APPLIED SIGNAL PROCESSING SADASIVAN PUTHUSSERYPADY Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 United States Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 ISBN: 978-1-68083-978-4 E-ISBN: 978-1-68083-979-1 DOI: 10.1561/9781680839791 Copyright © 2021 Sadasivan Puthusserypady Suggested citation: Sadasivan Puthusserypady. (2021). Applied Signal Processing. Boston–Delft: Now Publishers The work will be available online open access and governed by the Creative Commons “Attribution-Non Commercial” License (CC BY-NC), according to https://creativecommons.org/ licenses/by-nc/4.0/ Table of Contents Preface xv Acknowledgements xvii Notations, Symbols xix Glossary xxv Chapter 1 Introduction 1 1.1 What is a Signal?. 3 1.2 Classification of Signals . 6 1.2.1 Analog or Digital Signals . 6 1.2.2 Periodic and Aperiodic Signals . 9 1.2.3 Deterministic and Random Signals . 10 1.2.4 Real and Complex Signals . 10 1.3 Typical Real World Biomedical Signals. 11 1.3.1 Electroencephalogram (EEG) . 11 1.3.2 Electrocardiogram (ECG) . 14 1.3.3 Electrooculogram (EOG) . 14 1.3.4 Electroretinogram (ERG) . 15 1.3.5 Electromyogram (EMG) . 15 1.4 Concluding Remarks . 16 Chapter 2 Power and Energy 17 2.1 Power and Energy of Signals . 17 2.2 Energy Signals and Power Signals . 19 2.2.1 Single Real Sinusoid . 19 2.2.2 Single Complex Sinusoid. 20 iii iv Table of Contents 2.2.3 Sum of Complex Sinusoids . 21 2.2.4 Sum of Real Sinusoids . 22 2.3 Concluding Remarks . 23 Chapter 3 Fourier Series 27 3.1 Introduction to Fourier Series . 29 3.2 Why Fourier Series? . 30 3.3 Fourier Series: Definition and Interpretation . 31 3.4 Properties of Fourier Series . 34 3.4.1 Linearity . 34 3.4.2 Time Shifting . 35 3.4.3 Frequency Shifting or Modulation. 36 3.4.4 Time Reversal . 36 3.4.5 Even-Symmetric Signal . 36 3.4.6 Odd-Symmetric Signal . 37 3.4.7 Real Signal . 37 3.5 Real Fourier Series . 38 3.6 Examples of Fourier Series Evaluation. 40 3.7 Limitations of Fourier Series. 45 3.7.1 Gibbs Phenomenon . 45 3.7.2 Dirichlet Conditions . 45 3.8 Concluding Remarks . 46 Chapter 4 Fourier Transform 51 4.1 Introduction to Fourier Transform . 52 4.2 Fourier Transform: Development and Interpretation . 53 4.3 Properties of Fourier Transform. 57 4.3.1 Linearity . 58 4.3.2 Time Shifting . 58 4.3.3 Frequency Shifting or Modulation. 59 4.3.4 Time Reversal . 60 4.3.5 Time Scaling . 60 4.3.6 Time Scaling and Delay. 62 4.3.7 Time Differentiation . 64 4.3.8 Duality . 64 4.3.9 Symmetric Signals. 65 4.4 Fourier Transform of Periodic Signals . 66 4.4.1 Difficulty with Sinusoids . 67 4.4.2 Impulse Function . 68 4.4.3 Properties of Impulse Function . 69 Table of Contents v 4.4.4 Fourier Transform of Sinusoids . 70 4.4.5 Fourier Transform of Periodic Signals . 71 4.5 Dirichlet Conditions . 72 4.6 Fourier Transform Summary . 72 4.7 Concluding Remarks . 74 Chapter 5 Complex Signals 83 5.1 Introduction to Complex Signals . 83 5.1.1 Some Useful Rules and Identities . 85 5.1.2 Phasors . 85 5.2 Spectrum of Complex Signals. 87 5.2.1 Properties of the Fourier Transform of Complex Signals . 88 5.3 Linear Processing of Complex Signals . 89 5.4 Hilbert Transform . 90 5.4.1 Hilbert Transform as a Filter . 91 5.4.2 Properties of Hilbert Transform. 93 5.5 Analytic Signals . 94 5.5.1 Analytic Analog Signals . 94 5.5.2 Analytic Discrete-time Signals . 96 5.6 Instantaneous Amplitude and Frequency . 97 5.7 Concluding Remarks . 100 Chapter 6 Analog Systems 103 6.1 Classification of Systems . 104 6.1.1 Continuous-Time or Discrete-Time Systems . 104 6.1.2 Time-Invariant Systems . 104 6.1.3 Causal Systems . 104 6.1.4 Linear or Non-linear Systems . 104 6.2 Description of a System. 105 6.2.1 Input-Output Equation . 105 6.2.2 Frequency Response . 105 6.2.3 Impulse Response . 105 6.3 Why Study of Systems is Important?. 106 6.3.1 System Identification . 106 6.3.2 Output Evaluation . 107 6.3.3 Input Estimation. 107 6.4 Frequency Response . 107 6.5 Sinusoidal Input to Linear Systems . 110 6.5.1 Generalization to Multiple Sinusoids . 111 6.6 Partial Fraction Approach . 111 vi.