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Contents

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'Hopital's Rules 732 Leibnitz's Rule, 732 L'Hopital'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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