Signal Analysis: Time, Frequency, Scale, and Structure

Signal analysis from concept to application
Signal analysis, a method of arriving at a structural description of a signalso that later high–level algorithms can interpret its content, is a growingfield with an increasing number of applications. Signal Analysis: Time,Frequency, Scale, and Structure opens a window into the practice of signalanalysis by providing a gradual yet thorough introduction to the theory behindsignal analysis as well as the abstract mathematics and functional analysis whichmay be new to many readers.
Making the material accessible for readers of differentlevels of mathematical background, the authors clarify the basic concepts andtypes of signals and slowly build up to mastering the mathematics of Hilbertspaces, complex analysis, distributions, random signals, and analog Fouriertransforms. Chapters cover:Concepts of analog, discrete, and digital signalsDiscrete systems and signal spaces, including linear and convolutional systems and lp signal spacesTime domain signal analysis including segmentation and thresholding, filtering and enhancement, and edge and pattern detectionContinuous, discrete, and generalized Fourier transformsThe z–transform, time–frequency and time–scale signal transformsFrequency domain and mixed–domain signal analysis
Suitable as a professional reference, up–to–date review forpractitioners, and a resource of signal analysis techniques, Signal Analysis:Time, Frequency, Scale, and Structure serves up a well–rounded, mathematicalapproach to problems in signal interpretation using the latest time, frequency,and mixed–domain methods.

Spis treści:
Preface.
Acknowledgments.
1 Signals: Analog, Discrete, and Digital.
1.1 Introduction to Signals.
1.1.1 Basic Concepts.
1.1.2 Time–Domain Description of Signals.
1.1.3 Analysis in the Time– Frequency Plane.
1.1.4 Other Domains: Frequency and Scale.
1.2 Analog Signals.
1.2.1 Definitions and Notation.
1.2.2 Examples.
1.2.3 Special Analog Signals.
1.3 Discrete Signals.
1.3.1 Definitions and Notation.
1.3.2 Examples.
1.3.3 Special Discrete Signals.
1.4 Sampling and Interpolation.
1.4.1 Introduction.
1.4.2 Sampling Sinusoidal Signals.
1.4.3 Interpolation.
1.4.4 Cubic Splines.
1.5 Periodic Signals.
1.5.1 Fundamental Period and Frequency.
1.5.2 Discrete Signal Frequency.
1.5.3 Frequency Domain.
1.5.4 Time and Frequency Combined.
1.6 Special Signal Classes.
1.6.1 Basic Classes.
1.6.2 Summable and Integrable Signals.
1.6.3 Finite Energy Signals.
1.6.4 Scale Description.
1.6.5 Scale and Structure.
1.7 Signals and Complex Numbers.
1.7.1 Introduction.
1.7.2 Analytic Functions.
1.7.3 Complex Integration.
1.8 Random Signals and Noise.
1.8.1 Probability Theory.
1.8.2 Random Variables.
1.8.3 Random Signals.
1.9 Summary.
1.9.1 Historical Notes.
1.9.2 Resources.
1.9.3 Looking Forward.
1.9.4 Guide to Problems.
References.
Problems.
2 Discrete Systems and Signal Spaces.
2.1 Operations on Signals.
2.1.1 Operations on Signals and Discrete Systems.
2.1.2 Operations on Systems.
2.1.3 Types of Systems.
2.2 Linear Systems.
2.2.1 Properties.
2.2.2 Decomposition.
2.3 Translation Invariant Systems.
2.4 Convolutional Systems.
2.4.1 Linear, Translation–Invariant Systems.
2.4.2 Systems Defined by Difference Equations.
2.4.3 Convolution Properties.
2.4.4 Application: Echo Cancellation in Digital Telephony.
2.5 The l<sup>p</sup> Signal Spaces.
2.5.1 l<sup>p</sup> Signals.
2.5.2 Stable Systems.
2.5.3 Toward Abstract Signal Spaces.
2.5.4 Normed Spaces.
2.5.5 Banach Spaces.
2.6 Inner Product Spaces.
2.6.1 Defi nitions and Examples.
2.6.2 Norm and Metric.
2.6.3 Orthogonality.
2.7 Hilbert Spaces.
2.7.1 Definitions and Examples.
2.7.2 Decomposition and Direct Sums.
2.7.3 Orthonormal Bases.
2.8 Summary.
References.
Problems.
3 Analog Systems and Signal Spaces.
3.1 Analog Systems.
3.1.1 Operations on Analog Signals.
3.1.2 Extensions to the Analog World.
3.1.3 Cross–Correlation, Autocorrelation, and Convolution.
3.1.4 Miscellaneous Operations.
3.2 Convolution and Analog LTI Systems.
3.2.1 Linearity and Translation–Invariance.
3.2.2 LTI Systems, Impulse Response, and Convolution.
3.2.3 Convolution Properties.
3.2.4 Dirac Delta Properties.
3.2.5 Splines.
3.3 Analog Signal Spaces.
3.3.1 L<sup>p</sup> Spaces.
3.3.2 Inner Product and Hilbert Spaces.
3.3.3 Orthonormal Bases.
3.3.4 Frames.
3.4 Modern Integration Theory.
3.4.1 Measure Theory.
3.4.2 Lebesgue Integration.
3.5 Distributions.
3.5.1 From Function to Functional.
3.5.2 From Functional to Distribution.
3.5.3 The Dirac Delta.
3.5.4 Distributions and Convolution.
3.5.5 Distributions as a Limit of a Sequence.
3.6 Summary.
3.6.1 Historical Notes.
3.6.2 Looking Forward.
3.6.3 Guide to Problems.
References.
Problems.
4 Time–Domain Signal Analysis.
4.1 Segmentation.
4.1.1 Basic Concepts.
4.1.2 Examples.
4.1.3 Classification.
4.1.4 Region Merging and Splitting.
4.2 Thresholding.
4.2.1 Global Methods.
4.2.2 Histograms.
4.2.3 Optimal Thresholding.
4.2.4 Local Thresholding.
4.3 Texture.
4.3.1 Statistical Measures.
4.3.2 Spectral Methods.
4.3.3 Structural Approaches.
4.4 Filtering and Enhancement.
4.4.1 Convolutional Smoothing.
4.4.2 Optimal Filtering.
4.4.3 Nonlinear Filters.
4.5 Edge Detection.
4.5.1 Edge Detection on a Simple Step Edge.
4.5.2 Signal Derivatives and Edges.
4 .5.3 Conditions for Optimality.
4.5.4 Retrospective.
4.6 Pattern Detection.
4.6.1 Signal Correlation.
4.6.2 Structural Pattern Recognition.
4.6.3 Statistical Pattern Recognition.
4.7 Scale Space.
4.7.1 Signal Shape, Concavity, and Scale.
4.7.2 Gaussian Smoothing.
4.8 Summary.
References.
Problems.
5 Fourier Transforms of Analog Signals.
5.1 Fourier Series.
5.1.1 Exponential Fourier Series.
5.1.2 Fourier Series Convergence.
5.1.3 Trigonometric Fourier Series.
5.2 Fourier Transform.
5.2.1 Motivation and Definition.
5.2.2 Inverse Fourier Transform.
5.2.3 Properties.
5.2.4 Symmetry Properties.
5.3 Extension to L<sup>2</sup>(R).
5.3.1 Fourier Transforms in L<sup>1</sup>(R) ∩   L<sup>2</sup>(R).
5.3.2 Definition.
5.3.3 Isometry.
5.4 Summary.
5.4.1 Historical Notes.
5.4.2 Looking Forward.
References.
Problems.
6 Generalized Fourier Transforms of Analog Signals.
6.1 Distribution Theory and Fourier Transforms.
6.1.1 Examples.
6.1.2 The Generalized Inverse Fourier Transform.
6.1.3 Generalized Transform Properties.
6.2 Generalized Functions and Fourier Series Coefficients.
6.2.1 Dirac Comb: A Fourier Series Expansion.
6.2.2 Evaluating the Fourier Coefficients: Examples.
6.3 Linear Systems in the Frequency Domain.
6.3.1 Convolution Theorem.
6.3.2 Modulation Theorem.
6.4 Introduction to Filters.
6.4.1 Ideal Low–pass Filter.
6.4.2 Ideal High–pass Filter.
6.4.3 Ideal Bandpass Filter.
6.5 Modulation.
6.5.1 Frequency Translation and Amplitude Modulation.
6.5.2 Baseband Signal Recovery.
6.5.3 Angle Modulation.
6.6 Summary.
References.
Problems.
7 Discrete Fourier Transforms.
7.1 Discrete Fourier Transform.
7.1.1 Introduction.
7.1.2 The DFT’s Analog Frequency&#