Signals, Systems, & Transforms

Preface xvii
1 Introduction 1
1.1 Modeling 1
1.2 Continuous-Time Physical Systems 4
Electric Circuits, 4
Operational Amplifier Circuits, 6
Simple Pendulum, 9
DC Power Supplies, 10
Analogous Systems, 12
1.3 Samplers and Discrete-Time Physical Systems 14
Analog-to-Digital Converter, 14
Numerical Integration, 16
Picture in a Picture, 17
Compact Disks, 18
Sampling in Telephone Systems, 19
Data-Acquisition System, 21
1.4 MATLAB and Simulink 22

2 Continuous-Time Signals and Systems 23
2.1 Transformations of Continuous-Time Signals 24
Time Transformations, 24
Amplitude Transformations, 30
2.2 Signal Characteristics 32
Even and Odd Signals, 32
Periodic Signals, 34
2.3 Common Signals in Engineering 39
2.4 Singularity Functions 45
Unit Step Function, 45
Unit Impulse Function, 49
2.5 Mathematical Functions for Signals 54
2.6 Continuous-Time Systems 59
Interconnecting Systems, 61
Feedback System, 64
2.7 Properties of Continuous-Time Systems 65
Stability, 69
Linearity, 74
Summary 76
Problems 78
3 Continuous-Time Linear Time-Invariant Systems 90
3.1 Impulse Representation of Continuous-Time Signals 91
3.2 Convolution for Continuous-Time LTI Systems 92
3.3 Properties of Convolution 105
3.4 Properties of Continuous-Time LTI Systems 108
Memoryless Systems, 109
Invertibility, 109
Causality, 110
Stability, 111
Unit Step Response, 112
3.5 Differential-Equation Models 113
Solution of Differential Equations, 115
General Case, 117
Relation to Physical Systems, 119
3.6 Terms in the Natural Response 120
Stability, 121
3.7 System Response for Complex-Exponential Inputs 124
Linearity, 124
Complex Inputs for LTI Systems, 125
Impulse Response, 129
3.8 Block Diagrams 130
Direct Form I, 134
Direct Form II, 134
nth-Order Realizations, 134
Practical Considerations, 136
Summary 139
Problems 149

4 Fourier Series 154
4.1 Approximating Periodic Functions 155
Periodic Functions, 155
Approximating Periodic Functions, 156
4.2 Fourier Series 160
Fourier Series, 161
Fourier Coefficients, 162
4.3 Fourier Series and Frequency Spectra 165
Frequency Spectra, 166
4.4 Properties of Fourier Series 175
4.5 System Analysis 178
4.6 Fourier Series Transformations 185
Amplitude Transformations, 186
Time Transformations, 188
Summary 190
Problems 191

5 The Fourier Transform 201
5.1 Definition of the Fourier Transform 201
5.2 Properties of the Fourier Transform 210
Linearity, 211
Time Scaling, 212
Time Shifting, 214
Time Reversal, 215
Time Transformation, 216
Duality, 218
Convolution, 220
Frequency Shifting, 221
Time Integration, 224
Time Differentiation, 226
Frequency Differentiation, 231
Symmetry, 232
Summary, 233
5.3 Fourier Transforms of Time Functions 233
DC Level, 233
Unit Step Function, 233
Switched Cosine, 234
Pulsed Cosine, 234
Exponential Pulse, 236
Fourier Transforms of Periodic Functions, 236
Summary, 241
5.4 Application of the Fourier Transform 241
Frequency Response of Linear Systems, 241
Frequency Spectra of Signals, 250
Summary, 252
5.5 Energy and Power Density Spectra 253
Energy Density Spectrum, 253
Power Density Spectrum, 256
Power and Energy Transmission, 258
Summary, 260
Summary 262
Problems 263

6 Applications of the Fourier Transform 272
6.1 I deal Filters 272
6.2 Real Filters 279
RC Low-Pass Filter, 280
Butterworth Filter, 282
Bandpass Filters, 288
Active Filters, 289
Summary, 291
6.3 Bandwidth Relationships 291
6.4 Sampling Continuous-Time Signals 295
Impulse Sampling, 296
Shannon's Sampling Theorem, 299
Practical Sampling, 299
6.5 Reconstruction of Signals from Sample Data 300
Interpolating Function, 302
Digital-to-Analog Conversion, 304
Quantization Error, 306
6.6 Sinusoidal Amplitude Modulation 308
Frequency-Division Multiplexing, 317
6.7 Pulse-Amplitude Modulation 319
Time-Division Multiplexing, 321
Flat-Top PAM, 323
Summary 326
Problems 326

7 The Laplace Transform 336
7.1 Definitions of Laplace Transforms 337
7.2 Examples 340
7.3 Laplace Transforms of Functions 345
7.4 Laplace Transform Properties 349
Real Shifting, 350
Differentiation, 354
Integration, 356
7.5 Additional Properties 357
Multiplication by t, 357
Initial Value, 358
Final Value, 359
Time Transformation, 360
7.6 Response of LTI Systems 363
Initial Conditions, 363
Transfer Functions, 364
Convolution, 369
Transforms with Complex Poles, 371
Functions with Repeated Poles, 374
7.7 LTI Systems Characteristics 375
Causality, 375
Stability, 376
Invertibility, 378
Frequency Response, 379
Step Response, 380
7.8 Bilateral Laplace Transform 382
Region of Convergence, 384
Bilateral Transform from Unilateral Tables, 386
Inverse Bilateral Laplace Transform, 389
7.9 Relationship of the Laplace Transform to the Fourier Transform 391
Summary 392
Problems 393

8 State Variables for Continuous-Time Systems 401
8.1 State-Variable Modeling 402
8.2 Simulation Diagrams 406
8.3 Solution of State Equations 412
Laplace-Transform Solution, 412
Convolution Solution, 417
Infinite Series Solution, 418
8.4 Properties of the State-Transition Matrix 421
8.5 Transfer Functions 423
Stability, 425
8.6 Similarity Transformations 427
Transformations, 427
Properties, 433
Summary 435
Problems 437

9 Discrete-Time Signals and Systems 446
9.1 Discrete-Time Signals and Systems 448
Unit Step and Unit Impulse Functions, 450
Equivalent Operations, 452
9.2 Transformations of Discrete-Time Signals 453
Time Transformations, 454
Amplitude Transformations, 459
9.3 Characteristics of Discrete-Time Signals 462
Even and Odd Signals, 462
Signals Periodic in n, 465
Signals Periodic in