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with MATLAB® Applications Signals and Systems Steven T. Karris Orchard Publications www.orchardpublications.com Second Edition Includes step-by-step procedures for designing analog and digital filters Xm[] xn[]e j2π mn N – n0= N1– ∑ = Orchard Publications, Fremont, California Visit us on the Internet www.orchardpublications.com or email us: info@orchardpublications.com Signals and Systems with MATLAB® Applications Second Edition Steven T. Karris Students and working professionals will find Signals and Systems with MATLAB® Applications, Second Edition, to be a concise and easy-to-learn text. It provides complete, clear, and detailed expla- nations of the principal analog and digital signal processing concepts and analog and digital filter design illustrated with numerous practical examples. This text includes the following chapters and appendices: • Elementary Signals • The Laplace Transformation • The Inverse Laplace Transformation • Circuit Analysis with Laplace Transforms • State Variables and State Equations • The Impulse Response and Convolution • Fourier Series • The Fourier Transform • Discrete Time Systems and the Z Transform • The DFT and The FFT Algorithm • Analog and Digital Filters • Introduction to MATLAB • Review of Complex Numbers • Review of Matrices and Determinants Each chapter contains numerous practical applications supplemented with detailed instructions for using MATLAB to obtain quick solutions. Steven T. Karris is the president and founder of Orchard Publications. He earned a bachelors degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a mas- ters degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and has done post-master work at the latter. He is a registered professional engineer in California and Florida. He has over 30 years of professional engineering experience in industry. In addi- tion, he has over 25 years of teaching experience that he acquired at several educational insti- tutions as an adjunct professor. He is currently with UC Berkeley Extension . ISBN 0-9709511-8-3 $39.95 U.S.A. Signals and Systems with MATLAB® Applications Second Edition Steven T. Karris Orchard Publications www.orchardpublications.com Signals and Systems with MATLAB Applications, Second Edition Copyright © 2003 Orchard Publications. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. Direct all inquiries to Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538 Product and corporate names are trademarks or registered trademarks of the Microsoft™ Corporation and The MathWorks™ Inc. They are used only for identification and explanation, without intent to infringe. Library of Congress Cataloging-in-Publication Data Library of Congress Control Number: 2003091595 ISBN 0-9709511-8-3 Copyright TX 5-471-562 Preface This text contains a comprehensive discussion on continuous and discrete time signals and systems with many MATLAB® examples. It is written for junior and senior electrical engineering students, and for self-study by working professionals. The prerequisites are a basic course in differential and integral calculus, and basic electric circuit theory. This book can be used in a two-quarter, or one semester course. This author has taught the subject material for many years at San Jose State University, San Jose, California, and was able to cover all material in 16 weeks, with 2½ lecture hours per week. To get the most out of this text, it is highly recommended that Appendix A is thoroughly reviewed. This appendix serves as an introduction to MATLAB, and is intended for those who are not familiar with it. The Student Edition of MATLAB is an inexpensive, and yet a very powerful software package; it can be found in many college bookstores, or can be obtained directly from The MathWorks™ Inc., 3 Apple Hill Drive , Natick, MA 01760-2098 Phone: 508 647-7000, Fax: 508 647-7001 http://www.mathworks.com e-mail: info@mathwork.com The elementary signals are reviewed in Chapter 1 and several examples are presented. The intent of this chapter is to enable the reader to express any waveform in terms of the unit step function, and subsequently the derivation of the Laplace transform of it. Chapters 2 through 4 are devoted to Laplace transformation and circuit analysis using this transform. Chapter 5 discusses the state variable method, and Chapter 6 the impulse response. Chapters 7 and 8 are devoted to Fourier series and transform respectively. Chapter 9 introduces discrete-time signals and the Z transform. Considerable time was spent on Chapter 10 to present the Discrete Fourier transform and FFT with the simplest possible explanations. Chapter 11 contains a thorough discussion to analog and digital filters analysis and design procedures. As mentioned above, Appendix A is an introduction to MATLAB. Appendix B contains a review of complex numbers, and Appendix C discusses matrices. New to the Second Edition This is an refined revision of the first edition. The most notable changes are chapter-end summaries, and detailed solutions to all exercises. The latter is in response to many students and working professionals who expressed a desire to obtain the author’s solutions for comparison with their own. The author has prepared more exercises and they are available with their solutions to those instructors who adopt this text for their class. The chapter-end summaries will undoubtedly be a valuable aid to instructors for the preparation of presentation material. 2 The last major change is the improvement of the plots generated by the latest revisions of the MATLAB® Student Version, Release 13. Orchard Publications Fremont, California www.orchardpublications.com info@orchardpublications.com Signals and Systems with MATLAB Applications, Second Edition i Orchard Publications Table of Contents Chapter 1 Elementary Signals Signals Described in Math Form 1-1 The Unit Step Function 1-2 The Unit Ramp Function 1-10 The Delta Function 1-12 Sampling Property of the Delta Function 1-12 Sifting Property of the Delta Function 1-13 Higher Order Delta Functions 1-15 Summary 1-19 Exercises 1-20 Solutions to Exercises 1-21 Chapter 2 The Laplace Transformation Definition of the Laplace Transformation 2-1 Properties of the Laplace Transform 2-2 The Laplace Transform of Common Functions of Time 2-12 The Laplace Transform of Common Waveforms 2-23 Summary 2- 29 Exercises 2-34 Solutions to Exercises 2-37 Chapter 3 The Inverse Laplace Transformation The Inverse Laplace Transform Integral 3-1 Partial Fraction Expansion 3-1 Case where is Improper Rational Function ( ) 3-13 Alternate Method of Partial Fraction Expansion 3-15 Summary 3-18 Fs() mn≥ ii Signals and Systems with MATLAB Applications, Second Edition Orchard Publications Exercises 3-20 Solutions to Exercises 3-22 Chapter 4 Circuit Analysis with Laplace Transforms Circuit Transformation from Time to Complex Frequency 4-1 Complex Impedance 4-8 Complex Admittance 4-10 Transfer Functions 4-13 Summary 4-16 Exercises 4-18 Solutions to Exercises 4-21 Chapter 5 State Variables and State Equations Expressing Differential Equations in State Equation Form 5-1 Solution of Single State Equations 5- 7 The State Transition Matrix 5-9 Computation of the State Transition Matrix 5-11 Eigenvectors 5-18 Circuit Analysis with State Variables 5-22 Relationship between State Equations and Laplace Transform 5-28 Summary 5-35 Exercises 5-39 Solutions to Exercises 5-41 Chapter 6 The Impulse Response and Convolution The Impulse Response in Time Domain 6-1 Even and Odd Functions of Time 6-5 Convolution 6-7 Graphical Evaluation of the Convolution Integral 6-8 Circuit Analysis with the Convolution Integral 6-18 Summary 6-20 Zs() Y s() Signals and Systems with MATLAB Applications, Second Edition iii Orchard Publications Exercises 6-22 Solutions to Exercises 6-24 Chapter 7 Fourier Series Wave Analysis 7-1 Evaluation of the Coefficients 7-2 Symmetry 7-7 Waveforms in Trigonometric Form of Fourier Series 7-11 Gibbs Phenomenon 7-24 Alternate Forms of the Trigonometric Fourier Series 7-25 Circuit Analysis with Trigonometric Fourier Series 7-29 The Exponential Form of the Fourier Series 7-31 Line Spectra 7-35 Computation of RMS Values from Fourier Series 7-40 Computation of Average Power from Fourier Series 7-42 Numerical Evaluation of Fourier Coefficients 7-44 Summary 7-48 Exercises 7-51 Solutions to Exercises 7-53 Chapter 8 The Fourier Transform Definition and Special Forms 8-1 Special Forms of the Fourier Transform 8-2 Properties and Theorems of the Fourier Transform 8-9 Fourier Transform Pairs of Common Functions 8-17 Finding the Fourier Transform from Laplace Transform 8-25 Fourier Transforms of Common Waveforms 8-27 Using MATLAB to Compute the Fourier Transform 8-33 The System Function and Applications to Circuit Analysis 8-34 Summary 8-41 Exercises 8- 47 Solutions to Exercises 8-49 iv Signals and Systems with MATLAB Applications, Second Edition Orchard Publications Chapter 9 Discrete Time Systems and the Z Transform Definition and Special Forms 9-1 Properties and Theorems of the Z Tranform 9-3 The Z Transform of Common Discrete Time Functions 9-11 Computation of the Z transform with Contour Integration 9-20 Transformation Between and Domains 9-22 The Inverse Z Transform 9-24 The Transfer Function of Discrete Time Systems 9-38 State Equations for Discrete Time Systems 9-43 Summary 9-47 Exercises 9-52 Solutions to Exercises 9-54 Chapter 10 The DFT and the FFT Algorithm The Discrete Fourier Transform (DFT) 10-1 Even and Odd Properties of the DFT 10-8 Properties and Theorems of the DFT 10-10 The Sampling Theorem 10-13 Number of Operations Required to Compute the DFT 10-16 The Fast Fourier Transform (FFT) 10-17 Summary 10-28 Exercises 10-31 Solutions to Exercises 10-33 Chapter 11 Analog and Digital Filters Filter Types and Classifications 11-1 Basic Analog Filters 11-2 Low-Pass Analog Filters 11-7 Design of Butterworth Analog Low-Pass Filters 11-11 Design of Type I Chebyshev Analog Low-Pass Filters 11-22 Other Low-Pass Filter Approximations 11-34 High-Pass, Band-Pass, and Band-Elimination Filters 11-39 sz [...]... Exercises C-30 vi Signals and Systems with MATLAB Applications, Second Edition Orchard Publications Chapter 1 Elementary Signals his chapter begins with a discussion of elementary signals that may be applied to electric networks The unit step, unit ramp, and delta functions are introduced The sampling and sifting properties of the delta function are defined and derived Several examples for... as a function of the unit step function, and sketch the appropriate waveform Solution: For this example, the output voltage v out = 0 for t < T , and v out = v S for t > T Therefore, v out = v S u 0 ( t – T ) (1.7) and the waveform is shown in Figure 1.7 Signals and Systems with MATLAB Applications, Second Edition Orchard Publications 1-3 Chapter 1 Elementary Signals vS u0 ( t – T ) v out t T 0 Figure... waveform and definition are as shown in Figure 1.4 and relation (1.5) 1 u0 ( t – t0 ) t 0 t0 Figure 1.4 Waveform for u 0 ( t – t 0 ) * In some books, the unit step function is denoted as u ( t ) , that is, without the subscript 0 In this text, however, we will reserve the u ( t ) designation for any input when we discuss state variables in a later chapter 1-2 Signals and Systems with MATLAB Applications, ... C-9 Minors and Cofactors .C-12 Signals and Systems with MATLAB Applications, Second Edition Orchard Publications v Cramer’s Rule C-16 Gaussian Elimination Method C-19 The Adjoint of a Matrix C-20 Singular and Non-Singular Matrices C-21 The Inverse of a Matrix .C-21 Solution of Simultaneous Equations with Matrices... capacitor when charged with a constant current source * Since the initial condition for the capacitor voltage was not specified, we express this integral with – ∞ at the lower limit of integration so that any non-zero value prior to t < 0 would be included in the integration Signals and Systems with MATLAB Applications, Second Edition Orchard Publications 1-9 Chapter 1 Elementary Signals 1.3 The Unit... in the next section Signals and Systems with MATLAB Applications, Second Edition Orchard Publications 1-11 Chapter 1 Elementary Signals 1.4 The Delta Function δ ( t ) The unit impulse or delta function, denoted as δ ( t ) , is the derivative of the unit step u 0 ( t ) It is also defined as t ∫– ∞ δ ( τ ) d τ = u0 ( t ) (1.33) and δ ( t ) = 0 for all t ≠ 0 (1.34) To better understand the delta function... origin, with unbounded amplitude, zero width, and area equal to 1 Two useful properties of the delta function are the sampling property and the sifting property 1.5 Sampling Property of the Delta Function δ ( t ) The sampling property of the delta function states that f ( t )δ ( t – a ) = f ( a )δ ( t ) (1.35) f ( t )δ ( t ) = f ( 0 )δ ( t ) (1.36) or, when a = 0 , 1-12 Signals and Systems with MATLAB Applications, ... right side is zero for α < a , and therefore, we can replace the lower limit of integration a by α We can now rewrite (1.47) as b ∫a and letting f ( t )δ ( t – α ) dt = f ( b ) – b ∫ α f ′( t ) d t a → – ∞ and b → ∞ for any α < ∞ = f(b ) – f(b) + f(α) , we get ∞ ∫–∞ f ( t )δ ( t – α ) dt = f ( α ) (1.48) Sifting Property of δ ( t ) 1-14 Signals and Systems with MATLAB Applications, Second Edition Orchard... segments 1-6 and shown in Figure 1.13 Signals and Systems with MATLAB Applications, Second Edition Orchard Publications The Unit Step Function 2 t + 1 T 1 –T ⁄ 2 v(t) 2 – t + 1 T T⁄2 0 t Figure 1.13 Equations for the linear segments of Figure 1.12 For line segment , 2 T v 1 ( t ) = ⎛ t + 1⎞ u 0 ⎛ t + ⎞ – u 0 ( t ) ⎝T ⎠ ⎝ 2⎠ 2 T v 2 ( t ) = ⎛ – t + 1⎞ u 0 ( t ) – u 0 ⎛ t – ⎞ ⎝ T ⎠ ⎝ 2⎠ and for... Figure 1.2 Signals and Systems with MATLAB Applications, Second Edition Orchard Publications 1-1 Chapter 1 Elementary Signals v out vS 0 t Figure 1.2 Waveform for v out as defined in relation (1.3) The waveform of Figure 1.2 is an example of a discontinuous function A function is said to be discontinuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without . info@orchardpublications.com Signals and Systems with MATLAB Applications Second Edition Steven T. Karris Students and working professionals will find Signals and Systems with MATLAB . iv Signals and Systems with MATLAB Applications, Second Edition Orchard Publications Chapter 9 Discrete Time Systems and the Z Transform Definition and