# Fourier Analysis

ISBN: 978-1-119-38894-4

Jan 2017, Wiley-ISTE

266 pages

Select type: O-Book

## Description

This book aims to learn to use the basic concepts in signal processing. Each chapter is a reminder of the basic principles is presented followed by a series of corrected exercises. After resolution of these exercises, the reader can pretend to know those principles that are the basis of this theme. ""We do not learn anything by word, but by example.""

Preface xi

Chapter 1. Fourier Series  1

1.1. Theoretical background 1

1.1.1. Orthogonal functions  1

1.1.2. Fourier Series 3

1.1.3. Periodic functions  5

1.1.4. Properties of Fourier series 6

1.1.5. Discrete spectra. Power distribution 8

1.2. Exercises  9

1.2.1. Exercise 1.1. Examples of decomposition calculations  10

1.2.2. Exercise 1.2  11

1.2.3. Exercise 1.3  12

1.2.4. Exercise 1.4  12

1.2.5. Exercise 1.5  12

1.2.6. Exercise 1.6. Decomposing rectangular functions 13

1.2.7. Exercise 1.7. Translation and composition of functions  14

1.2.8. Exercise 1.8. Time derivation of a function 15

1.2.9. Exercise 1.9. Time integration of functions 15

1.2.10. Exercise 1.10  15

1.2.11. Exercise 1.11. Applications in electronic circuits 16

1.3. Solutions to the exercises  17

1.3.1. Exercise 1.1. Examples of decomposition calculations  17

1.3.2. Exercise 1.2  25

1.3.3. Exercise 1.3  26

1.3.4. Exercice 1.4  26

1.3.5. Exercise 1.5  27

1.3.6. Exercise 1.6 27

1.3.7. Exercise 1.7. Translation and composition of functions  29

1.3.8. Exercise 1.8. Time derivation of functions  31

1.3.9. Exercise 1.9. Time integration of functions  32

1.3.10. Exercise 1.10 32

1.3.11. Exercise 1.11 35

Chapter 2. Fourier Transform  39

2.1. Theoretical background  39

2.1.1. Fourier transform 39

2.1.2. Properties of the Fourier transform  42

2.1.3. Singular functions 46

2.1.4. Fourier transform of common functions  51

2.1.5. Calculating Fourier transforms using the Dirac impulse method  53

2.1.6. Fourier transform of periodic functions  54

2.1.7. Energy density 54

2.1.8. Upper limits to the Fourier transform 55

2.2. Exercises  56

2.2.1. Exercise 2.1  56

2.2.2. Exercise 2.2  57

2.2.3. Exercise 2.3  58

2.2.4. Exercise 2.4  59

2.2.5. Exercise 2.5  59

2.2.6. Exercise 2.6  59

2.2.7. Exercise 2.7  60

2.2.8. Exercise 2.8  60

2.2.9. Exercise 2.9  61

2.2.10. Exercise 2.10 62

2.2.11. Exercise 2.11 62

2.2.12. Exercise 2.12 63

2.2.13. Exercise 2.13 63

2.2.14. Exercise 2.14 64

2.2.15. Exercise 2.15 64

2.2.16. Exercise 2.16 65

2.2.17. Exercise 2.17 66

2.3. Solutions to the exercises 67

2.3.1. Exercise 2.1  67

2.3.2. Exercise 2.2  68

2.3.3. Exercise 2.3  74

2.3.4. Exercise 2.4  74

2.3.5. Exercise 2.5  76

2.3.6. Exercise 2.6  76

2.3.7. Exercise 2.7  77

2.3.8. Exercise 2.8  79

2.3.9. Exercise 2.9  82

2.3.10. Exercise 2.10  85

2.3.11 Exercise 2.11 86

2.3.12 Exercise 2.12 88

2.3.13 Exercise 2.13 91

2.3.14 Exercise 2.14 91

2.3.15 Exercice 2.15  92

2.3.16 Exercise 2.16 94

2.3.17 Exercise 2.17 95

Chapter 3. Laplace Transform 97

3.1. Theoretical background 97

3.1.1. Definition 97

3.1.2. Existence of the Laplace transform  98

3.1.3. Properties of the Laplace transform  98

3.1.4. Final value and initial value theorems  102

3.1.5. Determining reverse transforms  102

3.1.6. Approximation methods  105

3.1.7. Laplace transform and differential equations  107

3.1.8. Table of common Laplace transforms  108

3.1.9. Transient state and steady state  110

3.2. Exercise instruction  111

3.2.1. Exercise 3.1  111

3.2.2. Exercise 3.2  111

3.2.3. Exercise 3.3  112

3.2.4. Exercise 3.4  112

3.2.5. Exercise 3.5  112

3.2.6. Exercise 3.6  113

3.2.7. Exercise 3.7  113

3.2.8. Exercise 3.8  115

3.2.9. Exercise 3.9  115

3.2.10. Exercise 3.10  115

3.3. Solutions to the exercises  116

3.3.1. Exercise 3.1  116

3.3.2. Exercise 3.2  117

3.3.3. Exercise 3.3  121

3.3.4. Exercise 3.4  122

3.3.5. Exercise 3.5  130

3.3.6. Exercise 3.6  131

3.3.7. Exercise 3.7  132

3.3.8. Exercise 3.8  136

3.3.9. Exercise 3.9  138

3.3.10. Exercise 3.10 139

Chapter 4. Integrals and Convolution Product  143

4.1. Theoretical background  143

4.1.1. Analyzing linear systems using convolution integrals 143

4.1.2. Convolution properties  144

4.1.3. Graphical interpretation of the convolution product 145

4.1.4. Convolution of a function using a unit impulse 145

4.1.5. Step response from a system  147

4.1.6. Eigenfunction of a convolution operator 148

4.2. Exercises  149

4.2.1. Exercise 4.1  149

4.2.2. Exercise 4.2  150

4.2.3. Exercise 4.3  150

4.2.4. Exercise 4.4  151

4.2.5. Exercise 4.5  151

4.2.6. Exercise 4.6  152

4.3. Solutions to the exercises 153

4.3.1. Exercise 4.1  153

4.3.2. Exercise 4.2  156

4.3.3. Exercise 4.3  160

4.3.4. Exercise 4.4  163

4.3.5. Exercise 4.5  164

4.3.6. Exercise 4.6  165

Chapter 5. Correlation 169

5.1. Theoretical background  169

5.1.1. Comparing signals  169

5.1.2. Correlation function 170

5.1.3. Properties of correlation functions 172

5.1.4. Energy of a signal 176

5.2. Exercises  177

5.2.1. Exercise 5.1  177

5.2.2. Exercise 5.2  178

5.2.3. Exercise 5.3  178

5.2.4. Exercise 5.4  178

5.2.5. Exercice 5.5  179

5.2.6. Exercice 5.6  179

5.2.7. Exercise 5.7  179

5.2.8. Exercice 5.8  180

5.2.9. Exercise 5.9  180

5.2.10. Exercise 5.10  181

5.2.11. Exercise 5.11  181

5.2.12. Exercise 5.12  182

5.2.13. Exercise 5.13  182

5.2.14. Exercise 5.14  183

5.3. Solutions to the exercises  183

5.3.1. Exercise 5.1  183

5.3.2. Exercice 5.2  188

5.3.3. Exercise 5.3  191

5.3.4. Exercice 5.4  192

5.3.5. Exercise 5.5  193

5.3.6. Exercise 5.6  196

5.3.7. Exercise 5.7  197

5.3.8. Exercise 5.8  201

5.3.9. Exercise 5.9  204

5.3.10. Exercise 5.10  205

5.3.11 Exercise 5.11 206

5.3.12 Exercise 5.12 207

5.3.13 Exercise 5.13 208

5.3.14 Exercise 5.14 209

Chapter 6. Signal Sampling 213

6.1. Theoretical background 213

6.1.1. Sampling principle  213

6.1.2. Ideal sampling  214

6.1.3. Finite width sampling  218

6.1.4. Sample and hold (S/H) sampling 221

6.2. Exercises  225

6.2.1. Exercise 6.1  225

6.2.2. Exercise 6.2  225

6.2.3. Exercise 6.3  226

6.2.4. Exercise 6.4  226

6.2.5. Exercise 6.5  226

6.2.6. Exercise 5.6  227

6.2.7. Exercise 6.7  227

6.2.8. Exercice 6.8  228

6.3. Solutions to the exercises 229

6.3.1. Exercise 6.1  229

6.3.2. Exercise 6.2  229

6.3.3. Exercise 6.3  233

6.3.4. Exercice 6.4  235

6.3.5. Exercise 6.5  236

6.3.6. Exercise 6.6  238

6.3.7. Exercise 6.7  240

6.3.8. Exercise 6.8  242

Bibliography  245

Index 247