PREFACE
1 INTRODUCTION
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
1.5 Signals and Systems References 23
References 23
2 CONTINUOUS-TIME SIGNALS AND SYSTEMS
2.1 Transformations of Continuous-Time Signals 25
Time Transformations, 25
Amplitude Transformations, 31
2.2 Signal Characteristics 33
Even and Odd Signals, 33
Periodic Signals, 35
2.3 Common Signals in Engineering 40
2.4 Singularity Functions 46
Unit Step Function, 46
Unit Impulse Function, 50
2.5 Mathematical Functions for Signals 55
2.6 Continuous-Time Systems 60
Interconnecting Systems, 62
Feedback System, 64
2.7 Properties of Continuous-Time Systems 66
Stability 70
Linearity 75
Summary 77
References 79
Problems 79
3 CONTINUOUS-TIME LINEAR TIME-INVARIANT SYSTEMS
3.1 Impulse Representation of Continuous-Time Signals 90
3.2 Convolution for Continuous-Time LTI Systems 93
3.3 Properties of Convolution 105
3.4 Properties of Continuous-Time LTI Systems 109
Memoryless Systems, 110
Invertibility, 110
Causality, 111
Stability, 112
Unit Step Response, 113
3.5 Differential-Equation Models 114
Solution of Differential Equations, 116
General Case, 118
Relation to Physical Systems, 120
3.6 Terms in the Natural Response 121
Stability, 122
3.7 System Response for Complex-Exponential Inputs 125
Linearity, 125
Complex Inputs for LTI Systems, 126
Impulse Response, 130
3.8 Block Diagrams 131
Direct Form I, 135
Direct Form II, 135
nth-Order Realizations, 135
Practical Considerations, 137
Summary 139
References 141
Problems 141
4 FOURIER SERIES
4.1 Approximating Periodic Functions 153
Periodic Functions, 153
Approximating Periodic Functions, 154
4.2 Fourier Series 158
Fourier Series, 159
Fourier Coefficients, 160
4.3 Fourier Series and Frequency Spectra 163
Frequency Spectra, 164
4.4 Properties of Fourier Series 173
4.5 System Analysis 176
4.6 Fourier Series Transformations 183
Amplitude Transformations, 184
Time Transformations, 186
Summary 188
References 189
Problems 189
5 THE FOURIER TRANSFORM
5.1 Definition of the Fourier Transform 199
5.2 Properties of the Fourier Transform 208
Linearity, 208
Time Scaling, 210
Time Shifting, 213
Time Transformation, 214
Duality, 215
Convolution, 218
Frequency Shifting, 219
Time Differentiation, 221
Time Integration, 226
Frequency Differentiation, 229
Summary, 229
5.3 Fourier Transforms of Time Functions 230
DC Level, 230
Unit Step Function, 230
Switched Cosine, 231
Pulsed Cosine, 231
Exponential Pulse, 233
Fourier Transforms of Periodic Functions, 233
Summary, 239
5.4 Sampling Continuous-Time Signals 239
Impulse Sampling, 240
Shannon's Sampling Theorem, 242
Practical Sampling, 244
5.5 Application of the Fourier Transform 244
Frequency Response of Linear Systems, 244
Frequency Spectra of Signals, 253
Summary, 256
5.6 Energy and Power Density Spectra 256
Energy Density Spectrum, 256
Power Density Spectrum, 259
Power and Energy Transmission, 262
Summary, 264
Summary 265
References 267
Problems 267
6 APPLICATIONS OF THE FOURIER TRANSFORM
6.1 Ideal Filters 275
6.2 Real Filters 282
RC Low-Pass Filter, 283
Butterworth Filter, 285
Chebyschev and Elliptic Filters, 291
Bandpass Filters, 295
Summary, 296
6.3 Bandwidth Relationships 296
6.4 Reconstruction of Signals from Sample Data 300
Interpolating Function, 302
Digital-to-Analog Conversion, 304
6.5 Sinusoidal Amplitude Modulation 307
Frequency-Division Multiplexing, 316
6.6 Pulse-Amplitude Modulation 318
Time-Division Multiplexing, 320
Flat-Top PAM, 322
Summary 325
References 325
Problems 326
7 THE LAPLACE TRANSFORM
7.1 Definitions of Laplace Transforms 338
7.2 Examples 341
7.3 Laplace Transforms of Functions 346
7.4 Laplace Transform Properties 350
Real Shifting, 351
Differentiation, 355
Integration, 357
7.5 Additional Properties 358
Multiplication by t, 358
Initial Value, 359
Final Value, 360
Time Transformation, 361
7.6 Response of LTI Systems 364
Initial Conditions, 364
Transfer Functions, 365
Convolution, 370
Transforms with Complex Poles, 372
Functions with Repeated Poles, 375
7.7 LTI Systems Characteristics 376
Causality, 376
Stability, 377
Invertibility, 379
Frequency Response, 380
7.8 Bilateral Laplace Transform 382
Region of Convergence, 384
Bilateral Transform from Unilateral Tables, 386
Inverse Bilateral Laplace Transform, 388
7.9 Relationship of the Laplace Transform to the Fourier Transform 390
Summary 391
References 392
Problems 392
8 STATE VARIABLES FOR CONTINUOUS-TIME SYSTEMS 400
8.1 State-Variable Modeling 401
8.2 Simulation Diagrams 405
8.3 Solution of State Equations 410
Laplace-Transform Solution, 411
Convolution Solution, 416
Infinite Series Solution, 417
8.4 Properties of the State Transition Matrix 420
8.5 Transfer Functions 422
Stability, 424
8.6 Similarity Transformations 426
Transformations, 426
Properties, 432
Summary 434
References 436
Problems 436
9 DISCRETE-TIME SIGNALS AND SYSTEMS 445
9.1 Discrete-Time Signals and Systems 447
Unit Step and Unit Impulse Functions, 449
Equivalent Operations, 451
9.2 Transformations of Discrete-Time Signals 452
Time Transformations, 453
Amplitude Transformations, 458
9.3 Characteristics of Discrete-Time Signals 461
Even and Odd Signals, 461
Signals Periodic in n, 464
Signals Periodic in , 467
9.4 Common Discrete-Time Signals 468
9.5 Discrete-Time Systems 474
Interconnecting Systems, 475
9.6 Properties of Discrete-Time Systems 477
Systems with Memory, 477
Invertibility, 478
Inverse of a System, 479
Causality, 479
Stability, 480
Time Invariance, 480
Linearity, 481
Summary 483
References 485
Problems 485
10 DISCRETE-TIME LINEAR TIME-INVARIANT SYSTEMS 493
10.1 Impulse Representation of Discrete-Time Signals 494
10.2 Convolution for Discrete-Time Systems 495
Properties of Convolution, 504
10.3 Properties of Discrete-Time LTI Systems 507
Memory, 508
Invertibility, 508
Causality, 508
Stability, 509
Unit Step Response, 511
10.4 Difference-Equation Models 512
Difference-Equation Models, 512
Classical Method, 514
Solution by Iteration, 519
10.5 Terms in the Natural Response 520
Stability, 521
10.6 Block Diagrams 523
Two Standard Forms, 525
10.7 System Response for Complex-Exponential Inputs 529
Linearity, 530
Complex Inputs for LTI Systems, 530
Stability, 535
Sampled Signals, 535
Impulse Response, 535
Summary 537
Reference 538
Problems 538
11 THE z-TRANSFORM 547
11.1 Definitions of z-Transforms 547
11.2 Examples 550
Two z-Transforms, 550
Digital-Filter Example, 553
11.3 z-Transforms of Functions 555
Sinusoids, 557
11.4 z-Transform Properties 560
Real Shifting, 560
Initial and Final Values, 563
11.5 Additional Properties 565
Time Scaling, 565
Convolution in Time, 567
11.6 LTI System Applications 568
Transfer Functions, 569
Inverse z-Transform, 571
Complex Poles, 574
Causality, 575
Stability, 576
Invertibility, 579
11.7 Bilateral z-Transform 580
Bilateral Transforms, 585
Regions of Convergence, 586
Inverse Bilateral Transforms, 588
Summary 590
References 591
Problems 591
12 FOURIER TRANSFORMS OF DISCRETE-TIME SIGNALS 599
12.1 Discrete-Time Fourier Transform 600
z-Transform, 602
12.2 Properties of the Discrete-Time Fourier Transform 605
Periodicity, 605
Linearity, 606
Time Shift, 606
Frequency Shift, 607
Symmetry, 608
Time Reversal, 608
Convolution in Time, 609
Convolution in Frequency, 609
Multiplication by n, 610
Parseval's Theorem, 610
12.3 Discrete-Time Fourier Transform of Periodic Sequences 611
12.4 Discrete Fourier Transform 617
Shorthand Notation for the DFT, 620
Frequency Resolution of the DFT, 621
Validity of the DFT, 622
Summary, 626
12.5 Fast Fourier Transform 627
Decomposition-in-Time Fast Fourier Transform Algorithm, 627
Decomposition-in-Frequency Fast Fourier Transform, 632
Summary, 635
12.6 Applications of the Discrete Fourier Transform 635
Calculation of Fourier Transforms, 635
Convolution, 646
Filtering, 653
Correlation, 660
Energy Spectral Density Estimation, 666
Summary, 667
12.7 The Discrete Cosine Transform, 667
Summary 669
References 671
Problems 671
13 STATE VARIABLES FOR DISCRETE-TIME SYSTEMS 677
13.1 State-Variable Modeling 678
13.2 Simulation Diagrams 682
13.3 Solution of State Equations 688
Recursive Solution, 688
z-Transform Solution, 690
13.4 Properties of the State Transition Matrix 695
13.5 Transfer Functions 697
Stability, 699
13.6 Similarity Transformations 700
Properties, 704
Summary 705
References 706
Problems 707
APPENDICES 714
A. Integrals and Trigonometric Identities 714
Integrals, 714
Trigonometric Identities, 715
B. Leibnitz's and L'Hopital's Rules 716
Leibnitz's Rule, 716
L'Hopital's Rule, 717
C. Summation Formulas for Geometric Series 718
D. Complex Numbers and Euler's Relation 719
Complex-Number Arithmetic, 720
Euler's Relation, 723
Conversion Between Forms, 724
References, 725
E. Solution of Differential Equations 726
Complementary Function, 726
Particular Solution, 727
General Solution, 728
Repeated Roots, 728
Reference, 729
F. Partial-Fraction Expansions 730
Reference, 732
G. Review of Matrices 733
Algebra of Matrices, 737
Other Relationships, 738
References, 739
H. Answers to Selected Problems 740
INDEX 759
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