Signals, Systems, Transforms, and Digital Signal Processing ® with MATLAB This page intentionally left blank Signals, Systems, Transforms, and Digital Signal Processing ® with MATLAB Michael Corinthios École Polytechnique de Montréal Montréal, Canada MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Corinthios, Michael. Signals, systems, transforms, and digital signal processing with MATLAB / Michael Corinthios. p. cm. Includes bibliographical references and index. ISBN 978-1-4200-9048-2 (hard back : alk. paper) 1. Signal processing--Digital techniques. 2. System analysis. 3. Fourier transformations. 4. MATLAB. I. Title. TK5102.9.C64 2009 621.382’2--dc22 2009012640 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Maria, Angela, Gis`ele, John. v This page intentionally left blank Contents Preface xxv Acknowledgment xxvii 1 Continuous-Time and Discrete-Time Signals and Systems 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Continuous-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Unit Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Graphical Representation of Functions . . . . . . . . . . . . . . . . . . . . 5 1.6 Even and Odd Parts of a Function . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Dirac-Delta Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.8 Basic Properties of the Dirac-Delta Impulse . . . . . . . . . . . . . . . . . 8 1.9 Other Important Properties of the Impulse . . . . . . . . . . . . . . . . . 11 1.10 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.11 Causality, Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.12 Examples of Electrical Continuous-Time Systems . . . . . . . . . . . . . . 12 1.13 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.14 Transfer Function and Frequency Response . . . . . . . . . . . . . . . . . 14 1.15 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.16 A Right-Sided and a Left-Sided Function . . . . . . . . . . . . . . . . . . . 20 1.17 Convolution with an Impulse and Its Derivatives . . . . . . . . . . . . . . 21 1.18 Additional Convolution Properties . . . . . . . . . . . . . . . . . . . . . . 21 1.19 Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.20 Properties of the Correlation Function . . . . . . . . . . . . . . . . . . . . 22 1.21 Graphical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.22 Correlation of Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . 25 1.23 Average, Energy and Power of Continuous-Time Signals . . . . . . . . . . 25 1.24 Discrete-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.25 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.26 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.27 Even/Odd Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.28 Average Value, Energy and Power Sequences . . . . . . . . . . . . . . . . 29 1.29 Causality, Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.30 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.31 Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 Fourier Series Expansion 47 2.1 Trigonometric Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Exponential Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 Exponential versus Trigonometric Series . . . . . . . . . . . . . . . . . . . 50 2.4 Periodicity of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 51 vii ® viii Signals, Systems, Transforms and Digital Signal Processing with MATLAB 2.5 Dirichlet Conditions and Function Discontinuity . . . . . . . . . . . . . . 53 2.6 Proof of the Exponential Series Expansion . . . . . . . . . . . . . . . . . . 55 2.7 Analysis Interval versus Function Period . . . . . . . . . . . . . . . . . . . 55 2.8 Fourier Series as a Discrete-Frequency Spectrum . . . . . . . . . . . . . . 56 2.9 Meaning of Negative Frequencies . . . . . . . . . . . . . . . . . . . . . . . 58 2.10 Properties of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.10.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.10.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.10.3 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.10.4 Function Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.10.5 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.10.6 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.10.7 Half-Periodic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 65 2.10.8 Double Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.10.9 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.10.10 Differentiation Property . . . . . . . . . . . . . . . . . . . . . . . . 72 2.11 Differentiation of Discontinuous Functions . . . . . . . . . . . . . . . . . . 74 2.11.1 Multiplication in the Time Domain . . . . . . . . . . . . . . . . . . 74 2.11.2 Convolution in the Time Domain . . . . . . . . . . . . . . . . . . . 75 2.11.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.12 Fourier Series of an Impulse Train . . . . . . . . . . . . . . . . . . . . . . 77 2.13 Expansion into Cosine or Sine Fourier Series . . . . . . . . . . . . . . . . . 78 2.14 Deducing a Function Form from Its Expansion . . . . . . . . . . . . . . . 81 2.15 Truncated Sinusoid Spectral Leakage . . . . . . . . . . . . . . . . . . . . . 83 2.16 The Period of a Composite Sinusoidal Signal . . . . . . . . . . . . . . . . 86 2.17 Passage through a Linear System . . . . . . . . . . . . . . . . . . . . . . . 88 2.18 Parseval’s Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.19 Use of Power Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . 90 2.20 Inverse Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.21 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.22 Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . . . 100 3 Laplace Transform 105 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Bilateral Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3 Conditions of Existence of Laplace Transform . . . . . . . . . . . . . . . . 107 3.4 Basic Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.5 Notes on the ROC of Laplace Transform . . . . . . . . . . . . . . . . . . . 112 3.6 Properties of Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . 115 3.6.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.6.2 Differentiation in Time . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.6.3 Multiplication by Powers of Time . . . . . . . . . . . . . . . . . . . 116 3.6.4 Convolution in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.6.5 Integration in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.6.6 Multiplication by an Exponential (Modulation) . . . . . . . . . . . 118 3.6.7 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.6.8 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6.9 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6.10 Final Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6.11 Laplace Transform of Anticausal Functions . . . . . . . . . . . . . 120 3.6.12 Shift in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Table of Contents ix 3.7 Applications of the Differentiation Property . . . . . . . . . . . . . . . . . 122 3.8 Transform of Right-Sided Periodic Functions . . . . . . . . . . . . . . . . 123 3.9 Convolution in Laplace Domain . . . . . . . . . . . . . . . . . . . . . . . . 124 3.10 Cauchy’s Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.11 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.12 Case of Conjugate Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.13 The Expansion Theorem of Heaviside . . . . . . . . . . . . . . . . . . . . . 131 3.14 Application to Transfer Function and Impulse Response . . . . . . . . . . 132 3.15 Inverse Transform by Differentiation and Integration . . . . . . . . . . . . 133 3.16 Unilateral Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.16.1 Differentiation in Time . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.16.2 Initial and Final Value Theorem . . . . . . . . . . . . . . . . . . . 137 3.16.3 Integration in Time Property . . . . . . . . . . . . . . . . . . . . . 137 3.16.4 Division by Time Property . . . . . . . . . . . . . . . . . . . . . . 137 3.17 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.18 Table of Additional Laplace Transforms . . . . . . . . . . . . . . . . . . . 141 3.19 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.20 Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . . . 149 4 Fourier Transform 153 4.1 Definition of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 153 4.2 Fourier Transform as a Function of f . . . . . . . . . . . . . . . . . . . . 155 4.3 From Fourier Series to Fourier Transform . . . . . . . . . . . . . . . . . . 156 4.4 Conditions of Existence of the Fourier Transform . . . . . . . . . . . . . . 157 4.5 Table of Properties of the Fourier Transform . . . . . . . . . . . . . . . . . 158 4.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.5.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.5.3 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.5.4 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.5.5 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.5.6 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.5.7 Modulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.5.8 Initial Time Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.5.9 Initial Frequency Value . . . . . . . . . . . . . . . . . . . . . . . . 163 4.5.10 Differentiation in Time . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.5.11 Differentiation in Frequency . . . . . . . . . . . . . . . . . . . . . . 164 4.5.12 Integration in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.5.13 Conjugate Function . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.5.14 Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.5.15 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.6 System Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.7 Even–Odd Decomposition of a Real Function . . . . . . . . . . . . . . . . 167 4.8 Causal Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.9 Transform of the Dirac-Delta Impulse . . . . . . . . . . . . . . . . . . . . 169 4.10 Transform of a Complex Exponential and Sinusoid . . . . . . . . . . . . . 169 4.11 Sign Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.12 Unit Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.13 Causal Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.14 Table of Fourier Transforms of Basic Functions . . . . . . . . . . . . . . . 172 4.15 Relation between Fourier and Laplace Transforms . . . . . . . . . . . . . . 174 4.16 Relation to Laplace Transform with Poles on Imaginary Axis . . . . . . . 175