Wasyl Wasylkiwskyj (auth.) Signals and Transforms in Linear Sys....pdf

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Preface

Contents

Introduction

Chapter 1 Signals and Their Representations

1.1 Signal Spaces and the Approximation Problem

1.2 Inner Product, Norm and Representations by Finite Sumsof Elementary Functions

1.2.1 Inner Product and Norm

1.2.2 Orthogonality and Linear Independence

1.2.3 Representations by Sums of OrthogonalFunctions

1.2.4 Nonorthogonal Expansion Functionsand Their Duals

1.2.5 Orthogonalization Techniques

Orthogonalization via the Gram Matrix

The Gram–Schmidt Orthogonalization

1.3 The LMS Approximation and the Normal Equations

1.3.1 The Projection Theorem

1.3.2 The Normal Equations

1.3.3 Generalizations of the Normal Equations

1.3.4 LMS Approximation and Stochastic Processes*

1.4 LMS Solutions via the Singular Value Decomposition

1.4.1 Basic Theory Underlying the SVD

1.4.2 Solutions of the Normal EquationsUsing the SVD

1.4.3 Signal Extraction from Noisy Data

1.4.4 The SVD for the Continuum

1.4.5 Frames

1.4.6 Total Least Squares

1.4.7 Tikhonov Regularization

1.5 Finite Sets of Orthogonal Functions

1.5.1 LMS and Orthogonal Functions

1.5.2 Trigonometric Functions

1.5.3 Orthogonal Polynomials [1]

Legendre Polynomials

Laguerre Polynomials

Hermite Polynomials

1.6 Singularity Functions

1.6.1 The Delta Function

Definition of the Delta Function

Limiting Forms Leading to Delta Functions

1.6.2 Higher Order Singularity Functions

1.6.3 Idealized Signals

1.6.4 Representation of Functions with StepDiscontinuities

1.6.5 Delta Function with Functions as Arguments

1.7 Infinite Orthogonal Systems

1.7.1 Deterministic Signals

1.7.2 Stochastic Signals: Karhunen–Loeve Expansion

Chapter 2 Fourier Series and Integrals with Applications to SignalAnalysis

2.1 Fourier Series

2.1.1 Pointwise Convergence at Interior Pointsfor Smooth Functions

2.1.2 Convergence at Step Discontinuities

2.1.3 Convergence at Interval Endpoints

2.1.4 Delta Function Representation

2.1.5 The Fejer Summation Technique

2.1.6 Fundamental Relationships Between the Frequencyand Time Domain Representations

Parseval Formula

Time and Frequency Domain Convolution

Symmetries

2.1.7 Cosine and Sine Series

Sine Series

2.1.8 Interpolation with Sinusoids

Interpolation Using Exponential Functions

Interpolation Using Cosine Functions

2.1.9 Anharmonic Fourier Series

2.2 The Fourier Integral

2.2.1 LMS Approximation by Sinusoids Spanninga Continuum

2.2.2 Transition to an Infinite Observation Interval:The Fourier Transform

2.2.3 Completeness Relationship and Relation to FourierSeries

2.2.4 Convergence and the Use of CPV Integrals

2.2.5 Canonical Signals and Their Transforms

The Delta Function

The Unit Step Function

The Rectangular Pulse Function

Triangular Pulse Function

Exponential Functions

Gaussian Function

2.2.6 Basic Properties of the FT

Linearity

Symmetries

Time Shift and Frequency Shift

Differentiation

Inner Product Invariance

Convolution

Integration

Causal Signals and the Hilbert Transform [16]

Initial and Final Value Theorems

Fourier Series and the Poisson Sum Formula

2.2.7 Convergence at Discontinuities

2.2.8 Fejer Summation

2.3 Modulation and Analytic Signal Representation

2.3.1 Analytic Signals

2.3.2 Instantaneous Frequency and the Methodof Stationary Phase

2.3.3 Bandpass Representation

2.3.4 Bandpass Representation of Random Signals*

2.4 Fourier Transforms and Analytic Function Theory

2.4.1 Analyticity of the FT of Causal Signals

2.4.2 Hilbert Transforms and Analytic Functions

2.4.3 Relationships Between Amplitude and Phase

2.4.4 Evaluation of Inverse FT Using Complex VariableTheory

2.5 Time-Frequency Analysis

2.5.1 The Uncertainty Principle

2.5.2 The Short-Time Fourier Transform

2.6 Frequency Dispersion

2.6.1 Phase and Group Delay

2.6.2 Phase and Group Velocity

2.6.3 Effects of Frequency Dispersion on Pulse Shape

2.6.4 Another Look at the Propagation of a GaussianPulse When B"' (W0) =0

2.6.5 Effects of Finite Transmitter Spectral Line Width*

2.7 Fourier Cosine and Sine Transforms

Chapter 3 Linear Systems

3.1 Fundamental Properties

3.1.1 Single-valuedness, Reality, and Causality

3.1.2 Impulse Response

3.1.3 Step Response

3.1.4 Stability

3.1.5 Time-invariance

3.2 Characterizations in terms of Input/Output Relationships

3.2.1 LTI Systems

3.2.2 Time-varying Systems

3.3 Linear Systems Characterized by Ordinary DifferentialEquations

3.3.1 First-Order Differential Equations

3.3.2 Second-Order Differential Equations

Time-varying Systems

LTI System

3.3.3 N-th Order Differential Equations

Time-Varying Systems

Standard Form.

Equivalent N-Dimensional First-Order Vector Form.

Feedback Representation.

Time-invariance

Chapter 4 Laplace Transforms

4.1 Single-Sided Laplace Transform

4.1.1 Analytic Properties

4.1.2 Singularity Functions

4.1.3 Some Examples

4.1.4 Inversion Formula

4.1.5 Fundamental Theorems

Properties

LT of Polynomial and Exponential Signals

Periodic Signals

4.1.6 Evaluation of the Inverse LT

Rational Functions

Transcendental Functions

4.2 Double-Sided Laplace Transform

4.2.1 Definition and Analytic Properties

4.2.2 Inversion Formula

Example 1

Example 2

Example 3

Example 4

4.2.3 Relationships Between the FT and the Unilateral LT

Determination of the FT from the Unilateral Laplace Transform

Determination of the Unilateral Laplace Transform from the FT

Chapter 5 Bandlimited Functions Sampling and the Discrete FourierTransform

5.1 Bandlimited Functions

5.1.1 Fundamental Properties

5.1.2 The Sampling Theorem

5.1.3 Sampling Theorem for Stationary RandomProcesses*

5.2 Signals Defined by a Finite Number of Samples

5.2.1 Spectral Concentration of Bandlimited Signals

5.2.2 Aliasing

5.3 Sampling

5.3.1 Impulse Sampling

5.3.2 Zero-Order Hold Sampling and Reconstruction

5.3.3 BandPass Sampling

5.3.4 Sampling of Periodic Signals

5.4 The Discrete Fourier Transform

5.4.1 Fundamental Definitions

5.4.2 Properties of the DFT

Periodicity

Orthogonality

Matrix Properties

Parseval Identities

Time and Frequency Shift

Convolution

Chapter 6 The Z-Transform and Discrete Signals

6.1 The Z-Transform

6.1.1 From FS to the Z-Transform

6.1.2 Direct ZT of Some Sequences

6.1.3 Properties

Time Shift

Forward.

Backward Shift.

Time Convolution

Frequency Convolution

Initial Value Theorem

Differentiation

6.2 Analytical Techniques in the Evaluation of the Inverse ZT

6.3 Finite Difference Equations and Their Use in IIR and FIRFilter Design

6.4 Amplitude and Phase Relations Using the Discrete HilbertTransform

6.4.1 Explicit Relationship Between Real and ImaginaryParts of the FT of a Causal Sequence

6.4.2 Relationship Between Amplitude and Phaseof a Transfer Function

6.4.3 Application to Design of FIR Filters

Appendix A Introduction to Functions of a Complex Variable

A.1 Complex Numbers and Complex Variables

A.1.1 Complex Numbers

A.1.2 Function of a Complex Variable

A.2 Analytic Functions

A.2.1 Differentiation and the Cauchy–RiemannConditions

A.2.2 Properties of Analytic Functions

A.2.3 Integration

A.3 Taylor and Laurent Series

A.3.1 The Cauchy Integral Theorem

A.3.2 The Taylor Series

A.3.3 Laurent Series

A.4 Singularities of Functions and the Calculus of Residues

A.4.1 Classification of Singularities

Isolated Singularities

Definitions.

Laurent Series for Functions with Isolated Singularities.

Multivalued Functions and Extended Singularities.

A.4.2 Calculus of Residues

The Residue Theorem

Computation of Residues

Bibliography

Index

Signals and Transforms in Linear Systems Analysis

Wasyl Wasylkiwskyj Signals and Transforms in Linear Systems Analysis 123

Wasyl Wasylkiwskyj Professor of Engineering and Applied Science The George Washington University Washington, DC, USA ISBN 978-1-4614-3286-9 DOI 10.1007/978-1-4614-3287-6 Springer New York Heidelberg Dordrecht London ISBN 978-1-4614-3287-6 (eBook) Library of Congress Control Number: 2012956318 © Springer Science+Business Media, LLC 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface This book deals with aspects of mathematical techniques and models that con- stitute an important part of the foundation for the analysis of linear systems. The subject is classical and forms a signiﬁcant component of linear systems theory. These include Fourier, Z-transforms, Laplace, and related transforms both in their continuous and discrete versions. The subject is an integral part of electrical engineering curricula and is covered in many excellent textbooks. In light of this, an additional book dealing with the same topics would appear su- perﬂuous. What distinguishes this book is that the same topics are viewed from a distinctly diﬀerent perspective. Rather than dealing with diﬀerent transforms essentially in isolation, a methodology is developed that uniﬁes the classical portion of the subject and permits the inclusion of topics that usually are not considered part of the linear systems theory. The unifying principle here is the least mean square approximation, the normal equations, and their extensions to the continuum. This approach gives equal status to expansions in terms of special functions (that need not be orthogonal), Fourier series, Fourier integrals, and discrete transforms. As a by-product one also gains new insights. For ex- ample, the Gibbs phenomenon is a general property of LMS convergence at step discontinuities and is not limited to Fourier series. This book is suitable for a ﬁrst year graduate course that provides a tran- sition from the level the subject is presented in an undergraduate course in signals and systems to a level more appropriate as a prerequisite for graduate work in specialized ﬁelds. The material presented here is based in part on the notes used for a similar course taught by the author in the School of Electri- cal and Computer Engineering at The George Washington University. The six chapters can be covered in one semester with suﬃcient ﬂexibility in the choice of topics within each chapter. The exception is Chap. 1 which, in the spirit of the intended unity, sets the stage for the remainder of the book. It includes the mathematical foundation and the methodology applied in the chapters to follow. The prerequisites for the course are an undergraduate course in signals and systems, elements of linear algebra, and the theory of functions of a complex variable. Recognizing that frequently the preparation, if any, in the latter is sketchy, the necessary material is presented in the Appendix. Wasyl Wasylkiwskyj V

Contents 1 Signals and Their Representations 1.1 Signal Spaces and the Approximation Problem . . . . . . . . . . 1.2 Inner Product, Norm and Representations by Finite Sums of Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Inner Product and Norm . . . . . . . . . . . . . . . . . . 1.2.2 Orthogonality and Linear Independence . . . . . . . . . . 1.2.3 Representations by Sums of Orthogonal 1 1 4 4 7 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.4 Nonorthogonal Expansion Functions and Their Duals 14 16 19 19 21 22 25 27 27 30 32 35 37 39 43 44 44 45 47 52 52 59 VII . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Orthogonalization Techniques . . . . . . . . . . . . . . . . 1.3 The LMS Approximation and the Normal Equations . . . . . . . 1.3.1 The Projection Theorem . . . . . . . . . . . . . . . . . . . 1.3.2 The Normal Equations . . . . . . . . . . . . . . . . . . . . 1.3.3 Generalizations of the Normal Equations . . . . . . . . . 1.3.4 LMS Approximation and Stochastic Processes* . . . . . . 1.4 LMS Solutions via the Singular Value Decomposition . . . . . . . 1.4.1 Basic Theory Underlying the SVD . . . . . . . . . . . . . 1.4.2 Solutions of the Normal Equations Using the SVD . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Signal Extraction from Noisy Data . . . . . . . . . . . . . 1.4.4 The SVD for the Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Frames 1.4.6 Total Least Squares . . . . . . . . . . . . . . . . . . . . . 1.4.7 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 LMS and Orthogonal Functions . . . . . . . . . . . . . . . 1.5.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . 1.5.3 Orthogonal Polynomials [1] . . . . . . . . . . . . . . . . . 1.6 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 The Delta Function . . . . . . . . . . . . . . . . . . . . . 1.6.2 Higher Order Singularity Functions . . . . . . . . . . . . . 1.5 Finite Sets of Orthogonal Functions

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