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
Contents
Preface
Notation in Brief
1.
xiii
xv
The Time and Frequency Description of Signals
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
Introduction
Time Description of Signals
Frequency Description of Signals
Simple Calculation Tricks
Bandwidth Equation
AM and FM Contributions to the Bandwidth
Duration and Mean Time in Terms of the Spectrum
The Covariance of a Signal
The Fourier Transform of the Time and Frequency Densities
Nonadditivity of Spectral Properties
Classification of Signals
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Introduction
Reasons for the Complex Signal
The Analytic Signal
Calculating the Analytic Signal
Physical Interpretation of the Analytic Signal
The Quadrature Approximation
Instantaneous Frequency
Density of Instantaneous Frequency
2.
1
2
6
8
15
17
19
20
22
23
25
Instantaneous Frequency and the Complex Signal
3.
27
28
30
31
35
36
39
41
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
44
46
47
50
viii
Contents
4.
Densities and Characteristic Functions
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5.
Introduction
One Dimensional Densities
One Dimensional Characteristic Functions
Two Dimensional Densities
Local Quantities
Relation Between Local and Global Averages
Distribution of a New Variable
Negative Densities
53
53
56
59
63
64
65
69
The Need for Time-Frequency Analysis
5.1
5.2
5.3
5.4
6.
Introduction
Simple Analytic Examples
Real Signals
Why Spectra Change
70
71
75
80
Time-Frequency Distributions: Fundamental Ideas
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
7.
Introduction
Global Averages
Local Average
Time and Frequency Shift Invariance
Linear Scaling
Weak and Strong Finite Support
Uncertainty Principle
The Uncertainty Principle and Joint Distributions
Uncertainty Principle and Conditional Standard Deviation
The Basic Problems and Brief Historical Perspective
82
84
84
85
86
86
87
88
90
91
The Short-Time Fourier Transform
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
Introduction
The Short-Time Fourier Transform and Spectrogram
General Properties
Global Quantities
Local Averages
Narrowing and Broadening the Window
Group Delay
Examples
Inversion
Expansion in Instantaneous Frequency
Optimal Window
8.1
8.2
8.3
8.4
Introduction
The Wigner Distribution
General Properties
Global Averages
8.
93
94
96
99
100
101
102
103
108
109
110
The Wigner Distribution
113
114
117
118
Contents
ix
8.5
8.6
8.7
8.8
8.9
8.10
8.11
Local Averages
Examples
The Wigner Distribution of the Sum of Two Signals
Additional Properties
Pseudo Wigner Distribution
Modified Wigner Distributions and Positivity
Comparison of the Wigner Distribution with the Spectrogram
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 Keniel
Global Averages
Local Averages
Transformation Between Distributions
10.1
10.2
10.3
10.4
10.5
10.6
Introduction
Characteristic Function Method
Evaluation of the Characteristic Function
The General Class
Averages
The Moment Method
9.
119
120
124
127
130
132
133
General Approach and the Kernel Method
10.
136
136
140
141
146
147
149
Characteristic Function Operator Method
11.
152
152
154
156
157
158
Kernel Design for Reduced Interference
11.1
11.2
11.3
11.4
11.5
Introduction
Reduced Interference Distributions
Kernel Design for Product Kernels
Projection Onto Convex Sets
Baraniuk-Jones Optimal Kernel Design
12.1
12.2
12.3
12.4
12.5
12.6
Introduction
Choi-Williams Method
Zhao-Atlas-Marks Distribution
Born-Jordan Distribution
Complex Energy Spectrum
Running Spectrum
12.
162
162
165
166
166
Some Distributions
13.
168
168
172
174
174
175
Further Developments
13.1
13.2
13.3
Introduction
Instantaneous Bandwidth
Multicomponent Signals
178
178
182
Contents
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
13.12
13.13
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
Synthesis
Random Signals
Numerical Computation
Signal Analysis and Quantum Mechanics
14.1
14.2
14.3
Introduction
Positive Distributions
The Method of Loughlin, Pitton, and Atlas
14.
184
185
186
188
189
190
191
192
193
195
Positive Distributions Satisfying the Marginals
15.
198
198
201
The Representation of Signals
15.1
15.2
15.3
15.4
15.5
16.
Introduction
Orthogonal Expansion of Signals
Operator Algebra
Averages
The Uncertainty Principle for Arbitrary Variables
204
204
209
213
216
Density of a Single Variable
16.1
16.2
16.3
16.4
16.5
17.
Introduction
Density of a Single Variable
Mean Values
Bandwidth
Arbitrary Starting Representation
219
219
222
223
224
Joint Representations for Arbitrary Variables
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
17.10
17.11
17.12
17.13
17.14
Introduction
Marginals
Characteristic Function Operator Method
Methods of Evaluation
C
General Class for Arbitrary Variables
Transformation Between Distributions
Local Autocorrelation
Instantaneous Values
Local Values for Arbitrary Variable Pairs
The Covariance
Generalization of the Short-Time Fourier Transform
Unitary Transformation
Inverse Frequency
Appendix
225
225
225
226
229
229
230
231
232
233
234
235
238
240
Contents
18.
xi
Scale
18.1
18.2
18.3
18.4
18.5
18.6
18.7
18.8
18.9
18.10
18.11
19.
Introduction
The Scale and Compression Operator
The Scale Eigenfunctions
The Scale Transform
Signals with High Scale Content
Scale Characteristic Function
Mean Scale and Bandwidth
Instantaneous Scale
Uncertainty Principle for Scale
Frequency and Other Scaling
Appendix
242
242
244
245
248
249
250
251
251
252
253
Joint Scale Representations
19.1
19.2
19.3
19.4
19.5
19.6
Introduction
Joint Tune-Scale Representations
General Class of Time-Scale Representations
Joint Frequency-Scale Representations
Joint Representations of Time, Frequency, and Scale
Appendix
255
255
256
258
258
260
Bibliography
263
Index
:
291