CONTENTS
1. INTRODUCTION
1.1 Systems = 4
1.2 Classification of Systems = 6
1.2.1 Static and dynamic systems = 7
1.2.2 Linear and nonlinear systems = 7
1.2.3 Time-varying and time-invariant systems = 10
1.2.4 Other system types = 10
1.3 Continuous-Time and Discrete-Time systems = 11
1.4 The Input-Output Model of the Continuous-Time System = 13
1.4.1 The p Operator = 14
1.4.2 Electric circuits using the p operator = 18
1.4.3 The operational transfer function = 23
1.5 Simulation of the Continuous-Time System = 27
1.5.1 The direct programming technique = 30
1.5.2 Other programming techniques = 34
1.5.3 The role of time and initial conditions = 37
1.5.4 Physical implementation of simulations = 43
1.6 Diserete-Time Systems = 44
1.6.1 Discrete-time functions = 46
1.6.2 Special discrete-time functions = 48
1.6.3 Discrete-time systems as approximations = 49
1.6.4 Difference equations = 53
1.7 Modeling of the Discrete-Time System = 55
1.7.1 The E operator = 56
1.7.2 The operational transfer function = 57
1.8 Simulation of the Diserete-Time System =57
1.8.1 Programming techniques = 59
1.8.2 The role of time and initial conditions = 61
1.8.3 Iterative methods of solution = 66
Summary = 70
Problems = 71
2. THE CLASSICAL APPROACH TO SYSTEM ANALYSIS
2.1 The Continuous-Time System = 83
2.1.1 The complementary solution = 74
2.1.2 The particular solution = 87
2.1.3 The total solution for a SISO system = 91
2.1.4 complex s-plane analysis = 93
2.1.5 Frequency response of continuous-time systems = 96
2.2 The Discrete-Time system = 98
2.2.1 The complementary solution = 99
2.2.2 The particular solution = 105
2.2.3 The total solution for a SISO system = 113
2.2.4 Complex z-Plane analysis = 117
2.2.5 Frequency response of discrete-time systems = 119
2.3 The Charaeteristic Equation = 124
2,3.1 The continuous-time system = 125
2.3.2 The discrete-time system = 128
2.3.3 Stability of linear, time-invariant systems = 134
2.4 Zero-State and zero-Input Solutions = 137
2.4.1 The concept of state = 138
2.4.2 Zero-state and zero-input responses = 140
Summary = 147
Problems = 148
3. CONVOLUTION TECHNIQUES
3.1 The Diserete-Time System = 158
3.1.1 The impulse-response sequence = 160
3.1.2 Convolution = 162
3.1.3 Interpretation and implementation of the sum = 174
3.1.4 Determining the impulse response = 172
3.2 The Continuous-Time System = 178
3.2.1 The continuous-time impulse function = 178
3.2.2 The impulse response of a continuous-time system = 184
3.2.3 Continuous-time convolution = 186
3.2.4 Graphical evaluation of the convolution integral = 190
3.2.5 Determinlng the impulse response = 197
3.2.6 The step response of a continuous-time system = 205
3.2.7 Numerical evaluation of the convolution integral = 207
3.3 Discrete-Time Approximations to Continuous-Time Systems = 213
3.3.1 The impulse-response invariance method of approximation = 215
3.3.2 The transformation method of approximation = 219
3.4 Discrete-Time Integrators = 220
Summary = 232
Problems = 233
4. STATE VARIABLE METHODS
4.1 Introduction = 247
4.2 Writing the State Equations = 250
4.2.1 Simulation-defined state variables = 252
4.2.2 Natural state variables = 257
4.2.3 The state-variable formulation of electrical networks = 260
4.2.4 State equations for degenerate networks = 266
4.3 The Relation between the State-Variable Model and the Input-Output Model = 269
4.4. The Solution of the Discrete-Time State Equations = 273
4.4.1 The computation of \Ak : distinct eigenvalues = 277
4.4.2 The computation of \Ak : repeated eigenvalues = 281
4.5 The Solution of Continuous-Time State Equations = 284
4.5.1 The solution to source-free equations = 284
4.5.2 The general solution to state equations = 285
4.5.3 The evaluation of \eAt = 287
4.6 State Bquations in Diagonal Form = 292
4.7 Trajectories in State Space = 298
4.7.1 The zero-input response = 299
4.7.2 stability and the zero-input response = 304
4.7.3 suppressed modes = 307
4.8 controllability = 309
4.8.1 conditions for controllability = 312
4.8.2 Determining controllability from diagonal representations = 314
4.9 Observability = 319
4.5.1 Determining observability from diagonal forms = 322
4.9.2 A control algorithm = 324
4.10 Systems with common Poles and Zeros = 326
Summary = 331
Problems = 332
5. THE z-TRANBFORM
5.1 Introduction = 348
5.1.1 Closed-form representations = 350
5.1.2 Properties of the z-transform = 354
5.2 Solution of Difference Equations = 372
5.3 The Inverse Transform by Partial Fraction Expansion = 364
5.3.1 Distinct, real poles = 365
5.3.2 Distinct, complex poles = 368
5.3.3 Repeated poles = 371
5.4 Partitioning the Total Response = 375
5.4.1 The zero-state response = 376
5.4.2 The zero-input response = 378
5.5 The z-Transform of the State Equations = 383
5.6 The System Function and the z-Plane = 385
5.6.1 Discrete-time approximations = 385
5.6.2 The frequency response of discrete-time systems = 387
Summary = 391
Problems = 391
6. FOURIER ANALYSIS
6.1 Introduction = 398
6.2 Periodic Signals = 399
6.2.1 The sine-cosine form of the Fourier series = 399
6.2.2 Alternative forms of the Fourier series = 402
6.3 Systems with Periodic Inputs = 407
6.4 Some Characteristics of the Fourier Series = 410
6.5 The Truncated Fourier Series = 414
6.5.1 Quadratic content and RMS value = 414
6.5.2 Parseval's theorem = 417
6.6 Spectrum of a Periodic Signal = 420
6.7 The Fourier Transform = 426
6.8 Energy content of a signal = 432
6.9 Some Properties of the Fourier Transform = 436
6.10 Linear System Analysis and the Fourier Transform = 446
6.11 Power Signals = 465
6.12 Modulation and Demodulation = 460
6.13 The Sampling Theorem = 403
6.14 The Discrete Fourier Transform = 408
6.14.1 The numerical evaluation of the Fourier transform integral = 468
6.14.2 The frequency response of discrete-times system = 470
6.14.3 The numerical evaluation of the inverse Fourier transform integral = 470
6.14.4 The discrete-transform pair = 472
Summary = 480
Problems = 480
7. THE LAPLACE TRANSFORM
7.1 Definition of the Laplace Transform = 474
7.2 Properties of the Laplace Transform = 477
7.3 The Solution of Differential Equations = 506
7.4 Partial Fraction Expansions and the Inverse Laplace Transform = 508
7.5 SISO Systems = 513
7.5.1 The zero-state, zero-input transforms = 514
7.5.2 The forced-and-natural-response transforms = 520
7.6 Initial Conditions = 521
7.7 Circuit Analysis with the Laplace Transform = 524
7.8 The Initial-Value Theorem = 531
7.9 The Final-Value Theorem = 534
7.10 State Equations and the Laplace Transform = 535
7.11 Frequency Response of Linear Systems = 538
Summary = 546
Problems = 546
8. DIGITAL FILTERS
8.1 The Sampler = 557
8.2 The Reconstruction Device = 558
8.3 The Frequency-Domain Design Criterion = 560
8.4 Filter Design by Pole-Zero Placement = 562
8.5 Filter Design Using Discrete-Time Approximations = 573
8.5.1 Thetransformation methods = 574
8.5.2 The impulse-response-invariance methods = 578
Summary = 588
Problems = 589
APPENDIX Matrices and Vector Spaces = 593
ANSWERS TO SELECTED PROBLEMS = 623
REFERENCES = 631
INDEX = 637
