A Practical Approach to Signals and Systems

Concisely covers all the important concepts in an easy-to-understand way Gaining a strong sense of signals and systems fundamentals is key for general proficiency in any electronic engineering discipline, and critical for specialists in signal processing, communication, and control. At the same time, there is a pressing need to gain mastery of these concepts quickly, and in a manner that will be immediately applicable in the real word. Simultaneous study of both continuous and discrete signals and systems presents a much easy path to understanding signals and systems analysis. In A Practical Approach to Signals and Systems, Sundararajan details the discrete version first followed by the corresponding continuous version for each topic, as discrete signals and systems are more often used in practice and their concepts are relatively easier to understand. In addition to examples of typical applications of analysis methods, the author gives comprehensive coverage of transform methods, emphasizing practical methods of analysis and physical interpretations of concepts. Gives equal emphasis to theory and practice Presents methods that can be immediately applied Complete treatment of transform methods Expanded coverage of Fourier analysis Self-contained: starts from the basics and discusses applications Visual aids and examples makes the subject easier to understand End-of-chapter exercises, with a extensive solutions manual for instructors MATLAB software for readers to download and practice on their own Presentation slides with book figures and slides with lecture notes A Practical Approach to Signals and Systems is an excellent resource for the electrical engineering student or professional to quickly gain an understanding of signal analysis concepts - concepts which all electrical engineers will eventually encounter no matter what their specialization. For aspiring engineers in signal processing, communication, and control, the topics presented will form a sound foundation to their future study, while allowing them to quickly move on to more advanced topics in the area. Scientists in chemical, mechanical, and biomedical areas will also benefit from this book, as increasing overlap with electrical engineering solutions and applications will require a working understanding of signals. Compact and self contained, A Practical Approach to Signals and Systems be used for courses or self-study, or as a reference book.

Table of Contents:
Preface xiii Abbreviations xv 1 Introduction 1 1.1 The Organization of this Book 1 2 Discrete Signals 5 2.1 Classification of Signals 5 2.1.1 Continuous, Discrete and Digital Signals 5 2.1.2 Periodic and Aperiodic Signals 7 2.1.3 Energy and Power Signals 7 2.1.4 Even- and Odd-symmetric Signals 8 2.1.5 Causal and Noncausal Signals 10 2.1.6 Deterministic and Random Signals 10 2.2 Basic Signals 11 2.2.1 Unit-impulse Signal 11 2.2.2 Unit-step Signal 12 2.2.3 Unit-ramp Signal 13 2.2.4 Sinusoids and Exponentials 13 2.3 Signal Operations 20 2.3.1 Time Shifting 21 2.3.2 Time Reversal 21 2.3.3 Time Scaling 22 2.4 Summary 23 Further Reading 23 Exercises 23 3 Continuous Signals 29 3.1 Classification of Signals 29 3.1.1 Continuous Signals 29 3.1.2 Periodic and Aperiodic Signals 30 3.1.3 Energy and Power Signals 31 3.1.4 Even- and Odd-symmetric Signals 31 3.1.5 Causal and Noncausal Signals 33 3.2 Basic Signals 33 3.2.1 Unit-step Signal 33 3.2.2 Unit-impulse Signal 34 3.2.3 Unit-ramp Signal 42 3.2.4 Sinusoids 43 3.3 Signal Operations 45 3.3.1 Time Shifting 45 3.3.2 Time Reversal 46 3.3.3 Time Scaling 47 3.4 Summary 48 Further Reading 48 Exercises 48 4 Time-domain Analysis of Discrete Systems 53 4.1 Difference Equation Model 53 4.1.1 System Response 55 4.1.2 Impulse Response 58 4.1.3 Characterization of Systems by their Responses to Impulse and Unit-step Signals 60 4.2 Classification of Systems 61 4.2.1 Linear and Nonlinear Systems 61 4.2.2 Time-invariant and Time-varying Systems 62 4.2.3 Causal and Noncausal Systems 63 4.2.4 Instantaneous and Dynamic Systems 64 4.2.5 Inverse Systems 64 4.2.6 Continuous and Discrete Systems 64 4.3 Convolution–Summation Model 64 4.3.1 Properties of Convolution–Summation 67 4.3.2 The Difference Equation and Convolution–Summation 68 4.3.3 Response to Complex Exponential Input 69 4.4 System Stability 71 4.5 Realization of Discrete Systems 72 4.5.1 Decomposition of Higher-order Systems 73 4.5.2 Feedback Systems 74 4.6 Summary 74 Further Reading 75 Exercises 75 5 Time-domain Analysis of Continuous Systems 79 5.1 Classification of Systems 80 5.1.1 Linear and Nonlinear Systems 80 5.1.2 Time-invariant and Time-varying Systems 81 5.1.3 Causal and Noncausal Systems 82 5.1.4 Instantaneous and Dynamic Systems 83 5.1.5 Lumped-parameter and Distributed-parameter Systems 83 5.1.6 Inverse Systems 83 5.2 Differential Equation Model 83 5.3 Convolution-integral Model 85 5.3.1 Properties of the Convolution-integral 87 5.4 System Response 88 5.4.1 Impulse Response 88 5.4.2 Response to Unit-step Input 89 5.4.3 Characterization of Systems by their Responses to Impulse and Unit-step Signals 91 5.4.4 Response to Complex Exponential Input 92 5.5 System Stability 93 5.6 Realization of Continuous Systems 94 5.6.1 Decomposition of Higher-order Systems 94 5.6.2 Feedback Systems 95 5.7 Summary 96 Further Reading 97 Exercises 97 6 The Discrete Fourier Transform 101 6.1 The Time-domain and the Frequency-domain 101 6.2 Fourier Analysis 102 6.2.1 Versions of Fourier Analysis 104 6.3 The Discrete Fourier Transform 104 6.3.1 The Approximation of Arbitrary Waveforms with a Finite Number of Samples 104 6.3.2 The DFT and the IDFT 105 6.3.3 DFT of Some Basic Signals 107 6.4 Properties of the Discrete Fourier Transform 110 6.4.1 Linearity 110 6.4.2 Periodicity 110 6.4.3 Circular Shift of a Sequence 110 6.4.4 Circular Shift of a Spectrum 111 6.4.5 Symmetry 111 6.4.6 Circular Convolution of Time-domain Sequences 112 6.4.7 Circular Convolution of Frequency-domain Sequences 113 6.4.8 Parseval’s Theorem 114 6.5 Applications of the Discrete Fourier Transform 114 6.5.1 Computation of the Linear Convolution Using the DFT 114 6.5.2 Interpolation and Decimation 115 6.6 Summary 119 Further Reading 119 Exercises 119 7 Fourier Series 123 7.1 Fourier Series 123 7.1.1 FS as the Limiting Case of the DFT 123 7.1.2 The Compact Trigonometric Form of the FS 125 7.1.3 The Trigonometric Form of the FS 126 7.1.4 Periodicity of the FS 126 7.1.5 Existence of the FS 126 7.1.6 Gibbs Phenomenon 130 7.2 Properties of the Fourier Series 132 7.2.1 Linearity 133 7.2.2 Symmetry 133 7.2.3 Time Shifting 135 7.2.4 Frequency Shifting 135 7.2.5 Convolution in the Time-domain 136 7.2.6 Convolution in the Frequency-domain 137 7.2.7 Duality 138 7.2.8 Time Scaling 138 7.2.9 Time Differentiation 139 7.2.10 Time Integration 140 7.2.11 Parseval’s Theorem 140 7.3 Approximation of the Fourier Series 141 7.3.1 Aliasing Effect 142 7.4 Applications of the Fourier Series 144 7.5 Summary 145 Further Reading 145 Exercises 145 8 The Discrete-time Fourier Transform 151 8.1 The Discrete-time Fourier Transform 151 8.1.1 The DTFT as the Limiting Case of the DFT 151 8.1.2 The Dual Relationship Between the DTFT and the FS 156 8.1.3 The DTFT of a Discrete Periodic Signal 158 8.1.4 Determination of the DFT from the DTFT 158 8.2 Properties of the Discrete-time Fourier Transform 159 8.2.1 Linearity 159 8.2.2 Time Shifting 159 8.2.3 Frequency Shifting 160 8.2.4 Convolution in the Time-domain 161 8.2.5 Convolution in the Frequency-domain 162 8.2.6 Symmetry 163 8.2.7 Time Reversal 164 8.2.8 Time Expansion 164 8.2.9 Frequency-differentiation 166 8.2.10 Difference 166 8.2.11 Summation 167 8.2.12 Parseval’s Theorem and the Energy Transfer Function 168 8.3 Approximation of the Discrete-time Fourier Transform 168 8.3.1 Approximation of the Inverse DTFT by the IDFT 170 8.4 Applications of the Discrete-time Fourier Transform 171 8.4.1 Transfer Function and the System Response 171 8.4.2 Digital Filter Design Using DTFT 174 8.4.3 Digital Differentiator 174 8.4.4 Hilbert Transform 175 8.5 Summary 178 Further Reading 178 Exercises 178 9 The Fourier Transform 183 9.1 The Fourier Transform 183 9.1.1 The FT as a Limiting Case of the DTFT 183 9.1.2 Existence of the FT 185 9.2 Properties of the Fourier Transform 190 9.2.1 Linearity 190 9.2.2 Duality 190 9.2.3 Symmetry 191 9.2.4 Time Shifting 192 9.2.5 Frequency Shifting 192 9.2.6 Convolution in the Time-domain 193 9.2.7 Convolution in the Frequency-domain 194 9.2.8 Conjugation 194 9.2.9 Time Reversal 194 9.2.10 Time Scaling 194 9.2.11 Time-differentiation 195 9.2.12 Time-integration 197 9.2.13 Frequency-differentiation 198 9.2.14 Parseval’s Theorem and the Energy Transfer Function 198 9.3 Fourier Transform of Mixed Classes of Signals 200 9.3.1 The FT of a Continuous Periodic Signal 200 9.3.2 Determination of the FS from the FT 202 9.3.3 The FT of a Sampled Signal and the Aliasing Effect 203 9.3.4 The FT of a Sampled Aperiodic Signal and the DTFT 206 9.3.5 The FT of a Sampled Periodic Signal and the DFT 207 9.3.6 Approximation of a Continuous Signal from its Sampled Version 209 9.4 Approximation of the Fourier Transform 209 9.5 Applications of the Fourier Transform 211 9.5.1 Transfer Function and System Response 211 9.5.2 Ideal Filters and their Unrealizability 214 9.5.3 Modulation and Demodulation 215 9.6 Summary 219 Further Reading 219 Exercises 219 10 The z-Transform 227 10.1 Fourier Analysis and the z-Transform 227 10.2 The z-Transform 228 10.3 Properties of the z-Transform 232 10.3.1 Linearity 232 10.3.2 Left Shift of a Sequence 233 10.3.3 Right Shift of a sequence 234 10.3.4 Convolution 234 10.3.5 Multiplication by n 235 10.3.6 Multiplication by an 235 10.3.7 Summation 236 10.3.8 Initial Value 236 10.3.9 Final Value 237 10.3.10 Transform of Semiperiodic Functions 237 10.4 The Inverse z-Transform 237 10.4.1 Finding the Inverse z-Transform 238 10.5 Applications of the z-Transform 243 10.5.1 Transfer Function and System Response 243 10.5.2 Characterization of a System by its Poles and Zeros 245 10.5.3 System Stability 247 10.5.4 Realization of Systems 248 10.5.5 Feedback Systems 251 10.6 Summary 253 Further Reading 253 Exercises 253 11 The Laplace Transform 259 11.1 The Laplace Transform 259 11.1.1 Relationship Between the Laplace Transform and the z-Transform 262 11.2 Properties of the Laplace Transform 263 11.2.1 Linearity 263 11.2.2 Time Shifting 264 11.2.3 Frequency Shifting 264 11.2.4 Time-differentiation 265 11.2.5 Integration 267 11.2.6 Time Scaling 268 11.2.7 Convolution in Time 268 11.2.8 Multiplication by t 269 11.2.9 Initial Value 269 11.2.10 Final Value 270 11.2.11 Transform of Semiperiodic Functions 270 11.3 The Inverse Laplace Transform 271 11.4 Applications of the Laplace Transform 272 11.4.1 Transfer Function and System Response 272 11.4.2 Characterization of a System by its Poles and Zeros 273 11.4.3 System Stability 274 11.4.4 Realization of Systems 276 11.4.5 Frequency-domain Representation of Circuits 276 11.4.6 Feedback Systems 279 11.4.7 Analog Filters 282 11.5 Summary 285 Further Reading 285 Exercises 285 12 State-space Analysis of Discrete Systems 293 12.1 The State-space Model 293 12.1.1 Parallel Realization 297 12.1.2 Cascade Realization 299 12.2 Time-domain Solution of the State Equation 300 12.2.1 Iterative Solution 300 12.2.2 Closed-form Solution 301 12.2.3 The Impulse Response 307 12.3 Frequency-domain Solution of the State Equation 308 12.4 Linear Transformation of State Vectors 310 12.5 Summary 312 Further Reading 313 Exercises 313 13 State-space Analysis of Continuous Systems 317 13.1 The State-space Model 317 13.2 Time-domain Solution of the State Equation 322 13.3 Frequency-domain Solution of the State Equation 327 13.4 Linear Transformation of State Vectors 330 13.5 Summary 332 Further Reading 333 Exercises 333 Appendix A: Transform Pairs and Properties 337 Appendix B: Useful Mathematical Formulas 349 Answers to Selected Exercises 355 Index 377

About the Author :
D. Sundararajan is a consultant in digital signal processing at NewTech Software Pt. Ltd in Bangalore India. He has taught undergraduate and graduate classes in digital signal processing, engineering mathematics, programming, operating systems and digital logic design at Concordia University, Canada and Nanyang Technological Institute, Singapore. Sundararajan is the principal inventor of the latest family of Discrete Fourier Transform (DFT) algorithms, and has published two books on DFT and related areas. He also wrote a lab manual for the Digital Electronics Course at Concordia University, and has three patents in fast Fourier transforms. Over the course of his engineering career he has held positions at the National Aerospace Laboratory, Bangalore, the National Physical Laboratory, New Delhi, working on the design of digital and analog signal processing systems. Sundararajan holds a B.E. in Electrical Engineering from Madras University, an M.Tech in Electrical Engineering from IIT Chennai, and a Ph.D. in the same from Concordia University.