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
