Digital Signal Processing: Illustration Using Python

Preface....5

Motivation....5

Target Audience....6

Salient Features of the Book....6

Organization of the Book....7

Acknowledgments....9

Contents....10

About the Authors....15

Chapter 1: Generation of Continuous-Time Signals....17

1.1 Continuous-Time Signal....18

1.1.1 Continuous-Time Periodic Signal....19

1.1.2 Exponential Function....24

1.2 Non-stationary Signal....33

1.3 Non-sinusoidal Waveform....37

1.3.1 Square Waveform....37

1.3.2 Triangle and Sawtooth Waveform....38

1.3.3 Sinc Function....38

1.3.4 Pulse Signal....41

1.3.5 Gaussian Function....42

Bibliography....49

Chapter 2: Sampling and Quantization of Signals....50

2.1 Sampling of Signal....51

2.1.1 Violation of Sampling Theorem....54

2.1.2 Quantization of Signal....60

2.1.2.1 Mid-Tread Quantizer....61

2.1.3 Mid-Rise Quantizer....64

2.2 Non-uniform Quantization....70

2.3 Signal Reconstruction....74

2.3.1 Zero-Order Hold Interpolation....75

2.3.2 First-Order Hold Interpolation....77

2.3.3 Ideal or Sinc Interpolation....79

Bibliography....85

Chapter 3: Generation and Operation on Discrete-Time Sequence....86

3.1 Generation of Discrete-Time Signals....88

3.2 Mathematical Operation on Discrete-Time Signals....99

3.2.1 Amplitude Modification on DT Signal....99

3.2.1.1 Time Scaling Operation....106

3.2.1.2 Time Shifting Operation....108

3.2.1.3 Time Reversal Operation....110

3.3 Convolution....112

3.4 Correlation....126

Bibliography....137

Chapter 4: Discrete-Time Systems....138

4.1 Discrete-Time System....140

4.2 Representation of DT Systems....140

4.2.1 Difference Equation Representation of Discrete-Time Linear Time-Invariant System....140

4.2.2 State-Space Model of a Discrete-Time System....145

4.2.2.1 State-Space to Transfer Function....145

4.2.3 Impulse Response and Step Response of Discrete-Time System....149

4.2.4 Pole-Zero Plot of Discrete-Time System....154

4.3 Responses of Discrete-Time System....157

4.4 Different Representations and Response of Unit Delay Discrete-Time System....160

4.5 Properties of Discrete-Time System....164

4.5.1 Linearity Property....164

4.5.2 Time-Invariant and Time-Variant System....166

4.5.3 Causal and Non-causal System....168

4.5.4 Stability of Discrete-Time System....169

4.5.5 Invertibility of Discrete-Time System....173

Bibliography....180

Chapter 5: Transforms....181

5.1 Introduction to Transform....183

5.2 Z-Transform....183

5.2.1 Z-Transform of Standard Test Sequences....184

5.3 Inverse Z-Transform....187

5.4 Family of Fourier Series and Transforms....188

5.4.1 Continuous-Time Fourier Transform (CTFT)....189

5.4.2 Fourier Transform of Standard Test Signals....193

5.4.3 Discrete-Time Fourier Transform (DTFT)....205

5.4.4 Analysis of Discrete-Time LTI System Using DTFT....207

5.4.5 Discrete Fourier Transform....210

5.4.6 Properties of DFT....214

5.4.7 Limitations of Fourier Transform....218

5.5 Discrete Cosine Transform (DCT)....220

5.6 Short-Time Fourier Transform....224

5.6.1 Shortcoming of STFT....233

5.7 Continuous Wavelet Transform (CWT)....233

5.7.1 Continuous Wavelets Family....234

5.7.2 Drawback of CWT....235

5.8 Discrete Wavelet Transform....235

Bibliography....244

Chapter 6: Filter Design Using Pole-Zero Placement Method....245

6.1 First-Order IIR Filter....246

6.2 Moving Average filter....253

6.3 M-Point Exponentially Weighted Moving Average Filter (EWMA)....256

6.4 Digital Resonator....260

6.5 Notch Filter....262

6.6 All-Pass Filter....265

6.7 Comb Filter....269

6.7.1 Location of Poles and Zeros of Comb Filter....270

Bibliography....275

Chapter 7: FIR Filter Design....276

7.1 FIR Filter....277

7.2 Classification of FIR Filter....277

7.3 Design of FIR Filter....284

7.3.1 Steps in Window-Based FIR Filter Design....285

7.3.2 Window-Based FIR Lowpass Filter....286

7.3.3 Window-Based FIR Highpass Filter....289

7.3.4 Window-Based FIR Bandpass Filter....291

7.3.5 Window-Based FIR Band Reject Filter....293

7.3.6 Design of FIR Filter Using Built-In Function....295

7.3.7 Window Functions....301

7.4 Frequency Sampling-Based FIR Filter Design....304

7.5 Design of Optimal FIR filter....306

7.6 Applications of FIR Filter....307

Bibliography....315

Chapter 8: Infinite Impulse Response Filter....316

8.1 IIR Filter....317

8.2 Mapping Techniques in the Design of IIR Filter....319

8.2.1 Backward Difference Method....320

8.2.2 Impulse Invariant Technique....323

8.2.3 Bilinear Transformation Technique (BLT)....333

8.2.4 Matched Z-Transform Technique....336

8.3 Analog Frequency Transformation....339

8.4 Butterworth Filter....341

8.5 Chebyshev Filter....357

8.6 Chebyshev Type II IIR Filter....371

8.7 Elliptic Filter....375

Bibliography....389

Chapter 9: Quantization Effect of Digital Filter Coefficients....390

9.1 Number Representation....391

9.2 Fixed-Point Quantization....391

9.2.1 Fixed-Point Quantization by Rounding....392

9.2.2 Fixed-Point Quantization Using Two´s Complement Truncation....393

9.2.3 Fixed-Point Quantization Using Magnitude Truncation....394

9.3 Coefficient Quantization....396

9.4 Limit Cycle Oscillations....405

9.5 Cascade Form of a Higher Order Filters....410

Bibliography....414

Chapter 10: Multirate Signal Processing....416

10.1 Multirate Operators....418

10.1.1 Downsampling Operation....418

10.1.2 Upsampling Operation....422

10.2 Noble Identity....426

10.2.1 Noble Identity for Downsampling Operation....426

10.2.2 Noble Identity for Upsampling Operation....428

10.3 Polyphase Decomposition....430

10.4 Filter Bank....432

10.4.1 Two-Channel Filter Bank....433

10.4.2 Relationship Between Analysis and Synthesis Filters....434

10.4.3 Two-Channel Filter Bank Without Filters....437

10.4.4 Three-Channel Filter Bank Without Filters....439

10.5 Tree-Structured Filter Bank....441

10.6 Transmultiplexer....444

Bibliography....453

Chapter 11: Adaptive Signal Processing....455

11.1 Wiener Filter....456

11.1.1 Wiener Filter in Frequency Domain....459

11.2 Adaptive Filter....463

11.2.1 LMS Adaptive Filter....464

11.2.2 Normalized LMS Algorithm....469

11.2.3 Sign LMS Algorithm....471

11.3 RLS Algorithm....474

Bibliography....480

Chapter 12: Case Study....481

12.1 Case Study 1: Speech Recognition Using MFCC (Mel-Frequency Cepstral Coefficient)....481

12.1.1 Speaker Identification....482

12.1.2 Speaker Verification System....483

12.1.3 Mel-Frequency Cepstral Coefficient (MFCC) Feature....484

12.1.3.1 Pre-emphasis....484

12.1.3.2 Sampling and Windowing....484

12.1.3.3 Discrete Fourier Transform (DFT)....486

12.1.3.4 Mel-Frequency Bandpass Filter....488

12.1.3.5 Log Operation....490

12.1.3.6 Discrete Cosine Transform (DCT)....490

12.2 Case Study 2: QRS Detection in ECG Signal Using Pan-Tomkins Algorithm....491

12.2.1 ECG Signal Preprocessing....493

12.2.1.1 Bandpass filter....494

12.2.1.2 Derivative Process....495

12.2.1.3 Squaring Operation....495

12.2.2 Moving Window Integration....496

12.2.3 Fiducial Mark....496

12.2.4 Decision Rule Approach....498

12.3 Case Study 3: Power Quality Disturbance Detection....500

12.3.1 Generation of Power Quality Disturbance....501

12.3.2 Simulation of Power Quality Disturbance....503

12.3.3 Time-Frequency Representation of Power Quality Disturbance....504

12.3.4 Time-Scale Representation of Power Quality Disturbance....505

Bibliography....508

Appendix....509

Chapter 1: Generation Of Continuous-Time Signals....509

Answers to PreLab Questions....509

Answers to Objective Questions....511

Chapter 2: Sampling and Quantization of Signals....512

Answers to PreLab Questions....512

Answers to Objective Questions....513

Chapter 3: Generation and Operation on Discrete-Time Sequence....513

Answers to PreLab Questions....513

Answers to Objective Questions....515

Chapter 4: Discrete-Time Systems....515

Answers to PreLab Questions....515

Answers to Objective Questions....518

Chapter 5: Transforms....518

Answers to PreLab Questions....518

Answers to Objective Questions....520

Chapter 6: Filter Design Using Pole-Zero Placement Method....520

Answers to PreLab Questions....520

Answers to Objective Questions....521

Chapter 7: FIR Filter Design....521

Answers to PreLab Questions....521

Answers to Objective Questions....523

Chapter 8: Infinite Impulse Response Filter....523

Answers to PreLab Questions....523

Answers to Objective Questions....525

Chapter 9: Quantization Effect of Digital Filter Coefficients....525

Answers to PreLab Questions....525

Answer to Objective Questions....526

Chapter 10: Multirate Signal Processing....527

Answers to PreLab Questions....527

Answers to Objective Questions....529

Chapter 11: Adaptive Signal Processing....529

Answers to PreLab Questions....529

Answers to Objective Questions....530

Index....531

Digital signal processing deals with extraction of useful information from signals. Signal processing algorithms help observe, analyse and transform signals. The objective of this book is to develop signal processing algorithms using Python. Python is an interpreted, object-oriented high-level programming language widely used in various software development fields such as data science, machine learning, web development and more.

Digital Signal Laboratory is playing an important role in realizing signal processing algorithms, utilizing different software solutions. The intention of this textbook is to implement signal processing algorithms using Python. Since Python is an open-source language, students, researchers, and faculty can install and work with it without spending money, reducing the financial burden on institutions. Each chapter in this book begins with prelab questions, a set of Python examples to illustrate the concepts, exercises to strengthen the understanding of the concepts, and objective questions to help students prepare for competitive examinations.

This book serves as an undergraduate textbook, it can be used for individual study, and it can also be used as the textbook for related courses.