Signals, Systems, and Transforms: Concise Coverage from Theory to Application

Concise, linear textbook exploring the fundamentals of signals and systems analysis using Fourier tools and generalized Fourier tools Signals, Systems, and Transforms covers the fundamentals of analyzing analog and discrete signals and systems in various domains using Fourier and generalized Fourier tools. The book shows how these tool elements are interconnected and weaves them into a sequential coherent story, with each element leading to the next, helping readers more easily grasp newer material due to previously developed concepts. Practically, the book examines how the theory applies to various fields ranging from biomedical imaging to filter designs for audio and video signals. The book includes interesting examples of the theories presented in the textbook for hands-on learning, as well as a lab section in MATLAB, where the reader is shown simulated examples and asked to perform certain tasks using simple MATLAB codes and functions. This book condenses material usually expressed in 800-1200 pages into approximately one-fourth of that length by capitalizing on how the various Fourier transforms relate and by unifying the treatment of the analog and discrete transforms. Signals, Systems, and Transforms includes information on: Laplace transform (LT) and Z-transform (ZT) as generalized Fourier transform. It then uses the concepts of transforms in the analysis of linear systems with rational Laplace and Z-transform The discrete Fourier transform (DFT) and its fast computation using fast Fourier transform (FFT) as a sampler in the Fourier domain Sampling of double-sided lowpass and bandpass signals (double- and single-sided) using concepts of modulation and Hilbert transform (HT) Quantization of signals, covering optimal quantizers, uniform quantizers, and compandors It introduces Singular Value Decomposition (SVD) as a subcase of optimal quantizers Unitary (energy-preserving) transforms such as the suboptimal cosine transforms Signals, Systems, and Transforms provides all of the necessary knowledge for electrical and computer engineering students to master fundamental tools related to the subject and be able to branch into the fascinating fields of signal processing and modeling, signal and system control, and power.

Table of Contents:
Preface xiii Acknowledgments xix List of Symbols xxi 1 Signals and Systems 1 1.1 Different Types of Signals 1 1.1.1 Causal, Anticausal, and Noncausal Signals 2 1.2 Symmetry in Signals 2 1.3 Singularity Functions 3 1.3.1 Impulse Function 3 1.3.2 Step Function 4 1.3.3 Rectangular Function 5 1.3.4 Ramp Function 5 1.3.5 Triangular Function 6 1.4 Transforming a Signal 6 1.4.1 Time Transformation 6 1.4.2 Amplitude Transformation 7 1.5 Characterizing a System and System’s Properties 7 1.5.1 Linearity 7 1.5.2 Time Invariance 8 1.5.3 Invertibility 8 1.5.4 Causality 9 1.5.5 Bounded-Input Bounded-Output Stability 9 1.6 Impulse Response of an LTI System 9 1.6.1 Convolution Integral and Sum 10 1.6.2 Step and Ramp Responses 10 1.7 Impulse Response of LTI, RL, or RC Circuit 12 1.8 Second-Order Systems 13 1.9 Impulse Response of Rational Systems 15 1.10 Deconvolution 15 1.11 Chapter Summary 16 Problems 17 2 Fourier Series (FS) 25 2.1 Development of Fourier Series: A Historical Perspective 25 2.2 Vector and Function Spaces 26 2.2.1 Geometric Projection and Dot Product 26 2.2.2 Function Space and Dot Product 27 2.3 Sinusoidal Functions 28 2.4 Fourier Series Expansion 29 2.4.1 Existence of Fourier Series 30 2.5 Fourier Series of Certain Signals and Properties 34 2.5.1 Train of Impulse Function 34 2.5.2 Shifting Property 34 2.5.3 Derivative Property 34 2.6 Spectral Representation of Periodic Functions 37 2.7 Fourier Series for Discrete Periodic Signals 38 2.7.1 2D Discrete Periodic Signal 39 2.8 Signal and Image Compression 39 2.8.1 Lossy Compression 39 2.9 Fourier Descriptors for Boundary Representation 41 2.10 Chapter Summary 43 Problems 44 2.A Appendix 47 References 47 3 Fourier Transform 49 3.1 Development of Fourier Transform 49 3.2 Fourier Transform from Fourier Series 49 3.3 Inverse Fourier Transform 50 3.3.1 Existence of Fourier Transform 51 3.4 Fourier Transform of Certain Signals 51 3.4.1 FT of an Impulse Function 51 3.4.2 FT of an Exponential Function 51 3.5 Properties of FT 51 3.5.1 Shifting Property 51 3.5.2 Derivative Property 52 3.5.3 Convolution Property 52 3.5.4 Eigenfunction Eigenvalue (EE) Property 53 3.5.5 Integral Property 54 3.5.6 Duality Property 54 3.5.7 Modulation Property 54 3.5.8 Scaling Property 55 36 Parseval’s Theorem 56 3.7 FT for Discrete Signals (DTFT) 57 3.7.1 Inverse DTFT 58 3.7.2 Properties of DTFT 60 3.7.2.1 Shifting Property 60 3.7.2.2 Derivative Property 60 3.7.2.3 Summation Property 61 3.8 Parseval’s Theorem for DTFT 62 3.9 Discrete Rational System Approximation to a Continuous Rational System 62 3.10 Phasors and FT 64 3.11 FT in Medical Imaging 67 3.11.1 Projection Theorem 68 3.11.1.1 Projection Slice Theorem 68 3.12 Chapter Summary 69 Problems 69 References 74 4 DFT and FFT 75 4.1 Development of Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) 75 4.2 DFT as Samples on the DTFT 75 4.2.1 Conditions for Retrieving x(n) from y(n) 76 4.3 IDFT 76 4.4 Circular Convolution 77 4.5 FFT 80 4.5.1 Radix 2 FFT 81 4.5.2 8-FFT Using Two 4-DFTs or Four 2-DFTs 82 4.5.3 9-FFT Using Three 3-DFT Units 84 4.5.4 Radix m-FFT 86 4.5.5 Decimation in Frequency 88 4.6 Chapter Summary 89 Problems 89 References 92 5 Laplace Transform 93 5.1 Development of Laplace Transform (LT) 93 5.2 The LT as a Generalized FT 94 5.3 Relationship Between the FT and the LT 94 5.4 The ROC for Rational Signals 95 5.4.1 The ROC for Rational Causal Signal of Infinite duration 95 5.4.2 The ROC for Rational Anticausal Signal- of Infinite duration 96 5.4.3 ROC for Rational Noncausal Signals 97 5.5 Inverse LT (ILT) 97 5.6 Laplace Transform of Certain Signals 99 5.6.1 The LT of an Impulse Function 99 5.6.2 The LT of an Exponential Function 99 5.7 Properties of the LT 100 5.7.1 Shifting Property 100 5.7.2 Derivative Property 101 5.7.3 Convolution Property 101 5.7.4 Eigenfunction Eigenvalue Property 102 5.7.5 Integral Property 102 5.7.6 Scaling Property 102 5.7.7 Complex Shift and Duality Property 103 5.7.8 Initial and Final Values 106 5.8 Rational Systems 106 5.8.1 Poles and Zeros of a Rational System 108 5.8.2 The BIBO Stable System 108 5.8.3 Step and Ramp Responses as Functions of the Impulse Response h(t) 109 5.9 Chapter Summary 110 Problems 111 References 114 6 Z-Transform 115 6.1 Development of Z-transform (ZT) 115 6.2 The ZT as Generalized DTFT 115 6.3 Relationship Between DTFT and ZT 116 6.4 Region of Convergence (ROC) for Rational Signals 116 6.4.1 ROC for Rational Causal Signals of Infinite Duration 117 6.4.2 ROC for Rational Anticausal Signals 118 6.4.3 ROC for Rational Noncausal Signals 119 6.5 Inverse ZT (IZT) 119 6.5.1 Cauchy’s Residue Theorem for IZT 119 6.6 The ZT of Certain Signals 120 6.6.1 The ZT of a Dirac Delta Function 120 6.6.2 The ZT of an Exponent Function 121 6.7 Properties of the ZT 121 6.7.1 Shifting Property 121 6.7.2 Difference and Summation Property 121 6.7.3 Derivative Property 122 6.7.4 Convolution Property 122 6.7.5 Eigenfunction Eigenvalue Property 122 6.7.6 Scaling Property 123 6.7.7 Time Reversal 123 6.8 Initial and Final Values 124 6.9 Relationship Between LT and ZT 125 6.10 Rational Systems 127 6.10.1 Poles and Zeros of a Rational System 127 6.10.2 BIBO Stable System 127 6.10.3 Step and Ramp Responses as Functions of the Impulse Response h(n) 128 6.11 Connecting the Various FT and GFT for Analog and Discrete Signals 130 6.12 Chapter Summary 131 Problems 131 References 133 7 Sampler 135 7.1 Development of Digitizer – The Sampler and Quantizer 135 7.2 Sampler and Nyquist Rate 135 7.2.1 Sampling of Bandpass Signals 137 7.3 Sampling of Images 142 7.4 Nonuniform Sampling 143 7.5 Chapter Summary 145 Problems 146 References 148 8 Quantizer 149 8.1 Development of the Quantizer 149 8.2 Lloyd-Max Quantizer 149 8.2.1 Lloyd-Max Solution 150 8.3 Uniform Quantizer 151 8.4 Suboptimal Quantizer 152 8.5 Intuitive Quantizer 153 8.6 Compandor Quantizer 155 8.7 Dithering – Reducing Quantization Distortions 157 8.8 Halftoning 157 8.9 Chapter Summary 159 Problems 160 References 163 9 Unitary Transforms 165 9.1 Development of Unitary Transforms 165 9.2 Unitary Transforms in Vector Spaces 165 9.3 Compactness Property 166 9.4 Optimal Transform 168 9.5 DFT as a Unitary Transform 170 9.6 DCT as a Unitary Transform 171 9.7 Singular Value Decomposition (SVD) and Unitary Transforms 173 9.8 Chapter Summary 175 Problems 175 References 178 10 Applications 179 10.1 Renography 179 10.2 Filter Design 180 10.2.1 Ideal Low Pass Filter 181 10.2.2 Digital Filters 182 10.3 Signal and Image Restoration 185 10.4 Chapter Summary 188 Problems 188 MATLAB Exercise 194 Appendix A Complex Variable Calculus 195 Index 000

About the Author :
Fernand Cohen is a Professor in the Electrical and Computer Engineering Department at Drexel University. He joined Drexel University in 1987. Prior, he was an Assistant Professor in the Electrical Engineering Department at the University of Rhode Island. He received his PhD and MSc in Electrical Engineering from Brown University in 1980 and 1983, respectively.