Lecture notes on wavelet transforms 978-3-319-59433-0, 3319594338, 978-3-319-59432-3

Table of contents :
Front Matter ....Pages i-xii
The Fourier Transforms (Lokenath Debnath, Firdous A. Shah)....Pages 1-54
The Time-Frequency Analysis (Lokenath Debnath, Firdous A. Shah)....Pages 55-91
The Wavelet Transforms (Lokenath Debnath, Firdous A. Shah)....Pages 93-122
Construction of Wavelets via MRA (Lokenath Debnath, Firdous A. Shah)....Pages 123-153
Elongations of MRA-Based Wavelets (Lokenath Debnath, Firdous A. Shah)....Pages 155-209
Back Matter ....Pages 211-219

Citation preview

Compact Textbooks in Mathematics

Lokenath Debnath Firdous A. Shah

Lecture Notes on Wavelet Transforms

Compact Textbooks in Mathematics

Compact Textbooks in Mathematics This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures or seminars which are also suitable for self-study. The books provide students and teachers with new perspectives and novel approaches. They feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance. • • •

compact: small books presenting the relevant knowledge learning made easy: examples and exercises illustrate the application of the contents useful for lecturers: each title can serve as basis and guideline for a 2–3 hours course/lecture/ seminar

More information about this series at http://www.springer.com/series/11225

Lokenath Debnath • Firdous A. Shah

Lecture Notes on Wavelet Transforms

Lokenath Debnath School of Mathematical and Statistical Sciences University of Texas – Rio Grande Valley Edinburg, TX, USA

Firdous A. Shah Department of Mathematics University of Kashmir Anantnag, Jammu and Kashmir India

ISSN 2296-4568 ISSN 2296-455X (electronic) Compact Textbooks in Mathematics ISBN 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 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife Sadhana with love, gratitude, and respect Lokenath Debnath To my beautiful niece Aleeza Firdous A. Shah

Preface

“A teacher can never truly teach unless he is still learning himself. A lamp can never light another lamp unless it continues to burn its own flame. The teacher who has come to the end of this subject, who has no living traffic with his knowledge but merely repeats his lessons to his students, can only load their minds; he cannot quicken them.” Rabindranath Tagore Nobel Prize Winner for Literature (1913)

The origin of wavelet analysis can be traced to the classic theory of harmonic analysis and the seminal contributions of Joseph Fourier, Alfred Haar, and Paul Levy. Since the appearance of the pioneering work of Morlet and Grossman in the 1980s wavelet methodology has been introduced to the literature as a regular alternative for analyzing irregular situations where the data/signal contains scaling properties, discontinuities, sharp spikes, etc. These contributions were followed by the introduction of the general idea of multiresolution analysis by Mallat and Meyer 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 wavelets came through the efforts of mathematicians with a variety of backgrounds and specialties, and of engineers and scientists with an eye for better solutions and models in their applications. Nowadays, there is no doubt that the introduction of wavelet transform was one of the most important events in mathematics over the past few decades. They have fascinated the scientific, engineering, and mathematical community with their versatile applicability and are now considered as a nucleus of shared aspirations and ideas. The application areas for wavelets have been growing for the last two decades at a very rapid rate. They have been applied in a number of fields including signal processing, image processing, sampling theory, turbulence, approximation theory, geophysics, astrophysics, quantum mechanics, computer graphics, statistics, economics and finance, quality control, differential and integral equations, numerical analysis, neuroscience, medicine, neural networks, chemistry, nano-technology, and even in political time series. A consequence of this interest is the appearance of several books, journals, and a large volume of research articles on this subject. Currently, there are many books in the market, with more being written everyday, which treat the subject of wavelets from a wide range of perspectives and with several areas of a large spectrum of possible applications. Workers in the field judge 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 a systematic introduction to the subject of wavelet transforms. While teaching courses on integral transforms and wavelet transforms, the authors have had difficulty choosing textbooks to accompany lectures on wavelet transforms at the senior undergraduate and/or graduate levels. Many hours of study convinced us that there is a need for lecture notes on wavelet transforms for mathematicians, scientists and engineers that provide both a systematic exposition of the basic ideas and results of wavelet analysis. The selection, arrangement, and presentation of the material in these lecture notes have carefully been made based on our past and present teaching, research, and professional experience. In particular, drafts of these lecture notes have been used by us for regular teaching courses in wavelet transforms and their applications at the University of Texas–Pan American, USA, and the University of Kashmir, India. These notes differ from many textbooks with similar titles due to major emphasis placed on numerous topics and systematic development of the underlying theory before presenting applications and the inclusion of many new and modern topics such as fractional Fourier transforms, windowed canonical transforms, fractional wavelet transforms, fast wavelet transforms, spline wavelets, Daubechies wavelets, harmonic wavelets, and nonuniform wavelets. Therefore, our primary goal is to show how different types of wavelets can be constructed, illustrate why they provide us with a particularly powerful tool in mathematical analysis, and indicate how they can be used in applications. Our secondary goal is to develop required analytical knowledge and skills on the part of the reader, rather than focus on the importance of more abstract formulation with full mathematical rigor. Indeed, our major emphasis is to provide an accessible working knowledge of the analytical and 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 is as simple as possible, but to paraphrase Einstein “it should not be simpler.” We have attempted to make the monograph as self-contained as possible. Mathematics, science, and engineering students need to gain a sound knowledge of mathematical and computational skills by the systematic development of underlying theory with varied applications and provision of carefully selected fully worked-out examples combined with their extensions and refinements through addition of a large set of a wide variety of exercises at the end of each chapter. Numerous standard and challenging worked-out examples and exercises are included so that they stimulate research interest among senior undergraduates and graduate students. Another special feature of this book is to include sufficient modern topics which are vital 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 the monograph’s main ideas are developed. Wavelet transforms can be considered as a 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 the motivation of Fourier series and Fourier transforms in L1 .R/ and L2 .R/ followed by their basic properties. Several important results including the approximate identity theorem, general Parseval’s relation, and Plancherel theorem are discussed in some detail. Discrete Fourier transform, fast Fourier transform, and fractional Fourier transform are also discussed briefly for the purpose of comparing them with the continuous, discrete, and fractional wavelet transforms. Applications of the fractional Fourier transform in solving generalized nonhomogeneous differential equations including the generalized wave and heat equations are also given. Special attention is also given to the Heisenberg’s uncertainty principle. Chapter 2 is devoted to a fairly detailed examination of the joint time-frequency analysis of signals. The main goal here is to set the foundation for the development of continuous and discrete wavelet transforms. We begin with the time-frequency localization of signals which leads us to the windowed Fourier transform. This is followed by the Gabor transform and its basic properties, including the inversion formula. Special attention is also given to the Zak transform and its basic properties. Based on the relationship between the Fourier transform and linear canonical transform, a hybrid windowed transform, namely the windowed linear canonical transform, has been introduced. Its basic properties and several results including the orthogonality relation and inversion formula are also discussed. The heart of the wavelet theory is covered in Chapters 3 and 4 in a comprehensive approach. We start Chapter 3 with the introduction of wavelets and wavelet transforms with examples. The basic ideas and properties of wavelet transforms are discussed with special attention given to the use of different wavelets for resolution and synthesis of signals. This is followed by the discrete version of wavelet transform and the construction of orthonormal dyadic wavelet basis. Special attention is given to fairly exact mathematical treatment of the fractional wavelet transform, and several important results including Parseval’s formula and inversion theorem are proved. Chapter 4 contains an exposition of the general notion of a multiresolution analysis together with several examples. Special attention is given to properties of scaling functions and orthonormal wavelet bases. This is followed by a method of constructing orthonormal bases of wavelets from an MRA. In the end, the fast wavelet transform is briefly discussed. Chapter 5 is devoted to several generalizations and extensions of orthonormal wavelet bases in L2 .R/. To construct wavelets with greater degrees of smoothness and having compact support, we construct wavelets that are smooth and piecewise polynomials, usually known as spline wavelets. The well-known Franklin and Battle-Lemarié wavelets are the special cases of these wavelets. This is followed by Daubechies algorithm for the construction of compactly supported wavelets. Then, we discuss another intersecting class of orthonormal wavelets called harmonic wavelets. Finally, we present a novel and simple procedure for the construction of nonuniform 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 dilation is an even positive integer related to the given spectral pair. In preparing the monograph, the authors have been encouraged by and have benefited from the helpful comments and criticisms of a number of faculty and postdoctoral and doctoral students of several universities in the USA and India. The authors express their grateful thanks to these individuals for their interest in the book. The editor from Birkhäuser-Springer, Benjamin Levitt, deserves a special vote of thanks for his cooperation and for the exemplary patience he displayed. It goes without saying, however, that all responsibility for errors, imperfections and residual or outright mistakes, is shared jointly by both of us. However, we hope that these are both few and obvious and will cause minimum confusion. Edinburg, TX, USA Anantnag, Jammu and Kashmir, India

Lokenath Debnath Firdous A. Shah

Contents

1

The Fourier Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 The Fourier Transform in L1 .R/ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 The Fourier Transform in L2 .R/ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 The Fractional Fourier Transform .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Exercises.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 17 25 31 34 50 53

2 The Time-Frequency Analysis .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 The Time-Frequency Localization . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 The Gabor Transforms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 The Zak Transform.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 The Windowed Linear Canonical Transform . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Exercises.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

55 55 56 61 71 79 90

3 The Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Continuous Wavelet Transform . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 The Discrete Wavelet Transform . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Orthonormal Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 The Fractional Wavelet Transform . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Exercises.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93 93 95 106 109 111 120

4 Construction of Wavelets via MRA . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Multiresolution Analysis in L2 .R/ . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Construction of Mother Wavelet . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 The Fast Wavelet Transform .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Exercises.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

123 123 124 128 149 151

xi

xii

Contents

5 Elongations of MRA-Based Wavelets .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 The Spline Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 The Daubechies Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 The Harmonic Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 The Nonuniform Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Exercises.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

155 155 156 166 184 195 206

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 211 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 217

1

The Fourier Transforms

Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. Lord Kelvin Fourier was motivated by the study of heat diffusion, which is governed by a linear differential equation. However, the Fourier transform diagonalizes all linear time-invariant operators, which are building blocks of signal processing. It is therefore not only the starting point 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 of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis (See Fourier 1822). The integral formulas for the coefficients of the Fourier expansion were already known to Leonhard Euler (1707– 1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions. Thus, the Fourier series of a function f .t/ defined on the interval .L; L/ is given by f .t/ D

X n2Z

 cn exp

 int ; L

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

(1.1.1)

1

2

1 The Fourier Transforms

where the Fourier coefficients are 1 cn D 2L

Z

  int dt: f .t/ exp  L L L

(1.1.2)

In order to obtain a representation for a nonperiodic function defined for all real t, it seems desirable to take limit as L ! 1 that leads to the formulation of the famous Fourier integral theorem: f .t/ D

1 2

Z

1

ei!t d! 1

Z

1

ei!t f .t/ dt:

(1.1.3)

1

Mathematically, this is a continuous version of the completeness property of Fourier series. Physically, this form (1.1.3) can be resolved into an infinite number of !  and amplitude, harmonic components with continuously varying frequency 2 1 2

Z

1

ei!t f .t/ dt;

(1.1.4)

1

whereas the ordinary Fourier series represents a resolution of a given function into an infinite but discrete set of harmonic components. Thus, using the notation of an inner product, the Fourier transform of a continuous function f .t/ can be expressed as ˝ ˛ fO .!/ D f ; ei!t D

Z

1

ei!t f .t/ dt:

(1.1.5)

1

This transform decomposes a signal into orthogonal trigonometric basis functions of different frequencies and phases, and it is often called the Fourier spectrum. It is generally believed that the theory of Fourier series and Fourier transforms is one of the most remarkable discoveries in mathematical sciences and has wide spread applications in mathematics, physics, and engineering. This chapter deals with Fourier transforms in L1 .R/ and L2 .R/ and their basic properties. Several important results including the approximate identity theorem, general Parseval relation, and Plancherel theorem are proved. Discrete Fourier transform (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 solving generalized nonhomogeneous differential equations including the generalized wave and heat equations are also given. Special attention is also given to the Heisenberg uncertainty principle.

1.2 The Fourier Transform in L1 .R/

1.2

3

The Fourier Transform in L1 .R/

We begin by introducing some notation that will be used throughout this work. The set of natural numbers (positive integers) is denoted by N and the set of integers by 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 spaces of 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 functions f defined on R with a norm Z 1    ˇ ˇp 1=p f  D ˇf .t/ˇ dt < 1: (1.2.1) p 1

  The number f p is 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 understanding of wavelets in this introductory book, the reader needs to know some elementary properties 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 these spaces play a very special role in many mathematical investigations. The case p D 2 is of special interest: L2 is a Hilbert space. That is, there is an inner product relation on square-integrable functions, which extends the idea of the vector space dot product to analog signals. When p D 1, the space L1 .R/ is the collection of measurable 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 and Mikusinski, 1999). In particular, L1 .R/ is the space of all Lebesgue integrable functions defined on R with the L1 -norm given by Z 1   ˇ ˇ f  D ˇf .t/ˇ dt < 1: 1 1

Suppose f is a Lebesgue integrable function on R. Since ei!t is continuous ˇ ˇ and bounded, the product ei!t f .t/ is locally integrable for any ! 2 R. Also, ˇei!t ˇ  1 for all ! and t on R. Consider the inner product ˝ i!t ˛ D f;e

Z

1

f .t/ ei!t dt;

! 2 R:

(1.2.2)

1

Clearly, ˇZ ˇ ˇ ˇ

1 1

ˇ Z ˇ ei!t f .t/ dtˇˇ 

1

1

ˇ ˇ   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 following definition.

4

1 The Fourier Transforms

Definition 1.2.1 (The Fourier Transform in L1 .R/). The Fourier transform of any function f 2 L1 .R/ is defined by ˚ fO .!/ D F f .t/ D

Z

1

ei!t f .t/ dt:

(1.2.4)

1

Remarks. 1. Physically, the Fourier integral (1.2.4) measures oscillations of f at the frequency ! and fO .!/ is called the frequency spectrum of a signal or waveform f .t/. p 2. The factor 1=2 may be bundled with the Fourier transform or 1= 2 can appear in front of the transform and the inversion formula to provide a symmetric appearance. 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 vibrate at integer frequencies. By contrast the Fourier transform decomposes a signal defined on an infinite time interval into a !-frequency component, where ! can be any real number. 4. Another form for the Fourier transform of f used frequently in probability theory replaces the kernel exp.i!t/ by exp.i!t/. In this case if f is the probability density function of the random variable x, then Z

1

f .t/ eixt dt;

g.x/ D 1

is called the characteristic function of f . 5. In general, the Fourier transform fO .!/ is a complex function of a real variable !. From a physical point of view, the polar representation of the Fourier transform is often convenient. The Fourier transform fO .!/ can be expressed in the polar form fO .!/ D R.!/ C iX.!/ D A.!/ exp fi.!/g ;

(1.2.5)

ˇ ˇ ˇ ˇ where A.!/ D ˇfO .!/ˇ is called the amplitude spectrum of the signal f .t/, and n o .!/ D arg fO .!/ is called the phase spectrum of f .t/.

Example 1.2.1. The Gaussian function is one of the most important functions in probability theory and analysis of random analysis. It plays a central role in Gabor transform. The Gaussian function with unit amplitude is given by 2 2

f .t/ D ea t ;

a > 0:

(1.2.6)

1.2 The Fourier Transform in L1 .R/

5

f(t)

^

f (ω) 0.8

1 0.6 0.4

0.5

0.2

-4

-2

0 0

2

Fig. 1.1 Graphs of f .t/ D ea

t

4

2 t2

-4

-2

0 0

2

4

ω

and fO .!/ with a D 1

The Fourier transform of f is computed as fO .!/ D

Z

1

Z

 2 2 e i!tCa t dt D

1

De

! 2 =4a2

Z

1

e

 a2 tC

i! 2a2

2

2

 !2 4a

dt

1

1

e

a2 y2

1

p  ! 2 =4a2 e dy D ; a

(1.2.7)

    i! i! t C 2 is used. Even though 2a 2a2 is a complex number, the above result is correct. The change of variable can be justified by the method of complex analysis. The graphs of f .t/ and fO .!/ are drawn in Figure 1.1. in which the change of variable y D

It is interesting to note that the Fourier transform of a Gaussian function is also a Gaussian function (see Figure 1.1). The parameter a can be used to control the width of the Gaussian pulse. It is evident from relations (1.2.6) and (1.2.7) that large values of a produce a narrow pulse but its spectrum spreads wider on !-axis. In particular, when a2 D .a/ .b/

1 and a D 1, we obtain the following results 2 n 2 o p 2 F et =2 D 2 e! =2 ; n 2o p 2 F et D  e! =4 :

Example 1.2.2 (Characteristic Function). This function is defined by 1; a < t < a f .t/ D 0; otherwise.

(1.2.8) (1.2.9)

(1.2.10)

6

1 The Fourier Transforms

In science and engineering, this function is often called a rectangular pulse or gate function. Its Fourier transform is ˚ fO .!/ D F f .t/ D

  2 sin.a!/: !

(1.2.11)

We have fO .!/ D

Z

1

f .t/ ei!t dt D

Z

1

a

ei!t dt D

a

  2 sin.a!/: !

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

Example 1.2.3. Find the Fourier transform of     jtj jtj H 1 f .t/ D 1  a 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 1 0.2

-a

a

t

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

0 -15

-10

-5

0

5

10

15

ω

1.2 The Fourier Transform in L1 .R/

7

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 finite discontinuity at t D a: Then, it can easily be verified that  a!  sin2 fO .!/ D a   22 : a!

(1.2.13)

2

Example 1.2.4. Find the Fourier transform of f .t/ D eajtj ;

a > 0.

We have ˚ F eajtj D

Z

1

eajtji!t dt 1

Z

0

e.ai!/t dt C

D 1

D

Z

1

e.ai!/t dt 0

1 2a 1 C D 2 : a  i! a C i! a C !2

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

f(t)

^

f (ω) 1

1

0.8 0.6 0.5 0.4 0.2 -6

-4

-2

0 0

2

4

6

t

Fig. 1.3 Graphs of f .t/ D eajtj and fO .!/ with a D 1

-6

-4

-2

0 0

2

4

6

ω

8

1 The Fourier Transforms

n

1 o ˚  D eaj!j ; a > 0: Example 1.2.5. F f .t/ D F a2 C t2 a 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 the translation, modulation, and dilation operators respectively, by Ta f .t/ D f .t  a/

(Translation),

Mb f .t/ D eibt f .t/

(Modulation), t

1 Dc f .t/ D p f c jcj

(Dilation),

where a; b; c 2 R and c ¤ 0. Each of these operators is a unitary operator from L2 .R/ onto itself. The following results can easily be verified: Ta Mb f .t/ D eib.ta/ f .t  a/; Mb Ta f .t/ D eibt f .t  a/; t  a 1 Dc Ta f .t/ D p f ; c jcj t  a 1 ; Ta Dc f .t/ D p f c jcj   1 b Mb Dc f .t/ D p exp i t f c jcj   1 b Dc Mb f .t/ D p exp i t f c jcj

t c 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 ˇ F g.t/ : ˚ (b) Shifting: F nTa f .t/ oD Ma fO .!/; (c) Scaling: F D 1 f .t/ D Da fO .!/; a o n (d) Conjugation: F D1 f .t/ D fO .!/; ˚ (e) Modulation: F Ma f .t/ D Ta fO .!/:

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

1.2 The Fourier Transform in L1 .R/

9

If f .t/ 2 L1 .R/, then fO .!/ is continuous on R.

Theorem 1.2.2 (Continuity).

We shall discuss the derivative of the Fourier transform. As we know that smoother the function f , the more rapidly fO will decay at infinity and conversely. Therefore, more rapidly f decays at infinity, the smoother fO will be. There are various ways to measure the smoothness of a given function f but here we will measure the smoothness 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/, d O then f .!/ exists and is given by d! ˚ d O f .!/ D .i/ F tf .t/ : d!

(1.2.14)

Proof. We have i 1 hO d fO D lim f .! C h/  fO .!/ D lim h!0 h h!0 d!

Z

1

e

i!t

1

  eiht  1 dt : f .t/ h (1.2.15) 

Note that ˇ  ˇ ht ˇ ˇ ˇ ˇ ˇ sin ˇ ˇ iht  iht ˇ ˇ 1 iht ˇ ˇ ˇ 1 iht 2 ˇ ˇ 2  ˇ .e ˇ  jtj: ˇ ˇ e 2  e 2 ˇ D 2ˇ  1/ˇ D ˇe ˇh ˇ jhj h ˇ ˇ ˇ ˇ Also,  lim

h!0

eiht  1 h

 D it:

Thus, result (1.2.15) becomes dfO D d!

Z

e

i!t

f .t/ lim

h!0

1

Z

1

D .i/ 1

This proves the theorem.



1

 eiht  1 dt h

˚ tf .t/ ei!t dt D .i/ F tf .t/ :

10

1 The Fourier Transforms

Corollary 1.2.1. If f 2 L1 .R/ such that tn f .t/ is integrable for finite n 2 N, then the nth derivative of fO .!/ exists and is given by ˚ dn fO D .i/n F tn f .t/ : n d!

(1.2.16)

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

dn fO .!/ d! n

# D .i/

n

Z

1

tn f .t/ dt D .i/n mn ;

(1.2.17)

1

!D0

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

Theorem 1.2.4 (The Riemann-Lebesgue Lemma).

If f 2 L1 .R/, then

ˇ ˇ ˇ ˇ lim ˇfO .!/ˇ D 0:

(1.2.18)

j!j!1



Proof. Since ei!t D ei! .tC ! / , we have fO .!/ D 

Z

1



ei! .tC ! / f .t/ dt D 

Z

1

  dx: ei!x f x  ! 1 1

Thus, 1 fO .!/ D 2

D

1 2

Z

e 1

Z

Z

1 i!t

1

f .t/ dt 

e 1

i!t

   dt f t !

h   i dt: ei!t f .t/  f t  ! 1 1

Clearly, Z 1ˇ ˇ ˇ 1   ˇˇ ˇ ˇ ˇ lim ˇfO .!/ˇ  lim ˇ dt D 0: ˇf .t/  f t  j!j!1 2 j!j!1 1 ! This completes the proof.

1.2 The Fourier Transform in L1 .R/

11

Observe that the space C0 .R/ of all continuous functions on R which decay at infinity, that is, f .t/ ! 0 as jtj ! 1, is a normed space with respect to the norm defined by   ˇ ˇ f  D sup ˇf .t/ˇ:

(1.2.19)

t2R

It follows from above theorems that the Fourier transform is a continuous linear operator from L1 .R/ into C0 .R/. Theorem 1.2.4 gives a necessary condition for a function f to have a Fourier transform. However, that belonging to C0 .R/ is not a sufficient condition for being the Fourier transform of an integrable function. Theorem 1.2.5. (a) If f .t/ is a continuously differentiable function, lim f .t/ D 0 and both jtj!1

f ; f 0 2 L1 .R/, then ˚ ˚ F f 0 .t/ D i! F f .t/ D .i!/fO .!/:

(1.2.20)

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

for r D 1; 2; : : : ; n  1;

jtj!1

then ˚ ˚ F f .n/ .t/ D .i!/n F f .t/ D .i!/n fO .!/:

Proof. We have, by definition, ˚ F f 0 .t/ D

Z

1

ei!t f 0 .t/ dt; 1

which is, integrating by parts,

1 D ei!t f .t/ 1 C .i!/ D .i!/fO .!/: This proves part (a) of the theorem.

Z

1

ei!t f .t/ dt 1

(1.2.21)

12

1 The Fourier Transforms

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@r u ously n times differentiable and r ! 0 as jxj ! 1 for r D 1; 2; 3; : : : ; .n  1/, @x @n u then, the Fourier transform of n with respect to x is @x

F



@n u @xn



D .ik/n F fu.x; t/g D .ik/n uO .k; t/:

It also follows from the Definition 1.2.1 that 2   n  d uO @ u @u d2 uO d n uO @u D ; F F D D n: ; : : : ; F 2 2 n @t dt @t dt @t dt

(1.2.22)

(1.2.23)

Remark. This result has an important consequence: The smoothness of f is manifested in the rate of decay of its Fourier transform fO . We have already noted that the Fourier transform of a L1 .R/ function must decay to zero at large frequencies 1 fO .!/ ! 0 as ! ! 1. If the nth derivative f n is also and the derivatives  in L .R/ n vanish at infinity, then its Fourier transform F f .t/ D .i!/n fO .!/ must go to zero as ! ! 1. This requires that fO .!/ go to zero more rapidly than j!jn . Thus, the smoother f , the more rapid the decay of its Fourier transform. As a general rule of thumb, local features of f such as smoothness are manifested by global features of fO .!/, such as decay for large j!j. The symmetry principle implies that reverse is also true: global features of f correspond to local features of fO .!/. This local-global duality 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 on R, then the convolution is defined by

 f  g .t/ D

Z

1

f .t  / g./ d:

(1.2.24)

1

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

1 1

Z

1 1

ˇ ˇ ˇf .t  / g./ˇdt d D

Z

1

ˇ ˇ ˇg./ˇd

Z

1

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

1 1

ˇ ˇ    ˇf .t/ˇdt D g f  : 1 1

1.2 The Fourier Transform in L1 .R/

13

     f  g  g f  : 1 1 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/ D f .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 is continuous on R.

Proposition 1.2.2 (Young’s Inequality). 1=s D 1=p C 1=q  1, then

If the exponents p; q and s satisfy

      f  g  f  g : s p 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 fO .!/ gO .!/:

(1.2.26)

Proof. Since f  g 2 L1 .R/, we apply the definition of the Fourier transform to obtain Z 1 Z 1 ˚

 F f  g .t/ D ei!t dt f .t  / g./ d 1

Z

1

Z

1

D

1

g./ 1

Z

ei!t f .t  / dt d

1

1

D

ei! g./d

1

D fO .!/ gO .!/; in which Fubini’s theorem was utilized.

Z

1

ei!u f .u/ du; 1

.t   D u/

14

1 The Fourier Transforms

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

1

g.!/ ei!x d!; then

h.x/ D

(1.2.27)

1



 f  h .x/ D

Z

1

g.!/fO .!/ ei!x d!: 1

We now turn to the problem of inverting the Fourier transform. That is, we shall consider the question: Given the Fourier transform of an integrable function f .t/, how do we obtain f .t/ back again from fO .!/? The reader, who is familiar with the elementary theory of Fourier series and integrals, would expect 1 f .t/ D 2

Z

1

ei!t fO .!/ d!:

(1.2.28)

1

Unfortunately, it is not necessary that if f 2 L1 .R/, then its FourierRtransform fO also 1 belongs to L1 .R/ (see Example 1.2.2), so that the Fourier integral 1 fO .!/ ei!t d! may not exist as a Lebesgue integral. However, we can introduce a function K .!/ in the integrand and formulate general conditions on K .!/ and its Fourier transform so that the following result holds: Z lim



!1 

fO .!/K .!/ ei!t d! D f .t/

(1.2.29)

for almost every t. This kernel K .!/ is called a convergent factor or a summability kernel 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: Z (i) ZR

K .x/ dx D 1;

for all  > 0;

ˇ ˇ ˇK .x/ dxˇ  M; for all  > 0 and for a constant M; (ii) R Z ˇ ˇ ˇK .x/ˇdx D 0; for all ı > 0: (iii) lim !1 jxj>ı

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

1.2 The Fourier Transform in L1 .R/

15

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

If f 2 L1 .R/ and fK ,

 

 lim  f  K  f  D 0:

(1.2.30)

!1

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

Z

 f  K .t/ D

1

f .t  u/K .u/ du; 1

so that ˇZ ˇ ˇ ˇ

 ˇ f  K .t/  f .t/ˇ D ˇ ˇ ˇZ ˇ D ˇˇ Z

1 1 1 1 1



ˇ ˚ ˇ f .t  u/K .u/ du  f .t/ ˇˇ ˇ ˇ ˚ f .t  u/  f .t/ K .u/ duˇˇ ;

by Definition 1.2.2(i);

ˇ ˇˇ ˇ ˇK .u/ˇˇf .t  u/  f .t/ˇ du:

1

ˇ Given " > 0, we can choose ı > 0 such that if 0  juj < ı, then ˇf .t  u/ ˇ   " f .t/ˇ < , where K 1  M. Consequently, M  

  f  K .t/  f .t/ D

Z R

ˇ ˇ ˇf  K .t/  f .t/ˇ dt

Z

Z



1

dt R

Z

1

D

ˇ ˇˇ ˇ ˇK .u/ˇˇf .t  u/  f .t/ˇ du

1

ˇ ˇ ˇK .u/ˇ f .u/ du;

1

where Z f .u/ D

R

ˇ ˇ ˇf .t  u/  f .t/ˇ dt  C:

16

1 The Fourier Transforms

Thus,  

  f  K .t/  f .t/ 

Z

ˇ ˇ ˇK .u/ˇ f .u/ du C

jujı

Z

ˇ ˇ ˇK .u/ˇ du D ";

 "CC juj>ı

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 frequency variable. Theorem 1.2.8 (General Modulation). If F ff .t/g D fO .!/ and F fg.t/g D gO .!/, where fO and gO belong to L1 .R/, then  ˚ 1 O F f .t/ g.t/ D f  gO .!/: 2

(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

ei!t f .t/ g.t/ dt 1

D

D

D

D This completes the proof.

1 2 1 2 1 2

Z

1

ei!t g.t/ dt

Z

1

Z

eixt fO .x/ dx 1

1

fO .x/ dx 1

Z

1

Z

1

eit.!x/ g.t/ dt

1

1

fO .x/ gO .!  x/ dx 1

 1 O f  gO .!/: 2

1.3 The Fourier Transform in L2 .R/

1.3

17

The Fourier Transform in L2 .R/

In this section, we discuss the extension of the Fourier transform onto L2 .R/. It 2 2 O turns   outpthatiff 2 L .R/, then the Fourier transform f of f is also in L .R/ and O   f  D 2 f 2 ; where 2

  f  D 2

Z

1

ˇ ˇ2 ˇf .t/ˇ dt

 12 :

(1.3.1)

1

p The factor 2 involved in the above result can be avoided by defining the Fourier transform as Z 1 1 fO .!/ D p ei!t f .t/ dt: (1.3.2) 2 1

Theorem 1.3.1. Suppose f is a continuous function on R vanishing outside a bounded interval. Then, fO 2 L2 .R/ and      f  D  fO  : 2 2

(1.3.3)

Proof. We assume that f vanishes outside the interval Œ; . We use the Parseval formula for the orthonormal sequence of functions on Œ; , 1 n .t/ D p eint ; n D 0; ˙1; ˙2; : : : ; 2 to obtain ˇ ˇ2 Z 1 ˇ2 X ˇˇ  2 X ˇ 1 ˇ int f  D ˇp ˇ D O .n/ˇˇ : e f .t/ dt f ˇ ˇ 2 ˇ 2 1 n2Z n2Z  2  2 Since this result also holds for g.t/ D eixt f .t/ instead of f .t/, and f 2 D g2 , then ˇ2  2 X ˇˇ f  D O .n C x/ˇˇ : f ˇ 2 n2Z

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

18

1 The Fourier Transforms

Z  2 X f  D 2

1

ˇ2 ˇ XZ ˇ ˇO ˇf .n C x/ˇ dx D

n2Z

0

Z

ˇ  2 ˇ ˇ O ˇ2   ˇf .y/ˇ dy D fO  :

1

D 1

n2Z

nC1 n

ˇ ˇ ˇ O ˇ2 ˇf .y/ˇ dy;

.y D n C x/

2

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

1 O !  : f a a

Thus, it turns out that  2   f  D ag2 D a 2 2

ˇ ˇ Z 1ˇ  2 ˇ ˇ 1  ! ˇ2   ˇ O ˇ2 ˇ d! D ˇ fO ˇf .!/ˇ d! D fO  : ˇ ˇa 2 a 1 1

Z

1

This completes the proof. The space of all continuous functions on R with compact support is dense in L2 .R/. Theorem 1.3.1 shows that the Fourier transform is a continuous mapping from that space into L2 .R/. Since the mapping is linear, it has a unique extension to a linear mapping from L2 .R/ into itself. This extension will be called the Fourier transform on L2 .R/. Definition 1.3.1 (Fourier Transform in L2 .R/). If f 2 L2 .R/ and ffn g is a sequence functions with compact support convergent to f in L2 .R/,  of continuous  that is, f  fn  ! 0 as n ! 1, then the Fourier transform of f is defined by fO D lim fn ; n!1

(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 particular sequence approximating f . It is important to note that the convergence in L2 .R/ does not imply pointwise convergence, and therefore the Fourier transform of a square-integrable function is not defined at a point, unlike the Fourier transform of an integrable function. We can assert that the Fourier transform fO of f 2 L2 .R/ is defined almost everywhere on R and fO 2 L2 .R/. For this reason, we cannot say that

1.3 The Fourier Transform in L2 .R/

19

if f 2 L1 .R/ \ L2 .R/, the Fourier transform defined by (1.2.4) and the one defined by (1.3.4) are equal. An immediate consequence of Definition 1.2.2 and Theorem 1.3.1 leads to the following theorem. If f 2 L2 .R/, then

Theorem 1.3.2 (Parseval’s Identity).

     f  D  fO  : 2 2

(1.3.5)

Theorem 1.3.3. If f 2 L2 .R/, then 1 fO .!/ D lim p n!1 2

Z

n

ei!t f .t/ dt;

(1.3.6)

n

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 < n 0; for jtj  n:

(1.3.7)

      Clearly, f  fn 2 ! 0 and, hence, fO  fOn  ! 0 as n ! 1. 2

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

E Z f ; gNO D

Z

1

1

f .t/ gO .t/ dt D 1

D E fO .t/ g.t/ dt D fO ; gN :

1

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

Z

1

eixt fm .x/ dx; 1

(1.3.8)

20

1 The Fourier Transforms

we obtain Z

1

1 fOm .t/ gn .t/ dt D p 2 1

Z

Z

1

1

eixt fm .x/ dx dt:

gn .t/ 1

1

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

Z 1 Z 1 1 fOm .t/gn .t/ dt D p fm .x/ eixt gn .t/ dt dx 2 1 1 1 Z 1 fm .x/ gO n .x/ dx: D 1

1

    Since g  gn 2 ! 0 and gO  gO n 2 ! 0, letting n ! 1 with the continuity of the inner product yields Z

1

fOm .t/ gn .t/ dt D 1

Z

1

fm .t/ gO n .t/ dt: 1

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

Lemma 1.3.1. If f 2 L2 .R/ and g D fNO , then f D gNO .

Proof. In view of Theorems 1.3.2 and 1.3.3 and the assumption g D fNO , we find D E D E D E  2   2   f ; gNO D fO ; gN D fO ; fO D fO  D f 2 :

(1.3.9)

˝ ˛ D E  2 f ; gNO D fO ; fO D f 2 :

(1.3.10)

2    2  2  2 gO  D g D  O f   D f 2 : 2 2

(1.3.11)

2

Also, we have

Finally, by Parseval’s relation,

2

1.3 The Fourier Transform in L2 .R/

21

Using (1.3.9)–(1.3.11) gives the following:  2 ˝ ˛  2 ˝ ˛ ˝ ˛  2   f  gNO  D f  gNO ; f  gNO D f 2  f ; gNO  f ; gNO C gO 2 D 0: 2

This proves the result f D gNO .

Example 1.3.1 (The Haar Function).

The Haar function is defined by

8 ˆ ˆ ˆ < 1;

1 for 0  t < ; 2 f .t/ D 1; for 1  t < 1; ˆ ˆ 2 ˆ : 0; otherwise:

(1.3.12)

The Fourier transform of f .t/ is given as fO .!/ D

Z

"Z

1

e

i!t

f .t/ dt D

1

1 2

e

i!t

Z

#

1

dt 

e

i!t

dt

1 2

0

 1  i! 1  2e 2 C ei! i! i!  e 2  i! i! D e 2  2 C e 2 .i!/   2 ! sin i! 4 : D e 2 ! 4 D

(1.3.13)

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

^

f (ω)

f(t) 1

0

1

t ω

-1

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

22

1 The Fourier Transforms

^

f (ω) f(t)

0

4 - √2

0

ω

√2

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

Example 1.3.2 (The Second Derivative of the Gaussian Function).  2

f .t/ D 1  t2 et =2 ;

If (1.3.14)

then the Fourier transform of f .t/ can be computed as  2 o 1  t2 et =2 2  d t2 =2 D F e dt2 n o 2 D .i!/2 F et =2

fO .!/ D F

n

D ! 2 et

2 =2

:

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

Theorem 1.3.5 (Inversion Formula). If f 2 L2 .R/, then 1 f .t/ D lim n!1 2

Z

n

ei!t fO .!/ d!; n

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

(1.3.15)

1.3 The Fourier Transform in L2 .R/

23

Proof. If f 2 L2 .R/ and g D fNO , then, by Lemma 1.3.1, 1 f .t/ D gO .t/ D lim n!1 2 1 D lim n!1 2 1 D lim n!1 2 1 D lim n!1 2

Z

n

ei!t g.!/ d! n

Z

n

ei!t g.!/ d! n

Z

n

ei!t g.!/ d! n

Z

n

ei!t fO .!/ d!: n

formula (1.3.15) is called the inverse Fourier transform. If we use the factor  The p  1= 2 in the definition of the Fourier transform, then the Fourier transform and its inverse are symmetrical in form, that is, 1 fO .!/ D p 2

Z

1

1 f .t/ D p 2

ei!t f .t/ dt; 1

Theorem 1.3.6 (General Parseval’s Relation). ˝

˛ f;g D

Z

Z

1

1

f .t/ g.t/ dt D 1

Z

1

ei!t fO .!/ d!:

(1.3.16)

1

If f ; g 2 L2 .R/, then

˝ ˛ fO .!/ gO .!/ d! D fO ; gO ;

(1.3.17)

1

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

Proof. It follows from (1.3.3) that  2   f C g2 D  O C gO  f   : 2 2

Or, equivalently, Z

1

ˇ ˇ ˇf C gˇ2 dt D

1

Z

1 1

Z

1 1





ˇ ˇ2 ˇO ˇ ˇf C gO ˇ d!;

 f C g fN C gN dt D

Z

1 1

   fO C gO fNO C gNO d!:

24

1 The Fourier Transforms

Simplifying both sides gives Z

1

Z

ˇ ˇ2 ˇf ˇ dt C

1

1

 f gN C gfN dt C

Z

1

Z

1

D 1

1

ˇ ˇ2 ˇgˇ dt

1

Z ˇ ˇ2 ˇO ˇ ˇf ˇ d! C



1

Z  fO gNO C gO fNO d! C

1

1

jOgj2 d!:

1

Applying (1.3.3) to the above identity leads to Z

1



 f gN C gfN dt D

1

Z

  fO gNO C gO fNO d!:

1

(1.3.18)

1

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

1

 f .ig/ C .ig/fN dt D

Z

1

1

h   i fO iOg C .iOg/ fNO d!:

1

Or Z

Z

1

i

1

g fN dt D i

f gN dt C i 1

1

Z

1

fO gNO d! C i

Z

1

1

gO fNO d!;

1

which is, multiplying by i, Z

Z

1

1

g fN dt D

f gN dt  1

1

Z

1

fO gNO d! 

1

Z

1

gO fNO d!:

(1.3.19)

1

Adding (1.3.18) and (1.3.19) gives Z

Z

1

1

fO .!/ gO .!/ d!:

f .t/ g.t/ dt D 1

1

This completes the proof. The following theorem summaries the major results of this section and is usually called Plancherel theorem.

1.4 The Discrete Fourier Transform

25

Theorem 1.3.7 (Plancherel’s Theorem). For every f 2 L2 .R/, there exists fO 2 L2 .R/ such that (i) (ii) (iii) (iv) (v)

1 If f 2 L .R/ \ L .R/, then fO .!/ D p 2 Z n i!t Of .!/ D lim p1 e f .t/ dt; n!1 2Z n n 1 f .t/ D lim p ei!t fO .!/ d!, n!1 2 n ˝ ˛ ˝ ˛ f ; g D fO ;gO ;    f  D  fO  ; 2 1

2

Z

1

ei!t f .t/ dt; 1

2

(vi) The mapping f ! fO 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., functions which are defined at all values of the time t. However, for many applications, we require functions which are discrete in nature rather than continuous. In modern digital media audio, still images or video-continuous signals are sampled at discrete time intervals before being processed. Fourier analysis decomposes the sampled signal into its fundamental periodic constituents sines and cosines or, more conveniently, complex exponentials. The crucial fact, upon which all modern signal processing is based, is that the sampled complex exponentials form an orthogonal basis. To meet the needs of both the automated and experimental computations, the discrete Fourier transform (DFT) has been introduced. To motivate the idea behind the discrete Fourier transform, we take two approaches, one from the approximation point of view and other one from discrete point of view. Consider the function f with Fourier transform fO .!/ D

Z

1

f .t/ ei!t dt:

(1.4.1)

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 an interval Œa; b and approximate the integral for fO by a finite sum as fO .!/ D

N1 X kD0

 f tk ei!tk h:

26

1 The Fourier Transforms

Now, for sufficiently large a < 0 and b > 0 Z b f .t/ ei!t dt a

is a good approximation to fO . Therefore, in order to approximate this integral, we sample the signal f at a finite number of sample points, say t0 D a < t1 < t2 <    < tN1 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

ba indicates the sample rate. In signal processing applications, t where h D N represents time instead of space and tk are the times at which we sample the signal f . This sample rate can be very high, e.g., every 10  20 milliseconds in current speech recognition systems. Thus, the approximation g of fO is given by g.!/ D

N1 X

 f tk ei!tk h

kD0

D ei!a

N1 X

 f tk ei!.ba/k=N h:

kD0

We now take the time duration Œa; b into account by focusing attention on the points 2n (frequencies), !n D ; where n is an integer. Then, the approximation g of fO at ba these points becomes g.!n / D e

ia!n

N1 X

 f tk ei2nk=N h:

kD0

By neglecting the term eia!n h in the R.H.S. of the above expression and focusing attention on the N-periodic function fO W Z ! C, we obtain fO .n/ D

N1 X

 f tk ei2nk=N h;

n2Z

kD0

D

N1 X

 f tk wnk ;

(1.4.2)

kD0

where w D e2i=N : Equation (1.4.2) is known as discrete Fourier transform (DFT).

1.4 The Discrete Fourier Transform

27

From a discrete ˚ perspective, one is dealing with the values of f at only a finite number of points 0; 1; 2; : : : ; N  1 . Consider f as defined on the cyclic group of integers modulo the positive integer N:  Z

ZN D  Z modulo N N

 ˚ where N 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

 f k 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, L1 ZN D L2 ZN D CN is the collection of all functions f W ZN ! C. One gets the discrete Fourier transform for f W ZN ! C as fO .n/ D

N1 X

 f k ei2nk=N ;

n 2 ZN :

kD0

This formula for the discrete Fourier transform is analogous to the formula for the nth 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 fO can be viewed as vectors. 0

1 f .0/ B C :: f D@ A : f .N  1/

0

1 fO .0/ B C :: and fO D @ A : fO .N  1/

with 0 1 1 1 B B B1 e12i=N e22i=N B B B 1 e22i=N e222i=N WN D B B B B: :: :: B :: : : B @ .N1/2i=N 2.N1/2i=N 1e e



1

1

C C    e.N1/2i=N C C C 2.N1/2i=N C  e C: C C C :: :: C : : C 2 A    e.N1/ 2i=N

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

28

1 The Fourier Transforms

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, 10 1 0 1 4 1 C C B C B B1 i 1 i C B B C B C B1C B0C B C CDB C fO .n/ D W4 f D B C: B1 1 1 1C B B0C 1C CB B A @ A @ A @ 1 i 1 i 0 1 0

1 1

1

1

Here some properties are analogous to the corresponding properties for the Fourier transform given in Theorem 1.2.1. Theorem 1.4.1. The following properties hold for the discrete Fourier transform: (a) Time shift: fO .n j/ D fO .n/ e2ijn=N ;  ^

(b) Frequency shift: f .n/ e2ijn=N D fO .n  j/;   i 2nj ^ 1 hO (c) Modulation: f .n/ cos D f .n  j/ C fO .n C j/ : N 2 Proof. Let g.n/ D f .n  j/. Then, by using the fact that f is N-periodic, we obtain gO .n/ D D

N1 X kD0 N1 X

g.k/ e2nik=N f .k  j/ e2nik=N

kD0 N1j

X

D D D D

mDj 1 X

f .m/ e2ni.mCj/=N N1j

f .m/ e2ni.mCj/=N C

mDj N1 X

f .m/ e2ni.mCj/=N

mD0 N1j

f .m/ e2ni.mCj/=N C

mDNj N1 X

!

X

mD0

f .m/ e2nim=N e2nij=N

mD0

D fO .n/ e2nij=N : This proves part (a).

X

f .m/ e2ni.mCj/=N

1.4 The Discrete Fourier Transform

29

(b) Let g.n/ D f .n/ e2nij=N . Then gO .n/ D D D

N1 X kD0 N1 X kD0 N1 X

g.k/ e2nik=N f .k/ e2ijk=N  e2nik=N f .k/ e2ik.nj/=N D fO .n  j/:

kD0

Hence  ^ f .n/ e2ijn=N D fO .n  j/: (c) Since f .n/ cos

i 2in 1h D f .n/ e2ijn=N C f .n/ e2ijn=N N 2

the desired result is obtained by using part (b). Our next task is to compute the inverse of the discrete Fourier transform. We have already computed the inverse Fourier transform, since Theorem 1.3.5 tells how to recover the function f from its Fourier transform. The inverse discrete Fourier transform is analogous and it allows us to recover the original discrete signal f from its discrete Fourier transform fO .

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

N1 X

f .k/ e2ikn=N D

kD0

N1 X

f .k/ wkn

kD0

where w D e2i=N . Then f .n/ D

N1 N1 1 XO 1 XO f .k/ e2ikn=N D f .k/ wkn : N kD0 N kD0

(1.4.3)

Proof. In order to establish the result (1.4.3), we must show that N1 X kD0

.nj/k

w

D

1 if n D j 0 if n ¤ j

(1.4.4)

30

1 The Fourier Transforms

Since w D e2i=N ; wkn  wkj D wk.nj/ : Therefore, in order to sum this expression over k D 0; 1; 2; : : : ; N  1, we use the following elementary observation that

1 C x C x2 C    C xN1

8 < N if x D 1 D 1  xN : if x ¤ 1 1x

Set x D wnj and note that xN D 1 because wN D 1. Also, we note that wnj ¤ 1 unless n D j for 0  n; j  1. Thus N1 X

.nj/k

w

D

kD0

1 if n D j 0 if n ¤ j

Hence N1 N1 1 XO 1 XO f .k/ e2ikn=N D f .k/ wkn N kD0 N kD0

D

1 N

N1 X kD0

0 1 N1 X @ f .j/ wkj A wkn jD0

! N1 N1 X 1X .nj/k : D f .j/ w N jD0 kD0 Therefore, by using (1.4.4), we get N1 1 XO 1 f .k/ e2ikn=N D f .n/  N D f .n/: N kD0 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 2p=q, for some p; q 2 Z. If p D 1 and q D N, then ! D 2=N, and f .n/ is periodic on Œ0; N  1 with fundamental period N. Therefore, the functions of the form cos.2kn=N/ are also periodic on Œ0; N  1, but since cos.2kn=N/ D  cos 2.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.2kn=N/ D  sin 2.N  k/n=N .

1.5 The Fast Fourier Transform

31

Note that for k D 0; 1; : : : ; bN=2c; we have     1 1 2ink 2in.N  k/ C exp ; exp 2 N 2 N       2nk 1 2ink 1 2in.N  k/ sin D exp  exp : N 2i N 2i N 

cos

2nk N



D

(1.4.5) (1.4.6)

Equations (1.4.5) and (1.4.6) thereby imply that fO .k/ D fO .N  k/ D N=2 with fO .m/ D 0; for 0  m  N  1; m ¤ kI and gO .k/ D Og.N  k/ D .iN/=2 with gO .m/ D 0; for 0  m  N  1; m ¤ k: The factor of N in the expressions for fO .k/ and gO .k/ ensures that the inverse DFT relation (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 about the frequency components of a signal is extremely valuable, the huge computational effort involved meant that until the 1960’s, it was rarely used in practical applications. Two important advances have changed the situation completely. The first was the development of the digital computer with its ability to perform numerical calculations rapidly and accurately. The second was the discovery by Cooley and Tukey of a numerical algorithm which allows the DFT to be evaluated with a significant reduction in the amount of calculations required. This algorithm, called the fast Fourier transform (FFT), allows the DFT of a sampled signal to be obtained rapidly 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 to make it practical: the digital computer. Cooley and Tukey are honored because they discovered the FFT at the right time, the beginning of the computer revolution. The publication of the FFT algorithm by Cooley and Tukey in 1965 was the turning point in digital signal processing and in certain areas of numerical analysis. Nowadays, the FFT is used in many areas, from the identification of characteristic mechanical vibration frequencies to image enhancement. Standard routines are available to perform the FFT by computer in programming languages such as Pascal, Fortran, and C, and many spreadsheet and other software packages for the analysis of numerical 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).

32

1 The Fourier Transforms

The DFT of an N times sampled signal requires a total of N 2 multiplications and N.N  1/ additions. The FFT algorithm for N D 2k reduces the N 2 multiplications to something proportional to N log2 N. If the calculations are handmade, then N is necessarily small and this is not that significant, but in case N is large, the number of operations is drastically reduced. For example, a 3-minute song may contain N D .44; 000/  180 D 7; 920; 000 samples. Thus, the DFT would take 63; 000; 000; 000; 000 multiplications. The FFT on the other hand would only take approximately 90; 751; 593 multiplications. This result is a significant difference in computing time. How do we go about achieving this reduction in computing time? Where do we start? The idea is to look at even entries and odd separately, and we can then piece them back together. This, seemingly simple, idea will allow certain multiplications that are normally repeated to only be done once. Let N 2 N with N even and wN D e2i=N . If N D 2M; M 2 N, then w2N D e2i2=N D e2i=.N=2/ D e2i=M D wM :



 Suppose f 2 `2 ZN D CN ; define a; b 2 `2 ZM 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/ D a.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 fO .n/ D D D D D

N1 X kD0 N1 X kD0 M1 X kD0 M1 X kD0 M1 X kD0

f .k/ e2ink=N f .k/ wnk N f .2k/ w2kn N C

M1 X

.2kC1/n

f .2k C 1/ wN

kD0 M1 X

nk

nk a.k/ w2N C wnN b.k/ w2N

n a.k/ wnk M C wN

O D aO .n/ C wnN b.n/

kD0 M1 X

b.k/ wnk M

kD0

1.5 The Fast Fourier Transform

33

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

2i=N M D e2iM=N D ei D 1: wM N D e Therefore, if 0  n  M  1, then O fO .n/ D aO .n/ C wnN b.n/:

(1.5.1)

Moreover, if M  n  N  1, then .nM/ O b.n  M/ fO .n/ D aO .n  M/ C wN n O D aO .n/ C wM N  wN b.n/

O D aO .n/  wnN b.n/:

(1.5.2)

O Let us now look at the number

of  multiplications that it takes to compute f .n/ in this way when f 2 CN D `2 ZN and N D 2M. Computing aO and bO each takes N2 O we need b.n/ O wn . This M2 D multiplications. In addition, for each entry of b, N 4 gives an additional M D N=2 multiplication. Thus, if m.N/ denotes the number of multiplications required to compute an N-point DFT using the FFT algorithm, then our total is



 m.N/ D 2M 2 C M D 2m N=2 C N=2 :

(1.5.3)

Here we have essentially cut the half time. This is good but not great; half of the ten thousand years in a previous example is still a long time. The order of magnitude really has not changed. How can we get a more significant change? The big improvement is going to come when N D 2n ; n 2 N. Now, with this assumption N D 2n , the M defined above will also be even. Thus, we can use the same method O Therefore, we have the following theorem in recursively, when computing aO and b. this regard. Theorem 1.5.1. If N D 2n for some n 2 N, then

 m.N/ D N=2 log2 N:

(1.5.4)

34

1 The Fourier Transforms

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 the result holds for n D k  1. Then



 m 2k D 2  m 2k1 C 2k1

 D 2 2k2 .k  1/ C 2k1

 D 2k1 .k  1/ C 1  k  2

k1 log2 2k : D2 kD 2 This proves that the result holds for n D k also. Thus, by induction the result must hold for any n 2 N.

1.6

The Fractional Fourier Transform

Undoubtedly, one of the most recognized tools in signal and image processing is the Fourier transform which generally converts a signal from time versus amplitude to frequency versus amplitude. The classical Fourier transform can be visualized as a change in representation of the signal corresponding to a counterclockwise  rotation of the axis by an angle . Two successive rotations of the signal through 2  will result in an inversion of the time axis. In spite of some remarkable success, 2 Fourier transform seems to be inadequate for studying nonstationary signals for at least two reasons. First, the Fourier transform of a signal does not contain any local information in the sense that it does not reflect the change of wave number with space or of frequency with time. Second, the Fourier transform method enables us to investigate problems either in time (space) domain or the frequency (wave number) domain, but not simultaneously in both domains. These are probably the major weaknesses of the Fourier transform analysis. To overcome these problems, Victor Namias (1980) proposed the fractional Fourier transform (FrFT) as a generalization of the conventional Fourier transform to solve certain problems arising in quantum mechanics from classical quadratic Hamiltonians. It is also called rotational Fourier transform or angular Fourier transform since it depends on a parameter ˛ which is interpreted as a rotation by an angle ˛ in the time-frequency plane. Like the ordinary Fourier transform that corresponds to a rotation in the time-frequency plane over  an angle ˛ D 1  , the FrFT corresponds to a rotation over an arbitrary angle 2  ˛ D a  with a 2 R. 2 The fractional Fourier transforms are relatively recent developments that have fascinated 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 decades at a very rapid rate. They have been applied in a number of fields including signal processing, image processing, quantum mechanics, neural networks, differential equations, time-frequency distributions, optical systems, statistical optics, signal detectors and pattern recognition, radar, sonar, and communications. A comprehensive overview of FrFTs and their applications can also be found in Mendlovic and 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 Sejdi´c et al. (2011). Definition 1.6.1 (Fractional Fourier Transform). form with parameter ˛ of signal f .t/ is defined by ˚ F ˛ f .t/ .!/ D fO ˛ .!/ D

Z

The fractional Fourier trans-

1

K˛ .t; !/f .t/ dt;

(1.6.1)

1

where K˛ .t; !/ is the so-called kernel of the FrFT given by o n 8 cot ˛ ˆ  it! csc ˛ ; ˛ ¤ n; < C˛ exp i.t2 C ! 2 / 2 K˛ .t; !/ D ı.t  !/; ˛ D 2n; ˆ : ı.t C !/; ˛ D .2n ˙ 1/;

(1.6.2)

a denotes the rotation angle of the transformed signal for FrFT, the FrFT 2 operator is designated by F ˛ , and

˛ D

r C˛ D .2i sin ˛/

1=2 i˛=2

e

D

1  i cot ˛ : 2

(1.6.3)

The corresponding inversion formula is given by f .t/ D

1 2

Z

1

K˛ .t; !/ fO ˛ .!/ d!;

(1.6.4)

1

where  i.t2 C ! 2 / cot ˛ .2i sin ˛/1=2 ei˛=2  exp C it! csc ˛ sin ˛ 2  i.t2 C ! 2 / cot ˛ C it! csc ˛ D C˛ exp 2

K˛ .t; !/ D

D K˛ .t; !/

(1.6.5)

36

1 The Fourier Transforms

and .2i sin ˛/1=2 ei˛=2 C˛ D D 2 sin ˛

r

1 C i cot ˛ D C˛ : 2

(1.6.6)

Remarks: 1. It is important to p point out that when a D 1, the kernel K˛ .t; !/ given by (1.6.2) reduces to eit! = 2, corresponding to the ordinaryp Fourier transform, and that when a D 1, the kernel K˛ .t; !/ reduces to eit! = 2, 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 Fourier transform; for instance, F 2 corresponds to the Fourier transform of the Fourier transform. The order ˛ may assume any real value; however the operator F ˛ is periodic in ˛ with period 4, that F ˛C4n D F ˛ ; where n is any integer. This is because F 2 equals the parity operator P which maps f .t/ to f .t/ and F 4 equals the identity operator. Therefore, the range of a is usually restricted to Œ0; 4/. These facts can be restated in operator notation: F 0 D I ; F 1 D F ; F 2 D P ; F 3 D F P ; 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 rotation operator 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: R 0 D I; R 2 D I; R =2 D F ;

and R ˛Cˇ D R ˛  R ˇ :

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

1.6 The Fractional Fourier Transform

R R f .!/ D ˛

ˇ

37

Z

1

K˛ .t; !/ dt

1 Z 1

D

Z

Z

D

f . / Kˇ . ; t/ d 1

1

K˛ .t; !/ Kˇ . ; t/ dt

f . / d 1 Z 1

1

1

f . / K˛Cˇ . ; !/ d

1

D R ˛Cˇ f .!/: Thus, we have F ˛  F ˇ D F ˛Cˇ : From the index additive property, we can deduce the inverse of the ˛th-order fractional Fourier operator F ˛ as Z

1

F ˛ .!/ K˛ .t; !/ d! D 1

Z

1

Z

1 1

K˛ .t; !/ d! f .t0 / dt0

D 1 Z 1

D 1 Z 1

D

Z

Z

1

f .t0 / K˛ .t0 ; !/ dt0 1

1

K˛ .t; !/ K˛ .!; t0 / d! 1

f .t0 / K0 .t; t0 / dt0 f .t0 / ı.t  t0 / dt0

1

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 d r ˛˚ f .t/ .!/ D i! sin ˛ C cos ˛ F f .t/ .!/: dtr d!

(1.6.8)

2

Example 1.6.1. The FrFT of the Gaussian function f .t/ D ebt ; b > 0 is F

˛

n

e

bt2

o

r .!/ D

2  i! cot ˛ .! csc ˛/2 1  i cot ˛ exp  : 2b  i cot ˛ 2 2.2b  i cot ˛/

38

1 The Fourier Transforms

Example 1.6.2 (The Second Derivative of the Gaussian Function). This function is defined by f .t/ D .1  t2 / et

2 =2

D

d 2 t2 =2 e : dt2

(1.6.9)

Applying the definition of FrFT (1.6.1) and using (1.6.8) with r D 2, we obtain n o F ˛ f .t/ .!/ D F

˛



 d 2 t2 =2 .!/ e dt2

 d 2 ˛ n t2 =2 o .!/ D  i! sin ˛ C cos ˛ F e d!   cos ˛ csc2 ˛ i 2 ! 2 csc2 ˛ ! cot ˛  i sin ˛ C i cot ˛ cos ˛  D exp 2 2.1  i cot ˛/ .1  i cot ˛/    2 csc ˛  i! 2 sin ˛  cos ˛ 1 C ! 2 i cot ˛  .1  i cot ˛/  2 2 ! csc ˛ i 2 ! cot ˛  .1 C i cot ˛/ D exp 2 2.1 C cot2 ˛/  cos ˛ csc2 ˛.1 C i cot ˛/  i sin ˛ C i cot ˛ cos ˛  .1 C cot2 ˛/    csc2 ˛.1 C i cot ˛/ 2 2  i! sin ˛  cos ˛ 1 C ! i cot ˛  .1 C cot2 ˛/ n n o o 2 2 D e! =2 ! 2 sin2 ˛ C cos2 ˛.1  ! 2 / C ie! =2 sin ˛ cos ˛.2 ! 2  1/ :

Both f .t/ and fO ˛ .!/ 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 the FrFT. However, in order to discuss some properties of the FrFT, we shall first recall the definition of the Schwartz space S.R/. Definition 1.6.2. An infinitely differential complex-valued function f is member of S.R/ iff for every choice of ˇ and of nonnegative integers, it satisfies ˇ ˇ ˇ ˇ ˇ; .f / D sup ˇtˇ D f .t/ˇ < 1: t2R

(1.6.10)

1.6 The Fractional Fourier Transform

39

Re( f(t))

Re( f(w))

1.0

0.8

0.3

0.6 0.4

0.2

0.2 –4

–2

4

2

–0.2

t

–0.4

0.1 –4

–2

Re( f(w))

2

4

2

4

w

Re( f(w))

0.25 0.20

0.15

0.15

0.10

0.10 0.05

0.05 –4

–2

2

4

w

–4

–2

w

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

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



ˇ .f / D sup ˇtˇ t f .t/ˇ < 1;

ˇ; 2 N0

(1.6.11)

t2R

where



t D



d  it cot ˛ dt



:

(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 n o

r

rt K˛ .t; !/ D  i! csc ˛ K˛ .t; !/;

r 2 N0 :

(1.6.13)

40

1 The Fourier Transforms

Proof. Differentiating the kernel K˛ .t; !/ with respect to t, we obtain   d i.t2 C ! 2 / cot ˛ d K˛ .t; !/ D C˛ exp   it! csc ˛ dt dt 2 D K˛ .t; !/ i .t cos ˛  ! csc ˛/ so that 



 d  it cot ˛ K˛ .t; !/ D  i! csc ˛ K˛ .t; !/: dt

Continuing this process r-times, we obtain 

d  it cot ˛ dt

r

r K˛ .t; !/ D  i! csc ˛ K˛ .t; !/:

This completes the proof.

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

1 1

where 0t D 

rt K˛ .t; !/f .t/dt D 

Z

1 1

r K˛ .t; !/ 0t f .t/ dt;

(1.6.14)

 d C it cot ˛ : dt

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

1

t K˛ .t; !/f .t/ dt D

Z

1

1 1

K˛ .t; !/ . 0t /f .t/ dt:

Integrating by parts, we have Z

1 1



d  it cot ˛ dt



K˛ .t; !/ dt D

Z

1

K˛ .t; !/ 1



 d C it cot ˛ f .t/ dt: dt

Thus, we have Z

1

t K˛ .t; !/f .t/ dt D 1

Z

1 1

K˛ .t; !/ . 0t /f .t/ dt:

1.6 The Fractional Fourier Transform

41

In general, we can obtain Z

1

1

rt K˛ .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 n o

˚ r (a) F ˛ . 0t /r f .t/ D  i! csc ˛ F ˛ f .t/ n n o o (b) r! F ˛ f .t/ D F ˛ .it csc ˛/r f .t/ :

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

F

˛

. 0t /r

Z

o

1

f .t/ .!/ D 1 1

Z D

1

˚ K˛ .t; !/ . 0t /r f .t/ dt

rt K˛ .t; !/ f .t/ dt

D .i! csc ˛/

r

Z

1

K˛ .t; !/ f .t/ dt 1

˚

r D  i! csc ˛ 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), we obtain n o Z 1 r ˛

! F f .t/ D

r! K˛ .t; !/f .t/ dt Z

1 1

.itcsc ˛/r K˛ .t; !/f .t/ dt

D 1 Z 1

D

n o K˛ .t; !/ .itcsc ˛/r f .t/ dt

1

n o D F ˛ .it csc ˛/r f .t/ :

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

42

1 The Fourier Transforms

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

Proposition 1.6.5. If f 2 L1 .R/, then fO ˛ satisfies the following properties: (a) (b) (c)

fO ˛ .!/ 2 L1 .R/, fO ˛ .!/ continuous on R, fO ˛ .!/ ! 0 as ! ! ˙1:

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

1

   jC˛ j f 1 :

(b) For any h > 0; we have ˇ ˇ ˇ ˇ sup ˇfO ˛ .! C h/  fO ˛ .!/ˇ

!2R

ˇZ  ˇ 1 o n cot ˛ ˇ D sup ˇ  it.! C h/ csc ˛ C˛ exp i.t2 C .! C h/2 / 2 !2R ˇ 1 ˇ ˇ o n ˇ 2 2 cot ˛  it! csc ˛ f .t/ dtˇ   exp  i.t C ! / ˇ 2 ˇ ˇ   Z 1 ˇ ˇˇ ˇ h cot ˛ ˇ  sup C ! cot ˛  t csc ˛  1ˇˇˇf .t/ˇ dt: jC˛ j ˇexp ih 2 !2R 1

Since ˇ ˇ   ˇ ˇ ˇexp ih h cot ˛ C ! cot ˛  t csc ˛  1ˇˇ ! 0; ˇ 2

as h ! 0:

Therefore, it follows that the R.H.S of above inequality tends to zero as h ! 0, that is, ˇ ˇ ˇ ˇ lim sup ˇfO ˛ .! C h/  fO ˛ .!/ˇ D 0:

h!0 !2R

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

1.6 The Fractional Fourier Transform

43

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

o

˚

  F ˛ . 0t / f .t/ D  i! csc ˛ F ˛ f .t/ .!/ D  i! csc ˛ fO ˛ .!/

so that ˇ ˇ ˇ ˇO ˛ ˇf .!/ˇ D

ˇn oˇ 1 ˇ ˇ O˛ 0 ˇ F . t /f .t/ ˇ ! 0 as j! csc j

! ! ˙1:

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

1; t  0; 0; t < 0:

n it2 cot ˛ o  t h.t/ 2 L1 .R/, Then, it is easy to verify that the product f ˛ .t/ D exp  2 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;  2 i! cot ˛ C it! csc ˛ f .t/; M!˛ f .t/ D exp 2 Da f .t/ D f .at/;

(Translation), (Modulation), (Dilation),

where t; !; a 2 R and a ¤ 0. The following results can easily be verified: ˚ ˚ ˛ F ˛ f .t/ .!/ F ˛ T!˛ f .t/ .!/ D M! ˚ ˚ ˛ F ˛ f .t/ .!/ F ˛ M!˛ f .t/ .!/ D T! ˚ ˚ F ˛ D1 f .t/ .!/ D F ˛ f .t/ .!/ Z 1 Z 1 f .!/ gO ˛ .!/ d!: fO ˛ .t/ g.t/ dt D 1

1

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

44

1 The Fourier Transforms

Proof. For any f .t/ 2 S.R/  L1 .R/, we have ˚ F ˛ f .t/ .!/ D fO ˛ .!/ 2   2   p i! cot ˛ it cot ˛ D 2 C˛ exp F exp f .t/ .! csc ˛/ 2 2 n i! 2 cot ˛ o F ˛ .!/ (1.6.15) D C˛ exp 2 where F ˛ .!/ D

 2   p it cot ˛ 2 F exp f .t/ .! csc ˛/ 2 S.R/: 2

We have Dˇ! fO ˛ .!/

  2  i! cot ˛ O ˛ f .!/ exp D 2 !  ˇ n i! 2 cot ˛ o X ˇ 0 ˛ ˇ0 D! exp Dˇˇ D C˛ F .!/ ! 0 ˇ 2 0 C˛ Dˇ!

ˇ D0

D C˛

ˇ X ˇ 0 D0

!  2  i cot ˛  ˇ i! cot ˛ 0 ˛ 0 !; P Dˇˇ F .!/ exp ˇ ! 0 2 2 ˇ

 i cot ˛  is a polynomial. Thus where Pˇ0 !; 2 Dˇ! fO ˛ .!/ D C˛

ˇ X ˇ 0 D0

!  ˇ0 2 00 0 ˇ i! cot ˛ X aˇ00 .cot ˛/ ! ˇ Dˇˇ F ˛ .!/: exp ! ˇ0 2 00 ˇ D0

Therefore, we have ˇ ˇ ! ˇ0 ˇ 2  ˇ ˇ ˇˇ ˇ X X 00 ˇ i! ˇ cot ˛ 0 ˇ ˇ ˇ t ˇ O˛ ˇ Ct ˇˇ ˛ exp D! F .!/ˇˇ aˇ00 .cot ˛/ ! ˇ! D! f .!/ˇD ˇC˛ 0 ˇ 2 ˇ ˇ0 D0 ˇ 00 ˇ D0 ˇ ˇ ˇX ˇC˛ ˇ ˇ0 D0

ˇ ˇ0

!

ˇ0 ˇ ˇ ˇ 00 ˇ X 0 ˛ ˇˇ ˇ ˇ 00 F .!/ ˇ: ˇaˇ .cot ˛/ˇ ˇ! ˇ Ct Dˇˇ ! 00

ˇ D0

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

1.6 The Fractional Fourier Transform ˇ ˇ ˇ ˇX ˇ ˇ ˇ sup ˇ! t Dˇ! fO ˛ .!/ˇ  ˇC˛ ˇ

!2R

ˇ ˇ0

ˇ0 D0

45

!

ˇ0 ˇ ˇ ˇ ˇ 00 X 0 ˛ ˇ ˇ ˇ 00 ˇ F .!/ˇ < 1: ˇaˇ .cot ˛/ˇ sup ˇ! ˇ Ct Dˇˇ ! !2R

00

ˇ D0

(1.6.16)

Hence, fO˛ .!/ 2 S.R/: In view of (1.6.1) and (1.6.4), we see that for all f 2 S.R/, we have h i h i (1.6.17) f .t/ D F ˛ F ˛ f .t/ D F ˛ F ˛ f .!/ : Therefore, it follows that F ˛ is a one-one map from S.R/ onto itself. ˚ To show that F ˛ is continuous on R, assume that there exist a null sequence fn n2N in S.R/; then from (1.6.16), we infer that F ˛ ffn .t/g ! 0 in S.R/; and hence, the continuity of FrFT follows.

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

˛ f;g D

Z

1

E D fO ˛ .!/ gO ˛ .!/ d! D fO ˛ ; gO ˛ :

(1.6.18)

1

Proof. The proof of the theorem follows immediately from the Parseval formula of the 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-known differential equations by using the fractional Fourier transform method. We consider the nth-order linear nonhomogeneous ordinary differential equations 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 an1 . 0t /n1 C    C a1 . 0t / C a0

(1.6.20)

where an ; an1 ; : : : ; a1 ; a0 are constants, 0t is the same as defined in Proposition 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

K˛ .t; !/ Ly.t/ dt D 1

Z

1

1

K˛ .t; !/ f .t/ dt;

46

1 The Fourier Transforms

that is, i h a0 .i! csc ˛/n Can1 .i! csc ˛/n1 C  Ca1 .i! csc ˛/Ca0 yO ˛ .!/ D fO˛ .!/: Or, equivalently, P.i! csc ˛/ yO ˛ .!/ D fO˛ .!/ where P.z/ D

n X

am zm . Therefore, it follows that

mD0

yO ˛ .!/ D

fO˛ .!/ : P.i! csc ˛/

(1.6.21)

Applying the inverse FrFT to (1.6.21) gives the formal solution " y.t/ D F

˛

# fO˛ .!/ : P.i! csc ˛/

(1.6.22)

Example 1.6.3. Consider the generalized wave equation @2 y.x; t/ D k2 . 0x /2 y.x; t/; @t2 with the initial data y.x; 0/ D f .x/; is the same as in Proposition 1.6.2.

1 < x < 1; t > 0

(1.6.23)

@y.x; 0/ D g.x/, where k is a constant and 0x @t

Application of the fractional Fourier transform to (1.6.23) gives Z

1

K˛ .x; !/ 1

@2 y.x; t/ dx D k2 @t2

Z

1 1

K˛ .x; !/. 0x /2 y.x; t/ dx

so that @2 yO ˛ .!; t/ D k2 @t2

Z

1

. x /2 K˛ .x; !/ y.x; t/ dx D k2 ! 2 csc2 ˛ yO ˛ .!; t/: 1

1.6 The Fractional Fourier Transform

47

Therefore, it follows that ˚ ˛

 ˚

 F g .!/ yO ˛ .!; t/ D F ˛ f .!/ cos .c! csc ˛/t C sin .c! csc ˛/t : c! csc ˛ (1.6.24) This can readily be inverted by the inverse FrFT (1.6.4) to obtain h˚



i F ˛ f .!/ cos .c! csc ˛/t .x/ # "˚

 F ˛ g .!/ ˛ sin .c! csc ˛/t .x/ CF c! csc ˛

y.x; t/ D F ˛

D R.x; t/ C S.x; t/

(1.6.25)

R.x; t/ D F ˛

h˚

i F ˛ f .!/ cos .c! csc ˛/t .x/

(1.6.26)

S.x; t/ D F

# "˚

 F ˛ g .!/ sin .c! csc ˛/t .x/: c! csc ˛

where

and ˛

(1.6.27)

We now estimate R.x; t/ as  cot ˛ C ix! csc ˛ R.x; t/ D C˛ exp i.x C ! / 2 1 ˚ ˛ ˚ F f .!/ cos .c! csc ˛/t d!  Z 1 cot ˛ 1 exp i.x2 C ! 2 / C ix! csc ! D 2 sin ˛ 1 2   Z 1 ˚ 2 2 cot ˛  iz! csc ˛ f .z/ dz cos .c! csc ˛/t d!:  exp i.z C ! / 2 1 (1.6.28) Z



1

Setting H˛ .z/ D exp

n

iz2 cot ˛ 2

2

o

2

f .z/ and ! csc ˛ D , equation (1.6.28) becomes   Z 1 Z 1 ix2 cot ˛ 1 R.x; t/ D eix cos.c t/ d

eiz H˛ .z/ dz exp sin ˛ 2 sin ˛ 2 1 1   Z 1 h i ˚ 2 ix cot ˛ 1 exp D p ei.xCct/ C ei.xct/ F H˛ .z/ . / d

2 2 2 1

48

1 The Fourier Transforms

 i ix2 cot ˛ h 1 D exp H˛ .x C ct/ C H˛ .x  ct/ 2 2  i ix2 cot ˛ h i.xCct/2 cot ˛ 1 2 cot ˛ 2 f .x C ct/ C ei.xct/ 2 f .x  ct/ : e D exp 2 2 (1.6.29)

o n 2 ˛ g.z/, and again setting ! csc ˛ D , Similarly, assume that G˛ .z/ D exp iz cot 2 we obtain  Z 1 ix

1 e sin.c t/ ˚ ix2 cot ˛ S.x; t/ D p exp F G˛ .z/ . / d : (1.6.30) 2 c

2 1 Differentiating equation (1.6.30) with respect to t, we obtain the same result as that of R.x; t/, and then by integrating, we have S.x; t/ D

 Z xCct  2 1 ix2 cot ˛ i! cos ˛ exp g.!/ d!: exp 2c 2 2 xct

(1.6.31)

After substituting equations (1.6.29) and (1.6.31) in (1.6.25), the solution of the given problem (1.6.23) can be obtained in the form " 1 ix2 cot ˛ 2 cot ˛ 2 cot ˛ y.x; t/ D exp ei.xCct/ 2 f .x C ct/ C ei.xct/ 2 f .x  ct/C 2 2 1 c

Z

xCct xct

#  i! 2 cot ˛ g.!/d! : exp 2

Example 1.6.4. Consider the generalized heat equation

2 @y.x; t/ D 0x y.x; t/; @t

1 < x < 1; t > 0

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

K˛ .x; !/ 1

@y.x; t/ dx D @t

Z

1 1

K˛ .x; !/. 0x /2 y.x; t/ dx

so that @Oy˛ .!; t/ D @t

Z

1

. x /2 K˛ .x; !/ y.x; t/ dx D ! 2 csc2 ˛ yO ˛ .!; t/: 1

(1.6.32)

1.6 The Fractional Fourier Transform

49

Therefore, it follows that ˚ yO ˛ .!; t/ D C.!/ exp  .! 2 csc2 ˛/t

(1.6.33)

which gives yO ˛ .!; 0/ D C.!/. Since yO ˛ .!; 0/ D

Z

1

K˛ .x; !/ y.x; 0/ dx

1 Z 1

D

K˛ .x; !/f .x/ dx

1

˚ D F ˛ f .x/ .!/; hence, ˚ C.!/ D F ˛ f .x/ .!/:

(1.6.34)

Thus, equation (1.6.33) becomes ˚ yO ˛ .!; 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/ n ˚ o D F ˛ .F ˛ f /.!/ exp .! 2 csc2 ˛/t .x/  ˚ i.x2 C ! 2 / cot ˛ C ix! csc ˛ .F ˛ f /.!/ exp .! 2 csc2 ˛/t d! exp  2 1  Z 1 2 E˛ i.x C ! 2 / cot ˛ D C ix! csc ˛ exp  2 1 2   2  Z 1 ˚ i.z C ! 2 / cot ˛  iz! csc ˛ f .z/dz exp .! 2 csc2 ˛/t d! exp  C˛ 2 1 Z 1 ˚ ˚ 1 ix2 cot ˛ D exp exp ix! csc ˛ exp  .! 2 csc2 ˛/t d! 2 sin ˛ 2 1  2 Z 1 ˚ iz cot ˛ f .z/dz: exp  iz! csc ˛ exp  2 1 D

E˛ 2

Z

1

50

1 The Fourier Transforms

 iz2 cot ˛ f .z/; then Let us assume that H˛ .z/ D exp 2

y.x; t/ D

Z 1 Z 1 ix2 cot ˛ 1 2 2 eix! csc ˛ e.! csc ˛/t d! eiz! csc ˛ H˛ .z/dz: exp 2 sin ˛ 2 1 1

Setting ! csc ˛ D , the above relation becomes Z 1 Z 1 ˚ ix2 cot ˛ 1 exp exp fix g exp  2 t d

expfi zgH˛ .z/ dz 2 2 1 1 Z 1 n 2 o n o 1 ix2 cot ˛ exp fix g F ex =4t . /F H˛ .z/ . / d

D p exp 2 2 t 1 Z 1 i h 2 2 ix cot ˛ 1 exp fix g F ex =4t  H˛ .x/ . /d

p exp D p 2 2 t  2 1   Z 1 2 .x  !/2 1 ix cot ˛ H˛ .!/ d!: exp D p exp 2 4t 2 t 1

y.x; t/ D

Therefore, the solution of the generalized heat equation (1.6.32) is Z 1  2  1 i! cot ˛ .x  !/2 ix2 cot ˛ y.x; t/D p exp  exp f .!/ d!: exp 2 4t 2 2 t 1

1.7

The Uncertainty Principle

Heisenberg first formulated the uncertainty principle between the position and momentum in quantum mechanics. This principle has an important interpretation as an uncertainty of both the position and momentum of a particle described by a wave function 2 L2 .R/. In other words, it is impossible to determine the position and momentum of a particle exactly and simultaneously (See Heisenberg, 1948a,b). In signal processing, time and frequency concentrations of energy of a signal f are also governed by the Heisenberg uncertainty principle. The average or expectation values of time t and frequency ! are, respectively, defined by

1.7 The Uncertainty Principle

1 t D  2 f  

2

Z

1

51

ˇ ˇ2 tˇf .t/ˇ dt;

1 ! D  2 O f  

1

Z

1 1

ˇ ˇ2 ˇ ˇ ! ˇfO .!/ˇ d!;

(1.7.1)

2

where the energy of a signal f is well localized in time, and its Fourier transform fO has an energy concentrated in a small frequency domain. The variances around these average values are given respectively by 1 t2 D  2 f  2

Z

1

2 ˇ ˇ2 t  t ˇf .t/ˇ dt;

!2 D

1

1  2 O 2 f 

Z

1 1

ˇ2

2 ˇˇ ˇ !  !  ˇfO .!/ˇ d!:

2

If f .t/; tf .t/, and ! fO .!/ belong to

Theorem 1.7.1 p (Heisenberg’s Inequality). L2 .R/ and t jf .t/j ! 0 as jtj ! 1, then t2 !2 

(1.7.2)

1 ; 4

(1.7.3)

where t is defined as a measure of time duration of a signal f and ! is a measure of frequency dispersion (or bandwidth) of its Fourier transform fO . Equality in (1.7.3) holds only if f .t/ is a Gaussian signal given by f .t/ D 2 C ebt ; 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 ei! t f 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 fO  , we have 2

 4 2 2 f    D 1 2 t ! 2

Z

1

ˇ ˇ ˇtf .t/ˇ2 dt

Z

1

1

1

ˇ ˇ ˇ O ˇ2 ˇ! f .!/ˇ d!:

˚ Using i! fO .!/ D F f 0 .t/ and Parseval’s formula    0  f .t/ D 1  O .!/ f i!  ; 2 2 2

52

1 The Fourier Transforms

we obtain Z  4 2 2 f    D 2 t !

1 1

Z

1

ˇ 0 ˇ2 ˇf .t/ˇ d!

1

ˇZ ˇ  ˇˇ

1

ˇZ ˇ  ˇˇ

o ˇˇ2 1n 0 0 f .t/ f .t/ C f .t/f .t/ dtˇˇ t 2 1

1 D 4 D in which

ˇ ˇ ˇtf .t/ˇ2 dt

1

n o ˇˇ2 tf .t/ f 0 .t/ dtˇˇ ;

by Schwarz’s inequality

1

Z

 2 2 Z 1 d 2 1 h ˇˇ ˇˇ2 i1 2 jf j dt D t f .t/ t  jf j dt 1 dt 4 1 1 1



 1 f 4 ; 2 4

p 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 a 2 constant of proportionality, this leads to the Gaussian signal f .t/ D C ebt ; where a C is a constant of integration and b D  > 0. 2 Remarks. 1. In a time-frequency analysis of signals, the measure of the resolution of a signal f in the time or frequency domain is given by t and ! . Then, the joint resolution is given by the product .t /.! / which is governed by the Heisenberg uncertainty principle. In other words, the product .t /.! / cannot be arbitrarily small and is 1 always greater than the minimum value which is attained only for the Gaussian 2 signal. 2. In many applications in science and engineering, signals with a high concentration of energy in the time and frequency domains are of special interest. The uncertainty principle can also be interpreted as a measure of this concentration of the second moment of f 2 .t/ and its energy spectrum fO 2 .!/.

1.8 Exercises

1.8

53

Exercises

1. Find the Fourier transforms of each of the following functions: 2 (b) f .t/ D t eat ; a > 0; (a) f .t/ D t eajtj ; a > 0; 2 t2 2 at2 Cbt (c) f .t/ D t e ; (d) f .t/ D e ; 2 at sin (f) f .t/ D : (e) f .t/ D jtja1 ; at2  t Œa;a .t/; where a > 0, show that 2. If f .t/ D 1  a sin2 .a!/ fO .!/ D : a.!/2 3. Show that the Fourier transform of f .t/ D

cos bt ; a2 C t 2

a > 0;

for any constant b is i  h aj!bj e fO .!/ D C eaj!Cbj : 2a 4. If f .t/ has a finite discontinuity at a point t D a, prove that ˚ F f 0 .t/ D .i!/fO .!/  eia! 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 o n o n 2 2 (b) F tn eat : (a) F tn et =2 ; 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/ D f  g .t/;   (d) t f .t/  g.t/ D tf .t/  g.t/ C f .t/  tg.t/ 7. Show that 2 2 n 2 2o n 2 2o  e! =4c ; F ea t  F eb t D ab

where

1 D c2



1 1 C 2 2 a b

 :

54

1 The Fourier Transforms

1 t2 e , then show that the family of functions G .t/ D  G.t/,   > 0 forms a summability kernel on R. Moreover, show that

8. If G.t/ D

Z

1

e!

G .t/  f .t/ D

2 =42

fO .!/ ei!t d!:

1

9. Consider f W ZN ! C, such that f .n/ D ean , for some constant a. Show that the discrete Fourier transform of f is fO .n/ D

1  eaN : 1  ea2in=N

10. For positive integer N, let w D e2i=N . Show that (a) w w D 1; (b) wk D wk D wNCk D wNCk ; for any integer k, 2 (c) 1 C w C w C    C wN1 D 0; N > 1:

 11. Show that the fast Fourier transform of f .n/ D 1; 0; i; 2; i; 1; 0; i 2 l2 .Z8 / is  p p p p fO .n/ D 4 C i; 2  2 2 C i; i; 2 C i.1  2/; 2  i; 2 C 2 2 C i; 2  3i; p  p  2 C i.1 C 2/ : 12. Verify the equality of the uncertainty principle Gaussian function p for the 2 2 f .t/ D et =2 and its Fourier transform fO .!/ D 2 e! =2 . 13. Repeat the calculation of the previous exercise using the function 1 2 2 f .t/ D p et =4a and its Fourier transform fO .!/ D ea! . Show that 2 a equality of the Heisenberg inequality is preserved for all values of a.

2

The 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 timefrequency atoms as waveforms that have a minimal spread in a time-frequency plane. To measure time-frequency ‘information’ content, he proposed decomposing signals over these elementary atomic waveforms. By showing that such decompositions are closely related to our sensitivity to sounds, and that they exhibit important structures in speech and music recordings, Gabor demonstrated the importance of localized time-frequency signal processing. Stéphane Mallat

2.1

Introduction

Signals are in general nonstationary. A complete representation of nonstationary signals requires frequency analysis that is local in time, resulting in the time-frequency analysis of signals. The Fourier transform analysis has long been recognized as the great tool for the study of stationary signals and processes where the properties are statistically invariant over time. However, it cannot be used for the frequency analysis that is local in time because it requires all previous as well as future information 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, there has been major development of the time-frequency distributions approach in the last three decades. The basic idea of the method is to develop a joint function of time and frequency, known as a time-frequency distribution, that can describe the energy density of a signal simultaneously in both time and frequency. In principle, the

© Springer International Publishing AG 2017 L. Debnath, F.A. Shah, Lecture Notes on Wavelet Transforms, Compact Textbooks in Mathematics, DOI 10.1007/978-3-319-59433-0_2

55

56

2 The Time-Frequency Analysis

time-frequency distributions characterize phenomena in a two-dimensional timefrequency plane. Basically, there are two kinds of time-frequency representations. One is the quadratic method covering the time-frequency distributions, and the other is the linear approach including the Gabor transform, the Zak transform, the linear canonical transform, and the wavelet transform analysis. So, the timefrequency signal analysis deals with time-frequency representations of signals and with problems related to their definition, estimation, and interpretation, and it has evolved into a widely recognized applied discipline of signal processing. For more detailed information, we refer to Debnath (2001), Grochenig (2001), and Debnath and Shah (2015). This chapter is devoted to a fairly detailed examination of the joint timefrequency analysis of signals. We start with the time-frequency localization of signals which leads to the windowed Fourier transform. This is followed by the Gabor transform and its basic properties. Included are the Zak transform and its basic properties. Based on the relationship between the Fourier transform and linear canonical transform, a coupled windowed transform, namely, windowed linear canonical transform (WLCT) is introduced.

2.2

The Time-Frequency Localization

To achieve the time-frequency localization of spectral characteristics of a timevarying signal, a window function is introduced into the Fourier transform. A window function g.t/ is a function in L2 .R/ such that both g.t/ and gO .!/ have rapid decay, that is, g.t/ is well localized in time domain, while gO .!/ is well localized in frequency domain. Multiplying a signal f .t/ by a window function g.t/ before its Fourier transform has the effect of restricting the spectral information of the signal to the domain of influence of the window function. Using the translates of the window function on the time axis to cover the entire time domain, the signal is analyzed for spectral 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 magnitude jg.t/j2 has a second-order moment. Therefore, if g.t/ is a window function, then t1=2  g.t/ also belongs to L2 .R/. Writing g.t/ D .1 C jtj/1 .1 C jtj/g.t/ and applying the Schwartz inequality, we can obtain       g  .1 C jtj/1 .1 C jtj/ < 1; 1 2 2 which infer that g.t/ is integrable on R and, hence, gO .!/ is continuous. Although it follows from the Parseval identity that gO .!/ is also in L2 .R/, in general, it is not true

2.2 The Time-Frequency Localization

57

that gO 2 L2 .R/. In other words, it is possible that while g is a window function, gO is not. An example of such a window function is the Haar function: 8 < 1; g.t/ D 1; : 0;

0  t < 12 1 2 t < 1 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; 0;

jtj  a; jtj > a:

a>0

Its Fourier transform is gO .!/ D

eia!  eia! : i!

Although g.t/ is compactly supported, it gives a bad localization in frequency due to its discontinuous nature. Therefore, usually the more smooth functions are needed. 2. The triangular window or Fejer window is given by 8 ˆ ˆ 1, the reconstruction of f is, in general, impossible no matter how g is selected.

2.4

The Zak Transform

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

72

2 The Time-Frequency Analysis

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

 Definition 2.4.1 (The Zak Transform). The Zak transform Za f .t; !/ of a function f 2 L2 .R/ is defined by the series 

p X f .at C an/ e2in! ; Za f .t; !/ D a

(2.4.1)

n2Z

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 timefrequency representation of the signal f . It can also be considered as the discrete Fourier transform of f in which an infinite set of samples in the form f .at C an/ is used for n D 0;  ˙1; ˙2; : : : : Without loss of generality, we set a D 1 so that we can write Z f .t; !/ in the explicit form X

 Z f .t; !/ D F.t; !/ D f .t C n/ e2in! :

(2.4.2)

n2Z

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 e2i! Z f .t; !/;

(2.4.4)

and therefore the Zak transform Z f is completely determined by its values on the unit square S D Œ0; 1  Œ0; 1. It is easy to prove that the Zak transform of f can be expressed in terms of the Zak transform of its Fourier transform fO ./ D F ff .t/g. More precisely,



 Z f .t; !/ D e2i!t Z fO .!; t/: To prove this result, we define a function g for fixed t and ! by g.x/ D e2i!x f .x C t/:

(2.4.5)

2.4 The Zak Transform

73

Then, it follows that Z

1

g.x/ e2ix dx

gO ./ D 1 Z 1

f .x C t/ e2ix.C!/ dx

D 1

D e2i.C!/t

Z

1

f .u/ e2i.C!/u du 1

D e2i.C!/t fO . C !/: We next use the Poisson summation formula in the form X X g.n/ D gO .2n/: n2Z

Or, equivalently, X

n2Z

f .t C n/ e2i!n D e2i!t

n2Z

X

eŒ2i .2n/t fO .! C 2n/

n2Z

D e2i!t

X

fO .! C m/e2imt :

m2Z

This gives the desired result (2.4.5). The following results can be easily verified:





 Z F f .!; t/ D e2i!t Z f .t; !/;

(2.4.6)

  Z F 1 f .!; t/ D e2i!t Z f .t; !/:

(2.4.7)

If gm;n .t/ D e2imt g.t  n/, then 



Z gm;n .!; t/ D e2i.mtCn!/ Z g.!; t/ : We next observe that L2 .S/ is the set of all square-integrable complex-valued functions F on the unit square S, that is, Z

1

Z

0

1

ˇ ˇ ˇF.t; !/ˇ2 dt d! < 1:

0

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

˛ F; G D

Z

1

Z

1

F.t; !/ G.t; !/ dt d! 0

0

(2.4.8)

74

2 The Time-Frequency Analysis

and the norm   F  D 2

Z

1

Z

0

1

ˇ ˇ ˇF.t; !/ˇ2 dt d!

 1=2 :

0

The set n

o Mm;n D M2m;2n .t; !/ D e2i.mtCn!/ W m; n 2 Z

forms an orthonormal basis of L2 .S/. Example 2.4.1. If 1 m;nIa .t/ D p Tna M2m=a Œ0;a .t/; a where a > 0, then

 Za m;nIa .t; !/ D em .t/ en .!/;

where ek .t/ D e2ikt :

We have tna 1 m;nIa .t/ D p e2im. a / Œ0;a .t  na/ a

1 2imt D p e a Œna;.nC1/a .t/: a Thus, we obtain

X 2im  Za m;nIa .t; !/ D e a .atCak/ Œna;naCa .at C ak/ k2Z

D

X

em .t/ e2ik! Œnk;nC1k .t/

k2Z

D em .t/ en .!/: We shall now discuss the basic properties of continuous Zak transform.

(2.4.9)

2.4 The Zak Transform

75

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 .Ta f / .t; !/ D Z f .t   a; !/; 

 

b ibt ; (c) Modulation: Z .Mb f / .t; !/ D e Z f t; !  2

  (d) Translation and modulation: Z M2m Tn f .t; !/ D e2i.mtn!/ Z f .t; !/;



 (e) Conjugation: Z fN .t; !/ D Z f .t; !/; (f) Symmetry:

 Z f .t; !/ D



 Z f .t; !/;

  Z f .t; !/;

if f is even if f is odd

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

1

 Z f .t; !/ d!;

1

 exp.2i!t/ Z f .t; !/ dt;

1

 exp.2ixt/ Z fO .t; x/ dt

f .t/ D fO .!/ D

Z Z

0

0

f .x/ D 0

 

 !  (h) Dilation: Z D 1 f .t; !/ D Za f at; ; a 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 two variables t and !. The product of two Zak transforms is periodic in t and !.

Proof. We consider the product



 F.t; !/ D Z f .t; !/ Z g .t; !/ and find from (2.4.4) that



 Z g .t; !/ D e2i! Z g .t; !/:

76

2 The Time-Frequency Analysis

Therefore, it follows that



 F.t C 1; !/ D Z f .t; !/ Z g .t; !/ D F.t; !/;



 F.t; ! C 1/ D Z f .t; !/ Z g .t; !/ D F.t; !/: These show that F is periodic in t and !. Consequently, it can be expanded in a Fourier series on a unit square XX

F.t; !/ D

cm;n e2imt e2in! ;

(2.4.10)

m2Z n2Z

where Z

1

Z

1

cm;n D 0

F.t; !/ e2imt e2in! dt d!:

0

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

1

Z

1

(

cm;n D 0

Z

0 1

(

D 0

Z

1

(

D 0

D

)( f .t C r/ e

r2Z

X

)(

f .t C r/

r2Z

X

2ir!

X

) gN .t C s/ e

2is!

e2i.mtCn!/ dt d!

s2Z

X

)

gN .t C s/ e

s2Z

2imt

Z

1

e2i!.snr/ d!

dt 0

)

f .t C r/ gN .t C n C r/ e2imt dt

r2Z

XZ r2Z

Z

X

rC1

f .x/ gN .x C n/e2im.xr/ dx r

1

f .x/ gN .x C n/ e2imx dx

D 1

E D D f .x/; e2imx g.x C n/ ˛ ˝ D f ; M2m Tn g : Consequently, (2.4.10) becomes XX˝

 ˛

 f ; M2m Tn g e2i.mtCn!/ : Z f .t; !/ Z g .t; !/ D m2Z n2Z

This completes the proof.

(2.4.11)

2.4 The Zak Transform

77

Theorem 2.4.2. Suppose H is a function of two real variables t and s satisfying the condition Z 1



 H t C 1; s C 1 D H t; s ; and h.t/ D H.t; s/f .s/ ds; s; t 2 R; 1

where the integral is absolutely and uniformly convergent. Then,

 Z f .t; !/ D

Z

1

 Z f .s; !/ ˆ.t; s; !/ ds;

(2.4.12)

0

where ˆ is given by ˆ.t; s; !/ D

X

H.t C n; s/ e2in! ;

0  t; s; !  l:

(2.4.13)

n2Z

Proof. Using the definition of Zak transform, we have X X

 Z h .t; !/ D h.t C k/ e2ik! D e2ik! k2Z

D

k2Z

X

e

2ik!

X k2Z

Z

1

1

H.t C k; s/f .s/ ds 1

mC1

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

m2Z m

k2Z

D

XZ

Z

e

2ik!

(

D 0

XZ

1

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

m2Z 0

XX

)

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

2ik!

ds;

k2Z m2Z

which is, due to (2.4.11), Z

1

(

1

(

D 0

Z D 0

Z

) H.t C k  m; s/ f .s C m/ e2ik! ds

k2Z m2Z

XX

) H.t C n; s/ f .s C m/ e

2i.mCn/!

ds

m2Z n2Z 1

D

XX

 Z f .s; !/ ˆ.t; s; !/ ds:

0

This completes the proof.

(2.4.14)

78

2 The Time-Frequency Analysis

In particular, if H.t; s/ D H.t  s/, ˆ.t; s; !/ D

X

 H.t  s C n/ e2in! D Z H .t  s; !/:

n2Z

Consequently, Theorem 2.4.2 leads to the following convolution theorem. Theorem 2.4.3 (Convolution Theorem). Z

If

1

h.t/ D

H.t  s/f .s/ ds D .H  f /.t/; 1

then (2.4.12) reduces to the form

 Z h .t; !/ D

Z

1





 Z H .t  s/ Z f .s; !/ ds D Z .H  f /.t; !/:

(2.4.15)

0

Example 2.4.2. If H.t/ D

X

ak ı.t  k/; then

k2Z

 Z .H  f /.t; !/ D A.!/ Z f .t; !/;

(2.4.16)

where A.!/ D

X

ak e2ik! :

k2Z

Clearly, Z .H  f /.t; !/ D Z

Z (

DZ ( DZ

1

 H.t  s/f .s/ ds .t; !/

1

X k2Z

X k2Z

Z

)

1

ı.t  s  k/f .s/ ds .t; !/

ak 1

) ak f .t  k/ .t; !/

2.5 The Windowed Linear Canonical Transform

D D

X

ak

X

k2Z

n2Z

X

X

k2Z

ak

79

f .t C n  k/ e2in! f .t C m/ e2i!.mCk/

m2Z

 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 ˝ ˛ Za f ; Za g D a

Z

Z

1

0

Z

1 0

(

1

Da 0

D

n2Z 1

D

X

X

)( f .at C an/ e

n2Z

2in!

)

X

) g.at C am/ e

2im!

dt d!

m2Z

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

n2Z

XZ Z

(

.nC1/a

f .y/ g.y/ dy na

˝ ˛ f .y/ g.y/ dy D f ; g :

(2.4.17)

1

In particular, if f D g, we obtain from (2.4.17) that     Za f 2 D f 2 : 2 2

(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 basis of 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 and image processing. This transform has also been referred to by different names in

80

2 The Time-Frequency Analysis

the open literature such as quadratic-phase integrals (Bastiaans, 1979), generalized Huygens 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 Fourier transform (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 transforms such as the Fourier transform, the fractional Fourier transform (FrFT), the Fresnel transform, the Lorentz transform, and scaling operations. Thus, understanding the LCT may help to gain more insight into its special cases and to carry the knowledge gained 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 of conventional Fourier transform (see Healy and Sheridan, 2010). The LCT has found many applications in phase reconstruction, filter design, signal synthesis, pattern recognition, time-frequency analysis, optimal filtering, radar analysis, holographic three-dimensional television, quantum physics, and many others. However, the LCT cannot reveal the local LCT-frequency contents due to its global kernel. On the other hand, the windowed Fourier transform (WFT) with a local window function handles this kind of situation very well, but unfortunately, the WFT often performs unsatisfactorily for its low resolution. Therefore, in order to attain the local contents and high localization properties of a signal, it is desirable to develop a new transform by replacing the Fourier transform kernel with the LCT kernel in the windowed Fourier transform definition. This new transform was first introduced by Bultheel and Martinez-Sulbaran (2007) and is called the windowed linear canonical transform (WLCT) which offers a flexible local frequency content, eliminates cross term, and enjoys high resolution of a signal. For more about LCT and their applications to signal and image processing, the reader is to referred to Stankovic et 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).   ab Definition 2.5.1. Let A D be a unimodular matrix, i.e., det.A/ D ad cd 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

LA Œf .!/ D

8Z 1 ˆ ˆ f .t/ KA .t; !/ dt; ˆ ˆ < 1

b¤0

 ˆ ˆ p cd! 2 ˆ ˆ : d exp i f .d!/; b D 0 2

(2.5.1)

2.5 The Windowed Linear Canonical Transform

81

where the kernel KA .t; !/ of LCT is given by   2 2t! d! 2  1 i at KA .t; !/ D p  C  : exp 2 b b b 4 2b

(2.5.2)

For typographical convenience, we shall often denote the matrix A as A D .a; b; c; d/, in the text, but all operations have to be understood in the usual matrix sense. Moreover, we note that when b D 0, the LCT becomes a chirp multiplication. Therefore, we only consider the case of b ¤ 0, and without loss of generality, 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 LA Œf .!/ D

Z

1

f .t/ KA .t; !/ dt;

(2.5.3)

1

where p   ˇ i  KA .t; !/ D p exp

t2  2ˇt! C ˛! 2  ; 2 4 2

(2.5.4)

and the parameter matrix is given by      1 ab

=ˇ 1=ˇ ˛=ˇ 1=ˇ : AD D D cd ˇ C ˛ =ˇ ˛=ˇ ˇ  ˛ =ˇ =ˇ

(2.5.5)

2. The LCT given by (2.5.1) can be computed via Fourier transform as   2   d! 2  iat !  F exp f .t/ : b 2 2b b (2.5.6)   2  i at 1  f .t/, then equation exp Note that if we let h.t/ D p 2 b 2 2b (2.5.6) takes the form 1



i LA Œf .!/ D p exp 2 2b

exp



 ˚ !  id! 2 LA Œf .!/ D F h.t/ ; 2b 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.

82

2 The Time-Frequency Analysis

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˛=2 LA if A D R˛. 5. For the matrix A D .a; b; c; d/ D .1; b; 0; 1/, the LCT reduces to the Fresnel transform. 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/ D d1 ; 0; 0; d . Two interesting and important properties of LCT are the index additivity and reversibility. Index additivity means that the composition of two LCTs with parameter matrices A1 D .a1 ; b1 ; c1 ; d1 / and A2 D .a2 ; b2 ; c2 ; d2 /, respectively, equals to the LCT with parameter matrix A3 D A2 A1 , that is,

 LA1 LA1 Œf  D LA3 Œf  D LA2 A1 Œf :

(2.5.8)

The inverse of the LCT with parameter matrix A D .a; b; c; d/ is the LCT with parameter matrix of A1 D .d; b; c; a/, that is, LA1 .FA / .f / D f :

(2.5.9)

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

a2 d1 C b2 b1

 > 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 since A1 D .a; b; c; d/ and A2 D .d; b; c; a/ D A1 1 by invoking additive property (2.5.8). Another important property of LCT is the Parseval formula: ˝

E ˛ D f ; g D LA .f /; LA .g/ :

(2.5.11)

In particular, when f D g, we obtain the Plancherel formula for the LCT: 2  2   f  D  LA .f / : 2 2

(2.5.12)

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

2.5 The Windowed Linear Canonical Transform

83

Before we give the formal definition of WLCT, we first recall that the windowed Fourier transform (2.2.6) of any f 2 L2 .R/ with respect to the window function g 2 L2 .R/ is given by G Œf .u; !/ D

Z

1

˛ ˝ f .t/ g.t  u/ ei!t dt D f ; gu;! ;

(2.5.13)

1

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

(2.5.14)

is called a Gabor filter. The extension of the Gabor filter to the LCT domain is given by the following definition. Definition 2.5.2. For a window function g 2 L2 .R/n f0g, its window daughter function associated with LCT is defined by   1 i at2 2t! d! 2  gAu;! .t/ D p exp   C  g.t  u/: 2 b b b 4 2b

(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 function f 2 L2 .R/ with respect to a window function g 2 L2 .R/ is denoted by GA Œf .u; !/ and defined by GA Œf .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 recovers the standard definition of WFT (2.5.13). In fact, we have the following relation between WLCT and WFT:

84

2 The Time-Frequency Analysis

GA Œf .u;!/   d! 2  i at2 2t!  C  dt Dp f .t/ g.t  u/ exp 2 b b b 4 2b 1  2 Z 1  1 i at id! 2  Dp exp exp  f .t/g.t  u/ eit!=b dt 2b 2 b 2 2b 1 Z 1 id! 2 h.t/ g.t  u/ eit!=b dt D exp 2b 1   ! id! 2 G Œh u; : (2.5.17) D exp 2b b 1

Z

1



2. If we take the Gaussian function as a window function in (2.5.16), then we get the Gabor linear canonical transform (GLCT). 3. For a fixed u, WLCT (2.5.16) can be interpreted as the LCT of the product of a function f and a conjugate and translated window function g, that is, n o GA Œf .u; !/ D LA f .t/ g.t  u/ .!/:

(2.5.18)

4. Implementing the inverse LCT (2.5.9) to WLCT (2.5.16), we obtain   Z 1 d! 2  1 i at2 2t!  C  d!: f .t/ g.t  u/ D p GA f .u; !/ exp  2 b b b 4 2b 1 (2.5.19) 5. The energy density of the WLCT is defined by  2  ˇ2 ˇ2 ˇˇ 1 Z 1 ˇ ˇ i at 2t! d! 2  ˇ ˇ ˇ G Œf .u; !/ D p f .t/ g.t  u/ exp  C  dtˇˇ : ˇ ˇ ˇ A 2 b b b 4 2b 1 (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: GA Œ˛f C ˇh.u; !/ D ˛ GA Œf .u; !/ C ˇ GA Œh.u; !/, (b) Parity: GA ŒPf .u; !/ D GA Œf .u; !/, where Pg.t/ D g.t/,  iat02 c GA Œf .u  t0 ; !  at0 /, (c) Translation: GA ŒTt0 f .u; !/ D exp it0 !c  2  idb!02 GA Œf .t; !  !0 b/, (d) Modulation: GA ŒM!0 f .u; !/ D exp id!!0  2 (e) Conjugation: GA Œ f .u; !/ D GA1 Œf .u; !/.

2.5 The Windowed Linear Canonical Transform

85

Proof. (a). The proof of linearity easily follows from the definition of the WLCT. (b). For every f 2 L2 .R/, we have GA ŒPf .u; !/ 1

Z

1







i f .t/ g  .t  u/ exp Dp 2 2b 1 Z 1

 1 Dp f .t/ g  .t  u/ 2b 1

i  exp 2





at2 2t! d! 2   C  b b b 4

a.t/2 2.t/.!/ d.!/2   C  b b b 4

 dt

 dt

D GA Œf .u; !/: (c). From the definition of WLCT, we have GA ŒTt0 f .u; !/

(

!) i at2 2t! d! 2  D p f .t  t0 / g.t  u/ exp  C  dt 2 b b b 4 2b 1 ( !) Z 1

 i a.y C t0 /2 2.y C t0 /! d! 2  1 f .y/ g y  .u  t0 / exp  C  dy D p 2 b b b 4 2b 1 Z 1

 1 f .y/ g y  .u  t0 / D p 2b 1 ( !) i ay2 C at02 C 2ayt0 2y! 2t0 ! d! 2   exp   C  dy 2 b b b b 4 ( !) Z 1

 i ay2 2y.!  t0 a/ d! 2  1 f .y/ g y  .u  t0 / exp  C  D p 2 b b b 4 2b 1 !) ( i at02  2t0 ! dy  exp 2 b ( !) Z 1

 i ay2 2y.!  t0 a/ d.!  t0 a/2  1 f .y/ g y  .u  t0 / exp  C  D p 2 b b b 4 2b 1 1

Z

1

86

2 The Time-Frequency Analysis

(

!) ( !) 2d.!  t0 a/t0 a C dt02 a2 i at02  2t0 !  exp exp dy b 2 b !) ( !) (

 i at02  2t0 ! i 2d.!  t0 a/t0 a C dt02 a2 exp G A f u  t0 ; !  t0 a D exp 2 b 2 b ( )

 iat02 c D exp it0 !c  G A f u  t0 ; !  t0 a : 2 i 2

This completes the proof of part (c). (d). For every f 2 L2 .R/, we have GA ŒM!0 f .u; !/ D p

D p

D p

1

Z

f .t/ g.t  u/ exp

1

Z

(

1

f .t/ g.t  u/ exp

2b 1

( i!0 t

e

2b 1

1

1

Z

(

1

f .t/ g.t  u/ exp

2b

1

i 2

at2 2t! d! 2   C  b b b 4

!) dt !)

i 2

at2 2t!  d! 2  C C 2!0 t  b b b 4

i 2

at2 2t.!  !0 b/ d! 2   C  b b b 4

dt !) dt

(

!) i at2 2t.!  !0 b/ d.!  !0 bC!0 b/2  f .t/ g.t  u/ exp  C  dt Dp 2 b b b 4 2b 1 Z 1 1 f .t/ g.t  u/ D p 2b 1 !) ( i at2 2t.!  !0 b/ d.!  !0 b/2  2 2  C  C 2.!  !0 b/!0 b C !0 b dt  exp 2 b b b 4 ) ( idb!02 GA Œf .t; !  !0 b/: D exp id!!0  2 1

Z

1

(e). For every f 2 L2 .R/, we have  2  Z 1 i at 1 2t! d! 2  GA Œ f .u; !/ D p f .t/ g.t  u/ exp  C  dt 2 b b b 4 2b 1   Z 1 i at2 2t! d! 2  1 f .t/ g.t  u/ exp  C  dt D p 2 b b b 4 2b 1 D GA1 Œf .u; !/:

2.5 The Windowed Linear Canonical Transform

Theorem 2.5.2 (Orthogonality Relation). then the following formula hold: D

87

If f1 ; g1 ; f2 and g2 belong to L2 .R/,

E ˝ ˛˝ ˛ GA;g1 f1 ; GA;g2 f2 D f1 ; f2 g1 ; g2 :

(2.5.21)

Proof. Assume that both the window functions g1 ; g2 2 L1 .R/ \ L1 .R/; then it is obvious that for every u 2 R, f1 .t/ g1 .t  u/ 2 L2 .R/

and f2 .t/ g2 .t  u/ 2 L2 .R/:

from the Parseval formula (2.5.11) for the LCT that E DTherefore, it follows GA;g1 f1 ; GA;g2 f2 Z

1

Z

1

D 1 Z 1

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

1 Z 1

D 1

Z

1

Z

 n n o o LA f1 .t/ g1 .t  u/ .!/ LA f2 .t/ g2 .t  u/ .!/ d! du

1



1

f1 .t/ f2 .t/ g1 .t  u/ g2 .t  u/ exp

D 1

1

  iat2 dt du: b

(2.5.22)

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

GA;g1 f1 ; GA;g2 f2

E

 Z 1 iat2 dt D f1 .t/ f2 .t/ exp g1 .t  u/ g2 .t  u/ du b 1 1 ˛˝ ˛ ˝ D f1 ; f2 g1 ; g2 Z



1

where the extension to general g1 ; g2 2 L2 .R/ has been done by the standard density argument. 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 all g2 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

ˇ2 ˇ  2 ˇ ˇ ˇGA;g f .u; !/ˇ du d! D f 2 :

(2.5.23)

88

2 The Time-Frequency Analysis

Proof. Taking f1 D f2 D f and g1 D g2 D g in (2.5.21), we obtain 2 Z    GA;g f  D 2

1

Z

1

ˇ2 ˇ  2  2 ˇ ˇ ˇGA;g f .u; !/ˇ du d! D f 2 g2 :

1 1

(2.5.24)

 2 The desired result is obtained by taking g2 D 1 in (2.5.24), that is, Z

1 1

Z

1 1

ˇ ˇ2  2 ˇ ˇ ˇGA;g f .u; !/ˇ du d! D f 2 ;

for all f 2 L2 .R/:

(2.5.25)

Remark. 1. Suppose that kf k22 D 1. Then, equation (2.5.25) reduces to  2 Z   GA;g f  D 2

Z

1 1

1 1

ˇ ˇ2 ˇ ˇ ˇGA;g f .u; !/ˇ 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;g f . Furthermore, the condition ˛ 1 ˝ GA;g f .u; !/ D p f ; M! Tu g D 0; 2b

8 u; ! 2 R;

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

Theorem ˛ 2.5.3 (Inversion Theorem). Suppose that g1 ; g2 ˝ g1 ; g2 ¤ 0. Then for all f 2 L2 .R/, we have 1 ˛ f .t/ D ˝ g1 ; g2

Z

1 1

Z

2

L2 .R/ and

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 vectorvalued integral 1 ˛ fQ .t/ D ˝ g1 ; g2

Z

1

1

Z

1 1

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

2.5 The Windowed Linear Canonical Transform

89

is well defined in L2 .R/. Using the orthogonality relation (2.5.21), we observe that Z 1Z 1 ˝ ˛ ˝ ˛ 1 GA;g1 f .u; !/ h; M! Tu g2 du d! fQ ; h D p ˛ ˝ 2b g1 ; g2 1 1 E 1 D ˛ GA;g1 f ; GA;g2 h D˝ g1 ; g2 ˝ ˛ D f;h : Thus f D fQ . This proves the inversion theorem. Second Proof. For every h 2 L2 .R/, the inverse transform of the WFT (2.5.13) implies that 1 ˛ ˝ h.t/ D 2 g1 ; g2 1 ˝ ˛ D 2 g1 ; g2

Z

1

Z

1

Z

1

1 1

Z

1

Gg1 h.u; !/ ei!t g2 .t  u/ du d!

 ! ! ei!t=b g2 .t  u/ du d : Gg1 h u; b b 1 1

(2.5.28)

  2  1 i at  f .t/ and using the relation (2.5.17), If we let h.t/ D p exp 2 b 2 2b equation (2.5.28) becomes   2 1  i at  f .t/ p exp 2 b 2 2b 1 ˛ ˝ D 2 g1 ; g2

Z

1 1

Z



1

id! 2 exp 2b 1



! GA;g1 f .u; !/ ei!t=b g2 .t  u/ du d : b

Or, equivalently, 1 f .t/ D p ˛ ˝ 2b g1 ; g2

Z

1

1

Z

1 1

 exp 1 ˛ D ˝ g1 ; g2

Z

1 1

Z

GA;g1 f .u; !/

i 2



at2 d! 2 2t!  C   b b b 2



1

GA;g1 f .u; !/ KA .t; !/ g2 .t  u/ du d!: 1

This proves the inversion theorem.

g2 .t  u/ du d!

90

2.6

2 The Time-Frequency Analysis

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 gO .!/ D a cos

 a!   2

 1 1  :   a!  C a!

2. Consider the hat function g defined by

g.t/ D

8
0. c Proofs of the above properties are straightforward and are left as exercises.

 Theorem 3.2.3 (Parseval’s Formula). If 2 L2 .R/ and W 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 Z 1Z 1    ˝ ˛ 1 db da W f .a; b/ W g .a; b/ 2 ; f;g D (3.2.20) C 1 1 a where Z

1

C D 1

ˇ ˇ2 ˇO ˇ ˇ .!/ˇ ˇ ˇ d! < 1: ˇ! ˇ

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

Z  W f .a; b/ D

1

f .t/ jaj 1

˝ D f; D

a;b

 12

˛

1 DO O E f ; a;b 2



 tb dt a

(3.2.21)

3.2 The Continuous Wavelet Transform

Z

1 D 2

1

105 1 fO .!/ jaj 2 eib! O .a!/ d!:

(3.2.22)

1

Similarly, Z   W g .a; b/ D

1

g.t/ jaj

 12



1

Z

1 D 2

1

tb a

 dt

1

gO ./ jaj 2 eib O .a/ d:

(3.2.23)

1

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

Z

   db da W f .a; b/ W g .a; b/ 2 a 1 1 Z 1Z 1 Z Z ˚ 1 db da 1 1 D jaj fO .!/ gO . / O .a!/ O .a / exp ib.!   / d! d; 2 2 .2/ 1 1 a 1 1 1

1



which is, by interchanging the order of integration, D D

1 2 1 2

1 D 2

Z

1

1

Z

1

1

Z

1

1

da jaj da jaj da jaj

Z

1

Z

1

Z

1

1 fO .!/ gO . / O .a!/ O .a / d! d 2 1

1

Z

1

1

Z

1 1

Z

1

˚ exp ib.!   / db

1

fO .!/ gO . / O .a!/ O .a / ı.  !/ d! d

1

ˇ ˇ2 ˇ ˇ fO .!/ gO .!/ ˇ O .a!/ˇ d!

which is, again by interchanging the order of integration and putting a ! D x; ˇ2 ˇ Z 1 Z 1 ˇˇ O .x/ˇˇ 1 D dx fO .!/ gO .!/ d!  2 1 jxj 1 1 DO E DC  f ; gO : 2

Theorem 3.2.4 (Inversion Formula). by the formula 1 f .t/ D C

Z

1 1

Z

1

If f 2 L2 .R/, then f can be reconstructed

  W f .a; b/

1

where the equality holds almost everywhere.

a;b .t/

db da ; a2

(3.2.24)

106

3 The Wavelet Transforms

Proof. For any g 2 L2 .R/, we have, from Theorem 3.2.3, E ˝ ˛ D C f;g D W f;W g Z 1Z 1    db da W f .a; b/ W g .a; b/ 2 D a 1 1 Z 1Z 1 Z  1 db da W f .a; b/ D g.t/ a;b .t/ dt 2 a 1 1 1 Z 1Z 1Z 1  db da W f .a; b/ a;b .t/ 2 g.t/ dt D a 1 1 1  Z 1 Z 1   db da W f .a; b/ a;b .t/ 2 ; g : D a 1 1

(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

Z 1 ˇ ˇ2 da db   2 ˇ ˇ2 ˇ ˇ   ˇf .t/ˇ dt: W f f .a; b/ D C D C ˇ ˇ 2 2 a 1 1 1

(3.2.26)

Thus, we conclude that except for the factor C , the continuous wavelet transform W 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 in signal processing, data are represented by a finite number of values, so it is important and often useful to consider discrete versions of the continuous wavelet transform (3.2.13). However, unlike the discretized time and frequency axes shown earlier in Fourier analysis, here we take the discrete values of the scale parameter a and the translation parameter b in a different way. For convenience in the discretization, we restrict a to positive values only, so that the admissibility condition (3.2.1) becomes Z

1

ˇ ˇ ˇ O .!/ˇ2

1

j!j

Z d! D 2 0

1

ˇ ˇ ˇ O .!/ˇ2 j!j

d! < 1:

First, we choose a D am 0 , 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 integer multiples of one fixed b0 , where b0 is approximately chosen so that the .t  nb0 /

3.3 The Discrete Wavelet Transform

107

m=2 m  cover the whole line. For different values of m, the width a0 a0 t is am 0 times m the width of .t/, so that the choice b D nb0 a0 ; m; n 2 Z will ensure that the discretized wavelets at level m “cover” the whole line in the same way as the .t  m nb0 / do. Thus, we choose a D am 0 ; b D nb0 a0 ; where the two positive constants a0 and 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/

m=2

D a0



t  nb0 am 0 am 0



m=2

D a0



m a0 x  nb0 ;

(3.3.1)

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

 ˝ W f .m; n/ D f ;

m;n

˛

m=2

Z

D a0

1 1

f .t/ .am 0 t  nb0 / dt;

(3.3.2)

where both f and are continuous and 0;0 .t/ D .t/. Thus, the discrete wavelet transform represents a function by a countable set of wavelet coefficients, which correspond to points on a two-dimensional grid or˚ lattice of discrete points in the scale 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 an affine wavelet. Then, we can express any f 2 L2 .R/ as the superposition f .t/ D

XX˝

f;

m;n

˛

m;n .t/:

(3.3.3)

m2Z n2Z

Such complete sets are called frames. They are not yet a basis. Frames do not satisfy the Parseval theorem for the Fourier series, and the expansion in terms of frames is not unique. In fact, it can be shown that  2 X X ˇ˝ ˇ f; Af 2 

m;n

  ˛ˇ2 ˇ  Bf 2 ; 2

(3.3.4)

m2Z n2Z

˚ 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 information and 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;n2Z in the wavelet series expansion (3.3.3) of a function are nothing but  wavelet transform

the integral m of the function evaluated at certain discrete points am ; nb a 0 0 0 . No such relationship exists between Fourier series and Fourier transform which are applicable to different

108

3 The Wavelet Transforms

ω

σt

m = –3

σω m = –2 m = –1 m=0 n=0

n=1

n=2

t

Fig. 3.7 Dyadic sampling grid for the discrete wavelet transform

classes of functions. Fourier series applies to functions that are square integrable in Œ0; 2, whereas Fourier transform is for functions that are in L2 .R/. On the other hand, 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 that results lead to a binary dilation of 2m 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 2m=2

.2m t  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 grid diagram for the discrete wavelet transform. Each node corresponds to a wavelet basis function m;n .t/ with scale 2m and time shift n 2m . The answer to the preceding question is positive if the wavelets form a complete system 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;n i and hg; m;n i to be “close” if f and g are “close.” This will be guaranteed if there exists a B > 0 independent of f such that X X ˇ˝ ˇ f; m2Z n2Z

m;n

  ˛ˇ2 ˇ  Bf 2 : 2

(3.3.6)

3.4 Orthonormal Wavelets

109

Similarly, we want f and g to be “close” if hf ; m;n i and hg; m;n i are “close.” This is important because we want to be sure that when we neglect some small terms in the representation of f in terms of hf ; m;n i, the reconstructed function will not differ much from f . The representation will have this property if there exists an A > 0 independent of f , such that   2 X X ˇ˝ ˇ f; Af 2 

m;n

˛ˇ2 ˇ :

(3.3.7)

m2Z n2Z

Combining (3.3.6) with (3.3.7), we obtain the following equation:  2 X X ˇ˝ ˇ f; Af 2 

m;n

  ˛ˇ2 ˇ  Bf 2 : 2

(3.3.8)

m2Z n2Z

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

3.4

Orthonormal Wavelets

Since the discovery of wavelets, orthonormal wavelets with good time-frequency localization are found to play an important role in wavelet theory and have a great variety of applications. In general, the theory of wavelets begins with a single function 2 L2 .R/, and a family of functions m;n is generated from this single function by the operation of binary dilations (i.e., dilation by 2m ) and dyadic translation of n2m so that m;n .t/

D 2m=2



 2m t  n :

(3.4.1)

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

m;n ;

k;`

˛

Z

1

D

m;n .t/

k;` .t/ dt

D ım;k ın;` ;

8 m; n; k; ` 2 Z:

(3.4.2)

1

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

110

3 The Wavelet Transforms

and  2 L2 .R/, then

Lemma 3.4.1. If ˝

m;k ; m;`

˛

D

˝

n;k ; n;`

˛ ;

8 m; n; k; ` 2 Z:

Definition 3.4.1 (Orthonormal Wavelet). A wavelet orthonormal if the family of functions m;n generated from orthonormal.

(3.4.3)

2 L2 .R/ is called given by (3.4.1) is

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 XX

f .x/ D

m;n .t/;

cm;n

(3.4.4)

m2Z n2Z

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 and the same for m;n defined by (3.4.1). We have ˝

m;n ;

k;`

˛

Z

1 1

D2



2m=2

D k=2

2

Z



 2m x  n 2k=2 2k x  ` dx 1

 2km .t C n/  ` dt:

(3.4.6)

.t/ .t C n  `/ dt D ı0;n` D ın;` ;

(3.4.7)

m=2

.t/



1

For m D k, this result gives ˝

m;n ;

m;`

˛

Z

1

D 1

where .t/ ¤ 0 in Œ0; 1 and .t  `  n/ ¤ 0 in Œ`  n; 1 C `  n/, and these intervals 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 consider the case m > k: Putting r D m  k > 0 in (3.4.6), we can complete the proof by showing that, for k ¤ m, ˝

m;n ;

k;l

˛

D 2r=2

Z

1



.t/

 2r t C s dt D 0;

(3.4.8)

1

where s D 2r n  ` 2 Z. In view of the definition of the Haar wavelet , we must prove that the integral in (3.4.8) vanishes for k ¤ m. In other words, it suffices to show Z

1 2



 2r t C s dt 

Z

1



 2r t C s dt D 0:

1 2

0

Invoking a simple change of variables, 2r t C s D x, we find Z

Z

a

b

.x/ dx  s

.x/ dx D 0;

(3.4.9)

a

where a D s C 2r1 and b D s C 2r . A simple argument reveals that Œs; a contains the 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 is orthonormal.

3.5

The Fractional Wavelet Transform

Wavelet transforms serve as an important and powerful analyzing tool for timefrequency analysis and have been applied in a number of fields including signal processing, image processing, sampling theory, differential and integral equations, quantum mechanics, and medicine. However, the signal analysis capability of the wavelet transform is limited in the time-frequency plane as each wavelet component is 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 not well concentrated in the frequency domain. One of the examples of such signal is chirp-like signals. Recently, researchers have come up with the new mathematical transforms to analyze such signals, namely, fractional Fourier transform (FrFT), the short-time fractional Fourier transform (STFrFT), the Radon-Wigner transform

112

3 The Wavelet Transforms

(RWT), the ridgelet transform (RT), etc. Besides lot of advantages, the FrFT has one major drawback due to using global kernel i.e., the fractional Fourier representation only provides such FrFT spectral content with no indication about the time localization of the FrFT spectral components. Therefore, in order to analyze nonstationary signals whose FrFT spectral characteristics change with time, researchers have come up with the short-time FrFT which provides a joint representation of a signal in both time and FrFT domains, rather than just a FrFT domain representation. The basic idea behind this transform is segmenting the signal by using a time-localized window and then performing the FrFT spectral analysis for 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 signals having high spectral components for short durations and low spectral components for long durations. Hence, it is desirable to derive more general approach than the STFrFT in order to obtain joint signal representations in both time and FrFT domains. An important way to analyze time-varying FrFT spectra is the fractional wavelet transform (FrWT). The FrWT inherits the excellent mathematical properties of wavelet transform and FrFT along with some fascinating properties of its own. These properties make FrWT a useful mathematical tool in signal and image processing with numerous advantages over conventional wavelet transform. As a generalization of the wavelet transform, Mendlovic et al. (1997) first introduced the FrWT as a way to deal with optical signals. The idea behind this transform is deriving the fractional spectrum of the signal by using the FrFT and performing the wavelet transform of the fractional spectrum. Besides being a generalization of the wavelet transform, the FrWT can be interpreted as a rotation of the time-frequency plane and has been proved to relate to other time-varying signal analysis tools, which make it as a unified time-frequency transform. In recent years, this transform has been paid a considerable amount of attention, resulting in many applications in the areas of optics, quantum mechanics, pattern recognition, and signal processing. 2 L2 .R/ which satisfies the

Definition 3.5.1. A fractional wavelet is a function following condition:

C˛ D

ˇ ˇ2 n o ˇ Z ˇˇF ˛ ei.t!/2 =2 cot ˛ .!/ˇ R

j!j

d! < 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)

3.5 The Fractional Wavelet Transform

113

Its fractional Fourier transform O ˛ .!/ is given by n o F ˛ .t/  Z 1  cot ˛

exp i t2 C ! 2  it! csc ˛ .t/dt D C˛ 2 1  Z 1   i i 2 1 2 ! cot ˛  cot ˛ t C t .i! csc ˛  i˛/ dt: exp  D C˛ exp 2 2 2 1    i 2 .a  ! csc ˛/2 p 1  i cot ˛ 1=2 ! cot ˛ exp   D C˛ exp 2 2.1  i cot ˛/ 2    i 2 1 D exp !  .a sin ˛  !/2 cot ˛  .a sin ˛  !/2 : 2 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/ is defined by Z



1

f .t/ ˛ g.t/ D

f .!/g.t  !/ exp 1

  i  2 t  ! 2 cot ˛ d!; 2

(3.5.3)

where ˛ denotes the fractional convolution operator. For any two bounded integrable functions Q .t/ D eit2 cot ˛=2 .t/

.t/ and .t/, we set

Q D eit2 cot ˛=2 .t/: and .t/

Then, the FrFT of the function h.t/ D eit

2

cot ˛=2



 Q ˛ Q .t/

(3.5.4)

is given by 2 hO ˛ .!/ D ei! cot ˛=2 O ˛ .!/ O ˛ .!/:

(3.5.5)

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

114

3 The Wavelet Transforms

Proof. Since Z

1

1

Z ˇ  ˇˇ2 ˇ it2 cot ˛=2 Q ˛ Q .t/ˇ dt D ˇe

ˇ

 ˇ ˇ Q ˛ Q .t/ˇ2 dt

1

1

1

ˇZ ˇ ˇ ˇ

1

Z

Z

1

D Z

ˇ2 ˇ Q .t  !/ .!/ Q d! ˇˇ dt

1

1 1

ˇ ˇˇ ˇ ˇ Q .t  !/ˇ ˇ.!/ ˇ d! Q

 1

Z

Z

1

1

1

1

1

1

Z

1

Z

1



2 ˇ ˇˇ ˇ1=2 ˇ1=2 ˇ ˇ Q .t  !/ˇ ˇ.!/ ˇ d! dt ˇ ˇ.!/ Q Q

ˇ ˇ ˇ ˇ ˇ Q .t  !/ˇ2 ˇ.!/ ˇ d! Q

1

ˇ ˇ ˇ.!/ ˇ d! Q

1

D

ˇ ˇ ˇ.!/ ˇ d! Q

1

Z

1 1

2

cot ˛=2

ˇ n oˇˇ2 ˇ ˛ it2 cot ˛=2 Q ˛ Q ˇ ˇF e j!j Z

1

D



Z

1

2 Z

1

 ˇ ˇ ˇ.!/ ˇ d! dt Q

ˇ ˇ ˇ ˇ ˇ Q .t  !/ˇ2 ˇ.!/ ˇ d! dt Q

ˇ ˇ ˇ Q .t/ˇ2 dt < 1:

 Q ˛ Q .t/ 2 L2 .R/. Moreover,

d!

j!j

1

Z

1

1

ˇ2 ˇ ˇ ˇ i! 2 cot ˛=2 O ˛ .!/ O ˛ .!/ˇ ˇe

ˇ ˇ  2 ˇ sin ˛ ˇ

1

1

1

Therefore, it follows that eit

Z

1

Z

1

Z

dt

1

D Z 

2

1 1

d!

ˇ ˇ2 ˇ i! 2 cot ˛=2 O ˛ ˇ .!/ˇ ˇe j!j

ˇ ˇ2 ˇ ˇ d! sup ˇO ˛ .!/ˇ < 1:

Thus, the convolution function Q ˛ Q is a fractional wavelet. Example 3.5.2 (Fractional Mexican Hat Wavelet.). If   2 2 t2 it cot ˛ t .1 C 2i/ cot ˛ and .t/ D p : .t/ D exp exp 2 2 2 Then, in view of Theorem 3.5.1, the convolution function  Z 1

 i.t2  ! 2 / cot ˛  ˛ .t/ D d! .!/ .t  !/ exp 2 0  2

 t cot ˛ D tan3=2 ˛ 1  t2 cot ˛ exp 2 forms a fractional wavelet.

3.5 The Fractional Wavelet Transform

115

Analogous to the classical wavelets, the fractional wavelets can be obtained from a fractional mother wavelet 2 L2 .R/ by the combined action of translation and dilations as    i.t2  b2 / cot ˛ 1 tb ˛ exp (3.5.6) a;b .t/ D p a 2 a where a 2 RC and b 2 R are scaling and translation parameters, respectively. Note that if  

 ˛ 2 a;b 2

.t/ 2 L2 .R/, then D jaj

1

Z

˛ a;b .t/

2 L2 .R/ as

ˇ  ˇ Z 1 ˇ ˇ ˇ   t  b ˇˇ2 ˇ ˇ .y/ˇ2 dy D  2 : dt D ˇ ˇ 2 a 1 1 1

˛ a;b .t/

Moreover, the fractional Fourier transform of F˛ D

˚

p

˛ a;b .t/

is given by



 o n ia2 ! 2 cot ˛ i.b2 C ! 2 / cot ˛ 2  ib ! csc ˛  a exp F ˛ ei./ cot ˛=2 .a!/: 2 2 (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 ˛ defined 2 on L .R/ by ˝

W Œf .a; b/ D f ; ˛

˛ a;b

˛

1 Dp a

Z



1

f .t/ 1

 2  i.t  b2 / cot ˛ tb exp dt a 2 (3.5.8)

is called a continuous fractional wavelet transform (FrWT) of f .t/. This definition allows us to make the following comments:  , the FrWT coincides with the conventional continuous wavelet 1. For ˛ D 2 transform. 2. Using the Parseval relation of the fractional Fourier transform, it also follows from (3.5.8) that

116

3 The Wavelet Transforms

D E ˛ D fO ˛ ; O a;b  Z 1 p ia2 ! 2 cot ˛ i.b2 C ! 2 / cot ˛ C ib ! csc ˛ C D a exp 2 2 1 n o  F ˛ ei./2 cot ˛=2 .a!/fO ˛ .!/ d!: (3.5.9)

D W ˛ Œf .a; b/ D f ;

˛ a;b

E

3. The FrWT (3.5.8) can be expressed in terms of the ordinary convolution as   ib2 cot ˛  i./2 cot ˛=2 e (3.5.10) f  a .b/; W ˛ Œf .a; b/ D exp 2 whereas in terms of the fractional convolution, it takes the form   W ˛ Œf .a; b/ D f ˛ a .b/; a .t/

where

 t 

1 D p a

Theorem 3.5.2. If

a

1;

2

(3.5.11)

and ˛ denote the fractional convolution operator.

    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

Z

1

1 0

    ˝ ˛ dadb W ˛1 f .a; b/ W ˛2 g .a; b/ 2 D 2 sin ˛ C˛ 1 ; 2 f ; g ; a

(3.5.12)

where C

Z

˛ 1;

2

D

1

n F ˛ ei./2 cot ˛=2

0

o 1

n 2 .a/ F ˛ ei./ cot ˛=2

o 2

.a/

da < 1: a (3.5.13)

Proof. We have Z   p ˛ W 1 f .a; b/ D a

 ia2 ! 2 cot ˛ i.b2 C ! 2 / cot ˛ C ib ! csc ˛ C exp 2 2 1 n o  F ˛ ei./2 cot ˛=2 1 .a!/fO ˛ .!/ d! 1



and  ia2 2 cot ˛ i.b2 C 2 / cot ˛  ib  csc ˛  exp 2 2 1 n o 2  F ˛ ei./ cot ˛=2 2 .a/Og˛ ./ d:

Z   p W ˛2 g .a; b/ D a

1



3.5 The Fractional Wavelet Transform

117

Now Z

Z

1

1

Z



Z

1

0

1

   dadb W ˛1 f .a; b/ W ˛2 g .a; b/ 2 a

1

"Z

 ia2 ! 2 cot ˛ i.b2 C ! 2 / cot ˛ C ib ! csc ˛ C exp 2 2 1 # n o 2 cot ˛=2 ˛ ˛ i./ O F e 1 .a!/f .!/ d!

D 1

0

"Z

1

Z

1

 ia2 2 cot ˛ i.b2 C 2 / cot ˛  ib  csc ˛  exp 2 2 1 # n o dadb ˛ i./2 cot ˛=2 ˛ F e g ./ d 2 .a/O a



Z

1



1



Z



1

D

exp 1

1

0

n F ˛ ei./2 cot ˛=2 Z

1

Z

1

1

Z

i.2  ! 2 / cot ˛ C ia2 .! 2  2 / cot ˛ 2

o n 2 .a!/fO ˛ .!/  F ˛ ei./ cot ˛=2

1

D 2 sin ˛

exp 1

1

0

n F ˛ ei./2 cot ˛=2 Z

1

D 2 sin ˛ 1

1

o 2

 Z

1

 eib.!/ csc ˛ db

1

.a/Og˛ ./ d!d

da a

 i.2  ! 2 / cot ˛ C ia2 .! 2  2 / cot ˛ ı.!  / 2

o n 2 .a!/fO ˛ .!/  F ˛ ei./ cot ˛=2

o 2

.a/Og˛ ./ d!d

da a

 "Z 1  i! 2 .a2  1/ cot ˛ i.a2  1/2 cot ˛ exp exp 2 2 0 1 # o n o da g˛ ./ ı.!/ d F ˛ ei./2 cot ˛=2 1 .a!/fO ˛.!/ d! 2 .a/O a

Z

1



n 2 F ˛ ei./ cot ˛=2 Z 1 fO ˛ .!/ gO ˛ .!/ D 2 sin ˛ 1 " n o n R1 2  0 F ˛ ei./2 cot ˛=2 1 .a!/F ˛ ei./ cot ˛=2 Z 1 D 2 sin ˛ C˛ 1 ; 2 fO ˛ .!/ gO ˛ .!/ d! D1 E D 2 sin ˛ C˛ 1 ; 2 fO ˛ ; gO ˛ ˝ ˛ D 2 sin ˛ C˛ 1 ; 2 f ; g :

# o da 2 .a!/ a d!

118

3 The Wavelet Transforms

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

    ˝ ˛ dadb W ˛ f .a; b/ W ˛ g .a; b/ 2 D 2 sin ˛ C˛ f ; g : a

Proof. The proof of Corollary 3.5.1 can be easily deduced by setting in the Theorem 3.5.2.

(3.5.14)

1

D

2

D

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

1 1

Z

1 0

ˇ2 dadb ˇ    ˇ ˇ ˛  2 f 2: D 2 sin ˛ C ˇ W ˛ f .a; b/ˇ a2

Theorem 3.5.3 (Inversion Formula). using the formula f .t/ D

1 2 sin ˛ C˛

Z

1 1

Z

1

0

(3.5.15)

If f 2 L2 .R/, then f can be constructed by

  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 Z

Z

    dadb W ˛ f .a; b/ W ˛ g .a; b/ 2 a 1 0  Z Z 1 Z 1  1 dadb ˛ ˛ W f .a; b/ D g.t/ a;b .t/ dt a2 1 0 1  Z 1 Z 1 Z 1   dadb ˛ W ˛ f .a; b/ a;b D .t/ 2 g.t/ dt a 1 1 0 Z 1 Z 1    dadb ˛ W ˛ f .a; b/ a;b D .t/ 2 ; g.t/ : a 1 0

˝ ˛ 2 sin ˛ C f ; g D ˛

1

1

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

3.5 The Fractional Wavelet Transform

119

Remark. Using the fractional Fourier transform on both sides of (3.5.16), we obtain 1 2 sin ˛ C˛

fO ˛ .!/ D

Z

1 1

Z

1



0

2 L2 .R/ and

Theorem 3.5.4. If

 dadb ˛ W ˛ f .a; b/ O a;b .!/ 2 : a

(3.5.17)

    W ˛ f .a; b/ and W ˛ g .a; b/ are the

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

1

h    i ˛ ˝ W ˛ f .a; b/ W ˛ g .a; b/ db D 2a sin ˛ R˛ ; S˛

(3.5.18)

1

where R˛ and S˛ are, respectively, defined as  2 2 n o ia ! cot ˛ ˛ O F ˛ ei./2 cot ˛=2 .a!/; R˛ .!/ D f .!/ exp 2 S˛ .!/ D gO ˛ .!/ exp



(3.5.19)

 o n ia2 ! 2 cot ˛ F ˛ ei./2 cot ˛=2 .a!/: 2

(3.5.20)

Proof. We have Z

1

n    o W ˛ f .a; b/ W ˛ g .a; b/ db

1

Z

1

D

D

1

Z

1

˛ fO ˛ ; O a;b

Z

Z

1

"Z

1

D

˛ .!/ d! fO ˛ .!/ O a;b



1

a

exp

1

"Z

˛ ˛ gO ˛ ; O a;b db

1

D 1

E˝

1



1



exp 1

 Z

1 1

 ˛ gO ˛ ./ O a;b ./ d db

 ia2 ! 2 cot ˛ i.b2 C ! 2 / cot ˛ C ib! csc ˛ C 2 2 # n o  F ˛ ei./2 cot ˛=2 .a!/fO ˛ .!/ d!

ia2 2 cot ˛ i.b2 C 2 / cot ˛  ib csc ˛  2 2 n

 F˛ e

i./2 cot ˛=2

o

 #

.a/Og˛ ./ d db

120

3 The Wavelet Transforms

#  i.b2 C ! 2 / cot ˛ Da C˛ exp  ib! csc ˛  R˛ .!/ d! 2 1 C ˛ 1  Z 1 2   i.b C 2 / cot ˛  C˛ exp  ib csc ˛  S˛ ./ d db 2 Z 1  1    a D RO ˛ .b/ SO ˛ .b/ db C˛ C˛ 1 E D D 2a sin ˛ RO ˛ ; SO ˛ D E D 2a sin ˛ RO ˛ ; SO ˛ Z

1

1

"

Z

1



D 2a sin ˛ hR˛ ; S˛ i :

This completes the proof of the theorem. For more detailed information on fractional wavelet transforms and their applications, the reader is referred to Huang and Suter (1998), Chen and Zhao (2005), Dinc et 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    2  2 t2 t .t/ D 1 p 1  2 exp  2 a 2a  4 3a is r O .!/ D

 2 2 8 5=2 1=4 2 a ! a  ! exp  : 3 2

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

W Œf .a; b/ D f 

a



.b/;

3.6 Exercises

121

where a .t/

1 D p a

 t  : a

What is the physical significance of the convolution? 4. If f is a homogeneous function of degree n, show that     1 W f .a; b/ D nC 2 W f .a; b/: 5. Show that Z

1 1

 n  1 sin x sin .2x  n/  dx D sin : x .2x  n/ 2n 2

6. Show that the Fourier transform of one cycle of the sine function f .t/ D sin t;

jtj < I

is fO .!/ D

2i sin !: .! 2  1/

7. For the Shannon wavelet  t    sin 3t 2 .t/ D  t  cos ; 2 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   1 1 t2 f .t/ D p exp  2 cos !0 t 2 2 

122

3 The Wavelet Transforms

is    2 2 Of .!/ D 1 exp   .!  !0 /2 C exp   .! C !0 /2 : 2 2 2 Explain the physical features of fO .!/. 9. Show that the Fourier transform of 1 f .t/ D p a .t/ ei!0 t 2 is r fO .!/ D

 2 sin a.!  !0 / :  .!  !0 /

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

tb a



8 a ˆ ˆ < 1; b  t < b C 2 D 1; b C a  t < b C a ˆ 2 ˆ : 0; otherwise;

where a > 0, show that 1 W Œf .a; b/ D p a 11. Suppose

1

and

Z

bC a2 b

   a f .t/  f t C dt: 2

are two wavelets and the integral

2

Z

1

O 1 .!/ O 2 .!/

1

j!j

d! D C

1

2

< 1:

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

where f ; g 2 L2 .R/:

E W 1f ; W 2g D C

˝ 2 2

˛ f;g ;

4

Construction of Wavelets via MRA

Multiresolution analysis provides a natural framework for the understanding of wavelet bases, and for the construction of new examples. The history of the formulation of multiresolution analysis is a beautiful example of applications stimulating theoretical development. 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 other disciplines. The detour through signal and image processing was the most direct path leading from Haar basis to Daubechies’s wavelets. Yves Meyer

4.1

Introduction

The idea of multiresolution analysis (MRA) was proposed by Stéphane Mallat and Yves Meyer in 1986, and this can be considered as a rebirth of wavelet theory. This is a new and remarkable idea which deals with a general formalism for construction of an orthogonal basis of wavelets. Indeed, MRA is central to all constructions of wavelet bases. Mallat’s brilliant work (Mallat 1989a,b,c) has been the major source of 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 a limit of successive approximations, each of which is a finer version of the function f . These successive approximations correspond to different levels of resolutions. Thus, MRA is a formal approach to constructing orthogonal wavelet bases using a definite set of rules and procedures. The key feature of this analysis is to describe

© Springer International Publishing AG 2017 L. Debnath, F.A. Shah, Lecture Notes on Wavelet Transforms, Compact Textbooks in Mathematics, DOI 10.1007/978-3-319-59433-0_4

123

124

4 Construction of Wavelets via MRA

mathematically the process of studying signals or images at different scales. The basic principle of the MRA deals with the decomposition of the whole function space into individual subspaces Vm  VmC1 so that the space VmC1 consists of all rescaled functions in Vm . This essentially means a decomposition of each function into components of different scales so that an individual component of the original function f occurs in each subspace. This chapter deals with the idea of MRA with examples. Special attention is given to properties of scaling functions and orthonormal wavelet bases. This is followed by a method of constructing orthonormal bases of wavelets from an MRA. In the end, the fast wavelet transform (FWT) is briefly discussed.

Multiresolution Analysis in L2 .R/

4.2

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 the following properties: (i) VS for all m 2 ZI m  VmC1 ; (ii) Vm is dense in L2 .R/; m2Z T (iii) Vm D f0g I m2Z

(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 or father 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 .2k t/ 2 VmCk for all m; k 2 Z. In other words, f 2 Vm if and only if f .2m t/ 2 V0 for all m 2 Z. This shows that functions in Vm are obtained from those in V0 through a scaling 2m . If the scale m D 0 is associated with V0 , then the scale 2m is associated with Vm . Thus, subspaces Vm are just scaled versions of the central space V0 which is invariant under translation by integers, that is, Tn V0 D V0 for all n 2 Z. 2. It follows from Definition 4.2.1 that an MRA is completely determined by the scaling function , but not conversely. For a given  2 V0 , we first define ( V0 D f .x/ D

X n2Z

cn 0;n D

X n2Z

) 2

cn .t  n/ W fcn g 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 . P Then, V0 consists of all functions f .t/ D n2Z cn .t  n/ with finite energy  2 ˇ ˇ2 P f  D ˇcn ˇ < 1. Similarly, the space Vm has the orthonormal basis of n2Z 2 the form ˚

 m;n .t/ D 2m=2  2m t  n ; n 2 Z

(4.2.1)

so that fm .x/ is given by fm .x/ D

X

cm;n m;n .t/

(4.2.2)

n2Z

ˇ ˇ2  2 P with the finite energy fm 2 D n2Z ˇcm;n ˇ < 1: Thus, fm represents a typical function in ˚ the space Vm : It builds self-invariance and scale invariance through the basis m;n I m; n 2 Z . 3. Conditions (ii) and (iii) can be expressed in terms of the orthogonal projections Pm onto Vm , that is, for all f 2 L2 .R/, lim Pm f D 0 and

m!1

lim Pm f D f :

m!C1

(4.2.3)

The projection Pm f can be considered as an approximation of f at the scale 2m . Therefore, the successive approximations of a given function f are defined as the orthogonal projections Pm onto the space Vm : Pm f D

X˝

˛ f ; m;n m;n ;

(4.2.4)

n2Z

where fm;n .t/ W m; n 2 Zg given by (4.2.1) is an orthonormal basis for Vm . 4. Since V0  V1 , the scaling p function  that leads to a basis for V0 is also V1 . Since  2 V1 and 1;n .t/ D 2 .2t  n/ is an orthonormal basis for V1 ,  can be expressed in the form .t/ D

p X

 2 cn  2t  n ;

(4.2.5)

n2Z

˛ ˝ P where cn D ; 1;n and n2Z jcn j2 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-scale equation or refinement equation because it displays .t/ in the refined space V1 . The space V1 has the finer scale 21 and it contains .t/ which has scale 1.

126

4 Construction of Wavelets via MRA

All of the preceding facts reveal that MRA can be described at least three ways so 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 to construct an orthonormal basis for L2 .R/. In order to prove this statement, we first assume that fVm W m 2 Zg is an MRA. Since Vm  VmC1 , we define Wm as the orthogonal complement of Vm in VmC1 for every m 2 Z, so that we have VmC1 D Vm ˚ Wm 

D Vm1 ˚ Wm1 ˚ Wm D ::: D V0 ˚ W0 ˚ W1 ˚    ˚ Wm " m # M D V0 ˚ Wn

(4.2.6)

nD0

and Vn ? Wm for n ¤ m. Since as m ! 1 to obtain

S

m2Z

" V0 ˚

Vm is dense in L2 .R/, we may take the limit

1 M

# Wm D L2 .R/:

mD0

Similarly, we may go in the other direction to write V0 D V

1 ˚ W1  D V2 ˚ W2 ˚ W1 D ::: D Vm ˚ Wm ˚    ˚ W1 : We may again take the limit as m ! 1. Since Vm D f0g. Consequently, it turns out that M m2Z

T

Wm D L2 .R/:

m2Z

Vm D f0g, it follows that

(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 are also scaled versions of W0 and, for f 2 L2 .R/, f 2 Wm

if and only if

f .2m t/ 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

f .t  n/ 2 W0 ;

if and only if

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

for m ¤ k;

M

and

Wm D L2 .R/:

(4.2.9)

m2Z

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

m;n .t/

D 2m=2

o .2m t  n/ W n 2 Z

(4.2.10)

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

XX˝ f;

m;n

˛

m;n .t/;

(4.2.11)

m2Z n2Z

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 PmC1 f D Pm f C

X˝

f;

m;n

˛

m;n :

(4.2.12)

n2Z

Thus, Pm is the result of observing f at the resolution level m, and the difference Qm f D PmC1 f  Pm f D

X˝

f;

m;n

˛

m;n

(4.2.13)

n2Z

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

128

4 Construction of Wavelets via MRA

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

X

ck m;k

k2Z

) X ˇ ˇ2 ˇ ˇ ck < 1 ; W k2Z

m t  n/: The spaces Vm satisfy all the conditions of where m;n .t/ D 2m=2 .2 ˚ Definition 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 all  functions in L2 .R/ which are constant on intervals 2m n; 2m .nC1/ , where n 2 Z. Obviously, Vm  VmC1 because any function that is constant on intervals of length 2m is automatically constant on intervals of half that length. The space V0 contains all functions f .t/ in L2 .R/ that are constant on Œn; n C 1/ having jumps possibly at integer 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 2m are usually referred to as dyadic intervals. 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 its translates. 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 scaling function  as f .t/ D

X

cn .t  n/:

n2Z

Thus, the condition (v) is satisfied by the characteristic function Œ0;1 as the scaling function. 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 the scaling property f .t/ 2 Wm if and only if f .2t/ 2 WmC1 , so the family f 0;n .t/ D .t  n/ W n 2 Zg constitutes an orthonormal set of W0 if and only if the set f m;n .t/ W m; n 2 Zg defined by (4.2.10) forms an orthonormal basis of Wm , for all m 2 Z. Thus, our task reduces to finding 2 W0 such that the set f .t  n/ W n 2 Zg constitutes an orthonormal basis for W0 . To construct this , we first 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 equivalent: ˚ (a) The system 0;n .t  n/; n 2 Z is orthonormal. ˇ2 X ˇˇ O C 2k/ˇˇ D 1 almost everywhere (a.e.). (b) ˇ.! k2Z

Proof. Obviously, the Fourier transform of 0;n .t/ D .t  n/ is O O0;n .!/ D ein! .!/: In view of the general Parseval relation (1.3.17) for the Fourier transform, we have ˝ ˛ ˝ ˛ 0;n ; 0;m D 0;0 ; 0;mn D E 1 O D 0;0 ; O0;mn 2 Z 1 ˇ ˇ2 1 ˇO ˇ D ei.mn/! ˇ.!/ ˇ d! 2 1Z ˇ2 1 X 2.kC1/ i.mn/! ˇˇ O ˇ D e ˇ.!/ˇ d! 2 k2Z 2k Z 2 ˇ2 X ˇˇ 1 O C 2k/ˇˇ : D ei.mn/! d! ˇ.! 2 0 k2Z ˚ Thus, it follows from the completeness of ein! W n 2 Z in L2 .0; 2/ that ˝ ˛ 0;n ; 0;m D ın;m if and only if ˇ2 X ˇˇ O C 2k/ˇˇ D 1 ˇ.!

almost everywhere:

k2Z

˚ Proposition ; 2 L2 .R/, the sets of functions .t 4.3.1. ˚ For any two functions 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 2k/ O .! C 2k/ D 0; .!

almost everywhere:

130

4 Construction of Wavelets via MRA

The proof of this result can be obtained by applying arguments similar to those stated in the proof of Theorem 4.3.1. We next proceed to the construction of a mother wavelet by introducing an important generating function m O 0 .!/ 2 L2 Œ0; 2 in the following lemma. Lemma 4.3.1. The Fourier transform of the scaling function  satisfies the following conditions: ˇ2 X ˇˇ O C 2k/ˇˇ D 1 a.e; ˇ.!

(4.3.1)

k2Z

!  !  O ; .!/ Dm O0 O 2 2

(4.3.2)

1 X m O 0 .!/ D p cn ein! 2 n2Z

(4.3.3)

where

is a 2-periodic function and satisfies the so-called orthogonality condition ˇ ˇ2 ˇ ˇ2 ˇm O 0 .!/ˇ C ˇm O 0 .! C /ˇ D 1

a.e:

(4.3.4)

Proof. Condition (4.3.1) follows from Theorem 4.3.1. n p To establish (4.3.2), we first note that  2 V0  V1 and 1;n .t/ D 2 .2t  n/ W n 2 Zg is an orthonormal basis for V1 . Thus, the scaling function  has the following representation: .t/ D

p X 2 cn .2t  n/;

(4.3.5)

n2Z

where cn D h; 1;n i and

X ˇ ˇ2 ˇcn ˇ < 1: The Fourier transform of (4.3.5) gives n2Z

!  !  1 X i!n=2 O  !  O Dm O0 O ;  .!/ Dp cn e 2 2 2 2 n2Z

(4.3.6)

4.3 Construction of Mother Wavelet

131

1 X i!n where m O 0 .!/ D p cn e is the 2-periodic function and is called the low2 n2Z pass filter or discrete filter associated with the scaling function . This proves the functional equation (4.3.2). To verify the orthogonality condition (4.3.4), we substitute (4.3.2) in (4.3.1) so that condition (4.3.1) becomes ˇ2 X ˇˇ O C 2k/ˇˇ ˇ.! k2Z ˇ2 ˇ  ! ˇ2 X ˇˇ  ! ˇ ˇ ˇ O0 C k ˇ ˇO C k ˇ : D ˇm 2 2 k2Z

1D

This is true for any !, and hence, replacing ! by 2! gives 1D

Xˇ ˇ2 ˇm O 0 .! C k/ˇ

ˇ2 ˇ ˇ ˇO ˇ.! C k/ˇ :

(4.3.7)

k2Z

We now split the above infinite sum over k into even and odd integers and use the 2-periodic property of the function m O 0 to obtain 1D

ˇ2 Xˇ

Xˇ ˇ2 ˇˇ ˇˇ2 ˇ2 ˇˇ

ˇ ˇm ˇm O O 0 .!C2k/ˇ ˇ.!C2k/ O 0 !C.2k C 1/ ˇ ˇO !C.2k C 1/ ˇ ˇ C k2Z

Xˇ ˇ2 ˇm D O 0 .!/ˇ

k2Z

ˇ2 X ˇ ˇ2 ˇ ˇ2 ˇˇ ˇ ˇO ˇm O C  C 2k/ˇˇ O 0 .! C /ˇ ˇ.! ˇ.! C 2k/ˇ C

k2Z

k2Z

ˇ ˇ2 ˇ ˇ2 D ˇm O 0 .!/ˇ C ˇm O 0 .! C /ˇ

by (4.3.1) used in its original form and ! replaced by .! C /. This leads to the desired condition (4.3.4). ˇ ˇ O ˇ D 1 ¤ 0; m Remark. Since ˇ.0/ O 0 .0/ D 1 and m O 0 ./ D 0 . This implies that m O 0 can be considered as a low-pass filter because the transfer function passes the frequencies near ! D 0 and cuts off the frequencies near ! D .

Lemma 4.3.2. The Fourier transform of the scaling function  can be represented by an infinite product O .!/ D

1 Y kD1

m O0

!  2k

:

(4.3.8)

132

4 Construction of Wavelets via MRA

Proof. By the repeated applications of (4.3.2), we obtain !  !  O .!/ Dm O0 O 2 2 !   !   !  Dm O0 m O 0 2 O 2 2 2 2 !   !   !   !  Dm O0 O 0 3 O 3 m O0 2 m 2 2 2 2 :: : !  !  !  !  m O0 2 :::m O 0 k O k Dm O0 2 2 2 2 k     Y ! ! D m O 0 j O j : 2 2 jD1

(4.3.9)

O O Since .0/ D 1 and .!/ is continuous, we obtain ! O lim O k D .0/ D 1: k!1 2 The limit of (4.3.9) as k ! 1 gives (4.3.8). We next prove the following major technical lemma which provides the Fourier characteristics of the subspace W0 . Lemma 4.3.3. The Fourier transform of any function f 2 W0 can be expressed in the form !  !  C  O ; O0 fO .!/ D v.!/ O ei!=2 m 2 2

(4.3.10)

!  !  C  O is where v.!/ O is a 2-periodic function and the factor ei!=2 m O0 2 2 independent of f :

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

p X 2 cn .2t  n/; n2Z

(4.3.11)

4.3 Construction of Mother Wavelet

133

where cn D hf ; 1;n i: We use an argument similar to that in Lemma 4.3.1 to obtain the result !  !  1 X in!=2 O  !  fO .!/ D p Dm Of O ;  cn e 2 2 2 2 n2Z

(4.3.12)

where the function m O f is given by 1 X cn ein! : m O f .!/ D p 2 n2Z

(4.3.13)

Evidently, m O f is a 2- periodic function which belongs to L2 .0; 2/. By applying the Parseval identity and using the fact that f 2 W0 ? V0 , we obtain ˛ ˝ 1 DO O E 0 D f ; 0;n D f ; 0;n 2 Z 1 1 O D fO .!/ .!/ ein! d! 2 1 Z 1 X 2.kC1/ O O D ein! d! f .!/ .!/ 2 k2Z 2k # Z 2 "X 1 O C 2k/ ein! d!: D fO .! C 2k/ .! 2 0 k2Z

(4.3.14)

Consequently, X

O C 2k/ D 0: fO .! C 2k/ .!

(4.3.15)

k2Z

We now substitute (4.3.12) and (4.3.2) into (4.3.15) to obtain

0D

X k2Z

m Of

!   ˇ ! ˇ2 ˇ ˇ C k m C k ˇO C k ˇ ; O0 2 2 2

!

which is, by splitting the sum into even and odd integers k and then using the 2-periodic property of the function m O 0,

134

4 Construction of Wavelets via MRA

0D

X

m Of

 !  ˇ ! ˇ2 ˇ ˇ C 2k mO0 C 2k ˇO C 2k ˇ 2 2 2

!

k2Z

C

X

m Of

k2Z

Dm Of

!  2

mO0

 !  ˇ ! ˇ2 ˇ ˇ C  C 2k mO0 C  C 2k ˇO C  C 2k ˇ 2 2 2

!

ˇ2 !  X ˇ ! ˇ ˇO C 2k ˇ ˇ 2 2 k2Z

Cm Of

 !  X ˇ ! ˇ2 ˇ ˇO C  mO0 C C  C 2k ˇ ; ˇ 2 2 2 k2Z

!

˚ which is, due to orthonormality of the system 0;k .t/ W k 2 Z and (4.3.1),  ! o ! n !  !  C  mO0 C   1: mO0 D m Of Cm Of 2 2 2 2

(4.3.16)

Finally, replacing ! by 2! in (4.3.16) gives m O f .!/ mO0 .!/ C m O f .! C / mO0 .! C / D 0 a.e:

(4.3.17)

Or, equivalently, ˇ ˇ ˇ m O f .!/ mO0 .! C /ˇˇ ˇ D 0: ˇm O f .! C / mO0 .!/ ˇ  This can be interpreted as the linear dependence of two vectors m O f .!/;    m O f .! C / and mO0 .! C /; mO0 .!/ , and hence, there exists a function O such that O m O f .!/ D .!/ mO0 .! C /

a.e:

(4.3.18)

O Further, substitutO f are 2-periodic functions, so is . Since both mO0 and m ing (4.3.18) into (4.3.17) gives O O C / D 0 .!/ C .!

a.e:

(4.3.19)

Thus, there exists a 2- periodic function vO defined by O O .!/ D ei! v.2!/:

(4.3.20)

4.3 Construction of Mother Wavelet

135

Finally, a simple combination of (4.3.12), (4.3.18), and (4.3.20) gives the desired representation (4.3.10). This completes the proof of Lemma 4.3.3. t u Now, we return to the main problem of constructing˚a mother wavelet .t/ from an 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/ D

X

hn

0;n .t/

D

n2Z

where

P

n2Z

X

hn .t  n/;

(4.3.21)

n2Z

jhn j2 < 1: Application of the Fourier transform to (4.3.21) gives fO .!/ D

" X

# hn e

in!

O .!/ D h.!/ O O .!/;

(4.3.22)

n2Z

where the function hO is O h.!/ D

X

hn ein! ;

(4.3.23)

n2Z

and it is a square integrable and 2 -periodic function in Œ0; 2. When (4.3.22) is compared with (4.3.10), we see that O .!/ should be     !  !  O .!/ D ei!=2 mO0 ! C  O ! D m O1 O 2 2 2 2

(4.3.24)

where the function m O 1 is given by m O 1 .!/ D ei! mO0 .! C /:

(4.3.25)

Thus, the function m O 1 .!/ is called the filter conjugate to mO0 .!/, and hence, mO0 and m O 1 are called conjugate quadratic filters (CQF) in signal processing. Finally, substituting (4.3.3) into (4.3.24) gives !  X ! O .!/ D ei!=2  p1 cn ein. 2 C / O 2 2 n2Z !  i.nC1/! 1 X cn einC 2 O Dp 2 2 n2Z

136

4 Construction of Wavelets via MRA

which is, by putting n D .k C 1/ !  1 X Dp : ck1 .1/k eik!=2  O 2 2 k2Z

(4.3.26)

Invoking the inverse Fourier transform to (4.3.26) with k replaced by n gives the mother wavelet p X .t/ D 2 .1/n1 cn1 .2t  n/ (4.3.27) n2Z

p X D 2 dn .2t  n/;

(4.3.28)

n2Z

where the coefficients dn are given by dn D .1/n1 cn1 : Thus, the representation (4.3.28) of a mother wavelet that of the father wavelet  given by (4.3.5).

(4.3.29) has the same structure as

Remarks. 1. The mother wavelet

associated with a given MRA is not unique because dn D .1/n1 c2N1n

(4.3.30)

defines the same mother wavelet (4.3.27) with suitably selected N 2 Z. This wavelet with coefficients dn given by (4.3.30) has the Fourier transform     O .!/ D ei.2N1/!=2 mO0 ! C  O ! : 2 2 The nonuniqueness property of of (4.3.27), by .t/ D

(4.3.31)

allows us to define another form of

, instead

p X 2 dn .2t  n/;

(4.3.32)

n2Z

4.3 Construction of Mother Wavelet

137

where a slightly modified dn is dn D .1/n c1n :

(4.3.33)

In practice, any one of the preceding formulas for dn can be used to find a mother wavelet. 2. The orthogonality condition (4.3.4) together with (4.3.2) and (4.3.24) implies ˇ ˇ2 ˇ ˇ2 ˇ  ! ˇ2 ˇO ˇ ˇ ˇ ˇ ˇ ˇ.!/ˇ C ˇ O .!/ˇ D ˇO ˇ : 2

(4.3.34)

Or, equivalently, ˇ ˇ ˇ ˇ ˇ ˇ ˇ O m ˇ2 ˇ O m ˇ2 ˇ O m1 ˇ2 ˇ.2 !/ˇ C ˇ .2 !/ˇ D ˇ.2 !/ˇ :

(4.3.35)

Summing both sides of (4.3.35) from m D 1 to 1 leads to the result 1 ˇ ˇ2 X ˇ ˇ ˇ ˇO ˇ O m ˇ2 ˇ.!/ˇ D ˇ .2 !/ˇ :

(4.3.36)

mD1

3. If  has a compact support, the series (4.3.28) for the mother wavelet terminates, and consequently, is represented by a finite linear combination of 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/ D

p X 2 .1/n1 cn1 .2t  n/;

(4.3.37)

n2Z

where the coefficients cn are given by p Z cn D h; 1;n i D 2

1

.t/ .2t  n/ dt: 1

That is, the system

˚

m;n .t/

W m; n 2 Z is an orthonormal basis of L2 .R/.

(4.3.38)

138

4 Construction of Wavelets via MRA

Proof. First, we have to verify that f Indeed, we have Z

1

.t  k/ .t  `/ dt D 1

D

1 2 1 2

m;n .t/

Z

1 1

Z

2

W m; n 2 Zg is an orthonormal set.

ˇ ˇ2 ˇ ˇ ei!.k`/ ˇ O .!/ˇ d! ei!.k`/

0

ˇ2 X ˇˇ ˇ ˇ O .! C 2k/ˇ d! k2Z

ˇ2 X ˇ  ! ˇ2 ˇ  ! ˇ2 X ˇˇ ˇ ˇ ˇ ˇ ˇ C .k C 1/ ˇ ˇO C k ˇ ˇ O .! C 2k/ˇ D ˇmO0 2 2 k2Z k2Z which is, by splitting the sum into even and odd integers k, D

ˇ2 ˇ  ! ˇ2 X ˇˇ  ! ˇ ˇ ˇ C .2k C 1/ ˇ ˇ O C k ˇ O0 ˇm 2 2 k2Z C

ˇ2 ˇ  ! ˇ2 X ˇˇ  ! ˇ ˇ ˇ O0 C .2k C 2/ ˇ ˇ O C .2k C 1/ ˇ ˇm 2 2 k2Z

ˇ ! ˇ2 X ˇ  ! ˇ2 ˇ ˇ ˇ ˇO O0 C ˇ C 2k ˇ D ˇm ˇ 2 2 k2Z

ˇ  ! ˇ2 X ˇ  ! ˇ2 ˇ ˇ ˇO ˇ C .2k C 1/ ˇ C ˇm O0 ˇ ˇ 2 2 k2Z ˇ  ! ˇ2 ˇ  ! ˇ2 ˇ ˇ ˇ ˇ C ˇ D1 D ˇm O0 O0 ˇ C ˇm 2 2

by (4.3.4):

Thus, we find Z

1

.t  k/

.t  `/ dx D ık;l :

1

˚ 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, it suffices to show that function vO in (4.3.20) is square integrable over Œ0; 2: In fact,

4.3 Construction of Mother Wavelet

Z

2

ˇ2 ˇ ˇ d! D 2 ˇv.!/ O

Z

0



ˇ2 ˇ ˇ ˇO ˇ.!/ˇ d!



ˇ2 nˇ ˇ ˇ2 ˇ ˇ2 o ˇ ˇO O 0 .! C /ˇ C ˇm O 0 .!/ˇ d!; ˇ.!/ˇ ˇm

0

Z D2 0

Z

2

ˇ2 ˇ ˇ ˇ2 ˇ ˇO O 0 .! C /ˇ d! ˇ.!/ˇ ˇm

2

ˇ ˇ2 ˇm O f .!/ˇ d!;

D2 0

Z

139

D2

by (4.3.4)

by (4.3.18)

0

D 2

X ˇ ˇ2 ˇc n ˇ ;

˛ ˝ cn D f ; 1;n

n2Z

 2 D 2 f 2 < 1: This completes the proof. Example 4.3.1 (The Haar Wavelet). Example 4.2.2 shows that spaces of piecewise constant functions constitute an MRA with the scaling function .t/ D Œ0;1/ .t/. Moreover,  satisfies the dilation equation .t/ D

p X 2 cn .2t  n/;

(4.3.39)

n2Z

where the coefficients cn are given by cn D

p Z 2

1

.t/ .2t  n/ dt:

(4.3.40)

1

Evaluating this integral with  D Œ0;1/ gives cn as follows: 1 c0 D c1 D p 2

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.

140

4 Construction of Wavelets via MRA

1

1

0

1

t

1

0

1/2

ϕ(t)

t

1

0

1/2

1

t

ϕ(2t−1)

ϕ(2t)

Fig. 4.1 Two-scale relation of .t/ D .2t/ C .2t  1/

In view of (4.3.33), we obtain 1 d 0 D c1 D p 2

1 and d1 D c0 D  p : 2

Thus, the Haar mother wavelet is obtained from (4.3.32) as a simple two-scale relation .t/ D .2t/  .2t  1/ D Œ0;:5 .t/  Œ:5;1 .t/ 8 1 ˆ ˆ ˆ < C1; 0  t < 2 D 1; 1  t < 1 ˆ ˆ 2 ˆ : 0; otherwise: This two-scale relation (4.3.42) of

We consider the Fourier transform O of a

O .!/ D Œ; .!/ so that 1 2

(4.3.43)

is represented in Figure 4.2.

Example 4.3.2 (The Shannon Wavelet). scaling function  defined by

.t/ D

(4.3.42)

Z





ei!x d! D

sin t : t

4.3 Construction of Mother Wavelet

1

0

ψ(t)

141

φ(2t)

1

t

1 2

t

1 2

0

1

-1

t

0

-1

−φ(2t−1) Fig. 4.2 Two-scale relation of

.t/ D .2t/  .2t  1/

This is also known as the Shannon sampling function. Clearly, the Shannon scaling function does not have finite support. However, its Fourier transform has a finite support (band-limited) in the frequency domain and has good frequency localization. Evidently, the system 0;k .t/ D .t  k/ D

sin .t  k/ ; .t  k/

k2Z

is orthonormal because ˝

E ˛ 1 DO 0;k ; 0;` D 0;k ; O0;` 2 Z 1 1 D O 0;k .!/ O 0;` .!/ d! 2 1 D

1 2

Z

1

ei.k`/! d! D ık;` : 1

In general, we define, for m D 0, ( V0 D

) sin .t  k/ X ˇˇ ˇˇ2 ck < 1 ; W ck .t  k/ k2Z k2Z

X

and, for other m ¤ 0, m 2 Z, ( Vm D

) 2m=2 sin .2m t  k/ X ˇˇ ˇˇ2 W ck ck < 1 : .2m t  k/ k2Z k2Z

X

142

4 Construction of Wavelets via MRA

It is easy to check that all conditions of Definition 4.2.1 are satisfied. We next find out the coefficients ck defined by ˛ p ˝ ck D ; 1;n D 2

Z

1 1

sin .t/ sin .2t  k/  dx t .2t  k/

8 1 ˆ ˆ kD0

< sin2 = i!=2 4 !  D i e  ˆ > : ; 4

(4.3.48)

4.3 Construction of Mother Wavelet

145

8 ! 9 ˆ > < =

sin2  i!=2 4 !   ei! : D ie ˆ > : ; 4

(4.3.49)

This corresponds to the same Fourier transform (3.2.5) of the Haar wavelet (4.3.43) except for the factor ei! . This means that this factor induces a translation of the Haar wavelet to the left by one unit. Thus, we have chosen v.!/ O D ei! 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 Yves Meyer, and for that reason, it is often called the Meyer wavelet. However, Meyer called it the Littlewood-Paley wavelet. This wavelet is orthogonal and symmetric in nature; however, it does not have a finite support. Define  by 8 2 ˆ 1; j!j  ˆ ˆ < 3    O .!/ D cos  v 3 j!j  1 ; 2  j!j  4 ˆ ˆ 2 4 3 3 ˆ : 0; otherwise

(4.3.50)

where v is a smooth function .Ck or C1 / such that v.t/ D

0; t  0 1; t  1

and v.t/ C v.1  t/ D 1:

(4.3.51)

By using (4.3.51), it is easy to verify that ˇ2 X ˇˇ O C 2k/ˇˇ D 1; ˇ.! k2Z

and hence, the family of functions f0;k D .t  k/ W k 2 Zg forms an orthonormal system. We then define V0 to be the closed subspace spanned by this orthonormal set. Similarly, Vm is the closed space spanned by the m;n ; n 2 Z. Then, Vm satisfy the condition (i) of Definition 4.2.1 if and only if there exists a 2-periodic function m O 0 of the form X O m O 0 .!/ D .2! C 4k/: k2Z

146

4 Construction of Wavelets via MRA

Thus, we have !  !  X !    O O C 2k/O ! D .!/ O D O .! O .!/ Dm O0 2 2 2 2 k2Z !  O do not overlap if k ¤ 0. In fact, as the support of .! C 2k/ and O 2 !  O Using Theorem 4.3.3 with v.!/ D 1, we can get D 1; if ! 2 supp./: O 2 a mother wavelet of the form     O .!/ D ei!=2 mO0 ! C  O ! 2 2   X i!=2 O C 2 C 4k/O ! De .! 2 k2Z h i   O C 2/ C .! O  2/ O ! D ei!=2 .! 2 which gives

O .!/ D

8    3 2 4  ˆ i!=2 ˆ e v j!j  1 ;  j!j  sin ˆ ˆ < 2 2 3 3    ˆ ˆ 3 4 8  ˆ i!=2 ˆ v j!j  1 ;  j!j  : cos :e 2 2 3 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  and the wavelet function . In fact, Mallat’s multiresolution algorithm (fast wavelet transform) uses the cn to calculate the wavelet transform without explicit knowledge of . Furthermore, both  and can be reconstructed from the cn and this in fact is central to Daubechies’ wavelet analysis.

Theorem 4.3.4. If cn are coefficients of the scaling function  defined by (4.2.5), then X p cn D 2; (i) n2Z X .1/n cn D 0; (ii) n2Z

X 1 c2n D p D c2nC1 ; 2 n2Z n2Z X .1/n nm cn D 0; for m D 0; 1; 2; : : : ; .p  1/: (iv)

(iii)

X

n2Z

4.3 Construction of Mother Wavelet

147

O Proof. It follows from (4.3.2) and (4.3.3) that .0/ D 0 and m O 0 .0/ D 1. Putting ! D 0 in (4.3.3) gives (i). Since m O 0 .0/ D 1, (4.3.4) implies that m O 0 ./ 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 m O0 2 ::: .!/ Dm O0 2 2 and m O0

! 2k

1 X k Dp cn ein!=2 : 2 n2Z

Clearly, !  O .2/ Dm O 0 ./ m : O0 2 O According to Strang’s (1989) accuracy condition, .!/ must have zeros of the highest possible order when ! D 2; 4; 6; : : : : Thus, !  ! O .2/ Dm O 0 ./ m O0 m O0 2 :::; 2 2 and the first factor m O 0 .!/ will be zero of order p at ! D  if dm m O 0 .!/ D0 d! m

for m D 0; 1; 2; : : : .p  1/;

which gives X

cn .in/m ein D 0;

for m D 0; 1; 2; : : : .p  1/:

n2Z

Or, equivalently, X n2Z

.1/n nm cn D 0;

for m D 0; 1; 2; : : : .p  1/:

148

4 Construction of Wavelets via MRA

From the fact that the scaling function .t/ is orthonormal to itself in any translated position, we can show that X

c2n D 1:

(4.3.53)

n2Z

This can be seen by using .t/ from (4.3.5) to obtain Z

1 2

 .t/ dt D 2 1

XX

Z

1

.2t  m/ .2t  n/ dt

cm cn 1

m2Z n2Z

where the integral on the right-hand side vanishes due to orthonormality unless m D n, giving Z

1

 2 .t/ dt D 2 1

X

c2n

Z

n2Z

D2

X n2Z

c2n 

1

 2 .2t  n/ dt

1

1 2

Z

1

 2 .y/ dy 1

whence follows (4.3.53). Finally, we prove X

ck ckC2n D ı0;n :

(4.3.54)

k2Z

We use the scaling function  defined by (4.3.5) and the corresponding wavelet given by (4.3.28) with (4.3.30), that is, .t/ D

p X 2 .1/n1 c2N1n .2t  n/ n2Z

which is, by substituting 2N  1  n D k, D

p X 2 .1/k ck .2t C k  2N C 1/:

(4.3.55)

n2Z

We use the fact that mother wavelet .t/ is orthonormal to its own translate .t n/ so that Z

1

.t/ .t  n/ dt D ı0;n : 1

(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

.t/ .t  n/ dt 1

D2

XX

.1/

kCm

Z

1

.2t C k  2N C 1/ .2t C m  2N C 1  2n/ dt;

ck cm 1

k2Z m2Z

where the integral on the right-hand side is zero unless k D m  2n so that Z

1

.t/ .t  n/ dt D 2 1

Z X 1 1 2 .1/2.kCn/ ck ckC2n   .y/ dy: 2 1 k2Z

This means that X

ck ckC2n D 0;

for all n ¤ 0:

n2Z

4.4

The Fast Wavelet Transform

In this section, we introduce the basic algorithm, derived by Mallat (1989b) for fast computation of the discrete wavelet transform, commonly known as the fast wavelet transform. Recall that the successive approximations of a given function f can be obtained by orthogonal projections Pm onto the scaled spaces Vm as Pm f D

X

cm;n m;n ;

(4.4.1)

n2Z

where ˛ ˝ cm;n D f ; m;n :

(4.4.2)

Then, its projection on the detailed spaces Wm is given by Qm f D PmC1 f  Pm f D

X

dm;n

m;n

(4.4.3)

n2Z

where ˝ dm;n D f ;

m;n

˛ :

(4.4.4)

150

4 Construction of Wavelets via MRA

˚ Since 0;n .t/ D .t  n/ W n 2 Z is an orthonormal basis of V0 and V1  V0 , t hence we can decompose  2 V1 as 2 1 t X  D hn .t C n/: 2 2

(4.4.5)

n2Z

Therefore, we have

 m1;n .t/ D 2m1=2  2m1 t  n X

 D 2m1=2 hk  2m1 t  2n C k k2Z

p X hk m;2nk .t/: D 2

(4.4.6)

k2Z

Similarly, one can decompose ˚ 0;n .t/ W n 2 Z of V0 as 1 2

t

2 W1  V0 in terms of the basis function

2 t 2

D

X

gn .t C n/:

(4.4.7)

n2Z

Therefore, it follows by computations similar to those above that m1;n .t/

m1  2 tn X

 gk  2m1 t  2n C k D 2m1=2

D 2m1=2

k2Z

p X D 2 gk m;2nk .t/:

(4.4.8)

k2Z

If we denote the low-pass and high-pass filters associated with  and tively, by m O 0 .!/ D

X

hn ei!n ;

and m O 1 .!/ D

n2Z

X

gn ei!n :

n2Z

Then, it follows from (4.3.2) and (4.3.24) that O O .2!/ Dm O 0 .!/.!/;

and

O .2!/ D m O O 1 .!/.!/:

, respec-

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 m O 1 .!/ D ei! m O 0 .! C /, that is, X

X .1/nC1 h1n ei!n ;

gn ei!n D

n2Z

n2Z

which implies that gn D .1/nC1 h1n :

(4.4.9)

Substituting (4.4.6) into (4.4.2), we obtain p X 2 hk cm;2nk :

cm1;n D

(4.4.10)

k2Z

Similarly, substituting (4.4.8) into (4.4.4), we obtain dm1;n D

p X 2 gk cm;2nk :

(4.4.11)

k2Z

Equations (4.4.10) and (4.4.11) are the general recursions of the decomposition algorithm, whereas the recursions for the reconstruction algorithm can be obtained from the equations (4.4.3), (4.4.6), and (4.4.8). This gives cm;n D

p X  h2kn cm1;k C g2kn dm1;k : 2

(4.4.12)

k2Z

4.5

Exercises 2

1. Show that the Gaussian function .t/ D et cannot be the scaling function of an MRA. 2. Suppose (4.2.5) has a finite sum and that  2 L1 .R/ Z 1that the dilation equationX with .t/ dt ¤ 0. Show that cn D 2. 1

n2Z

3. Prove that the function f .t/ D

8
= 4 2  4 2  :   p p 1 1 > c2 D p 3  3 ; c3 D p 1  3 > ; 4 2 4 2

(5.3.21)

Consequently, the Daubechies scaling function 2 .t/ takes the form, dropping the subscript, .t/ D

i p h 2 c0 .2t/ C c1 .2t  1/ C c2 .2t  2/ C c3 .2t  3/ :

(5.3.22)

Using (4.3.30) with N D 2, we obtain the Daubechies wavelet 2 .t/, dropping the subscript, i p h 2 d0 .2t/ C d1 .2t  1/ C d2 .2t  2/ C d3 .2t  3/ i p h D 2  c3 .2t/ C c2 .2t  1/  c1 .2t  2/ C c0 .2t  3/ ;

.t/ D

(5.3.23) where the coefficients in (5.3.23) are the same as for the scaling function .t/, but in reverse order and with alternate terms having their signs changed from plus to minus.

5.3 The Daubechies Wavelets

171

On the other hand, the use of (4.3.28) with (4.3.33) also gives the Daubechies wavelet 2 .t/ in the form 2

.t/ D

i p h 2  c0 .2t  1/ C c1 .2t/  c2 .2t C 1/ C c3 .2t C 2/ :

(5.3.24)

The wavelet has the same coefficients as given in (5.3.23) except that the wavelet 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 four coefficients. However, in general, c’s (some positive and some negative) in (5.3.22) are numerical 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 an iterative algorithm in which each new approximation m .t/ is computed from the previous approximation m1 .t/ by the scheme m .t/ D

i p h 2 c0 m1 .2t/ C c1 m1 .2t  1/ C c2 m1 .2t  2/ C c3 m1 .2t  3/ : (5.3.25)

This iteration process can be continued until m .t/ becomes indistinguishable from m1 .t/. This iterative algorithm is briefly described below starting from the characteristic function 1; 0  t < 1 Œ0;1 .t/ D (5.3.26) 0; otherwise. After one iteration the characteristic function over 0  t < 1 assumes the shape of 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 factor 1 p in each coefficient so that it produces the Daubechies scaling function .t/ 2 given by (5.3.22) and the orthonormal p Daubechies wavelet .t/ (or D4 wavelet) given by (5.3.23) without the factor 2. We represent the characteristic function by the ordinate 1 at t D 0. The first iteration 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 another four ordinates c20 ; c0 c1 ; c1 c2 ; c1 c3 at t D 0:00; 0:25; 0:75; and so on. After completing the second iteration process, there are ten new ordinates c20 ; c0 c1 ; c0 c1 C c1 c0 ; c0 c3 Cc21 ; c1 c2 Cc2 c0 ; c1 c3 Cc2 c1 ; c22 Cc3 c0 ; c2 c3 Cc3 c1 c3 c2 ; c23 at t D 0:25; 0:50; 0:75; 1:00; : : : ; 2:25: This iteration process can be described by the matrix scheme

172

5 Elongations of MRA-Based Wavelets

2

3

c0

6 6 c1 6 6 c2 6 6 6 c3 6 6

 6  D6 6 2 6 6 6 6 6 6 6 6 4

c0 c1 c2 c3

7 7 7 7 7 72 3 7 c0 7 76 7 c0 7 6c1 7 7 6 7 Œ1 D M2 M1 Œ1; 6 7 c1 7 7 4c2 5 7 c2 c0 7 7 c3 7 c3 c1 7 7 c2 7 5 c3

(5.3.27)



 where Mn represents the matrix of the order 2nC1 C 2n  2  2n C 2n1  2 in which each column has a submatrix of the coefficients c0 ; c1 ; c2 ; c3 located two places below the submatrix to its left. We also use the same matrix scheme for developing the Daubechies wavelet p .t/ from .t/ which is given by (5.3.22) without the factor 2. For simplicity, 2 2 we assume that only one iteration process gives the final 2 .t/, so this can be described by four ordinates c0 ; c1 ; c2 ; c3 at t D 0:0; 0:50; 1:0; 1:50: In view of (5.3.23), these four ordinates produce ten new ordinates spaced 0:25 apart. The term 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 other terms, so that the new ten ordinates for the wavelet are given by c3 c0 ; c3 c1 ; c3 c2 Cc2 c0 ; c23 Cc2 c1 ; c22 c1 c0 ; c2 c3 c21 ; c1 c2 Cc20 ; c1 c3 Cc0 c1 ; c0 c2 ; c0 c3 : These ordinates are generated by the matrix scheme

2

2 c3 6 6 0 6 6 c2 6 6 6 0 6 6  6c1 D6 6 0 6 6 6 c0 6 6 6 6 6 4

3 c3 0 c3 c2

0

0

c2

c1 0 0 c1 c0

Or, alternatively, by the matrix scheme

0 c0

7 7 7 7 7 72 3 c3 7 7 c0 76 7 0 7 6c1 7 7 6 7 Œ1: 6 7 c2 7 7 4c2 5 7 0 7 7 c3 7 c1 7 7 0 7 5 c0

(5.3.28)

5.3 The Daubechies Wavelets

173

2

2

3

c0

6 6 c1 6 6 c2 6 6 6 c3 6 6  6 D6 6 6 6 6 6 6 6 6 6 4

c0 c1 c2 c3

7 7 7 7 7 72 3 7 c3 7 7 76 c0 7 6 c2 7 7 Œ1: 76 7 6 c1 7 7 4c1 5 7 c2 c0 7 7 c0 7 c3 c1 7 7 c2 7 5 c3

(5.3.29)

Making reference to Newland (1993b), it can be verified that 3 .t/ can be described by the matrix scheme

3

2 3 c3 6 7 6 c2 7  7 D M3 M2 6 6c1 7 Œ1; 4 5 c0

(5.3.30)

where the matrix M3 is of order 22  10 with ten submatrices Œc0 c1 c2 c3 T , each organized two places below its left-hand neighboring matrix. The matrix scheme (5.3.30) is used to generate wavelets in the inverse discrete wavelet 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. After eight steps leading to 766 ordinates as before, the resulting wavelet is very close to that 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 simple algebraic relation in terms of elementary or special functions. She also demonstrated that  satisfies several algebraic relations that play a major role in computational analysis.

174

5 Elongations of MRA-Based Wavelets

a

2ψ(t)

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

b

0.5

1

1.5

2

2.5

3

2

2.5

3

2φ(t)

1.5

1

0.5

0

-0.5

0

0.5

1

1.5

t

Fig. 5.6 (a) The Daubechies wavelet 2 .t/. (b) The Daubechies scaling function 2 .t/

Replacing t by

t in (5.3.22) gives 2 

t 2

D

3 p X 2 ck .t  k/

(5.3.32)

kD0

which can be found exactly if .t/; .t1/; .t2/; .t3/ are all known. Suppose that 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 

      3 5 1 ;  ;  : 2 2 2

5.3 The Daubechies Wavelets

175

Again, by using (5.3.32) and these new values, we can calculate 

            3 5 7 9 11 1 ;  ;  ;  ;  ;  ; 4 4 4 4 4 4

and so on. In order to carry out this recursive process, we set initial values .0/ D 0;

.1/ D

p  1  1C 3 ; 2

.2/ D

p  1  1 3 ; 2

.3/ D 0: (5.3.33)

For example, for t D 1, we obtain from (5.3.32) that 

  i p h 1 D 2 c0 .1/ C c1 .0/ C c2 .1/ C c3 .2/ 2

which is, by (5.3.21) and (5.3.31),  p 2 1C 3 p  p 1 D 2C 3 : D 2 c0 .1/ D 8 4     5 3 ;  so that Similarly, we can calculate  2 2 tD .t/ D

p  1 2C 3 ; 4

0;

1 ; 2

3 ; 2

5 ; 2

p  1 2 3 ; 4

and .t  3/ D 0:

A similar calculation gives the values of  at multiples of tD p 5C3 3 .t/ D ; 16

1 ; 4

p 9C5 3 ; 16

3 ; 4

5 ; 4

7 ; 4

 p  2 1C 3 16

1 as given below: 4 9 ; 4

 p  2 1 3 ;

16

;

p 95 3 : 16

The Daubechies wavelet .t/ is given by (5.3.24). In view of (5.3.31), it turns out that .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,

176

5 Elongations of MRA-Based Wavelets

.0/ D D

i p h 2 c3 .2/  c2 .1/ C c1 .0/  c0 .1/ i p h 2 c3 .2/  c3 .1/ p ! p ! 1 3 1 3  4 2

D

D

Consequently,

p ! p ! 3 3 1C 3 4 2

p  1  1 3 : 2

1 3 .t/ at t D 1;  ; 0; 1; is given as follows: 2 2 t D 1;

.t/ D 0;

1  ; 4

1  ; 2

0;

p  1 1 3 ; 2

1; 

3 ; 2

p  1 1C 3 ; 2

Both Daubechies’ scaling function  and Daubechies’ wavelet shown in Figure 5.6a, b, respectively.

1  : 4 for N D 2 are

In view of its fractal shape, the Daubechies wavelet 2 .t/ given in Figure 5.7a has 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 D4 wavelet represents a linear function f .t/ D at exactly, where a is a constant. Six wavelet coefficients 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 with terms 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 wavelet coefficients generate longer wavelets. As N increases, wavelets lose their irregular shape and become increasingly smooth with a Gaussian harmonic waveform. For N D 10; the frequency of the waveform is not constant and some minor irregularities still persist on the right. Each of the wavelets in Figure 5.7a, b represents the basis for a family of wavelets of different levels and different locations along the x-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 on the 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 The Daubechies Wavelets

a

ψ(t)

2

177 N=3

0

-2

0

5

ψ(t)

2

t

N=5

0

-2

b

0

ψ(t)

2

t

9 N=7

0

-2

0

13

ψ(t)

2

N = 10

0

-2

0

19

t

Fig. 5.7 (a) Wavelets for N D 3; 5 drawn using the Daubechies algorithm. (b) Wavelets for N D 7; 10 drawn using the Daubechies algorithm

178

5 Elongations of MRA-Based Wavelets

of 2. These values are sufficient for equally spaced samples from a signal. Due to the importance of such powers of 2, the idea of a dyadic number and related notation and terminology seem to be useful in wavelet algorithms. Definition 5.3.1. (Dyadic Number). A number m is called a dyadic number if and only 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 by Dn for n 2 N. A dyadic number has a finite binary expansion, and a dyadic number in Dn has a binary expansion with at most n binary digits past the binary point. Definition 5.3.2. The set of all linear of 1 and i hp combinations 3 so that coefficients p; q 2 D is denoted by D D

p 3 with dyadic

o hp i n p 3 D p C q 3 W p; q 2 D :

For every integer n, we consider combinations with coefficients in Dn so that Dn

hp i n o p 3 D p C q 3 W p; q 2 Dn :

We define the conjugate m of m by  p   p  pCq 3 D pq 3 : hp i 3 is an integer ring under ordinary addition and multiplication. The set D In terms of two quantities aD

p  1 1C 3 4

and a D

p  1 1 3 ; 4

(5.3.34)

the scaling function 2  can be written as 2 .t/

D

X p 2N1 2 ck .2t  k/;

.N D 2/

kD0

D a .2t/ C .1  a/ .2t  1/ C .1 C a/ .2t  2/ C a .2t  3/: (5.3.35) If 0  m  2N  1,(5.3.35) can be rewritten as

5.3 The Daubechies Wavelets

179

.m/ D

X p 2N1 2 c2mk .k/:

(5.3.36)

kD0

This system of equations can be written in the matrix form 3 3 2 2 32 a 0 0 0 .0/ .0/ 7 7 6 6 76 0 7 6.1/7 6.1/7 61  a 1  a a 7 7D6 6 76 6.2/7 6 0 7 6.2/7 : a 1  a 1  a 5 5 4 4 54 .3/ .3/ a 0 0 0

(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 be calculated for all of D by (5.3.35). Finally, we conclude this section by including the Daubechies scaling function and the Daubechies wavelet 3 .t/ for N D 3. In this case, (5.3.10) gives

3 .t/

P.t/ D P3 .t/ D 1 C 3t C 6t2 ;

(5.3.39)

where  1 i! ! D e and C 2  ei! 2 4  1 2i! e t2 D C 4 C e2i!  4ei!  4ei! C 2 : 16 t D sin2

Consequently, (5.3.12) gives the result ˇ ˇ2 3 9 19 19 i! 3 2i! ˇO ˇ  e C e : ˇL.!/ˇ D e2i!  ei! C 8 4 4 4 8

(5.3.40)

In this case, A.!/ D b0 C b1 ei! C b2 e2i! ; so that

(5.3.41)

180

5 Elongations of MRA-Based Wavelets

ˇ2 ˇ ˇ ˇO ˇL.!/ˇ    DA.!/A.!/ D b0 C b1 ei! C b2 e2i! b0 C b1 ei! C b2 e2i!   D b20 C b21 C b22 C ei! .b0 b1 C b2 b1 / C ei! .b0 b1 C b1 b2 / C b0 b2 e2i! C b0 b2 e2i! : (5.3.42)

Equating the coefficients in (5.3.40) and (5.3.42) gives b20 C b21 C b22 D

19 ; 4

9 b1 b0 C b2 b1 D  ; 4

b2 b0 D

3 : 8

(5.3.43)

ˇ ˇ ˇ O ˇ2 In view of the fact that ˇL.0/ ˇ 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 requirement b0 C b1 C b2 D 1: Eliminating b1 from this equation and the second equation in (5.3.43) gives b21  b1 

9 D0 4

so that b1 D

p  1  1 ˙ 10 : 2

(5.3.44)

Consequently,

b0 C b2 D

p  1  1  10 : 2

(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) so that b1 D

p  1  1  10 : 2

(5.3.46)

Obviously,

b0 C b2 D

p  1  1 C 10 2

and b0 b2 D

3 8

5.3 The Daubechies Wavelets

181

lead to the fact that b0 and b2 satisfy t2 

p  1  3 1 C 10 t C D 0: 2 8

(5.3.47)

Thus, q  p  p  1  .b0 ; b2 / D 1 C 10 ˙ 5 C 2 10 : 4

(5.3.48)

Consequently, A.!/ is explicitly known and, hence, mO0 .!/ becomes mO0 .!/ D

 1 b0 C .3b0 C b1 /ei! C .3b0 C 3b1 C b2 /e2i! 8

 C .b0 C 3b1 C 3b2 /e3i! C .b1 C 3b2 /e4i! C b2 e5i! ;

(5.3.49)

which is equal to (4.3.3) Equating the coefficients of (4.3.3) and (5.3.49) gives all six ck ’s as p c0 D

c1 D

c2 D

c3 D

c4 D

c5 D

p  q  p  p 2 2 b0 D 1 C 10 C 5 C 2 10 ; 8 32 p p  q  p  p 2 2 .3 b0 C b1 / D 5 C 10 C 3 5 C 2 10 ; 8 32 p p  q  p  p 2 2 .3 b0 C 3 b1 C b2 / D 5  10 C 5 C 2 10 ; 8 32 p p  q  p  p 2 2 .b0 C 3 b1 C 3 b2 / D 5  10  5 C 2 10 ; 8 32 p p  q  p  p 2 2 .b1 C 3 b2 / D 5 C 10  3 5 C 2 10 ; 8 32 p p  q  p  p 2 2 b2 D 1 C 10  5 C 2 10 : 8 32

182

5 Elongations of MRA-Based Wavelets

Fig. 5.8 (a) The Daubechies scaling function 3 .t/ for N D 3. (b) The Daubechies wavelet 3 .t/ for N D 3

a

3φ(t)

1

2

x

0 1

b

3

4

5

3

4

5

3ψ(t)

1

x

0 1

2

Evidently, the Daubechies scaling function 3 .t/ and the Daubechies wavelet 3 .t/ (or simply D6 wavelet) can be rewritten as 3 .t/

D

5 p X 2 ck .2t  k/and

(5.3.50)

kD0

3

.t/ D

5 p X 2 dk .2t  k/; respectively;

(5.3.51)

kD0

where ck and dk are explicitly known. Figure 5.8a, b exhibits the scaling function and the wavelet 3 .t/.

3 .t/

With a given even number of wavelet coefficients ck ; k D 0; 1; : : : ; 2N  1, we can define the scaling function  by .t/ D

X p 2N1 2 ck .2t  k/ kD0

and the corresponding wavelet by

(5.3.52)

5.3 The Daubechies Wavelets

.t/ D

183

X p 2N1 2 .1/k ck .2t C k  2N C 1/;

(5.3.53)

kD0

where the coefficients ck satisfy the following conditions: 2N1 X

ck D

2N1 X

p 2;

kD0

.1/k km ck D 0;

(5.3.54)

kD0

where m D 0; 1; 2; : : : ; N  1, and 2N1 X

ck ckC2m D 0;

m ¤ 0;

(5.3.55)

kD0

where m D 0; 1; 2; : : : ; N  1, and 2N1 X

c2k D 1:

(5.3.56)

kD0

When N D 1, two coefficients c0 and c1 satisfy the following equations: c0 C c1 D

p 2;

c0  c1 D 0;

c20 C c21 D 1

1 which admit solutions c0 D c1 D p : They give the classic Haar scaling function 2 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 D

p 2;

c0 c2 C c1 c3 D 0;

c0  c1 C c2  c3 D 0; c20 C c21 C c22 C c23 D 1:

These give solutions p  1  c0 D p 1 C 3 ; 4 2 p  1  c2 D p 3 3 ; 4 2

1 c1 D p 4 2 1 c3 D p 4 2

 p  3C 3 ;  p  1 3 :

These coefficients constitute the Daubechies scaling function (5.3.22) and the Daubechies D4 wavelet (5.3.23) or (5.3.24).

184

5.4

5 Elongations of MRA-Based Wavelets

The Harmonic Wavelets

So far, all wavelets have been constructed from dilation equations with real coefficients. However, many wavelets cannot always be expressed in functional form. As the number of coefficients in the dilation equation increases, wavelets get increasingly longer and the Fourier transforms of wavelets become more tightly confined to an octave band of frequencies. It turns out that the spectrum of a wavelet with n coefficients becomes more boxlike as n increases. This fact led (Newland, 1993a,b) to introduce a new harmonic wavelet .t/ whose spectrum is exactly like a box, so that the magnitude of its Fourier transform O .!/ is zero except for an octave band of frequencies. Furthermore, he generalized the concept of the harmonic wavelet to describe a family of mixed wavelets with the simple mathematical structure. It is also shown that this family provides a complete set of orthonormal basis functions for signal analysis. These musical wavelets provide greater frequency discrimination than is possible with harmonic wavelets whose frequency band is always an octave. A major advantage for all harmonic wavelets is that they can be computed by an effective parallel algorithm rather than by the series algorithm needed for the dilation wavelet transform (See Mouri and Kubotani, 1995; Newland, 1993a,b, 1994). Newland (1993a) introduced a real even function is defined by 8 < 1 O e .!/ D 4 for  4  ! < 2; : 0; otherwise

e .t/

whose Fourier transform

and 2  ! < 4

;

(5.4.1)

where the Fourier transform is defined by 1 fO .!/ D 2

Z

1

f .t/ ei!t dt:

(5.4.2)

1

The inverse Fourier transform of O e .!/ gives Z e .t/

1

D 1

 O e .!/ ei!t d! D 1 sin 4t  sin 2t : 2t

On the other hand, the Fourier transform O 0 .!/ of a real odd function defined by 8 i ˆ ˆ ˆ < 4 for  4  ! < 2 i O 0 .!/ D for 2  ! < 4 ˆ ˆ 4 ˆ : 0; otherwise:

(5.4.3)

0 .t/

is

(5.4.4)

5.4 The Harmonic Wavelets

185

Then, the inverse Fourier transform gives Z 0 .t/

1

D 1

The harmonic wavelet form

 O 0 .!/ ei!t d! D 1 cos 2t  cos 4t : 2t

(5.4.5)

.t/ is then defined by combining (5.4.3) and (5.4.5) in the

.t/ D

e .t/

Ci

0 .t/

 e4it  e2it : D 2it 

(5.4.6)

The real and imaginary parts of .t/ are shown in Figure 5.9a, b Clearly, 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 harmonic wavelet .t/ 8 < 1 O .!/ D 2 ; 2  ! < 4 : 0; otherwise:

(5.4.8)

For the general harmonic wavelet .t/ at level m and translated in k steps of size 2m , we define   8 1 m i!k ˆ ˆ 2 exp  m ; 22m  ! < 42m < 2 O .!/ D 2 ; (5.4.9) ˆ ˆ : 0; otherwise where m and k are integers. The inverse Fourier transform of (5.4.9) gives



m

m tk/

e4i.2 tk/  e2i.2

 2 tk D 2i 2m t  k m

where m is a nonnegative integer and k is an integer.

;

(5.4.10)

186

5 Elongations of MRA-Based Wavelets

a

1 0.8 0.6 0.4 0.2

0 −0.2 −0.4 −0.6 −0.8 −1 −8

Re{ψ(t)}

−6

−4

−2

0

2

4

6

8

−4

−2

0

2

4

6

8

t

b

1 0.8 0.6 Im{ψ(t )} 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −8 −6

Fig. 5.9 (a) Real part of

.t/ and (b) imaginary part of

t

.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 8 < 1 i!k ; 0  ! < 2 O .!/ D 2 e ; (5.4.11) : 0; otherwise so that the inverse Fourier transform gives the so-called harmonic scaling function

 e2i.tk/  1

 :  tk D 2i t  k

(5.4.12)

5.4 The Harmonic Wavelets

187

^

ψ (ω) m = −1

1/2π m=0 m=1

1/4π

m=2 m=3

1/8π

m=4

1/16π 0 2π 4π

8π

16π

32π

ω

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 to O be appropriate in the sense that they form an orthogonal set.

mIf .!/  is the Fourier transform of .t/, then the Fourier transform of g.t/ D 2 t  k is gO .!/ D 2

m

  i!k O m exp  m .2 !/ : 2

(5.4.13)

Clearly, the Fourier transforms of successive levels of harmonic wavelets decrease in 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 the general Parseval relation (1.3.17) in the form Z

Z

1

1

fO .!/ gO .!/ d!;

f .t/ g.t/ dt D 2 1

(5.4.14)

1

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

Z

1

1

fO .!/ gO .!/ d!:

f .t/ g.t/ dt D 2 1

1

(5.4.15)

188

5 Elongations of MRA-Based Wavelets

This result follows from the following formal calculation: Z

Z

1

Z

1

f .t/ g.t/ dt D

1

fO .!1 / d!1

dt

1

1

Z

1 1

D 2 1 Z 1

D 2 1 Z 1

D 2

fO .!1 / d!1

Z

Z

1

gO .!2 / d!2 ei.!1 C!2 /t 1

1

gO .!2 / ı.!1 C !2 / d!2 1

fO .!1 / gO .!1 / d!1 fO .!/ gO .!/ d!;

.! D !1 /:

1

Theorem 5.4.1. The family of harmonic wavelets set.



 2m t  k forms an orthogonal

Proof. To prove this theorem, it suffices to show orthogonality conditions: Z

1

.t/



 2m t  k dt D 0;

for all m; k;

(5.4.16)

.t/



for m ¤ 0:

(5.4.17)

1

Z

1

 2m t  k dt D 0;

1



We put g.t/ D

 2m t  k so that its Fourier transform is   i!k gO .!/ D 2m exp  m O .2m !/ 2

(5.4.18)

and then apply (5.4.15) to obtain Z

Z

1

1

O .!/ gO .!/ d!:

.t/ g.t/ dt D 2 1

(5.4.19)

1

If .t/ and g.t/ are two harmonic wavelets, they have the one-sided Fourier transforms as shown in Figure 5.10, so that the product O .!/ gO .!/ must always vanish. Thus, the right-hand side of (5.4.19) is always zero for all k and m, that is, Z

1

.t/ 1



 2m t  k dt D 0;

for all m; k:

(5.4.20)

5.4 The Harmonic Wavelets

189

To prove (5.4.17), we apply (5.4.14) so that Z

Z

1

1

O .!/ gO .!/ d!:

.t/ g.t/ dt D 2 1

(5.4.21)

1

Clearly, wavelets of different levels are always orthogonal to each other because their Fourier transforms occupy different frequency bands so that the product O .!/ gO .!/ is zero for m ¤ 0. On the other hand, at the same level .m D 0/, we have gO .!/ D ei!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

1 .t/ .t  k/ dt D 2

Z

4

ei!k d! D 0;

(5.4.23)

2

provided e4ik D e2ik ; k ¤ 0. This gives e2ik D 1 for k ¤ 0. Thus, all wavelets 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 other levels except that the unit interval is now that for the wavelet level concerned. For instance, for level m, the unit interval is 2m and translation is equal to any multiple of 2m . 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) are always orthogonal, and wavelets at the same level are orthogonal if one is translated with 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 2

m  2 t  k dt D 0:

(5.4.24)

1

Theorem 5.4.2 (Normalization of Harmonic Wavelets). Let f m;k g be the family of general harmonic wavelets defined by (5.4.10). Then, we have Z

1 1

ˇ

ˇˇ2 ˇ ˇ 2m t  k ˇ dt D 2m :

(5.4.25)

190

5 Elongations of MRA-Based Wavelets

Proof. It follows from (5.4.21) that Z

Z

1

1

O .!/ O .!/ d!:

.t/ .t/ dt D 2 1

(5.4.26)

1

Using (5.4.18) in (5.4.26) gives Z

1



2m t  k



 2m t  k dt D 222m

1

Z

1

O .2m !/ O .2m !/ d!: 1

(5.4.27)

1 It follows from (5.4.8) that O .2m !/ D for 22m  ! < 42m so 2 that (5.4.27) becomes Z

1 1

Z ˇ

ˇˇ2 ˇ m 2m ˇ 2 t  k ˇ dt D 22

42m 22m

1  d! D 2m : .2/2

This implies the property of normality. We next investigate some properties of harmonic scaling functions by virtue of the Fourier transforms. Newland (1993a) first introduced the even Fourier transform 8 < 1 ; 2  ! < 2 Oe .!/ D 4 (5.4.28) : 0; otherwise to define an even scaling function e .x/ D

sin 2x : 2x

(5.4.29)

Similarly, the odd Fourier transform given by 8 i ˆ ˆ ˆ < 4 ; 2  ! <  O0 .!/ D  i ; 0  ! < 2 ˆ ˆ ˆ : 4 0; otherwise;

(5.4.30)

gives an odd scaling function 0 .t/ D

.1  cos 2t/ : 2x

(5.4.31)

5.4 The Harmonic Wavelets

191

All of these results allow us to define a complex scaling function .t/ by .t/ D e .t/ C i0 .t/

(5.4.32)

so that .t/ D

e2it1 : 2it

(5.4.33)

O Its Fourier transform .!/ is given by 8 < 1 ; 0  ! < 2 O .!/ D 2 : 0; otherwise.

(5.4.34)

The real and imaginary parts of the harmonic scaling function (5.4.33) are shown in 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. ˚ O O Proof. We substitute Fourier transforms .!/ and F .t  k/ D ei!k .!/ in (5.4.15) to obtain Z

Z

1

1 i!k O O d! D 0: .!/ .!/e

.t/ .t  k/ dt D 2 1

(5.4.35)

1

O The right-hand side is always zero for all k because .!/ is the one-sided Fourier transform given by (5.4.34). ˚ O O On the other hand, we substitute .!/ and F .tk/ D ei!k .!/ in (5.4.15) to obtain Z 1 Z 1 O O .!/ .!/ ei!k d! .t/ .t  k/ dt D 2 1

1

D D

1 2

Z

2

ei!k d!;

by (5.4.34)

0

 1 2ik e 1 D0 2ik

for k ¤ 0:

(5.4.36)

This shows that .t/ and .t  k/ are orthogonal for all integers k except k D 0.

192

5 Elongations of MRA-Based Wavelets

Fig. 5.11 (a) Real part of the scaling function  and (b) imaginary part of the scaling function 

a 1 0.8 0.6 Re{φ(t)} 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −8

−6

−4

−2

0

2

4

6

8

t

b 1 0.8 Im{φ(t)}

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −8

−6

It can also be shown that Z 1

m 

 2 t  k  t  n dt D 0

−4

−2

0

2

4

6

8

t

for all m; k; n .m  0/;

(5.4.37)

for all m; k; n .m  0/:

(5.4.38)

1

Z

1





 2m t  k  t  n dt D 0

1

Theorem 5.4.4 (Normalization of Scaling Function). Z

1 1

ˇ

ˇ ˇ t  k ˇ2 dt D 1:

(5.4.39)

5.4 The Harmonic Wavelets

193

Proof. We have the identity (5.4.14) so that Z

1



  t  k  t  k dt D 2

Z

1

1

O O .!/ .!/ d!: 1

Thus, Z

1

ˇ

ˇ ˇ t  k ˇ2 dt D 2

1

Z

2 0

1 d! D 1 .2/2

by (5.3.7):

This completes the proof. The rest of this section is devoted to wavelet expansions and Parseval’s formula for harmonic wavelets. Any arbitrary (real or complex) function f .t/ can be expanded in terms of complex harmonic wavelets in the form f .t/ D

XXh



am;k

 2m t  k C aQ m;k



i 2m t  k ;

(5.4.40)

m2Z k2Z

where the complex coefficients am;k and aQ m;k are defined by am;k D 2m

Z

1

f .t/



 2m t  k dt;

(5.4.41)

f .t/



 2m t  k dt:

(5.4.42)

1

aQ m;k D 2m

Z

1 1

In terms of these coefficients, the contribution of a single complex wavelet to the function f .t/ is given by am;k



 2m t  k C aQ m;k



 2m t  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

hˇ XX ˇ ˇ2 ˇ2 ˇ ˇ2 i ˇf .t/ˇ dt D 2m ˇam;k ˇ C ˇaQ m;k ˇ : m2Z k2Z

(5.4.44)

194

5 Elongations of MRA-Based Wavelets

Multiplying (5.4.40) by fN .t/ and then integrating the result from 1 to 1 term by term give Z

1

f .t/ f .t/ dt 1

D

Z XX am;k

1



f .t/

 2m t  k dt C aQ m;k

Z

1

m2Z k2Z

1

f .t/



m  2 t  k dt :

1

(5.4.45) We use (5.4.41) and (5.4.42) to replace the integrals on the right-hand side of (5.4.46) by am;k and aQ m;k so that (5.4.46) becomes Z

1 1

hˇ XX ˇ ˇ2 ˇ2 ˇ ˇ2 i ˇf .t/ˇ dt D 2m ˇam;k ˇ C ˇaQ m;k ˇ : m2Z k2Z

It may be noted that, for real functions f .t/; am;k D aQ m;k so that the expansion (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 Z

1

a;k D

 f .t/  t  k dt;

(5.4.46)

 f .t/  t  k dt:

(5.4.47)

1

Z

1

aQ ;k D 1

m  In view of the orthogonality and normalization properties of 2 t  k and  t  k , it can be shown that any arbitrary function f .t/ can be expanded in the form Xh



i X Xh



i a;k  t  k CQa;k  t  k C am;k 2m t  k CQam;k 2m t  k : f .t/D k2Z

m2Z k2Z

(5.4.48) Newland (1993a) proved that this expansion (5.4.48) is equivalent to (5.4.40).

5.5 The Nonuniform Wavelets

5.5

195

The Nonuniform Wavelets

The previous concepts of MRA are developed on regular lattices, that is, the translation set is always a group. Recently, Gabardo and Nashed (1998a) considered a generalization of Mallat’s celebrated theory of MRA based on spectral pairs, in which the translation set acting on the scaling function associated with the MRA to generate the subspace V0 is no longer a group but is the union of Z and a translate of Z. More precisely, this set is of the form ƒ D f0; r=Ng C 2Z, where N  1 is an integer, 1  r  2N  1; andr is an odd integer relatively prime to N. They call this a nonuniform multiresolution analysis (NUMRA). In this theory, the translation set ƒ is chosen so that for some measurable set A  with 0 < jAj < 1; .A; ƒ/ forms a spectral pair, i.e., the collection ˚ 1=2R2i! A e A .!/ 2ƒ forms an orthonormal basis for L2 .A/, where A .!/ is the characteristic 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 a measurable subset of R with 0 < jAj < 1. Then .A; ƒ/ is a spectral pair if and only if there exist an integer N  1 and an odd integer r, with 1  r  2N  1 and r and N relatively prime, such that a D r=N, and N1 X jD0

ıj=2 

X

ınN  A D 1;

(5.5.1)

n2Z

where  denotes the usual convolution product of Schwartz distributions and ıc is the Dirac measure at c. The following is the definition of NUMRA associated with the translation set ƒ on 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 is an 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 S T 2 (ii) m2Z Vm is dense in L .R/ and m2Z Vm 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 the collection f.t  / W  2 ƒg is a complete orthonormal system for V0 :

196

5 Elongations of MRA-Based Wavelets

It is worth noticing that, when N D 1, one recovers from the definition above the standard definition of a one-dimensional multiresolution analysis with dilation factor 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 and translation 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 ˚

mM1 M

Wmk ;

(5.5.3)

kD0

where all these subspaces are orthogonal. By virtue of condition (ii) in the Definition 5.5.1, this implies L2 .R/ D

M

Wm ;

(5.5.4)

m2Z

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 by elements of ƒ and dilations by the integral powers of 2N form an orthonormal basis for L2 .R/. Definition 5.5.2. A set of functions f 1 ; 1 ; : : : ; 2N1 g in L2 .R/ is said to be a set of basic wavelets associated with the NUMRA fVm W m 2 Zg if the family of functions 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 ; : : : ; 2N1 g in W0 such that .2N/m=2 ` .2N/m t   W 1  `  2N  1;  2 ƒ constitutes an orthonormal basis of Wm .˚ By means  of NUMRA, this task can be reduced to 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 the f1; g2ƒ is an orthonormal basis in V1 , we have

5.5 The Nonuniform Wavelets

.t/ D

X

197

a 1; .t/ D

2ƒ

X

 a .2N/1=2  .2N/t   ;

(5.5.5)

2ƒ

with Z a D h; 1; i D

R

.t/ 1; .t/ dt

and

X

ja j2 < 1:

(5.5.6)

2ƒ

Equation (5.5.5) can be written in frequency domain as O O .2N!/ D m O 0 .!/ .!/; where m O 0 .!/ D

P

2ƒ

(5.5.7)

a e2i! 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 V1  V0 , it follows from property (iv) of Definition 5.5.1 that for each `; 0  `  ˇ ˇ2 ˚ P 2N  1, there exists a sequence a` W  2 ƒ with 2ƒ ˇa` ˇ < 1 such that 1 2N

 t  X D a` '.t  /: ` 2N

(5.5.8)

2ƒ

Taking Fourier transform, we get O ` .2N!/ D m O O ` .!/ .!/;

(5.5.9)

where m O ` .!/ D

X

a` e2i! :

(5.5.10)

2ƒ

The functions m O ` ; 0  `  2N  1, are locally L2 functions. In view of the specific form of ƒ, we observe that m O ` .!/ D m O 1` .!/ C e2ir!=N m O 2` .!/;

0  `  2N  1;

(5.5.11)

O 2` are locally L2 ; 1=2-periodic functions. where m O 1` and m We are now in a position to establish the completeness of the system f ` .t  /g1`2N1;2ƒ in V1 , and in fact, we will find two equivalent conditions to the orthonormality of the system by means of the periodic functions m O ` as defined in (5.5.11).

198

5 Elongations of MRA-Based Wavelets

Lemma 5.5.1. Let  be a scaling function of the given NUMRA as in Definition 5.5.1. Suppose that there exist 2N  1 functions ` ; 1  `  2N  1, in V1 such that the family of functions f ` .t  /g0`2N1;2ƒ forms an orthonormal system in V1 . Then the system is complete in V1 .

Proof. By the orthonormality of domain ˝

k .t

 /;

` .t

2 L2 .R/; 0  `  2N  1, we have in the time

`

Z

˛

 / D R

k .t

 /

` .t

 / dx D ık;` ı; ;

where ;  2 ƒ and k; ` 2 f0; 1; 2; : : : ; 2N  1g. Equivalently, in the frequency domain, we have Z O k .!/ O ` .!/ e2i!. / d!: ık;` ı; D R

Taking  D 2m;  D 2n where m; n 2 Z, we have Z ık;` ım;n D

R

O k .!/ O ` .!/ e2i!2.mn/ d!

Z

e4i!.mn/

D Œ0;N/

X

O k .! C Nj/ O ` .! C Nj/ d!:

j2Z

Let hk;` .!/ D

X

O k .! C Nj/ O ` .! C Nj/:

j2Z

Then, we have Z

e4i!.mn/ hk;` .!/ d!

ık;` ım;n D Œ0;N/

2

Z

2N1 X

e4i!.mn/ 4

D Œ0;1=2/

pD0

hk;`



3 p 5 d!; !C 2

and 2N1 X pD0

 p D 2ık;` : hk;` ! C 2

(5.5.12)

5.5 The Nonuniform Wavelets

Also on taking  D Z

199

r C 2m and  D 2n, where m; n 2 Z, we have N

1

e4i!.mn/ e2i!r=N O k .!/ O ` .!/ d! Z1 X O k .! C Nj/ O ` .! C Nj/ d! e4i!.mn/ e2i!r=N D

0D

Z

Œ0;N/

D

j2Z 4i!.mn/ 2i!r=N

hk;` .!/ d! 2 3 Z 2N1   X p 5 d!: e4i!.mn/ e2i!r=N 4 eipr=N hk;` ! C D 2 Œ0;1=2/ pD0 e

e

Œ0;N/

Thus, we conclude that 2N1 X pD0

 p D 0; ˛ p hk;` ! C 2

where ˛ D eir=N :

(5.5.13)

Now we will express the conditions (5.5.12) and (5.5.13) in terms of m O ` as follows: hk;` .2N!/ D

X j2Z

D

X j2Z

D

X j2Z

      O k 2N ! C j O ` 2N ! C j 2 2         j j O j O j  !C  !C m Ok ! C m O` ! C 2 2 2 2   ˇ  ˇ  j j ˇˇ O j ˇˇ2 m Ok ! C m O` ! C  ! C 2 2 ˇ 2 ˇ

ˇ2 h i X ˇˇ  ˇ ˇO ! C j ˇ D m O 1k .!/ m O 1` .!/ C m O 2k .!/m O 2` .!/ ˇ 2 ˇ j2Z 2 C 4m O 2` .!/ O 1k .!/ m

X j2Z

2 O 2` .!/ C 4m O 2k .!/ m

X j2Z

3 ˇ  ˇ2 ˇ ˇ j ˇ 5 e2i.!Cj=2/r=N ˇˇO ! C 2 ˇ 3 ˇ  ˇ2 ˇ ˇ j ˇ5 e2i.!Cj=2/r=N ˇˇO ! C : 2 ˇ

200

5 Elongations of MRA-Based Wavelets

Therefore,   h i 2N1 X j O 1k .!/m O 1` .!/ C m O 2k .!/m O 2` .!/ h0;0 ! C hk;` .2N!/ D m 2 jD0 2 C 4m O 1k .!/m O 2` .!/ e2i!r=N

2N1 X jD0

2 C 4m O 2` .!/ e2i!r=N O 2k .!/m

3   j 5 ˛ j h0;0 ! C 2

2N1 X jD0

3   j 5 ˛ j h0;0 ! C 2

h i O 1` .!/ C m O 2k .!/m O 2` .!/ : D2 m O 1k .!/m

By using the last identity and equations (5.5.12) and (5.5.13), we obtain 2N1 X pD0

m O 1k

   p  1 p  p  2 p  2 !C Cm Ok ! C D ık;` ; m O` ! C m O` ! C 4N 4N 4N 4N (5.5.14)

and 2N1 X pD0

    p  1 p  p  2 p  Cm O 2k ! C D 0; O 1k ! C ˛p m m O` ! C m O` ! C 4N 4N 4N 4N (5.5.15)

for 0  k; `  2N  1, where ˛ D eir=N : Both of these conditions together are equivalent to the orthonormality of the system f ` .t  / W 0  `  2N  1;  2 ƒg : The completeness of this system in ˚1

.t=2N/  / W 0  `  V1 is equivalent to the completeness of the system 2N ` 2N  1;  2 ƒg in V0 . For a given arbitraryPfunction f 2 V0 , by assumption, there P exists a unique function m.!/ O of the form 2ƒ b e2i! , where 2ƒ jb j2 < O 1 such that fO .!/ D m.!/ O .!/: Therefore, in order to prove the claim, it is enough to show that the system of functions n o P D e4iN! m O ` .!/ A .!/ W 0  `  2N  1;  2 ƒ

5.5 The Nonuniform Wavelets

201

is complete in L2 .A/, where A  R with 0 < jAj < 1. Since the collection ˚ 2i! e A .!/ 2ƒ is an orthonormal basis for L2 .A/, therefore there exist locally L2 functions g1 and g2 such that h i g.!/ D g1 .!/ C e2i!r=N g2 .!/ 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 Z

e4iN! m O ` .!/ g.!/ d!

0D Z

A

i h e4iN! m O ` .! C N=2/ g.! C N=2/ d! O ` .!/ g.!/ C m

D Z

Œ0;1=2/

D Œ0;1=2/

h i e4iN! m O 1` .!/ g1 .!/ C m O 2` .!/ g2 .!/ d!:

(5.5.16)

Taking  D 2m, where m 2 Z, and defining O 1` .!/ g1 .!/ C m O 2` .!/ g2 .!/; w` .!/ D m

0  `  2N  1;

we obtain Z

e2i!.4N/m w` .!/ d!

0D Œ0;1=2/

Z D

e Œ0;1=4N/

2i!.4N/m

2N1 X jD0

  j d!: w` ! C 4N

Since this equality holds for all m 2 Z, therefore 2N1 X jD0

  j D0 w` ! C 4N

for a.e. !

Similarly, on taking  D 2m C r=N, where m 2 Z, we obtain Z

e2i!.4N/m e2i2r! w` .!/ d!

0D Œ0;1=2/

Z

e2i!.4N/m e2i2r!

D Œ0;1=4N/

2N1 X jD0

  j d!: ˛ j w` ! C 4N

(5.5.17)

202

5 Elongations of MRA-Based Wavelets

Hence, we deduce that 2N1 X jD0

  j D 0 for a.e. !; ˛ j w` ! C 4N

which proves our claim. If 0 ; 1 ; : : : ; 2N1 2 V1 are as in Lemma 5.5.1, one can obtain from them an orthonormal basis for L2 .R/ by following the standard procedure for construction of wavelets from a given MRA (see Chapter 4). It can be easily checked that for every m 2 Z, the collection n o

 F.`; m; / D `;m; .t/ W .2N/m=2 ` .2N/m t   W 0  `  2N  1;  2 ƒ (5.5.18) is a complete orthonormal system for VmC1 . Therefore, it follows immediately from (5.5.4) that the collection F.`; m; / forms a complete orthonormal system for L2 .R/. The following theorem proves the necessary and sufficient condition for the existence of associated set of wavelets to nonuniform multiresolution analysis. Theorem 5.5.1. Consider a NUMRA with associated parameters N and r as in Definition 5.5.1, such that the corresponding space V0 has an orthonormal system of the form f.t  / W  2 ƒg, where ƒ D f0; r=Ng C 2Z and O satisfies the twoscale relation O O .2N!/ D m O 0 .!/ .!/;

(5.5.19)

m O 0 .!/ D m O 10 .!/ C e2i!r=N m O 20 .!/;

(5.5.20)

where m O 0 is of the form

O 10 and m O 20 . Define M0 as for some locally L2 functions m ˇ 1 ˇ2 ˇ 2 ˇ2 O 0 .!/ˇ C ˇm O 0 .!/ˇ : M0 .!/ D ˇm

(5.5.21)

Then a necessary and sufficient condition for the existence of associated wavelets 1 ; : : : ; 2N1 is that M0 satisfies the identity   1 D M0 .!/: M0 ! C 4

(5.5.22)

5.5 The Nonuniform Wavelets

203

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 of Lemma 5.5.1: 2N1 X ˇ

  p ˇˇ2 ˇˇ 2  p ˇˇ2 ˇ 1 O0 ! C O0 ! C ˇm ˇ C ˇm ˇ D 1; 4N 4N

(5.5.23)

ˇ   p ˇˇ2 ˇˇ 2  p ˇˇ2 ˇ 1 O0 ! C O0 ! C ˛ p ˇm ˇ C ˇm ˇ D 0; 4N 4N

(5.5.24)

pD0

and 2N1 X pD0

where ˛ D eir=N . Similarly, if f ` g`D1;:::;2N1 is a set of wavelets associated with the given NUMRA, then it satisfies the relation (5.5.9), and the orthonormality of the collection f ` g`D0;1;:::;2N1 in V1 is equivalent to the identities 2N1 X pD0

m O 1k

   p  1 p  p  2 p  2 !C Cm Ok ! C D ık;` ; m O` ! C m O` ! C 4N 4N 4N 4N (5.5.25)

and 2N1 X pD0

    p  1 p  p  2 p  ˛p m m O` ! C m O` ! C Cm O 2k ! C O 1k ! C D 0; 4N 4N 4N 4N (5.5.26)

for 0  k; `  2N  1.   p  p  O 1` ! C O 2` ! C ; b` .p/ D m If ! 2 Œ0; 1=4N is fixed and a` .p/ D m 4N 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    p  .p C N/ D M0 ! C ; M0 ! C 4N 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 to Gabardo and Nashed (1998b).

204

5 Elongations of MRA-Based Wavelets

We note here that the function M0 in the above theorem can also be written in terms of the filter m O 0 as hˇ

i ˇ2 ˇm O 0 ! C N2 ˇ C jm O 0 .!/j2 : M0 .!/ D 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 condition from wavelet analysis jm O 0 .! C 1=2/j2 C jm O 0 .!/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 not imply 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, and choosing  D AN ; where AN D

N1 [ mD0



 2m 2m C 1 ; ; N N

we have  D Œ0;1=N/ 

N1 X

ı2m=N :

mD0

We now define V0 as the closed linear span of f.t  /g2ƒ , i.e., V0 Dspan f.t/ W  2 ƒg and Vj , for each integer m, by the relation f .t/ 2 Vm if and only if f .t=.2N/m / 2 V0 . Then, the condition (i) of the Definition 5.5.1 is verified by fact that N1 1  t  1 X  D Œ0;2/  ı4m 2N 2N 2N mD0 N1 

1 X D ı0 C ı1=N    ı4m : 2N mD0

(5.5.27)

Equation (5.5.26) can be written in the frequency domain as O O .2N!/ Dm O 0 .!/ .!/;

(5.5.28)

5.5 The Nonuniform Wavelets

205

where "N1 # X  1

1 C e2i!=N m O 0 .!/ D e8i!k : 2N kD0 Furthermore, we have m O 10 .!/ D m O 20 .!/ D

N1 1 X 8i!k e : 2N kD0

(5.5.29)

Here, both the functions m O 10 and m O 20 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 m O 10 .!/ D m O 20 .!/ D 1=2, then the corresponding wavelet 1 is given by the identity 2i! 1 O O 1 .2!/ D e .!/: 2

Or, equivalently, 1

D Œ0;1=2/ C Œ1=2;1/ ;

O 20 which is the classical Haar wavelet. For N D 2, the periodic functions m O 10 and m are given by m O 10 .!/ D m O 20 .!/ D

e4i! cos.4!/ ; 2

and thus M0 .!/ D cos2 .4i!/=2. In this case, the associated wavelets can easily O be computed using the relation O ` .4!/ D m O ` .!/ .!/; ` D 1; 2; 3. Therefore, we have 1 2

3

D Œ0;1=2/  Œ1;3=2/ ; D Œ8=8;7=8/ C Œ7=8;6=8/  Œ6=8;5=8/ C Œ5=8;4=8/ Œ0;1=8/ C Œ1=8;2=8/  Œ2=8;3=8/ C Œ3=8;4=8/ ; D Œ8=8;7=8/ C Œ7=8;6=8/  Œ6=8;5=8/ C Œ5=8;4=8/ CŒ0;1=8/  Œ1=8;2=8/ C Œ2=8;3=8/  Œ3=5;4=8/ :

206

5 Elongations of MRA-Based Wavelets

5.6

Exercises

1. Show that the nth-order B-spline Nn .t/ and its integer translates form a partition of unity, that is, X

Nn .t  k/ D 1;

for all t 2 R:

k2Z

2. Show that cardinal B-spline Nn .t/ is symmetric about t D n=2, that is, Nn

  n C t D Nn t ; 2 2

n

for all t 2 R:

3. Use induction on n to prove that h

tNn1 .t/ C .n  t/Nn1 .t  1/

Nn .t/ D

i

n1

:

4. Show that the two-scale equation associated with the linear spline function N1 .t/ D

1  jtj; 0 < jtj < 1 0; otherwise

is N1 .t/ D

1 1 N1 .2t C 1/ C N1 .2t/ C N1 .2t  1/: 2 2

Hence, show that ˇ2   X ˇˇ O C 2k/ˇˇ D 1  2 sin2 ! : ˇ.! 3 2 k2Z

5. Using result (5.2.10), prove that n  1 C ei!=2 NO n .!/ !  D : 2 NO n 2 Hence, derive the following: !  n ik! O  !  1 X n O (a) Nn .!/ D n exp  ; Nn 2 kD0 k 2 2

5.6 Exercises

207

! n 1 X n Nn .2t  k/: (b) Nn .t/ D n1 k 2 kD0 6. Use the Fourier transform formula (5.2.27) for O .!/ of the Franklin wavelet to show that satisfies the following properties: Z 1 (a) O .0/ D .t/ dt D 0; Z

1 1

t .t/ dt D 0;

(b) 1

1 (c) is symmetric with respect to t D  : 2 7. From an expression (5.2.26) for the filter, show that

m.!/ O D

2 C 3 cos ! C cos2 !

 1 C 2 cos2 !



and, hence, deduce  2  O .2!/ D exp.i!/ 2  cos ! C cos ! .!/: O 1 C 2 cos2 ! 8. Obtain a solution of (5.3.22) for the following cases: 1 (a) c0 D c1 D p ; c2 D c3 D 0; 2 1 1 (b) c0 D c2 D p ; c1 D p ; c3 D 0; 2 p 2 2 (c) c0 D 2; 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 Daubechies scaling function (5.3.50) and the Daubechies D6 wavelet (5.3.51). 10. Prove the following results (Newland, 1993a): Z 1

m  n  (a) 2 tk 2 t  ` dt D 0 for all m; k; n; ` .m; n  0/: Z1 1

m  n  (b) 2 tk 2 t  ` dt D 0 for all m; k; n; ` .m; n  0/: 1

(c) When m D n and k D `, the above result 1(b) becomes Z

1

ˇ m ˇ ˇ 2 t  k ˇ2 dt D 2m :

1

Z

1

.t  m/ .t  n/ dt D 0

(d) 1

for all m; n; m ¤ n:

208

5 Elongations of MRA-Based Wavelets

Z

1

(e) Z1 1 (f)





 2m t  k  t  ` dt D 0 for all m; k; ` .m  0/:





 2m t  k  t  ` dt D 0 for all m; k; ` .m  0/:

1

11. Show that (Newland, Z 1 1993a) O (a) a;k D 2 ei!k d!: fO .!/ .!/ (b)

1 X

1



 a;k  t  k D

kD1 1 Z X

D 2

Z

1

kD1 1

Z

1

O 1 / .! O 2 / ei!2 t ei.!1 !2 /k d!2 fO .!1 / .!

d!1 1

2

fO .!/ ei!t d!:

D 0

12. Prove that (Newland, Z 1 1993a) m (a) am;k D 2 fO .!/ O .!2m / ei!k2 d!: Z1 1 m fO .!/ O .!2m / ei!k2 d!: (b) aQ m;k D 2 1

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