Signal Computing: Digital Signals in the Software Domain Michael Stiber Bilin Zhang Stiber Eric C. Larson Signal Computing: Digital Signals in the Software Domain Spring 2020 Michael Stiber Bilin Zhang Stiber University of Washington Bothell 18115 Campus Way NE Bothell, Washington 98011 Eric C. Larson Southern Methodist University Lyle School of Engineering 3145 Dyer Street Dallas, TX 75205 Copyright © 2002–2020 by Michael and Bilin Stiber and Eric C. Larson This material is based upon work supported by the National Science Foundation under Grant No. 0443118. Cover art by Eduardo Sainos. This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/. Contents Preface xiii Objectives ........................................ xiv Prerequisites ...................................... xiv About This Book .................................... xvi Typographical Conventions .............................. xvi Further Reading .................................... xvii 1 Signals in the Physical World 1 1.1 Multimedia and Sensation ............................ 1 1.2 Sensation and Perception ............................. 2 1.3 Computer Video Displays ............................ 4 1.4 Multimedia System Operation .......................... 5 1.5 Vibrations and Sound ............................... 6 1.6 Phasors ...................................... 9 1.7 Spectra ...................................... 14 1.7.1 Interlude: Vectors ............................. 15 1.7.2 Derivation of the Fourier Series ..................... 17 1.8 Problems ...................................... 25 1.9 Further Reading .................................. 29 2 Signals in the Computer 31 2.1 From the physical to the digital ......................... 31 2.2 Measuring Noise .................................. 32 2.3 Sampling ...................................... 33 2.3.1 Aliasing .................................. 35 2.4 Quantization ................................... 38 2.5 Dynamic Range .................................. 41 2.6 Periodic and Aperiodic Signals .......................... 41 2.7 Problems ...................................... 42 Signal Computing i 2.8 Further Reading .................................. 43 3 Filtering and Feedforward Filters 45 3.1 Introduction .................................... 45 3.2 Feedforward Filters ................................ 46 3.2.1 Delaying a phasor ............................. 46 3.2.2 A simple feedforward filter ........................ 47 3.2.3 Digital Filters ............................... 50 3.2.4 Delay as an Operator ........................... 52 3.2.5 The z-plane ................................ 55 3.2.6 Phase Response .............................. 59 3.2.7 Implementing Digital Filters ....................... 62 3.3 Problems ...................................... 64 3.4 Further Reading .................................. 65 4 The Z-Transform and Convolution 67 4.1 Domains ...................................... 67 4.2 The z-transform .................................. 68 4.2.1 Example: z-transform of an impulse ................... 69 4.2.2 Example: z-transform of exponential signal ............... 70 4.3 Convolution .................................... 75 4.3.1 Example of Convolution ......................... 75 4.3.2 Implementing Convolution ........................ 76 4.4 Properties of the Z-Transform .......................... 79 4.4.1 Example: Time Shifting ......................... 80 4.4.2 Example: Convolution .......................... 80 4.5 Impulse Response and the Transfer Function .................. 82 4.6 Problems ...................................... 83 4.7 Further Reading .................................. 83 5 Feedback Filters 85 5.1 Introduction .................................... 85 5.1.1 Poles .................................... 85 5.1.2 Example: Computing Transfer Function and Impulse Response .... 88 5.1.3 Stability .................................. 88 5.1.4 Resonance and Bandwidth ........................ 91 5.2 Mixing Feedback and Feedforward Filters .................... 95 5.3 Implementation .................................. 95 5.3.1 Avoiding Complex Numbers ....................... 96 5.3.2 Limitations of Numerical Accuracy ................... 97 5.4 Problems ...................................... 99 5.5 Further Reading .................................. 99 Signal Computing ii 6 Spectral Analysis 101 6.1 The Fourier Transform .............................. 101 6.1.1 Example: Fourier transform of a rectangular pulse ........... 102 6.2 The Discrete Fourier Transform ......................... 104 6.2.1 Derivation of the IDFT [Optional] .................. 106 6.2.2 Finite vs. Infinite Signals ......................... 107 6.2.3 Properties of the DFT .......................... 107 6.2.4 Computing the DFT Directly ...................... 111 6.2.5 The Fast Fourier Transform Algorithm ................. 112 6.3 The inverse DFT ................................. 118 6.3.1 Example: Sum of Two Sinusoids ..................... 119 6.4 Power Leakage [Optional] ........................... 120 6.5 Tradeo↵Between Time and Frequency Resolution [Optional] ........ 124 6.6 Windowing [Optional] ............................. 128 6.7 Problems ...................................... 134 6.8 Further Reading .................................. 135 7 Compression 137 7.1 Signals and Information ............................. 137 7.2 Entropy (Lossless) Compression ......................... 141 7.2.1 Repetitive Sequence Compression .................... 141 7.2.2 Statistical Compression .......................... 142 7.3 Source (Lossy) Compression ........................... 144 7.3.1 Di↵erential Compression ......................... 145 7.3.2 Transform Compression ......................... 146 7.4 Problems ...................................... 147 7.5 Further Reading .................................. 148 8 Audio & Video Compression and Coding 149 8.0.1 Issues in Coding Method Selection ................... 149 8.1 Audio Coding Standards ............................. 150 8.1.1 Speech Coding for Telephony ...................... 150 8.1.2 High-Quality Audio Coding ....................... 151 8.2 Still Image Coding Standards .......................... 153 8.2.1 JPEG ................................... 153 8.3 Video Coding Standards ............................. 159 8.3.1 MPEG Coding .............................. 159 8.4 Problems ...................................... 161 8.5 Further Reading .................................. 162 Signal Computing iii 9 Review and Conclusions 163 9.1 A Generic Digital Multimedia System ...................... 163 9.2 Compact Discs .................................. 163 9.2.1 Data Encoding .............................. 165 9.2.2 CD System Signal Processing ...................... 168 9.3 Conclusion ..................................... 170 9.4 Further Reading .................................. 170 A Answers to Self-Test Exercises 171 A.1 Chapter 1:SignalsinthePhysicalWorld.................... 171 A.2 Chapter 2:SignalsintheComputer....................... 173 A.3 Chapter 3: Filtering and Feedforward Filters .................. 174 A.4 Chapter 4:TheZ-TransformandConvolution ................. 176 A.5 Chapter 5: Feedback Filters ........................... 178 A.6 Chapter 6: Spectral Analysis ........................... 178 A.7 Chapter 7:Compression ............................. 181 A.8 Chapter 8: Audio & Video Compression and Coding ............. 182 A.9 Chapter 9:ReviewandConclusions....................... 182 Index 183 Signal Computing iv List of Figures 0.1 Course conceptual road map ........................... xv 1.1 Schematic diagram of a CRT. .......................... 4 1.2 Interlaced and non-interlaced raster scanning .................. 5 1.3 Real periodic signals ............................... 7 1.4 Components of a phasor are sinusoids. ..................... 9 1.5 Graphical summation of vectors. ......................... 10 1.6 Phasor representation of beating. ........................ 13 1.7 Amplitude vs. time representation of beating. ................. 13 1.8 A train of periodic rectangular pulses ...................... 21 1.9 The sinc function. ................................. 22 1.10 First 12 terms in the Fourier series for a periodic sequence of rectangular pulses 24 1.11 First 100 terms in the Fourier series for a periodic sequence of rectangular pulses 25 1.12 Spectrum of a periodic sequence of rectangular pulses ............. 26 1.13 Spectrum of a periodic sequence of rectangular pulses; varying pulse width . 27 1.14 Spectrum of a periodic sequence of rectangular pulses; varying period .... 28 2.1 Block diagram of the data acquisition process. ................. 31 2.2 Sample and hold output ............................. 34 2.3 Sampling and apparent frequency ........................ 35 2.4 Error in output for 8-bit ADC .......................... 39 3.1 Four frequency components ........................... 46 3.2 Filter frequency response ............................. 47 3.3 A filtered signal .................................. 47 3.4 Unfiltered vs. filtered signal ........................... 48 3.5 A basic feedforward filter block diagram. ................... 48 3.6 Treat transfer function as a black box. ..................... 55 3.7 Two zero filter r =0.9,! ˆ = ⇡/2. ....................... 58 0 ± 3.8 Two zero filter r =0.9,! ˆ = 5⇡/6. .....................