovery by Cooley andTukey of a numerical algorithm which allows the DFT to be evaluated with asignificant reduction in the amount of calculations required. This algorithm, calledthe fast Fourier transform (FFT), allows the DFT of a sampled signal to be obtainedrapidly and efficiently.

Actually the idea of this algorithm goes back to Carl Friedrich Gauss (1777–1855) in 1805, but this early work was forgotten because it lacked the tool tomake it practical: the digital computer. Cooley and Tukey are honored because theydiscovered the FFT at the right time, the beginning of the computer revolution. Thepublication of the FFT algorithm by Cooley and Tukey in 1965 was the turning pointin digital signal processing and in certain areas of numerical analysis. Nowadays,the FFT is used in many areas, from the identification of characteristic mechanicalvibration frequencies to image enhancement. Standard routines are available toperform the FFT by computer in programming languages such as Pascal, Fortran,and C, and many spreadsheet and other software packages for the analysis ofnumerical data allow the FFT of a set of data values to be determined readily.For more about FFTs and their applications, we refer to the monographs (Brigham,1998; Bracewell, 2000; Butz, 2006; Duhamel and Vetterli, 1990; Rao et al., 2010).


The DFT of an N times sampled signal requires a total of N2 multiplications andN.N � 1/ additions. The FFT algorithm for N D 2k reduces the N2 multiplicationsto something proportional to N log2 N. If the calculations are handmade, then Nis necessarily small and this is not that significant, but in case N is large, thenumber of operations is drastically reduced. For example, a 3-minute song maycontain N D .44; 000/ � 180 D 7; 920; 000 samples. Thus, the DFT would take63; 000; 000; 000; 000multiplications. The FFT on the other hand would only takeapproximately 90; 751; 593multiplications. This result is a significant difference incomputing time.

How do we go about achieving this reduction in computing time? Where do westart? The idea is to look at even entries and odd separately, and we can then piecethem back together. This, seemingly simple, idea will allow certain multiplicationsthat are normally repeated to only be done once.

Let N 2 N with N even and wN D e�2� i=N . If N D 2M; M 2 N, then

w2N D e�2� i2=N D e�2� i=.N=2/ D e�2� i=M D wM:

Suppose f 2 `2ZN�

D CN ; define a; b 2 `2ZM�

D CM D CN=2 by

a.k/ D f .2k/ for k D 0; 1; 2; : : : ;M � 1;

b.k/ D f .2k C 1/ for k D 0; 1; 2; : : : ;M � 1:

Here a is the vector of the even entries of f and b is the vector of the odd entries.Thus

f .k/ Da.0/; b.0/; a.1/; b.1/; : : : ; a.M � 1/; b.M � 1/

�:

Then, breaking the definition of the DFT of f into even and its odd parts, we obtain

Of .n/ D

N�1X

kD0

f .k/ e�2� ink=N

D

N�1X

kD0

f .k/wnkN

D

M�1X

kD0

f .2k/w2knN C

M�1X

kD0

f .2k C 1/w.2kC1/nN

D

M�1X

kD0

a.k/w2N�nk

C wnN

M�1X

kD0

b.k/w2N�nk

D

M�1X

kD0

a.k/wnkM C wn

N

M�1X

kD0

b.k/wnkM

D Oa.n/C wnN

Ob.n/

1.5 The Fast Fourier Transform 33

where Oa.n/ and Oa.n/ are the M-points, DFTs of a and b, respectively. But

w�MN D

e�2� i=N

��MD e2� iM=N D e� i D �1:

Therefore, if 0 � n � M � 1, then

Of .n/ D Oa.n/C wnN

Ob.n/: (1.5.1)

Moreover, if M � n � N � 1, then

Of .n/ D Oa.n � M/C w.n�M/N

Ob.n � M/

D Oa.n/C w�MN � wn

NOb.n/

D Oa.n/� wnN

Ob.n/: (1.5.2)

Let us now look at the number of multiplications that it takes to compute Of .n/in this way when f 2 CN D `2

ZN�

and N D 2M. Computing Oa and Ob each takes

M2 DN2

4multiplications. In addition, for each entry of Ob, we need Ob.n/wn

N . This

gives an additional M D N=2 multiplication. Thus, if m.N/ denotes the number ofmultiplications required to compute an N-point DFT using the FFT algorithm, thenour total is

m.N/ D 2M2 C M D 2mN=2

�CN=2

�: (1.5.3)

Here we have essentially cut the half time. This is good but not great; half ofthe ten thousand years in a previous example is still a long time. The order ofmagnitude really has not changed. How can we get a more significant change? Thebig improvement is going to come when N D 2n; n 2 N. Now, with this assumptionN D 2n, the M defined above will also be even. Thus, we can use the same methodrecursively, when computing Oa and Ob. Therefore, we have the following theorem inthis regard.

Theorem 1.5.1. If N D 2n for some n 2 N, then

m.N/ DN=2

�log2 N: (1.5.4)


Proof. We prove the theorem by using the method of induction on n. Let n D 2.Then, m.22/ D m.4/ D 4 and hence the result holds for n D 2. Assume that theresult holds for n D k � 1. Then

m2k�

D 2 � m2k�1

�C 2k�1

D 22k�2.k � 1/

�C 2k�1

D 2k�1.k � 1/C 1

�

D 2k�1k D

�2k

2

� log2 2

k�:

This proves that the result holds for n D k also. Thus, by induction the result musthold for any n 2 N.

1.6 The Fractional Fourier Transform

Undoubtedly, one of the most recognized tools in signal and image processing isthe Fourier transform which generally converts a signal from time versus amplitudeto frequency versus amplitude. The classical Fourier transform can be visualizedas a change in representation of the signal corresponding to a counterclockwise

rotation of the axis by an angle�

2. Two successive rotations of the signal through

�

2will result in an inversion of the time axis. In spite of some remarkable success,

Fourier transform seems to be inadequate for studying nonstationary signals for atleast two reasons. First, the Fourier transform of a signal does not contain any localinformation in the sense that it does not reflect the change of wave number withspace or of frequency with time. Second, the Fourier transform method enables us toinvestigate problems either in time (space) domain or the frequency (wave number)domain, but not simultaneously in both domains. These are probably the majorweaknesses of the Fourier transform analysis. To overcome these problems, VictorNamias (1980) proposed the fractional Fourier transform (FrFT) as a generalizationof the conventional Fourier transform to solve certain problems arising in quantummechanics from classical quadratic Hamiltonians. It is also called rotational Fouriertransform or angular Fourier transform since it depends on a parameter ˛ which isinterpreted as a rotation by an angle ˛ in the time-frequency plane. Like the ordinaryFourier transform that corresponds to a rotation in the time-frequency plane over

an angle ˛ D 1 ��

2, the FrFT corresponds to a rotation over an arbitrary angle

˛ D a ��

2with a 2 R.

The fractional Fourier transforms are relatively recent developments that havefascinated the scientific, engineering, and mathematics community with their versa-

1.6 The Fractional Fourier Transform 35

tile applicability. The application areas for FrFT have been growing for two decadesat a very rapid rate. They have been applied in a number of fields including signalprocessing, image processing, quantum mechanics, neural networks, differentialequations, time-frequency distributions, optical systems, statistical optics, signaldetectors and pattern recognition, radar, sonar, and communications. A compre-hensive overview of FrFTs and their applications can also be found in Mendlovicand Ozaktas (1993a,b), Almeida (1994), Zayed (1998), Atakishiyev et al. (1999),Candan et al. (2000), Ozaktas et al. (2000, 2010), Ran et al. (2006), Tao et al. (2009),and Sejdic et al. (2011).

Definition 1.6.1 (Fractional Fourier Transform). The fractional Fourier trans-form with parameter ˛ of signal f .t/ is defined by

F ˛˚f .t/

.!/ D Of ˛.!/ D

Z 1

�1

K˛.t; !/f .t/ dt; (1.6.1)

where K˛.t; !/ is the so-called kernel of the FrFT given by

K˛.t; !/ D

8<

:

C˛ expni.t2 C !2/

cot˛

2� it! csc ˛

o; ˛ ¤ n�;

ı.t � !/; ˛ D 2n�;ı.t C !/; ˛ D .2n ˙ 1/�;

(1.6.2)

˛ Da�

2denotes the rotation angle of the transformed signal for FrFT, the FrFT

operator is designated by F ˛ , and

C˛ D .2�i sin ˛/�1=2 ei˛=2 D

r1 � i cot˛

2�: (1.6.3)

The corresponding inversion formula is given by

f .t/ D1

2�

Z 1

�1

K˛.t; !/ Of ˛.!/ d!; (1.6.4)

where

K˛.t; !/ D.2�i sin ˛/1=2 e�i˛=2

sin ˛� exp

��i.t2 C !2/ cot˛

2C it! csc ˛

�

D C˛ exp

��i.t2 C !2/ cot˛

2C it! csc ˛

�

D K�˛.t; !/ (1.6.5)


and

C˛ D.2�i sin˛/1=2e�i˛=2

2� sin ˛D

r1C i cot˛

2�D C�˛: (1.6.6)

Remarks:

1. It is important to point out that when a D 1, the kernel K˛.t; !/ given by (1.6.2)reduces to e�it!=

p2� , corresponding to the ordinary Fourier transform, and that

when a D �1, the kernel K˛.t; !/ reduces to eit!=p2� , corresponding to the

ordinary inverse Fourier transform.2. The zeroth-order FrFT operator F 0 is equal to the identity operator I, whereas

the integer values of ˛ correspond to repeated application of the Fouriertransform; for instance, F 2 corresponds to the Fourier transform of the Fouriertransform. The order ˛ may assume any real value; however the operator F ˛ isperiodic in ˛ with period 4, that F ˛C4n D F ˛; where n is any integer. This isbecause F 2 equals the parity operatorP which maps f .t/ to f .�t/ and F 4 equalsthe identity operator. Therefore, the range of a is usually restricted to Œ0; 4/. Thesefacts can be restated in operator notation:

F 0 D I;F 1 D F ;F 2 D P ;F 3 D FP ;F 4 D F 0 D I;F ˛C4n D F ˛C4m;m; n 2 Z:

3. Let F ˛.R/ denote the class of all FrFT on R parameterized by the parameter ˛;then it has a group structure called the elliptic group. If R˛ denotes a rotationoperator whose action is governed by

R˛f D F ˛; (1.6.7)

where

R˛ D

�cos˛ sin ˛

� sin ˛ cos˛

�; ˛ D

a�

2:

Then, R˛ is required to satisfy the following properties:

R0 D I; R2� D I; R�=2 D F ; and R˛Cˇ D R˛ � Rˇ:

The first two of these requirements involve the identity operator and they followdirectly from the definition of K˛.t; !/, whereas the third one is obvious. Thefourth one is the index additive property which can be obtained as follows:


R˛ Rˇf .!/ D

Z 1

�1

K˛.t; !/ dtZ 1

�1

f ./Kˇ.; t/ d

D

Z 1

�1

f ./ dZ 1

�1

K˛.t; !/Kˇ.; t/ dt

D

Z 1

�1

f ./K˛Cˇ.; !/ d

D R˛Cˇf .!/:

Thus, we have

F ˛ � F ˇ D F ˛Cˇ:

From the index additive property, we can deduce the inverse of the ˛th-orderfractional Fourier operator F�˛ as

Z 1

�1

F ˛.!/K�˛.t; !/ d! D

Z 1

�1

K�˛.t; !/ d!Z 1

�1

f .t0/K˛.t0; !/ dt0

D

Z 1

�1

f .t0/ dt0Z 1

�1

K�˛.t; !/K˛.!; t0/ d!

D

Z 1

�1

f .t0/K0.t; t0/ dt0

D

Z 1

�1

f .t0/ ı.t � t0/ dt0

D f .t/;

where it has been assumed that the integration order can be inverted.4. For any nonnegative integer r and f 2 L1.R/, we have

F ˛

�dr

dtrf .t/

�.!/ D

�i! sin ˛ C cos˛

d

d!

� r

F ˛˚f .t/

.!/: (1.6.8)

Example 1.6.1. The FrFT of the Gaussian function f .t/ D e�bt2 ; b > 0 is

F ˛ne�bt2

o.!/ D

r1 � i cot˛

2b � i cot˛exp

�i!2 cot˛

2�

.! csc ˛/2

2.2b � i cot˛/

�:


Example 1.6.2 (The Second Derivative of the Gaussian Function). This functionis defined by

f .t/ D .1 � t2/ e�t2=2 D �d2

dt2e�t2=2: (1.6.9)

Applying the definition of FrFT (1.6.1) and using (1.6.8) with r D 2, we obtain

F ˛nf .t/

o.!/

D �F ˛

�d2

dt2e�t2=2

�.!/

D �

�i! sin ˛ C cos˛

d

d!

� 2F ˛

ne�t2=2

o.!/

D exp

�i

2!2 cot˛ �

!2 csc2 ˛

2.1� i cot˛/

� �i sin ˛ C i cot˛ cos˛ �

cos˛ csc2 ˛

.1 � i cot˛/

�

�

��i!2 sin ˛ � cos˛

�1C !2

�i cot˛ �

csc2˛

.1 � i cot˛/

��

D exp

�i

2!2 cot˛ �

!2 csc2 ˛

2.1C cot2 ˛/.1C i cot˛/

�

�

�i sin˛ C i cot˛ cos˛ �

cos˛ csc2 ˛.1C i cot˛/

.1C cot2 ˛/

�

�

��i!2 sin ˛ � cos˛

�1C !2

�i cot˛ �

csc2 ˛.1C i cot˛/

.1C cot2 ˛/

��

D e�!2=2n!2 sin2 ˛ C cos2 ˛.1 � !2/

oC ie�!2=2

nsin˛ cos˛.2 !2 � 1/

o:

Both f .t/ and Of ˛.!/ are plotted in Figure 1.6.The FrFT is a generalization of the Fourier transform, so most of the properties

of the Fourier transforms have their corresponding generalization versions of theFrFT. However, in order to discuss some properties of the FrFT, we shall first recallthe definition of the Schwartz space S.R/.

Definition 1.6.2. An infinitely differential complex-valued function f is member ofS.R/ iff for every choice of ˇ and of nonnegative integers, it satisfies

�ˇ; .f / D supt2R

ˇˇtˇD f .t/

ˇˇ < 1: (1.6.10)


–0.4–0.2

–2 2

Re( f(t))

Re( f(w)) Re( f(w))

w w

Re( f(w))

t

w

4

2 4

–4

–2 –22 2

0.050.05

0.15

0.100.10

0.15

0.20

0.25

4 4–4 –4

–2–4

0.20.40.60.8

0.3

0.2

0.1

1.0

Fig. 1.6 Graphs of f .t/ and Of ˛.!/ with ˛ D �=6; �=4 and �=2

Definition 1.6.3. The space S˛.R/ consists of all infinitely differentiable functionsf .t/ that vanish at infinity and satisfying

�

ˇ.f / D supt2R

ˇˇtˇ

t f .t/ˇˇ < 1; ˇ; 2 N0 (1.6.11)

where

t D

�d

dt� it cot˛

�: (1.6.12)

Proposition 1.6.1. Let K˛.t; !/ be the kernel of FrFT (1.6.1) and rt be as in

(1.6.12); then

rt

nK˛.t; !/

oD

� i! csc ˛�r K˛.t; !/; r 2 N0: (1.6.13)


Proof. Differentiating the kernel K˛.t; !/ with respect to t, we obtain

d

dtK˛.t; !/ D C˛

d

dt

�exp

��

i.t2 C !2/ cot˛

2� it! csc ˛

��

D K˛ .t; !/ i .t cos˛ � ! csc ˛/

so that�

d

dt� it cot˛

�K˛.t; !/ D

� i! csc ˛

�K˛.t; !/:

Continuing this process r-times, we obtain

�d

dt� it cot˛

�r

K˛.t; !/ D

� i! csc ˛�r K˛.t; !/:


Proposition 1.6.2. If f 2 S.R/ � L1.R/; then

Z 1

�1

rtK˛.t; !/f .t/dt D

Z 1

�1

K˛.t; !/ 0

t

�rf .t/ dt; (1.6.14)

where 0t D �

�d

dtC it cot˛

�:

Proof. We shall first prove the result for r D 1, that is,

Z 1

�1

t K˛.t; !/f .t/ dt D

Z 1

�1

K˛.t; !/ . 0t/f .t/ dt:

Integrating by parts, we have

Z 1

�1

�d

dt� it cot˛

�K˛.t; !/ dt D

Z 1

�1

K˛.t; !/

�d

dtC it cot˛

�f .t/ dt:

Thus, we have

Z 1

�1

t K˛.t; !/f .t/ dt D

Z 1

�1

K˛.t; !/ . 0t/f .t/ dt:


In general, we can obtain

Z 1

�1

rtK˛.t; !/f .t/ dt D

Z 1

�1

K˛.t; !/. 0t/

r f .t/ dt:

Proposition 1.6.3. For all f 2 S.R/ � L1.R/ and r 2 N0, we have

(a)nF ˛. 0

t/r f .t/

oD

� i! csc ˛�r

F ˛˚f .t/

(b)n r!F ˛f .t/

oD F ˛

n.�it csc ˛/rf .t/

o:

Proof.

(a) By invoking Propositions 1.6.1 and 1.6.2, we obtain

nF ˛. 0

t/r f .t/

o.!/ D

Z 1

�1

K˛.t; !/˚. 0

t/rf .t/

dt

D

Z 1

�1

rt K˛.t; !/ f .t/ dt

D .�i! csc ˛/rZ 1

�1

K˛.t; !/ f .t/ dt

D

� i! csc ˛�r

F ˛˚f .t/

.!/:

(b) Since f 2 S.R/ and the integral defining the FrFT is uniformly convergent for! 2 R, we can differentiate within the integral sign, and using (1.6.13), weobtain

n r!F ˛f .t/

oD

Z 1

�1

r! K˛.t; !/f .t/ dt

D

Z 1

�1

.�itcsc ˛/r K˛.t; !/f .t/ dt

D

Z 1

�1

K˛.t; !/n.�itcsc ˛/rf .t/

odt

D F ˛n.�it csc ˛/rf .t/

o:

Proposition 1.6.4. The mapping F ˛ W S.R/ ! S˛.R/ is linear and continuous.


Proof. The proof is left to the reader as an exercise.

Proposition 1.6.5. If f 2 L1.R/, then Of ˛ satisfies the following properties:

(a) Of ˛.!/ 2 L1.R/,(b) Of ˛.!/ continuous on R,(c) Of ˛.!/ ! 0 as ! ! ˙1:

Proof. Part (a) follows directly by using the definition of FrFT and the fact that

��Of ˛��

1� jC˛j

��f��1:

(b) For any h > 0; we have

sup!2R

ˇˇOf ˛.! C h/� Of ˛.!/

ˇˇ

D sup!2R

ˇˇˇ

Z 1

�1

C˛

�exp

ni.t2 C .! C h/2/

cot˛

2� it.! C h/ csc ˛

o

� � expn

� i.t2 C !2/cot˛

2� it! csc ˛

o�f .t/ dt

ˇˇˇ

� sup!2R

Z 1

�1

jC˛j

ˇˇexp

�ih

�h cot˛

2C ! cot˛ � t csc ˛

�� 1

ˇˇˇf .t/

ˇdt:

Since

ˇˇexp

�ih

�h cot˛

2C ! cot˛ � t csc ˛

�� 1

ˇˇ ! 0; as h ! 0:

Therefore, it follows that the R.H.S of above inequality tends to zero as h ! 0,that is,

limh!0

sup!2R

ˇˇOf ˛.! C h/� Of ˛.!/

ˇˇ D 0:

This shows that Of ˛.!/ is continuous on R.


(c) By taking r D 1 in Proposition 1.6.3(a), we have

nF ˛. 0

t/ f .t/o

D

� i! csc ˛�F ˛

˚f .t/

.!/ D

� i! csc ˛

�Of ˛.!/

so that

ˇˇOf ˛.!/

ˇˇ D

1

j! csc � j

ˇˇn

OF ˛. 0t/f .t/

oˇˇ ! 0 as ! ! ˙1:

Remark. As we have seen in Proposition 1.6.5(c) that Of ˛.!/ ! 0 as ! ! ˙1 forevery f 2 L1.R/, it does not necessarily mean that Of ˛.!/ 2 L1.R/. For example,consider the Heaviside unit step function given by

h.t/ D

�1; t � 0;

0; t < 0:

Then, it is easy to verify that the product f ˛.t/ D expn

�it2 cot˛

2� toh.t/ 2 L1.R/,

but its FrFT is not in L1.R/.

Before we discuss basic properties of FrFT, we define the fractional translation,modulation, and dilation operators, respectively, by

T˛! f .t/ D f .t C !/ expfit! cot˛g; (Translation),

M˛! f .t/ D exp

�i!2 cot˛

2C it! csc ˛

�f .t/; (Modulation),

Daf .t/ D f .at/; (Dilation),

where t; !; a 2 R and a ¤ 0. The following results can easily be verified:

F ˛˚T˛! f .t/

.!/ D M�˛

�!F ˛˚f .t/

.!/

F ˛˚M˛! f .t/

.!/ D T�˛

�!F ˛˚f .t/

.!/

F ˛˚D�1f .t/

.!/ D F ˛

˚f .t/

.�!/

Z 1

�1

Of ˛.t/ g.t/ dt D

Z 1

�1

f .!/ Og˛.!/ d!:

Theorem 1.6.1. The fractional Fourier transform F ˛ is a continuous linearoperator from S.R/ onto itself.


Proof. For any f .t/ 2 S.R/ � L1.R/, we have

F ˛˚f .t/

.!/ D Of ˛.!/

Dp2� C˛ exp

�i!2 cot˛

2

�F

�exp

�it2 cot˛

2

�f .t/

�.! csc ˛/

D C˛ expn i!2 cot˛

2

oF˛.!/ (1.6.15)

where

F˛.!/ Dp2�F

�exp

�it2 cot˛

2

�f .t/

�.! csc ˛/ 2 S.R/:

We have

Dˇ!

Of ˛.!/ D C˛ Dˇ!

�exp

�i!2 cot˛

2

�Of ˛.!/

�

D C˛

ˇX

ˇ0D0

ˇ

ˇ0

!Dˇ0

!

�exp

n i!2 cot˛

2

o�Dˇ�ˇ0

! F˛.!/

D C˛

ˇX

ˇ0D0

ˇ

ˇ0

!exp

�i!2 cot˛

2

�Pˇ0

�!;

i cot˛

2

�Dˇ�ˇ0

! F˛.!/

where Pˇ0

�!;

i cot˛

2

�is a polynomial. Thus

Dˇ!

Of ˛.!/ D C˛

ˇX

ˇ0D0

ˇ

ˇ0

!exp

�i!2 cot˛

2

� ˇ0

X

ˇ00

D0

aˇ00 .cot˛/!ˇ00

Dˇ�ˇ

0

! F˛.!/:

Therefore, we have

ˇˇ! t Dˇ

!Of ˛.!/

ˇˇD

ˇˇˇC˛

ˇX

ˇ0

D0

ˇ

ˇ0

!ˇ0X

ˇ00

D0

exp�

i!2 cot ˛

2

�aˇ00 .cot ˛/!ˇ

00

Ct Dˇ�ˇ0

! F˛.!/

ˇˇˇ

�ˇC˛ˇ ˇX

ˇ0

D0

ˇ

ˇ0

!ˇ0X

ˇ00

D0

ˇˇaˇ00 .cot ˛/

ˇˇˇˇ!ˇ

00

CtDˇ�ˇ0

! F˛.!/ˇˇ:

Taking supremum over ! on both sides of above inequality and using the fact thatF˛.!/ 2 S.R/, we obtain


sup!2R

ˇˇ! t

Dˇ!

Of ˛.!/ˇˇ �

ˇC˛ˇ ˇX

ˇ0

D0

ˇ

ˇ0

!ˇ0X

ˇ00

D0

ˇˇaˇ00 .cot ˛/

ˇˇ sup!2R

ˇˇ!ˇ

00

CtDˇ�ˇ0

! F˛.!/ˇˇ < 1:

(1.6.16)

Hence, Of ˛.!/ 2 S.R/: In view of (1.6.1) and (1.6.4), we see that for all f 2 S.R/,we have

f .t/ D F�˛hF ˛f .t/

iD F ˛

hF�˛f .!/

i: (1.6.17)

Therefore, it follows that F ˛ is a one-one map from S.R/ onto itself. To show thatF ˛ is continuous on R, assume that there exist a null sequence

˚fn

n2Nin S.R/;

then from (1.6.16), we infer that F ˛ ffn.t/g ! 0 in S.R/; and hence, the continuityof FrFT follows.

Theorem 1.6.2 (Parseval’s Identity for FrFT). If f ; g 2 L1.R/, then

˝f ; g˛D

Z 1

�1

Of ˛.!/ Og˛.!/ d! DDOf ˛; Og˛

E: (1.6.18)

Proof. The proof of the theorem follows immediately from the Parseval formula ofthe Fourier transforms in L1.R/ and equation (1.6.15).

The rest of this section is devoted to find out the solution of some well-knowndifferential equations by using the fractional Fourier transform method.

We consider the nth-order linear nonhomogeneous ordinary differential equa-tions with constant coefficients:

Ly.t/ D f .t/ (1.6.19)

where L is the nth-order differential operator given by

L D an. 0t/

n C an�1. 0t/

n�1 C � � � C a1. 0t/C a0 (1.6.20)

where an; an�1; : : : ; a1; a0 are constants, 0t is the same as defined in Proposi-

tion 1.6.2, and f .t/ is a given function.

Application of the fractional Fourier transform to both sides of (1.6.19) gives

Z 1

�1

K˛.t; !/ Ly.t/ dt D

Z 1

�1

K˛.t; !/ f .t/ dt;


that is,

ha0.�i! csc ˛/n Can�1.�i! csc ˛/n�1C� � �Ca1.�i! csc ˛/Ca0

iOy˛.!/ D Of ˛.!/:

Or, equivalently,

P.�i! csc ˛/ Oy˛.!/ D Of ˛.!/

where P.z/ D

nX

mD0

amzm. Therefore, it follows that

Oy˛.!/ DOf ˛.!/

P.�i! csc ˛/: (1.6.21)

Applying the inverse FrFT to (1.6.21) gives the formal solution

y.t/ D F�˛

"Of ˛.!/

P.�i! csc ˛/

#: (1.6.22)

Example 1.6.3. Consider the generalized wave equation

@2y.x; t/

@t2D k2. 0

x/2y.x; t/; �1 < x < 1; t > 0 (1.6.23)

with the initial data y.x; 0/ D f .x/;@y.x; 0/

@tD g.x/, where k is a constant and 0

x

is the same as in Proposition 1.6.2.

Application of the fractional Fourier transform to (1.6.23) gives

Z 1

�1

K˛.x; !/@2y.x; t/

@t2dx D k2

Z 1

�1

K˛.x; !/. 0x/2y.x; t/ dx

so that

@2 Oy˛.!; t/

@t2D k2

Z 1

�1

. x/2K˛.x; !/ y.x; t/ dx D �k2!2 csc2 ˛ Oy˛.!; t/:


Therefore, it follows that

Oy˛.!; t/ D˚F ˛f

.!/ cos

.c! csc ˛/t

�C

˚F ˛g

.!/

c! csc ˛sin .c! csc ˛/t

�:

(1.6.24)

This can readily be inverted by the inverse FrFT (1.6.4) to obtain

y.x; t/ D F�˛h˚

F ˛f.!/ cos

.c! csc ˛/t

�i.x/

C F�˛

"˚F ˛g

.!/


�#.x/

D R.x; t/C S.x; t/ (1.6.25)

where

R.x; t/ D F�˛h˚

F ˛ f.!/ cos

.c! csc ˛/t

�i.x/ (1.6.26)

and

S.x; t/ D F�˛

"˚F ˛g

.!/


�#.x/: (1.6.27)

We now estimate R.x; t/ as

R.x; t/ D C˛

Z 1

�1

exp

��i.x2 C !2/

cot˛

2C ix! csc ˛

�

˚F ˛f

.!/ cos

˚.c! csc ˛/t

d!

D1

2� sin˛

Z 1

�1

exp

��i.x2 C !2/

cot˛

2C ix! csc!

�

�

�Z 1

�1

exp

�i.z2 C !2/

cot˛

2� iz! csc ˛

�f .z/ dz

�cos

˚.c! csc ˛/t

d!:

(1.6.28)

Setting H˛.z/ D expn

iz2 cot˛2

of .z/ and ! csc ˛ D , equation (1.6.28) becomes

R.x; t/ D1

2� sin˛exp

��ix2 cot ˛

2

�sin˛

�Z1

�1

eix cos.c t/ dZ

1

�1

e�izH˛.z/ dz

�

D1

2p2�

exp�

�ix2 cot˛

2

� �Z1

�1

hei.xCct/ C ei.x�ct/

iF˚H˛.z/

./ d

�


D1

2exp

��ix2 cot ˛

2

� hH˛.x C ct/C H˛.x � ct/

i

D1

2exp

��ix2 cot ˛

2

� hei.xCct/2 cot˛

2 f .x C ct/C ei.x�ct/2 cot˛2 f .x � ct/

i:

(1.6.29)

Similarly, assume that G˛.z/ D expn

iz2 cot˛2

og.z/, and again setting ! csc ˛ D ,

we obtain

S.x; t/ D1

p2�

exp

��ix2 cot˛

2

� Z 1

�1

eix sin.c t/

cF˚G˛.z/

./ d: (1.6.30)

Differentiating equation (1.6.30) with respect to t, we obtain the same result as thatof R.x; t/, and then by integrating, we have

S.x; t/ D1

2cexp

��ix2 cot˛

2

� Z xCct

x�ctexp

�i!2 cos˛

2

�g.!/ d!: (1.6.31)

After substituting equations (1.6.29) and (1.6.31) in (1.6.25), the solution of thegiven problem (1.6.23) can be obtained in the form

y.x; t/ D1

2exp

��ix2 cot˛

2

� "ei.xCct/2 cot ˛

2 f .x C ct/C ei.x�ct/2 cot ˛2 f .x � ct/C

1

c

Z xCct

x�ctexp

�i!2 cot˛

2

�g.!/d!

#:

Example 1.6.4. Consider the generalized heat equation

@y.x; t/

@tD 0

x

�2y.x; t/; �1 < x < 1; t > 0 (1.6.32)

where 0x is the same as given in Proposition 1.6.2 and y.x; 0/ D f .x/.

Application of the fractional Fourier transform to (1.6.32) gives

Z 1

�1

K˛.x; !/@y.x; t/

@tdx D

Z 1

�1

K˛.x; !/. 0x/2 y.x; t/ dx

so that

@Oy˛.!; t/

@tD

Z 1

�1

. x/2K˛.x; !/ y.x; t/ dx D �!2 csc2 ˛ Oy˛.!; t/:



Oy˛.!; t/ D C.!/ exp˚

� .!2csc2˛/t

(1.6.33)

which gives Oy˛.!; 0/ D C.!/. Since

Oy˛.!; 0/ D

Z 1

�1

K˛.x; !/ y.x; 0/ dx

D

Z 1

�1

K˛.x; !/f .x/ dx

D F ˛˚f .x/

.!/;

hence,

C.!/ D F ˛˚f .x/

.!/: (1.6.34)

Thus, equation (1.6.33) becomes

Oy˛.!; t/ D fF ˛ f g .!/ exp˚�.!2 csc2 ˛/t

: (1.6.35)

Applying the inverse FrFT on both sides of (1.6.35), we obtain

y.x; t/

D F�˛n.F ˛ f /.!/ exp

˚�.!2 csc2 ˛/t

o.x/

DE˛2�

Z1

�1

exp�

�i.x2 C !2/ cot ˛

2C ix! csc ˛

�.F ˛ f /.!/ exp

˚�.!2 csc2 ˛/t

d!

DE˛2�

Z1

�1

exp

��

i.x2 C !2/ cot ˛

2C ix! csc ˛

�

�

�C˛

Z1

�1

exp�

i.z2 C !2/ cot ˛

2� iz! csc ˛

�f .z/dz

�exp

˚�.!2 csc2 ˛/t

d!

D1

2� sin ˛exp

��ix2 cot˛

2

� Z1

�1

exp˚ix! csc ˛

exp

˚� .!2 csc2 ˛/t

d!

�

Z1

�1

exp˚

� iz! csc ˛

exp�

iz2 cot˛

2

�f .z/dz:


Let us assume that H˛.z/ D exp

�iz2 cot˛

2

�f .z/; then

y.x; t/

D1

2� sin ˛exp

��ix2 cot˛

2

� Z1

�1

eix! csc ˛e�.!2 csc2 ˛/td!Z

1

�1

e�iz! csc ˛H˛.z/dz:

Setting ! csc ˛ D , the above relation becomes

y.x; t/ D1

2�exp

��ix2 cot˛

2

� Z1

�1

exp fixg exp˚�2t

dZ

1

�1

expf�izgH˛.z/ dz

D1

2p� t

exp

��ix2 cot˛

2

� Z1

�1

exp fixg Fne�x2=4t

o./F

nH˛.z/

o./ d

D1

2p� t �

p2�

exp

��ix2 cot˛

2

� Z1

�1

exp fixg Fhe�x2=4t � H˛.x/

i./d

D1

2p� t

exp

��ix2 cot˛

2

� Z1

�1

exp

��.x � !/2

4t

�H˛.!/ d!:

Therefore, the solution of the generalized heat equation (1.6.32) is

y.x; t/D1

2p�t

exp

��ix2 cot˛

2

� Z 1

�1

exp

��.x � !/2

4t

�� exp

�i!2 cot˛

2

�f .!/ d!:

1.7 The Uncertainty Principle

Heisenberg first formulated the uncertainty principle between the position andmomentum in quantum mechanics. This principle has an important interpretationas an uncertainty of both the position and momentum of a particle described by awave function 2 L2.R/. In other words, it is impossible to determine the positionand momentum of a particle exactly and simultaneously (See Heisenberg, 1948a,b).

In signal processing, time and frequency concentrations of energy of a signalf are also governed by the Heisenberg uncertainty principle. The average orexpectation values of time t and frequency ! are, respectively, defined by

1.7 The Uncertainty Principle 51

t� D1

��f��22

Z 1

�1

tˇf .t/

ˇ2dt; !� D

1��Of��2

2

Z 1

�1

!ˇˇOf .!/

ˇˇ2

d!; (1.7.1)

where the energy of a signal f is well localized in time, and its Fourier transform Ofhas an energy concentrated in a small frequency domain.

The variances around these average values are given respectively by

�2t D1��f��22

Z 1

�1

t � t�

�2ˇf .t/

ˇ2dt; �2! D

1

2��Of��2

2

Z 1

�1

! � !�

�2 ˇˇOf .!/ˇˇ2

d!:

(1.7.2)

Theorem 1.7.1 (Heisenberg’s Inequality). If f .t/; tf .t/, and ! Of .!/ belong toL2.R/ and

pt jf .t/j ! 0 as jtj ! 1, then

�2t �2! �

1

4; (1.7.3)

where �t is defined as a measure of time duration of a signal f and �! is a measureof frequency dispersion (or bandwidth) of its Fourier transform Of .

Equality in (1.7.3) holds only if f .t/ is a Gaussian signal given by f .t/ D

C e�bt2 ; b > 0.

Proof. If the average time and frequency localization of a signal f are hti and h!i,then the average time and frequency location of e�i!� tf

t C t�

�is zero. Hence, it is

sufficient to prove the theorem around the zero mean values, that is, hti D h!i D 0:

Since��f��2

D��Of��2, we have

��f��42�2t �

2! D

1

2�

Z 1

�1

ˇtf .t/

ˇ2dtZ 1

�1

ˇˇ! Of .!/

ˇˇ2

d!:

Using i! Of .!/ D F˚f 0.t/

and Parseval’s formula

��f 0.t/��2

D1

2�

��i! Of .!/��2;


we obtain

��f��42�2t �

2! D

Z 1

�1

ˇtf .t/

ˇ2dtZ 1

�1

ˇf 0.t/

ˇ2d!

�

ˇˇZ 1

�1

ntf .t/ f 0.t/

odt

ˇˇ2

; by Schwarz’s inequality

�

ˇˇZ 1

�1

t �1

2

nf 0.t/ f .t/C f 0.t/f .t/

odt

ˇˇ2

D1

4

�Z 1

�1

t

�d

dtjf j2

�dt

�2D1

4

�htˇf .t/

ˇ2i1

�1�

Z 1

�1

jf j2dt

� 2

D1

4

��f��42;

in whichp

t f .t/ ! 0 as jtj ! 1 was used to eliminate the integrated term.

This completes the proof of inequality (1.7.3).

If we assume f 0.t/ is proportional to tf .t/, that is, f 0.t/ D atf .t/, where a is aconstant of proportionality, this leads to the Gaussian signal f .t/ D C e�bt2 ; where

C is a constant of integration and b D �a

2> 0.

Remarks.

1. In a time-frequency analysis of signals, the measure of the resolution of a signal fin the time or frequency domain is given by �t and �! . Then, the joint resolutionis given by the product .�t/.�!/ which is governed by the Heisenberg uncertaintyprinciple. In other words, the product .�t/.�!/ cannot be arbitrarily small and is

always greater than the minimum value1

2which is attained only for the Gaussian

signal.2. In many applications in science and engineering, signals with a high concentra-

tion of energy in the time and frequency domains are of special interest. Theuncertainty principle can also be interpreted as a measure of this concentrationof the second moment of f 2.t/ and its energy spectrum Of 2.!/.

1.8 Exercises 53

1.8 Exercises

1. Find the Fourier transforms of each of the following functions:(a) f .t/ D t e�ajtj; a > 0; (b) f .t/ D t e�at2 ; a > 0;(c) f .t/ D t2 e�t22; (d) f .t/ D e�at2Cbt;

(e) f .t/ D jtja�1; (f) f .t/ Dsin2 at

�at2:

2. If f .t/ D�1 �

t

a

��Œ�a;a�.t/; where a > 0, show that

Of .!/ Dsin2.a�!/

a.�!/2:

3. Show that the Fourier transform of

f .t/ Dcos bt

a2 C t2; a > 0;

for any constant b is

Of .!/ D�

2a

he�aj!�bj C e�aj!Cbj

i:

4. If f .t/ has a finite discontinuity at a point t D a, prove that

F˚f 0.t/

D .i!/Of .!/ � e�ia!Œf �a;

where Œf �a D f .a C 0/� f .a � 0/. Generalize this result for F˚f .n/.t/

.

5. Use result (1.2.17) to find

(a) Fntn e�t2=2

o; (b) F

ntn e�at2

o:

6. Prove the following convolution properties:(a) h.t C a C b/ D f .t C a/ � g.t C b/; h D f � g(b) h.at C b C c/ D jaj

f .at C b/ � g.at C c/

�; h D f � g

(c) h.t � t0/ D f .t � t0/ � g.t/; h D f � g

(d)f .t/ � g.t/

�Df � g

�.t/;

(d) t f .t/ � g.t/

�D tf .t/ � g.t/

�C f .t/ � tg.t/

�

7. Show that

Fne�a2t2

o� F

ne�b2t2

oD� e�!2=4c2

ab; where

1

c2D

�1

a2C1

b2

�:


8. If G.t/ D1

�e�t2 , then show that the family of functions G�.t/ D �G.�t/,

� > 0 forms a summability kernel on R. Moreover, show that

G�.t/ � f .t/ D

Z 1

�1

e�!2=4�2 Of .!/ ei!td!:

9. Consider f W ZN ! C, such that f .n/ D e�an, for some constant a. Show thatthe discrete Fourier transform of f is

Of .n/ D1 � e�aN

1� e�a�2� in=N:

10. For positive integer N, let w D e2� i=N . Show that(a) w w D 1; (b) wk D w�k D wNCk D wNCk; for any integer k,(c) 1C w C w2 C � � � C wN�1 D 0; N > 1:

11. Show that the fast Fourier transform of f .n/ D1; 0; i; 2;�i; 1; 0; i

�2 l2.Z8/ is

Of .n/ D�4C i; 2 � 2

p2C i;�i;

p2C i.1 �

p2/;�2 � i; 2C 2

p2C i; 2 � 3i;

�p2C i.1C

p2/�:

12. Verify the equality of the uncertainty principle for the Gaussian functionf .t/ D e�t2=2 and its Fourier transform Of .!/ D

p2� e�!2=2.

13. Repeat the calculation of the previous exercise using the function

f .t/ D1

2p�a

e�t2=4a and its Fourier transform Of .!/ D e�a!2 . Show that

equality of the Heisenberg inequality is preserved for all values of a.

2The Time-Frequency Analysis

What we know is not much. What we do not know is immense.

Pierre-Simon Laplace

Motivated by ‘quantum mechanics’, in 1946 the physicist Gabor defined elementary time-frequency atoms as waveforms that have a minimal spread in a time-frequency plane.To measure time-frequency ‘information’ content, he proposed decomposing signals overthese elementary atomic waveforms. By showing that such decompositions are closelyrelated to our sensitivity to sounds, and that they exhibit important structures in speech andmusic recordings, Gabor demonstrated the importance of localized time-frequency signalprocessing.

Stéphane Mallat

2.1 Introduction

Signals are in general nonstationary. A complete representation of nonstationary sig-nals requires frequency analysis that is local in time, resulting in the time-frequencyanalysis of signals. The Fourier transform analysis has long been recognized as thegreat tool for the study of stationary signals and processes where the propertiesare statistically invariant over time. However, it cannot be used for the frequencyanalysis that is local in time because it requires all previous as well as futureinformation about the signal to evaluate its spectral density at a single frequency !.Although time-frequency analysis of signals had its origin almost 60 years ago, therehas been major development of the time-frequency distributions approach in the lastthree decades. The basic idea of the method is to develop a joint function of timeand frequency, known as a time-frequency distribution, that can describe the energydensity of a signal simultaneously in both time and frequency. In principle, the


55

56 2 The Time-Frequency Analysis

time-frequency distributions characterize phenomena in a two-dimensional time-frequency plane. Basically, there are two kinds of time-frequency representations.One is the quadratic method covering the time-frequency distributions, and theother is the linear approach including the Gabor transform, the Zak transform,the linear canonical transform, and the wavelet transform analysis. So, the time-frequency signal analysis deals with time-frequency representations of signals andwith problems related to their definition, estimation, and interpretation, and it hasevolved into a widely recognized applied discipline of signal processing. For moredetailed information, we refer to Debnath (2001), Grochenig (2001), and Debnathand Shah (2015).

This chapter is devoted to a fairly detailed examination of the joint time-frequency analysis of signals. We start with the time-frequency localization ofsignals which leads to the windowed Fourier transform. This is followed by theGabor transform and its basic properties. Included are the Zak transform and itsbasic properties. Based on the relationship between the Fourier transform and linearcanonical transform, a coupled windowed transform, namely, windowed linearcanonical transform (WLCT) is introduced.

2.2 The Time-Frequency Localization

To achieve the time-frequency localization of spectral characteristics of a time-varying signal, a window function is introduced into the Fourier transform. A win-dow function g.t/ is a function in L2.R/ such that both g.t/ and Og.!/ have rapiddecay, that is, g.t/ is well localized in time domain, while Og.!/ is well localizedin frequency domain. Multiplying a signal f .t/ by a window function g.t/ before itsFourier transform has the effect of restricting the spectral information of the signal tothe domain of influence of the window function. Using the translates of the windowfunction on the time axis to cover the entire time domain, the signal is analyzed forspectral information in localized neighborhoods in time.

Definition 2.2.1 (Window Function). If g.t/ 2 L2.R/; kgk2 ¤ 0, and t � g.t/ 2

L2.R/, then g.t/ is called a window function.

It is important to note that g.t/ is a window function when its squared magnitudejg.t/j2 has a second-order moment. Therefore, if g.t/ is a window function, thent1=2 �g.t/ also belongs to L2.R/. Writing g.t/ D .1Cjtj/�1.1Cjtj/g.t/ and applyingthe Schwartz inequality, we can obtain

��g��1

��.1C jtj/

��1

2

��.1C jtj/��2< 1;

which infer that g.t/ is integrable on R and, hence, Og.!/ is continuous. Although itfollows from the Parseval identity that Og.!/ is also in L2.R/, in general, it is not true

2.2 The Time-Frequency Localization 57

that Og 2 L2.R/. In other words, it is possible that while g is a window function, Og isnot. An example of such a window function is the Haar function:

g.t/ D

8<

:

1; 0 � t < 12

�1; 12

� t < 10; otherwise:

(2.2.1)

Example 2.2.1 (Examples of Window Functions).

1. The simplest window function is the rectangular function given by

g.t/ D

�1; jtj � a; a > 00; jtj > a:

Its Fourier transform is

Og.!/ Deia! � e�ia!

i!:

Although g.t/ is compactly supported, it gives a bad localization in frequencydue to its discontinuous nature. Therefore, usually the more smooth functionsare needed.

2. The triangular window or Fejer window is given by

g.t/ D

8ˆ<

ˆ:

1Ct

a; �a � t < 0

1 �t

a; 0 � t < a

0; jtj > a:

It can be easily verified that

Og.!/ Da sin2

�a!

2

�

�a!

2

�2 :

This function provides a good localization in frequency as its spectrum decay

at the rate of1

!2which is faster than the decay of

1

!exhibited by the Fourier

transform of rectangular function.


3. The Hanning window is given by

g.t/ D

(cos2

��t

a

�; �a=2 � t � a=2

0; elsewhere.

The corresponding Fourier transform is

Og.!/ Da

4sin�a!

2

� � 1

� � a!=2C

2

a!=2�

1

� C a!=2

�:

4. The Hamming window is given by

g.t/ D

(b C .1 � b/ cos2

��t

a

�; �a=2 � t � a=2

0; elsewhere.

The Fourier transform of this window is

Og.!/ Da

4sin�a!

2

� � 1 � b

� � a!=2C2.1C b/

a!=2�

1 � b

� C a!=2

�:

5. The Blackman-Harris window is given by

g.t/ D

8<

:

3X

kD0

ak cos

�2�kt

a

�; �a=2 � t � a=2

0; elsewhere.

The Fourier transform of this function is

Og.!/ D a sin�a!

2

� 3X

kD0

ak.�1/k

�1

2�k C a!�

1

2�k � a!

�:

The most two important parameters for a window function are its center andradius which are defined as below.

Definition 2.2.2 (Center and Radius). If g.t/ is a window function, then thecenter t� and the root mean square radius �t for g.t/ are given by

t� D1

kgk22

Z 1

�1

tˇg.t/

ˇ2dt (2.2.2)

2.2 The Time-Frequency Localization 59

and

�t D1

kgk2

�Z 1

�1

.t � t�/2ˇg.t/

ˇ2dt

� 1=2; (2.2.3)

respectively. The diameter or width of a windowing function g.t/ is 2�g.

It is immediate from the definition 2.2.2 that the center and standard width ofthe rectangular window function g.t/ D �Œa;�a�.t/ are zero and 2a, respectively. Thefunction g described above with finite �g is called a time window. Similarly, wecan have a frequency window Og.!/ with center !� and (RMS) radius �! definedanalogous to relations (2.2.2) and (2.2.3) as

!� D1

kOgk22

Z 1

�1

!ˇOg.!/

ˇ2d! (2.2.4)

�! D1

kOgk2

�Z 1

�1

.! � !�/2ˇOg.!/

ˇ2d!

� 1=2: (2.2.5)

For a window function g to be useful in time-frequency analysis, it is necessary thatboth g and Og are window functions. Henceforth, we will assume that both g and Og arewindow functions with rapid decay in time and frequency, respectively. As we haveindicated in the beginning, we could obtain the approximate frequency contents ofa signal f in the neighbourhood of some desired location in time, say t D b, byfirst windowing function g to produce the window function fb.t/ D f .t/ g.t � b/and then taking the Fourier transform of fb.t/. This is called the windowed Fouriertransform or short-time Fourier transform (STFT), or sometimes referred to asrunning-windowed Fourier transform.

Formally, we define the STFT of a function f 2 L2.R/ with respect to thewindow function g evaluated at the location .b; !/ in the time-frequency plane as

Fgf .b; !/ D

Z 1

�1

f .t/ g.t � b/ e�i!tdt: (2.2.6)

Unlike the case of Fourier transform in which the function f must be known forthe entire time axis before its spectral component at any single frequency can becomputed, STFT needs to know f .t/ only in the interval in which g.t � b/ is non-zero.

Moreover, equation (2.2.6) gives the localized spectral information of f .t/ inthe time window

ht� C b � �t; t

� C b C �t

i: (2.2.7)


Fig. 2.1 Time-frequencyplane for the WFT ω

σt

σω

t

To derive the corresponding window in the frequency domain, apply Parsevalidentity to equation (2.2.6), so that we obtain

Fgf .b; !/ D

Z 1

�1

f .t/ g.t � b/ e�i!tdt

D1

2�

Z 1

�1

Of ./Og. � !/ eibd

D e�i!bF�1hOf ./Og. � !/

i.b/ (2.2.8)

where F�1 is the inverse Fourier transform. If !� is the center and �! is the radiusof the window function Og, then, (2.2.6) also gives the localized spectral informationof f in the frequency window

h!� C ! � �!; !

� C ! C �!

i: (2.2.9)

Thus, we have a time-frequency window

ht� C b � �t; t

� C b C �t

i�h!� C ! � �!; !

� C ! C �!

i(2.2.10)

centered at .t� C b; !C!�/ in the time-frequency plane, with width 2�t and height2�! as shown in Figure 2.1. The width and height of the time-frequency windoware constant for all time and frequency values and have a constant area 4 �t�! .

2.3 The Gabor Transforms 61

From the above discussion, we conclude that in order to achieve a highdegree of localization in time and frequency, we need to choose a window functionwith sufficiently narrow time and frequency windows. However, the Heisenberguncertainty principle (Theorem 1.7.1) imposes a theoretical lower bound on the areaof the time-frequency window of any window function g and is given by

�t �! �1

2(2.2.11)

where the equality holds only when g is a Gaussian function.

Example 2.2.2. A sinusoidal wave f .t/ D ei!0t whose Fourier transform is a Diracdelta function given by Of .!/ D 2� ı.! � !0/ has a windowed Fourier transform

Fgf .b; !/ D e�ib.!�!0/ Og.! � !0/

Its energy is spread over the frequency interval

h!0 �

�!

2; !0 C

�!

2

i:

Example 2.2.3. The windowed Fourier transform of a Dirac delta function definedby f .t/ D f .t � b0/ is

Fgf .b; !/ D e�i!b0g.b0 � b/:

Its energy is spread over the time interval

hb0 �

�t

2; b0 C

�t

2

i:

2.3 The Gabor Transforms

It has already been stated in previous section that decomposition of a signalinto a small number of elementary waveforms that are localized in time andfrequency plays a remarkable role in signal processing. Such a decompositionreveals important structures in analyzing nonstationary signals such as speech andmusic. In order to incorporate both time and frequency localization properties inone single transform function, Gabor (1946) first introduced the windowed Fouriertransform (or the Gabor transform) by using a Gaussian distribution function as awindow function. His major idea was to use a time localization window functionga.t � b/ for extracting local information from the Fourier transform of a signal,where the parameter a measures the width of the window and the parameter b isused to translate the window in order to cover the whole time domain. The idea isto use this window function in order to localize the Fourier transform, then shift


the window to another position, and so on. This remarkable property of the Gabortransform provides the local aspect of the Fourier transform with time resolutionequal to the size of the window. In fact, Gabor used

gt;!.�/ D ei!�g.� � t/ D M! Tt g.�/; (2.3.1)

as the window function by translating and modulating a function g, where g.t/ D

�� 14 e�t2=2, which is called canonical coherent states in quantum physics. The

energy associated with the function gt;! is localized in the neighborhood of t in aninterval of size �t measured by the standard deviation of jgj2. Evidently, the Fouriertransform of gt;!.�/ with respect to � is given by

Ogt;!.�/ D e�it.��!/ Og.� � !/: (2.3.2)

Obviously, the energy of Ogt;! is concentrated near the frequency ! in an interval ofsize �! which measures the frequency dispersion of Ogt;! . In a time-frequency .t; !/plane, the energy spread of the Gabor atom Ogt;! can be represented by the rectanglewith the center at

t�; !�

�and sides �t (along the time axis) and �! (along the

frequency axis). According to the Heisenberg uncertainty principle, the area of the

rectangle is at least1

2; that is, �t �! �

1

2. This area is minimum when g is a Gaussian

function, and the corresponding gt;! is called the Gabor function or Gabor wavelet.

Gabor transform has effectively been applied in many fields of science andengineering, such as image analysis and image compression, object and patternrecognition, computer vision, optics, and filter banks. Since medical signal analysisand medical signal processing play a crucial role in medical diagnostics, the Gabortransform has also been used for the study of brain functions, ECG signals, andother medical signals.

Definition 2.3.1 (The Continuous Gabor Transform). The continuous Gabortransform of a function f 2 L2.R/ with respect to a window function g 2 L2.R/ isdenoted by G Œf �.t; !/ D Qfg.t; !/ and defined by

G Œf �.t; !/ D Qfg.t; !/ D

Z 1

�1

f .�/ g.� � t/ e�i!�d� D˝f ; gt;!

˛; (2.3.3)

where gt;!.�/ D g.� � t/ei!� , so��gt;!

��2

D��g��2

and, hence, gt;! 2 L2.R/.


We next discuss the following consequences of the preceding definition:

1. If the window g is real and symmetric with g.�/ D g.��/ and if g is normalizedso that

��g�� D 1 and

��gt;!

�� D��g.� � t/

�� D 1 for any .t; !/ 2 R2, then the

Gabor transform of f 2 L2.R/ becomes

G Œf �.t; !/ D˝f ; gt;!

˛D

Z 1

�1

f .�/ g.� � t/ e�i!�d�: (2.3.4)

This can be interpreted as the windowed Fourier transform because the multipli-cation by g.�� t/ induces localization of the Fourier integral in the neighborhoodof � D t. Application of the Schwarz inequality to (2.3.4) gives

ˇˇG Œf �.t; !/

ˇˇ D

ˇ˝f ; gt;!

˛ˇ��f�� gt;!

�� D��f��g

��:

This shows that the Gabor transform G Œf �.t; !/ is bounded.2. It follows from definition (2.3.1) with a fixed ! that

G Œf �.t; !/ D e�i!tZ 1

�1

f .�/ g.� � t/ ei!.t��/d� D e�i!tf � g!

�.t/; (2.3.5)

where g!.�/ D ei!�g.�/ and g.��/ D g.�/. Furthermore, by the Parsevalformula, we find

G Œf �.t; !/ D˝f ; gt;!

˛DDOf ; Ogt;!

ED ei!t

Z 1

�1

Of .�/ Og.� � !/ e�i�td�: (2.3.6)

Note that except for the factor exp.i!t/, result (2.3.6) is almost identical with(2.3.3), but the time variable t is replaced by the frequency variable ! and thetime window g.� � t/ is replaced by the frequency window Og.� � !/.

3. For a fixed !, the Fourier transform of G Œf �.t; !/ with respect to t is given by thefollowing:

F fG Œf �.t; !/g D Of .� C !/ Og.�/: (2.3.7)

This follows from the Fourier transform of (2.3.5) with respect to t

FnG Œf �.t; !/

oD F

ne�i!t

f � g!

�.t/o

D Of .� C !/ Og.�/:

4. If g.t/ D e�t2=4, then

G Œf �.t; !/ Dp2 e.i!t�!2/

Wf�.t C 2i!/; (2.3.8)


where W represents the Weierstrass transformation of f .x/ defined by

W f .x/

�D

1

2p2

Z 1

�1

f .x/e�.t�x/2=4dx: (2.3.9)

5. The time width �t around t and the frequency spread �! around! are independentof t and !. In fact, we have

�2t D

Z 1

�1

.� � t/2ˇgt;!.�/

ˇ2d� D

Z 1

�1

.� � t/2ˇg.� � t/

ˇ2d� D

Z 1

�1

�2ˇg.�/

ˇ2d�:

Similarly, we obtain, by (2.3.2),

�2! D1

2�

Z 1

�1

.� � !/2ˇOgt;!.�/

ˇ2d� D

1

2�

Z 1

�1

.� � !/2ˇOg.�/

ˇ2d�

D1

2�

Z 1

�1

�2ˇOg.�/

ˇ2d�:

Thus, both �t and �! are independent of t and !. The energy spread of gt;!.�/

can be represented by the Heisenberg rectangle centered at .t; !/ with the area�t�! which is independent of t and !. This means that the Gabor transform hasthe same resolution in the time-frequency plane.

Example 2.3.1. Consider the function f defined by

f .�/ D e�a2�2 ; with g.�/ D 1:

Then, the Gabor transform of f is

G Œf �.t; !/ D

Z 1

�1

e�

a2�2Ci!��d� D

p�

ae�!2=4a2 :

Example 2.3.2. Obtain the Gabor transform of function

f .�/ D e�i�� :

We have

G Œf �.t; !/ D

Z 1

�1

e�i�.!C�/g.� � t/ d� D e�it.�C!/ Og.� C !/:


Example 2.3.3. Find the Gabor transform of functions

(a) f .�/ D 1; (b) f .�/ D ı.�/; (c) f .�/ D ı.� � t0/:

Next, we discuss some basic properties of continuous Gabor transform.

Theorem 2.3.1. Let f ; g; h 2 L2.R/ and a; b be any two arbitrary constants. Then,the following results hold:

(a) Linearity: G af C bh

�.t; !/ D a G

f�.t; !/C b G

h�.t; !/;

(b) Translation: G Taf�.t; !/ D e�i!a G

f�.t � a; !/;

(c) Modulation: G Maf

�.t; !/ D G

f�.t; ! � a/;

(d) Conjugation: G Nf�.t; !/ D G

f�.�;�!/:

The proof easily follows from the definition of the Gabor transform and is leftas an exercise.

Theorem 2.3.2. If two signals f ; g 2 L2.R/, then

Z 1

�1

Z 1

�1

ˇˇG Œf �.t; !/

ˇˇ2

dt d! D��f��22

��g��22:

Proof. The left-hand side of the above result is equal to

Z 1

�1

Z 1

�1

ˇˇG Œf �.t; !/

ˇˇ2

dt d!

D

Z 1

�1

Z 1

�1

ˇˇZ 1

�1

f .�/ g.� � t/ e�i!�d�

ˇˇ2

dt d!

D

Z 1

�1

Z 1

�1

ˇˇZ 1

�1

ht.�/ e�i!�d�

ˇˇ2

dt d!; ht.�/ D f .�/ g.� � t/

D

Z 1

�1

dtZ 1

�1

ˇˇOht.!/

ˇˇ2

d!

D

Z 1

�1

��Oht.!/��2

dt


D

Z 1

�1

��ht.�/��2dt; by Plancherel’s theorem

D

Z 1

�1

dtZ 1

�1

ˇf .�/

ˇ2ˇg.� � t/

ˇ2d�

D

Z 1

�1

ˇf .�/

ˇ2d�Z 1

�1

ˇg.x/

ˇ2dx D

��f��22

��g��22:


Theorem 2.3.3 (Parseval’s Formula). If the Gabor transforms of the twofunctions f and h exist with respect to a window function g, then

˝Qfg; Qhg

˛D��g��22

˝f ; h˛: (2.3.10)

In particular, if��g��2

D 1, then the Gabor transformation is an isometry from L2.R/into L2.R2/.

Proof. We first note that, for a fixed t,

Qfg.t; !/ D F˚ft.�/

D F

˚f .�/gt.�/

;

where gt.�/ D g.� � t/:

Thus, the Parseval formula (1.3.17) for the Fourier transform gives

Z 1

�1

Qfg.t; !/ Qhg.t; !/ d! DDF˚fgt;F

˚hgtE

D hfgt; hgti D

Z 1

�1

f .�/ g.� � t/ h.�/ g.� � t/ d�

D

Z 1

�1

f .�/ h.�/ˇg.� � t/

ˇ2d�:


Integrating this result with respect to t from �1 to 1 gives

˝Qfg; Qhg

˛D

Z 1

�1

Z 1

�1

Qfg.t; !/ Qhg.t; !/ dt d!

D

Z 1

�1

f .�/ h.�/ d�Z 1

�1

ˇg.� � t/

ˇ2dt

D

Z 1

�1

f .�/ h.�/ d�Z 1

�1

ˇg.x/

ˇ2dx .� � t D x/

D��g��22

˝f ; h˛:

This proves the result.

If��g��2

D 1, then (2.3.10) shows isometry from L2.R/ into L2.R2/.

Theorem 2.3.4 (Inversion Formula). If a function f 2 L2.R/, then

f .�/ D1

2�

1��g��22

Z 1

�1

Z 1

�1

Qfg.t; !/ g.� � t/ ei!� d! dt: (2.3.11)

Proof. We apply the inverse Fourier transform of f .�/ and the Parseval formula to

replace��g��22

by1

2�

��Og��22

so that

f .�/��g��22

D1

2�

Z 1

�1

ei!� Of .!/ d! �1

2�

��Og��22

D1

2�

Z 1

�1

ei!� Of .!/ d! �1

2�

Z 1

�1

ˇOg.�/

ˇ2d�:

Since the integral is true for any arbitrary !, we replace ! by ! C � to obtain

f .�/��g��22

D1

2�

Z 1

�1

ei�.!C�/ Of .! C �/d! �1

2�

Z 1

�1

Og.�/Og.�/d�

D1

2�

Z 1

�1

ei!�d! �1

2�

Z 1

�1

ei��hOf .! C �/ Og.�/

iOg.�/ d�


D1

2�

Z 1

�1

ei!�d! �

�1

2�

Z 1

�1

ei�� OQfg.! C �/ Og.�/ d�

�; by (2.3.7)

D1

2�

Z 1

�1

ei!�d! � Qfg.�; !/ � g.�/

�; by (1.2.26)

D1

2�

Z 1

�1

ei!�d!Z 1

�1

Qfg.t; !/ g.� � t/ dt

D1

2�

Z 1

�1

Z 1

�1

ei!� Qfg.t; !/ g.� � t/ dt d!:

This proves the inversion theorem.

Theorem 2.3.5 (Conservation of Energy). If f 2 L2.R/, then

��f��22

D1

2�

Z 1

�1

Z 1

�1

ˇQfg.t; !/

ˇ2dt d!; (2.3.12)

where g is a normalized window function��g

��2

D 1�:

Proof. Using (2.3.7) dealing with the Fourier transform of Qfg.t; !/ with respect to t,we apply the Plancherel formula to the right-hand side of .2:3:12/ to obtain

1

2�

Z 1

�1

Z 1

�1

ˇQfg.t; !/

ˇ2dt d!

D1

2�

Z 1

�1

d!1

2�

Z 1

�1

ˇF˚Qfg.t; !/

ˇ2d�

D1

2�

Z 1

�1

d!1

2�

Z 1

�1

ˇˇOf .! C �/

ˇˇ2

jOg.�/j2 d�

D1

2�

Z 1

�1

d!1

2�

Z 1

�1

ˇˇOf .!/

ˇˇ2

jOg.�/j2 d�


D1

2�

Z 1

�1

ˇˇOf .!/

ˇˇ2

d!; since��Og�� D

1

2�

Z 1

�1

jOg.�/j2 d� D 1

D

Z 1

�1

jf .�/j2 d� D��Of��22:


In many applications to physical and engineering problems, it is more important,at least from a computational viewpoint, to work with discrete transforms rather thancontinuous ones. In sampling theory, the sample points are defined by � D m!0 and� D nt0, where m; n are integers and t0 and !0 are positive quantities. The discreteGabor functions are defined by

gm;n.t/ D e2�m!0t g.t � nt0/ D M2�m!0 Tnt0 g.t/; (2.3.13)

where g 2 L2.R/ is a fixed function and t0 and !0 are the time shift and thefrequency shift parameters, respectively. A typical set of Gabor functions is shownin Figure 2.2.

Definition 2.3.2 (Discrete Gabor Transform). The discrete Gabor transform isdefined by

Qf .m; n/ D

Z 1

�1

f .t/ gm;n.t/dt D˝f ; gm;n

˛: (2.3.14)

The double series

X

m2Z

X

n2Z

Qf .m; n/ gm;n.t/ DX

m2Z

X

n2Z

˝f ; gm;n

˛gm;n.t/ (2.3.15)

is called the Gabor series of f . It is of special interest to find the inverse of thediscrete Gabor transform so that f 2 L2.R/ can be determined by the formula

Qf .mt0; n!0/ D

Z 1

�1

f .t/ gm;n.t/ dt D˝f ; gm;n

˛: (2.3.16)

The set of sample points˚

mt0; n!0�1

m;nD�1is called the Gabor lattice. The

answer to the question of finding the inverse is in the affirmative if the set offunctions

˚gm;n.t/

forms an orthonormal basis or, more generally, if the set is a


Fig. 2.2 The Gabor elementary functions gm;n.t/

frame for L2.R/. A system˚gm;n.t/

D˚M2�m!0 Tnt0 g.t/

is called a Gabor frame

or Weyl-Heisenberg frame in L2.R/ if there exist two constants A;B > 0 such that

A��f��22

�X

m2Z

X

n2Z

ˇ˝f ; gm;n

˛ˇ2� B

��f��22

(2.3.17)

holds for all f 2 L2.R/. For a Gabor frame˚gm;n.t/

, the analysis operator Tg is

defined by

2.4 The Zak Transform 71

Tgf Dn˝

f ; gm;n˛o

m;n; (2.3.18)

and its synthesis operator T�

g is defined by

T�

g cm;n DX

m2Z

X

n2Z

cm;n gm;n; (2.3.19)

where cm;n 2 `2.Z/. Both Tg and T�

g are bounded linear operators and in fact areadjoint operators with respect to the inner product h ; i: The Gabor frame operatorSg is defined by Sg D T

�

g Tg. More explicitly,

Sgf DX

m2Z

X

n2Z

˝f ; gm;n

˛gm;n: (2.3.20)

If˚gm;n W m; n 2 Z

constitutes a Gabor frame for L2.R/, any function f 2 L2.R/

can be expressed as

f .t/ DX

m2Z

X

n2Z

˝f ; gm;n

˛g�

m;n DX

m2Z

X

n2Z

˝f ; g�

m;n

˛gm;n; (2.3.21)

where˚g�

m;n

is called the dual frame given by g�

m;n D S�1g gm;n. Equation (2.3.21)

provides an answer for constructing f from its Gabor transform˝f ; gm;n

˛for a given

window function g.

Finding the conditions on t0; !0, and g under which the Gabor series of fdetermines f or converges to it, is known as the Gabor representation problem.For an appropriate function g, the answer is positive provided that 0 < !0t0 < 1.If 0 < !0t0 < 1, the reconstruction is stable and g can have a good time andfrequency localization. This is in contrast with the case when !0t0 D 1, where theconstruction is unstable and g cannot have a good time and frequency localization.For the case when !0t0 > 1, the reconstruction of f is, in general, impossible nomatter how g is selected.

2.4 The Zak Transform

Historically, the Zak transform (ZT), known as the Weil-Brezin transform inharmonic analysis, was introduced by Gelfand (1950) in his famous paper on eigen-function expansions associated with Schrödinger operators with periodic potentials.This transform was also known as the Gelfand mapping in the Russian mathematicalliterature. However, Zak (1967, 1968) independently rediscovered it as the k � qtransform in solid state physics to study a quantum-mechanical representation ofthe motion of electrons in the presence of an electric or magnetic field. Although


the Gelfand-Weil-Brezin-Zak transform seems to be a more appropriate name forthis transform, there is a general consensus among scientists to name it as the Zaktransform since Zak himself first recognized its deep significance and usefulnessin a more general setting. In recent years, the Zak transform has been widely usedin time-frequency signal analysis, in the coherent states representation in quantumfield theory, and also in mathematical analysis of Gabor systems.

Definition 2.4.1 (The Zak Transform). The Zak transformZaf

�.t; !/ of a

function f 2 L2.R/ is defined by the series

Zaf

�.t; !/ D

paX

n2Z

f .at C an/ e�2� in!; (2.4.1)

where a > 0 is a fixed parameter, t and ! are real.

If f .t/ represents a signal, then its Zak transform can be treated as the joint time-frequency representation of the signal f . It can also be considered as the discreteFourier transform of f in which an infinite set of samples in the form f .at C an/ isused for n D 0;˙1;˙2; : : : : Without loss of generality, we set a D 1 so that wecan write

Z f

�.t; !/ in the explicit form

Z f

�.t; !/ D F.t; !/ D

X

n2Z

f .t C n/ e�2� in!: (2.4.2)

This transform satisfies the periodic relation

Z f

�.t; ! C 1/ D

Z f

�.t; !/; (2.4.3)

and the following quasiperiodic relation

Z f

�.t C 1; !/ D e2� i!

Z f

�.t; !/; (2.4.4)

and therefore the Zak transform Z f is completely determined by its values on theunit square S D Œ0; 1� � Œ0; 1�.

It is easy to prove that the Zak transform of f can be expressed in terms of theZak transform of its Fourier transform Of .�/ D F ff .t/g. More precisely,

Z f

�.t; !/ D e2� i!t

Z Of

�.!;�t/: (2.4.5)

To prove this result, we define a function g for fixed t and ! by

g.x/ D e�2� i!x f .x C t/:


Then, it follows that

Og.�/ D

Z 1

�1

g.x/ e�2� ix� dx

D

Z 1

�1

f .x C t/ e�2� ix.�C!/ dx

D e2� i.�C!/tZ 1

�1

f .u/ e�2� i.�C!/u du

D e2� i.�C!/t Of .� C !/:

We next use the Poisson summation formula in the formX

n2Z

g.n/ DX

n2Z

Og.2�n/:

Or, equivalently,X

n2Z

f .t C n/ e�2� i!n D e2� i!tX

n2Z

eŒ2� i .2n�/t�Of .! C 2�n/

D e2� i!tX

m2Z

Of .! C m/e2� imt:

This gives the desired result (2.4.5).

The following results can be easily verified:

Z F f

�.!; t/ D e2� i!t

Z f

�.�t; !/; (2.4.6)

Z F�1f

�.!; t/ D e2� i!t

Z f

�.�t; !/: (2.4.7)

If gm;n.t/ D e�2� imtg.t � n/, thenZ gm;n

�.!; t/ D e�2� i.mtCn!/

Z g.!; t/

�:

We next observe that L2.S/ is the set of all square-integrable complex-valuedfunctions F on the unit square S, that is,

Z 1

0

Z 1

0

ˇF.t; !/

ˇ2dt d! < 1:

It is easy to check that L2.S/ is a Hilbert space with the inner product

˝F;G

˛D

Z 1

0

Z 1

0

F.t; !/G.t; !/ dt d! (2.4.8)


and the norm

��F��2

D

�Z 1

0

Z 1

0

ˇF.t; !/

ˇ2dt d!

� 1=2:

The set

nMm;n D M2�m;2�n.t; !/ D e2� i.mtCn!/ W m; n 2 Z

o(2.4.9)

forms an orthonormal basis of L2.S/.

Example 2.4.1. If

�m;nIa.t/ D1

pa

Tna M2�m=a �Œ0;a�.t/;

where a > 0, then

Za�m;nIa

�.t; !/ D em.t/ en.!/; where ek.t/ D e2� ikt:

We have

�m;nIa.t/ D1

pa

e2� im. t�naa /�Œ0;a�.t � na/

D1

pa

e2�imt

a �Œna;.nC1/a�.t/:

Thus, we obtain

Za�m;nIa

�.t; !/ D

X

k2Z

e2�im

a .atCak/�Œna;naCa�.at C ak/

DX

k2Z

em.t/ e�2� ik! �Œn�k;nC1�k�.t/

D em.t/ en.!/:

We shall now discuss the basic properties of continuous Zak transform.


Theorem 2.4.1. Let f ; g 2 L2.R/ and a; b be any two arbitrary constants. Then,the following results hold:

(a) Linearity: Z .af C bg/

�.t; !/ D a

Z f

�.t; !/C b

Z g

�.t; !/;

(b) Translation: Z .Taf /

�.t; !/ D

Z f

�.t � a; !/;

(c) Modulation: Z .Mbf /

�.t; !/ D eibt

Z f

� �t; ! �

b

2�

�;

(d) Translation and modulation: Z M2�mTnf

�.t; !/ D e2� i.mt�n!/

Z f

�.t; !/;

(e) Conjugation:Z Nf

�.t; !/ D

Z f

�.t;�!/;

(f) Symmetry:

Z f

�.t; !/ D

� Z f

�.�t;�!/; if f is even

�Z f

�.�t;�!/; if f is odd

(g) Inversion: For t; ! 2 R,

f .t/ D

Z 1

0

Z f

�.t; !/ d!;

Of .!/ D

Z 1

0

exp.�2�i!t/Z f

�.t; !/ dt;

f .x/ D

Z 1

0

exp.�2�ixt/Z Of

�.t; x/ dt

(h) Dilation:�Z D 1

af�.t; !/ D

Zaf

��at;!

a

�;

(i) Product and convolution of Zak transforms.

Results (2.4.3) and (2.4.4) show that the Zak transform is not periodic in the twovariables t and !. The product of two Zak transforms is periodic in t and !.

Proof. We consider the product

F.t; !/ DZ f

�.t; !/

Z g

�.t; !/

and find from (2.4.4) that

Z g

�.t; !/ D e�2� i!

Z g

�.t; !/:



F.t C 1; !/ DZ f

�.t; !/

Z g

�.t; !/ D F.t; !/;

F.t; ! C 1/ DZ f

�.t; !/

Z g

�.t; !/ D F.t; !/:

These show that F is periodic in t and !. Consequently, it can be expanded in aFourier series on a unit square

F.t; !/ DX

m2Z

X

n2Z

cm;n e2� imt e2� in!; (2.4.10)

where

cm;n D

Z 1

0

Z 1

0

F.t; !/ e�2� imt e�2� in! dt d!:

If we assume that the series involved are uniformly convergent, we can interchangethe summation and integration to obtain

cm;n D

Z 1

0

Z 1

0

(X

r2Z

f .t C r/ e�2� ir!

) (X

s2Z

Ng.t C s/ e2� is!

)e�2� i.mtCn!/ dt d!

D

Z 1

0

(X

r2Z

f .t C r/

) (X

s2Z

Ng.t C s/

)e�2� imt dt

Z 1

0

e2� i!.s�n�r/ d!

D

Z 1

0

(X

r2Z

f .t C r/ Ng.t C n C r/

)e�2� imt dt

DX

r2Z

Z rC1

rf .x/ Ng.x C n/e�2� im.x�r/ dx

D

Z 1

�1

f .x/ Ng.x C n/ e�2� imx dx

DDf .x/; e2� imxg.x C n/

E

D˝f ;M2�m T�ng

˛:

Consequently, (2.4.10) becomes

Z f

�.t; !/

Z g

�.t; !/ D

X

m2Z

X

n2Z

˝f ;M2�m T�ng

˛e2� i.mtCn!/: (2.4.11)



Theorem 2.4.2. Suppose H is a function of two real variables t and s satisfying thecondition

Ht C 1; s C 1

�D H

t; s�; and h.t/ D

Z 1

�1

H.t; s/f .s/ ds; s; t 2 R;

where the integral is absolutely and uniformly convergent. Then,

Z f

�.t; !/ D

Z 1

0

Z f

�.s; !/ˆ.t; s; !/ ds; (2.4.12)

where ˆ is given by

ˆ.t; s; !/ DX

n2Z

H.t C n; s/ e�2� in!; 0 � t; s; ! � l: (2.4.13)

Proof. Using the definition of Zak transform, we have

Z h

�.t; !/ D

X

k2Z

h.t C k/ e�2� ik! DX

k2Z

e�2� ik!Z 1

�1

H.t C k; s/f .s/ ds

DX

k2Z

e�2� ik!X

m2Z

Z mC1

mH.t C k; s/f .s/ ds

DX

k2Z

e�2� ik!X

m2Z

Z 1

0

H.t C k; s C m/f .s C m/ ds

D

Z 1

0

(X

k2Z

X

m2Z

H.t C k; s C m/ f .s C m/ e�2� ik!

)ds;

which is, due to (2.4.11),

D

Z 1

0

(X

k2Z

X

m2Z

H.t C k � m; s/ f .s C m/ e�2� ik!

)ds

D

Z 1

0

(X

m2Z

X

n2Z

H.t C n; s/ f .s C m/ e�2� i.mCn/!

)ds

D

Z 1

0

Z f

�.s; !/ˆ.t; s; !/ ds: (2.4.14)



In particular, if H.t; s/ D H.t � s/,

ˆ.t; s; !/ DX

n2Z

H.t � s C n/ e�2� in! DZ H

�.t � s; !/:

Consequently, Theorem 2.4.2 leads to the following convolution theorem.

Theorem 2.4.3 (Convolution Theorem). If

h.t/ D

Z 1

�1

H.t � s/f .s/ ds D .H � f /.t/;

then (2.4.12) reduces to the form

Z h

�.t; !/ D

Z 1

0

Z H

�.t � s/

Z f

�.s; !/ ds D Z .H � f /.t; !/: (2.4.15)

Example 2.4.2. If H.t/ DX

k2Z

ak ı.t � k/; then

Z .H � f /.t; !/ D A.!/Z f

�.t; !/; (2.4.16)

where

A.!/ DX

k2Z

ak e�2� ik!:

Clearly,

Z .H � f /.t; !/ D Z

�Z 1

�1

H.t � s/f .s/ ds

�.t; !/

D Z

(X

k2Z

ak

Z 1

�1

ı.t � s � k/f .s/ ds

).t; !/

D Z

(X

k2Z

ak f .t � k/

).t; !/

2.5 The Windowed Linear Canonical Transform 79

DX

k2Z

ak

X

n2Z

f .t C n � k/ e�2� in!

DX

k2Z

ak

X

m2Z

f .t C m/ e�2� i!.mCk/

D A.!/Z f

�.t; !/:

Theorem 2.4.4. The Zak transformation is a unitary mapping from L2.R/ to L2.S/.

Proof. It follows from the definition of the inner product (2.4.8) in L2.S/ that

˝Zaf ;Zag

˛D a

Z 1

0

Z 1

0

(X

n2Z

f .at C an/ e�2� in!

) (X

m2Z

g.at C am/ e2� im!

)dt d!

D aZ 1

0

(X

n2Z

f .at C an/ g.at C an/

)dt

DX

n2Z

Z .nC1/a

naf .y/ g.y/ dy

D

Z 1

�1

f .y/ g.y/ dy D˝f ; g˛: (2.4.17)

In particular, if f D g, we obtain from (2.4.17) that

��Zaf��22

D��f��22: (2.4.18)

This means that the Zak transform is an isometry from L2.R/ to L2.S/.

Further, Example 2.4.1 shows that˚�m;nIa.t/ W m; n 2 Z

is an orthonormal basis

of L2.R/. Hence, the Zak transform is a one-to-one mapping of an orthonormal basisof L2.R/ onto an orthonormal basis of L2.S/. This completes the proof of theorem.

2.5 The Windowed Linear Canonical Transform

In early 1970s, a promising linear integral transform with three free parameters,namely, linear canonical transform (LCT), was proposed by Moshinsky and Quesne(1971a,b) which is considered as one of the most powerful tools for signal andimage processing. This transform has also been referred to by different names in


the open literature such as quadratic-phase integrals (Bastiaans, 1979), generalizedHuygens integrals (Siegman, 1986), the affine Fourier transform (Abe and Sheridan,1994a,b), ABCD transform (Bernardo, 1996), the generalized Fresnel transform(James and Agarwal, 1996; Palma and Bagini, 1997), the extended fractional Fouriertransform (Hua et al., 1997), and Moshinsky-Quesne transform (Healy et al., 2016).Therefore, we can say that the LCT is a generalization of many optical transformssuch as the Fourier transform, the fractional Fourier transform (FrFT), the Fresneltransform, the Lorentz transform, and scaling operations. Thus, understanding theLCT may help to gain more insight into its special cases and to carry the knowledgegained from one subject to others.

With more degrees of freedom compared to the Fourier transform and the FrFT,the LCT is more flexible in nature but with similar computation cost as that ofconventional Fourier transform (see Healy and Sheridan, 2010). The LCT has foundmany applications in phase reconstruction, filter design, signal synthesis, patternrecognition, time-frequency analysis, optimal filtering, radar analysis, holographicthree-dimensional television, quantum physics, and many others. However, theLCT cannot reveal the local LCT-frequency contents due to its global kernel.On the other hand, the windowed Fourier transform (WFT) with a local windowfunction handles this kind of situation very well, but unfortunately, the WFT oftenperforms unsatisfactorily for its low resolution. Therefore, in order to attain the localcontents and high localization properties of a signal, it is desirable to develop a newtransform by replacing the Fourier transform kernel with the LCT kernel in thewindowed Fourier transform definition. This new transform was first introduced byBultheel and Martinez-Sulbaran (2007) and is called the windowed linear canonicaltransform (WLCT) which offers a flexible local frequency content, eliminatescross term, and enjoys high resolution of a signal. For more about LCT and theirapplications to signal and image processing, the reader is to referred to Stankovicet al. (2003), Koc et al. (2008), Tao et al. (2010), Kou and Xu (2012), Shi et al.(2014), Bahri and Ashino (2016), and Healy et al. (2016).

We shall start here with the formal definition of the linear canonical transform(LCT).

Definition 2.5.1. Let A D

�a bc d

�be a unimodular matrix, i.e., det.A/ D ad

�bc D 1; a; b; c; d 2 R or in C. Then, the continuous linear canonical transform(LCT) with parameter A of any function f 2 L2.R/ is defined by

LAŒf �.!/ D

8ˆ<

ˆ:

Z 1

�1

f .t/KA.t; !/ dt; b ¤ 0

pd exp

�icd!2

2

�f .d!/; b D 0

(2.5.1)


where the kernel KA.t; !/ of LCT is given by

KA.t; !/ D1

p2�b

exp

�i

2

�at2

b�2t!

bC

d!2

b��

4

��: (2.5.2)

For typographical convenience, we shall often denote the matrix A asA D .a; b; c; d/, in the text, but all operations have to be understood in the usualmatrix sense. Moreover, we note that when b D 0, the LCT becomes a chirpmultiplication. Therefore, we only consider the case of b ¤ 0, and without loss ofgenerality, we assume b > 0 throughout this section.

The above definition allows us to make the following comments:

1. Actually, the LCT has three free parameters; if we let a D =ˇ; b D 1=ˇ;

c D �ˇ C ˛=ˇ; d D ˛=ˇ, then the LCT of f .t/ can be rewritten as

LAŒf �.!/ D

Z 1

�1

f .t/KA.t; !/ dt; (2.5.3)

where

KA.t; !/ D

pˇ

p2�

exp

�i

2

� t2 � 2ˇt! C ˛!2 �

�

4

��; (2.5.4)

and the parameter matrix is given by

A D

�a bc d

�D

�=ˇ 1=ˇ

�ˇ C ˛=ˇ ˛=ˇ

�D

�˛=ˇ �1=ˇ

ˇ � ˛=ˇ =ˇ

��1

: (2.5.5)

2. The LCT given by (2.5.1) can be computed via Fourier transform as

LAŒf �.!/ D1

p2�b

exp

�i

2

�d!2

b��

2

��F

�exp

�iat2

2b

�f .t/

� �!b

�:

(2.5.6)

Note that if we let h.t/ D1

p2�b

exp

�i

2

�at2

b��

2

��f .t/, then equation

(2.5.6) takes the form

exp

��id!2

2b

�LAŒf �.!/ D F

˚h.t/

�!b

�; (2.5.7)

3. As a special case, when A D .a; b; c; d/ D .0; 1;�1; 0/, the LCT definition(2.5.1) reduces to the classical Fourier transform.


4. For the parameter matrix A D .a; b; c; d/ D .cos˛; sin ˛;� sin ˛; cos˛/ D R˛ ,the LCT multiplied by ei˛=2 coincides with the FrFT, i.e., F ˛ D ei˛=2LA ifA D R˛.

5. For the matrix A D .a; b; c; d/ D .1; b; 0; 1/, the LCT reduces to the Fresneltransform.

6. Multiplication by a Gaussian or chirp function is obtained with A D .a; b; c; d/ D

.1; 0; c; 1/.7. The scaling operator can be viewed as a special case of the LCT with

A D .a; b; c; d/ Dd�1; 0; 0; d

�.

Two interesting and important properties of LCT are the index additivity andreversibility. Index additivity means that the composition of two LCTs withparameter matrices A1 D .a1; b1; c1; d1/ and A2 D .a2; b2; c2; d2/, respectively,equals to the LCT with parameter matrix A3 D A2A1, that is,

LA1

LA1

�Œf � D LA3Œf � D LA2A1 Œf �: (2.5.8)

The inverse of the LCT with parameter matrix A D .a; b; c; d/ is the LCT withparameter matrix of A�1 D .d;�b;�c; a/, that is,

LA�1 .FA/ .f / D f : (2.5.9)

In case the parameter matrices A1 and A2 contain complex numbers, then theadditivity property (2.5.8) holds if

Im

�a2b2

Cd1b1

�> 0: (2.5.10)

However, if Im.a2=b2 C d1=b1/ D 0, then both of b1 and b2 must be real.Combining with the inverse property (2.5.9), then b must be real sinceA1 D .a; b; c; d/ and A2 D .d;�b;�c; a/ D A�1

1 by invoking additive property(2.5.8).

Another important property of LCT is the Parseval formula:

˝f ; g˛DDLA.f /;LA.g/

E: (2.5.11)

In particular, when f D g, we obtain the Plancherel formula for the LCT:

��f��22

D��LA.f /

��2

2: (2.5.12)

Following the idea of windowed Fourier transform, we shall try to generalize theLCT to a new transform, namely, windowed linear canonical transform (WLCT).


Before we give the formal definition of WLCT, we first recall that the windowedFourier transform (2.2.6) of any f 2 L2.R/ with respect to the window functiong 2 L2.R/ is given by

G Œf �.u; !/ D

Z 1

�1

f .t/ g.t � u/ e�i!t dt D˝f ; gu;!

˛; (2.5.13)

where gu;!.t/ D ei!tg.t � u/ D M!Tug.t/: Note that the optimal window for time-frequency localization can be achieved only if g is a Gaussian function. Moreover,for fixed ! D !0,

gu;!0.t/ D ei!0tg.t � u/; (2.5.14)

is called a Gabor filter. The extension of the Gabor filter to the LCT domain is givenby the following definition.

Definition 2.5.2. For a window function g 2 L2.R/n f0g, its window daughterfunction associated with LCT is defined by

gAu;!.t/ D

1p2�b

exp

��

i

2

�at2

b�2t!

bC

d!2

b��

4

��g.t � u/: (2.5.15)

This function is also called the linear canonical windowed Fourier kernel.

We are now in a position to introduce the basic definition of WLCT.

Definition 2.5.3. The windowed linear canonical transform (WLCT) of a functionf 2 L2.R/ with respect to a window function g 2 L2.R/ is denoted by GAŒf �.u; !/and defined by

GAŒf �.u; !/ D

Z 1

�1

f .t/ gAu;!.t/ dt; (2.5.16)

where u; ! 2 R, A D .a; b; c; d/ with det.A/ D 1 and gAu;!.t/ is given by (2.5.15).

We next discuss the following consequences of the proceeding definition.

1. It is worth noticing that, when A D .a; b; c; d/ D .0; 1;�1; 0/, one recoversthe standard definition of WFT (2.5.13). In fact, we have the following relationbetween WLCT and WFT:


GAŒf �.u;!/

D1

p2�b

Z 1

�1

f .t/ g.t � u/ exp

�i

2

�at2

b�2t!

bC

d!2

b��

4

��dt

D1

p2�b

exp

�id!2

2b

� Z 1

�1

exp

�i

2

�at2

b��

2

��f .t/g.t � u/ e�it!=bdt

D exp

�id!2

2b

� Z 1

�1

h.t/ g.t � u/ e�it!=bdt

D exp

�id!2

2b

�G Œh�

�u;!

b

�: (2.5.17)

2. If we take the Gaussian function as a window function in (2.5.16), then we getthe Gabor linear canonical transform (GLCT).

3. For a fixed u, WLCT (2.5.16) can be interpreted as the LCT of the product of afunction f and a conjugate and translated window function g, that is,

GAŒf �.u; !/ D LA

nf .t/ g.t � u/

o.!/: (2.5.18)

4. Implementing the inverse LCT (2.5.9) to WLCT (2.5.16), we obtain

f .t/ g.t � u/ D1

p2�b

Z1

�1

GAf .u; !/ exp

��

i

2

�at2

b�2t!

bC

d!2

b��

4

��d!:

(2.5.19)

5. The energy density of the WLCT is defined by

ˇˇGAŒf �.u; !/

ˇˇ2

D

ˇˇ 1p2�b

Z1

�1

f .t/ g.t � u/ exp

�i

2

�at2

b�2t!

bC

d!2

b��

4

��dt

ˇˇ2

:

(2.5.20)

We now investigate some basic properties of WLCT.

Theorem 2.5.1. Let f ; g; h 2 L2.R/ and ˛; ˇ be any two arbitrary constants. Then,the following results hold:

(a) Linearity: GAŒ˛f C ˇh�.u; !/ D ˛ GAŒf �.u; !/C ˇ GAŒh�.u; !/,(b) Parity: GAŒPf �.u; !/ D GAŒf �.�u;�!/, where Pg.t/ D g.�t/,

(c) Translation: GAŒTt0 f �.u; !/ D exp

�it0!c �

iat20c

2

�GAŒf �.u � t0; ! � at0/,

(d) Modulation: GAŒM!0 f �.u; !/ D exp

�id!!0 �

idb!202

�GAŒf �.t; ! � !0b/,

(e) Conjugation: GAŒ f �.u; !/ D G �1A Œf �.u; !/.


Proof.

(a). The proof of linearity easily follows from the definition of the WLCT.(b). For every f 2 L2.R/, we have

GAŒPf �.u; !/

D1

p2�b

Z 1

�1

f .�t/ g

� .t � u/�

exp

�i

2

�at2

b�2t!

bC

d!2

b��

4

��dt

D1

p2�b

Z 1

�1

f .�t/ g

� .t � u/�

� exp

�i

2

�a.�t/2

b�2.�t/.�!/

bC

d.�!/2

b��

4

��dt

D GAŒf �.�u;�!/:

(c). From the definition of WLCT, we have

GAŒTt0 f �.u; !/

D1

p2�b

Z 1

�1f .t � t0/ g.t � u/ exp

(i

2

at2

b�2t!

bC

d!2

b��

4

!)dt

D1

p2�b

Z 1

�1f .y/ g

y � .u � t0/

�exp

(i

2

a.y C t0/2

b�2.y C t0/!

bC

d!2

b��

4

!)dy

D1

p2�b

Z 1

�1f .y/ g

y � .u � t0/

�

� exp

(i

2

ay2 C at20 C 2ayt0

b�2y!

b�2t0!

bC

d!2

b��

4

!)dy

D1

p2�b

Z 1

�1f .y/ g

y � .u � t0/

�exp

(i

2

ay2

b�2y.! � t0a/

bC

d!2

b��

4

!)

� exp

(i

2

at20 � 2t0!

b

!)dy

D1

p2�b

Z 1

�1f .y/ g

y � .u � t0/

�exp

(i

2

ay2

b�2y.! � t0a/

bC

d.! � t0a/2

b��

4

!)


� exp

(i

2

2d.! � t0a/t0a C dt20a2

b

!)exp

(i

2

at20 � 2t0!

b

!)dy

D exp

(i

2

2d.! � t0a/t0a C dt20a2

b

!)exp

(i

2

at20 � 2t0!

b

!)GAf

u � t0; ! � t0a

�

D exp

(it0!c �

iat20c

2

)GAf

u � t0; ! � t0a

�:

This completes the proof of part (c).(d). For every f 2 L2.R/, we have

GAŒM!0 f �.u; !/

D1

p2�b

Z 1

�1ei!0 tf .t/ g.t � u/ exp

(i

2

at2

b�2t!

bC

d!2

b��

4

!)dt

D1

p2�b

Z 1

�1f .t/ g.t � u/ exp

(i

2

at2

b�2t!

bC

d!2

bC 2!0t �

�

4

!)dt

D1

p2�b

Z 1


(i

2

at2

b�2t.! � !0b/

bC

d!2

b��

4

!)dt

D1

p2�b

Z 1


(i

2

at2

b�2t.! � !0b/

bC

d.! � !0bC!0b/2

b��

4

!)dt

D1

p2�b

Z 1

�1f .t/ g.t � u/

� exp

(i

2

at2

b�2t.! � !0b/

bC

d.! � !0b/2

b��

4C 2.! � !0b/!0b C !20b2

!)dt

D exp

(id!!0 �

idb!202

)GAŒf �.t; ! � !0b/:

(e). For every f 2 L2.R/, we have

GAŒ f �.u; !/ D1

p2�b

Z1

�1

f .t/ g.t � u/ exp�

i

2

�at2

b�2t!

bC

d!2

b��

4

��dt

D1

p2�b

Z1

�1

f .t/ g.t � u/ exp�

�i

2

�at2

b�2t!

bC

d!2

b��

4

��dt

D G �1A Œf �.u; !/:


Theorem 2.5.2 (Orthogonality Relation). If f1; g1; f2 and g2 belong to L2.R/,then the following formula hold:

DGA;g1 f1;GA;g2 f2

ED˝f1; f2

˛ ˝g1; g2

˛: (2.5.21)

Proof. Assume that both the window functions g1; g2 2 L1.R/ \ L1.R/; then it isobvious that for every u 2 R,

f1.t/ g1.t � u/ 2 L2.R/ and f2.t/ g2.t � u/ 2 L2.R/:

Therefore, it follows from the Parseval formula (2.5.11) for the LCT thatDGA;g1 f1;GA;g2 f2

E

D

Z 1

�1

Z 1

�1

GA;g1 f1.u; !/GA;g2 f2.u; !/ du d!

D

Z 1

�1

�Z 1

�1

LA

nf1.t/ g1.t � u/

o.!/LA

nf2.t/ g2.t � u/

o.!/ d!

�du

D

Z 1

�1

�Z 1

�1

f1.t/ f2.t/ g1.t � u/ g2.t � u/ exp

��iat2

b

�dt

�du: (2.5.22)

By virtue of Fubini theorem, we can interchange the order of integration in (2.5.22)to get

DGA;g1 f1;GA;g2 f2

ED

Z 1

�1

f1.t/ f2.t/ exp

��iat2

b

�dtZ 1

�1

g1.t � u/ g2.t � u/ du

D˝f1; f2

˛ ˝g1; g2

˛

where the extension to general g1; g2 2 L2.R/ has been done by the standard densityargument. As it is easy to verify that a fixed g1 2 L1.R/ \ L1.R/, the mapping

g2 !˝GA;g1 f1;GA;g2 f2

˛L2.R2/

is a linear functional that coincides with˝f1; f2

˛ ˝g1; g2

˛

on a dense subset L1.R/\L1.R/ of L2.R/. It is therefore bounded and extends to allg2 2 L2.R/. Similarly, for arbitrary f1; f2 and g2 2 L2.R/, the conjugate linear

functional g1 !˝GA;g1 f1;GA;g2 f2

˛L2.R2/

coincides with˝f1; f2

˛ ˝g1; g2

˛on L1.R/ \

L1.R/ and extends to all functions of L2.R/.

Corollary 2.5.1 (Conservation of Energy). If f 2 L2.R/, then

Z 1

�1

Z 1

�1

ˇˇGA;gf .u; !/

ˇˇ2

du d! D��f��22: (2.5.23)


Proof. Taking f1 D f2 D f and g1 D g2 D g in (2.5.21), we obtain

��GA;gf��2

2D

Z 1

�1

Z 1

�1

ˇˇGA;gf .u; !/

ˇˇ2

du d! D��f��22

��g��22: (2.5.24)

The desired result is obtained by taking��g��22

D 1 in (2.5.24), that is,

Z 1

�1

Z 1

�1

ˇˇGA;gf .u; !/

ˇˇ2

du d! D��f��22; for all f 2 L2.R/: (2.5.25)

Remark.

1. Suppose that kf k22 D 1. Then, equation (2.5.25) reduces to

��GA;gf��2

2D

Z 1

�1

Z 1

�1

ˇˇGA;gf .u; !/

ˇˇ2

du d! D 1: (2.5.26)

This relation is known as the radar uncertainty principle in the WLCT domain.2. It follows from (2.5.25) that f is completely determined by GA;gf . Furthermore,

the condition

GA;gf .u; !/ D1

p2�b

˝f ;M!Tug

˛D 0; 8 u; ! 2 R;

implies f D 0, which means that for each fixed g 2 L2.R/, the setfM!Tug W u; ! 2 Rg spans a dense subspace of L2.R/. Therefore, it is interestingto see how f can be recovered from GA;gf . In this regard, we present two proofsfor the remarkable inversion formula.

Theorem 2.5.3 (Inversion Theorem). Suppose that g1; g2 2 L2.R/ and˝g1; g2

˛¤ 0. Then for all f 2 L2.R/, we have

f .t/ D1˝

g1; g2˛Z 1

�1

Z 1

�1

GA;g1 f .u; !/KA.t; !/ g2.t � u/ du d!: (2.5.27)

First Proof. Since GA;g1 f 2 L2.R/, it follows by Corollary 2.5.1 that the vector-valued integral

Qf .t/ D1˝

g1; g2˛Z 1

�1

Z 1

�1

GA;g1 f .u; !/KA.t; !/ g2.t � u/ du d!


is well defined in L2.R/. Using the orthogonality relation (2.5.21), we observe that

˝Qf ; h˛D

1p2�b

˝g1; g2

˛Z 1

�1

Z 1

�1

GA;g1 f .u; !/˝h;M!Tug2

˛du d!

D1˝

g1; g2˛DGA;g1 f ;GA;g2h

E

D˝f ; h˛:

Thus f D Qf . This proves the inversion theorem.

Second Proof. For every h 2 L2.R/, the inverse transform of the WFT (2.5.13)implies that

h.t/ D1

2�˝g1; g2

˛Z 1

�1

Z 1

�1

Gg1h.u; !/ ei!t g2.t � u/ du d!

D1

2�˝g1; g2

˛Z 1

�1

Z 1

�1

Gg1h�

u;!

b

�ei!t=b g2.t � u/ du d

!

b: (2.5.28)

If we let h.t/ D1

p2�b

exp

�i

2

�at2

b��

2

��f .t/ and using the relation (2.5.17),

equation (2.5.28) becomes

1p2�b

exp

�i

2

�at2

b��

2

��f .t/

D1

2�˝g1; g2

˛Z 1

�1

Z 1

�1

exp

��id!2

2b

�GA;g1 f .u; !/ ei!t=b g2.t � u/ du d

!

b:

Or, equivalently,

f .t/ D1

p2�b

˝g1; g2

˛Z 1

�1

Z 1

�1

GA;g1 f .u; !/

� exp

��i

2

�at2

bC

d!2

b�2t!

b��

2

��g2.t � u/ du d!

D1˝

g1; g2˛Z 1

�1

Z 1

�1

GA;g1 f .u; !/KA.t; !/ g2.t � u/ du d!:

This proves the inversion theorem.


2.6 Exercises

1. For the cosine window

g.t/ D

(cos

��t

a

�; �a=2 � t � a=2

0; elsewhere

show that its Fourier transform is

Og.!/ D a cos�a!

2

� � 1

� � a!�

1

� C a!

�:

2. Consider the hat function g defined by

g.t/ D

8<

:

t; 0 � t � 1

2 � t; 1 � t � 2

0; otherwise;

and compute the time-frequency window for g.3. Use definition 2.2.1 to show that the triplet function

g.t/ D

(e��jtj cos2

��t

a

�; �a=2 � t � a=2

0; elsewhere;

is a window function.4. Let f .t/ D sin.�t/ and the window function g.t/ be the function:

g.t/ D

8<

:

1C t; �1 � t < 01 � t; 0 � t < 1

0; otherwise.

Determine the windowed Fourier transform for f .t/.5. Let f .t/ D sin.�t/ and the window function g.t/ be the symmetrical hat

function:

g.t/ D

�1; �1 � t < 10; otherwise.

Show that the windowed Fourier transform of f .t/ is

Fg.t; !/ D i

�sin.� C !/

� C !�

sin.� � !/

� � !

�:

2.6 Exercises 91

6. For the Gaussian function g.t/ D1

p4�a

e�t2=4a, show that

(a)Z 1

�1

G Œf �.t; !/ dt D Of .!/; (b) Og.�/ D e�a�2 :

7. Find the Gabor transform of the following functions:(a) f .t/ D e�a2t2 with g.t/ D 1; (b) f .t/ D e�itb:

8. Suppose gt;!.�/ D g.� � t/ei!� where g is a Gaussian window defined inExercise 6, and show that

(a) Ogt;!.�/ D e�i.��!/t�a.��!/2 ; (b) G Œf �.t; !/ D1

2�

˝Of ; Ogt;!

˛:

9. For the Gaussian window defined in Exercise 6, introduce

�2t D1��g��2

�Z 1

�1

�2g2.�/ d�

� 1=2:

Show that the radius of the window function isp

a and the width of the windowis twice the radius.

10. Show that the marginals of the Zak transform are given by

Z 1

0

Z f

�.t; !/ d! D f .t/; and

Z 1

0

e�2� i!tZ f

�.t; !/ dt D Of .!/:

11. If f .t/ is time-limited to �a � t � a and band-limited to �b � ! � b, where

0 � a; b �1

2, then the following results hold:

(a)Z f

�.t; !/ D f .�/; j� j � 1

2; ! 2 R;

(b)Z f

�.t; !/ D e2� i!� Of .!/; j!j � 1

2; � 2 R:

Show that the second of the above results gives the Shannon sampling formula

f .t/ DX

n2Z

sin 2�b.n � t/

�.n � t/; t 2 R:

3The Wavelet Transforms

Wavelets are without doubt an exciting and intuitive concept. The concept brings with it anew way of thinking, which is absolutely essential and was entirely missing in previouslyexisting algorithms.

Yves Meyer

Wavelet transform has provided not only a wealth of new mathematical results, but also acommon language and rallying call for researchers in a remarkably wide variety of fields:mathematicians working in harmonic analysis because of the special properties of waveletbases; mathematical physicists because of the implications for time-frequency or phase-space analysis and relationships to concepts of renormalization; digital signal processorsbecause of connections with multirate filtering, quadrature mirror filters, and subbandcoding; image processors because of applications in pyramidal image representation andcompression; researchers in computer vision who have used scale-space for some time;researchers in stochastic processes interested in self-similar processes, 1=f noise, andfractals; speech processors interested in efficient representation, event extraction andmimicking the human auditory system. And the list goes on.

Ingrid Daubechies

3.1 Introduction

Historically, the concept of wavelets started to appear more frequently only in theearly 1980s. This new concept can be viewed as a synthesis of various ideas orig-inating from different disciplines including mathematics, physics, and engineering.One of the main reasons for the discovery of wavelets and wavelet transformsis that the Fourier transform analysis does not contain the local information ofsignals. So the Fourier transform cannot be used for analyzing signals in a jointtime and frequency domain. In 1982, Jean Morlet, in collaboration with a group


93

94 3 The Wavelet Transforms

of French engineers, first introduced the idea of wavelets as a family of functionsconstructed by using translation and dilation of a single function, called the motherwavelet, for the analysis of nonstationary signals (Morlet et al., 1982). Wavelettransforms are relatively recent developments that have fascinated the scientific,engineering, and mathematics community with their versatile applicability. Theapplication areas for wavelets have been growing for the last 20 years at a veryrapid rate. They have been applied in a number of fields including signal and imageprocessing, sampling theory, turbulence, differential equations, statistics, qualitycontrol, computer graphics, economics and finance, medicine, neural networks,geophysics, astrophysics, quantum mechanics, neuroscience, and chemistry. Formore information about the history and applications of wavelet transforms, thereader is referred to Daubechies (1992, 1993), Chui (l992), Meyer (1993a,b), Kaiser(1994), Cohen (1995), Hubbard (1996), Strang and Nguyen (1996), Burrus et al.(1997), Wojtaszczyk (1997), Debnath (1998a,b,c, 2001), Mallat (1998), Pinsky(2001), Ali et al. (2015), Gomes and Velho (2015), and Debnath and Shah (2015).

As we have already seen in Section 2.2. we cannot reduce the size of the time-frequency window beyond than that of Gabor transform whose frequency windowis rigid and does not vary over time or frequency. In other words, it means that anywindowed Fourier transform is not suitable for analyzing signals with both veryhigh and very low frequencies. Therefore, we have to modify the windowed Fouriertransform in a fundamental different way to achieve varying time and frequencywindows. The only way one can vary the size of the time window for differentdegrees of localization is by reciprocally varying the size of the frequency windowat the same time, so as to keep the area of the window constant. This means a trade-off between time and frequency localization. Thus, we must have a window functionwhose radius increases in time (reduces in frequency), while resolving the low-frequency contents, and decreases in time (increases frequency), while resolvinghigh-frequency contents of a signal. This is achieved by directly windowing thesignal instead of its Fourier transform and its Fourier transform instead of the inverseFourier transform and by scaling the window function appropriately to change itstime window width. This is the so-called wavelet transform.

This chapter is devoted to wavelets and wavelet transforms with examples. Thebasic ideas and properties of wavelet transforms are discussed with special attentiongiven to the use of different wavelets for resolution and synthesis of signals. Thisis followed by the discrete version of wavelet transform and the construction oforthonormal dyadic wavelets. Special attention is given to fairly exact mathematicaltreatment to the fractional wavelet transform (FrWT), and several important resultsincluding Parseval’s formula and inversion theorem are proved.

3.2 The Continuous Wavelet Transform 95

3.2 The Continuous Wavelet Transform

We first give the formal definition of a wavelet.

Definition 3.2.1 (Wavelet). A wavelet is a function 2 L2.R/ which satisfies thecondition

C

Z 1

�1

ˇˇ O .!/

ˇˇ2

j!jd! < 1; (3.2.1)

where O .!/ is the Fourier transform of .t/.

Condition (3.2.1) is called the admissibility condition which guarantees theexistence of the inversion formula for the continuous wavelet transform. From thiscondition it follows that O .!/ ! 0 as ! ! 0. Indeed, if O .!/ is continuous, thenO .0/ D 0, that is,

R1

�1 .t/ dt D 0, which implies that must be an oscillatoryfunction with zero mean. Thus, a wavelet is by necessity an oscillating function,real or complex-valued, and, in fact, should have good time localization propertiesso that it looks like a “small wave.” That is why is named by wavelet.

In addition to the admissibility condition, there are other properties that may beuseful in particular applications. For instance, restrictions on the support of andof O or may be required to have a certain number of vanishing moments thatrepresent the regularity of the wavelet functions and ability of a wavelet transformto capture localized information. A wavelet .t/ has n-vanishing moments if thefollowing condition is satisfied:

mk D

Z 1

�1

tk .t/ dt D 0; k D 0; 1; : : : ; n: (3.2.2)

Or, equivalently,

"dk O .!/

d!k

#

!D0

D 0; for k D 0; 1; : : : ; n: (3.2.3)

The number of vanishing moments is directly related to the regularity of the wavelet.Thus, a more regular wavelet has a greater number of vanishing moments. Anotherdesirable property of wavelets is the so-called localization property that helps tocapture the localized effects of a signal in the time domain as well as the frequencydomain. Localization and regularity (vanishing moments) are inversely related toeach other. Thus, wavelets with a large number of vanishing moments result in moreflatness when frequency ! is small.


Example 3.2.1 (The Haar Wavelet). The Haar wavelet is the first known waveletproposed by Hungarian mathematician Alfred Haar in 1910 (Haar, 1910). The Haarwavelet, being an odd rectangular pulse pair, is the simplest and oldest orthonormalwavelet with compact support. It is defined by

.t/ D

8ˆ<

ˆ:

1; 0 � t <1

2

�1;1

2� t < 1

0; otherwise:

(3.2.4)

It is obvious thatZ 1

�1

.t/ dt D 0;

Z 1

�1

ˇ .t/

ˇ2dt D 1:

This wavelet is very well localized in the time domain, but it is not continuous.Its Fourier transform O .!/ is calculated as follows:

O .!/ D

Z 12

0

e�i!tdt �

Z 1

12

e�i!tdt

D1

.�i!/

� e�i!t

� 12

0� e�i!t

�112

�

D

�i

!

��2e� i!

2 � 1 � e�i!�

Dsin2

�!4

�

�!4

� e

i.� � !/

2

D i e�

i!

2sin2

�!4

�

�!4

� (3.2.5)

and

Z 1

�1

ˇˇ O .!/

ˇˇ2

j!jd! D 16

Z 1

�1

j!j�3ˇˇsin

!

4

ˇˇ4

d! < 1: (3.2.6)

Hence, condition (3.2.1) is satisfied, so the Haar function defined by (3.2.4) is awavelet. Both .t/ and O .!/ are plotted in Figure 3.1. These figures indicate thatthe Haar wavelet has good time localization but poor frequency localization. The

functionˇˇ O .!/

ˇˇ is even, attains its maximum at the frequency !0 4:662, and


1

1

0 0 4p 8p 16p0.5t

-1

w

y(w)

‹

y (t )

Fig. 3.1 The Haar wavelet and its Fourier transform

decays slowly as !�1 as ! ! 1, which means that it does not have compactsupport in the frequency domain. Indeed, the discontinuity of causes a slowdecay of O as ! ! 1. Its discontinuous nature is a serious weakness in manyapplications. However, the Haar wavelet is one of the most fundamental examplesthat illustrate major features of the general wavelet theory.

The following theorem provides a method for constructing a new wavelet.

Theorem 3.2.1. If is a wavelet and � is a bounded integrable function, then theconvolution function � � is a wavelet.

Proof. Since

Z 1

�1

ˇ � �.x/

ˇ2dx D

Z 1

�1

ˇˇZ 1

�1

.x � u/ �.u/ du

ˇˇ2

dx

�

Z 1

�1

�Z 1

�1

ˇ .x � u/

ˇˇ�.u/

ˇdu

�2dx

D

Z 1

�1

�Z 1

�1

ˇ .x � u/

ˇˇ�.u/

ˇ 12ˇ�.u/

ˇ 12 du

�2dx

�

Z 1

�1

�Z 1

�1

ˇ .x � u/

ˇ2ˇ�.u/

ˇduZ 1

�1

ˇ�.u/

ˇdu

�dx

�

Z 1

�1

ˇ�.u/

ˇduZ 1

�1

Z 1

�1

ˇ .x � u/

ˇ2ˇ�.u/

ˇdx du

D

�Z 1

�1

ˇ�.u/

ˇdu

�2 Z 1

�1

ˇ .x/

ˇ2dx < 1;


we have � � 2 L2.R/. Moreover,

Z 1

�1

ˇF˚ � �

ˇ2ˇ!ˇ d! D

Z 1

�1

ˇˇ O .!/ O�.!/

ˇˇ2

ˇ!ˇ d!

D

Z 1

�1

ˇˇ O .!/

ˇˇ2

ˇ!ˇ

ˇˇ O�.!/

ˇˇ2

d!

�

Z 1

�1

ˇˇ O .!/

ˇˇ2

ˇ!ˇ d! sup

ˇˇ O�.!/

ˇˇ2

< 1:

Thus, the convolution function � � is a wavelet.

Example 3.2.2.

This example illustrates how to generate other wavelets by using Theorem 3.2.1. Forexample, if we take the Haar wavelet and convolute it with the following function

�.t/ D

8<

:

0; t < 0;1; 0 � t � 1

0; t � 1

; (3.2.7)

we obtain a simple wavelet, as shown in Figure 3.2.

Fig. 3.2 The wavelet � �

�.t/


Exercise 3.1. Show that the convolution of the Haar wavelet with �.t/ D e�t2

generates a wavelet.

Based on the idea of wavelets as a family of functions constructed from translationand dilation of a single function , called the mother wavelet, we define waveletsby

a;b.t/ D1pjaj

�t � b

a

�; a; b 2 R; a ¤ 0; (3.2.8)

where a is called a scaling parameter which measures the degree of compressionor scale and b is a translation parameter which determines the time location of thewavelet. Clearly, wavelets a;b.t/ generated by the mother wavelet are somewhatsimilar to the Gabor wavelets gt;!.�/ which can be considered as musical notes thatoscillate at the frequency! inside the envelope defined by

ˇg.�� t/

ˇas a function of

� . If jaj < 1, the wavelet (3.2.8) is the compressed version (smaller support in timedomain) of the mother wavelet and corresponds mainly to higher frequencies. Thus,wavelets have time widths adapted to their frequencies. It may be noted that theresolution of wavelets at different scales varies in the time and frequency domainsas governed by the Heisenberg uncertainty principle.

We sketch a typical mother wavelet with a compact support Œ�T;T� inFigure 3.3a. Different values of the parameter b represent the time localizationcenter, and each of a;b.t/ is localized around the center t D b. As scaleparameter a varies, wavelet a;b.t/ covers different frequency ranges. Small valuesof jaj.0 < jaj << 1/ result in very narrow windows and correspond to highfrequencies or very fine scales a;b as shown in Figure 3.3b, whereas very largevalues of jaj

jaj >> 1

�result in very wide windows and correspond to small

frequencies or very coarse scales a;b as shown in Figure 3.3c. Different shapes ofthe wavelets are plotted in Figure 3.3b, c.

Note that if 2 L2.R/, then a;b.t/ 2 L2.R/, and

�� a;b

��22

Dˇaˇ�1

Z 1

�1

ˇˇ �

t � b

a

�ˇˇ2

dt D

Z 1

�1

ˇ .x/

ˇ2dx D

�� 22: (3.2.9)

Moreover, the Fourier transform of a;b.t/ is given by

O a;b.!/ Dˇaˇ� 1

2

Z 1

�1

e�i!t

�t � b

a

�dt D

ˇaˇ 12 e�ib! O .a!/: (3.2.10)


ψa,b(t) ψa,b(t) ψa,b(t)

00 0

-T Tt

a b c

t t

Fig. 3.3 (a) Typical mother wavelet. (b) Compressed and translated wavelet a;b.t/ with0 < jaj << 1; b > 0. (c) Magnified and translated wavelet a;b.t/ with jaj >> 1; b > 0

Example 3.2.3 (The Mexican Hat Wavelet). The Mexican hat wavelet is definedby the second derivative of a Gaussian function as

.t/ D1� t2

�e�t2=2 D �

d2

dt2

�e�t2=2

�D 1;0.t/: (3.2.11)

Its Fourier transform is given by

O .!/ D O 1;0.!/ Dp2� !2e�!2=2: (3.2.12)

In contrast to the Haar wavelet, the Mexican hat wavelet is a C1-function. It hastwo vanishing moments. The Mexican hat wavelet 1;0.t/ and its Fourier transformare shown in Figure 3.4a, b. This wavelet has excellent localization in time andfrequency domains and clearly satisfies the admissibility condition.

Two other wavelets, 32 ;�2

and 14 ;

p2, from the mother wavelet (3.2.11) can

be obtained. These three wavelets 1;0.t/; 32 ;�2

.t/, and 14 ;

p2.t/ are shown in

Figure 3.5i, ii, iii, respectively.

Similar wavelets, with more vanishing moments, are obtained by taking higherderivatives of the Gaussian function:

.m/.t/ D

�1

i

d

dt

�m

e�t2=2; b .m/.!/ Dp2� !me�!2=2:

Definition 3.2.2 (Continuous Wavelet Transform). If 2 L2.R/ and a;b.t/ isgiven by (3.2.8), then the integral transformation W defined on L2.R/ by

W

f�.a; b/ D

˝f ; a;b

˛D

Z 1

�1

f .t/ a;b.t/ dt (3.2.13)

is called a continuous wavelet transform of f .t/.


3

2 2

3–

– 0

0

ω

ψ 1.0 (t)

ψ1.0 (ω)ˆ

a

b

t

Fig. 3.4 (a) The Mexican hat wavelet 1:0.t/ and (b) its Fourier transform O 1:0.!/

Fig. 3.5 Three wavelets 1;0.t/; 3

2 ;�2.t/, and 1

4 ;p

2.t/

ψa,b(t)

2

1.5

1 (ii) (i) (iii)

0.5

0

-0.5

-1

-6 -4 -2 0 2

t


This definition allows us to make the following comments:

1. The kernel a;b.t/ in (3.2.13) plays the same role as the kernel e�i!t in the Fouriertransform. Like the Fourier transformation, the continuous wavelet transforma-tion W is linear. However, unlike the Fourier transformation, the continuouswavelet transform is not a single transform but any transform obtained in thisway.

2. The continuous wavelet transform W is similar to the Fourier transform in thesense that it is based on a single function and that this function is scaled. But,unlike the Fourier transform, we also shift the function, thus generating a two-parameter family of functions a;b.t/ defined by (3.2.8).

3. As a function of b for a fixed scaling parameter a; W

f�.a; b/ represents the

detailed information contained in the signal f .t/ at the scale a.4. If the mother wavelet possesses n-vanishing moments, then expanding the

function f .t/ in (3.2.13) in a Taylor series at t D b, one obtains

W

f�.a; b/

D

"1pjaj

f .b/Z 1

�1

�t � b

a

�dt C f 0.b/

Z 1

�1.t � b/

�t � b

a

�dtC

f .2/.b/

2Š

Z 1

�1.t � b/2

�t � b

a

�dtC : : :C

f .n/.b/

nŠ

Z 1

�1.t � b/n

�t � b

a

�dt C : : :

#:

(3.2.14)

By moment condition (3.2.2), it follows that the first n-terms of (3.2.14)vanish, and as a consequence, they do not contribute to W

f�.a; b/.

5. In quantum mechanics, quantities such asˇ .t/

ˇ2and

ˇˇ O .!/

ˇˇ2

are interpreted as

the probability density functions in the time and frequency domains, respectively,with mean values defined by

t� D

Z 1

�1

tˇ .t/

ˇ2dt and !� D

1

2�

Z 1

0

!ˇˇ O .!/

ˇˇ2

d!: (3.2.15)

Then, the time resolution and frequency resolution associated with a motherwavelet around the mean values are defined by

�2t D

Z 1

�1

t � t�

�2ˇ .t/

ˇ2dt; and �2! D

1

2�

Z 1

0

! � !�

�2 ˇˇ O .!/ˇˇ2

d!;

(3.2.16)

respectively. Thus, the continuous wavelet transform W gives the localizedspectral information of f .t/ in the time window

hat� C b � a�t; at� C b C a�t

i: (3.2.17)


By applying the Parseval identity formula of the Fourier transform to (3.2.13)and using (3.2.10), we obtain

W

f�.a; b/ D

˝f ; a;b

˛D

1

2�

DOf ; O a;b

ED

1

2�

Z 1

�1

�pjaj Of .!/ O .a!/

�eib!d!:

This means that the continuous wavelet transform W also gives the localinformation of Of .!/ with frequency window

�!�

a��!

a;!�

aC�!

a

�: (3.2.18)

Thus, the time-frequency window for analyzing finite energy analog signals withwidth 2a�t is defined by

hat� C b � a�t; at� C b C a�t

i�

�!�

a��!

a;!�

aC�!

a

�: (3.2.19)

This width 2a�t of the time-frequency window is inversely proportional to thecenter frequency a�1!�, while the height (frequency window size) 2a�1�!is directly propositional to the center frequency. Thus, the area of the time-frequency window is equal to 4�t�! and is governed by the Heisenberg uncer-

tainty principle, that is, �t �! �1

2(See Figure 3.6).

Fig. 3.6 Time-frequency plane for the wavelet transform


The following theorem gives several properties of continuous wavelet transforms.

Theorem 3.2.2. If and � are wavelets and f and g are functions which belong toL2.R/, then

(i) Linearity: W

˛f Cˇg

�.a; b/ D ˛

W f

�.a; b/Cˇ

W g

�.a; b/; ˛; ˇ 2 R;

(ii) Translation:W .Tcf /

�.a; b/ D

W f

�.a; b � c/;

(iii) Dilation:�W .Dcf /

�.a; b/ D

1p

c

W f

� �a

c;

b

c

�; c > 0;

(iv) Symmetry:W f

�.a; b/ D

Wf

� �1a;�

b

a

�; a ¤ 0;

(v) Parity:WP Pf

�.a; b/ D

W f

�.a;�b/;

(vi) Antilinearity:W˛ Cˇ� f

�.a; b/ D ˛

W f

�.a; b/C ˇ

W� f

�.a; b/;

(vii)WTc f

�.a; b/ D

W f

�.a; b C ca/;

(viii)WDc f

�.a; b/ D

1p

c

W f

�.ac; b/; c > 0.

Proofs of the above properties are straightforward and are left as exercises.

Theorem 3.2.3 (Parseval’s Formula). If 2 L2.R/ andW f

�.a; b/ is the

wavelet transform of f defined by (3.2.13), then, for any functions f ; g 2 L2.R/,we obtain

˝f ; g˛D

1

C

Z 1

�1

Z 1

�1

�W f

�.a; b/

�W g

�.a; b/

db da

a2; (3.2.20)

where

C D

Z 1

�1

ˇˇ O .!/

ˇˇ2

ˇ!ˇ d! < 1: (3.2.21)

Proof. By Parseval’s relation (1.3.17) for the Fourier transforms, we have

�W f

�.a; b/ D

Z 1

�1

f .t/ jaj�12

�t � b

a

�dt

D˝f ; a;b

˛

D1

2�

DOf ; O a;b

E


D1

2�

Z 1

�1

Of .!/ jaj12 eib! O .a!/ d!: (3.2.22)

Similarly,

�W g

�.a; b/ D

Z 1

�1

g.t/ jaj�12

�t � b

a

�dt

D1

2�

Z 1

�1

Og.�/ jaj12 e�ib� O .a�/ d�: (3.2.23)

Substituting (3.2.22) and (3.2.23) in the left-hand side of (3.2.20) givesZ

1

�1

Z1

�1

�W f

�.a; b/

�W g

�.a; b/

db da

a2

D1

.2�/2

Z1

�1

Z1

�1

db da

a2

Z1

�1

Z1

�1

jaj Of .!/ Og.�/ O .a!/ O .a�/ exp˚ib.! � �/

d! d�;

which is, by interchanging the order of integration,

D1

2�

Z1

�1

da

jaj

Z1

�1

Z1

�1

Of .!/ Og.�/ O .a!/ O .a�/ d! d�1

2�

Z1

�1

exp˚ib.! � �/

db

D1

2�

Z1

�1

da

jaj

Z1

�1

Z1

�1

Of .!/ Og.�/ O .a!/ O .a�/ ı.� � !/ d! d�

D1

2�

Z1

�1

da

jaj

Z1

�1

Of .!/ Og.!/ˇˇ O .a!/

ˇˇ2

d!

which is, again by interchanging the order of integration and putting a! D x;

D1

2�

Z1

�1

Of .!/ Og.!/ d! �

Z1

�1

ˇˇ O .x/

ˇˇ2

jxjdx

D C �1

2�

DOf ; OgE:

Theorem 3.2.4 (Inversion Formula). If f 2 L2.R/, then f can be reconstructedby the formula

f .t/ D1

C

Z 1

�1

Z 1

�1

�W f

�.a; b/ a;b.t/

db da

a2; (3.2.24)

where the equality holds almost everywhere.


Proof. For any g 2 L2.R/, we have, from Theorem 3.2.3,

C ˝f ; g˛DDW f ;W g

E

D

Z 1

�1

Z 1

�1

�W f

�.a; b/

�W g

�.a; b/

db da

a2

D

Z 1

�1

Z 1

�1

�W f

�.a; b/

Z 1

�1

g.t/ a;b.t/ dtdb da

a2

D

Z 1

�1

Z 1

�1

Z 1

�1

�W f

�.a; b/ a;b.t/

db da

a2g.t/ dt

D

�Z 1

�1

Z 1

�1

�W f

�.a; b/ a;b.t/

db da

a2; g

�: (3.2.25)

Since g is an arbitrary element of L2.R/, the inversion formula (3.2.24) follows.

If f D g in (3.2.20), then

Z 1

�1

Z 1

�1

ˇˇ�W f

�.a; b/

ˇˇ2 da db

a2D C

��f��22

D C

Z 1

�1

ˇf .t/

ˇ2dt: (3.2.26)

Thus, we conclude that except for the factor C , the continuous wavelet transformW is an isometry from L2.R/ to L2.R2/.

3.3 The Discrete Wavelet Transform

It has been stated in the last section that the continuous wavelet transform (3.2.13)is a two-parameter representation of a function. In many applications, especially insignal processing, data are represented by a finite number of values, so it is importantand often useful to consider discrete versions of the continuous wavelet transform(3.2.13). However, unlike the discretized time and frequency axes shown earlier inFourier analysis, here we take the discrete values of the scale parameter a and thetranslation parameter b in a different way. For convenience in the discretization, werestrict a to positive values only, so that the admissibility condition (3.2.1) becomes

Z 1

�1

ˇO .!/

ˇ2

j!jd! D 2

Z 1

0

ˇO .!/

ˇ2

j!jd! < 1:

First, we choose a D am0 , where m 2 Z and the dilation step a0 ¤ 1 is fixed. Then,

for m D 0, it becomes natural as well to discretize b by taking only the integermultiples of one fixed b0, where b0 is approximately chosen so that the .t � nb0/

3.3 The Discrete Wavelet Transform 107

cover the whole line. For different values of m, the width a�m=20

a�m0 t

�is am

0 timesthe width of .t/, so that the choice b D nb0am

0 ;m; n 2 Z will ensure that thediscretized wavelets at level m “cover” the whole line in the same way as the .t �

nb0/ do. Thus, we choose a D am0 ; b D nb0am

0 ; where the two positive constants a0and b0 are fixed. With these choices of a and b, the continuous family of wavelets a;b as defined in (3.2.8) becomes

m;n.x/ D a�m=20

�t � nb0am

0

am0

�D a�m=2

0 a�m0 x � nb0

�; (3.3.1)

where both m and n 2 Z. Then, for f 2 L2.R/, we calculate the discretewavelet coefficients hf ; m;ni. The fundamental question is whether it is possibleto determine f completely by its wavelet coefficients or discrete wavelet transform(DWT) which is defined by

�W f

�.m; n/ D

˝f ; m;n

˛D a�m=2

0

Z 1

�1

f .t/ .a�m0 t � nb0/ dt; (3.3.2)

where both f and are continuous and 0;0.t/ D .t/. Thus, the discrete wavelettransform represents a function by a countable set of wavelet coefficients, whichcorrespond to points on a two-dimensional grid or lattice of discrete points in thescale time domain indexed by m and n. If the set

˚ m;n.t/ W m; n 2 Z

defined by

(3.3.1) is complete in L2.R/ for some choice of ; a; and b, then the set is called anaffine wavelet. Then, we can express any f 2 L2.R/ as the superposition

f .t/ DX

m2Z

X

n2Z

˝f ; m;n

˛ m;n.t/: (3.3.3)

Such complete sets are called frames. They are not yet a basis. Frames do not satisfythe Parseval theorem for the Fourier series, and the expansion in terms of frames isnot unique. In fact, it can be shown that

A��f��22

�X

m2Z

X

n2Z

ˇ˝f ; m;n

˛ˇ2� B

��f��22; (3.3.4)

where A and B are constants. The set˚ m;n.t/ W m; n 2 Z

constitutes a frame if .t/

satisfies the admissibility condition and 0 < A < B < 1. For further informationand recent developments, the reader is referred to Chui and Shi (1993), Shah (2013,2016), Shah and Abdullah (2014a,b), and Shah and Debnath (2011a,b, 2012, 2013).

It is important to note that the wavelet coefficients˚˝

f ; m;n˛

m;n2Zin the wavelet

series expansion (3.3.3) of a function are nothing but the integral wavelet transformof the function evaluated at certain discrete points

am0 ; nb0am

0

�. No such relationship

exists between Fourier series and Fourier transform which are applicable to different


ωσt

σω

t

m = –3

m = –2

m = –1

m = 0n = 0 n = 1 n = 2

Fig. 3.7 Dyadic sampling grid for the discrete wavelet transform

classes of functions. Fourier series applies to functions that are square integrable inŒ0; 2��, whereas Fourier transform is for functions that are in L2.R/. On the otherhand, both wavelet series and wavelet transform are applicable to functions in L2.R/.

For computational efficiency, a0 D 2 and b0 D 1 are commonly used so thatresults lead to a binary dilation of 2�m and a dyadic translation of n 2m. Therefore,a practical sampling lattice is a D 2m and b D n 2m in (3.3.1) so that

m;n.t/ D 2�m=2 .2�mt � n/ : (3.3.5)

With this octave time scale and dyadic translation, the sampled values of .a; b/ D

.2m; n2m/ are shown in Figure 3.7, which represents the dyadic sampling griddiagram for the discrete wavelet transform. Each node corresponds to a waveletbasis function m;n.t/ with scale 2�m and time shift n 2�m.

The answer to the preceding question is positive if the wavelets form a completesystem in L2.R/. The problem is whether there exists another function g 2 L2.R/such that

˝f ; m;n

˛D˝g; m;n

˛for all m; n 2 Z

implies f D g. In practice, we expect much more than that: we want hf ; m;ni andhg; m;ni to be “close” if f and g are “close.” This will be guaranteed if there existsa B > 0 independent of f such that

X

m2Z

X

n2Z

ˇ˝f ; m;n

˛ˇ2� B

��f��22: (3.3.6)

3.4 Orthonormal Wavelets 109

Similarly, we want f and g to be “close” if hf ; m;ni and hg; m;ni are “close.” This isimportant because we want to be sure that when we neglect some small terms in therepresentation of f in terms of hf ; m;ni, the reconstructed function will not differmuch from f . The representation will have this property if there exists an A > 0

independent of f , such that

A��f��22

�X

m2Z

X

n2Z

ˇ˝f ; m;n

˛ˇ2: (3.3.7)

Combining (3.3.6) with (3.3.7), we obtain the following equation:

A��f��22

�X

m2Z

X

n2Z

ˇ˝f ; m;n

˛ˇ2� B

��f��22: (3.3.8)

This ensures that the DWT of a signal f .t/ can be obtained. Equation (3.3.8) iscalled a wavelet frame. The largest A and the smallest B for which (3.3.8) holds arecalled wavelet frame bounds. A wavelet frame is a tight wavelet frame if A and Bare chosen such that A D B and a normalized tight frame if A D B D 1. The valuesof the wavelet frame bounds, A and B, depend on both the scale parameter a and thetranslation parameter b that are chosen for analysis and the base wavelet functionused. Detailed results may be found in Daubechies (1992), Chui and Shi (1993), andDaubechies et al. (1986).

3.4 Orthonormal Wavelets

Since the discovery of wavelets, orthonormal wavelets with good time-frequencylocalization are found to play an important role in wavelet theory and have a greatvariety of applications. In general, the theory of wavelets begins with a singlefunction 2 L2.R/, and a family of functions m;n is generated from this singlefunction by the operation of binary dilations (i.e., dilation by 2m) and dyadictranslation of n2�m so that

m;n.t/ D 2m=2 2mt � n

�: (3.4.1)

A situation of interest in applications is to deal with an orthonormal family˚ m;n W m; n 2 Z

, that is,

˝ m;n; k;`

˛D

Z 1

�1

m;n.t/ k;`.t/ dt D ım;k ın;`; 8 m; n; k; ` 2 Z: (3.4.2)

To show how the inner products behave in this formalism, we prove the followinglemma.


Lemma 3.4.1. If and � 2 L2.R/, then

˝ m;k; �m;`

˛D˝ n;k; �n;`

˛; 8 m; n; k; ` 2 Z: (3.4.3)

Definition 3.4.1 (Orthonormal Wavelet). A wavelet 2 L2.R/ is calledorthonormal if the family of functions m;n generated from given by (3.4.1) isorthonormal.

As in the classical Fourier series, the wavelet series for a function f 2 L2.R/based on a given orthonormal wavelet is given by

f .x/ DX

m2Z

X

n2Z

cm;n m;n.t/; (3.4.4)

where the wavelet coefficients cm;n are given by

cm;n D˝f ; m;n

˛(3.4.5)

and the double wavelet series (3.4.4) converges to the function f in the L2-norm.

The simplest example of an orthonormal wavelet is the classic Haar wavelet(3.2.4). To prove this fact, we note that the norm of defined by (3.2.4) is one andthe same for m;n defined by (3.4.1). We have

˝ m;n; k;`

˛D

Z 1

�1

2m=2 2mx � n

�2k=2

2kx � `

�dx

D 2k=2 2�m=2Z 1

�1

.t/ 2k�m.t C n/� `

�dt: (3.4.6)

For m D k, this result gives

˝ m;n; m;`

˛D

Z 1

�1

.t/ .t C n � `/ dt D ı0;n�` D ın;`; (3.4.7)

where .t/ ¤ 0 in Œ0; 1� and .t � ` � n/ ¤ 0 in Œ` � n; 1 C ` � n/, and theseintervals are disjoint from each other unless n D `.

3.5 The Fractional Wavelet Transform 111

We now consider the case m ¤ k. In view of symmetry, it suffices to considerthe case m > k: Putting r D m � k > 0 in (3.4.6), we can complete the proof byshowing that, for k ¤ m,

˝ m;n; k;l

˛D 2r=2

Z 1

�1

.t/ 2rt C s

�dt D 0; (3.4.8)

where s D 2rn � ` 2 Z.

In view of the definition of the Haar wavelet , we must prove that the integralin (3.4.8) vanishes for k ¤ m. In other words, it suffices to show

Z 12

0

2rt C s

�dt �

Z 1

12

2rt C s

�dt D 0:

Invoking a simple change of variables, 2rt C s D x, we find

Z a

s .x/ dx �

Z b

a .x/ dx D 0; (3.4.9)

where a D s C 2r�1 and b D s C 2r. A simple argument reveals that Œs; a� containsthe support [0,1] of so that the first integral in (3.4.9) is identically zero. Similarly,the second integral is also zero. This completes the proof that the Haar wavelet isorthonormal.

3.5 The Fractional Wavelet Transform

Wavelet transforms serve as an important and powerful analyzing tool for time-frequency analysis and have been applied in a number of fields including signalprocessing, image processing, sampling theory, differential and integral equations,quantum mechanics, and medicine. However, the signal analysis capability of thewavelet transform is limited in the time-frequency plane as each wavelet componentis actually a differently scaled bandpass filter in the frequency domain, and hence,it does not serve as an efficient tool for processing those signals whose energy is notwell concentrated in the frequency domain. One of the examples of such signal ischirp-like signals. Recently, researchers have come up with the new mathematicaltransforms to analyze such signals, namely, fractional Fourier transform (FrFT),the short-time fractional Fourier transform (STFrFT), the Radon-Wigner transform


(RWT), the ridgelet transform (RT), etc. Besides lot of advantages, the FrFThas one major drawback due to using global kernel i.e., the fractional Fourierrepresentation only provides such FrFT spectral content with no indication aboutthe time localization of the FrFT spectral components. Therefore, in order toanalyze nonstationary signals whose FrFT spectral characteristics change withtime, researchers have come up with the short-time FrFT which provides a jointrepresentation of a signal in both time and FrFT domains, rather than just a FrFTdomain representation. The basic idea behind this transform is segmenting the signalby using a time-localized window and then performing the FrFT spectral analysisfor each segment. Although STFrFT has rectified almost all the limitations of FrFT,still in some cases STFrFT is also not applicable as in the case of real signalshaving high spectral components for short durations and low spectral componentsfor long durations. Hence, it is desirable to derive more general approach thanthe STFrFT in order to obtain joint signal representations in both time and FrFTdomains. An important way to analyze time-varying FrFT spectra is the fractionalwavelet transform (FrWT). The FrWT inherits the excellent mathematical propertiesof wavelet transform and FrFT along with some fascinating properties of its own.These properties make FrWT a useful mathematical tool in signal and imageprocessing with numerous advantages over conventional wavelet transform.

As a generalization of the wavelet transform, Mendlovic et al. (1997) firstintroduced the FrWT as a way to deal with optical signals. The idea behind thistransform is deriving the fractional spectrum of the signal by using the FrFTand performing the wavelet transform of the fractional spectrum. Besides being ageneralization of the wavelet transform, the FrWT can be interpreted as a rotationof the time-frequency plane and has been proved to relate to other time-varyingsignal analysis tools, which make it as a unified time-frequency transform. In recentyears, this transform has been paid a considerable amount of attention, resulting inmany applications in the areas of optics, quantum mechanics, pattern recognition,and signal processing.

Definition 3.5.1. A fractional wavelet is a function 2 L2.R/ which satisfies thefollowing condition:

C˛ D

Z

R

ˇˇF ˛

ne�i.t�!/2=2 cot˛

o.!/

ˇˇ2

j!jd! < 1; (3.5.1)

where F ˛ denotes the FrFT operator.

Example 3.5.1 (Morlet Wavelet.). This function is defined by

.t/ D exp˚iat � t2=2

: (3.5.2)


Its fractional Fourier transform O ˛.!/ is given by

F ˛n .t/

o

D C˛

Z1

�1

exp

��it2 C !2

� cot˛

2� it! csc ˛

� .t/dt

D C˛ exp

�i

2!2 cot˛

� Z1

�1

exp

��

�1

2�

i

2cot˛

�t2 C t .i! csc ˛ � i˛/

�dt:

D C˛ exp�

i

2!2 cot˛

�exp

��.a � ! csc ˛/2

2.1 � i cot˛/

�p�

�1 � i cot˛

2

��1=2

D exp�

i

2

�!2 � .a sin ˛ � !/2

�cot˛ �

1

2.a sin ˛ � !/2

�:

It can easily be verified that .t/ satisfies the admissibility condition (3.5.1).

Definition 3.5.2. The fractional convolution of two functions f ; g 2 L1.R/ isdefined by

f .t/ �˛ g.t/ D

Z 1

�1

f .!/g.t � !/ exp

��i

2

�t2 � !2

�cot˛

�d!; (3.5.3)

where �˛ denotes the fractional convolution operator.

For any two bounded integrable functions .t/ and �.t/, we set

Q .t/ D eit2 cot˛=2 .t/ and Q�.t/ D eit2 cot˛=2�.t/:

Then, the FrFT of the function

h.t/ D e�it2 cot˛=2 Q �˛ Q��.t/ (3.5.4)

is given by

Oh˛.!/ D e�i!2 cot˛=2 O ˛.!/ O�˛.!/: (3.5.5)

Theorem 3.5.1. If Q is a fractional wavelet and Q� is a fractional boundedintegrable function, then the convolution Q �˛ Q� is a fractional wavelet.


Proof. SinceZ

1

�1

ˇˇe�it2 cot˛=2

Q �˛ Q�

�.t/ˇˇ2

dt D

Z1

�1

ˇQ �˛ Q�

�.t/ˇ2

dt

D

Z1

�1

ˇˇZ

1

�1

Q .t � !/ Q�.!/ d!

ˇˇ2

dt

�

Z1

�1

�Z1

�1

ˇQ .t � !/

ˇ ˇQ�.!/

ˇd!

� 2dt

D

Z1

�1

�Z1

�1

ˇQ .t � !/

ˇ ˇQ�.!/

ˇ1=2 ˇ Q�.!/ˇ1=2

d!

� 2dt

�

Z1

�1

�Z1

�1

ˇQ .t � !/

ˇ2 ˇ Q�.!/ˇd!

Z1

�1

ˇQ�.!/

ˇd!

�dt

�

Z1

�1

ˇQ�.!/

ˇd!

Z1

�1

Z1

�1

ˇQ .t � !/

ˇ2 ˇ Q�.!/ˇd! dt

D

�Z1

�1

ˇQ�.!/

ˇd!

� 2 Z1

�1

ˇQ .t/

ˇ2dt < 1:

Therefore, it follows that e�it2 cot˛=2

Q �˛ Q��.t/ 2 L2.R/. Moreover,

Z 1

�1

ˇˇF ˛

ne�it2 cot˛=2

Q �˛ Q�

�oˇˇ2

j!jd!

D

Z 1

�1

ˇˇe�i!2 cot˛=2 O ˛.!/ O�˛.!/

ˇˇ2

j!jd!

� 2�ˇsin ˛

ˇ Z 1

�1

ˇˇe�i!2 cot˛=2 O ˛.!/

ˇˇ2

j!jd! sup

ˇˇ O�˛.!/

ˇˇ2

< 1:

Thus, the convolution function Q �˛ Q� is a fractional wavelet.

Example 3.5.2 (Fractional Mexican Hat Wavelet.). If

.t/ D exp

�it2 cot˛

2

�and �.t/ D

t2p2�

exp

��t2.1C 2i/ cot˛

2

�:

Then, in view of Theorem 3.5.1, the convolution function

� �˛

�.t/ D

Z 1

0

�.!/ .t � !/ exp

��i.t2 � !2/ cot˛

2

�d!

D tan3=2 ˛1 � t2 cot˛

�exp

��t2 cot˛

2

�

forms a fractional wavelet.


Analogous to the classical wavelets, the fractional wavelets can be obtained froma fractional mother wavelet 2 L2.R/ by the combined action of translation anddilations as

˛a;b.t/ D1

pa

�t � b

a

�exp

��i.t2 � b2/ cot˛

2

�(3.5.6)

where a 2 RC and b 2 R are scaling and translation parameters, respectively.

Note that if .t/ 2 L2.R/, then ˛a;b.t/ 2 L2.R/ as

�� ˛a;b��22

D jaj�1Z 1

�1

ˇˇ �

t � b

a

�ˇˇ2

dt D

Z 1

�1

ˇ .y/

ˇ2dy D

�� 22:

Moreover, the fractional Fourier transform of ˛a;b.t/ is given by

F ˛˚ ˛a;b.t/

Dp

a exp

�i.b2 C !2/ cot˛

2� ib! csc ˛ �

ia2!2 cot˛

2

�F ˛

ne�i.�/2 cot˛=2

o.a!/:

(3.5.7)

Definition 3.5.3 (Continuous Fractional Wavelet Transform). If .t/ 2 L2.R/and ˛a;b.t/ 2 L2.R/ is given by (3.5.6), then the integral transformation W ˛

definedon L2.R/ by

W ˛ Œf �.a; b/ D

˝f ; ˛a;b

˛D

1p

a

Z 1

�1

f .t/

�t � b

a

�exp

�i.t2 � b2/ cot˛

2

�dt

(3.5.8)

is called a continuous fractional wavelet transform (FrWT) of f .t/.

This definition allows us to make the following comments:

1. For ˛ D�

2, the FrWT coincides with the conventional continuous wavelet

transform.2. Using the Parseval relation of the fractional Fourier transform, it also follows

from (3.5.8) that


W ˛ Œf �.a; b/ D

Df ; ˛a;b

EDDOf ˛; O ˛a;b

E

Dp

aZ 1

�1

exp

��i.b2 C !2/ cot˛

2C ib! csc ˛ C

ia2!2 cot˛

2

�

� F ˛


o.a!/Of ˛.!/ d!: (3.5.9)

3. The FrWT (3.5.8) can be expressed in terms of the ordinary convolution as

W ˛ Œf �.a; b/ D exp

��ib2 cot˛

2

� �e�i.�/2 cot˛=2f � a

�.b/; (3.5.10)

whereas in terms of the fractional convolution, it takes the form

W ˛ Œf �.a; b/ D

�f �˛ a

�.b/; (3.5.11)

where a.t/ D1

pa ��t

a

�and �˛ denote the fractional convolution operator.

Theorem 3.5.2. If 1; 2 2 L2.R/ and�W ˛ 1

f�.a; b/ and

�W ˛ 2

g�.a; b/ denote

the continuous fractional wavelet transforms of f ; g 2 L2.R/, respectively, then

Z 1

�1

Z 1

0

�W ˛ 1

f�.a; b/

�W ˛ 2

g�.a; b/

dadb

a2D 2� sin ˛ C˛

1; 2

˝f ; g˛; (3.5.12)

where

C˛ 1; 2

D

Z 1

0

F ˛

ne�i.�/2 cot˛=2 1

o.a/F ˛

ne�i.�/2 cot˛=2 2

o.a/

da

a< 1:

(3.5.13)

Proof. We have

�W ˛ 1

f�.a; b/ D

paZ 1

�1

exp

��i.b2 C !2/ cot˛

2C ib! csc ˛ C

ia2!2 cot˛

2

�

� F ˛

ne�i.�/2 cot˛=2 1

o.a!/Of ˛.!/ d!

and

�W ˛ 2

g�.a; b/ D

paZ 1

�1

exp

�i.b2 C �2/ cot˛

2� ib � csc ˛ �

ia2�2 cot˛

2

�

� F ˛ne�i.�/2 cot˛=2 2

o.a�/Og˛.�/ d�:


NowZ

1

�1

Z1

0

�W ˛ 1

f�.a; b/

�W ˛ 2

g�.a; b/

dadb

a2

D

Z1

�1

Z1

0

" Z1

�1

exp

��i.b2 C !2/ cot˛

2C ib! csc ˛ C

ia2!2 cot˛

2

�

� F ˛ne�i.�/2 cot ˛=2 1

o.a!/Of ˛.!/ d!

#

�

" Z1

�1

exp

�i.b2 C �2/ cot˛

2� ib � csc ˛ �

ia2�2 cot˛

2

�

� F ˛ne�i.�/2 cot ˛=2 2

o.a�/Og˛.�/ d�

#dadb

a

D

Z1

�1

Z1

0

Z1

�1

exp

�i.�2 � !2/ cot˛ C ia2.!2 � �2/ cot˛

2

� �Z1

�1

eib.!��/ csc ˛db

�

�F ˛

ne�i.�/2 cot ˛=2 1

o.a!/Of ˛.!/ � F ˛

ne�i.�/2 cot ˛=2 2

o.a�/Og˛.�/ d!d�

da

a

D 2� sin˛Z

1

�1

Z1

0

Z1

�1

exp

�i.�2 � !2/ cot˛ C ia2.!2 � �2/ cot˛

2

�ı.! � �/

�F ˛ne�i.�/2 cot ˛=2 1

o.a!/Of ˛.!/ � F ˛

ne�i.�/2 cot ˛=2 2

o.a�/Og˛.�/ d!d�

da

a

D 2� sin˛Z

1

�1

Z1

0

exp

�i!2.a2 � 1/ cot˛

2

� "Z1

�1

exp

��i.a2 � 1/�2 cot˛

2

�

�F ˛ne�i.�/2 cot ˛=2 2

o.a�/Og˛.�/ ı.!��/ d�

#F ˛

ne�i.�/2 cot ˛=2 1

o.a!/Of ˛.!/ d!

da

a

D 2� sin˛Z

1

�1

Of ˛.!/ Og˛.!/

�

"R

1

0F ˛

ne�i.�/2 cot ˛=2 1

o.a!/F ˛

ne�i.�/2 cot ˛=2 2

o.a!/ da

a

#d!

D 2� sin˛ C˛ 1; 2

Z1

�1

Of ˛.!/ Og˛.!/ d!

D 2� sin˛ C˛ 1; 2

DOf ˛; Og˛

E

D 2� sin˛ C˛ 1; 2

˝f ; g˛:


Corollary 3.5.1. If is a wavelet and�W ˛ f�.a; b/ and

�W ˛ g�.a; b/ are the

continuous fractional wavelet transforms of f ; g 2 L2.R/, then

Z 1

�1

Z 1

0

�W ˛ f�.a; b/

�W ˛ g�.a; b/

dadb

a2D 2� sin ˛ C˛

˝f ; g˛: (3.5.14)

Proof. The proof of Corollary 3.5.1 can be easily deduced by setting 1 D 2 D

in the Theorem 3.5.2.

Remark. If f D g in (3.5.14), then

Z 1

�1

Z 1

0

ˇˇ�W ˛ f�.a; b/

ˇˇ2 dadb

a2D 2� sin ˛ C˛

��f��22: (3.5.15)

Theorem 3.5.3 (Inversion Formula). If f 2 L2.R/, then f can be constructed byusing the formula

f .t/ D1

2� sin ˛ C˛

Z 1

�1

Z 1

0

�W ˛ f�.a; b/ ˛a;b.t/

dadb

a2: (3.5.16)

Proof. For any f ; g 2 L2.R/, we have from Corollary 3.5.1

2� sin ˛C˛

˝f ; g˛D

Z 1

�1

Z 1

0

�W ˛ f�.a; b/

�W ˛ g�.a; b/

dadb

a2

D

Z 1

�1

Z 1

0

�W ˛ f�.a; b/

�Z 1

�1

g.t/ ˛a;b.t/ dt

�dadb

a2

D

Z 1

�1

�Z 1

�1

Z 1

0

�W ˛ f�.a; b/ ˛a;b.t/

dadb

a2

�g.t/ dt

D

�Z 1

�1

Z 1

0

�W ˛ f�.a; b/ ˛a;b.t/

dadb

a2; g.t/

�:

Since g is an arbitrary element of L2.R/, the inversion formula (3.5.16) follows. ut


Remark. Using the fractional Fourier transform on both sides of (3.5.16), we obtain

Of ˛.!/ D1

2� sin ˛ C˛

Z 1

�1

Z 1

0

�W ˛ f�.a; b/ O ˛a;b.!/

dadb

a2: (3.5.17)

Theorem 3.5.4. If 2 L2.R/ and�W ˛ f�.a; b/ and

�W ˛ g�.a; b/ are the

continuous fractional wavelet transforms of f ; g 2 L2.R/, then

Z 1

�1

h �W ˛ f�.a; b/

�W ˛ g�.a; b/

idb D 2�a sin˛

˝R˛; S˛

˛(3.5.18)

where R˛ and S˛ are, respectively, defined as

R˛.!/ D Of ˛.!/ exp

�ia2!2 cot˛

2

�F ˛


o.a!/; (3.5.19)

S˛.!/ D Og˛.!/ exp

��ia2!2 cot˛

2

�F ˛


o.a!/: (3.5.20)

Proof. We have

Z 1

�1

n �W ˛ f�.a; b/

�W ˛ g�.a; b/

odb

D

Z 1

�1

DOf ˛; O ˛a;b

E ˝Og˛; O ˛a;b

˛db

D

Z 1

�1

�Z 1

�1

Of ˛.!/ O ˛a;b.!/ d!

� �Z 1

�1

Og˛.�/ O ˛a;b.�/ d�

�db

D

Z 1

�1

a

"Z 1

�1

exp

��i.b2 C !2/ cot˛

2C ib! csc ˛ C

ia2!2 cot˛

2

�

� F ˛


o.a!/Of ˛.!/ d!

#

�

"Z 1

�1

exp

�i.b2 C �2/ cot˛

2� ib� csc ˛ �

ia2�2 cot˛

2

�

� F ˛ne�i.�/2 cot˛=2

o.a�/Og˛.�/ d�

#db


D aZ

1

�1

1

C ˛

"C˛

Z1

�1

exp

�i.b2 C !2/ cot ˛

2� ib! csc ˛

�� R˛.!/ d!

#

�

�C˛

Z1

�1

exp

�i.b2 C �2/ cot ˛

2� ib� csc ˛

�� S˛.�/ d�

�db

Da

C˛C˛

Z1

�1

�OR˛.b/

� �OS˛.b/

�db

D 2�a sin ˛DOR˛; OS˛

E

D 2�a sin˛DOR˛; OS˛

E

D 2�a sin˛ hR˛; S˛i :

This completes the proof of the theorem.

For more detailed information on fractional wavelet transforms and their applica-tions, the reader is referred to Huang and Suter (1998), Chen and Zhao (2005), Dincet al. (2011), Shi et al. (2012, 2013), Prasad and Mahato (2012), Bhatnagar et al.(2013), Prasad et al. (2014), and Prasad and Kumar (2015).

3.6 Exercises

1. Discuss the scaled and translated versions of the mother wavelet .t/ D t exp

�t2

�.

2. Show that the Fourier transform of the normalized Mexican hat wavelet

.t/ D2

�14

p3a

�1 �

t2

a2

�exp

��

t2

2a2

�

is

O .!/ D

r8

3a5=2 �1=4 !2 exp

��

a2!2

2

�:

3. Show that the continuous wavelet transform can be expressed as a convolution,that is,

W Œf �.a; b/ Df � a

�.b/;

3.6 Exercises 121

where

a.t/ D1

pa ��

t

a

�:

What is the physical significance of the convolution?

4. If f is a homogeneous function of degree n, show that

�W f

�.�a; �b/ D �nC 1

2

�W f

�.a; b/:

5. Show that

Z 1

�1

sin�x

�x�

sin�.2x � n/

�.2x � n/dx D

1

2�nsin�n�

2

�:

6. Show that the Fourier transform of one cycle of the sine function

f .t/ D sin t; jtj < �I

is

Of .!/ D2i

.!2 � 1/sin�!:

7. For the Shannon wavelet

.t/ Dsin��t

2

�

��t

2

� cos

�3�t

2

�;

show that its Fourier transform is

O .!/ D

�1; � < j!j < 2�

0; otherwise

8. Show that the Fourier transform of the wave train

f .t/ D1

p2�

1

�exp

��

t2

2�2

�cos!0t


is

Of .!/ D1

2

�exp

��2

2.! � !0/

2

�C exp

��2

2.! C !0/

2

��:

Explain the physical features of Of .!/.9. Show that the Fourier transform of

f .t/ D1

p2�

�a.t/ ei!0t

is

Of .!/ D

r2

�

sin a.! � !0/

�

.! � !0/:

Explain the features of Of .!/.10. If

�t � b

a

�D

8ˆ<

ˆ:

1; b � t < b Ca

2�1; b C

a

2� t < b C a

0; otherwise;

where a > 0, show that

W Œf �.a; b/ D1

pa

Z bC a2

b

�f .t/ � f

�t C

a

2

� �dt:

11. Suppose 1 and 2 are two wavelets and the integral

Z 1

�1

O 1.!/ O 2.!/

j!jd! D C 1 2 < 1:

If W 1Œf �.a; b/ and W 2Œf �.a; b/ denote wavelet transforms, show that

DW 1 f ;W 2g

ED C 2 2

˝f ; g˛;

where f ; g 2 L2.R/:

4Construction of Wavelets via MRA

Multiresolution analysis provides a natural framework for the understanding of waveletbases, and for the construction of new examples. The history of the formulation ofmultiresolution analysis is a beautiful example of applications stimulating theoreticaldevelopment.

Ingrid Daubechies

Today the boundaries between mathematics and signal and image processing have faded,and mathematics has benefited from the rediscovery of wavelets by experts from otherdisciplines. The detour through signal and image processing was the most direct pathleading from Haar basis to Daubechies’s wavelets.

Yves Meyer

4.1 Introduction

The idea of multiresolution analysis (MRA) was proposed by Stéphane Mallat andYves Meyer in 1986, and this can be considered as a rebirth of wavelet theory. Thisis a new and remarkable idea which deals with a general formalism for constructionof an orthogonal basis of wavelets. Indeed, MRA is central to all constructions ofwavelet bases. Mallat’s brilliant work (Mallat 1989a,b,c) has been the major sourceof many new developments in wavelet analysis and its wide variety of applications.

Mathematically, the fundamental idea of MRA is to represent a function f as alimit of successive approximations, each of which is a finer version of the functionf . These successive approximations correspond to different levels of resolutions.Thus, MRA is a formal approach to constructing orthogonal wavelet bases using adefinite set of rules and procedures. The key feature of this analysis is to describe


123

124 4 Construction of Wavelets via MRA

mathematically the process of studying signals or images at different scales. Thebasic principle of the MRA deals with the decomposition of the whole functionspace into individual subspaces Vm � VmC1 so that the space VmC1 consists of allrescaled functions in Vm. This essentially means a decomposition of each functioninto components of different scales so that an individual component of the originalfunction f occurs in each subspace.

This chapter deals with the idea of MRA with examples. Special attention isgiven to properties of scaling functions and orthonormal wavelet bases. This isfollowed by a method of constructing orthonormal bases of wavelets from an MRA.In the end, the fast wavelet transform (FWT) is briefly discussed.

4.2 Multiresolution Analysis in L2.R/

Definition 4.2.1 (Multiresolution Analysis). A multiresolution analysis (MRA)of L2.R/ is a sequence fVm W m 2 Zg of closed subspaces of L2.R/ satisfying thefollowing properties:

(i) Vm � VmC1; for all m 2 ZI

(ii)S

m2Z

Vm is dense in L2.R/;

(iii)T

m2Z

Vm D f0g I

(iv) f .t/ 2 Vm if and only if f .2t/ 2 VmC1 for all m 2 Z;(v) there is a function � in V0 such that the system f�.t � n/ W n 2 Zg forms an

orthonormal basis for V0:

The function � whose existence is asserted in (v) is called a scaling function orfather wavelet of the given MRA.

Consequences of Definition 4.2.1

1. A repeated application of condition (iv) implies that f 2 Vm if and only if f .2kt/ 2

VmCk for all m; k 2 Z. In other words, f 2 Vm if and only if f .2�mt/ 2 V0 for allm 2 Z. This shows that functions in Vm are obtained from those in V0 througha scaling 2�m. If the scale m D 0 is associated with V0, then the scale 2�m isassociated with Vm. Thus, subspaces Vm are just scaled versions of the centralspace V0 which is invariant under translation by integers, that is, Tn V0 D V0 forall n 2 Z.

2. It follows from Definition 4.2.1 that an MRA is completely determined by thescaling function �, but not conversely. For a given � 2 V0, we first define

V0 D

(f .x/ D

X

n2Z

cn �0;n DX

n2Z

cn �.t � n/ W fcng 2 `2.Z/

):

4.2 Multiresolution Analysis in L2.R/ 125

Condition (iv) implies that V0 has an orthonormal basis˚�.t � n/ W n 2 Z

.

Then, V0 consists of all functions f .t/ DP

n2Z cn�.t � n/ with finite energy��f��22

DP

n2Z

ˇcn

ˇ2< 1. Similarly, the space Vm has the orthonormal basis of

the form

˚�m;n.t/ D 2m=2�

2mt � n

�; n 2 Z

(4.2.1)

so that fm.x/ is given by

fm.x/ DX

n2Z

cm;n �m;n.t/ (4.2.2)

with the finite energy��fm��22

DP

n2Z

ˇcm;n

ˇ2< 1: Thus, fm represents a typical

function in the space Vm: It builds self-invariance and scale invariance throughthe basis

˚�m;nI m; n 2 Z

.

3. Conditions (ii) and (iii) can be expressed in terms of the orthogonal projectionsPm onto Vm, that is, for all f 2 L2.R/,

limm!�1

Pmf D 0 and limm!C1

Pmf D f : (4.2.3)

The projection Pmf can be considered as an approximation of f at the scale 2�m.

Therefore, the successive approximations of a given function f are defined as theorthogonal projections Pm onto the space Vm:

Pmf DX

n2Z

˝f ; �m;n

˛�m;n; (4.2.4)

where f�m;n.t/ W m; n 2 Zg given by (4.2.1) is an orthonormal basis for Vm.

4. Since V0 � V1, the scaling function � that leads to a basis for V0 is also V1. Since� 2 V1 and �1;n.t/ D

p2 �.2t � n/ is an orthonormal basis for V1, � can be

expressed in the form

�.t/ Dp2X

n2Z

cn �2t � n

�; (4.2.5)

where cn D˝�; �1;n

˛and

Pn2Z jcnj2 D 1: Equation (4.2.5) is called the dilation

equation. It involves both t and 2t and is often referred to as the two-scaleequation or refinement equation because it displays �.t/ in the refined space V1.The space V1 has the finer scale 2�1 and it contains �.t/ which has scale 1.


All of the preceding facts reveal that MRA can be described at least three waysso that we can specify:

(a) the subspaces Vm,(b) the scaling function �,(c) the coefficient cn in the dilation equation (4.2.5).

The real importance of an MRA lies in the simple fact that it enables us toconstruct an orthonormal basis for L2.R/. In order to prove this statement, we firstassume that fVm W m 2 Zg is an MRA. Since Vm � VmC1, we define Wm as theorthogonal complement of Vm in VmC1 for every m 2 Z, so that we have

VmC1 D Vm ˚ Wm

D Vm�1 ˚ Wm�1

�˚ Wm

D : : :

D V0 ˚ W0 ˚ W1 ˚ � � � ˚ Wm

D V0 ˚

"mM

nD0

Wn

#(4.2.6)

and Vn ? Wm for n ¤ m. SinceS

m2Z Vm is dense in L2.R/, we may take the limitas m ! 1 to obtain

V0 ˚

"1M

mD0

Wm

#D L2.R/:

Similarly, we may go in the other direction to write

V0 D V�1 ˚ W�1

D V�2 ˚ W�2

�˚ W�1

D : : :

D V�m ˚ W�m ˚ � � � ˚ W�1:

We may again take the limit as m ! 1. SinceT

m2Z Vm D f0g, it follows thatV�m D f0g. Consequently, it turns out that

M

m2Z

Wm D L2.R/: (4.2.7)

4.2 Multiresolution Analysis in L2.R/ 127

It follows from conditions (i) to (v) in Definition 4.2.1 that the spaces Wm arealso scaled versions of W0 and, for f 2 L2.R/,

f 2 Wm if and only if f .2�mt/ 2 W0 for all m 2 Z; (4.2.8)

and they are translation-invariant for the discrete translations n 2 Z, that is,

f 2 W0 if and only if f .t � n/ 2 W0;

and they are mutually orthogonal spaces generating all of L2.R/,

Wm ? Wk for m ¤ k; andM

m2Z

Wm D L2.R/: (4.2.9)

Moreover, there exists a function 2 W0 such that f .t � n/ W n 2 Zg constitutesan orthonormal basis for W0. It follows from (4.2.8) that

n m;n.t/ D 2m=2 .2mt � n/ W n 2 Z

o(4.2.10)

constitutes an orthonormal basis for Wm. Thus, it follows from (4.2.7) that thefamily f m;n.t/g represents an orthonormal basis of wavelets for L2.R/. It is calledan orthonormal wavelet basis with mother wavelet . Therefore, the waveletexpansion of any function f 2 L2.R/ can be written as

f .t/ DX

m2Z

X

n2Z

˝f ; m;n

˛ m;n.t/; (4.2.11)

where the series converges in L2-sense under suitable restriction on � and .

Recall that Pm is the projection on Vm, so for any f 2 L2.R/, we have

PmC1f D Pmf CX

n2Z

˝f ; m;n

˛ m;n: (4.2.12)

Thus, Pm is the result of observing f at the resolution level m, and the difference

Qmf D PmC1f � Pmf DX

n2Z

˝f ; m;n

˛ m;n (4.2.13)

is the additional detail required to pass from the resolution level m to the higherlevel m C 1. This operator Qm is called a detailed operator on Wm.


Example 4.2.1 (Characteristic Function). We assume that � D �Œ0;1� is thecharacteristic function of the interval Œ0; 1�. Define spaces Vm by

Vm D

(X

k2Z

ck �m;k WX

k2Z

ˇck

ˇ2< 1

);

where �m;n.t/ D 2�m=2 �.2�mt � n/: The spaces Vm satisfy all the conditions ofDefinition 4.2.1, and so,

˚Vm W m 2 Z

is an MRA.

Example 4.2.2 (Piecewise Constant Function). Consider the space Vm of allfunctions in L2.R/which are constant on intervals

2�mn; 2�m.nC1/

�, where n 2 Z.

Obviously, Vm � VmC1 because any function that is constant on intervals of length2�m is automatically constant on intervals of half that length. The space V0 containsall functions f .t/ in L2.R/ that are constant on Œn; n C 1/ having jumps possibly atinteger values, and V1 consists of constant functions on the intervals Œn=2; n C 1=2/

of length 1=2, and so on. Intervals of length 2�m are usually referred to as dyadicintervals.

Clearly, the piecewise constant function space Vm satisfies the conditions (i)–(iv)of an MRA. It is easy to guess a scaling function � in V0 which is orthogonal to itstranslates. The simplest choice for � is the characteristic function so that �.t/ D

�Œ0;1�.t/. Therefore, any function f 2 V0 can be expressed in terms of the scalingfunction � as

f .t/ DX

n2Z

cn �.t � n/:

Thus, the condition (v) is satisfied by the characteristic function �Œ0;1� as the scalingfunction. As we shall see later, this MRA is related to the classical Haar wavelet.

4.3 Construction of Mother Wavelet

In the previous section, we have seen that the subspaces Wm have thescaling property f .t/ 2 Wm if and only if f .2t/ 2 WmC1, so the familyf 0;n.t/ D .t � n/ W n 2 Zg constitutes an orthonormal set of W0 if and only ifthe set f m;n.t/ W m; n 2 Zg defined by (4.2.10) forms an orthonormal basis ofWm, for all m 2 Z. Thus, our task reduces to finding 2 W0 such that the setf .t � n/ W n 2 Zg constitutes an orthonormal basis for W0. To construct this , wefirst study some interesting properties of the father wavelet � (and W0).

4.3 Construction of Mother Wavelet 129

Theorem 4.3.1. For any function � 2 L2.R/, the following conditions are equiva-lent:

(a) The system˚�0;n �.t � n/; n 2 Z

is orthonormal.

(b)X

k2Z

ˇˇ O�.! C 2k�/

ˇˇ2

D 1 almost everywhere (a.e.).

Proof. Obviously, the Fourier transform of �0;n.t/ D �.t � n/ is

O�0;n.!/ D e�in! O�.!/:

In view of the general Parseval relation (1.3.17) for the Fourier transform, we have˝�0;n; �0;m

˛D˝�0;0; �0;m�n

˛

D1

2�

DO�0;0; O�0;m�n

E

D1

2�

Z 1

�1

e�i.m�n/!ˇˇ O�.!/

ˇˇ2

d!

D1

2�

X

k2Z

Z 2�.kC1/

2�ke�i.m�n/!

ˇˇ O�.!/

ˇˇ2

d!

D1

2�

Z 2�

0

e�i.m�n/!d!X

k2Z

ˇˇ O�.! C 2�k/

ˇˇ2

:

Thus, it follows from the completeness of˚e�in! W n 2 Z

in L2.0; 2�/ that

˝�0;n; �0;m

˛D ın;m

if and only if

X

k2Z

ˇˇ O�.! C 2�k/

ˇˇ2

D 1 almost everywhere:

Proposition 4.3.1. For any two functions�; 2 L2.R/, the sets of functions˚�.t�

n/ W n 2 Z

and˚ .t � m/ W m 2 Z

are biorthogonal, that is,

˝�0;n; 0;m

˛D 0; for all n;m 2 Z;

if and only if

X

k2Z

O�.! C 2�k/ O .! C 2�k/ D 0; almost everywhere:


The proof of this result can be obtained by applying arguments similar to thosestated in the proof of Theorem 4.3.1.

We next proceed to the construction of a mother wavelet by introducing animportant generating function Om0.!/ 2 L2Œ0; 2�� in the following lemma.

Lemma 4.3.1. The Fourier transform of the scaling function � satisfies the follow-ing conditions:

X

k2Z

ˇˇ O�.! C 2�k/

ˇˇ2

D 1 a.e; (4.3.1)

O�.!/ D Om0

�!2

�O��!2

�; (4.3.2)

where

Om0.!/ D1

p2

X

n2Z

cn e�in! (4.3.3)

is a 2�-periodic function and satisfies the so-called orthogonality condition

ˇOm0.!/

ˇ2CˇOm0.! C �/

ˇ2D 1 a.e: (4.3.4)

Proof. Condition (4.3.1) follows from Theorem 4.3.1.

To establish (4.3.2), we first note that � 2 V0 � V1 andn�1;n.t/ D

p2 �.2t � n/ W

n 2 Zg is an orthonormal basis for V1. Thus, the scaling function� has the followingrepresentation:

�.t/ Dp2X

n2Z

cn �.2t � n/; (4.3.5)

where cn D h�; �1;ni andX

n2Z

ˇcn

ˇ2< 1: The Fourier transform of (4.3.5) gives

O�.!/ D1

p2

X

n2Z

cne�i!n=2 O��!2

�D Om0

�!2

�O��!2

�; (4.3.6)


where Om0.!/ D1

p2

X

n2Z

cne�i!n is the 2�-periodic function and is called the low-

pass filter or discrete filter associated with the scaling function �. This proves thefunctional equation (4.3.2).

To verify the orthogonality condition (4.3.4), we substitute (4.3.2) in (4.3.1) sothat condition (4.3.1) becomes

1 DX

k2Z

ˇˇ O�.! C 2�k/

ˇˇ2

DX

k2Z

ˇˇ Om0

�!2

C k��ˇˇ2 ˇˇ O��!2

C k��ˇˇ2

:

This is true for any !, and hence, replacing ! by 2! gives

1 DX

k2Z

ˇOm0.! C k�/

ˇ2 ˇˇ O�.! C k�/ˇˇ2

: (4.3.7)

We now split the above infinite sum over k into even and odd integers and use the2�-periodic property of the function Om0 to obtain

1DX

k2Z

ˇOm0.!C2�k/

ˇ2 ˇˇ O�.!C2�k/ˇˇ2

CX

k2Z

ˇOm0

!C.2k C 1/�

�ˇ2 ˇˇ O�!C.2k C 1/�

�ˇˇ2

DX

k2Z

ˇOm0.!/

ˇ2 ˇˇ O�.! C 2�k/ˇˇ2

CX

k2Z

ˇOm0.! C �/

ˇ2 ˇˇ O�.! C � C 2k�/ˇˇ2

DˇOm0.!/

ˇ2CˇOm0.! C �/

ˇ2

by (4.3.1) used in its original form and ! replaced by .! C �/. This leads to thedesired condition (4.3.4).

Remark. SinceˇO�.0/

ˇD 1 ¤ 0; Om0.0/ D 1 and Om0.�/ D 0 . This implies that

Om0 can be considered as a low-pass filter because the transfer function passes thefrequencies near ! D 0 and cuts off the frequencies near ! D � .

Lemma 4.3.2. The Fourier transform of the scaling function � can be representedby an infinite product

O�.!/ D

1Y

kD1

Om0

�!2k

�: (4.3.8)


Proof. By the repeated applications of (4.3.2), we obtain

O�.!/ D Om0

�!2

�O��!2

�

D Om0

�!2

�Om0

� !22

�O�� !22

�

D Om0

�!2

�Om0

� !22

�Om0

� !23

�O�� !23

�

:::

D Om0

�!2

�Om0

� !22

�: : : Om0

�!2k

�O�� !2k

�

D

kY

jD1

Om0

�!2j

�O��!2j

�: (4.3.9)

Since O�.0/ D 1 and O�.!/ is continuous, we obtain

limk!1

O�� !2k

�D O�.0/ D 1:

The limit of (4.3.9) as k ! 1 gives (4.3.8).

We next prove the following major technical lemma which provides the Fouriercharacteristics of the subspace W0.

Lemma 4.3.3. The Fourier transform of any function f 2 W0 can be expressed inthe form

Of .!/ D Ov.!/ ei!=2 Om0

�!2

C ��

O��!2

�; (4.3.10)

where Ov.!/ is a 2�-periodic function and the factor ei!=2 Om0

�!2

C ��

O��!2

�is

independent of f :

Proof. Since f 2 W0, it follows from V1 D V0 ˚ W0 that f 2 V1 and is orthogonalto V0. Thus, the function f can be expressed in the form

f .t/ Dp2X

n2Z

cn �.2t � n/; (4.3.11)


where cn D hf ; �1;ni: We use an argument similar to that in Lemma 4.3.1 to obtainthe result

Of .!/ D1

p2

X

n2Z

cne�in!=2 O��!2

�D Omf

�!2

�O��!2

�; (4.3.12)

where the function Omf is given by

Omf .!/ D1

p2

X

n2Z

cn e�in!: (4.3.13)

Evidently, Omf is a 2�- periodic function which belongs to L2.0; 2�/. By applyingthe Parseval identity and using the fact that f 2 W0 ? V0, we obtain

0 D˝f ; �0;n

˛D

1

2�

DOf ; O�0;n

E

D1

2�

Z 1

�1

Of .!/ O�.!/ ein! d!

D1

2�

X

k2Z

Z 2�.kC1/

2�k

Of .!/ O�.!/ ein! d!

D1

2�

Z 2�

0

"X

k2Z

Of .! C 2�k/ O�.! C 2�k/

#ein! d!: (4.3.14)

Consequently,

X

k2Z

Of .! C 2�k/ O�.! C 2�k/ D 0: (4.3.15)

We now substitute (4.3.12) and (4.3.2) into (4.3.15) to obtain

0 DX

k2Z

Omf

�!2

C �k�

Om0

�!2

C �k� ˇˇ O��!2

C �k�ˇˇ2

;

which is, by splitting the sum into even and odd integers k and then using the2�-periodic property of the function Om0,


0 DX

k2Z

Omf

�!2

C 2�k�

Om0

�!2

C 2�k� ˇˇ O��!2

C 2�k�ˇˇ2

CX

k2Z

Omf

�!2

C � C 2�k�

Om0

�!2

C � C 2�k� ˇˇ O��!2

C � C 2�k�ˇˇ2

D Omf

�!2

�Om0

�!2

�X

k2Z

ˇˇ O��!2

C 2�k�ˇˇ2

C Omf

�!2

C ��

Om0

�!2

C ��X

k2Z

ˇˇ O��!2

C � C 2�k�ˇˇ2

;

which is, due to orthonormality of the system˚�0;k.t/ W k 2 Z

and (4.3.1),

Dn

Omf

�!2

�Om0

�!2

�C Omf

�!2

C ��

Om0

�!2

C ��o

� 1: (4.3.16)

Finally, replacing ! by 2! in (4.3.16) gives

Omf .!/ Om0.!/C Omf .! C �/ Om0.! C �/ D 0 a.e: (4.3.17)

Or, equivalently,

ˇˇ Omf .!/ Om0.! C �/

� Omf .! C �/ Om0.!/

ˇˇ D 0:

This can be interpreted as the linear dependence of two vectors�

Omf .!/;

� Omf .! C �/�

and�

Om0.! C �/; Om0.!/�

, and hence, there exists a function O�

such that

Omf .!/ D O�.!/ Om0.! C �/ a.e: (4.3.18)

Since both Om0 and Omf are 2�-periodic functions, so is O�. Further, substitut-ing (4.3.18) into (4.3.17) gives

O�.!/C O�.! C �/ D 0 a.e: (4.3.19)

Thus, there exists a 2�- periodic function Ov defined by

O�.!/ D ei! Ov.2!/: (4.3.20)


Finally, a simple combination of (4.3.12), (4.3.18), and (4.3.20) gives the desiredrepresentation (4.3.10). This completes the proof of Lemma 4.3.3. ut

Now, we return to the main problem of constructing a mother wavelet .t/ froman MRA. Suppose that there is a function such that

˚ 0;n W n 2 Z

is a basis for

the space W0. Then, every function f 2 W0 has a series representation

f .t/ DX

n2Z

hn 0;n.t/ DX

n2Z

hn .t � n/; (4.3.21)

whereP

n2Z jhnj2 < 1: Application of the Fourier transform to (4.3.21) gives

Of .!/ D

"X

n2Z

hn e�in!

#O .!/ D Oh.!/ O .!/; (4.3.22)

where the function Oh is

Oh.!/ DX

n2Z

hn e�in!; (4.3.23)

and it is a square integrable and 2� -periodic function in Œ0; 2��. When (4.3.22) iscompared with (4.3.10), we see that O .!/ should be

O .!/ D ei!=2 Om0

�!2

C ��

O��!2

�D Om1

�!2

�O��!2

�(4.3.24)

where the function Om1 is given by

Om1.!/ D ei! Om0.! C �/: (4.3.25)

Thus, the function Om1.!/ is called the filter conjugate to Om0.!/, and hence, Om0 andOm1 are called conjugate quadratic filters (CQF) in signal processing.

Finally, substituting (4.3.3) into (4.3.24) gives

O .!/ D ei!=2 �1

p2

X

n2Z

cn ein. !2 C�/ O��!2

�

D1

p2

X

n2Z

cn ein�Ci.nC1/!

2 O��!2

�


which is, by putting n D �.k C 1/

D1

p2

X

k2Z

c�k�1.�1/k e�ik!=2 � O�

�!2

�: (4.3.26)

Invoking the inverse Fourier transform to (4.3.26) with k replaced by n gives themother wavelet

.t/ Dp2X

n2Z

.�1/n�1 c�n�1�.2t � n/ (4.3.27)

Dp2X

n2Z

dn �.2t � n/; (4.3.28)

where the coefficients dn are given by

dn D .�1/n�1 c�n�1: (4.3.29)

Thus, the representation (4.3.28) of a mother wavelet has the same structure asthat of the father wavelet � given by (4.3.5).

Remarks.

1. The mother wavelet associated with a given MRA is not unique because

dn D .�1/n�1 c2N�1�n (4.3.30)

defines the same mother wavelet (4.3.27) with suitably selected N 2 Z. Thiswavelet with coefficients dn given by (4.3.30) has the Fourier transform

O .!/ D ei.2N�1/!=2 Om0

�!2

C ��

O��!2

�: (4.3.31)

The nonuniqueness property of allows us to define another form of , insteadof (4.3.27), by

.t/ Dp2X

n2Z

dn �.2t � n/; (4.3.32)


where a slightly modified dn is

dn D .�1/n c1�n: (4.3.33)

In practice, any one of the preceding formulas for dn can be used to find a motherwavelet.

2. The orthogonality condition (4.3.4) together with (4.3.2) and (4.3.24) implies

ˇˇ O�.!/

ˇˇ2

Cˇˇ O .!/

ˇˇ2

Dˇˇ O��!2

�ˇˇ2

: (4.3.34)

Or, equivalently,

ˇˇ O�.2m!/

ˇˇ2

Cˇˇ O .2m!/

ˇˇ2

Dˇˇ O�.2m�1!/

ˇˇ2

: (4.3.35)

Summing both sides of (4.3.35) from m D 1 to 1 leads to the result

ˇˇ O�.!/

ˇˇ2

D

1X

mD1

ˇˇ O .2m!/

ˇˇ2

: (4.3.36)

3. If � has a compact support, the series (4.3.28) for the mother wavelet terminates, and consequently, is represented by a finite linear combinationof translated versions of �.2t/.

Finally, all of the above results lead to the main theorem of this section.

Theorem 4.3.2. If˚Vm W m 2 Z

is an MRA with the scaling function �, then there

is a mother wavelet given by

.t/ Dp2X

n2Z

.�1/n�1 c�n�1 �.2t � n/; (4.3.37)

where the coefficients cn are given by

cn D h�; �1;ni Dp2

Z 1

�1

�.t/ �.2t � n/ dt: (4.3.38)

That is, the system˚ m;n.t/ W m; n 2 Z

is an orthonormal basis of L2.R/.


Proof. First, we have to verify that f m;n.t/ W m; n 2 Zg is an orthonormal set.Indeed, we have

Z 1

�1

.t � k/ .t � `/ dt D1

2�

Z 1

�1

e�i!.k�`/ˇˇ O .!/

ˇˇ2

d!

D1

2�

Z 2�

0

e�i!.k�`/X

k2Z

ˇˇ O .! C 2�k/

ˇˇ2

d!

X

k2Z

ˇˇ O .! C 2�k/

ˇˇ2

DX

k2Z

ˇˇ Om0

�!2

C .k C 1/��ˇˇ2 ˇˇ O��!2

C k��ˇˇ2

which is, by splitting the sum into even and odd integers k,

DX

k2Z

ˇˇ Om0

�!2

C .2k C 1/��ˇˇ2 ˇˇ O �!2

C k��ˇˇ2

CX

k2Z

ˇˇ Om0

�!2

C .2k C 2/��ˇˇ2 ˇˇ O �!2

C .2k C 1/��ˇˇ2

Dˇˇ Om0

�!2

C ��ˇˇ2X

k2Z

ˇˇ O��!2

C 2k��ˇˇ2

Cˇˇ Om0

�!2

�ˇˇ2X

k2Z

ˇˇ O��!2

C .2k C 1/��ˇˇ2

Dˇˇ Om0

�!2

�ˇˇ2

Cˇˇ Om0

�!2

C ��ˇˇ2

D 1 by (4.3.4):

Thus, we find

Z 1

�1

.t � k/ .t � `/ dx D ık;l:

This shows that˚ m;n W m; n 2 Z

is an orthonormal system. In view of

Lemma 4.3.2 and our discussion preceding this theorem, to prove that it is a basis, itsuffices to show that function Ov in (4.3.20) is square integrable over Œ0; 2��: In fact,


Z 2�

0

ˇOv.!/

ˇ2d! D 2

Z �

0

ˇˇ O�.!/

ˇˇ2

d!

D 2

Z �

0

ˇˇ O�.!/

ˇˇ2 nˇ

Om0.! C �/ˇ2

CˇOm0.!/

ˇ2od!; by (4.3.4)

D 2

Z 2�

0

ˇˇO�.!/

ˇˇ2 ˇ

Om0.! C �/ˇ2

d!

D 2

Z 2�

0

ˇOmf .!/

ˇ2d!; by (4.3.18)

D 2�X

n2Z

ˇcn

ˇ2; cn D

˝f ; �1;n

˛

D 2��f��22< 1:


Example 4.3.1 (The Haar Wavelet). Example 4.2.2 shows that spaces of piecewiseconstant functions constitute an MRA with the scaling function �.t/ D �Œ0;1/.t/.Moreover, � satisfies the dilation equation

�.t/ Dp2X

n2Z

cn �.2t � n/; (4.3.39)

where the coefficients cn are given by

cn Dp2

Z 1

�1

�.t/ �.2t � n/ dt: (4.3.40)

Evaluating this integral with � D �Œ0;1/ gives cn as follows:

c0 D c1 D1

p2

and cn D 0 for n ¤ 0; 1:

Consequently, the dilation equation becomes

�.t/ D �.2t/C �.2t � 1/: (4.3.41)

This means that �.t/ is a linear combination of the even and odd translates of �.2t/and satisfies a very simple two-scale relation (4.3.41), as shown in Figure 4.1.


ϕ(t) ϕ(2t)

t t t1

1 1 1

0 10 1/2ϕ(2t−1)

10 1/2

Fig. 4.1 Two-scale relation of �.t/ D �.2t/C �.2t � 1/

In view of (4.3.33), we obtain

d0 D c1 D1

p2

and d1 D �c0 D �1

p2:

Thus, the Haar mother wavelet is obtained from (4.3.32) as a simple two-scalerelation

.t/ D �.2t/� �.2t � 1/

D �Œ0;:5�.t/ � �Œ:5;1�.t/ (4.3.42)

D

8ˆ<

ˆ:

C1; 0 � t <1

2

�1;1

2� t < 1

0; otherwise:

(4.3.43)

This two-scale relation (4.3.42) of is represented in Figure 4.2.

Example 4.3.2 (The Shannon Wavelet). We consider the Fourier transform O� of ascaling function � defined by

O�.!/ D �Œ��;��.!/

so that

�.t/ D1

2�

Z �

��

ei!xd! Dsin�t

�t:


ψ(t)

t t

φ(2t)

−φ(2t−1)

1 1

-1

1

-1

0 0t

012

12

Fig. 4.2 Two-scale relation of .t/ D �.2t/� �.2t � 1/

This is also known as the Shannon sampling function. Clearly, the Shannonscaling function does not have finite support. However, its Fourier transform hasa finite support (band-limited) in the frequency domain and has good frequencylocalization. Evidently, the system

�0;k.t/ D �.t � k/ Dsin�.t � k/

�.t � k/; k 2 Z

is orthonormal because

˝�0;k; �0;`

˛D

1

2�

DO�0;k; O�0;`

E

D1

2�

Z 1

�1

O�0;k.!/ O�0;`.!/ d!

D1

2�

Z 1

�1

e�i.k�`/! d! D ık;`:

In general, we define, for m D 0,

V0 D

(X

k2Z

cksin�.t � k/

�.t � k/WX

k2Z

ˇck

ˇ2< 1

);

and, for other m ¤ 0, m 2 Z,

Vm D

(X

k2Z

ck2m=2 sin�.2mt � k/

�.2mt � k/WX

k2Z

ˇck

ˇ2< 1

):


It is easy to check that all conditions of Definition 4.2.1 are satisfied. We nextfind out the coefficients ck defined by

ck D˝�; �1;n

˛D

p2

Z 1

�1

sin�.t/

�t�

sin�.2t � k/

�.2t � k/dx

D

8ˆ<

ˆ:

1p2; k D 0

p2

�ksin

��k

2

�; k ¤ 0:

Consequently, we can use the formula (4.3.37) to find the Shannon mother wavelet

.t/ DX

n2Z

.�1/n�1c�n�1 �.2t � n/

D1

p2

sin�.2t C 1/

�.2t C 1/C

p2

�

X

n¤�1

.�1/n�1

.n C 1/cos

�n�

2

� sin�.2t � n/

�.2t � n/:

Obviously, the system˚ m;n W m; n 2 Z

is an orthonormal basis in L2.R/. It is

known as the Shannon system.

The following theorem gives the characterization of the mother wavelet interms of the scaling function and of the low-pass filter in the frequency domain.

Theorem 4.3.3. If � is the scaling function for an MRA and Om0.!/ is the associatedlow-pass filter, then a function 2 W0 is an orthonormal wavelet for L2.R/ if andonly if

O .!/ D ei!=2 Ov.!/ Om0

�!2

C ��

O��!2

�(4.3.44)

for some 2�-periodic function Ov such thatˇOv.!/

ˇD 1:

Proof. It is enough to prove that all orthonormal wavelets 2 W0 can berepresented by (4.3.44). For any 2 W0, by Lemma 4.3.3, there must be a2�-periodic function Ov such that

O .!/ D ei!=2 Ov.!/ Om0

�!2

C ��

O��!2

�:


If is an orthonormal wavelet, then the orthonormality of˚ .t�k/; k 2 Z

leads to

1 DX

k2Z

ˇˇ O .! C 2�k/

ˇˇ2

DˇOv.!/

ˇ2X

k2Z

ˇˇ Om0

�!2

C k� C ��ˇˇ2 ˇˇ O��!2

C �k�ˇˇ2

which is, by splitting the sum into even and odd integers k,

DˇOv.!/2

ˇ(X

k2Z

ˇˇ Om0

�!2

C 2k� C ��ˇˇ2 ˇˇ O��!2

C 2k��ˇˇ2

CX

k2Z

ˇˇ Om0

�!2

C .2k C 1/� C ��ˇˇˇˇ O��!2

C .2k C 1/��ˇˇ2

)

which is, through (4.3.1) and the 2�-periodic property of Om0,

DˇOv.!/

ˇ2� ˇˇ Om0

�!2

�ˇˇ2

Cˇˇ Om0

�!2

C ��ˇˇ2�

DˇOv.!/

ˇ2; by (4.3.4):


We now use the Theorem 4.3.3 to construct the following orthonormal wavelets.

Example 4.3.3. The Haar wavelet can be obtained from the Fourier transform of thescaling function � D �Œ0;1� so that

O�.!/ D O�Œ0;1�.!/ D e�i!=2sin�!2

�

�!2

�

D e�i!=4 cos�!4

�:e�i!=4

sin�!4

�

�!4

�

D Om0

�!2

�O��!2

�; (4.3.45)


where the associated filter Om0.!/ and its complex conjugate are given by

Om0.!/ D e�i!=2 cos�!2

�D1

2

1C e�i!

�; (4.3.46)

Om0.!/ D ei!=2 cos�!2

�D1

2

1C ei!

�: (4.3.47)

Thus, the Haar wavelet can be obtained from (4.3.24) or (4.3.44) and is given by

O .!/ D Ov.!/ � ei!=2 Om0

�!2

C ��

O��!2

�

D Ov.!/ � ei!=2 �1

2

1 � ei!=2

�O��!2

�

where Ov.!/ D �ie�i! is chosen to find the exact result (4.3.42). Using this valuefor Ov.!/, we obtain

O .!/ D1

2O��!2

��1

2e�i!=2 O�

�!2

�

so that the inverse Fourier transform gives the exact result (4.3.42) as

.t/ D �.2t/� �.2t � 1/:

On the other hand, using (4.3.24) also gives the Haar wavelet as

O .!/ D ei!=2 Om0

�!2

C ��

O��!2

�

D ei!=2 �1

2

1 � ei!=2

��

1 � e�i!=2

��

i!

2

� (4.3.48)

Di

!

1 � ei!=2

�2

Di

!

�ei!=4 � e�i!=4 � ei!=4 � ei!=4

�2

D �i ei!=2 �

8<

:

sin2�!4

�

�!4

�

9>=

>;


D

8<

:i e�i!=2

sin2�!4

�

�!4

�

9>=

>;

� e�i!

�: (4.3.49)

This corresponds to the same Fourier transform (3.2.5) of the Haar wavelet (4.3.43)except for the factor �e�i! . This means that this factor induces a translation of theHaar wavelet to the left by one unit. Thus, we have chosen Ov.!/ D �e�i! in (4.3.44)to find the same value (4.3.43) for the classic Haar wavelet.

Example 4.3.4 (The Meyer Wavelet). This example was first reported by YvesMeyer, and for that reason, it is often called the Meyer wavelet. However, Meyercalled it the Littlewood-Paley wavelet. This wavelet is orthogonal and symmetric innature; however, it does not have a finite support. Define � by

O�.!/ D

8ˆ<

ˆ:

1; j!j �2�

3

cos

��

2v

�3

4�j!j � 1

��;2�

3� j!j �

4�

3

0; otherwise

(4.3.50)

where v is a smooth function .Ck or C1/ such that

v.t/ D

�0; t � 0

1; t � 1and v.t/C v.1 � t/ D 1: (4.3.51)

By using (4.3.51), it is easy to verify that

X

k2Z

ˇˇ O�.! C 2k�/

ˇˇ2

D 1;

and hence, the family of functions f�0;k D �.t � k/ W k 2 Zg forms an orthonormalsystem. We then define V0 to be the closed subspace spanned by this orthonormalset. Similarly, Vm is the closed space spanned by the �m;n; n 2 Z. Then, Vm satisfythe condition (i) of Definition 4.2.1 if and only if there exists a 2�-periodic functionOm0 of the form

Om0.!/ DX

k2Z

O�.2! C 4k�/:


Thus, we have

O�.!/ D Om0

�!2

�O��!2

�DX

k2Z

O�.! C 2k�/ O��!2

�D O�.!/ O�

�!2

�

as the support of O�.! C 2k�/ and O��!2

�do not overlap if k ¤ 0. In fact,

O��!2

�D 1; if ! 2 supp. O�/: Using Theorem 4.3.3 with v.!/ D 1, we can get

a mother wavelet of the form

O .!/ D ei!=2 Om0

�!2

C ��

O��!2

�

D ei!=2X

k2Z

O�.! C 2� C 4k�/ O��!2

�

D ei!=2h

O�.! C 2�/C O�.! � 2�/i

O��!2

�

which gives

O .!/ D

8ˆ<

ˆ:

ei!=2 sin

��

2v

�3

2�j!j � 1

��;2�

3� j!j �

4�

3

ei!=2 cos

��

2v

�3

2�j!j � 1

��;4�

3� j!j �

8�

3:

(4.3.52)

Finally, we describe some properties of the coefficients of the scaling function�. The coefficients cn determine all the properties of the scaling function � andthe wavelet function . In fact, Mallat’s multiresolution algorithm (fast wavelettransform) uses the cn to calculate the wavelet transform without explicit knowledgeof . Furthermore, both � and can be reconstructed from the cn and this in fact iscentral to Daubechies’ wavelet analysis.

Theorem 4.3.4. If cn are coefficients of the scaling function � defined by (4.2.5),then

(i)X

n2Z

cn Dp2;

(ii)X

n2Z

.�1/ncn D 0;

(iii)X

n2Z

c2n D1

p2

DX

n2Z

c2nC1;

(iv)X

n2Z

.�1/n nm cn D 0; for m D 0; 1; 2; : : : ; .p � 1/:


Proof. It follows from (4.3.2) and (4.3.3) that O�.0/ D 0 and Om0.0/ D 1. Putting! D 0 in (4.3.3) gives (i).

Since Om0.0/ D 1, (4.3.4) implies that Om0.�/ D 0 which gives (ii).

Then, (iii) is a simple consequence of (i) and (ii).

To prove (iv), we recall (4.3.8) and (4.3.3) so that

O�.!/ D Om0

�!2

�Om0

� !22

�: : :

and

Om0

� !2k

�D

1p2

X

n2Z

cn e�in!=2k:

Clearly,

O�.2�/ D Om0.�/ Om0

�!2

�:

According to Strang’s (1989) accuracy condition, O�.!/ must have zeros of thehighest possible order when ! D 2�; 4�; 6�; : : : : Thus,

O�.2�/ D Om0.�/ Om0

�!2

�Om0

� !22

�: : : ;

and the first factor Om0.!/ will be zero of order p at ! D � if

dm Om0.!/

d!mD 0 for m D 0; 1; 2; : : : .p � 1/;

which gives

X

n2Z

cn.�in/m e�in� D 0; for m D 0; 1; 2; : : : .p � 1/:

Or, equivalently,

X

n2Z

.�1/n nmcn D 0; for m D 0; 1; 2; : : : .p � 1/:


From the fact that the scaling function �.t/ is orthonormal to itself in anytranslated position, we can show that

X

n2Z

c2n D 1: (4.3.53)

This can be seen by using �.t/ from (4.3.5) to obtain

Z 1

�1

�2.t/ dt D 2X

m2Z

X

n2Z

cmcn

Z 1

�1

�.2t � m/ �.2t � n/ dt

where the integral on the right-hand side vanishes due to orthonormality unlessm D n, giving

Z 1

�1

�2.t/ dt D 2X

n2Z

c2n

Z 1

�1

�2.2t � n/ dt

D 2X

n2Z

c2n �1

2

Z 1

�1

�2.y/ dy

whence follows (4.3.53).

Finally, we prove

X

k2Z

ck ckC2n D ı0;n: (4.3.54)

We use the scaling function � defined by (4.3.5) and the corresponding waveletgiven by (4.3.28) with (4.3.30), that is,

.t/ Dp2X

n2Z

.�1/n�1c2N�1�n �.2t � n/

which is, by substituting 2N � 1 � n D k,

Dp2X

n2Z

.�1/kck �.2t C k � 2N C 1/: (4.3.55)

We use the fact that mother wavelet .t/ is orthonormal to its own translate .t�n/so that

Z 1

�1

.t/ .t � n/ dt D ı0;n: (4.3.56)

4.4 The Fast Wavelet Transform 149

Substituting (4.3.55) to the left-hand side of (4.3.56) gives

Z 1

�1

.t/ .t � n/ dt

D 2X

k2Z

X

m2Z

.�1/kCmck cm

Z 1

�1

�.2t C k � 2N C 1/ �.2t C m � 2N C 1 � 2n/ dt;

where the integral on the right-hand side is zero unless k D m � 2n so that

Z 1

�1

.t/ .t � n/ dt D 2X

k2Z

.�1/2.kCn/ck ckC2n �1

2

Z 1

�1

�2.y/ dy:

This means that

X

n2Z

ck ckC2n D 0; for all n ¤ 0:

4.4 The Fast Wavelet Transform

In this section, we introduce the basic algorithm, derived by Mallat (1989b) for fastcomputation of the discrete wavelet transform, commonly known as the fast wavelettransform. Recall that the successive approximations of a given function f can beobtained by orthogonal projections Pm onto the scaled spaces Vm as

Pmf DX

n2Z

cm;n �m;n; (4.4.1)

where

cm;n D˝f ; �m;n

˛: (4.4.2)

Then, its projection on the detailed spaces Wm is given by

Qmf D PmC1f � Pmf DX

n2Z

dm;n m;n (4.4.3)

where

dm;n D˝f ; m;n

˛: (4.4.4)


Since˚�0;n.t/ D �.t � n/ W n 2 Z

is an orthonormal basis of V0 and V�1 � V0,

hence we can decompose �� t

2

�2 V�1 as

1

2�� t

2

�DX

n2Z

hn �.t C n/: (4.4.5)

Therefore, we have

�m�1;n.t/ D 2m�1=2�2m�1t � n

�

D 2m�1=2X

k2Z

hk �2m�1t � 2n C k

�

Dp2X

k2Z

hk �m;2n�k.t/: (4.4.6)

Similarly, one can decompose � t

2

�2 W�1 � V0 in terms of the basis function

˚�0;n.t/ W n 2 Z

of V0 as

1

2 � t

2

�DX

n2Z

gn �.t C n/: (4.4.7)

Therefore, it follows by computations similar to those above that

m�1;n.t/ D 2m�1=2 2m�1t � n

�

D 2m�1=2X

k2Z

gk �2m�1t � 2n C k

�

Dp2X

k2Z

gk �m;2n�k.t/: (4.4.8)

If we denote the low-pass and high-pass filters associated with � and , respec-tively, by

Om0.!/ DX

n2Z

hn ei!n; and Om1.!/ DX

n2Z

gn ei!n:

Then, it follows from (4.3.2) and (4.3.24) that

O�.2!/ D Om0.!/ O�.!/; and O .2!/ D Om1.!/ O�.!/:

4.5 Exercises 151

In view of Theorem 4.3.3, we can make a particular choice of the wavelet corresponding to v.!/ D 1 such that Om1.!/ D ei! Om0.! C �/, that is,

X

n2Z

gn ei!n DX

n2Z

.�1/nC1 h1�n ei!n;

which implies that

gn D .�1/nC1 h1�n: (4.4.9)

Substituting (4.4.6) into (4.4.2), we obtain

cm�1;n Dp2X

k2Z

hk cm;2n�k: (4.4.10)

Similarly, substituting (4.4.8) into (4.4.4), we obtain

dm�1;n Dp2X

k2Z

gk cm;2n�k: (4.4.11)

Equations (4.4.10) and (4.4.11) are the general recursions of the decompositionalgorithm, whereas the recursions for the reconstruction algorithm can be obtainedfrom the equations (4.4.3), (4.4.6), and (4.4.8). This gives

cm;n Dp2X

k2Z

h2k�n cm�1;k C g2k�n dm�1;k

�: (4.4.12)

4.5 Exercises

1. Show that the Gaussian function �.t/ D e�t2 cannot be the scaling function ofan MRA.

2. Suppose that the dilation equation (4.2.5) has a finite sum and that � 2 L1.R/

withZ 1

�1

�.t/ dt ¤ 0. Show thatX

n2Z

cn D 2.

3. Prove that the function

f .t/ D

8<

:1; 0 � t �

3

40; otherwise

does not satisfy the refinement equation (4.2.5).


4. If the generating function is defined by (4.3.3), then show that

(a)X

n2Z

cn Dp2; (b)

X

n2Z

c2n DX

n2Z

c2nC1 D1

p2

(c)X

n2Z

c2n D 1:

5. If the function Om0.!/ D1C e�3i!

2satisfies the orthogonality condition (4.3.4)

with Om0.0/ D 1, then show that the corresponding system f�.t � n/ W n 2 Zg isnot orthonormal. Why is this the case?

6. Using the Strang (1989) accuracy condition that O�.!/must have zeros of n when! D 2�; 4�; 6�; : : : , show that

X

k2Z

.�1/k km ck D 0; m D 0; 1; 2; : : : ; .n � 1/:

7. Using the properties of Om0 and Om1, prove that

(a) O��!2

�D

�Om0

�!2

�C Om0

�!2

C ��

O�.!/C

�Om1

�!2

�C Om1

�!2

C ��

O .!/

(b) exp

��

i!

2

�O��!2

�

D

�exp

��

i!

2

�Om0

�!2

�� exp

��

i!

2

�Om0

�!2

C ��

O�.!/

C

�exp

��

i!

2

�Om1

�!2

��exp

��

i!

2

�Om0

�!2

C ��

O .!/:

8. If Om0.!/ D1

2

1C e�i!

� 1 � e�i! C e�2i!

�D e� 3i!

2 cos

�3!

2

�; show that it

satisfies the condition (4.3.4) and Om0.0/ D 1. Hence, derive the following results:

(a) O�.!/ D exp

��3i!

2

� sin

�3!

2

�

�3!

2

� ;

(b)X

k2Z

ˇˇ O�.! C 2�k/

ˇˇ2

D1

9

3C 4 cos! C 2 cos 2!

�;

(c) �.t/ D

8<

:

1

3; 0 � t � 3

0; otherwise;

(d) cn D

Z 1

�1

�.t/ �.t � n/ dt D1

3

Z 3

0

�.t � n/ dt D1

3

Z nC3

n�.t/ dt:

4.5 Exercises 153

9. Show that, for any t 2 Œ0; 1�,(a)

X

k2Z

�.t � k/ D 1 and (b)X

k2Z

.c C k/ �.t � k/ D t;

where c D1

2

�3 �

p3�: Hence, using (a) and (b), show that

(c) 2�.t/C �.t C 1/ D t C 2 � c;

(d) �.t C 1/C 2�.t C 2/ D c � t;

(e) �.t/� �.t C 2/ D t C c Cp3 � 2

�:

5Elongations of MRA-Based Wavelets

The wavelets arrive in succession, and each wavelet eventually dies out. The wavelets allhave the same basic form and shape, but the strength or impetus of each wavelet is randomand uncorrelated with the strength of the other wavelets. Despite the fore-ordained deathof any individual wavelet, the time-series does not die. The reason is that a new wavelet isborn each day to take the place of the one that does die on any given day, the time-series iscomposed of many living wavelets, all of a different age, some young, others old.

Ender A. Robinson

In most sciences one generation tears down what another has built, and what one hasestablished, another undoes. In mathematics alone each generation adds a new storey tothe old structure.

Hermann Hankel

5.1 Introduction

In the previous chapter, we have introduced the concept of an MRA by which wehave constructed several types of orthonormal wavelets in L2.R/. However, theonly example we have seen so far of a compactly supported wavelet has been theHaar wavelet. Recall that the Haar space V0 was generated by the Haar scalingfunction �.t/ D �Œ0;1�.t/ (see Example 4.2.2). Although this scaling function hasmany desirable properties such as short support, symmetry about the line t D 1=2,and orthogonal to its translates, it is not continuous and its derivative is zeroalmost everywhere. Moreover, we saw that the analytic expression for the scalingfunction and wavelet is, in general, not available. Therefore, it is desirable toconstruct wavelets with greater degrees of smoothness and having compact support.In Section 5.2, we construct wavelets that are smooth and piecewise polynomial.


155

156 5 Elongations of MRA-Based Wavelets

Specifically, we construct a wavelet that is Cn�1 on R and that is piecewisepolynomial of degree n. These wavelets are called spline wavelets and the well-known Franklin and Battle-Lemarié wavelets are the special cases of these wavelets.Although B-splines are continuous and compactly supported, they fail to form anorthonormal basis. In Section 5.3, we develop the tools to construct orthonormalwavelets whose scaling functions are both differentiable and compactly supported.These wavelets were first constructed by Daubechies (1988b) that created a lot ofexcitement in the study of wavelets. Compactly supported wavelets possess certaindesirable properties such as compact support, orthogonality, symmetry, smoothness,high order of vanishing moments, and so on. In Section 5.4, we construct anotherintersecting class of orthonormal wavelets called harmonic wavelets. Harmonicwavelets are complex functions and band-limited in the frequency domain, so thatthey can be used to analyze frequency changes as well as oscillations in a smallinterval of time. They are closely related to Shannon wavelets: their real part isan even function which is identical to the Shannon wavelet and their imaginarypart is a kin but an odd function. Harmonic wavelets are also referred to asphysical family of wavelets because they were proposed for the analysis of physicalproblems, particularly in the fields of vibration and acoustic analysis (see Newland,1993a). In the end, we present a novel and simple procedure for the construction ofnonuniform wavelets associated with nonuniform MRA. In this nonstandard setting,the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair, and the associated dilationis an even positive integer related to the given spectral pair.

5.2 The Spline Wavelets

Spline functions consist of piecewise polynomials joined together smoothly at thebreak points (knots: t1; t2; : : : /where the degree of smoothness depends on the orderof spline. Polynomial splines with uniform knots were introduced by Schoenberg(1946) in his landmark paper, which sets the theoretical foundations for the subject.These functions now play a central role in approximation theory and numericalanalysis. Splines have also had a significant impact on the early development ofthe theory of the wavelet transform (see Strömberg, 1983). Spline wavelets areclassified in four categories: orthogonal, semi-orthogonal, shift orthogonal, andbiorthogonal. Orthogonal spline wavelets are usually symmetric or antisymmetricin nature. The simplest example of an orthonormal spline wavelet basis is the Haarbasis. The orthonormal spline wavelet bases of higher-order spline wavelets weregiven by Battle (1987) and Lemarié (1988) by using different methods. Chui andWang (1991, 1992) and Micchelli (1991) independently studied semi-orthogonalwavelets. The former authors constructed B-spline wavelets using linear splines.Then, they used the B-spline wavelets without orthogonalization to construct the

5.2 The Spline Wavelets 157

semi-orthogonal B-spline wavelets. The only difference between Chui and Wang’sand Battle’s constructions lies in the orthogonal property of the scaling function.Another advantage of the cardinal B-spline wavelet is that the wavelet spaces arekept orthogonal and the wavelets are still symmetric or antisymmetric. For moreabout spline wavelets and their applications to signal and image processing, werefer to the monograph (Averbuch et al., 2014).

The cardinal B-splines Nn.t/ of order n can be constructed from the n-foldconvolution of the indicator function in the unit interval:

Nn.t/ D N1.x/ � N1.t/ � � � � � N1.t/„ ƒ‚ …n�times

D N1.x/ � Nn�1.t/; .n � 2/; (5.2.1)

where

N1.t/ D �Œ0;1/.t/ D

�1; 0 � t < 10; otherwise:

(5.2.2)

Obviously,

Nn.t/ D

Z 1

�1

Nn�1.t � x/N1.x/ dx D

Z 1

0

Nn�1.t � x/ dx D

Z t

t�1Nn�1.x/ dx:

(5.2.3)Using the formula (5.2.3), we can obtain the explicit representation of splines

N2.t/;N3.t/; and N4.t/ as follows:

N2.t/ D

Z t

t�1N1.x/ dx D

Z t

t�1�Œ0;1�.x/ dx:

Evidently, it turns out that

N2.t/ D

8ˆ<

ˆ:

0; t � 0

t; 0 � t � 1

2 � t; 1 � t � 2

0; otherwise:

Or, equivalently,

N2.t/ D t�Œ0;1�.t/C .2 � t/ �Œ1;2�.t/: (5.2.4)


1N1(t)

N2(t)N3(t)

N4(t)

0 1 2 3 4

Fig. 5.1 Cardinal B-spline functions

Similarly, we find

N3.t/ D

8ˆˆˆ<

ˆˆˆ:

0; t � 0

t2

2; 0 � t � 1

6t � 2t2 � 3

2; 1 � t � 2

.t � 3/2

2; 2 � t � 3

0; otherwise:

Or, equivalently,

N3.t/ Dt2

2�Œ0;1�.t/C

1

2.6t � 2t2 � 3/ �Œ1;2�.t/C

1

2.t � 3/2 �Œ2;3�.t/: (5.2.5)

In a similar fashion, one can obtain fourth-order spline N4.t/ as

N4.t/ D1

6t3 �Œ0;1�.t/C

1

3

2�6tC6t2�t3

��Œ1;2�.t/C

1

2

t3�2t2C20t�13

��Œ2;3�.t/:

(5.2.6)

The graphical representation of Nn.x/ and their filter characteristicsˇˇ ONn.!/

ˇˇ are

shown in Figures 5.1 and 5.2.

In order to obtain the two-scale relation for the B-spline Nn.t/ of order n, weapply the Fourier transform of (5.2.2) so that

ON1.!/ D e�i!=2 sin!=2

!=2D e�i!=2 sinc

�!2

�D

1

i!

1 � e�i!

�(5.2.7)


N1(ω)

ω ω0 0

ω ω0 0

^

N2(ω)^

N3(ω)^ N4(ω)

^

Fig. 5.2 Fourier transforms of cardinal B-spline functions

where the sine function sinc.t/ is defined by

sinc.x/ D

8<

:

sin t

t; t ¤ 0

1; t D 0(5.2.8)

We can also express (5.2.7) in terms of z D e�i!=2 as

ON1.!/ D

�1C z

2

�ON1�!2

�: (5.2.9)

Application of the convolution theorem of the Fourier transform to (5.2.1) gives

ONn.!/ Dn

ON1.!/on

D ON1.!/ ONn�1.!/

D

�1C z

2

�n nON1�!2

�on

D

�1C z

2

�n

ONn

�!2

�

D OMn

�!2

�ONn

�!2

�; (5.2.10)


where the associated filter OMn is given by

OMn

�!2

�D

�1C z

2

�n

D

�1C e�i!=2

2

�n

D e�i!n=2�

cos!

2

�n

D1

2n

nX

kD0

n

k

!e�i!n=2

D1

p2

nX

kD0

cn;k e�i!k=2: (5.2.11)

Obviously, the coefficients cn;k are given by

cn;k D

8<

:

p2

2n

n

k

!; 0 � k � n

0; otherwise:

(5.2.12)

Therefore, the spline function in the time domain is

Nn.t/ Dp2

1X

kD0

cn;k �.2t � k/ D1

2n�1

nX

kD0

n

k

!Nn.2t � k/: (5.2.13)

This may be referred to as the two-scale relation for the B-splines of order n.

In view of (5.2.7), it follows that

ˇˇ ONn.!/

ˇˇ D

ˇˇsinc

�!2

�ˇˇn; (5.2.14)

where sinc.t/ is defined by (5.2.8). Thus, for each n � 1; ONn.!/ is a first-orderButterworth filter which satisfies the following conditions:

ˇˇ ONn.0/

ˇˇ D 1;

�d

d!

ˇˇ ONn.!/

ˇˇ�

!D0

D 0; and

�d2

d!2

ˇˇ ON2

n.!/ˇˇ�

!D0

¤ 0:

(5.2.15)

It is evident from (5.2.14) that


ˇˇ ONn.! C 2�k/

ˇˇ2

Dsin2n

�!2

C �k�

�!2

C �k�2n

Dsin2n

�!2

�

�!2

C �k�2n

: (5.2.16)

We replace ! by 2! in (5.2.16) and then sum the result over all integers k to obtain

X

k2Z

ˇˇ ONn.2! C 2�k/

ˇˇ2

D sin2n !

1X

kD�1

1

.! C �k/2n: (5.2.17)

It is well known in complex analysis that

X

k2Z

1

.! C �k/D cot!; (5.2.18)

which leads to the following result after differentiating .2n � 1/ times:

X

k2Z

1

.! C �k/2n D �1

.2n � 1/Š

d2n�1

d!2n�1.cot!/: (5.2.19)

Substituting this result in (5.2.17) yields

X

k2Z

ˇˇ ONn.2! C 2�k/

ˇˇ2

D �sin2n.!/

.2n � 1/Š

d2n�1

d!2n�1.cot!/: (5.2.20)

These results are used to find the Franklin wavelets and the Battle-Lemarié wavelets.

When n D 1, (5.2.19) gives another useful identity:

X

k2Z

1

.! C 2�k/2D1

4cosec2

�!2

�: (5.2.21)

Summing (5.2.16) over all integers k and using (5.2.21) lead to

X

k2Z

ˇˇ ON1.! C 2�k/

ˇˇ2

D 4 sin2�!2

�X

k2Z

1

.! C 2�k/2D 1: (5.2.22)

This shows that the first-order B-spline N1.t/ defined by (5.2.2) is a scaling functionthat generates the classic Haar wavelet.


Example 5.2.1 (The Franklin Wavelet). The Franklin wavelet is generated by thesecond-order .n D 2/ splines.

Differentiating (5.2.21) .2n � 2/ times gives the result

X

k2Z

1

.! C 2�k/2nD

1

4.2n � 1/Š

d2n�2

d!2n�2

�cosec2

�!2

� �: (5.2.23)

When n D 2, (5.2.23) yields the identity

X

k2Z

1

.! C 2�k/4D

1�2 sin

!

2

�4 �

�1 �

2

3sin2

�!2

��:

For n D 2, we sum (5.2.16) over all integers k so that

X

k2Z

ˇˇ ON2.! C 2�k/

ˇˇ2

D 16 sin4�!2

�X

k2Z

1

.! C 2�k/4D

�1 �

2

3sin2

!

2

�:

(5.2.24)

Or, equivalently,

"�1 �

2

3sin2

!

2

�� 12

#2X

k2Z

ˇˇ ON2.! C 2�k/

ˇˇ2

D 1:

Thus, the condition of orthonormality (5.2.20) ensures that the scaling function �has the Fourier transform:

O�.!/ D

0B@

sin!

2!

2

1CA

2 �1 �

2

3sin2

!

2

��1=2

: (5.2.25)

Thus, the filter associated with this scaling function � is obtained from (4.3.2) sothat

Om0.!/ DO�.2!/

O�.!/D

0B@

sin!

2 sin!

2

1CA

20B@1 �

2

3sin2

!

2

1 �2

3sin2 !

1CA

1=2


D cos2�!2

�0

B@1 �

2

3sin2

!

2

1 �2

3sin2 !

1

CA

1=2

: (5.2.26)

Finally, the Fourier transform of the orthonormal wavelet is obtainedfrom (4.3.24) so that

O .2!/ D Om1.!/ O�.!/ D ei! Om0.! C �/ O�.!/

D ei!

0

B@1 �

2

3sin2

!

2

1 �2

3sin2 !

1

CA

1=20

B@sin2

!

2!

2

1

CA

2 �1 �

2

3sin2

!

2

��1=2

D ei!sin4

�!2

�

�!2

�2

8ˆ<

ˆ:

1 �2

3cos2

!

4�1 �

2

3sin2

!

2

��1 �

2

3sin2

!

4

�

9>>=

>>;

1=2

: (5.2.27)

This is known as the Franklin wavelet generated by the second-order spline functionN2.t/. The scaling function � for the Franklin wavelet and the magnitude of its

Fourier transform,ˇˇ O�.!/

ˇˇ, and the Franklin wavelet and the magnitude of its

Fourier transform,ˇˇ O .!/

ˇˇ, are shown in Figures 5.3 and 5.4, respectively.

Fig. 5.3 (a) Scaling function of the Franklin wavelet �. (b) The Fourier transform j O�j

Fig. 5.4 (a) The Franklin wavelet . (b) The Fourier transform j O j


Example 5.2.2 (The Battle-Lemarié Wavelet). The Fourier transform O�.!/ associ-ated with the nth-order spline function Nn.t/ is

O�.!/ DONn.!/

(X

k2Z

ˇˇ ONn.! C 2k�/

ˇˇ2

) 1=2 ; (5.2.28)

where ONn.!/ is given by (5.2.10), and

ˇˇ ONn.! C 2k�/

ˇˇ2

D

8ˆ<

ˆ:

sin�!2

C k��

�! C 2k�

2

�

9>>=

>>;

2n

;

and

(X

k2Z

ˇˇ ONn.! C 2k�/

ˇˇ2

) 1=2D2n sinn

�!2

�

qOS2n.!/

;

with

OS2n.!/ DX

k2Z

1

.! C 2k�/2n : (5.2.29)

Consequently, (5.2.28) can be expressed in the form

O�.!/ D

��

i"!

2

�

!n

qOS2n.!/

; (5.2.30)

where " D 1 when n is odd or " D 0 when n is even, and OS2n.!/ can be computedby using the formula (5.2.23).

In particular, when n D 4, corresponding to the cubic spline of order four, O�.!/

is calculated from (5.2.30) by inserting

OS8.!/ DX

k2Z

1

.! C 2k�/8D

ON1.!/C ON2.!/

.105/�2 sin

!

2

�8 ; (5.2.31)


where

ON1.!/ D 5C 30 cos2�!2

�C 30

�sin

!

2cos

!

2

�2; (5.2.32)

and

ON2.!/ D 70 cos4�!2

�C 2 sin4

�!2

�cos2

�!4

�C2

3sin6

�!2

�: (5.2.33)

Finally, the Fourier transform of the Battle-Lemarié wavelet can be found byusing the same formula (4.3.24). The Battle-Lemarié scaling function � and theBattle-Lemarié wavelet are displayed in Figure 5.5a, b.

Fig. 5.5 (a) TheBattle-Lemarie scalingfunction. (b) TheBattle-Lemarié wavelet

t−4 −2 2

1

0.5

4−1/2

−0.5

ψ(t )

x−2 4

1

0.5

62

φ(x)a

b


Inspired by the pioneer work of Yves Meyer and stimulated by the excitingdevelopments in wavelets, Daubechies (1988a,b, 1992) first developed the theoryand construction of orthonormal wavelets with compact support. Compact supportis undoubtedly one of the wavelet properties that is given the greatest weight both intheory and applications. Wavelets with compact support can be constructed to havea given number of derivatives and to have a given number of vanishing moments.It is usually believed to be essential for two main reasons: to have fast numericalalgorithms and good time or space localization properties.

To construct orthonormal wavelet basis made up of compactly supported basisfunctions, it is fruitful to start with the periodic function:

Om0.!/ D1

p2

X

n2Z

cn e�i!n: (5.3.1)

To ensure that the wavelet basis functions are compactly supported, it is necessaryand sufficient to ensure that the basic wavelet .t/ is compactly supported. Theeasiest way to do this is to choose a scaling function �.t/ that is compactly supportedsuch that f�.t � n/ W n 2 Zg is an orthonormal system of functions. Then, it followsfrom the definition of cn W

cn D˝�; �1;n

˛D

p2

Z 1

�1

�.t/ �.2t � n/ dt; 8 n 2 Z (5.3.2)

that only finitely many cn are non-zero so that wavelet equation (4.3.37) reducesto finite linear combination of compactly supported functions. Thus, if the scalingfunction �.t/ has a compact support, then the associated generating function Om0.!/

defined in (5.3.1) is a trigonometric polynomial and it satisfies the orthogonalitycondition (4.3.4) with special values Om0.0/ D 1 and Om0.�/ D 0. If coefficientscn are real, then the corresponding scaling function as well as the mother wavelet .t/ will also be real-valued. The mother wavelet corresponding to � is given

by the formula (4.3.24) withˇˇ O�.0/

ˇˇ D 1. The Fourier transform O .!/ of order N

is N-times continuously differentiable and it satisfies the moment condition (3.2.3),that is,

O .k/.0/ D 0 for k D 0; 1; : : : ;m: (5.3.3)

It follows that 2 Cm implies that Om0 has a zero at ! D � of order .mC1/. In otherwords,

Om0 .!/ D

�1C e�i!

2

�mC1

OL.!/; (5.3.4)

5.3 The Daubechies Wavelets

5.3 167

where OL is a trigonometric polynomial. In addition to the orthogonality condi-tion (4.3.4), we assume

Om0 .!/ D

�1C e�i!

2

�N

OL.!/; N � 1 (5.3.5)

where OL.!/ is 2�-periodic and OL 2 CN�1. Evidently,

ˇOm0.!/

ˇ2D Om0.!/ Om0.�!/ D

�1C e�i!

2

�N �1C ei!

2

�N

OL.!/ OL.�!/

D�

cos2!

2

�N ˇˇ OL.!/

ˇˇ2

; (5.3.6)

whereˇˇ OL.!/

ˇˇ2

is a polynomial in cos!, that is,

ˇˇ OL.!/

ˇˇ2

D Q.cos!/:

Since cos! D 1 � 2 sin2�!2

�, it is convenient to introduce t D sin2

�!2

�so that

(5.3.6) reduces to the form

ˇOm0.!/

ˇ2D�

cos2!

2

�NQ.1 � 2t/ D .1 � t/N P.t/; (5.3.7)

where P.t/ is a polynomial in t. We next use the fact that

cos2�! C �

2

�D sin2

�!2

�D t

and

ˇˇ OL.! C �/

ˇˇ2

D Q.� cos!/ D Q.2t � 1/

D Q1 � 2.1� t/

�D P.1 � t/ (5.3.8)

to express the identity (4.3.4) in terms of t so that (4.3.4) becomes

.1 � t/NP.t/C tNP.1 � t/ D 1: (5.3.9)

The Daubechies Wavelets


Since .1 � t/N and tN are two polynomials of degree N which are relatively prime,then, by Bezout’s theorem (see Daubechies, 1992), there exists a unique polynomialPN of degree � N � 1 such that (5.3.9) holds. An explicit solution for PN.t/ isgiven by

PN.t/ D

N�1X

kD0

N C k � 1

k

!tk; (5.3.10)

which is positive for 0 < t < 1 so that PN.t/ is at least a possible candidate forˇˇ OL.!/

ˇˇ2

. There also exist higher degree polynomial solutions PN.t/ of (5.3.9) which

can be written as

PN.t/ D

N�1X

kD0

N C k � 1

k

!tk C tNR

�t �

1

2

�; (5.3.11)

where R is an odd polynomial. Since PN.t/ is a possible candidate forˇˇ OL.!/

ˇˇ2

and

OL.!/ OL.�!/ Dˇˇ OL2.!/

ˇˇ2

D Q.cos!/ D Q.1 � 2t/ D PN.t/; (5.3.12)

the next problem is how to find out OL.!/. This can be done by the following lemma:

Lemma 5.3.1 (Riesz Spectral Factorization). If

OA.!/ D

nX

kD0

ak cosk !; (5.3.13)

where ak 2 R and an ¤ 0, and if OA.!/ � 0 for real ! with OA.0/ D 0, then thereexists a trigonometric polynomial

OL.!/ D

nX

kD0

bk e�ik! (5.3.14)

with real coefficients bk with OL.0/ D 1 such that

OA.!/ D OL.!/ OL.�!/ Dˇˇ OL.!/

ˇˇ2

(5.3.15)

is identically satisfied for !.

5.3 169

We refer to Daubechies (1992) for a proof of the Riesz lemma 5.3.1. We alsopoint out that the factorization of OA.!/ given in (5.3.15) is not unique.

For a given N, if we select P D PN , then OA.!/ becomes a polynomial of degreeN � 1 in cos! and OL.!/ is a polynomial of degree .N � 1/ in exp.�i!/. Therefore,the generating function Om0.!/ given by (5.3.5) is of degree .2N � 1/ in exp.�i!/.The interval Œ0; 2N � 1� becomes the support of the corresponding scaling functionN�. The mother wavelet N obtained from N� is called the Daubechies wavelet.

Example 5.3.1 (The Haar Wavelet). For N D 1, it follows from (5.3.10) thatP1.t/ 1, and this in turn leads to the fact that Q.cos!/ D 1; OL.!/ D 1 sothat the generating function is

Om0.!/ D1

2

1C e�i!

�: (5.3.16)

This corresponds to the generating function (4.3.46) for the Haar wavelet

Example 5.3.2 (The Daubechies Wavelet). For N D 2, it follows from (5.3.10)that

P2.t/ D

1X

kD0

k C 1

k

!tk D 1C 2t

and hence (5.3.12) gives

ˇˇ OL2.!/

ˇˇ2

D P2.x/ D P2�

sin2!

2

�D 1C 2 sin2

!

2D .2 � cos!/:

Using (5.3.14) in Lemma 5.3.1, we obtain that OL.!/ is a polynomial of degreeN � 1 D 1 and

OL.!/ OL.�!/ D 2 �1

2

ei! C e�i!

�:

It follows from (5.3.14) that

b0 C b1e

�i!�

b0 C b1ei!�

D 2 �1

2

ei! C e�i!

�: (5.3.17)



Equating the coefficients in this identity gives

b20 C b21 D 1 and 2b0b1 D �1: (5.3.18)

These equations admit solutions as

b0 D1

2

�1C

p3�

and b1 D1

2

�1 �

p3�: (5.3.19)

Consequently. the generating function (5.3.5) takes the form

Om0.!/ D

�1C e�i!

2

�2 b0 C b1e

�i!�

D1

8

� �1C

p3�

C�3C

p3�

e�i! C�3 �

p3�

e�2i! C�1 �

p3�

e�3i!

�

(5.3.20)

with Om0.0/ D 1: Comparing coefficients of (5.3.20) and (4.3.3) gives cn as

c0 D1

4p2

�1C

p3�; c1 D

1

4p2

�3C

p3�

c2 D1

4p2

�3 �

p3�; c3 D

1

4p2

�1 �

p3�

9>>=

>>;: (5.3.21)

Consequently, the Daubechies scaling function 2�.t/ takes the form, dropping thesubscript,

�.t/ Dp2hc0 �.2t/C c1 �.2t � 1/C c2 �.2t � 2/C c3 �.2t � 3/

i: (5.3.22)

Using (4.3.30) with N D 2, we obtain the Daubechies wavelet 2 .t/, dropping thesubscript,

.t/ Dp2hd0 �.2t/C d1 �.2t � 1/C d2 �.2t � 2/C d3 �.2t � 3/

i

Dp2h

� c3 �.2t/C c2 �.2t � 1/� c1 �.2t � 2/C c0 �.2t � 3/i;

(5.3.23)

where the coefficients in (5.3.23) are the same as for the scaling function �.t/, butin reverse order and with alternate terms having their signs changed from plus tominus.

5.3 171

On the other hand, the use of (4.3.28) with (4.3.33) also gives the Daubechieswavelet 2 .t/ in the form

2 .t/ Dp2h

�c0 �.2t �1/Cc1 �.2t/�c2 �.2t C1/Cc3 �.2t C2/i: (5.3.24)

The wavelet has the same coefficients as given in (5.3.23) except that thewavelet is reversed in sign and runs from t D �1 to 2 instead of starting from t D 0.It is often referred to as the Daubechies D4 wavelet since it is generated by fourcoefficients.

However, in general, c’s (some positive and some negative) in (5.3.22) arenumerical constants. Except for a very simple case, it is not easy to solve (5.3.22)directly to find the scaling function �.t/. The simplest approach is to set up aniterative algorithm in which each new approximation �m.t/ is computed from theprevious approximation �m�1.t/ by the scheme

�m.t/ Dp2hc0 �m�1.2t/C c1 �m�1.2t � 1/C c2 �m�1.2t � 2/C c3 �m�1.2t � 3/

i:

(5.3.25)

This iteration process can be continued until �m.t/ becomes indistinguishablefrom �m�1.t/. This iterative algorithm is briefly described below starting from thecharacteristic function

�Œ0;1�.t/ D

�1; 0 � t < 10; otherwise.

(5.3.26)

After one iteration the characteristic function over 0 � t < 1 assumes the shapeof a staircase function over the interval 0 � t < 2. In order to describe the algorithm,we select the set of four coefficients c0; c1; c2; c3 given in (5.3.21), deleting the factor1

p2

in each coefficient so that it produces the Daubechies scaling function �.t/

given by (5.3.22) and the orthonormal Daubechies wavelet .t/ (or D4 wavelet)given by (5.3.23) without the factor

p2.

We represent the characteristic function by the ordinate 1 at t D 0. The firstiteration generates a new set of four ordinates c0; c1; c2; c3 at t D 0:0; 0:5; 1:0; 1:5.The second iteration with ordinate c0 at t D 0 produces a new set of anotherfour ordinates c20; c0 c1; c1 c2; c1 c3 at t D 0:00; 0:25; 0:75; and so on. Aftercompleting the second iteration process, there are ten new ordinates c20; c0 c1; c0 c1Cc1 c0; c0 c3Cc21; c1 c2Cc2 c0; c1 c3Cc2 c1; c22Cc3 c0; c2 c3Cc3 c1 c3 c2; c23 att D 0:25; 0:50; 0:75; 1:00; : : : ; 2:25: This iteration process can be described by thematrix scheme



2��

D

2

666666666666666666664

c0

c1

c2 c0

c3 c1

c2 c0

c3 c1

c2 c0

c3 c1

c2

c3

3

777777777777777777775

266664

c0

c1

c2

c3

377775Œ1� D M2M1Œ1�; (5.3.27)

where Mn represents the matrix of the order2nC1 C 2n � 2

��2n C 2n�1 � 2

�

in which each column has a submatrix of the coefficients c0; c1; c2; c3 located twoplaces below the submatrix to its left.

We also use the same matrix scheme for developing the Daubechies wavelet2 .t/ from 2�.t/ which is given by (5.3.22) without the factor

p2. For simplicity,

we assume that only one iteration process gives the final 2�.t/, so this canbe described by four ordinates c0; c1; c2; c3 at t D 0:0; 0:50; 1:0; 1:50: In viewof (5.3.23), these four ordinates produce ten new ordinates spaced 0:25 apart. Theterm �c3�.2t/ in (5.3.23) gives �c3 c0;�c3 c1;�c3 c2: � c23; the term c2 �.2t � 1/

gives c2 c0; c2 c1; c22; c2 c3 shifted two places to the right; and so on for the otherterms, so that the new ten ordinates for the wavelet are given by �c3 c0;�c3 c1;�c3 c2Cc2 c0;�c23Cc2 c1; c22�c1 c0; c2 c3�c21;�c1 c2Cc20;�c1 c3Cc0 c1; c0 c2; c0 c3:These ordinates are generated by the matrix scheme

2 �

D

2666666666666666666664

�c3

0 �c3

c2 0 �c3

0 c2 0 �c3

�c1 0 c2 0

0 �c1 0 c2

c0 0 �c1 0

c0 0 �c1

c0 0

c0

3777777777777777777775

2

66664

c0

c1

c2

c3

3

77775Œ1�: (5.3.28)

Or, alternatively, by the matrix scheme

5.3 173

2 �

D

2

666666666666666666664

c0

c1

c2 c0

c3 c1

c2 c0

c3 c1

c2 c0

c3 c1

c2

c3

3

777777777777777777775

266664

�c3

c2

�c1

c0

377775Œ1�: (5.3.29)

Making reference to Newland (1993b), it can be verified that 3 .t/ can be describedby the matrix scheme

3 �

D M3M2

266664

�c3

c2

�c1

c0

377775Œ1�; (5.3.30)

where the matrix M3 is of order 22 � 10 with ten submatrices Œc0 c1 c2 c3�T , eachorganized two places below its left-hand neighboring matrix.

The matrix scheme (5.3.30) is used to generate wavelets in the inverse discretewavelet transform (IDWT). All subsequent steps of the iteration use the matrices Mr

consisting of submatrices Œc0 c1 c2 c3�T staggered vertically two places each. Aftereight steps leading to 766 ordinates as before, the resulting wavelet is very close tothat in Figure 5.6a.

In order to analyze or synthesize a part of a signal by wavelets, Daubechies (1992)considered the scaling function � defined by (5.3.22) as a building block so that

�.t/ D 0 when t � 0 or t � 3: (5.3.31)

Daubechies (1992) proved that the scaling function � does not admit any simplealgebraic relation in terms of elementary or special functions. She also demonstratedthat � satisfies several algebraic relations that play a major role in computationalanalysis.



2ψ(t)

2φ(t)

t

2

a

b

1.5

0.5

-0.5

-1

-1.5

-20 1 2 30.5 1.5 2.5

0 1 2 30.5 1.5 2.5

0

1

1.5

0.5

-0.5

0

1

Fig. 5.6 (a) The Daubechies wavelet 2 .t/. (b) The Daubechies scaling function 2�.t/

Replacing t byt

2in (5.3.22) gives

�� t

2

�D

p2

3X

kD0

ck�.t � k/ (5.3.32)

which can be found exactly if �.t/; �.t�1/; �.t�2/; �.t�3/ are all known. Supposethat we can find �.0/; �.1/; �.3/. It is known that �.�1/; �.4/, etc., are all zero.Then, by using (5.3.32), we can calculate

�

�1

2

�; �

�3

2

�; �

�5

2

�:

5.3 175

Again, by using (5.3.32) and these new values, we can calculate

�

�1

4

�; �

�3

4

�; �

�5

4

�; �

�7

4

�; �

�9

4

�; �

�11

4

�;

and so on. In order to carry out this recursive process, we set initial values

�.0/ D 0; �.1/ D1

2

�1C

p3�; �.2/ D

1

2

�1 �

p3�; �.3/ D 0:

(5.3.33)

For example, for t D 1, we obtain from (5.3.32) that

�

�1

2

�D

p2hc0 �.1/C c1 �.0/C c2 �.�1/C c3 �.�2/

i

which is, by (5.3.21) and (5.3.31),

Dp2 c0 �.1/ D

�1C

p3�2

8D1

4

�2C

p3�:

Similarly, we can calculate �

�3

2

�; �

�5

2

�so that

t D1

2;

3

2;

5

2;

�.t/ D1

4

�2C

p3�; 0;

1

4

�2 �

p3�; and �.t � 3/ D 0:

A similar calculation gives the values of � at multiples of1

4as given below:

t D1

4;

3

4;

5

4;

7

4;

9

4;

�.t/ D5C 3

p3

16;

9C 5p3

16;

2�1C

p3�

16;

2�1 �

p3�

16;

9 � 5p3

16:

The Daubechies wavelet .t/ is given by (5.3.24). In view of (5.3.31), it turns outthat .t/ D 0 if 2t C 2 � 0 or 2t � 1 � 3, that is, .t/ D 0 for t � �1 or t � 2.Hence, can be computed from (5.3.24) with (5.3.21) and (5.3.33). For example,



.0/ Dp2hc3 �.2/ � c2 �.1/C c1 �.0/� c0 �.�1/

i

Dp2hc3 �.2/ � c3 �.1/

i

D

1 �

p3

4

! 1 �

p3

2

!�

3 �

p3

4

! 1C

p3

2

!

D1

2

�1 �

p3�:

Consequently, .t/ at t D �1;�1

2; 0; 1;

3

2is given as follows:

t D �1; �1

2; 0; 1;

3

2;

.t/ D 0; �1

4;

1

2

�1 �

p3�; �

1

2

�1C

p3�; �

1

4:

Both Daubechies’ scaling function � and Daubechies’ wavelet for N D 2 areshown in Figure 5.6a, b, respectively.

In view of its fractal shape, the Daubechies wavelet 2 .t/ given in Figure 5.7ahas received tremendous attention so that it can serve as a basis for signal analysis.According to Strang’s (1989) analysis, a wavelet expansion based on the D4waveletrepresents a linear function f .t/ D at exactly, where a is a constant. Six waveletcoefficients are needed to represent f .t/ D at C bt2, where a and b are constants.In general, more wavelet coefficients are necessary to represent a polynomial withterms like tn. Figure 5.7a, b exhibits wavelets with N D 3; 5; 7; and 10 coefficients.The range of these wavelets is always .2N � 1/ unit intervals so that more waveletcoefficients generate longer wavelets. As N increases, wavelets lose their irregularshape and become increasingly smooth with a Gaussian harmonic waveform. ForN D 10; the frequency of the waveform is not constant and some minor irregularitiesstill persist on the right. Each of the wavelets in Figure 5.7a, b represents thebasis for a family of wavelets of different levels and different locations along thex-axis. The only difference is that a wavelet with 2N coefficients occupies .2N � 1/

unit intervals with the exception of the Haar wavelet which occupies one interval.Wavelets at each level overlap one another and the amount of overlap depends onthe number of wavelet coefficients involved.

The recursive method just described above yields the values of the building block�.t/ and the wavelet .t/ only at integral multiples of positive or negative powers

5.3 177

ψ(t)

ψ(t)

t

N = 3

N = 5

2a

-2

0

2

-2

0

0 5

t0 9

ψ(t)

ψ(t)

N = 7

N = 10

2b

-2

0

2

-2

0

0 13

t0 19

Fig. 5.7 (a) Wavelets for N D 3; 5 drawn using the Daubechies algorithm. (b) Wavelets forN D 7; 10 drawn using the Daubechies algorithm



of 2. These values are sufficient for equally spaced samples from a signal. Due tothe importance of such powers of 2, the idea of a dyadic number and related notationand terminology seem to be useful in wavelet algorithms.

Definition 5.3.1. (Dyadic Number). A number m is called a dyadic number if andonly if it is an integral multiple of an integral power of 2.

We denote the set of all dyadic numbers by D and the set of all integral multiples byDn for n 2 N. A dyadic number has a finite binary expansion, and a dyadic numberin Dn has a binary expansion with at most n binary digits past the binary point.

Definition 5.3.2. The set of all linear combinations of 1 andp3 with dyadic

coefficients p; q 2 D is denoted by D

hp3i

so that

D

hp3i

Dnp C q

p3 W p; q 2 D

o:

For every integer n, we consider combinations with coefficients in Dn so that

Dn

hp3i

Dnp C q

p3 W p; q 2 Dn

o:

We define the conjugate m of m by

�p C q

p3�

D�

p � qp3�:

The set D

hp3i

is an integer ring under ordinary addition and multiplication.

In terms of two quantities

a D1

4

�1C

p3�

and a D1

4

�1 �

p3�; (5.3.34)

the scaling function 2� can be written as

2�.t/ Dp2

2N�1X

kD0

ck �.2t � k/; .N D 2/

D a �.2t/C .1 � a/ �.2t � 1/C .1C a/ �.2t � 2/C a�.2t � 3/:

(5.3.35)

If 0 � m � 2N � 1,(5.3.35) can be rewritten as

5.3 179

�.m/ Dp2

2N�1X

kD0

c2m�k �.k/: (5.3.36)

This system of equations can be written in the matrix form

266664

�.0/

�.1/

�.2/

�.3/

377775

D

266664

a 0 0 0

1 � a 1 � a a 0

0 a 1 � a 1 � a

0 0 0 a

377775

266664

�.0/

�.1/

�.2/

�.3/

377775: (5.3.37)

This system (5.3.37) has exactly one solution:

�.0/ D 0; �.1/ D 2a; �.2/ D 2a; �.3/ D 0: (5.3.38)

We set �.k/ D 0 for all remaining values of k 2 Z: Then, � can recursively becalculated for all of D by (5.3.35).

Finally, we conclude this section by including the Daubechies scaling function3�.t/ and the Daubechies wavelet 3 .t/ for N D 3. In this case, (5.3.10) gives

P.t/ D P3.t/ D 1C 3t C 6t2; (5.3.39)

where

t D sin2!

2D1

4

�e�i! C 2 � ei!

�and

t2 D1

16

e�2i! C 4C e2i! � 4e�i! � 4ei! C 2

�:

Consequently, (5.3.12) gives the result

ˇˇ OL.!/

ˇˇ2

D3

8e�2i! �

9

4e�i! C

19

4�19

4ei! C

3

8e2i!: (5.3.40)

In this case,

A.!/ D b0 C b1 e�i! C b2 e�2i!; (5.3.41)

so that



ˇˇ OL.!/

ˇˇ2

DA.!/A.�!/ D�

b0 C b1e�i! C b2e�2i!

� �b0 C b1ei! C b2e2i!

�

D�

b20 C b21 C b22

�C e�i! .b0 b1 C b2 b1/C ei! .b0 b1 C b1 b2/C b0 b2e2i! C b0 b2e

�2i! :

(5.3.42)

Equating the coefficients in (5.3.40) and (5.3.42) gives

b20 C b21 C b22 D19

4; b1 b0 C b2 b1 D �

9

4; b2 b0 D

3

8: (5.3.43)

In view of the fact thatˇˇ OL.0/

ˇˇ2

D 1 and P.0/ D 1, the Riesz lemma 5.3.1 ensures

that there are real solutions .b0; b1; b2/ that satisfy the additional requirementb0 C b1 C b2 D 1: Eliminating b1 from this equation and the second equationin (5.3.43) gives

b21 � b1 �9

4D 0

so that

b1 D1

2

�1˙

p10�: (5.3.44)

Consequently,

b0 C b2 D1

2

�1�

p10�: (5.3.45)

The plus and the minus signs in these equations result in complex roots for b0 and

b2. This means that the real root for b1 corresponds to the minus sign in (5.3.44) sothat

b1 D1

2

�1 �

p10�: (5.3.46)

Obviously,

b0 C b2 D1

2

�1C

p10�

and b0 b2 D3

8

5.3 181

lead to the fact that b0 and b2 satisfy

t2 �1

2

�1C

p10�

t C3

8D 0: (5.3.47)

Thus,

.b0; b2/ D1

4

��1C

p10�

˙

q5C 2

p10

�: (5.3.48)

Consequently, A.!/ is explicitly known and, hence, Om0.!/ becomes

Om0.!/ D1

8

�b0 C .3b0 C b1/e

�i! C .3b0 C 3b1 C b2/e�2i!

C .b0 C 3b1 C 3b2/e�3i! C .b1 C 3b2/e

�4i! C b2e�5i!

�; (5.3.49)

which is equal to (4.3.3) Equating the coefficients of (4.3.3) and (5.3.49) gives allsix ck’s as

c0 D

p2

8b0 D

p2

32

� �1C

p10�

C

q5C 2

p10

�;

c1 D

p2

8.3 b0 C b1/ D

p2

32

� �5C

p10�

C 3

q5C 2

p10

�;

c2 D

p2

8.3 b0 C 3 b1 C b2/ D

p2

32

� �5 �

p10�

C

q5C 2

p10

�;

c3 D

p2

8.b0 C 3 b1 C 3 b2/ D

p2

32

� �5 �

p10�

�

q5C 2

p10

�;

c4 D

p2

8.b1 C 3 b2/ D

p2

32

� �5C

p10�

� 3

q5C 2

p10

�;

c5 D

p2

8b2 D

p2

32

� �1C

p10�

�

q5C 2

p10

�:



Fig. 5.8 (a) The Daubechiesscaling function 3�.t/ forN D 3. (b) The Daubechieswavelet 3 .t/ for N D 3

3φ(t)

x

x

0

1

a

b

0

1

1 2 3 4 5

1 3 4 5

2

3ψ(t)

Evidently, the Daubechies scaling function 3�.t/ and the Daubechies wavelet 3 .t/(or simply D6 wavelet) can be rewritten as

3�.t/ Dp2

5X

kD0

ck �.2t � k/and (5.3.50)

3 .t/ Dp2

5X

kD0

dk �.2t � k/; respectively; (5.3.51)

where ck and dk are explicitly known. Figure 5.8a, b exhibits the scaling function3�.t/ and the wavelet 3 .t/.

With a given even number of wavelet coefficients ck; k D 0; 1; : : : ; 2N � 1, wecan define the scaling function � by

�.t/ Dp2

2N�1X

kD0

ck �.2t � k/ (5.3.52)

and the corresponding wavelet by

5.3 183

.t/ Dp2

2N�1X

kD0

.�1/k ck �.2t C k � 2N C 1/; (5.3.53)

where the coefficients ck satisfy the following conditions:

2N�1X

kD0

ck Dp2;

2N�1X

kD0

.�1/k km ck D 0; (5.3.54)

where m D 0; 1; 2; : : : ;N � 1, and

2N�1X

kD0

ck ckC2m D 0; m ¤ 0; (5.3.55)

where m D 0; 1; 2; : : : ;N � 1, and

2N�1X

kD0

c2k D 1: (5.3.56)

When N D 1, two coefficients c0 and c1 satisfy the following equations:

c0 C c1 Dp2; c0 � c1 D 0; c20 C c21 D 1

which admit solutions c0 D c1 D1

p2: They give the classic Haar scaling function

and the Haar wavelet.When N D 2, four coefficients c0; c1; c2; c3 satisfy the following equations:

c0 C c1 C c2 C c3 Dp2; c0 � c1 C c2 � c3 D 0;

c0 c2 C c1 c3 D 0; c20 C c21 C c22 C c23 D 1:

These give solutions

c0 D1

4p2

�1C

p3�; c1 D

1

4p2

�3C

p3�;

c2 D1

4p2

�3�

p3�; c3 D

1

4p2

�1 �

p3�:

These coefficients constitute the Daubechies scaling function (5.3.22) and theDaubechies D4 wavelet (5.3.23) or (5.3.24).



5.4 The Harmonic Wavelets

So far, all wavelets have been constructed from dilation equations with realcoefficients. However, many wavelets cannot always be expressed in functionalform. As the number of coefficients in the dilation equation increases, wavelets getincreasingly longer and the Fourier transforms of wavelets become more tightlyconfined to an octave band of frequencies. It turns out that the spectrum of awavelet with n coefficients becomes more boxlike as n increases. This fact led(Newland, 1993a,b) to introduce a new harmonic wavelet .t/ whose spectrumis exactly like a box, so that the magnitude of its Fourier transform O .!/ is zeroexcept for an octave band of frequencies. Furthermore, he generalized the conceptof the harmonic wavelet to describe a family of mixed wavelets with the simplemathematical structure. It is also shown that this family provides a complete setof orthonormal basis functions for signal analysis. These musical wavelets providegreater frequency discrimination than is possible with harmonic wavelets whosefrequency band is always an octave. A major advantage for all harmonic wavelets isthat they can be computed by an effective parallel algorithm rather than by the seriesalgorithm needed for the dilation wavelet transform (See Mouri and Kubotani, 1995;Newland, 1993a,b, 1994).

Newland (1993a) introduced a real even function e.t/ whose Fourier transformis defined by

O e.!/ D

8<

:

1

4�for � 4� � ! < �2�; and 2� � ! < 4�

0; otherwise; (5.4.1)

where the Fourier transform is defined by

Of .!/ D1

2�

Z 1

�1

f .t/ e�i!t dt: (5.4.2)

The inverse Fourier transform of O e.!/ gives

e.t/ D

Z 1

�1

O e.!/ ei!t d! D1

2�t

sin 4�t � sin 2�t

�: (5.4.3)

On the other hand, the Fourier transform O 0.!/ of a real odd function 0.t/ isdefined by

O 0.!/ D

8ˆ<

ˆ:

i

4�for � 4� � ! < �2�

�i

4�for 2� � ! < 4�

0; otherwise:

(5.4.4)

5.4 The Harmonic Wavelets 185

Then, the inverse Fourier transform gives

0.t/ D

Z 1

�1

O 0.!/ ei!t d! D1

2�t

cos 2�t � cos 4�t

�: (5.4.5)

The harmonic wavelet .t/ is then defined by combining (5.4.3) and (5.4.5) in theform

.t/ D e.t/C i 0.t/

D

�e4� it � e2� it

2�it

�: (5.4.6)

The real and imaginary parts of .t/ are shown in Figure 5.9a, bClearly, the Fourier transform of .t/ is given by

O .!/ D O e.!/C i O 0.!/ (5.4.7)

so that, from (5.4.1) and (5.4.4), we obtain the Fourier transform of the harmonicwavelet .t/

O .!/ D

8<

:

1

2�; 2� � ! < 4�

0; otherwise:(5.4.8)

For the general harmonic wavelet .t/ at level m and translated in k steps of size2�m, we define

O .!/ D

8ˆ<

ˆ:

1

2�2�m exp

��

i!k

2m

�; 2�2m � ! < 4�2m

0; otherwise

; (5.4.9)

where m and k are integers.

The inverse Fourier transform of (5.4.9) gives

2mt � k

�D

e4� i.2mt�k/ � e2� i.2mt�k/

2�i2mt � k

� ; (5.4.10)

where m is a nonnegative integer and k is an integer.


−8−1

−0.8−0.6−0.4−0.2

0.20.40.60.8

−6 −4 −2 2 4 6 80

−8 −6 −4 −2 2 4 6 80

0

1

−1−0.8−0.6−0.4−0.2

0.20.40.60.8

0

1

Re{ψ(t)}

Im{ψ(t )}

a

b

t

t

Fig. 5.9 (a) Real part of .t/ and (b) imaginary part of .t/

The level of the wavelet is determined by the value of m so that, at the level.m D 0/, the Fourier transform (5.4.9) of the wavelet occupies bandwidth 2� to 4� ,as shown in (5.4.8). At level m D �1 with bandwidth 0 to 2� , we define

O .!/ D

8<

:

1

2�e�i!k; 0 � ! < 2�

0; otherwise; (5.4.11)

so that the inverse Fourier transform gives the so-called harmonic scaling function

�t � k

�D

e2� i.t�k/ � 1

2�it � k

� : (5.4.12)


ψ^

(ω)

ω

m = −1

m = 0

m = 1

m = 2

m = 3m = 4

1/2π

1/4π

1/8π

1/16π

32π16π8π2π0 4π

Fig. 5.10 Fourier transforms of harmonic waveletsat levels m D 0; 1; 2; 3; 4

Evidently, the choice of the harmonic wavelet and the scaling function seem tobe appropriate in the sense that they form an orthogonal set. If O .!/ is the Fouriertransform of .t/, then the Fourier transform of g.t/ D

2mt � k

�is

Og.!/ D 2�m exp

��

i!k

2m

�O .2�m!/ : (5.4.13)

Clearly, the Fourier transforms of successive levels of harmonic wavelets decreasein proportion to their increasing bandwidth, as shown in Figure 5.10. For ! < 0,they are always zero.

In order to prove orthogonality of wavelets and scaling functions, we need thegeneral Parseval relation (1.3.17) in the form

Z 1

�1

f .t/ g.t/ dt D 2�

Z 1

�1

Of .!/ Og.!/ d!; (5.4.14)

where f ; g 2 L2.R/ and the factor 2� is present due to definition (5.4.2).

For f ; g 2 L2.R/, we also need another similar result of the form

Z 1

�1

f .t/ g.t/ dt D 2�

Z 1

�1

Of .!/ Og.�!/ d!: (5.4.15)


This result follows from the following formal calculation:

Z 1

�1

f .t/ g.t/ dt D

Z 1

�1

dtZ 1

�1

Of .!1/ d!1

Z 1

�1

Og.!2/ d!2 ei.!1C!2/t

D 2�

Z 1

�1

Of .!1/ d!1

Z 1

�1

Og.!2/ ı.!1 C !2/ d!2

D 2�

Z 1

�1

Of .!1/ Og.�!1/ d!1

D 2�

Z 1

�1

Of .!/ Og.�!/ d!; .! D !1/:

Theorem 5.4.1. The family of harmonic wavelets 2mt � k

�forms an orthogonal

set.

Proof. To prove this theorem, it suffices to show orthogonality conditions:

Z 1

�1

.t/ 2mt � k

�dt D 0; for all m; k; (5.4.16)

Z 1

�1

.t/ 2mt � k

�dt D 0; for m ¤ 0: (5.4.17)

We put g.t/ D 2mt � k

�so that its Fourier transform is

Og.!/ D 2�m exp

��

i!k

2m

�O .2�m!/ (5.4.18)

and then apply (5.4.15) to obtain

Z 1

�1

.t/ g.t/ dt D 2�

Z 1

�1

O .!/ Og.�!/ d!: (5.4.19)

If .t/ and g.t/ are two harmonic wavelets, they have the one-sided Fouriertransforms as shown in Figure 5.10, so that the product O .!/ Og.�!/ must alwaysvanish. Thus, the right-hand side of (5.4.19) is always zero for all k and m, that is,

Z 1

�1

.t/ 2mt � k

�dt D 0; for all m; k: (5.4.20)


To prove (5.4.17), we apply (5.4.14) so that

Z 1

�1

.t/ g.t/ dt D 2�

Z 1

�1

O .!/ Og.!/ d!: (5.4.21)

Clearly, wavelets of different levels are always orthogonal to each other becausetheir Fourier transforms occupy different frequency bands so that the productO .!/ Og.!/ is zero for m ¤ 0.

On the other hand, at the same level .m D 0/, we have

Og.!/ D e�i!k O .!/: (5.4.22)

Substituting this result in (5.4.21) and the value of O .!/ from (5.4.8) gives

Z 1

�1

.t/ .t � k/ dt D1

2�

Z 4�

2�

ei!k d! D 0; (5.4.23)

provided e4� ik D e2� ik; k ¤ 0. This gives e2� ik D 1 for k ¤ 0. Thus, allwavelets translated by any number of unit intervals are orthogonal to each other.Although (5.4.23) is true for m D 0, the same result (5.4.23) is also true for otherlevels except that the unit interval is now that for the wavelet level concerned. Forinstance, for level m, the unit interval is 2�m and translation is equal to any multipleof 2�m. The upshot of this analysis is that the set of wavelets defined by (5.4.10)forms an orthogonal set. Wavelets of different levels (different values of m) arealways orthogonal, and wavelets at the same level are orthogonal if one is translatedwith respect to the other by a unit interval (different values of k).

In view of (5.4.20), it can be shown that

Z 1

�1

22mt � k

�dt D 0: (5.4.24)

Theorem 5.4.2 (Normalization of Harmonic Wavelets). Let f m;kg be thefamily of general harmonic wavelets defined by (5.4.10). Then, we have

Z 1

�1

ˇˇ 2mt � k

�ˇˇ2

dt D 2�m: (5.4.25)


Proof. It follows from (5.4.21) that

Z 1

�1

.t/ .t/ dt D 2�

Z 1

�1

O .!/ O .!/ d!: (5.4.26)

Using (5.4.18) in (5.4.26) gives

Z 1

�1

2mt � k

� 2mt � k

�dt D 2�2�2m

Z 1

�1

O .2�m!/ O .2�m!/ d!:

(5.4.27)

It follows from (5.4.8) that O .2�m!/ D1

2�for 2�2�m � ! < 4�2m so

that (5.4.27) becomes

Z 1

�1

ˇˇ 2mt � k

�ˇˇ2

dt D 2�2�2mZ 4�2m

2�2m

1

.2�/2� d! D 2�m:

This implies the property of normality.

We next investigate some properties of harmonic scaling functions by virtue ofthe Fourier transforms. Newland (1993a) first introduced the even Fourier transform

O�e.!/ D

8<

:

1

4�; �2� � ! < 2�

0; otherwise(5.4.28)

to define an even scaling function

�e.x/ Dsin 2�x

2�x: (5.4.29)

Similarly, the odd Fourier transform given by

O�0.!/ D

8ˆ<

ˆ:

i

4�; �2� � ! < �

�i

4�; 0 � ! < 2�

0; otherwise;

(5.4.30)

gives an odd scaling function

�0.t/ D.1 � cos 2�t/

2�x: (5.4.31)


All of these results allow us to define a complex scaling function �.t/ by

�.t/ D �e.t/C i�0.t/ (5.4.32)

so that

�.t/ De2� it�1

2�it: (5.4.33)

Its Fourier transform O�.!/ is given by

O�.!/ D

8<

:

1

2�; 0 � ! < 2�

0; otherwise.(5.4.34)

The real and imaginary parts of the harmonic scaling function (5.4.33) are shownin Figure 5.11a, b.

Theorem 5.4.3 (Orthogonality of Scaling Functions). The scaling functions �.t/and �.t � k/ are orthogonal for all integers k except k D 0.

Proof. We substitute Fourier transforms O�.!/ and F˚�.t � k/

D e�i!k O�.!/ in

(5.4.15) to obtain

Z 1

�1

�.t/ �.t � k/ dt D 2�

Z 1

�1

O�.!/ O�.�!/ei!k d! D 0: (5.4.35)

The right-hand side is always zero for all k because O�.!/ is the one-sided Fouriertransform given by (5.4.34).

On the other hand, we substitute O�.!/ and F˚�.t�k/

D e�i!k O�.!/ in (5.4.15)

to obtainZ 1

�1

�.t/ �.t � k/ dt D 2�

Z 1

�1

O�.!/ O�.!/ ei!k d!

D1

2�

Z 2�

0

ei!k d!; by (5.4.34)

D1

2�ik

e2� ik � 1

�D 0 for k ¤ 0: (5.4.36)

This shows that �.t/ and �.t � k/ are orthogonal for all integers k except k D 0.


Fig. 5.11 (a) Real part ofthe scaling function � and (b)imaginary part of the scalingfunction �

1

a

b

0.8

0.6

Im{φ(t)}

Re{φ(t)}

0.4

0.2

0

−0.2

−0.4

−0.6

−4 −2 0 2 4 6 8t

−8 −6

−0.8

−1

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−4 −2 0 2 4 6 8t

−8 −6

−0.8

−1

It can also be shown thatZ 1

�1

2mt � k

��t � n

�dt D 0 for all m; k; n .m � 0/; (5.4.37)

Z 1

�1

2mt � k

��t � n

�dt D 0 for all m; k; n .m � 0/: (5.4.38)

Theorem 5.4.4 (Normalization of Scaling Function).

Z 1

�1

ˇ�t � k

�ˇ2dt D 1: (5.4.39)


Proof. We have the identity (5.4.14) so that

Z 1

�1

�t � k

��t � k

�dt D 2�

Z 1

�1

O�.!/ O�.!/ d!:

Thus,

Z 1

�1

ˇ�t � k

�ˇ2dt D 2�

Z 2�

0

1

.2�/2d! D 1 by (5.3.7):


The rest of this section is devoted to wavelet expansions and Parseval’s formulafor harmonic wavelets.

Any arbitrary (real or complex) function f .t/ can be expanded in terms ofcomplex harmonic wavelets in the form

f .t/ DX

m2Z

X

k2Z

ham;k

2mt � k

�C Qam;k

2mt � k

�i; (5.4.40)

where the complex coefficients am;k and Qam;k are defined by

am;k D 2mZ 1

�1

f .t/ 2mt � k

�dt; (5.4.41)

Qam;k D 2mZ 1

�1

f .t/ 2mt � k

�dt: (5.4.42)

In terms of these coefficients, the contribution of a single complex wavelet to thefunction f .t/ is given by

am;k 2mt � k

�C Qam;k

2mt � k

�: (5.4.43)

Adding all these terms gives the expansion (5.4.40).

We next give a formal proof of the Parseval formula:

Z 1

�1

ˇf .t/

ˇ2dt D

X

m2Z

X

k2Z

2�mhˇ

am;k

ˇ2CˇQam;k

ˇ2i: (5.4.44)


Multiplying (5.4.40) by Nf .t/ and then integrating the result from �1 to 1 term byterm give

Z 1

�1

f .t/ f .t/ dt

DX

m2Z

X

k2Z

�am;k

Z 1

�1

f .t/ 2mt � k

�dt C Qam;k

Z 1

�1

f .t/ 2mt � k

�dt

�:

(5.4.45)

We use (5.4.41) and (5.4.42) to replace the integrals on the right-hand sideof (5.4.46) by am;k and Qam;k so that (5.4.46) becomes

Z 1

�1

ˇf .t/

ˇ2dt D

X

m2Z

X

k2Z

2�mhˇ

am;k

ˇ2CˇQam;k

ˇ2i:

It may be noted that, for real functions f .t/; am;k D Qam;k so that the expan-sion (5.4.40) can be simplified.

Another interesting proof of the Parseval formula (5.4.45) is given by Newland(1993a) without making any assumption of the wavelet expansion (5.4.40).

We next define complex coefficients in terms of the scaling function in the form

a�;k D

Z 1

�1

f .t/ �t � k

�dt; (5.4.46)

Qa�;k D

Z 1

�1

f .t/ �t � k

�dt: (5.4.47)

In view of the orthogonality and normalization properties of 2mt � k

�and

�t � k

�, it can be shown that any arbitrary function f .t/ can be expanded in the

form

f .t/DX

k2Z

ha�;k �

t � k

�CQa�;k �

t � k

�iCX

m2Z

X

k2Z

ham;k

2mt � k

�CQam;k

2mt � k

�i:

(5.4.48)

Newland (1993a) proved that this expansion (5.4.48) is equivalent to (5.4.40).

5.5 The Nonuniform Wavelets 195

5.5 The Nonuniform Wavelets

The previous concepts of MRA are developed on regular lattices, that is, thetranslation set is always a group. Recently, Gabardo and Nashed (1998a) considereda generalization of Mallat’s celebrated theory of MRA based on spectral pairs, inwhich the translation set acting on the scaling function associated with the MRA togenerate the subspace V0 is no longer a group but is the union of Z and a translateof Z. More precisely, this set is of the formƒ D f0; r=Ng C 2Z, where N � 1 is aninteger, 1 � r � 2N � 1; andr is an odd integer relatively prime to N. They call thisa nonuniform multiresolution analysis (NUMRA).

In this theory, the translation set ƒ is chosen so that for some measurable setA � R with 0 < jAj < 1; .A; ƒ/ forms a spectral pair, i.e., the collection˚A�1=2e2� i!��A.!/

�2ƒ

forms an orthonormal basis for L2.A/, where �A.!/ is thecharacteristic function of A. The notion of spectral pairs was introduced by Fuglede(1974). The following proposition is proved in Gabardo and Nashed (1998a).

Proposition 5.5.1. Let ƒ D f0; ag C 2Z, where 0 < a < 2, and let A be ameasurable subset of R with 0 < jAj < 1. Then .A; ƒ/ is a spectral pair if andonly if there exist an integer N � 1 and an odd integer r, with 1 � r � 2N � 1 andr and N relatively prime, such that a D r=N, and

N�1X

jD0

ıj=2 �X

n2Z

ınN � �A D 1; (5.5.1)

where � denotes the usual convolution product of Schwartz distributions and ıc isthe Dirac measure at c.

The following is the definition of NUMRA associated with the translation set ƒon R introduced by Gabardo and Nashed (1998a).

Definition 5.5.1. Let N be an integer, N � 1, and ƒ D f0; r=Ng C 2Z, where r isan odd integer relatively prime to N with 1 � r � 2N �1. A sequence fVm W m 2 Zg

of closed subspaces of L2.R/ will be called a nonuniform multiresolution analysis(NUMRA) associated with ƒ if the following conditions are satisfied:

(i) Vm � VmC1 for all m 2 ZI

(ii)S

m2ZVm is dense in L2.R/ andT

m2ZVm D f0gI

(iii) f .t/ 2 Vm if and only if f .2Nt/ 2 VmC1 for all m 2 Z;

(iv) There exists a function � in V0, called the scaling function, such that thecollection f�.t � �/ W � 2 ƒg is a complete orthonormal system for V0:


It is worth noticing that, when N D 1, one recovers from the definition abovethe standard definition of a one-dimensional multiresolution analysis with dilationfactor equal to 2. When N > 1, the dilation factor of 2N ensures that 2Nƒ �

2Z � ƒ. However, the existence of associated wavelets with the dilation 2N andtranslation set ƒ is no longer guaranteed as is the case in the standard setting.

For every m 2 Z, define Wm to be the orthogonal complement of Vm in VmC1.Then we have

VmC1 D Vm ˚ Wm and Wk ? W` if k ¤ `: (5.5.2)

It follows that for m > M,

Vm D VM ˚

m�M�1M

kD0

Wm�k; (5.5.3)

where all these subspaces are orthogonal. By virtue of condition (ii) in the

Definition 5.5.1, this implies

L2.R/ DM

m2Z

Wm; (5.5.4)

a decomposition of L2.R/ into mutually orthogonal subspaces.

Observe that the dilation factor in the NUMRA is 2N. As in the standard case,one expects the existence of 2N � 1 number of functions so that their translation byelements ofƒ and dilations by the integral powers of 2N form an orthonormal basisfor L2.R/.

Definition 5.5.2. A set of functions f 1; 1; : : : ; 2N�1g in L2.R/ is said to be aset of basic wavelets associated with the NUMRA fVm W m 2 Zg if the family offunctions f `.� � �/ W 1 � ` � 2N � 1; � 2 ƒg forms an orthonormal basis for W0.

In the following, our task is to find a set of wavelet functions f 1; 1; : : : ; 2N�1g

in W0 such that˚.2N/m=2 `

.2N/mt � �

�W 1 � ` � 2N � 1; � 2 ƒ

constitutes

an orthonormal basis of Wm. By means of NUMRA, this task can be reducedto find ` 2 W0 such that

˚ `t � �

�W 1 � ` � 2N � 1; � 2 ƒ

constitutes an

orthonormal basis of W0.

Let � be a scaling function of the given NUMRA. Since � 2 V0 � V1, and thef�1;�g�2ƒ is an orthonormal basis in V1, we have


�.t/ DX

�2ƒ

a��1;�.t/ DX

�2ƒ

a�.2N/1=2�.2N/t � �

�; (5.5.5)

with

a� D h�; �1;�i D

Z

R

�.t/ �1;�.t/ dt andX

�2ƒ

ja�j2 < 1: (5.5.6)

Equation (5.5.5) can be written in frequency domain as

O� .2N!/ D Om0.!/ O�.!/; (5.5.7)

where Om0.!/ DP

�2ƒ a�e�2� i��! is called the symbol of �.x/.

We denote 0 D � the scaling function and consider 2N � 1 functions `; 1 �

` � 2N � 1, in W0, as possible candidates for wavelets. Since .1=2N/ `.t=2N/ 2

V�1 � V0, it follows from property (iv) of Definition 5.5.1 that for each `; 0 � ` �

2N � 1, there exists a sequence˚a`� W � 2 ƒ

with

P�2ƒ

ˇa`�ˇ2< 1 such that

1

2N `

� t

2N

�DX

�2ƒ

a`� '.t � �/: (5.5.8)

Taking Fourier transform, we get

O ` .2N!/ D Om`.!/ O�.!/; (5.5.9)

where

Om`.!/ DX

�2ƒ

a`� e�2� i�!: (5.5.10)

The functions Om`; 0 � ` � 2N � 1, are locally L2 functions. In view of the specificform of ƒ, we observe that

Om`.!/ D Om1`.!/C e�2� ir!=N Om2

`.!/; 0 � ` � 2N � 1; (5.5.11)

where Om1` and Om2

` are locally L2; 1=2-periodic functions.

We are now in a position to establish the completeness of the systemf `.t � �/g1�`�2N�1;�2ƒ in V1, and in fact, we will find two equivalent conditionsto the orthonormality of the system by means of the periodic functions Om` as definedin (5.5.11).


Lemma 5.5.1. Let � be a scaling function of the given NUMRA as in Defini-tion 5.5.1. Suppose that there exist 2N � 1 functions `; 1 � ` � 2N � 1, in V1such that the family of functions f `.t � �/g0�`�2N�1;�2ƒ forms an orthonormalsystem in V1. Then the system is complete in V1.

Proof. By the orthonormality of ` 2 L2.R/; 0 � ` � 2N � 1, we have in the timedomain

˝ k.t � �/; `.t � �/

˛D

Z

R

k.t � �/ `.t � �/ dx D ık;`ı�;� ;

where �; � 2 ƒ and k; ` 2 f0; 1; 2; : : : ; 2N � 1g. Equivalently, in the frequencydomain, we have

ık;`ı�;� D

Z

R

O k.!/ O `.!/ e�2� i!.��/d!:

Taking � D 2m; � D 2n where m; n 2 Z, we have

ık;`ım;n D

Z

R

O k.!/ O `.!/ e�2� i!2.m�n/d!

D

Z

Œ0;N/e�4� i!.m�n/

X

j2Z

O k.! C Nj/ O `.! C Nj/ d!:

Let

hk;`.!/ DX

j2Z

O k.! C Nj/ O `.! C Nj/:

Then, we have

ık;`ım;n D

Z

Œ0;N/e�4� i!.m�n/hk;`.!/ d!

D

Z

Œ0;1=2/

e�4� i!.m�n/

2

42N�1X

pD0

hk;`

�! C

p

2

�3

5 d!;

and

2N�1X

pD0

hk;`

�! C

p

2

�D 2ık;`: (5.5.12)


Also on taking � Dr

NC 2m and � D 2n, where m; n 2 Z, we have

0 D

Z 1

�1

e�4� i!.m�n/ e�2� i!�r=N O k.!/ O `.!/ d!

D

Z

Œ0;N/e�4� i!.m�n/ e�2� i!�r=N

X

j2Z

O k.! C Nj/ O `.! C Nj/ d!

D

Z

Œ0;N/e�4� i!.m�n/ e�2� i!�r=Nhk;`.!/ d!

D

Z

Œ0;1=2/

e�4� i!.m�n/ e�2� i!�r=N

2

42N�1X

pD0

e�� ipr=N hk;`

�! C

p

2

�3

5 d!:

Thus, we conclude that

2N�1X

pD0

˛p hk;`

�! C

p

2

�D 0; where ˛ D e�� ir=N : (5.5.13)

Now we will express the conditions (5.5.12) and (5.5.13) in terms of Om` as follows:

hk;`.2N!/ DX

j2Z

O k

�2N

�! C

j

2

��O `

�2N

�! C

j

2

��

DX

j2Z

Omk

�! C

j

2

�O�

�! C

j

2

�Om`

�! C

j

2

�O�

�! C

j

2

�

DX

j2Z

Omk

�! C

j

2

�Om`

�! C

j

2

� ˇˇ O�

�! C

j

2

�ˇˇ2

Dh

Om1k.!/ Om1

`.!/C Om2k.!/ Om2

`.!/iX

j2Z

ˇˇ O�

�! C

j

2

�ˇˇ2

C

2

4 Om1k.!/ Om2

`.!/X

j2Z

e2� i.!Cj=2/r=N

ˇˇ O�

�! C

j

2

�ˇˇ2

3

5

C

2

4 Om2k.!/ Om2

`.!/X

j2Z

e�2� i.!Cj=2/r=N

ˇˇ O�

�! C

j

2

�ˇˇ2

3

5 :


Therefore,

hk;`.2N!/ Dh

Om1k.!/ Om1

`.!/C Om2k.!/ Om2

`.!/i 2N�1X

jD0

h0;0

�! C

j

2

�

C

2

4 Om1k.!/ Om2

`.!/ e2� i!r=N2N�1X

jD0

˛�jh0;0

�! C

j

2

�3

5

C

2

4 Om2k.!/ Om2

`.!/ e�2� i!r=N2N�1X

jD0

˛jh0;0

�! C

j

2

�3

5

D 2h

Om1k.!/ Om1

`.!/C Om2k.!/ Om2

`.!/i:

By using the last identity and equations (5.5.12) and (5.5.13), we obtain

2N�1X

pD0

�Om1

k

�! C

p

4N

�Om1`

�! C

p

4N

�C Om2

k

�! C

p

4N

�Om2`

�! C

p

4N

��D ık;`;

(5.5.14)

and

2N�1X

pD0

˛p

�Om1

k

�! C

p

4N

�Om1`

�! C

p

4N

�C Om2

k

�! C

p

4N

�Om2`

�! C

p

4N

��D 0;

(5.5.15)

for 0 � k; ` � 2N � 1, where ˛ D e�� ir=N :

Both of these conditions together are equivalent to the orthonormality of thesystem f `.t � �/ W 0 � ` � 2N � 1; � 2 ƒg : The completeness of this system inV1 is equivalent to the completeness of the system

˚12N `

.t=2N/� �/ W 0� ` �

2N � 1; � 2 ƒg in V0. For a given arbitrary function f 2 V0, by assumption, thereexists a unique function Om.!/ of the form

P�2ƒ b� e�2� i�! , where

P�2ƒ jb�j2 <

1 such that Of .!/ D Om.!/ O�.!/: Therefore, in order to prove the claim, it is enoughto show that the system of functions

P Dne�4� iN!� Om`.!/ �A.!/ W 0 � ` � 2N � 1; � 2 ƒ

o


is complete in L2.A/, where A � R with 0 < jAj < 1. Since the collection˚e2� i!� �A.!/

�2ƒ

is an orthonormal basis for L2.A/, therefore there exist locallyL2 functions g1 and g2 such that

g.!/ Dhg1.!/C e�2� i!r=Ng2.!/

i�A.!/:

Assuming that g is orthogonal to all functions in P , we then have for any � 2 ƒ and` 2 f0; 1; : : : ; 2N � 1g, that

0 D

Z

Ae�4� iN!� Om`.!/ g.!/ d!

D

Z

Œ0;1=2/

e�4� iN!�h

Om`.!/ g.!/C Om`.! C N=2/ g.! C N=2/i

d!

D

Z

Œ0;1=2/

e�4� iN!�h

Om1`.!/ g1.!/C Om2

`.!/ g2.!/i

d!: (5.5.16)

Taking � D 2m, where m 2 Z, and defining

w`.!/ D Om1`.!/ g1.!/C Om2

`.!/ g2.!/; 0 � ` � 2N � 1;

we obtain

0 D

Z

Œ0;1=2/

e�2� i!.4N/mw`.!/ d!

D

Z

Œ0;1=4N/e�2� i!.4N/m

2N�1X

jD0

w`

�! C

j

4N

�d!:

Since this equality holds for all m 2 Z, therefore

2N�1X

jD0

w`

�! C

j

4N

�D 0 for a.e.! (5.5.17)

Similarly, on taking � D 2m C r=N, where m 2 Z, we obtain

0 D

Z

Œ0;1=2/

e�2� i!.4N/m e�2� i2r!w`.!/ d!

D

Z

Œ0;1=4N/e�2� i!.4N/m e�2� i2r!

2N�1X

jD0

˛jw`

�! C

j

4N

�d!:


Hence, we deduce that

2N�1X

jD0

˛jw`

�! C

j

4N

�D 0 for a.e.!;

which proves our claim.

If 0; 1; : : : ; 2N�1 2 V1 are as in Lemma 5.5.1, one can obtain from them anorthonormal basis for L2.R/ by following the standard procedure for construction ofwavelets from a given MRA (see Chapter 4). It can be easily checked that for everym 2 Z, the collection

F.`;m; �/ Dn `;m;�.t/ W .2N/m=2 `

.2N/mt � �

�W 0 � ` � 2N � 1; � 2 ƒ

o

(5.5.18)

is a complete orthonormal system for VmC1. Therefore, it follows immediatelyfrom (5.5.4) that the collection F.`;m; �/ forms a complete orthonormal systemfor L2.R/.

The following theorem proves the necessary and sufficient condition for theexistence of associated set of wavelets to nonuniform multiresolution analysis.

Theorem 5.5.1. Consider a NUMRA with associated parameters N and r as inDefinition 5.5.1, such that the corresponding space V0 has an orthonormal systemof the form f�.t � �/ W � 2 ƒg, where ƒ D f0; r=Ng C 2Z and O� satisfies the two-scale relation

O� .2N!/ D Om0.!/ O�.!/; (5.5.19)

where Om0 is of the form

Om0.!/ D Om10.!/C e�2� i!r=N Om2

0.!/; (5.5.20)

for some locally L2 functions Om10 and Om2

0. Define M0 as

M0.!/ DˇOm10.!/

ˇ2CˇOm20.!/

ˇ2: (5.5.21)

Then a necessary and sufficient condition for the existence of associated wavelets 1; : : : ; 2N�1 is that M0 satisfies the identity

M0

�! C

1

4

�D M0.!/: (5.5.22)


Proof. The orthonormality of the collection of functions f�.t � �/ W � 2 ƒg,which satisfies (5.5.19), implies the following identities as shown in the proof ofLemma 5.5.1:

2N�1X

pD0

�ˇˇ Om1

0

�! C

p

4N

�ˇˇ2

Cˇˇ Om2

0

�! C

p

4N

�ˇˇ2�

D 1; (5.5.23)

and

2N�1X

pD0

˛p

�ˇˇ Om1

0

�! C

p

4N

�ˇˇ2

Cˇˇ Om2

0

�! C

p

4N

�ˇˇ2�

D 0; (5.5.24)

where ˛ D e�� ir=N . Similarly, if f `g`D1;:::;2N�1 is a set of wavelets associated withthe given NUMRA, then it satisfies the relation (5.5.9), and the orthonormality ofthe collection f `g`D0;1;:::;2N�1 in V1 is equivalent to the identities

2N�1X

pD0

�Om1

k

�! C

p

4N

�Om1`

�! C

p

4N

�C Om2

k

�! C

p

4N

�Om2`

�! C

p

4N

��D ık;`;

(5.5.25)

and

2N�1X

pD0

˛p

�Om1

k

�! C

p

4N

�Om1`

�! C

p

4N

�C Om2

k

�! C

p

4N

�Om2`

�! C

p

4N

��D 0;

(5.5.26)

for 0 � k; ` � 2N � 1.

If ! 2 Œ0; 1=4N� is fixed and a`.p/ D Om1`

�! C

p

4N

�; b`.p/ D Om2

`

�! C

p

4N

�

are vectors in C2N for p D 0; 1; : : : ; 2N � 1, where 0 � ` � 2N � 1, then the

solvability of system of equations (5.5.25) and (5.5.26) is equivalent to

M0

�! C

.p C N/

4N

�D M0

�! C

p

4N

�; ! 2 Œ0; 1=4N� ; p D 0; 1; : : : ; 2N � 1;

which is equivalent to (5.5.22). For the proof of this result, the reader is referred toGabardo and Nashed (1998b).


We note here that the function M0 in the above theorem can also be written interms of the filter Om0 as

M0.!/ D

hˇOm0

! C N

2

�ˇ2C j Om0.!/j

2i

2:

When N D 1, we have r D 1 and ˛ D �1 so that the equations (5.5.23) and (5.5.24)reduce to M0.!/ D 1=2, or the more familiar quadrature mirror filter conditionfrom wavelet analysis j Om0 .! C 1=2/j

2 C j Om0.!/j2 D 1, and, in particular, M0 is

automatically 1=4-periodic. When N D 2, we must have r D 1 or 3, so that ˛ D ˙i.In that case, the 1=4-periodicity of M0 follows again automatically from (5.5.23)and (5.5.24). When N � 3, we note that the conditions (5.5.23) and (5.5.24) do notimply the 1=4-periodicity of the function M0 (see Gabardo and Nashed, 1998a,b).

Example 5.5.1 (Haar NUMRA). If we take r D 1, then ƒ D f0; 1=Ng C 2Z, andchoosing � D �AN ; where

AN D

N�1[

mD0

�2m

N;2m C 1

N

�;

we have

� D �Œ0;1=N/ �

N�1X

mD0

ı2m=N :

We now define V0 as the closed linear span of f�.t � �/g�2ƒ, i.e., V0Dspan f�.t��/ W

� 2 ƒg and Vj, for each integer m, by the relation f .t/ 2 Vm if and only iff .t=.2N/m/ 2 V0. Then, the condition (i) of the Definition 5.5.1 is verified byfact that

1

2N�� t

2N

�D �Œ0;2/ �

1

2N

N�1X

mD0

ı4m

Dı0 C ı1=N

��

1

2N

N�1X

mD0

ı4m: (5.5.27)

Equation (5.5.26) can be written in the frequency domain as

O�.2N!/ D Om0.!/ O�.!/; (5.5.28)


where

Om0.!/ D1

2N

1C e�2� i!=N

�"

N�1X

kD0

e�8� i!k

#:

Furthermore, we have

Om10.!/ D Om2

0.!/ D1

2N

N�1X

kD0

e�8� i!k: (5.5.29)

Here, both the functions Om10 and Om2

0 are 1=4-periodic and so is M0. Therefore,Theorem 5.5.1 can be applied to show the existence of the associated wavelets.Hence, when N D 1; � D �Œ0;1/ and Om1

0.!/ D Om20.!/ D 1=2, then the

corresponding wavelet 1 is given by the identity

O 1.2!/ De�2� i! � 1

2O�.!/:

Or, equivalently,

1 D ��Œ0;1=2/ C �Œ1=2;1/;

which is the classical Haar wavelet. For N D 2, the periodic functions Om10 and Om2

0

are given by

Om10.!/ D Om2

0.!/ De�4� i! cos.4�!/

2;

and thus M0.!/ D cos2.4�i!/=2. In this case, the associated wavelets can easilybe computed using the relation O `.4!/ D Om`.!/ O�.!/; ` D 1; 2; 3. Therefore, wehave

1 D �Œ0;1=2/ � �Œ1;3=2/;

2 D ��Œ�8=8;�7=8/ C �Œ�7=8;�6=8/ � �Œ�6=8;�5=8/ C �Œ�5=8;�4=8/

��Œ0;1=8/ C �Œ1=8;2=8/ � �Œ2=8;3=8/ C �Œ3=8;4=8/;

3 D ��Œ�8=8;�7=8/ C �Œ�7=8;�6=8/ � �Œ�6=8;�5=8/ C �Œ�5=8;�4=8/C�Œ0;1=8/ � �Œ1=8;2=8/ C �Œ2=8;3=8/ � �Œ3=5;4=8/:


5.6 Exercises

1. Show that the nth-order B-spline Nn.t/ and its integer translates form a partitionof unity, that is,

X

k2Z

Nn.t � k/ D 1; for all t 2 R:

2. Show that cardinal B-spline Nn.t/ is symmetric about t D n=2, that is,

Nn

�n

2C t�

D Nn

�n

2� t�; for all t 2 R:

3. Use induction on n to prove that

Nn.t/ D

htNn�1.t/C .n � t/Nn�1.t � 1/

i

n � 1:

4. Show that the two-scale equation associated with the linear spline function

N1.t/ D

�1 � jtj; 0 < jtj < 10; otherwise

is

N1.t/ D1

2N1.2t C 1/C N1.2t/C

1

2N1.2t � 1/:

Hence, show that

X

k2Z

ˇˇ O�.! C 2�k/

ˇˇ2

D 1 �2

3sin2

�!2

�:

5. Using result (5.2.10), prove that

ONn.!/

ONn

�!2

� D

�1C e�i!=2

2

�n

:

Hence, derive the following:

(a) ONn.!/ D1

2n

nX

kD0

n

k

!exp

��

ik!

2

�ONn

�!2

�;

5.6 Exercises 207

(b) Nn.t/ D1

2n�1

nX

kD0

n

k

!Nn.2t � k/:

6. Use the Fourier transform formula (5.2.27) for O .!/ of the Franklin wavelet to show that satisfies the following properties:

(a) O .0/ D

Z 1

�1

.t/ dt D 0;

(b)Z 1

�1

t .t/ dt D 0;

(c) is symmetric with respect to t D �1

2:

7. From an expression (5.2.26) for the filter, show that

Om.!/ D

2C 3 cos! C cos2 !

�1C 2 cos2 !

�

and, hence, deduce

O .2!/ D exp.�i!/

�2 � cos! C cos2 !

1C 2 cos2 !

�O�.!/:

8. Obtain a solution of (5.3.22) for the following cases:

(a) c0 D c1 D1

p2; c2 D c3 D 0;

(b) c0 D c2 D1

2p2; c1 D

1p2; c3 D 0;

(c) c0 Dp2; c1 D c2 D c3 D 0:

9. Given six wavelet coefficients ck .N D 6/; write down six equations from(5.3.50) to (5.3.52). Show that these six equations generate the Daubechiesscaling function (5.3.50) and the Daubechies D6 wavelet (5.3.51).

10. Prove the following results (Newland, 1993a):

(a)Z 1

�1

2mt � k

� 2nt � `

�dt D 0 for all m; k; n; ` .m; n � 0/:

(b)Z 1

�1

2mt � k

� 2nt � `

�dt D 0 for all m; k; n; ` .m; n � 0/:

(c) When m D n and k D `, the above result 1(b) becomes

Z 1

�1

ˇ 2mt � k

�ˇ2dt D 2�m:

(d)Z 1

�1

�.t � m/ �.t � n/ dt D 0 for all m; n;m ¤ n:


(e)Z 1

�1

2mt � k

��t � `

�dt D 0 for all m; k; ` .m � 0/:

(f)Z 1

�1

2mt � k

��t � `

�dt D 0 for all m; k; ` .m � 0/:

11. Show that (Newland, 1993a)

(a) a�;k D 2�

Z 1

�1

Of .!/ O�.!/ ei!k d!:

(b)1X

kD�1

a�;k �t � k

�D

D 2�

1X

kD�1

Z 1

�1

d!1

Z 1

�1

Of .!1/ O�.!1/ O�.!2/ ei!2t ei.!1�!2/kd!2

D

Z 2�

0

Of .!/ ei!t d!:

12. Prove that (Newland, 1993a)

(a) am;k D 2�

Z 1

�1

Of .!/ O .!2�m/ ei!k2�md!:

(b) Qam;k D 2�

Z 1

�1

Of .�!/ O .!2�m/ e�i!k2�md!:

13. Show that the wavelet expansion (5.4.40) is equivalent to that of (5.4.48)14. Prove Parseval’s formula (5.4.44) without making any assumption of the

wavelet expansion (5.4.40).15. If the Fourier transform of a wavelet r;n.t/ is

O r;n.!/ D

8<

:

1

2�.n � r/; 2�r � ! < 2�n

0; otherwise;

show that

r;n.x/ De2� int � e2� irt

2�i.n � r/t:

16. Introducing translation of the wavelet by s D k.n � r/�1, generalize the resultof Exercise 15 in the form

r;n.t � s/ De2� in.t�s/ � e2� ir.t�s/

2�i.n � r/.t � s/;

5.6 Exercises 209

where

O r;n.!/ D

8<

:

exp.�i!s/

2�.n � r/; 2�r � ! < 2�n

0; otherwise:

If r D 2m and n D 2mC1, show that the wavelet r;n.x/ reduces to that givenby (5.4.10).

17. Prove the following results for r;n.t/ in Exercise 16 with s1 D k1.n � r/�1 ands2 D k2.n � r/�1:

(a)Z 1

�1

r;n.t � s1/ r;n.t � s2/ dt D 0 for any k1 and k2;

(b)Z 1

�1

r;n.t � s1/ r;n.t � s2/ dt D 0 for k1 ¤ k2;

(c)Z 1

�1

ˇ r;n.t � s/

ˇ2dt D .n � r/�1 for s D k.n � r/�1:

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Index

AAffine wavelet, 107Amplitude spectrum, 4Analysis operator, 70Antilinearity, 104Approximate identity theorem, 15

BB-spline wavelet, 156Battle-Lemarié scaling function, 165Battle-Lemarié wavelet, 164Bezout’s theorem, 168Building block, 173Butterworth filter, 160

CCanonical coherent state, 62Center and Radius, 58Characteristic function, 4, 5, 128Conjugate quadratic filters, 135Conjugation, 8, 65, 75, 84Conservation of energy, 68Construction of mother wavelet, 128Construction of wavelets via MRA, 123Continuity, 9Continuous Fractional Wavelet Transform, 115Continuous Gabor transform, 62Continuous wavelet transform, 100Convolution theorem, 13, 78Cooley, 31

DDaubechies, 93, 166, 173Daubechies D4 wavelet, 171Daubechies algorithm, 176Daubechies scaling, 176

Daubechies scaling function, 179Daubechies Wavelet, 166Daubechies wavelet, 169Derivatives of Fourier transform, 9Dilation, 8, 75, 104Dilation equation, 125Dilation operator, 8Discrete filter, 131Discrete Fourier transform, ix, 2, 25, 26Discrete Gabor function, 69Discrete Gabor transform, 69Discrete wavelet transform, 106, 107Dual frame, 71Dyadic interval, 128Dyadic number, 178

EElongations of MRA based wavelet, 155Euler, 1Expectation value, 50

FFast Fourier transform, ix, 2, 31Fast wavelet transform, 149fast wavelet transform, ix, 124FFT algorithm, 33Filter conjugate, 135Fourier, 1Fourier coefficient, 2Fourier integral theorem, 2Fourier series, 1Fourier transform, 4, 17Fourier transform in L2.R/, 17Fourier transforms in L1.R/, 3Fractional Fourier Transform, 34Fractional Fourier transform, 35fractional Fourier transform, ix, 2

© Springer International Publishing AG 2017L. Debnath, F.A. Shah, Lecture Notes on Wavelet Transforms, Compact Textbooksin Mathematics, DOI 10.1007/978-3-319-59433-0

217

218 Index

Fractional Wavelet Transform, 111Frames, 107Franklin wavelet, ix, 156, 162, 163Frequency resolution, 102Frequency shift, 28Frequency spectrum, 4Fuglede, 195

GGabardo, 195Gabor filter, 83Gabor frame, 70, 71Gabor frame operator, 71Gabor function, 62Gabor lattice, 69Gabor representation problem, 71Gabor series, 69Gabor transform, 61, 62Gabor wavelet, 62Gate function, 6Gauss, 31Gaussian function, 22Gelfand, 71Gelfand mapping, 71Gelfand-Weil-Brezin-Zak transform, 72General modulation, 16General Parseval’s relation, 23

HHaar function, 21Haar NUMRA, 204Haar scaling function, 183Haar wavelet, 96, 111, 139, 143, 169,

183Harmonic wavelet, 184, 185harmonic wavelet, 156Harmonic wavelet expansions, 193Heaviside function, 6Heisenberg uncertainty principle, 50Heisenberg’s inequality, 51

IInverse Fourier transform, 23Inversion, 75Inversion Formula, 118Inversion formula, 22, 29, 67, 105

KKernel, 102

LLebesgue integrable function, 3Linear canonical transform , 80Linearity, 8, 65, 75, 84Linearty, 104Lord Kelvin, 1Low pass filter, 131Lp-norm, 3

MMallat, 1, 55, 123Measurable set, 195Mexican hat wavelet, 100, 120Meyer, 93, 123Meyer wavelet, 145Micchelli, 156Modulation, 8, 28, 65, 75, 84Modulation operator, 8Mother wavelet, 99MuItiresolution analysis, 124Multiresolution analysis, 123

NNashed, 195Newland, 184Non-uniform multiresolution analysis,

195Nonuniform wavelet, 195nonuniform wavelets, ix, 156Norm, 3Normalization of Harmonic, 189Normalization of scaling function, 192

OOrthogonal projection, 125Orthogonality condition, 130Orthogonality relation, 87Orthonormal system, 138Orthonormal wavelet, 109, 110orthonormal wavelet basis, 127

PParity, 84Parseval’s formula, 66, 104, 193Parseval’s formula for LCT, 82Parseval’s Identity for FrFT, 45Parseval’s relation, 19Phase spectrum, 4Piecewise constant function, 128Plancheral’s formula, 82

Index 219

Plancherel’s theorem, 25Product and convolution of Zak transform, 75

QQuasiperiodic relation, 72

Rradar uncertainty principle, 88Rectangular pulse, 6Refinement equation, 125Riemann-Lebesgue lemma, 10Riesz spectral factorization, 168

SScaling, 8Scaling parameter, 99Set of basic wavelet, 196Shannon’s sampling function, 141Shannon’s system, 142Shannon’s wavelet, 140Shifting, 8Short-time Fourier transform, 59Sine function, 159Spectral pairs, 195Spline wavelet, 156Strang, 147, 152Summability kernel, 14Symmetry, 75, 104symmetry, 155Synthesis operator, 71

TThe time-frequency analysis, 55

The windowed Fourier transform, 56Time resolution, 102Time shift, 28Translation, 65, 75, 84, 104Translation and Modulation, 75Translation operator, 8Translation parameter, 99Translation set, 196Tukey, 31Two-scale equation, 125Two-scale relation, 139

UUncertainty principle, 50uncertainty principle, ix, 2

WWang, 156wave equation, 46Wavelet, 95Weierstrass transformation, 64Weil-Brezin transform, 71Weyl-Heisenberg frame, 70window daughter function, 83Window function, 56windowed Fourier transform, 63Windowed Linear Canonical Transform, 79windowed linear canonical transform, 83

ZZak, 71Zak transform, 71, 72

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Lecture notes on wavelet transforms