Signal Analysis Time, Frequency, Scale, and Structure

by Allen, Ronald L.; Mills, Duncan

• ISBN13: 9780471234418
• ISBN10: 0471234419
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-01-02
• Publisher: Wiley-IEEE Press

Summary

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 Biography

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 in 1975, and his MS and PhD in Computer Science from the University of Texas at Arlington in 1990 and 1993, respectively. <p> 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 in 1992.

Table of Contents

Preface	xvii
Acknowledgments	xxi
Signals: Analog, Discrete, and Digital	1	(108)
Introduction to Signals	4	(17)
Basic Concepts	4	(7)
Time-Domain Description of Signals	11	(7)
Analysis in the Time-Frequency Plane	18	(2)
Other Domains: Frequency and Scale	20	(1)
Analog Signals	21	(14)
Definitions and Notation	22	(1)
Examples	23	(9)
Special Analog Signals	32	(3)
Discrete Signals	35	(5)
Definitions and Notation	35	(2)
Examples	37	(2)
Special Discrete Signals	39	(1)
Sampling and Interpolation	40	(11)
Introduction	40	(2)
Sampling Sinusoidal Signals	42	(1)
Interpolation	42	(4)
Cubic Splines	46	(5)
Periodic Signals	51	(12)
Fundamental Period and Frequency	51	(4)
Discrete Signal Frequency	55	(1)
Frequency Domain	56	(6)
Time and Frequency Combined	62	(1)
Special Signal Classes	63	(7)
Basic Classes	63	(2)
Summable and Integrable Signals	65	(1)
Finite Energy Signals	66	(1)
Scale Description	67	(1)
Scale and Structure	67	(3)
Signals and Complex Numbers	70	(8)
Introduction	70	(1)
Analytic Functions	71	(4)
Complex Integration	75	(3)
Random Signals and Noise	78	(14)
Probability Theory	79	(5)
Random Variables	84	(7)
Random Signals	91	(1)
Summary	92	(17)
Historical Notes	93	(2)
Resources	95	(1)
Looking Forward	96	(1)
Guide to Problems	96	(1)
References	97	(3)
Problems	100	(9)
Discrete Systems and Signal Spaces	109	(64)
Operations on Signals	110	(12)
Operations on Signals and Discrete Systems	111	(10)
Operations on Systems	121	(1)
Types of Systems	121	(1)
Linear Systems	122	(5)
Properties	124	(1)
Decomposition	125	(2)
Translation Invariant Systems	127	(1)
Convolutional Systems	128	(8)
Linear, Translation-Invariant Systems	128	(2)
Systems Defined by Difference Equations	130	(1)
Convolution Properties	131	(2)
Application: Echo Cancellation in Digital Telephony	133	(3)
The lp Signal Spaces	136	(13)
lp Signals	137	(1)
Stable Systems	138	(1)
Toward Abstract Signal Spaces	139	(3)
Normed Spaces	142	(5)
Banach Spaces	147	(2)
Inner Product Spaces	149	(9)
Definitions and Examples	149	(2)
Norm and Metric	151	(2)
Orthogonality	153	(5)
Hilbert Spaces	158	(10)
Definitions and Examples	158	(1)
Decomposition and Direct Sums	159	(4)
Orthonormal Bases	163	(5)
Summary	168	(5)
References	169	(1)
Problems	170	(3)
Analog Systems and Signal Spaces	173	(100)
Analog Systems	174	(3)
Operations on Analog Signals	174	(1)
Extensions to the Analog World	174	(1)
Cross-Correlation, Autocorrelation, and Convolution	175	(1)
Miscellaneous Operations	176	(1)
Convolution and Analog LTI Systems	177	(14)
Linearity and Translation-Invariance	177	(2)
LTI Systems, Impulse Response, and Convolution	179	(5)
Convolution Properties	184	(2)
Dirac Delta Properties	186	(2)
Splines	188	(3)
Analog Signal Spaces	191	(34)
lp Spaces	191	(14)
Inner Product and Hilbert Spaces	205	(6)
Orthonormal Bases	211	(5)
Frames	216	(9)
Modern Integration Theory	225	(16)
Measure Theory	226	(6)
Lebesgue Integration	232	(9)
Distributions	241	(18)
From Function to Functional	241	(1)
From Functional to Distribution	242	(5)
The Dirac Delta	247	(3)
Distributions and Convolution	250	(2)
Distributions as a Limit of a Sequence	252	(7)
Summary	259	(14)
Historical Notes	260	(1)
Looking Forward	260	(1)
Guide to Problems	260	(1)
References	261	(2)
Problems	263	(10)
Time-Domain Signal Analysis	273	(110)
Segmentation	277	(11)
Basic Concepts	278	(2)
Examples	280	(3)
Classification	283	(3)
Region Merging and Splitting	286	(2)
Thresholding	288	(12)
Global Methods	289	(1)
Histograms	289	(3)
Optimal Thresholding	292	(7)
Local Thresholding	299	(1)
Texture	300	(14)
Statistical Measures	301	(7)
Spectral Methods	308	(6)
Structural Approaches	314	(1)
Filtering and Enhancement	314	(12)
Convolutional Smoothing	314	(2)
Optimal Filtering	316	(5)
Nonlinear Filters	321	(5)
Edge Detection	326	(12)
Edge Detection on a Simple Step Edge	328	(4)
Signal Derivatives and Edges	332	(2)
Conditions for Optimality	334	(3)
Retrospective	337	(1)
Pattern Detection	338	(13)
Signal Correlation	338	(4)
Structural Pattern Recognition	342	(4)
Statistical Pattern Recognition	346	(5)
Scale Space	351	(18)
Signal Shape, Concavity, and Scale	354	(3)
Gaussian Smoothing	357	(12)
Summary	369	(14)
References	369	(6)
Problems	375	(8)
Fourier Transforms of Analog Signals	383	(57)
Fourier Series	385	(18)
Exponential Fourier Series	387	(4)
Fourier Series Convergence	391	(6)
Trigonometric Fourier Series	397	(6)
Fourier Transform	403	(21)
Motivation and Definition	403	(5)
Inverse Fourier Transform	408	(3)
Properties	411	(9)
Symmetry Properties	420	(4)
Extension to L2(R)	424	(8)
Fourier Transforms in L1(R) ∩ L2(R)	425	(2)
Definition	427	(2)
Isometry	429	(3)
Summary	432	(8)
Historical Notes	432	(1)
Looking Forward	433	(1)
References	433	(1)
Problems	434	(6)
Generalized Fourier Transforms of Analog Signals	440	(42)
Distribution Theory and Fourier Transforms	440	(11)
Examples	442	(1)
The Generalized Inverse Fourier Transform	443	(1)
Generalized Transform Properties	444	(7)
Generalized Functions and Fourier Series Coefficients	451	(8)
Dirac Comb: A Fourier Series Expansion	452	(2)
Evaluating the Fourier Coefficients: Examples	454	(5)
Linear Systems in the Frequency Domain	459	(3)
Convolution Theorem	460	(1)
Modulation Theorem	461	(1)
Introduction to Filters	462	(6)
Ideal Low-pass Filter	465	(1)
Ideal High-pass Filter	465	(1)
Ideal Bandpass Filter	465	(3)
Modulation	468	(7)
Frequency Translation and Amplitude Modulation	469	(1)
Baseband Signal Recovery	470	(1)
Angle Modulation	471	(4)
Summary	475	(7)
References	476	(1)
Problems	477	(5)
Discrete Fourier Transforms	482	(72)
Discrete Fourier Transform	483	(27)
Introduction	484	(11)
The DFT's Analog Frequency-Domain Roots	495	(2)
Properties	497	(4)
Fast Fourier Transform	501	(9)
Discrete-Time Fourier Transform	510	(28)
Introduction	510	(19)
Properties	529	(5)
LTI Systems and the DTFT	534	(4)
The Sampling Theorem	538	(9)
Band-Limited Signals	538	(2)
Recovering Analog Signals from Their Samples	540	(3)
Reconstruction	543	(2)
Uncertainty Principle	545	(2)
Summary	547	(7)
References	548	(1)
Problems	549	(5)
The z-Transform	554	(31)
Conceptual Foundations	555	(11)
Definition and Basic Examples	555	(2)
Existence	557	(4)
Properties	561	(5)
Inversion Methods	566	(7)
Contour Integration	566	(1)
Direct Laurent Series Computation	567	(2)
Properties and z-Transform Table Lookup	569	(2)
Application: Systems Governed by Difference Equations	571	(2)
Related Transforms	573	(4)
Chirp z-Transform	573	(2)
Zak Transform	575	(2)
Summary	577	(8)
Historical Notes	578	(1)
Guide to Problems	578	(1)
References	578	(2)
Problems	580	(5)
Frequency-Domain Signal Analysis	585	(127)
Narrowband Signal Analysis	586	(22)
Single Oscillatory Component: Sinusoidal Signals	587	(1)
Application: Digital Telephony DTMF	588	(16)
Filter Frequency Response	604	(1)
Delay	605	(3)
Frequency and Phase Estimation	608	(12)
Windowing	609	(2)
Windowing Methods	611	(2)
Power Spectrum Estimation	613	(5)
Application: Interferometry	618	(2)
Discrete filter design and implementation	620	(23)
Ideal Filters	621	(2)
Design Using Window Functions	623	(1)
Approximation	624	(2)
Z-Transform Design Techniques	626	(6)
Low-Pass Filter Design	632	(7)
Frequency Transformations	639	(1)
Linear Phase	640	(3)
Wideband Signal Analysis	643	(7)
Chirp Detection	643	(3)
Speech Analysis	646	(4)
Problematic Examples	650	(1)
Analog Filters	650	(36)
Introduction	651	(1)
Basic Low-Pass Filters	652	(2)
Butterworth	654	(10)
Chebyshev	664	(6)
Inverse Chebyshev	670	(6)
Elliptic Filters	676	(9)
Application: Optimal Filters	685	(1)
Specialized Frequency-Domain Techniques	686	(14)
Chirp-z Transform Application	687	(1)
Hilbert Transform	688	(6)
Perfect Reconstruction Filter Banks	694	(6)
Summary	700	(12)
References	701	(3)
Problems	704	(8)
Time-Frequency Signal Transforms	712	(90)
Gabor Transforms	713	(23)
Introduction	715	(2)
Interpretations	717	(1)
Gabor Elementary Functions	718	(5)
Inversion	723	(7)
Applications	730	(5)
Properties	735	(1)
Short-Time Fourier Transforms	736	(11)
Window Functions	736	(2)
Transforming with a General Window	738	(2)
Properties	740	(1)
Time-Frequency Localization	741	(6)
Discretization	747	(13)
Transforming Discrete Signals	747	(2)
Sampling the Short-Time Fourier Transform	749	(2)
Extracting Signal Structure	751	(3)
A Fundamental Limitation	754	(3)
Frames of Windowed Fourier Atoms	757	(2)
Status of Gabor's Problem	759	(1)
Quadratic Time-Frequency Transforms	760	(11)
Spectrogram	761	(1)
Wigner--Ville Distribution	761	(8)
Ambiguity Function	769	(1)
Cross-Term Problems	769	(1)
Kernel Construction Method	770	(1)
The Balian--Low Theorem	771	(16)
Orthonormal Basis Decomposition	772	(5)
Frame Decomposition	777	(10)
Avoiding the Balian--Low Trap	787	(1)
Summary	787	(15)
Historical Notes	789	(1)
Resources	790	(1)
Looking Forward	791	(1)
References	791	(3)
Problems	794	(8)
Time-Scale Signal Transforms	802	(71)
Signal Scale	803	(1)
Continuous Wavelet Transforms	803	(18)
An Unlikely Discovery	804	(1)
Basic Theory	804	(11)
Examples	815	(6)
Frames	821	(11)
Discretization	822	(2)
Conditions on Wavelet Frames	824	(1)
Constructing Wavelet Frames	825	(4)
Better Localization	829	(3)
Multiresolution Analysis and Orthogonal Wavelets	832	(31)
Multiresolution Analysis	835	(12)
Scaling Function	847	(5)
Discrete Low-Pass Filter	852	(5)
Orthonormal Wavelet	857	(6)
Summary	863	(10)
References	865	(2)
Problems	867	(6)
Mixed-Domain Signal Analysis	873	(56)
Wavelet Methods for Signal Structure	873	(20)
Discrete Wavelet Transform	874	(1)
Wavelet Pyramid Decomposition	875	(8)
Application: Multiresolution Shape Recognition	883	(10)
Mixed-Domain Signal Processing	893	(7)
Filtering Methods	895	(2)
Enhancement Techniques	897	(3)
Biophysical Applications	900	(4)
David Marr's Program	900	(1)
Psychophysics	900	(4)
Discovering Signal Structure	904	(9)
Edge Detection	905	(3)
Local Frequency Detection	908	(4)
Texture Analysis	912	(1)
Pattern Recognition Networks	913	(4)
Coarse-to-Fine Methods	913	(2)
Pattern Recognition Networks	915	(1)
Neural Networks	916	(1)
Application: Process Control	916	(1)
Signal Modeling and Matching	917	(1)
Hidden Markov Models	917	(1)
Matching Pursuit	918	(1)
Applications	918	(1)
Afterword	918	(11)
References	919	(6)
Problems	925	(4)
Index	929