Orthogonal Transforms for Digital Signal Processing

273 Pages·1975·8.141 MB·English

by  Nasir Ahmed, Kamisetty Ramamohan Rao (auth.)

Orthogonal Transforms for Digital Signal Processing
N. Ahmed . 1(. R. Rao
Orthogonal Transforms
for Digital
Signal Processing
Springer-Verlag
Berlin· Heidelberg· New York 1975
Nasir Ahmed
Associate Professor
Department of Electrical Engineering,
Kansas State University, Manhattan, Kansas
Kamisetty Ramamohan Rao
Professor
Department of Electrical Engineering,
University of Texas at Arlington, Arlington, Texas
With 129 Figures
ISBN-13: 978-3-642-45452-3 e-ISBN-13: 978-3-642-45450-9
001: 10.1007/978-3-642-45450-9
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is coucerned, specifically those of translation, repriuting, re-use of illustrations, broadcasting, repro
duction by photocopying machine or simllar means, and storage in data banks.
Under § 54 of the German Copyright Law, where copies are made for other than private use, a fee is
payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin· Heidelberg 1975
Softcover reprint of the hardcover 1st edition 1975
Library of Congress Catalog Card Number 73-18912.
To the memory of
my mother and grandmother
N.Ahmed
Preface
This book is intended for those wishing to acquire a working knowledge of
orthogonal transforms in the area of digital signal processing. The authors
hope that their introduction will enhance the opportunities for interdiscipli
nary work in this field.
The book consists of ten chapters. The first seven chapters are devoted
to the study of the background, motivation and development of orthogonal
transforms, the prerequisites for which are a basic knowledge of Fourier
series transform (e.g., via a course in differential equations) and matrix al
gebra. The last three chapters are relatively specialized in that they are di
rected toward certain applications of orthogonal transforms in digital signal
processing. As such, a knowlegde of discrete probability theory is an essential
additional prerequisite. A basic knowledge of communication theory would
be helpful, although not essential.
Much of the material presented here has evolved from graduate level
courses offered by the Departments of Electrical Engineering at Kansas State
University and the University of Texas at Arlington, during the past five
years. With advanced graduate students, all the material was covered in one
semester. In the case of first year graduate students, the material in the first
seven chapters was covered in one semester. This was followed by a prob
lems project-oriented course directed toward specific applications, using
the material in the last three chapters as a basis.
N. lilIMED • K. R. RAO
Acknowledgements
Since this book has evolved from course material, the authors are grateful for
the interest and cooperation of their graduate students, and in particular to
Mr. T. Natarajan, Department of Electrical Engineering, Kansas State Uni
versity. The authors thank the Departments of Electrical Engineering at
Kansas State University and the University of Texas at Arlington for their
encouragement. In this connection, the authors are grateful to Dr. W. W.
Koepsel, Chairman of the Department of Electrical Engineering, Kansas
State University. Special thanks are due to Dr. A. E. Salis, Dean, and
Dr. R. L. Tucker, Associate Dean, College of Engineering, and Dr. F. L. Cash,
Chairman, Department of Electrical Engineering, University of Texas at Ar
lington, for providing financial, technical, and secretarial support throughout
the development of this book. Thanks are also due to Ms. Dorothy Bridges,
Ann Woolf, Marsha Pierce, Linda Dusch, Eugenie Joe, Eva Hooper, Dana
Kays, Kay Morrison, and Sharon Malden for typing various portions of the
manuscript.
Finally, personal notes of gratitude go to our wives, Esther and Karuna,
without whose encouragement, patience, and understanding this book could
not have been written.
N. AHMED· K. R. RAO
Contents
Chapter One
Introduction
1.1 General Remarks . . . . . . . . . . . . . . . 1
1.2 Terminology . . . . . . . . . . . . . . . . . 1
1.3 Signal Representation Using Orthogonal Functions. 2
1.4 Book Outline 4
References 5
Problems .. 7
Chapter Two
Fourier Representation of Signals
2.1 Fourier Representation. . . . . 9
2.2 Power, Amplitude, and Phase Spectra 12
2.3 Fourier Transform. . . . . . . . . 16
2.4 Relation Between the Fourier Series and the Fourier Transform . 18
2.5 Crosscorrelation, Autocorrelation, and Convolution. 19
2.6 Sampling Theorem. 25
2.7 Summary. 28
References 29
Problems. 29
Chapter Three
Fourier Representation of Sequences
3.1 Definition of the Discrete Fourier Transform 31
3.2 Properties of the DFT . . . . . . . . . . 32
3.3 Matrix Representation of Correlation and Convolution 37
3.4 Relation Between the DFT and the Fourier Transform Series 39
3.5 Power, Amplitude, and Phase Spectra 41
3.6 2-dimensional DFT . . . . . 42
3.7 Time-varying Fourier Spectra. 44
3.8 Summary. . 49
Appendix 3.1 50
References 51
Problems .. 52
X Contents
Chapter Four
Fast Fourier Transform
4.1 Statement of the Problem. 54
4.2 Motivation to Search for an Algorithm 54
4.3 Key to Developing the Algorithm 56
4.4 Development of the Algorithm 57
4.5 Illustrative Examples . . . . . 64
4.6 Shuffling. . . . . . . . . . . 69
4.7 Operations Count and Storage Requirements 70
4.8 Some Applications. . . . . . . . . . . 71
4.9 Summary. . . . . . . . . . . . . . . 78
Appendix 4.1 An FFT Computer Program 78
References 78
Problems .............. . 81
Chapter Five
A Class of Orthogonal Functions
5.1 Definition of Sequency . . . . . 85
5.2 Notation .......... . 86
5.3 Rademacher and Haar Functions 87
5.4 Walsh Functions ....... . 89
5.5 Summary. . . . . . . . . . . 94
Appendix 5.1 Elements of the Gray Code 94
References 95
Problems ............. . 96
Chapter Six
Walsh-Hadamard Transform
6.1 Walsh Series Representation ............... . 99
6.2 Hadamard Ordered Walsh-Hadamard Transform (WHT)h .. . 102
6.3 Fast Hadamard Ordered Walsh·Hadamard Transform (FWHT)h 105
6.4 Walsh Ordered Walsh·Hadamard Transform (WHT)w ... 109
6.5 Fast Walsh Ordered Walsh·Hadamard Transform (FWHT)w 111
6.6 Cyclic and Dyadic Shifts. 115
6.7 (WHT)w Spectra ................. . 117
6.8 (WHT)h Spectra . . . . . . . . . . . . . . . . . . 119
6.9 Physical Interpretations for the (WHT)h Power Spectrum 123
6.10 Modified Walsh-Hadamard Transform (MWHT). 131
6.11 Cyclic and Dyadic Correlation/Convolution. 137
6.12 Multidimensional (WHT)h and (WHT)w . 139
6.13 Summary . . . . . . . . . . . . . . 141
Appendix 6.1 WHT Computer Program. 141
References. 143
Problems ............. . 145
Contents XI
Chapter Seven
Miscellaneous Orthogonal Transforms
7.1 Matrix Factorization. 153
7.2 Generalized Transform 156
7.3 Haar Transform ... 160
7.4 .Algorithms to Compute the HT 160
7.5 Slant Matrices. . . . . . . . 163
7.6 Definition of the Slant Transform (ST) 167
7.7 Discrete Cosine Transform (DCT) . . 169
7.8 2-dimensional Transform Considerations 171
7.9 Summary ........... . 172
.Appendix 7.1 Kronecker Products. 172
.Appendix 7.2 Matrix Factorization. 173
References 175
Problems ........... . 176
Chapter Eight
Generalized Wiener Filtering
8.1 Some Basic Matrix Operations. 180
8.2 Mathematical Model . . . . 181
8.3 Filter Design . . . . . . . 183
8.4 Suboptimal Wiener Filtering 186
8.5 Optimal Diagonal Filters . . 189
8.6 Suboptimal Diagonal Filters. 191
8.7 2-dimensional Wiener Filtering Considerations. 194
8.8 Summary. . . . . . . . . . . . . . . . . 194
.Appendix 8.1 Some Terminology and Definitions. 195
References 197
Problems ................. . 197
Chapter Nine
Data Compression
9.1 Search for the Optimum Transform . . . . . . 200
9.2 Variance Criterion and the Variance Distribution 203
9.3 Electrocardiographic Data Compression 205
9.4 Image Data Compression Considerations 211
9.5 Image Data Compression Examples 213
9.6 .Additional Considerations. . . . . 219
9.7 Summary ........... . 220
.Appendix 9.1 Lagrange Multipliers 221
References 222
Problems ........... . 222
XII Contents
Chapter Ten
Feature Selection in Pattern Recognition
10.1 Introduction. . . . . . 225
10.2 The Concept of Training. . . . 226
10.3 d·Dimensional Patterns . . . . 228
10.4 The 3·Class Problem . . . . . 229
10.5 Image Classification Experiment 232
10.6 Least.Squares Mapping Technique 235
10.7 Augmented Feature Space .... 237
10.8 3·Class Least·Squares Minimum. Distance Classifier 238
10.9 K·Class Least·Squares Minimum. Distance Classifier . 245
10.10 Quadratic Classifiers. . . . . . . 248
10.11 An ECG Classification Experiment 249
10.12 Summary . 253
References . 253
Problems 254
Author Index 259
Subject Index . 261