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Signals and Systems.pdf

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    Cover
    Title Page
    Copyright
    Dedication
    Contents
    Preface
    Chapter 1 Introduction
    1.1 Signals and Systems Defined
    1.2 Types of Signals
    1.3 Examples of Systems
    A Mechanical System
    A Fluid System
    A Discrete-Time System
    Feedback Systems
    1.4 A Familiar Signal and System Example
    1.5 Use of MATLAB®
    Chapter 2 Mathematical Description of Continuous-Time Signals
    2.1 Introduction and Goals
    2.2 Functional Notation
    2.3 Continuous-Time Signal Functions
    Complex Exponentials and Sinusoids
    Functions with Discontinuities
    2.4 Combinations of Functions
    2.5 Shifting and Scaling
    Amplitude Scaling
    Time Shifting
    Time Scaling
    Simultaneous Shifting and Scaling
    2.6 Differentiation and Integration
    2.7 Even and Odd Signals
    Combinations of Even and Odd Signals
    Derivatives and Integrals of Even and Odd Signals
    2.8 Periodic Signals
    2.9 Signal Energy and Power
    Signal Energy
    Signal Power
    2.10 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 3 Discrete-Time Signal Description
    3.1 Introduction and Goals
    3.2 Sampling and Discrete Time
    3.3 Sinusoids and Exponentials
    Sinusoids
    Exponentials
    3.4 Singularity Functions
    The Unit-Impulse Function
    The Unit-Sequence Function
    The Signum Function
    The Unit-Ramp Function
    The Unit Periodic Impulse Function or Impulse Train
    3.5 Shifting and Scaling
    Amplitude Scaling
    Time Shifting
    Time Scaling
    3.6 Differencing and Accumulation
    3.7 Even and Odd Signals
    Combinations of Even and Odd Signals
    Symmetrical Finite Summation of Even and Odd Signals
    3.8 Periodic Signals
    3.9 Signal Energy and Power
    Signal Energy
    Signal Power
    3.10 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 4 Description of Systems
    4.1 Introduction and Goals
    4.2 Continuous-Time Systems
    System Modeling
    System Properties
    Dynamics of Second-Order Systems
    Complex Sinusoid Excitation
    4.3 Discrete-Time Systems
    System Modeling
    System Properties
    4.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 5 Time-Domain System Analysis
    5.1 Introduction and Goals
    5.2 Continuous Time
    Impulse Response
    Continuous-Time Convolution
    5.3 Discrete Time
    Impulse Response
    Discrete-Time Convolution
    5.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 6 Continuous-Time Fourier Methods
    6.1 Introduction and Goals
    6.2 The Continuous-Time Fourier Series
    Conceptual Basis
    Orthogonality and the Harmonic Function
    The Compact Trigonometric Fourier Series
    Convergence
    Minimum Error of Fourier-Series Partial Sums
    The Fourier Series of Even and Odd Periodic Functions
    Fourier-Series Tables and Properties
    Numerical Computation of the Fourier Series
    6.3 The Continuous-Time Fourier Transform
    Extending the Fourier Series to Aperiodic Signals
    The Generalized Fourier Transform
    Fourier Transform Properties
    Numerical Computation of the Fourier Transform
    6.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 7 Discrete-Time Fourier Methods
    7.1 Introduction and Goals
    7.2 The Discrete-Time Fourier Series and the Discrete Fourier Transform
    Linearity and Complex-Exponential Excitation
    Orthogonality and the Harmonic Function
    Discrete Fourier Transform Properties
    The Fast Fourier Transform
    7.3 The Discrete-Time Fourier Transform
    Extending the Discrete Fourier Transform to Aperiodic Signals
    Derivation and Definition
    The Generalized DTFT
    Convergence of the Discrete-Time Fourier Transform
    DTFT Properties
    Numerical Computation of the Discrete-Time Fourier Transform
    7.4 Fourier Method Comparisons
    7.5 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 8 The Laplace Transform
    8.1 Introduction and Goals
    8.2 Development of the Laplace Transform
    Generalizing the Fourier Transform
    Complex Exponential Excitation and Response
    8.3 The Transfer Function
    8.4 Cascade-Connected Systems
    8.5 Direct Form II Realization
    8.6 The Inverse Laplace Transform
    8.7 Existence of the Laplace Transform
    Time-Limited Signals
    Right- and Left-Sided Signals
    8.8 Laplace Transform Pairs
    8.9 Partial-Fraction Expansion
    8.10 Laplace Transform Properties
    8.11 The Unilateral Laplace Transform
    Definition
    Properties Unique to the Unilateral Laplace Transform
    Solution of Differential Equations with Initial Conditions
    8.12 Pole-Zero Diagrams and Frequency Response
    8.13 MATLAB System Objects
    8.14 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 9 The z Transform
    9.1 Introduction and Goals
    9.2 Generalizing the Discrete-Time Fourier Transform
    9.3 Complex Exponential Excitation and Response
    9.4 The Transfer Function
    9.5 Cascade-Connected Systems
    9.6 Direct Form II System Realization
    9.7 The Inverse z Transform
    9.8 Existence of the z Transform
    Time-Limited Signals
    Right- and Left-Sided Signals
    9.9 z-Transform Pairs
    9.10 z-Transform Properties
    9.11 Inverse z-Transform Methods
    Synthetic Division
    Partial-Fraction Expansion
    Examples of Forward and Inverse z Transforms
    9.12 The Unilateral z Transform
    Properties Unique to the Unilateral z Transform
    Solution of Difference Equations
    9.13 Pole-Zero Diagrams and Frequency Response
    9.14 MATLAB System Objects
    9.15 Transform Method Comparisons
    9.16 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 10 Sampling and Signal Processing
    10.1 Introduction and Goals
    10.2 Continuous-Time Sampling
    Sampling Methods
    The Sampling Theorem
    Aliasing
    Time-Limited and Bandlimited Signals
    Interpolation
    Sampling Bandpass Signals
    Sampling a Sinusoid
    Band-Limited Periodic Signals
    Signal Processing Using the DFT
    10.3 Discrete-Time Sampling
    Periodic-Impulse Sampling
    Interpolation
    10.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 11 Frequency Response Analysis
    11.1 Introduction and Goals
    11.2 Frequency Response
    11.3 Continuous-Time Filters
    Examples of Filters
    Ideal Filters
    Bode Diagrams
    Practical Filters
    11.4 Discrete-Time Filters
    Notation
    Ideal Filters
    Practical Filters
    11.5 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 12 Communication System Analysis
    12.1 Introduction and Goals
    12.2 Continuous Time Communication Systems
    Need for Communication Systems
    Frequency Multiplexing
    Analog Modulation and Demodulation
    12.3 Discrete-Time Sinusoidal-Carrier Amplitude Modulation
    12.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 13 Laplace System Analysis
    13.1 Introduction and Goals
    13.2 System Representations
    13.3 System Stability
    13.4 System Connections
    Cascade and Parallel Connections
    The Feedback Connection
    13.5 System Analysis Using MATLAB
    13.6 System Responses to Standard Signals
    Unit-Step Response
    Sinusoid Response
    13.7 Standard Realizations of Systems
    Cascade Realization
    Parallel Realization
    13.8 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 14 z-Transform System Analysis
    14.1 Introduction and Goals
    14.2 System Models
    Difference Equations
    Block Diagrams
    14.3 System Stability
    14.4 System Connections
    14.5 System Responses to Standard Signals
    Unit-Sequence Response
    Response to a Causal Sinusoid
    14.6 Simulating Continuous-Time Systems with Discrete-Time Systems
    z-Transform-Laplace-Transform Relationships
    Impulse Invariance
    Sampled-Data Systems
    14.7 Standard Realizations of Systems
    Cascade Realization
    Parallel Realization
    14.8 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 15 Filter Analysis and Design
    15.1 Introduction and Goals
    15.2 Analog Filters
    Butterworth Filters
    Chebyshev, Elliptic and Bessel Filters
    15.3 Digital Filters
    Simulation of Analog Filters
    Filter Design Techniques
    15.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Chapter 16 State-Space Analysis
    16.1 Introduction and Goals
    16.2 Continuous-Time Systems
    System and Output Equations
    Transfer Functions
    Alternate State-Variable Choices
    Transformations of State Variables
    Diagonalization
    MATLAB Tools for State-Space Analysis
    16.3 Discrete-Time Systems
    System and Output Equations
    Transfer Functions and Transformations of State Variables
    MATLAB Tools for State-Space Analysis
    16.4 Summary of Important Points
    Exercises
    Exercises with Answers
    Exercises without Answers
    Appendix A: Useful Mathematical Relations
    Appendix B: Continuous-Time Fourier Series Pairs
    Appendix C: Discrete Fourier Transform Pairs
    Appendix D: Continuous-Time Fourier Transform Pairs
    Appendix E: Discrete-Time Fourier Transform Pairs
    Appendix F: Tables of Laplace Transform Pairs
    Appendix G: z Transform Pairs
    Bibliography
    Index
    This page intentionally left blank
    ␦ ( ) t ⎧ 1 ⎨ ⎩⎩ 0 = = , , 0 , ≠ 0 t < < 0 t t 2 1 otherwise ␦ ( ) t dt t 2 ∫ t 1 = u( ) t ⎧ 1 ⎪ ⎨ 1 2 / ⎩⎪ 0 > = < 0 0 0 t t t , , , sgn( ) t = ⎧ 1 ⎪ ⎨ 0 ⎩⎪ − 1 ramp( ) t = ⎧ ⎨ ⎩ t 0 > = < 0 0 0 t t t ≥ < 0 0 t t , , , , , ␦ T ( ) t = ∞∑ ␦ ( =−∞ n − t nT ) ... rect( ) t = ⎧ 1 ⎪ ⎨ 1 2 / ⎩⎪ 0 | , | t | , | t | | , t < = >> 1 2 / 1 2 / 1 2/ tri( ) t = − | t | ⎧ 1 ⎨ ⎩ 0 < ≥ 1 1 | | t t | | , , sinc( ) t ) = ␲ sin( t ␲ t drcl( , t N ) = sin( N ␲ sin( ) Nt ␲ ) t δ(t) 1 u(t) 1 sgn(t) 1 -1 ramp(t) 1 ... t 1 δ (t)T 1 -2T -T T 2T rect(t) 1 1 2 1 2 tri(t) 1 −1 1 sinc(t) 1 −5 −4 −3 −2 −1 1 2 3 4 5 drcl(t,7) 1 ... -1 1 ␦ ␦ ␦ t t t t t t t ... t
    ␦ ␦ ␦ ␦ ␦[ ] n = ⎧ 1 ⎨ ⎩ 0 , , = ≠ 0 0 n n u[ ] n = ⎧ 1 ⎨ ⎩ 0 , , ≥ < 0 0 n n sgn[ ]n = ⎧ 1 ⎪ ⎨ 0 ⎩⎪ − 1 , , , > = < 0 0 0 n n n δ[n] 1 u[n] 1 n ... n sgn[n] 1 -1 ... n ... ... ramp[ ] n = ⎧ ⎨ ⎩ n 0 , , ≥ < n n ⎫ =0 ⎬ ⎭ 0 [ ] u n n ramp[n] 8 4 ... ... n 4 8 ␲ ␲ ␲ ␲ ␦ N [ ] n = ∞∑ ␦ [ =−∞ m − n mN ] δ [n] N 1 ... -N ... 2N n N
    Signals and Systems Analysis Using Transform Methods and MATLAB® Second Edition Michael J. Roberts Professor, Department of Electrical and Computer Engineering University of Tennessee
    SIGNALS AND SYSTEMS: ANALYSIS USING TRANSFORM METHODS AND MATLAB®, SECOND EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Previous edition © 2004. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on recycled, acid-free paper containing 10% postconsumer waste. 1 2 3 4 5 6 7 8 9 0 QDQ/QDQ 1 0 9 8 7 6 5 4 3 2 1 ISBN 978-0-07-338068-1 MHID 0-07-338068-7 Vice President & Editor-in-Chief: Marty Lange Vice President EDP/Central Publishing Services: Kimberly Meriwether David Publisher: Raghothaman Srinivasan Senior Sponsoring Editor: Peter E. Massar Senior Marketing Manager: Curt Reynolds Development Editor: Darlene M. Schueller Project Manager: Melissa M. Leick Cover Credit: © Digital Vision/Getty Images Buyer: Sandy Ludovissy Design Coordinator: Margarite Reynolds Media Project Manager: Balaji Sundararaman Compositor: Glyph International Typeface: 10.5/12 Times Roman Printer: Quad/Graphics Library of Congress Cataloging-in-Publication Data p. cm. Roberts, Michael J., Dr. Signals and systems: analysis using transform methods and MATLAB / Michael J. Roberts.—2nd ed. Includes bibliographical references and index. ISBN-13: 978-0-07-338068-1 (alk. paper) ISBN-10: 0-07-338068-7 (alk. paper) TK5102.9.R63 2012 621.382’2–dc22 1. Signal processing. 2. System analysis. 3. MATLAB. I. Title. 2010048334 www.mhhe.com
    To my wife Barbara for giving me the time and space to complete this effort and to the memory of my parents, Bertie Ellen Pinkerton and Jesse Watts Roberts, for their early emphasis on the importance of education.
    CONTENTS Preface, xii Chapter 1 Introduction, 1 1.1 Signals and Systems Defi ned, 1 1.2 Types of Signals, 3 1.3 Examples of Systems, 8 A Mechanical System, 9 A Fluid System, 9 A Discrete-Time System, 11 Feedback Systems, 12 1.4 A Familiar Signal and System Example, 14 1.5 Use of MATLAB®, 18 Chapter 2 Mathematical Description of Continuous-Time Signals, 19 2.1 2.2 Functional Notation, 20 2.3 Continuous-Time Signal Functions, 20 Introduction and Goals, 19 Complex Exponentials and Sinusoids, 21 Functions with Discontinuities, 23 The Signum Function, 24 The Unit-Step Function, 24 The Unit-Ramp Function, 26 The Unit Impulse, 27 The Impulse, the Unit Step and Generalized Derivatives, 29 The Equivalence Property of the Impulse, 30 The Sampling Property of the Impulse, 31 The Scaling Property of the Impulse, 31 The Unit Periodic Impulse or Impulse Train, 32 A Coordinated Notation for Singularity Functions, 33 The Unit-Rectangle Function, 33 2.4 Combinations of Functions, 34 2.5 Shifting and Scaling, 36 Amplitude Scaling, 36 Time Shifting, 37 iv Time Scaling, 39 Simultaneous Shifting and Scaling, 43 2.6 Differentiation and Integration, 47 2.7 Even and Odd Signals, 49 Combinations of Even and Odd Signals, 51 Derivatives and Integrals of Even and Odd Signals, 53 2.8 Periodic Signals, 53 2.9 Signal Energy and Power, 56 Signal Energy, 56 Signal Power, 57 2.10 Summary of Important Points, 60 Exercises, 60 Exercises with Answers, 60 Signal Functions, 60 Scaling and Shifting, 61 Derivatives and Integrals, 65 Even and Odd Signals, 66 Periodic Signals, 68 Signal Energy and Power, 69 Exercises without Answers, 70 Signal Functions, 70 Scaling and Shifting, 71 Generalized Derivative, 74 Derivatives and Integrals, 74 Even and Odd Signals, 75 Periodic Signals, 75 Signal Energy and Power, 76 Chapter 3 Discrete-Time Signal Description, 77 3.1 3.2 Sampling and Discrete Time, 78 3.3 Sinusoids and Exponentials, 80 Introduction and Goals, 77 Sinusoids, 80 Exponentials, 83 3.4 Singularity Functions, 84 The Unit-Impulse Function, 84 The Unit-Sequence Function, 85 The Signum Function, 85
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