 | Stankovic L. Time-frequency signal analysis with applications / L.Stankovic, M.Dakovic, T.Thayaparan. - Boston; London: Artech House, 2013. - xii, 659 p.: ill. - Bibliogr.: p.625-652. - Ind.: p.655-659. - Пер. заг.: Частотно-временной анализ сигналов и его применения. - ISBN 978-1-60807-651-2
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Preface ........................................................ xi
Chapter 1 Introduction to Fourier Analysis ...................... 1
1.1 Continuous-Time Signals .................................... 2
1.1.1 Periodic Signals and Fourier Series ................. 5
1.1.2 Linear Systems ...................................... 8
1.1.3 Fourier Transform .................................. 11
1.1.4 Relationship Between the Fourier Series and the
Fourier Transform .................................. 21
1.2 Discrete-Time Signals and Systems ......................... 22
1.2.1 Fourier Transform of Discrete-Time Signals ......... 26
1.2.2 Sampling Theorem in the Time Domain ................ 29
1.2.3 Discrete Fourier Transform ......................... 33
1.2.4 Analysis of a Sinusoid by Using the DFT ............ 39
1.2.5 Laplace and z-Transform ............................ 46
1.3 Discrete-Time Random Signals .............................. 49
1.3.1 First-Order Statistics ............................. 49
1.3.2 Second-Order Statistics ............................ 51
1.3.3 Noise .............................................. 53
1.3.4 Linear Systems and Random Signals .................. 56
1.3.5 Discrete Fourier Transform of Noisy Signals ........ 57
1.4 Two-Dimensional Signals ................................... 60
1.5 Problems .................................................. 63
1.6 Solutions ................................................. 68
Chapter 2 Linear Time-Frequency Representations ................ 81
2.1 Short-Time Fourier Transform .............................. 82
2.1.1 Windows ............................................ 85
2.1.2 Continuous STFT Inversion .......................... 91
2.1.3 Spectrogram ........................................ 94
2.1.4 STFT of Multicomponent Signals ..................... 95
2.2 Duration Measures and Uncertainty Principle ............... 96
2.3 Discrete Form and Realizations of the STFT ................ 99
2.3.1 Recursive STFT Implementation ..................... 100
2.3.2 Filter Bank STFT Implementation ................... 102
2.3.3 Time-Frequency Plane Lattice ...................... 103
2.4 Gabor Transform .......................................... 121
2.5 Stationary-Phase Method .................................. 123
2.6 Instantaneous Frequency .................................. 125
2.7 Local Polynomial Fourier Transform ....................... 130
2.8 Fractional Fourier Transform with Relation to the LPFT ... 135
2.9 Relation Between the STFT and the Continuous Wavelet
Transform ................................................ 136
2.9.1 Constant Q-Factor Transform ....................... 139
2.9.2 Affine Transforms ................................. 139
2.9.3 Filter Bank Formulation ........................... 140
2.9.4 Generalized Time-Frequency Varying Lattice ........ 142
2.9.5 S-Transform ....................................... 143
2.10 Chirplet Transform ....................................... 144
2.11 Generalization ........................................... 146
2.12 Parameter Optimization ................................... 148
2.12.1 Adaptive Analysis ................................. 151
2.13 Problems ................................................. 153
2.14 Solutions ................................................ 158
Chapter 3 Quadratic Time-Frequency Distributions .............. 177
3.1 Rihaczek Distribution .................................... 179
3.2 Wigner Distribution ...................................... 181
3.2.1 Introducing the Wigner Distribution Based on
the IF Representation ............................. 185
3.2.2 Signal Reconstruction and Inversion ............... 187
3.2.3 Properties of the Wigner Distribution ............. 189
3.2.4 Linear Coordinate Transforms ...................... 194
3.3 Quantum Mechanics Wigner Distribution Review ............. 201
3.3.1 Spreading Factor .................................. 204
3.3.2 Uncertainty Principle and the Wigner
Distribution ...................................... 204
3.3.3 Pseudo Quantum Signal Representation .............. 206
3.3.4 Instantaneous Frequency, Bandwidth, and Moments ... 207
3.4 Implementation of the Wigner distribution ................ 215
3.4.1 Pseudo Wigner Distribution ........................ 215
3.4.2 Smoothed Wigner Distribution ...................... 215
3.4.3 Discrete Pseudo Wigner Distribution ............... 218
3.4.4 Wigner Distribution-Based Inversion and
Synthesis ......................................... 227
3.4.5 Auto-Terms and Cross-Terms ........................ 229
3.4.6 Inner Interferences in the Wigner Distribution .... 231
3.5 Ambiguity Function ....................................... 232
3.6 Cohen Class of Distributions ............................. 238
3.6.1 Properties of the Cohen Class of Distributions .... 242
3.6.2 Reduced Interference Distributions ................ 243
3.6.3 Optimal Kernel Design ............................. 247
3.6.4 Auto-Term Form in the Cohen Class of
Distributions ..................................... 251
3.7 Kernel Decomposition-Based Calculation ................... 253
3.7.1 Spectrograms in the Cohen Class of Distributions .. 253
3.7.2 The Cohen Class of Distributions Decomposition .... 255
3.8 S-Method ................................................. 256
3.8.1 Discrete Realization of the S-Method .............. 260
3.8.2 Smoothed Spectrogram Versus S-Method as
a Principle of Composition ........................ 268
3.8.3 Decomposition of Multicomponent Signals ........... 270
3.8.4 Empirical Mode Decomposition ...................... 274
3.9 Reassignment in Time-Frequency Analysis .................. 277
3.10 Affine Class of Time-Frequency Representations ........... 285
3.11 Problems ................................................. 287
3.12 Solutions ................................................ 292
Chapter 4 Higher-Order Time-Frequency Representations ......... 317
4.1 Third-Order Time-Frequency Representations ............... 318
4.1.1 Second-Order Moment and Spectrum .................. 318
4.1.2 Third-Order Moment and Bispectrum ................. 320
4.1.3 The Wigner Bispectrum ............................. 324
4.2 Wigner Higher-Order Spectra .............................. 328
4.2.1 Instantaneous Frequency in the Wigner Higher-
Order Spectra ..................................... 329
4.2.2 Wigner Multitime Distribution ..................... 333
4.3 The L-Wigner Distribution ................................ 337
4.4 The Polynomial Wigner-Ville Distribution ................. 341
4.5 Phase Derivative Estimation .............................. 342
4.5.1 Quadratic Distributions ........................... 343
4.5.2 Higher-Order Distributions ........................ 344
4.5.3 Real-Time Causal Distributions .................... 347
4.5.4 Instantaneous Rate Estimation ..................... 348
4.6 Complex-Lag Distributions ................................ 348
4.7 S-Method-Based Realization ............................... 353
4.7.1 The L-Wigner Distribution Realization ............. 354
4.7.2 Real-Time Causal Distribution Realization ......... 356
4.7.3 Polynomial Wigner-Ville Distribution Realization .. 360
4.8 Local Polynomial Wigner Distribution ..................... 362
4.9 Higher-Order Ambiguity Functions ......................... 364
4.9.1 Monocomponent Polynomial Phase Signals ............ 364
4.9.2 Multicomponent Polynomial Phase Signals ........... 367
4.10 Problems ................................................. 371
4.11 Solutions ................................................ 375
Chapter 5 Analysis of Noisy Signals .......................... 391
5.1 Short-Time Fourier Transform of Noisy Signals ............ 392
5.2 Wigner Distribution of Noisy Signals ..................... 394
5.2.1 Pseudo Wigner Distribution Bias ................... 396
5.2.2 Pseudo Wigner Distribution Variance ............... 397
5.2.3 On the Optimal Window Width ....................... 398
5.3 Wigner Distribution-Based Instantaneous Frequency
Estimation ............................................... 399
5.3.1 Estimation Error .................................. 401
5.3.2 Instantaneous Frequency Estimation Bias ........... 407
5.3.3 Instantaneous Frequency Estimation Variance ....... 408
5.4 Adaptive Algorithm ....................................... 410
5.4.1 Parameters in the Adaptive Algorithm ............. 413
5.5 Influence of High Noise on the Instantaneous Frequency ... 422
5.5.1 Estimation Error .................................. 423
5.5.2 Mean Square Error ................................. 427
5.6 Noise in Quadratic Time-Frequency Distributions .......... 429
5.6.1 Complex Stationary and Nonstationary White Noise .. 431
5.6.2 Colored Stationary Noise .......................... 431
5.6.3 Analytic Noise .................................... 433
5.6.4 Real-Valued Noise ................................. 433
5.6.5 Noisy Signals ..................................... 434
5.7 Robust Time-Frequency Analysis ........................... 442
5.7.1 Robust Short-Time Fourier Transform ............... 443
5.7.2 Robust Wigner Distribution ........................ 450
5.7.3 L-Estimation ...................................... 451
5.7.4 Resulting Noise Distribution in the Local
Auto-Correlation Function ......................... 454
5.8 Sparse Signal Analysis in Time-Frequency ................. 455
5.9 Compressive Sensing and Robust Time-Frequency Analysis ... 463
5.9.1 Compressive Sensing-Based Processing of the
L-Estimated Time-Frequency Representations ........ 465
5.9.2 CS-Based Separation of Signals in Time-Frequency
Domain ............................................ 469
5.9.3 Compressive Sensing and Signal Inversion in
Overlapping STFT .................................. 472
5.9.4 Compressive Sensing Formulation with Frequency-
Varying Windows (Wavelets) ........................ 477
5.10 Wigner Spectrum and Time-Varying Filtering ............... 478
5.11 Problems ................................................. 482
5.12 Solutions ................................................ 487
Chapter 6 Applications of Time-Frequency Analysis ............ 511
6.1 Radar Signal Processing .................................. 511
6.1.1 Analytic CW Radar Signal Model .................... 512
6.1.2 Signal and Resolution in the Doppler Domain ....... 517
6.1.3 Nonuniform Target Motion .......................... 518
6.1.4 ISAR Basic Definitions and Model .................. 521
6.1.5 SAR Setup ......................................... 529
6.1.6 Micro-Doppler Effects in ISAR/SAR Imaging ......... 531
6.1.7 Micro-Doppler Description in SAR .................. 535
6.1.8 Time-Frequency Analysis and L-Statistics .......... 536
6.2 Interference Rejection in Spread Spectrum Communication
Systems .................................................. 553
6.2.1 Direct Sequence Spread Spectrum Model ............. 554
6.2.2 Filtering and Reconstruction ...................... 555
6.3 Car Engine Signal Analysis ............................... 562
6.3.1 Car Engine Signal Models and Analysis ............ 563
6.4 Estimation of Time-Varying Velocities in Video ........... 572
6.5 Time-Frequency-Based Detection of Deterministic Signals .. 579
6.5.1 Signal Detection by Using the Fourier Transform ... 581
6.5.2 Parametric Extension of the Fourier Transform ..... 583
6.5.3 Detection in the Time-Frequency Domain ............ 585
6.5.4 Real Radar Data Analysis .......................... 591
6.6 Multidimensional Space-Spatial Frequency Analysis ........ 594
6.6.1 Multidimensional Short-Time Fourier Transform ..... 597
6.6.2 Multidimensional Wigner Distribution .............. 598
6.6.3 Cohen Class of Distributions ...................... 598
6.6.4 Multicomponent n-Dimensional Signals .............. 599
6.7 Array Processing Based on Time-Frequency Distributions ... 603
6.8 High-Resolution Time-Frequency Techniques ................ 608
6.9 Watermarking in the Space/Spatial-Frequency Domain ....... 614
6.10 Hardware Design for Time-Frequency Analysis .............. 617
6.11 Seismic Signal Analysis .................................. 622
6.12 Biomedical Signal Analysis ............................... 623
6.13 Time-Frequency Analysis of Speech Signals ................ 623
Bibliography ............................................. 625
About the Authors ........................................ 653
Index ......................................................... 655
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