Signal analysis : time, frequency, scale, and structure

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Title:

Personal Author:

Allen, Ronald L.

Publication Information:

Hoboken, N.J. : IEEE Press ; Wiley-Interscience, 2004

ISBN:

9780471234418

Subject Term:

Signal processing

Added Author:

Mills, Duncan W.

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010063956	TK5102.9 A44 2004	Open Access Book	Book	Searching... Unknown

Offers a well-rounded, mathematical approach to problems in signal interpretation using the latest time, frequency, and mixed-domain methods Equally useful as a reference, an up-to-date review, a learning tool, and a resource for signal analysis techniques Provides a gradual introduction to the mathematics so that the less mathematically adept reader will not be overwhelmed with instant hard analysis Covers Hilbert spaces, complex analysis, distributions, random signals, analog Fourier transforms, and more

Author Notes

Ronald L. Allen received his BA in mathematics from the University of California, Berkeley in 1973, his MA in mathematics from the University of California, Los Angeles, and his MS and PhD in Computer Science from the University of Texas at Arlington
Duncan W. Mills received his BA in Physics from Wesleyan University, his MS in Electrical Engineering from George Washington University, and his PhD in Electrical Engineering from University of Texas at Dallas

Preface	p. xvii
Acknowledgments	p. xxi
1 Signals: Analog, Discrete, and Digital	p. 1
1.1 Introduction to Signals	p. 4
1.1.1 Basic Concepts	p. 4
1.1.2 Time-Domain Description of Signals	p. 11
1.1.3 Analysis in the Time-Frequency Plane	p. 18
1.1.4 Other Domains: Frequency and Scale	p. 20
1.2 Analog Signals	p. 21
1.2.1 Definitions and Notation	p. 22
1.2.2 Examples	p. 23
1.2.3 Special Analog Signals	p. 32
1.3 Discrete Signals	p. 35
1.3.1 Definitions and Notation	p. 35
1.3.2 Examples	p. 37
1.3.3 Special Discrete Signals	p. 39
1.4 Sampling and Interpolation	p. 40
1.4.1 Introduction	p. 40
1.4.2 Sampling Sinusoidal Signals	p. 42
1.4.3 Interpolation	p. 42
1.4.4 Cubic Splines	p. 46
1.5 Periodic Signals	p. 51
1.5.1 Fundamental Period and Frequency	p. 51
1.5.2 Discrete Signal Frequency	p. 55
1.5.3 Frequency Domain	p. 56
1.5.4 Time and Frequency Combined	p. 62
1.6 Special Signal Classes	p. 63
1.6.1 Basic Classes	p. 63
1.6.2 Summable and Integrable Signals	p. 65
1.6.3 Finite Energy Signals	p. 66
1.6.4 Scale Description	p. 67
1.6.5 Scale and Structure	p. 67
1.7 Signals and Complex Numbers	p. 70
1.7.1 Introduction	p. 70
1.7.2 Analytic Functions	p. 71
1.7.3 Complex Integration	p. 75
1.8 Random Signals and Noise	p. 78
1.8.1 Probability Theory	p. 79
1.8.2 Random Variables	p. 84
1.8.3 Random Signals	p. 91
1.9 Summary	p. 92
1.9.1 Historical Notes	p. 93
1.9.2 Resources	p. 95
1.9.3 Looking Forward	p. 96
1.9.4 Guide to Problems	p. 96
References	p. 97
Problems	p. 100
2 Discrete Systems and Signal Spaces	p. 109
2.1 Operations on Signals	p. 110
2.1.1 Operations on Signals and Discrete Systems	p. 111
2.1.2 Operations on Systems	p. 121
2.1.3 Types of Systems	p. 121
2.2 Linear Systems	p. 122
2.2.1 Properties	p. 124
2.2.2 Decomposition	p. 125
2.3 Translation Invariant Systems	p. 127
2.4 Convolutional Systems	p. 128
2.4.1 Linear, Translation-Invariant Systems	p. 128
2.4.2 Systems Defined by Difference Equations	p. 130
2.4.3 Convolution Properties	p. 131
2.4.4 Application: Echo Cancellation in Digital Telephony	p. 133
2.5 The l[superscript p] Signal Spaces	p. 136
2.5.1 l[superscript p] Signals	p. 137
2.5.2 Stable Systems	p. 138
2.5.3 Toward Abstract Signal Spaces	p. 139
2.5.4 Normed Spaces	p. 142
2.5.5 Banach Spaces	p. 147
2.6 Inner Product Spaces	p. 149
2.6.1 Definitions and Examples	p. 149
2.6.2 Norm and Metric	p. 151
2.6.3 Orthogonality	p. 153
2.7 Hilbert Spaces	p. 158
2.7.1 Definitions and Examples	p. 158
2.7.2 Decomposition and Direct Sums	p. 159
2.7.3 Orthonormal Bases	p. 163
2.8 Summary	p. 168
References	p. 169
Problems	p. 170
3 Analog Systems and Signal Spaces	p. 173
3.1 Analog Systems	p. 174
3.1.1 Operations on Analog Signals	p. 174
3.1.2 Extensions to the Analog World	p. 174
3.1.3 Cross-Correlation, Autocorrelation, and Convolution	p. 175
3.1.4 Miscellaneous Operations	p. 176
3.2 Convolution and Analog LTI Systems	p. 177
3.2.1 Linearity and Translation-Invariance	p. 177
3.2.2 LTI Systems, Impulse Response, and Convolution	p. 179
3.2.3 Convolution Properties	p. 184
3.2.4 Dirac Delta Properties	p. 186
3.2.5 Splines	p. 188
3.3 Analog Signal Spaces	p. 191
3.3.1 L[superscript p] Spaces	p. 191
3.3.2 Inner Product and Hilbert Spaces	p. 205
3.3.3 Orthonormal Bases	p. 211
3.3.4 Frames	p. 216
3.4 Modern Integration Theory	p. 225
3.4.1 Measure Theory	p. 226
3.4.2 Lebesgue Integration	p. 232
3.5 Distributions	p. 241
3.5.1 From Function to Functional	p. 241
3.5.2 From Functional to Distribution	p. 242
3.5.3 The Dirac Delta	p. 247
3.5.4 Distributions and Convolution	p. 250
3.5.5 Distributions as a Limit of a Sequence	p. 252
3.6 Summary	p. 259
3.6.1 Historical Notes	p. 260
3.6.2 Looking Forward	p. 260
3.6.3 Guide to Problems	p. 260
References	p. 261
Problems	p. 263
4 Time-Domain Signal Analysis	p. 273
4.1 Segmentation	p. 277
4.1.1 Basic Concepts	p. 278
4.1.2 Examples	p. 280
4.1.3 Classification	p. 283
4.1.4 Region Merging and Splitting	p. 286
4.2 Thresholding	p. 288
4.2.1 Global Methods	p. 289
4.2.2 Histograms	p. 289
4.2.3 Optimal Thresholding	p. 292
4.2.4 Local Thresholding	p. 299
4.3 Texture	p. 300
4.3.1 Statistical Measures	p. 301
4.3.2 Spectral Methods	p. 308
4.3.3 Structural Approaches	p. 314
4.4 Filtering and Enhancement	p. 314
4.4.1 Convolutional Smoothing	p. 314
4.4.2 Optimal Filtering	p. 316
4.4.3 Nonlinear Filters	p. 321
4.5 Edge Detection	p. 326
4.5.1 Edge Detection on a Simple Step Edge	p. 328
4.5.2 Signal Derivatives and Edges	p. 332
4.5.3 Conditions for Optimality	p. 334
4.5.4 Retrospective	p. 337
4.6 Pattern Detection	p. 338
4.6.1 Signal Correlation	p. 338
4.6.2 Structural Pattern Recognition	p. 342
4.6.3 Statistical Pattern Recognition	p. 346
4.7 Scale Space	p. 351
4.7.1 Signal Shape, Concavity, and Scale	p. 354
4.7.2 Gaussian Smoothing	p. 357
4.8 Summary	p. 369
References	p. 369
Problems	p. 375
5 Fourier Transforms of Analog Signals	p. 383
5.1 Fourier Series	p. 385
5.1.1 Exponential Fourier Series	p. 387
5.1.2 Fourier Series Convergence	p. 391
5.1.3 Trigonometric Fourier Series	p. 397
5.2 Fourier Transform	p. 403
5.2.1 Motivation and Definition	p. 403
5.2.2 Inverse Fourier Transform	p. 408
5.2.3 Properties	p. 411
5.2.4 Symmetry Properties	p. 420
5.3 Extension to L[superscript 2] (R)	p. 424
5.3.1 Fourier Transforms in L[superscript 1] (R) [intersection] L[superscript 2] (R)	p. 425
5.3.2 Definition	p. 427
5.3.3 Isometry	p. 429
5.4 Summary	p. 432
5.4.1 Historical Notes	p. 432
5.4.2 Looking Forward	p. 433
References	p. 433
Problems	p. 434
6 Generalized Fourier Transforms of Analog Signals	p. 440
6.1 Distribution Theory and Fourier Transforms	p. 440
6.1.1 Examples	p. 442
6.1.2 The Generalized Inverse Fourier Transform	p. 443
6.1.3 Generalized Transform Properties	p. 444
6.2 Generalized Functions and Fourier Series Coefficients	p. 451
6.2.1 Dirac Comb: A Fourier Series Expansion	p. 452
6.2.2 Evaluating the Fourier Coefficients: Examples	p. 454
6.3 Linear Systems in the Frequency Domain	p. 459
6.3.1 Convolution Theorem	p. 460
6.3.2 Modulation Theorem	p. 461
6.4 Introduction to Filters	p. 462
6.4.1 Ideal Low-pass Filter	p. 465
6.4.2 Ideal High-pass Filter	p. 465
6.4.3 Ideal Bandpass Filter	p. 465
6.5 Modulation	p. 468
6.5.1 Frequency Translation and Amplitude Modulation	p. 469
6.5.2 Baseband Signal Recovery	p. 470
6.5.3 Angle Modulation	p. 471
6.6 Summary	p. 475
References	p. 476
Problems	p. 477
7 Discrete Fourier Transforms	p. 482
7.1 Discrete Fourier Transform	p. 483
7.1.1 Introduction	p. 484
7.1.2 The DFT's Analog Frequency-Domain Roots	p. 495
7.1.3 Properties	p. 497
7.1.4 Fast Fourier Transform	p. 501
7.2 Discrete-Time Fourier Transform	p. 510
7.2.1 Introduction	p. 510
7.2.2 Properties	p. 529
7.2.3 LTI Systems and the DTFT	p. 534
7.3 The Sampling Theorem	p. 538
7.3.1 Band-Limited Signals	p. 538
7.3.2 Recovering Analog Signals from Their Samples	p. 540
7.3.3 Reconstruction	p. 543
7.3.4 Uncertainty Principle	p. 545
7.4 Summary	p. 547
References	p. 548
Problems	p. 549
8 The z-Transform	p. 554
8.1 Conceptual Foundations	p. 555
8.1.1 Definition and Basic Examples	p. 555
8.1.2 Existence	p. 557
8.1.3 Properties	p. 561
8.2 Inversion Methods	p. 566
8.2.1 Contour Integration	p. 566
8.2.2 Direct Laurent Series Computation	p. 567
8.2.3 Properties and z-Transform Table Lookup	p. 569
8.2.4 Application: Systems Governed by Difference Equations	p. 571
8.3 Related Transforms	p. 573
8.3.1 Chirp z-Transform	p. 573
8.3.2 Zak Transform	p. 575
8.4 Summary	p. 577
8.4.1 Historical Notes	p. 578
8.4.2 Guide to Problems	p. 578
References	p. 578
Problems	p. 580
9 Frequency-Domain Signal Analysis	p. 585
9.1 Narrowband Signal Analysis	p. 586
9.1.1 Single Oscillatory Component: Sinusoidal Signals	p. 587
9.1.2 Application: Digital Telephony DTMF	p. 588
9.1.3 Filter Frequency Response	p. 604
9.1.4 Delay	p. 605
9.2 Frequency and Phase Estimation	p. 608
9.2.1 Windowing	p. 609
9.2.2 Windowing Methods	p. 611
9.2.3 Power Spectrum Estimation	p. 613
9.2.4 Application: Interferometry	p. 618
9.3 Discrete filter design and implementation	p. 620
9.3.1 Ideal Filters	p. 621
9.3.2 Design Using Window Functions	p. 623
9.3.3 Approximation	p. 624
9.3.4 Z-Transform Design Techniques	p. 626
9.3.5 Low-Pass Filter Design	p. 632
9.3.6 Frequency Transformations	p. 639
9.3.7 Linear Phase	p. 640
9.4 Wideband Signal Analysis	p. 643
9.4.1 Chirp Detection	p. 643
9.4.2 Speech Analysis	p. 646
9.4.3 Problematic Examples	p. 650
9.5 Analog Filters	p. 650
9.5.1 Introduction	p. 651
9.5.2 Basic Low-Pass Filters	p. 652
9.5.3 Butterworth	p. 654
9.5.4 Chebyshev	p. 664
9.5.5 Inverse Chebyshev	p. 670
9.5.6 Elliptic Filters	p. 676
9.5.7 Application: Optimal Filters	p. 685
9.6 Specialized Frequency-Domain Techniques	p. 686
9.6.1 Chirp-z Transform Application	p. 687
9.6.2 Hilbert Transform	p. 688
9.6.3 Perfect Reconstruction Filter Banks	p. 694
9.7 Summary	p. 700
References	p. 701
Problems	p. 704
10 Time-Frequency Signal Transforms	p. 712
10.1 Gabor Transforms	p. 713
10.1.1 Introduction	p. 715
10.1.2 Interpretations	p. 717
10.1.3 Gabor Elementary Functions	p. 718
10.1.4 Inversion	p. 723
10.1.5 Applications	p. 730
10.1.6 Properties	p. 735
10.2 Short-Time Fourier Transforms	p. 736
10.2.1 Window Functions	p. 736
10.2.2 Transforming with a General Window	p. 738
10.2.3 Properties	p. 740
10.2.4 Time-Frequency Localization	p. 741
10.3 Discretization	p. 747
10.3.1 Transforming Discrete Signals	p. 747
10.3.2 Sampling the Short-Time Fourier Transform	p. 749
10.3.3 Extracting Signal Structure	p. 751
10.3.4 A Fundamental Limitation	p. 754
10.3.5 Frames of Windowed Fourier Atoms	p. 757
10.3.6 Status of Gabor's Problem	p. 759
10.4 Quadratic Time-Frequency Transforms	p. 760
10.4.1 Spectrogram	p. 761
10.4.2 Wigner-Ville Distribution	p. 761
10.4.3 Ambiguity Function	p. 769
10.4.4 Cross-Term Problems	p. 769
10.4.5 Kernel Construction Method	p. 770
10.5 The Balian-Low Theorem	p. 771
10.5.1 Orthonormal Basis Decomposition	p. 772
10.5.2 Frame Decomposition	p. 777
10.5.3 Avoiding the Balian-Low Trap	p. 787
10.6 Summary	p. 787
10.6.1 Historical Notes	p. 789
10.6.2 Resources	p. 790
10.6.3 Looking Forward	p. 791
References	p. 791
Problems	p. 794
11 Time-Scale Signal Transforms	p. 802
11.1 Signal Scale	p. 803
11.2 Continuous Wavelet Transforms	p. 803
11.2.1 An Unlikely Discovery	p. 804
11.2.2 Basic Theory	p. 804
11.2.3 Examples	p. 815
11.3 Frames	p. 821
11.3.1 Discretization	p. 822
11.3.2 Conditions on Wavelet Frames	p. 824
11.3.3 Constructing Wavelet Frames	p. 825
11.3.4 Better Localization	p. 829
11.4 Multiresolution Analysis and Orthogonal Wavelets	p. 832
11.4.1 Multiresolution Analysis	p. 835
11.4.2 Scaling Function	p. 847
11.4.3 Discrete Low-Pass Filter	p. 852
11.4.4 Orthonormal Wavelet	p. 857
11.5 Summary	p. 863
References	p. 865
Problems	p. 867
12 Mixed-Domain Signal Analysis	p. 873
12.1 Wavelet Methods for Signal Structure	p. 873
12.1.1 Discrete Wavelet Transform	p. 874
12.1.2 Wavelet Pyramid Decomposition	p. 875
12.1.3 Application: Multiresolution Shape Recognition	p. 883
12.2 Mixed-Domain Signal Processing	p. 893
12.2.1 Filtering Methods	p. 895
12.2.2 Enhancement Techniques	p. 897
12.3 Biophysical Applications	p. 900
12.3.1 David Marr's Program	p. 900
12.3.2 Psychophysics	p. 900
12.4 Discovering Signal Structure	p. 904
12.4.1 Edge Detection	p. 905
12.4.2 Local Frequency Detection	p. 908
12.4.3 Texture Analysis	p. 912
12.5 Pattern Recognition Networks	p. 913
12.5.1 Coarse-to-Fine Methods	p. 913
12.5.2 Pattern Recognition Networks	p. 915
12.5.3 Neural Networks	p. 916
12.5.4 Application: Process Control	p. 916
12.6 Signal Modeling and Matching	p. 917
12.6.1 Hidden Markov Models	p. 917
12.6.2 Matching Pursuit	p. 918
12.6.3 Applications	p. 918
12.7 Afterword	p. 918
References	p. 919
Problems	p. 925
Index	p. 929

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