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Cohen L. Time Frequency Analysis.pdf
Contents
Preface
Notation in Brief
1. The Time and Frequency Description of Signals
1.1 Introduction
1.2 Time Description of Signals
1.3 Frequency Description of Signals
1.4 Simple Calculation Tricks
1.5 Bandwidth Equation
1.6 AM and FM Contributions to the Bandwidth
1.7 Duration and Mean Time in Terms of the Spectrum
1.8 The Covariance of a Signal
1.9 The Fourier Transform of the Time and Frequency Densities
1.10 Nonadditivity of Spectral Properties
1.11 Classification of Signals
2. Instantaneous Frequency and the Complex Signal
2.1 Introduction
2.2 Reasons for the Complex Signal
2.3 The Analytic Signal
2.4 Calculating the Analytic Signal
2.5 Physical Interpretation of the Analytic Signal
2.6 The Quadrature Approximation
2.7 Instantaneous Frequency
2.8 Density of Instantaneous Frequency
3. The Uncertainty Principle
3.1 Introduction
3.2 The Uncertainty Principle
3.3 Proof of the Uncertainty Principle
3.4 The Uncertainty Principle for the Short-Time Fourier Transform
4. Densities and Characteristic Functions
4.1 Introduction
4.2 One Dimensional Densities
4.3 One Dimensional Characteristic Functions
4.4 Two Dimensional Densities
4.5 Local Quantities
4.6 Relation Between Local and Global Averages
4.7 Distribution of a New Variable
4.8 Negative Densities
5. The Need for Time-Frequency Analysis
5.1 Introduction
5.2 Simple Analytic Examples
5.3 Real Signals
5.4 Why Spectra Change
6. Time-Frequency Distributions: Fundamental Ideas
6.1 Introduction
6.2 Global Averages
6.3 Local Average
6.4 Time and Frequency Shift Invariance
6.5 Linear Scaling
6.6 Weak and Strong Finite Support
6.7 Uncertainty Principle
6.8 The Uncertainty Principle and Joint Distributions
6.9 Uncertainty Principle and Conditional Standard Deviation
6.10 The Basic Problems and Brief Historical Perspective
7. The Short-Time Fourier Transform
7.1 Introduction
7.2 The Short-Time Fourier Transform and Spectrogram
7.3 General Properties
7.4 Global Quantities
7.5 Local Averages
7.6 Narrowing and Broadening the Window
7.7 Group Delay
7.8 Examples
7.9 Inversion
7.10 Expansion in Instantaneous Frequency
7.11 Optimal Window
8. The Wigner Distribution
8.1 Introduction
8.2 The Wigner Distribution
8.3 General Properties
8.4 Global Averages
8.5 Local Averages
8.6 Examples
8.7 The Wigner Distribution of the Sum of Two Signals
8.8 Additional Properties
8.9 Pseudo Wigner Distribution
8.10 Modified Wigner Distributions and Positivity
8.11 Comparison of the Wigner Distribution with the Spectrogram
9. General Approach and the Kernel Method
9.1 Introduction
9.2 General Class
9.3 The Kernel Method
9.4 Basic Properties Related to the Kerrkl
9.5 Global Averages
9.6 Local Averages
9.7 Transformation Between Distributions
10. Characteristic Function Operator Method
10.1 Introduction
10.2 Characteristic Function Method
10.3 Evaluation of the Characteristic Function
10.4 The General Class
10.5 Averages
10.6 The Moment Method
11. Kernel Design for Reduced Interference
11.1 Introduction
11.2 Reduced Interference Distributions
11.3 Kernel Design for Product Kernels
11.4 Projection Onto Convex Sets
11.5 Baraniuk-Jones Optimal Kernel Design
12. Some Distributions
12.1 Introduction
12.2 Choi-Williams Method
12.3 Zhao-Atlas-Marks Distribution
12.4 Born-Jordan Distribution
12.5 Complex Energy Spectrum
12.6 Running Spectrum
13. Further Developments
13.1 Introduction
13.2 Instantaneous Bandwidth
13.3 Multicomponent Signals
13.4 Spatial/ Spatial-Frequency Distributions
13.5 Delta Function Distribution for FM Signals
13.6 Gabor Representation and Time-Frequency Distributions
13.7 Expansion in Spectrograms
13.8 Spectrogram in Terms of Other Distributions
13.9 Singular Value Decomposition of Distributions
13.10 Synthesis
13.11 Random Signals
13.12 Numerical Computation
13.13 Signal Analysis and Quantum Mechanics
14. Positive Distributions Satisfying the Marginals
14.1 Introduction
14.2 Positive Distributions
14.3 The Method of Loughlin, Pitton, and Atlas
15. The Representation of Signals
15.1 Introduction
15.2 Orthogonal Expansion of Signals
15.3 Operator Algebra
15.4 Averages
15.5 The Uncertainty Principle for Arbitrary Variables
16. Density of a Single Variable
16.1 Introduction
16.2 Density of a Single Variable
16.3 Mean Values
16.4 Bandwidth
16.5 Arbitrary Starting Representation
17. Joint Representations for Arbitrary Variables
17.1 Introduction
17.2 Marginals
17.3 Characteristic Function Operator Method
17.4 Methods of Evaluation
17.5 General Class for Arbitrary Variables
17.6 Transformation Between Distributions
17.7 Local Autocorrelation
17.8 Instantaneous Values
17.9 Local Values for Arbitrary Variable Pairs
17.10 The Covariance
17.11 Generalization of the Short-Time Fourier Transform
17.12 Unitary Transformation
17.13 Inverse Frequency
17.14 Appendix
18. Scale
18.1 Introduction
18.2 The Scale and Compression Operator
18.3 The Scale Eigenfunctions
18.4 The Scale Transform
18.5 Signals with High Scale Content
18.6 Scale Characteristic Function
18.7 Mean Scale and Bandwidth
18.8 Instantaneous Scale
18.9 Uncertainty Principle for Scale
18.10 Frequency and Other Scaling
18.11 Appendix
19. Joint Scale Representations
19.1 Introduction
19.2 Joint Tune-Scale Representations
19.3 General Class of Tune-Scale Representations
19.4 Joint Frequency-Scale Representations
19.5 Joint Representations of Time, Frequency, and Scale
19.6 Appendix
Bibliography
Index
TIME-FREQUENCY
ANALYSIS
Leon Cohen
Hunter College and Graduate Center of
The City University of New York
Prentice Hall PTR, Upper Saddle River, New Jersey 07458
Library of Congress Cataloging-in-Publication Data
Cohen, Leon
Time-frequency analysis I Leon Cohen
p
cm
Includes bibliographical references and index
ISBN 0-13-594532-I
1 Signal processing 2 Time-series analysis
3
Frequency
spectra
I Title
TK5102 9 C557
621 382 23--dc2G
1995
94-39843
CIP
Editorial/production supervision book'sorks
Manufacturing buyer Alexis R Heydt
© 1995 by Prentice-Hall P'IR
A Pearson Education Company
Upper Saddle River, NJ 07458
The publisher offers discounts on this book when ordered in hulk quantities. For
more information, contact: Corporate Sales Department. Prentice Hall PTR.
I Lake Street, Upper Saddle River, New Jersey, 07694. Phone 800-382-3419.
Fax 201-236-7141
E-mail: corporatesales@prenhall.com
All rights reserved. No part of this book may he reproduced,
in any form or by any means, without permission in writing from the publisher
Printed in the United States of America
10 9 8 7 6 5 4
ISBN 0-13-594532-1
Prentice-Hall International (UK) Limited, London
Prentice-Hall of Australia Pty Limited, Sydney
Prentice-Hall Canada Inc., Toronto
Prentice-Hall Hispanoamericana, S.A.. Mexico
Prentice-Hall of India Private Limited, New Delhi
Prentice-Hall of Japan. Inc., Tokyo
Pearson Education Asia Pte. Ltd., Singapore
Editoria Prentice-Hall do Brasil, Ltda, Rio De Janeiro
To Carol, Valerie, Ken,
Livia, and Douglas
Contents
Preface
Notation in Brief
1.
The Time and Frequency Description of Signals
1.1
1.2
1.3
1.4
Introduction
Time Description of Signals ...........................................
Frequency Description of Signals .....................................
Simple Calculation Tricks .............................................
1.5
Bandwidth Equation .................................................
1.6 AM and FM Contributions to the Bandwidth ..........................
Duration and Mean Time in Terms of the Spectrum ....................
1.7
The Covariance of a Signal ...........................................
1.8
The Fourier Transform of the Time and Frequency Densities ............
1.9
1.10 Nonadditivity of Spectral Properties ..................................
1.11 Classification of Signals ...............................................
2.
Instantaneous Frequency and the Complex Signal
2.1
Introduction .........................................................
2.2 Reasons for the Complex Signal .......................................
The Analytic Signal ..................................................
Calculating the Analytic Signal .......................................
Physical Interpretation of the Analytic Signal ..........................
The Quadrature Approximation ......................................
Instantaneous Frequency .............................................
2.8 Density of Instantaneous Frequency ..................................
2.3
2.4
2.5
2.6
2.7
3.
The Uncertainty Principle
3.1
3.2
3.3
3.4
Introduction .........................................................
The Uncertainty Principle ............................................
Proof of the Uncertainty Principle .....................................
The Uncertainty Principle for the Short-Time Fourier Transform ........
vii
1
2
6
8
15
17
19
20
22
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Viii
4.
Densities and Characteristic Functions
Contents
4.1
Introduction .....................................................
4.2 One Dimensional Densities .......................................
One Dimensional Characteristic Functions .. ......................
4.3
Two Dimensional Densities .......................................
4.4
4.5
Relation Between Local and Global Averages ......................
4.6
4.7 Distribution of a New Variable ....................................
Negative Densities ...............................................
Local Quantities ..................................................
4.8
5.
6.
5.1
5.2
5.3
5.4
6.1
6.2
The Need for Time-Frequency Analysis
Introduction .....................................................
Simple Analytic Examples ........................................
Real Signals ......................................................
Why Spectra Change .............................................
Time-Frequency Distributions: Fundamental Ideas
Introduction .....................................................
Global Averages ..................................................
6.3
6.4
6.5
Local Average ....................................................
Time and Frequency Shift Invariance ..............................
Linear Scaling ....................................................
6.6 Weak and Strong Finite Support ..................................
Uncertainty Principle .............................................
6.7
The Uncertainty Principle and Joint Distributions ..................
6.8
Uncertainty Principle and Conditional Standard Deviation .........
6.9
6.10 The Basic Problems and Brief Historical Perspective ................
7.
The Short-Time Fourier Transform
7.1
7.2
7.3
7.4
Introduction .....................................................
The Short-Time Fourier Transform and Spectrogram ...............
General Properties ...............................................
Global Quantities .................................................
7.5
Local Averages ...................................................
7.6 Narrowing and Broadening the Window ..........................
7.7
7.8
Group Delay
Examples ........................................................
7.9
Inversion ........................................................
7.10 Expansion in Instantaneous Frequency ............................
Optimal Window .................................................
7.11
8.
The Wigner Distribution
8.1
8.2
8.3
8.4
Introduction .....................................................
The Wigner Distribution ..........................................
General Properties ...............................................
Global Averages ..................................................
53
53
56
59
63
64
65
69
70
71
75
80
82
84
84
85
86
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Contents
ix
Local Averages ............................................
8.5
8.6 Examples ..................................................
The Wigner Distribution of the Sum of Two Signals ..........
8.7
8.8 Additional Properties ......................................
Pseudo Wigner Distribution ................................
8.9
8.10 Modified Wigner Distributions and Positivity ................
8.11
119
120
124
127
130
132
Comparison of the Wigner Distribution with the Spectrogram 133
9.
General Approach and the Kernel Method
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Introduction ...............................................
General Class ..............................................
The Kernel Method ........................................
Basic Properties Related to the Kerrkl .......................
Global Averages ...........................................
Local Averages ............................................
Transformation Between Distributions ......................
10.
Characteristic Function Operator Method
10.1
Introduction ...............................................
10.2 Characteristic Function Method ............................
10.3 Evaluation of the Characteristic Function ...................
The General Class .........................................
10.4
10.5
10.6
Averages ..................................................
The Moment Method ......................................
11.
12.
Kernel Design for Reduced Interference
Introduction ...............................................
11.1
11.2 Reduced Interference Distributions .........................
11.3 Kernel Design for Product Kernels ..........................
11.4 Projection Onto Convex Sets ...............................
11.5 Baraniuk-Jones Optimal Kernel Design .....................
Some Distributions
Introduction ...............................................
12.1
12.2 Choi-Williams Method .....................................
Zhao-Atlas-Marks Distribution .............................
12.3
12.4 Born-Jordan Distribution ...................................
12.5 Complex Energy Spectrum .................................
12.6 Running Spectrum ........................................
13.
Further Developments
13.1
13.2
Introduction ...............................................
Instantaneous Bandwidth ..................................
13.3 Multicomponent Signals ...................................
136
136
140
141
146
147
149
152
152
154
156
157
158
162
162
165
166
166
168
168
172
174
174
175
178
178
182
x
Contents
Spatial/ Spatial-Frequency Distributions ......................
Delta Function Distribution for FM Signals ....................
Gabor Representation and Time-Frequency Distributions ......
Expansion in Spectrograms ..................................
Spectrogram in Terms of Other Distributions ..................
Singular Value Decomposition of Distributions ................
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
Synthesis ...................................................
Random Signals .............................................
13.12 Numerical Computation .....................................
13.13 Signal Analysis and Quantum Mechanics .....................
14.
Positive Distributions Satisfying the Marginals
14.1
14.2
14.3
Introduction ................................................
Positive Distributions ........................................
The Method of Loughlin, Pitton, and Atlas ....................
15.
The Representation of Signals
15.1
Introduction ................................................
15.2 Orthogonal Expansion of Signals .............................
Operator Algebra ............................................
15.3
15.4
15.5
Averages ....................................................
The Uncertainty Principle for Arbitrary Variables ..............
16.
Density of a Single Variable
16.1
Introduction ................................................
16.2 Density of a Single Variable ..................................
Mean Values ................................................
16.3
16.4
16.5
Bandwidth ..................................................
Arbitrary Starting Representation ............................
17.
Joint Representations for Arbitrary Variables
17.1
Introduction ........................................ .......
Marginals ...................................................
17.2
Characteristic Function Operator Method .....................
17.3
17.4 Methods of Evaluation .......................................
General Class for Arbitrary Variables ..........................
17.5
Transformation Between Distributions ........................
17.6
Local Autocorrelation ........................................
17.7
Instantaneous Values ........................................
17.8
Local Values for Arbitrary Variable Pairs .......................
17.9
The Covariance ..............................................
17.10
17.11 Generalization of the Short-Time Fourier Transform ...........
17.12 Unitary Transformation ......................................
17.13
17.14
Inverse Frequency ...........................................
Appendix ......................................... .........
184
185
186
188
189
190
191
192
193
195
198
198
201
204
204
209
213
216
219
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