Time-Frequency Representations

288 Pages·1996·9.01 MB·English

by  Richard Tolimieri

Applied and Numerical Harmonic Analysis
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University of Maryland
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Washington University
Tillle-Frequency
Representations
Richard Tolimieri
and
MyoungAn
1998
Birkhiiuser
Boston • Basel • Berlin
Richard Tolimieri MyoungAn
Department of Electrical Engineering A. J. Devaney Associates
City College of New York Boston, Massachussets 02115
New York, New York 10037 USA
USA
Library of Congress Cataloging-in-PubJication Data
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Printed on acid-free paper i5
© 1998 Birkhauser Boston Birkhiiuser
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ISBN-13: 978-1-4612-8676-9 e-ISBN-13: 978-1-4612-4152-2
DOl: 10.1007/978-1-4612-4152-2
Camera-ready text prepared by the Author in TEX.
Printed and bound by Hamilton Printing Company, Renn
9 8 7 6 5 4 3 2 I
Preface
The aim of this work is to present several topics in time-frequency analysis
as subjects in abelian group theory. The algebraic point of view pre
dominates as questions of convergence are not considered. Our approach
emphasizes the unifying role played by group structures on the development
of theory and algorithms.
This book consists of two main parts. The first treats Weyl-Heisenberg
representations over finite abelian groups and the second deals with mul
tirate filter structures over free abelian groups of finite rank. In both,
the methods are dimensionless and coordinate-free and apply to one and
multidimensional problems.
The selection of topics is not motivated by mathematical necessity but
rather by simplicity. We could have developed Weyl-Heisenberg theory over
free abelian groups of finite rank or more generally developed both topics
over locally compact abelian groups. However, except for having to dis
cuss conditions for convergence, Haar measures, and other standard topics
from analysis the underlying structures would essentially be the same. A re
cent collection of papers [17] provides an excellent review of time-frequency
analysis over locally compact abelian groups.
A further reason for limiting the scope of generality is that our results
can be immediately applied to the design of algorithms and codes for time
frequency processing.
Chapter 1 presents an overview of abelian group theory. Groups, sub
groups and homomorphisms are defined. Special emphasis is placed on the
concept of coset-decompositions. Such decompositions underlie divide-and
conquer algorithms for computing the finite Fourier transform. They play
an equally important role in extending critical sampling results to inte
ger and rational over-sampling results in Weyl-Heisenberg theory and in
defining polyphase decompositions in multirate filter theory.
Two fundamental theorems are proved in Chapter 1: the structure theo
rem which characterizes finite abelian groups as the direct product of cyclic
groups, and the basis theorem which describes the relationship between
VI
subgroups of free abelian groups of finite rank and is used to characterize
quotients of such groups.
Chapter 2 presents some elementary definitions and results from linear
algebra. Much of the notation used in the text is introduced here.
Chapters 3 and 4 review Fourier analysis over finite abelian groups. The
character group is defined in Chapter 3. Character formulas, generalizing
the classical result that the sum of the complex N -th roots of unity van
ishes, are proved and used in Chapter 4 to prove the Poisson summation
formula, perhaps the most important result in Fourier theory. The Poisson
summation formula is a powerful tool for structuring results throughout
the first part.
Chapter 5 introduces the Zak transform over subgroups of finite abelian
groups. The Zak transform maps signal space onto joint time-frequency
Zak space. Most results and algorithms are described in Zak space.
In Chapter 6, we define Weyl-Heisenberg systems and discuss signal
expansions over these systems. The rest of the first part discusses prop
erties of such systems and develops several approaches for computing
Weyl-Heisenberg expansions.
In Chapter 7, the Zak transform is applied to the study of Weyl
Heisenberg systems. Two fundamental formulas are derived. These formulas
are used to characterize Weyl-Heisenberg systems in Zak space. Zak space
representation is the basis of the algorithms designed in Chapter 8 for
computing Weyl-Heisenberg expansions.
Chapter 9 is devoted to the orthogonal projection theorem which
describes an algorithm for orthogonal projection of Weyl-Heisenberg ex
pansions. This result is based on periodization in Zak space and is the
analogue of orthogonal projection of Fourier expansions by periodization in
signal space. The orthogonal projection theorem is the basis of an iterative
algorithm for computing Weyl-Heisenberg expansions.
A second approach to computing Weyl-Heisenberg expansions is based
on the construction of biorthogonals or dual functions to Weyl-Heisenberg
systems.
In Chapter 13, algorithms are derived for computing the Raz-Wexler
biorthogonal or dual function.
In Chapter 14, the Raz-Wexler dual function is compared with dual
functions derived from the frame operator, using the powerful formulas
developed in Chapter 11 and the orthonormal Weyl-Heisenberg systems
constructed in Chapter 12.
In Chapter 15, we describe the representation of algorithms in terms of
the tensor product algebra and develop representations for the essential
data readdressing.
Chapters 16 and 17 develop multirate filter structures over free abelian
groups of finite rank. The algebra is developed first and applied in Chapter
17 to polyphase representations and reduction theorems. The unifying con
cept of a short exact sequence is introduced to structure the main results.
vii
Two theories are considered, a matrix-free version that relates decimation
operations to coset-decompositions and a version which connects decimator
and expander matrix related operators to such coset-decompositions. The
basis theorem for subgroups of free abelian groups plays a significant role
throughout these chapters.
An application to financial modeling written by Professors Sudeshna
Adak and Abhinanda Sarkar is given in Chapter 18.
This book is a synthesis of results derived during the last ten years by
many researchers from several disciplines. The new part of this effort is
the unification of these results into a theory of algorithms based on several
classical concepts from finite abelian group theory.
We close each chapter with a discussion of relevant works, but as there
has been substantial overlapping of efforts, our list is by no means complete.
Indeed, we suggest that the reader consult those works of interest to find
how researchers viewed the main influence to their work.
We wish to thank AFOSR for its support in giving us the time in which
ideas in this book have been developed and motivated by applications to
electromagnetics, multispectral imaging, and imaging through turbulence.
Richard Tolimieri
Myoung An
Contents
Preface v
1 Review of algebra 1
1.1 Introduction ............ . 1
1.2 Definitions and examples of groups 1
1.3 Subgroups, cosets, and quotients . 5
1.4 Ideals ............... . 8
1.5 Mappings.............. 8
1.6 Finitely generated abelian groups 11
1.6.1 Cyclic groups .... 12
1.6.2 Free abelian groups 15
References 18
2 Linear algebra and abelian groups 19
2.1 Introduction ... 19
2.2 Vector space L(A) 19
3 Fourier transform over A 25
3.1 Introduction .... 25
3.2 Character groups .. 26
3.3 Character formulas . 29
3.4 Duality theory ... 31
3.5 Character group basis 36
3.6 Fourier transform .. 37
3.7 Shift and multiplication operators 42
References 45
Problems ................. . 45
4 Poisson summation formula 47
4.1 Introduction .............. . 47
4.2 Statement and proof .......... . 47
4.3 Fourier transform of periodic functions 53
x
4.4 Periodization-decimation operators 54
References 55
Problems ... 55
5 Zak transform 57
5.1 Introduction ....... . 57
5.2 Fourier analysis on A x A* 57
5.3 Zak transform ...... . 60
5.4 Functional equation ... . 65
5.5 Fourier and Zak transform 70
5.6 Isometry .......... . 72
5.7 Algorithm for computing Zak transform 73
References 74
Problems ......... . 74
6 Weyl-Heisenberg systems 77
6.1 Introduction 77
6.2 Translates ... 77
6.3 W-H systems . 82
6.4 Sampling rates 86
6.5 Divide-and-conquer 89
References 91
Problems ......... . 91
7 Zak transform and W-H systems 93
7.1 Introduction ..... . 93
7.2 Basic results ....... . 94
7.3 Fundamental formulas . . . 96
7.4 Zak space characterization of W-H systems. 100
7.4.1 Critical sampling subgroup ... 100
7.4.2 Integer over-sampling subgroup 101
7.4.3 General sampling subgroup. 101
7.5 Zero set characterization ....... . 103
7.5.1 Critical sampling subgroup .. . 104
7.5.2 Integer over-sampling subgroup 105
Problems. 112
8 Algoritluns for W -H systems 117
8.1 Introduction ..... . 117
8.2 Critical sampling algorithm ... 117
8.3 Integer over-sampling algorithm 120
8.3.1 Reducing the problem .. 123
8.4 General over-sampling algorithm. 125
8.4.1 Reducing the problem. 129
References ................ . 131
xi
Problems ............ . 131
9 Orthogonal projection theorem 135
9.1 Introduction .............. . 135
9.2 Orthogonal projection algorithm ... . 135
9.3 Iterative W-H coefficient set algorithm 139
10 Cross-ambiguity f1lllction 141
10.1 Introduction .. 141
10.2 Basic results ... 142
10.3 Direct algorithm . 145
10.4 Critical sampling algorithm. 146
10.5 Integer over-sampling algorithm 147
10.6 General divide-and-conquer algorithm. 148
References 149
Problems ..... . 150
11 Ambiguity surfaces 151
11.1 Introduction ............... . 151
11.2 Fourier transform of ambiguity surfaces. 151
11.3 Formulas D1 and D2 . 152
References ......... . 153
12 Orthonormal W-H systems 155
12.1 Introduction ...... . 155
12.2 Orthonormal W-H systems ... . 156
12.2.1 Critical sampling subgroup 156
12.2.2 Integer over-sampling subgroup 157
12.2.3 Over-sampling subgroup Ll . 160
References 167
Problems 168
13 Duality 169
13.1 Introduction ............... . 169
13.2 Biorthogonal ............... . 169
13.3 Algorithms for computing biorthogonals 174
13.3.1 Ll-periodization ........ . 174
13.3.2 Critical sampling subgroup ... . 175
13.3.3 Integer over-sampling subgroup . 176
13.3.4 General over-sampling subgroup. 179
References 184
Problems 184
14 Frames 187
14.1 Introduction 187