Contents 1 Overview of signals and systems 1 1.0 Introduction 1 1.1 Basic definitions and classification of signals 2 1.1.1 Concepts 2 1.1.2 Description of signals 2 1.1.3 Classification of signals 2 1.1.4 Representation and plotting of signals with MATLAB 8 1.2 Basic operations of signals 9 1.2.1 Operations “+”,“-”and “×”of signals 9 1.2.2 Signal transformations in the time domain 9 1.3 Elementary signals 13 1.3.1 The continuous-time unit step function 13 1.3.2 The continuous-time unit impulse function 14 1.3.3 Properties of the CT unit impulse function 16 1.3.4 The discrete-time unit step and impulse sequences 19 1.4 Basic definitions and classification of systems 21 1.4.1 Introduction to systems 21 1.4.2 Classifications of systems 22 1.5 Framework of analytical methods 29 1.5.1 Analytical methods for LTI systems 29 1.5.2 Key issues to study 29 1.5.3 Framework of all chapters 31 1.6 Summary 32 2 Time-domain analysis of LTIC systems 37 2.0 Introduction 37 2.1 Representation of the LTIC system 38 2.1.1 Analytical description based on mathematical models 38 2.1.2 Description based on the block diagram 38 2.2 Classical solution of the differential equation 40 2.2.1 Classical solution of the direct method 40 2.2.2 Initial value of the system 43 2.2.3 Zero-input response and zero-state response 44 2.2.4 Response calculation with MATLAB 47 2.3 The impulse response and step response 48 2.3.1 CT impulse response 48 2.3.2 CT step response 49 2.3.3 Solution by MATLAB 51 2.4 Convolution integral 52 2.4.1 Signal decomposition in the time domain 52 2.4.2 Definition of the convolution integral 53 2.4.3 Graphical method for evaluating the convolution integral 56 2.4.4 Properties of the convolution integral 58 2.4.5 Comprehensive application instances 63 2.4.6 Convolution computation with MATLAB 67 2.5 Summary 68 3 Time-domain analysis of LTID systems 73 3.0 Introduction 73 3.1 Representation of an LTID system 74 3.1.1 Analytical description based on mathematical models 74 3.1.2 Description based on the block diagram 74 3.1.3 General form of the difference equation 75 3.2 Classical solution of the difference equation 76 3.2.1 Classical solution of the direct method 76 3.2.2 Zero-input response and zero-state response 79 3.2.3 Response calculation with MATLAB 83 3.3 Impulse response and step response 84 3.3.1 Basic discrete-time sequence 84 3.3.2 Unit impulse response and step response of an LTID system 87 3.3.3 Calculation with MATLAB 91 3.4 Convolution sum 92 3.4.1 Representation of sequences using Dirac delta functions 92 3.4.2 Convolution sum 92 3.4.3 Graphical method for evaluating the convolution sum 95 3.4.4 The carry-less multiplication method 97 3.4.5 Properties of the convolution sum 98 3.4.6 Convolution calculation with MATLAB 100 3.4.7 Application of the convolution sum 101 3.5 Summary 103 4 Frequency-domain analysis of LTICs ystems 107 4.0 Introduction 107 4.1 CTFS of periodic signals 108 4.1.1 Trigonometric CTFS 108 4.1.2 Symmetry of waveform and harmonic characteristics 112 4.1.3 Exponential Fourier series 112 4.1.4 Parseval’s power theorem 115 4.2 Fourier spectrum of periodic signals 116 4.2.1 Definition of the Fourier spectrum 116 4.2.2 Characteristics of the spectrum of periodic signals 118 4.2.3 Application of the Fourier series 121 4.3 Continuous-time Fourier transforms 123 4.3.1 Definition of CTFT 123 4.3.2 CTFT pairs for elementary CT signals 124 4.3.3 Properties of CTFT 126 4.3.4 Fourier transforms of real-valued even and odd functions 134 4.3.5 Parseval’s energy theorem 135 4.3.6 CTFT of periodic functions 136 4.4 LTIC systems analysis using CTFT and CTFS 138 4.4.1 Response of the LTIC system to the complex exponential function 138 4.4.2 Response of the LTIC system to an arbitrary signal 139 4.4.3 The Fourier transfer function of an LTIC system 139 4.4.4 Steps of calculating the response with CTFT 141 4.4.5 Steps of calculating the response with CTFS 142 4.4.6 Response computation with MATLAB 144 4.5 Applications of transmission and filtering 145 4.5.1 The undistorted transmission system 145 4.5.2 Frequency characteristics of an ideal low-pass filter 147 4.5.3 Impulse and step response of an ideal low-pass filter 147 4.5.4 Conditions of physically realizable systems 149 4.5.5 Nonideal low-pass filter 149 4.5.6 Application of the amplitude modulation system 150 4.6 Sampling theorem 152 4.6.1 Model of ideal impulse-train sampling 152 4.6.2 CTFT of the sampled signal 153 4.6.3 Sampling theorem 154 4.6.4 Reconstruction of a band-limited signal from its samples 154 4.6.5 Sampling with MATLAB 157 4.7 Summary 158 5 Laplace transform and complex frequency-domain analysis 163 5.0 Introduction 163 5.1 Analytical development 163 5.1.1 From CTFT to the bilateral laplace transform 163 5.1.2 Region of convergence 164 5.1.3 Unilateral Laplace transform 166 5.1.4 Relationship between CTFT and the Laplace transform 167 5.2 Basic pairs and properties of the Laplace transform 168 5.2.1 Laplace transform pairs for several elementary CT signals 168 5.2.2 Properties of the Laplace transform 169 5.3 Inverse Laplace transformation 173 5.3.1 Characteristic roots of the Laplace transform 173 5.3.2 Real-valued and first-order poles 173 5.3.3 Complex-valued and first-order poles 174 5.3.4 Real-valued and repeated poles 176 5.3.5 Calculation with MATLAB 177 5.4 Application of the Laplace transform in circuit analysis 177 5.4.1 S-domain models of circuit 178 5.4.2 Analysis in the S-domain of the circuit system 180 5.5 Application of solutions of differential equations 181 5.5.1 Analysis of computing zero-input and zero-state response 181 5.5.2 Analysis of computing the overall response 183 5.6 Laplace transfer function 184 5.6.1 Definition of the Laplace transfer function 184 5.6.2 Characteristic equation, zeros and poles 186 5.6.3 Nature of the shape of impulse response for different poles 186 5.6.4 Stability conditions in the S-plane 188 5.6.5 Laplace transfer function with the frequency response function 189 5.6.6 Calculation with MATLAB 190 5.7 Signal-flow graph and LTIC system simulation 193 5.7.1 Block diagram representation 193 5.7.2 Model of basic components of LTIC systems 194 5.7.3 The signal-flow graph of LTIC systems 195 5.7.4 Mason’s rule 196 5.7.5 Simulation of the LTIC system 198 5.8 Summary 201 6 The Z-transform and Z-domain analysis 209 6.0 Introduction 209 6.1 Analytical development 209 6.1.1 From the Laplace transform to the Z-transform 209 6.1.2 Region of convergence 210 6.1.3 Unilateral Z-transform 213 6.2 Basic pairs and properties of the Z-transform 213 6.2.1 Z-transform pairs for several elementary DT signals 213 6.2.2 Properties of the Z-transform 214 6.3 Inverse Z-transform 218 6.3.1 Power series method 218 6.3.2 Characteristic roots of the Z-transform 220 6.3.3 Real-valued and first-order poles 221 6.3.4 Complex-valued and first-order poles 222 6.3.5 Real-valued and repeated poles 224 6.3.6 Calculation with MATLAB 226 6.4 Relationship between the Laplace and Z-transforms 226 6.4.1 Mapping relation between S-plane and Z-plane 226 6.4.2 Conversion from Z-transform to Laplace transform 227 6.4.3 Conversion from the Laplace transform to the Z-transform 228 6.5 Solution of difference equations with the Z-transform 230 6.5.1 Analysis of computing zero-input and zero-state response 230 6.5.2 Analysis of computing overall response 231 6.6 Z-transfer function of LTID systems 232 6.6.1 Definition of the Z-transfer function 232 6.6.2 Characteristic equation, zeros and poles 234 6.6.3 Nature of the shape of the impulse response for different poles 234 6.6.4 Stability analysis in the Z-domain 236 6.7 Signal flow graph and LTID system simulation 237 6.7.1 Block diagram representation 237 6.7.2 Model of basic components of LTID systems 238 6.7.3 Signal flow graph of LTID systems 239 6.7.4 Simulation of LTID systems 240 6.8 Characteristics of frequency response 243 6.8.1 Response of LTID systems to the complex exponential sequence 243 6.8.2 Response of LTID systems to the sinusoidal sequence 243 6.8.3 Definition of frequency response of LTID systems 244 6.8.4 Calculation with MATLAB 247 6.9 Summary 248 7 State-space analysis of systems 255 7.0 Introduction 255 7.1 Basic concepts of the state space 255 7.1.1 State variables of systems 255 7.1.2 State equations of continuous-time and discrete-time systems 256 7.1.3 Output equations of continuous-time and discrete-time systems 257 7.2 State-space description of CT systems 258 7.2.1 State-space description for electrical circuit systems 258 7.2.2 State-space description from differential equations 259 7.2.3 State-space description from the system diagram and the flow graph 261 7.2.4 State-space description with MATLAB 262 7.3 State-space description of DT systems 263 7.3.1 State-space description from difference equations 263 7.3.2 State-space description from system diagrams and flow graphs 265 7.4 Solution of state-space equations of LTIC systems 266 7.4.1 Laplace transform solution of state equations 266 7.4.2 Laplace transform solution of output equations 267 7.4.3 Calculation with MATLAB 269 7.5 Solution of state-space equations of LTID systems 270 7.5.1 Z-transform solution of state equations 270 7.5.2 Z-transform solution of output equations 271 7.5.3 Calculation with MATLAB 273 7.6 Stability analysis from the transfer function matrix 275 7.6.1 Stability condition 275 7.6.2 Calculation with MATLAB 276 7.7 Summary 277 8 Applications of system analysis 283 8.0 Introduction 283 8.1 Application of the Fourier transform in communication systems 283 8.1.1 Double-sideband suppressed-carrier amplitude modulation (DSB-SC-AM) 284 8.1.2 Amplitude modulation (AM) 285 8.1.3 Pulse-amplitude modulation (PAM) 286 8.2 Fast Fourier transform 288 8.2.1 Discrete-time Fourier series (DTFS) 288 8.2.2 Discrete-time Fourier transform (DTFT) 289 8.2.3 Discrete Fourier transform (DFT) 290 8.2.4 Relationship between Fourier transforms 290 8.2.5 Fast Fourier transform (FFT) 291 8.3 Application of the Laplace transform in control systems 292 8.3.1 Diagram of the closed-loop feedback system 292 8.3.2 Analysis of an automatic position control system 292 8.4 Digital filters 295 8.4.1 Filter classification 295 8.4.2 FIR and IIR filters 296 8.4.3 IIR filter design using the impulse-invariance method 297 8.4.4 IIR filter design using bilinear transformation 299 8.4.5 FIR filter design using the windowing method 301 8.5 Controllability and observability of linear systems 303 8.5.1 Controllability of linear systems 304 8.5.2 Observability of linear systems 305 8.5.3 Calculation with MATLAB 307 8.6 Applications of the Kalman filter 307 8.6.1 Basic principles of the Kalman filter 307 8.6.2 Temperature prediction simulation with MATLAB 308 8.7 Applications of convolution in image processing 310 8.8 Summary 311 References 315 Index 317
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