# Signals, Systems, & Transforms, 5th edition

• Charles L. Phillips
• John Parr
• Eve Riskin

## Overview

For sophomore/junior-level signals and systems courses in Electrical and Computer Engineering departments.  This book is also suitable for electrical and computer engineers.

Signals, Systems, and Transforms, Fifth Edition is ideal for electrical and computer engineers. The text provides a clear, comprehensive presentation of both the theory and applications in signals, systems, and transforms. It presents the mathematical background of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrete-time and the discrete Fourier transforms, and the z-transform. The text integrates MATLAB examples into the presentation of signal and system theory and applications.

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 , 468
9.4 Common Discrete-Time Signals 469
9.5 Discrete-Time Systems 475
Interconnecting Systems, 476
9.6 Properties of Discrete-Time Systems 478
Systems with Memory, 478
Invertibility, 479
Inverse of a System, 480
Causality, 480
Stability, 481
Time Invariance, 481
Linearity, 482
Summary 484
Problems 486

10 Discrete-Time Linear Time-Invariant Systems 495
10.1 Impulse Representation of Discrete-Time Signals 496
10.2 Convolution for Discrete-Time Systems 497
Properties of Convolution, 506
10.3 Properties of Discrete-Time LTI Systems 509
Memory, 510
Invertibility, 510
Causality, 510
Stability, 511
Unit Step Response, 513
10.4 Difference-Equation Models 514
Difference-Equation Models, 514
Classical Method, 516
Solution by Iteration, 521
10.5 Terms in the Natural Response 522
Stability, 523
10.6 Block Diagrams 525
Two Standard Forms, 527
10.7 System Response for Complex-Exponential Inputs 531
Linearity, 532
Complex Inputs for LTI Systems, 532
Stability, 537
Sampled Signals, 537
Impulse Response, 537
Summary 539
Problems 540

11 The z-Transform 552
11.1 Definitions of z-Transforms 552
11.2 Examples 555
Two z-Transforms, 555
Digital-Filter Example, 558
11.3 z-Transforms of Functions 560
Sinusoids, 561
11.4 z-Transform Properties 565
Real Shifting, 565
Initial and Final Values, 568
11.5 Additional Properties 570
Time Scaling, 570
Convolution in Time, 572
11.6 L TI System Applications 573
Transfer Functions, 573
Inverse z-Transform, 575
Complex Poles, 578
Causality, 580
Stability, 581
Invertibility, 584
Frequency Response, 585
11.7 Bilateral z-Transform 588
Bilateral Transforms, 592
Regions of Convergence, 594
Inverse Bilateral Transforms, 595
Summary 598
Problems 599

12 Fourier Transforms of Discrete-Time Signals 609

12.1 Discrete-Time Fourier Transform 610
z-Transform, 612
12.2 Properties of the Discrete-Time Fourier Transform 617
Periodicity, 618
Linearity, 619
Time Shift, 619
Frequency Shift, 620
Symmetry, 620
Time Reversal, 621
Convolution in Time, 621
Convolution in Frequency, 622
Multiplication by n, 623
Parseval’s Theorem, 623
12.3 Discrete-Time Fourier Transform of Periodic Sequences 624
12.4 Discrete Fourier Transform 630
Shorthand Notation for the DFT, 632
Frequency Resolution of the DFT, 632
Validity of the DFT, 634
Summary, 638
12.5 Fast Fourier Transform 638
Decomposition-in-Time Fast Fourier Transform Algorithm, 638
Decomposition-in-Frequency Fast Fourier Transform, 643
Summary, 646
12.6 Applications of the Discrete Fourier Transform 646
Calculation of Fourier Transforms, 646
Convolution, 654
Filtering, 663
Correlation, 671
Energy Spectral Density Estimation, 677
Summary, 678
12.7 The Discrete Cosine Transform, 678
Summary 683
Problems 684

13 State Variables for Discrete-Time Systems 692
13.1 State-Variable Modeling 693
13.2 Simulation Diagrams 697
13.3 Solution of State Equations 703
Recursive Solution, 703
z-Transform Solution, 705
13.4 Properties of the State Transition Matrix 710
13.5 Transfer Functions 712
Stability, 714
13.6 Similarity Transformations 715
Properties, 719
Summary 720
Problems 721

Appendices 718
A. Integrals and Trigonometric Identities 730
Integrals, 730
Trigonometric Identities, 731
B. Leibnitz’s and L’Hôpital’s Rules 732
Leibnitz’s Rule, 732
L’Hôpital’s Rule, 733
C. Summation Formulas for Geometric Series 734
D. Complex Numbers and Euler’s Relation 735
Complex-Number Arithmetic, 736
Euler’s Relation, 739
Conversion Between Forms, 740
E. Solution of Differential Equations 742
Complementary Function, 742
Particular Solution, 743
General Solution, 744
Repeated Roots, 744
F. Partial-Fraction Expansions 746
G.  Review of Matrices 749
Algebra of Matrices, 753
Other Relationships, 754
H. Answers to Selected Problems 756
I. Signals and Systems References 770
Index

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