Table Of ContentTIME-FREQUENCY ANALYSIS
AND SYNTHESIS OF
LINEAR SIGNAL SPACES
Time-Frequency Filters,
Signal Detection and Estimation,
and Range-Doppler Estimation
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
TIME-FREQUENCY ANALYSIS
AND SYNTHESIS OF
LINEAR SIGNAL SPACES
Time-Frequency Filters,
Signal Detection and Estimation,
and Range-Doppler Estimation
by
Franz Hlawatsch
Vienna University of Technology
111....
"
Springer-Science+Business Media, B.V.
Library of Congress Cataloging-in-Publication Data
A c.I.P. Catalogue record for this book is available
from the Library of Congress.
ISBN 978-1-4419-5032-1 ISBN 978-1-4757-2815-6 (eBook)
DOI 10.1007/978-1-4757-2815-6
Copyright © 1998 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998.
Softcover reprint of the hardcover 1s t edition 1998
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer-Science+Business Media, B.Y.
Printed on acid-free paper.
Contents
Preface IX
Acknowledgments XIII
1. INTRODUCTION AND OUTLINE 1
1.1 Linear Signal Spaces 2
1.2 Quadratic TF Signal Representations 4
1.3 Outline and Summary of Results 6
2. THE WIGNER DISTRIBUTION OF A LINEAR SIGNAL SPACE 9
2.1 Definitions and Expressions 10
2.1.1 Deterministic Definition 10
2.1.2 Stochastic Definition 11
2.1.3 Expression in Terms of the Projection Operator 12
2.1.4 Expression in Terms of a Basis 13
2.2 Quadratic Space Representations 13
2.2.1 Definitions and Expressions 13
2.2.2 Examples 16
2.3 Properties 17
2.4 Some Signal Spaces 23
2.5 Energetic Interpretation 29
2.6 Extensions 33
2.6.1 Cross-Wigner Distribution 34
2.6.2 Discrete-Time Wigner Distribution 36
2.7 Minimization/Maximization of TF Integrals 38
2.7.1 Unconstrained Minimization/Maximization 40
2.7.2 Subspace-Constrained Minimization/Maximization 42
Appendix 2.A: Proof of Theorem 2.1 45
Appendix 2.B: Proof of Theorem 2.2 46
vi TIME-FREQUENCY ANALYSIS AND SYNTHESIS OF SIGNAL SPACES
Appendix 2.C: Proof of Theorem 2.3 47
3. TIME-FREQUENCY LOCALIZATION OF LINEAR SIGNAL SPACES 49
3.1 Shape of the Wigner Distribution 50
3.1.1 Local Averages and Height Bounds 50
3.1.2 Sophisticated Spaces 52
3.2 TF Disjointness and TF Affiliation 55
3.2.1 Affiliation and Orthogonality 55
3.2.2 Orthogonality and TF Disjointness 57
3.2.3 Affiliation and TF Affiliation 61
3.3 Uncertainty Relations 63
3.3.1 TF Moments 63
3.3.2 Uncertainty Relations 64
3.4 Uncertainty Relation for Band-limited Spaces 66
3.4.1 Temporal Concentration 66
3.4.2 Uncertainty Relation for Band-limited Spaces 67
3.5 Localization in a TF Region 69
3.5.1 Regional TF Concentration 69
3.5.2 Time-Frequency Localization Error 71
3.5.3 Affiliation Inequality 72
Appendix 3.A: Proof of Theorem 3.1 74
Appendix 3.B: Proof of Theorem 3.2 77
4. TIME-FREQUENCY SYNTHESIS OF LINEAR SIGNAL SPACES 79
4.1 Optimum TF Synthesis 80
4.1.1 Optimality Criteria 80
4.1.2 Eigenspaces of a TF Region 81
4.2 Properties of Eigenspaces 84
4.2.1 Concentration Bounds 85
4.2.2 Asymptotic Properties 86
4.2.3 Covariance Property 87
4.2.4 Complement Property 89
4.3 Subspace-Constrained Optimum TF Synthesis 91
4.4 Discrete-Time Synthesis Algorithms 94
4.4.1 The Discrete-Time Synthesis Problem 94
4.4.2 Basis Method 95
4.4.3 Basis-Free Method 97
4.5 Simulation Results 98
4.6 Hermite and Prolate Spheroidal Spaces Reconsidered 98
4.6.1 Hermite Spaces 98
4.6.2 Prolate Spheroidal Spaces 102
4.7 Extension to Other Space Representations 102
Contents vii
Appendix 4.A: Proof of Theorem 4.7 103
Appendix 4.B: Proof of Theorem 4.8 104
5. TIME-FREQUENCY FILTERS AND EXPANSIONS 105
5.1 Motivation and Background 106
5.1.1 A Classical Example and Its Generalization 107
5.1.2 TF Filters 108
5.1.3 Related Work 108
5.2 TF Projections 109
5.2.1 TF Projection Filters 109
5.2.2 TF Signal Expansions 110
5.2.3 Pass/Reject Analysis 111
5.2.4 Noise Analysis 111
5.2.5 Simulation Results 112
5.3 TF Filter Banks 116
5.3.1 Orthogonal Space Partitions 116
5.3.2 Sequential Partitioning 118
5.3.3 Binary Partitioning 119
5.3.4 Simulation Results 121
5.4 Discussion 122
6. SIGNAL ESTIMATION AND SIGNAL DETECTION 125
6.1 Signal Estimation 126
6.1.1 Optimum Projection Filter for Signal Estimation 126
6.1.2 Two Special Cases 130
6.1.3 TF Design 132
6.1.4 Simulation Results 134
6.1.5 Estimation of TF Localized Signals 136
6.2 Signal Detection 138
6.2.1 Optimum Projection Filter for Signal Detection 139
6.2.2 Two Special Cases 142
6.2.3 TF Design 144
6.2.4 Simulation Results 145
6.2.5 Detection of TF Localized Signals 145
Appendix 6.A: Proof of Theorem 6.1 148
Appendix 6.B: Proof of Theorem 6.2 150
7. THE AMBIGUITY FUNCTION OF A LINEAR SIGNAL SPACE 153
7.1 Definitions and Expressions 154
7.2 Properties 155
7.3 Some Signal Spaces 160
7.4 Interpretations 165
7.4.1 Deterministic Interpretation 165
viii TIME-FREQUENCY ANALYSIS AND SYNTHESIS OF SIGNAL SPACES
7.4.2 Stochastic Interpretation 167
7.5 Extensions 169
7.5.1 Cross-Ambiguity Function 169
7.5.2 Discrete-Time Ambiguity Function 170
8. RANGE-DOPPLER ESTIMATION 173
8.1 The Single-Pulse Estimator and Its Performance 174
8.1.1 Local Accuracy 175
8.1.2 Global Accuracy 176
8.1.3 The Radar Uncertainty Principle 177
8.2 The Multipulse Estimator and Its Performance 177
8.2.1 The Multipulse Estimator 177
8.2.2 Local Accuracy 179
8.2.3 Global Accuracy 180
8.2.4 Volume Bounds 181
8.3 Optimum Pulse Sets and the AF of the Pulse Space 182
8.3.1 The Case of Orthogonal. Equal-Energy Pulses 182
8.3.2 Hermite Spaces and Prolate Spheroidal Spaces Reconsidered 183
Appendix 8.A: Proof of Theorem 8.1 185
Appendix 8.B: Proof of Theorem 8.2 189
Appendix 8.C: Proof of Theorem 8.3 193
9. CONCLUSIONS 195
References 199
Preface
It is now nearly 20 years ago that the Wigner distribution (also known as
Wigner-Ville distribution) was given a prominent place in the signal process
ing community's spectrum of research interests. Recognition was given almost
immediately for the high potential of the Wigner distribution as a tool for dis
playing and analyzing signal characteristics in the time-frequency plane that are
observable only indirectly from the temporal or spectral description of the sig
nal. This assessment was soundly supported by the long list of desirable math
ematical properties (such as correct marginal and support properties) satisfied
by the Wigner distribution. Moreover, unlike, for instance, the spectrogram,
the resolution of the Wigner distribution is not limited by the temporal or
spectral properties of a window, simply because such a window is not required
in the definition of the Wigner distribution.
On the other hand, due to its high resolution and its quadratic nature, the
Wigner distribution locally exhibits oscillations between positive and negative
values to an extent that smoothing, in a certain sense in accordance with the
Heisenberg uncertainty principle, is necessary to make it non-negative every
where. As a consequence, a pointwise interpretation of the Wigner distribution
as a true probability density function is not possible. This circumstance has
kept the discussions in the signal processing community quite animated, yield
ing all sorts of modified Wigner distributions through the design of smoothing
kernels and distributions involving the signals in a non-quadratic way. Thus,
to a large extent, the interest and research effort of the community has been
directed at redefining and re-interpreting the Wigner distribution, with the
purpose of obtaining a clear and useful display and analysis tool.
Besides being a useful display and analysis tool, the Wigner distribution
can be of great help in the design of time-frequency filtering methods. In this
book such methods are designed for the enhancement, decomposition, estima-
x TIME-FREQUENCY ANALYSIS AND SYNTHESIS OF SIGNAL SPACES
tion and detection of noisy deterministic and stochastic signals. These methods
are based on the assumption that there exists information about the regions
in the time-frequency plane where the signals and noise manifest themselves.
The central idea here is that the linear operators involved in the solutions of
many signal processing problems can be compressed to finite-dimensional lin
ear spaces, the dimensions of which are approximately equal to the area of
the region in the time-frequency plane to which the signals are supposed to be
restricted. As a consequence, a major obstruction, viz. the composition and
inversion of these operators, is removed since the linear spaces on which the op
erators have to be considered are finite dimensional. Besides that, formulation
of the available information in terms of regions in the time-frequency plane has
the advantage of yielding a direct interpretation of the effect of adding or delet
ing information and of changing parameters in the signal processing methods.
In a sense, the information and the signal processing tasks are brought to life
in the time-frequency plane.
In this approach to designing signal processing methods, the concept of the
Wigner distribution of a linear signal space is a crucial notion. This concept
can be defined simply as the sum of all Wigner distributions of the elements of
an arbitrary orthonormal base of the linear space. The key step then consists
of finding for a given region in the time-frequency plane that linear space whose
Wigner distribution yields the best approximation to the indicator function of
the considered region. It turns out that this amounts to rounding to 0 or 1 the
eigenvalues of the linear operator that is associated with the indicator function
via Weyl's correspondence principle.
The idea of defining the Wigner distribution of a linear space can be extended
to more general quadratic signal representations than the Wigner distribution.
This is worked out in this book for the ambiguity function, which is well-known
in radar analysis. The sort of applications of the concept obtained in this way
is quite different from the ones earlier mentioned, since Wigner distributions
aim at describing energy distributions over time and frequency, whereas ambi
guity functions are meant for revealing correlative structures. The concept of
ambiguity function of a linear space thus appears to be quite instrumental in
solving the problem of optimal design of sets of narrowband pulses to be used
in the estimation of the range and Doppler shift of a slowly fluctuating point
target.
The book consists of 9 chapters. A brief description of each of them can be
found in Section 1.3. Chapters 2-6 form a rather coherent part of the book, with
Chapter 4 being the cornerstone of the approach of designing signal processing
methods for which time-frequency considerations provide the guiding principles.
In this Chapter 4 the problem of synthesizing linear signal spaces is solved:
with a given region in the time-frequency plane there is associated the linear
