Table of Contents
Preface v
1 Signals 1
1.1 The concept of signals 1
1.2 Classification of signals 3
1.2.1 Continuous time and discrete time signals 3
1.2.2 Periodic and aperiodic signals 7
1.2.3 Energy and power signals 8
1.2.4 Deterministic and random signals 9
1.2.5 Causal and anticausal signals 10
1.3 Basic continuous-time signals 10
1.3.1 Direct current signals 11
1.3.2 Sinusoidal signals 11
1.3.3 Exponential signals 12
1.3.4 Complex exponential signals 13
1.3.5 Signum signal 14
1.3.6 Unit step signal 14
1.3.7 Unit ramp signal 16
1.3.8 Unit impulse signal 16
1.3.9 Unit doublet signal 20
1.3.10 Unit gate signal 22
1.3.11 Bell shaped pulse signal 23
1.4 Operations of continuous signals 24
1.4.1 Arithmetic operations 24
1.4.2 Operations of even and odd signals 25
1.4.3 Time shifting 25
1.4.4 Time reversal 26
1.4.5 Time scaling 26
1.4.6 Differentiation and integration 28
1.4.7 Decomposition and synthesis 28
1.4.8 Convolution integral 29
1.4.9 Plotting 36
1.5 Solved questions 37
1.6 Learning tips 41
1.7 Problems 41
2 Systems 45
2.1 The concept of a system 45
2.2 Excitation, response and system state 46
2.3 Classification of systems 49
2.3.1 Simple and complex systems 49
2.3.2 Continuous-time and discrete-time systems 50
2.3.3 Linear and nonlinear systems 50
2.3.4 Time-variant and time-invariant systems 55
2.3.5 Causal and noncausal systems 57
2.3.6 Dynamic and static systems 58
2.3.7 Open-loop and closed-loop systems 58
2.3.8 Stable and unstable systems 59
2.3.9 Lumped and distributed parameter systems 59
2.3.10 Invertible and nonreversible systems 60
2.4 Models of LTI systems 61
2.4.1 Mathematical models 61
2.4.2 Mathematical modeling 62
2.4.3 Block models 65
2.5 Analysis methods for LTI systems 67
2.6 Solved questions 68
2.7 Learning tips 70
2.8 Problems 71
3 Analysis of continuous-time systems in the time domain 73
3.1 Analysis methods with differential equations 74
3.1.1 The classical analysis method 74
3.1.2 Response decomposition analysis method 78
3.2 Impulse and step responses 88
3.2.1 Impulse response 88
3.2.2 Step response 90
3.3 The operator analysis method 94
3.3.1 Differential and transfer operators 94
3.3.2 Determining impulse response by the transfer operator 99
3.4 The convolution analysis method 100
3.5 Judgment of dynamics, reversibility and causality 103
3.5.1 Judgment of dynamics 104
3.5.2 Judgment of reversibility 105
3.5.3 Judgment of causality 105
3.6 Solved questions 105
3.7 Learning tips 107
3.8 Problems 107
4 Analysis of continuous-time systems excited by periodic signals in the real frequency domain 111
4.1 Orthogonal functions 112
4.1.1 The orthogonal function set 112
4.1.2 Trigonometric function set 114
4.1.3 Imaginary exponential function set 114
4.2 Fourier series 115
4.2.1 Trigonometric form of Fourier series 115
4.2.2 Relations between function symmetries and Fourier coefficients 118
4.2.3 Exponential form of the Fourier series 122
4.2.4 Properties of the Fourier series 125
4.3 Frequency spectrum 129
4.3.1 Concept of frequency spectrum 129
4.3.2 Properties of the frequency spectrum 132
4.4 Fourier series analysis 138
4.4.1 System function 138
4.4.2 Analysis method 139
4.5 Solved questions 142
4.6 Learning tips 144
4.7 Problems 144
5 Analysis of continuous-time systems excited by nonperiodic signals in the real frequency domain 147
5.1 The concept of Fourier transform 147
5.2 Fourier transforms of typical aperiodic signals 152
5.2.1 Gate signals 152
5.2.2 Unilateral exponential signals 153
5.2.3 Bilateral exponential signals 153
5.2.4 Unit DC signals 154
5.2.5 Unit impulse signals 155
5.2.6 Signum signals 156
5.2.7 Unit step signals 157
5.3 Properties of the Fourier transform 158
5.3.1 Linearity 158
5.3.2 Time shifting 159
5.3.3 Frequency shifting 160
5.3.4 Time scaling 161
5.3.5 Symmetry 163
5.3.6 Properties of convolution 165
5.3.7 Differentiation in the time domain 167
5.3.8 Integration in the time domain 169
5.3.9 Modulation 170
5.3.10 Conservation of energy 171
5.4 Fourier transforms of periodic signals 172
5.5 Solutions for the inverse Fourier transform 174
5.6 System analysis methods for aperiodic signals 175
5.6.1 Analysis method from system models 175
5.6.2 Analysis with the system function 176
5.6.3 Analysis with signal decomposition 177
5.7 System analysis methods for periodic signals 181
5.8 The Hilbert transform 182
5.9 Advantages and disadvantages of Fourier transform analysis 185
5.10 Solved questions 185
5.11 Learning tips 188
5.12 Problems 189
6 Analysis of continuous-time systems in the complex frequency domain 193
6.1 Concept of the Laplace transform 193
6.2 Laplace transforms of common signals 198
6.3 Laplace transforms of periodic signals 198
6.4 Properties of the Laplace transform 199
6.4.1 Linearity 199
6.4.2 Time shifting 200
6.4.3 Complex frequency shifting 202
6.4.4 Time scaling 202
6.4.5 Differentiation in the time domain 203
6.4.6 Integration in the time domain 204
6.4.7 Convolution theorem 205
6.4.8 Initial value theorem 207
6.4.9 Final value theorem 207
6.4.10 Differentiation in the s domain 209
6.4.11 Integration in the s domain 210
6.5 Solutions for the inverse Laplace transform 211
6.6 Analysis method of the system function in the s domain 216
6.6.1 System function 216
6.6.2 Analysis method with the system function 219
6.7 Analysis methods with system models in the s domain 221
6.7.1 Analysis with mathematic models 221
6.7.2 Analysis with a circuit model 224
6.8 Analysis method from signal decomposition in the s domain 230
6.9 Relationship among the time, frequency and complex frequency domain methods 231
6.10 Solved questions 233
6.11 Learning tips 236
6.12 Problems 236
7 Simulation and stability analysis of continuous-time systems 241
7.1 System simulation 241
7.1.1 Basic arithmetic units 241
7.1.2 Simulating system with block diagrams 242
7.1.3 Simulating systems with flow graphs 245
7.2 System stability analysis 255
7.2.1 System stability 255
7.2.2 Pole-zero analysis of the system function H(s) 257
7.2.3 Relationships between stability and ROC, and poles 264
7.2.4 Stability judgment based on the R-H criterion 264
7.3 Controllability and observability of a system 270
7.4 Solved questions 272
7.5 Learning tips 276
7.6 Problems 276
A Reference answers 281
Bibliography 303
Index 305
