Signals and Systems Using MATLAB

Second Edition

Luis F. Chaparro Department of Electrical and Computer Engineering

University of Pittsburgh

Pittsburgh, PA, USA

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • I'ARIS • SAN DIEGO • SAN FRANCISCO

SINGAPORE • SYDNEY • TOKYO

Academic Press is an imprint of Elsevier

Contents

PREFACE xv ACKNOWLEDGMENTS xxiii

. •; l t .

From the Ground Up! 3 Introduction 3 Examples of Signal Processing Applications 4

Compact-Disk (CD) Player 4 Software-Defined Radio and Cognitive Radio 6 Computer-Control Systems 6

Continuous or Discrete? 8 Continuous and Discrete Representations 9 Derivatives and Finite Differences 10 Integrals and Summations 13 Differential and Difference Equations 15

Complex or Real? 19 Complex Numbers and Vectors 20 Functions of a Complex Variable 26 Phasors and Sinusoidal Steady State 27 The Phasor Connection 29

Soft Introduction to MATLAB 32 Numerical Computations 33 Symbolic Computations 43

Problems 51 Basic Problems 51 Problems Using MATLAB 56

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Cont" -time 5ig:: a.Is and ":v ,1

3 ." 1 Continuous-time Signals 65 Introduction 65 Classification of Time-Dependent Signals 66

Con ten t s

Continuous-time Signals 67 Basic Signal Operations—Time-Shifting

and Reflection 72 Even and Odd Signals 76 Periodic and Aperiodic Signals 79 Finite-Energy and Finite-Power Signals 81

Representation of Continuous-time Signals Using Basic Signals 87 Complex Exponentials 87 Unit-step, Unit-impulse, and Ramp Signals 90 Generic Representation of Signals 102

Special Signals—The Sampling and the Sine Signals 104 Basic Signal Operations—Time Scaling, Frequency

Shifting, and Windowing 106 What Have We Accomplished? Where Do We

Go from Here? 109 Problems 110

Basic Problems 110 Problems Using MATLAB 118

Continuous-time Systems 123 Introduction 123 System Concept and Classification 124 Linear Time-Invariant (LTI) Continuous-time Systems 126

Linearity 126 Time-invariance 131 The Convolution Integral 142

Causality 155 Graphical Computation of Convolution Integral 157

Bounded-Input Bounded-Output (BIBO) Stability 161 What Have We Accomplished? Where Do We Go

from Here? 166 Problems 166

Basic Problems 166 Problems Using MATLAB 175

CJ! The Laplace Transform 179 Introduction 179 The Two-Sided Laplace Transform 181

Eigenfunctions of LTI Systems 181 Region of Convergence 185

The One-sided Laplace Transform 190

C o n t e n t s

Properties of the One-sided Laplace Transform 199 Linearity 200 Differentiation 204 Integration 207 Time-shifting 208 Convolution Integral 212

Inverse Laplace Transform 213 Inverse of One-sided Laplace Transforms 213 Inverse of Functions Containing e - ^ Terms 225 Inverse of Two-sided Laplace Transforms 227

Analysis of LTI Systems 230 LTI Systems Represented by Ordinary Differential

Equations 230 Computation of the Convolution Integral 237

What Have We Accomplished? Where Do We Go from Here? 244

Problems 244 Basic Problems 244 Problems Using MATLAB 256

Frequency Analysis: The Fourier Series 263 Introduction 263 Eigenfunctions Revisited 264 Complex Exponential Fourier Series 271

Line Spectrum—Power Distribution over Frequency 274

Trigonometric Fourier Series 277 Fourier Coefficients from Laplace 282 Reflection and Even and Odd Periodic Signals 283 Convergence of the Fourier Series 296 Time and Frequency Shifting 301

Response of LTI Systems to Periodic Signals 303 Filtering of Periodic Signals 308

Operations Using Fourier Series 311 Sum of Periodic Signals 311 Multiplication of Periodic Signals 312 Derivatives and Integrals of Periodic Signals 314

What Have We Accomplished? Where Do We Go from Here? 317

Problems 318 Basic Problems 318 Problems Using MATLAB 327

Con ten t s

Frequency Analysis: The Fourier Transform 333 Introduction 333 From the Fourier Series to the Fourier Transform 334 Existence of the Fourier Transform 336 Fourier Transforms from the Laplace Transform 336 Linearity, Inverse Proportionality, and Duality 338

Linearity 338 Inverse Proportionality of Time and Frequency 339 Duality 343

Spectral Representation 346 Signal Modulation 346 Fourier Transform of Periodic Signals 349 Parseval's Energy Relation 352 Symmetry of Spectral Representations 354

Convolution and Filtering 359 Basics of Filtering 361 Ideal Filters 363 Frequency Response from Poles and Zeros 369 The Spectrum Analyzer 375

Additional Properties 376 Time-shifting 377 Differentiation and Integration 378

What Have We Accomplished? What is Next? 382 Problems 383

Basic Problems 383 Problems Using MATLAB 393

Application of Laplace Analysis to Control 397 Introduction 397 System Connections and Block Diagrams 398 Application to Classical Control 402

Stability and Stabilization 407 Transient Analysis of First- and Second-Order

Control Systems 410 State-Variable Representation of LTI Systems 416

Canonical Realizations 424 Complete Solution from State and Output

Equations 431 External and Internal Representation of

Systems 435 What Have We Accomplished? What is Next? 438

Problems 439 Basic Problems 439 Problems Using MATLAB 445

Fourier Analysis in Communications and Filtering 449 Introduction 449 Application to Communications 450

AM Suppressed Carrier (AM-SC) 450 Commercial AM 452 AM Single Side-band 454 Quadrature AM and Frequency Division

Multiplexing 455 Angle Modulation 457

Analog Filtering 461 Filtering Basics 462 Butterworth Lowpass Filter Design 464 Chebyshev Lowpass Filter Design 468 Frequency Transformations 475 Filter Design with MATLAB 477

What Have We Accomplished? What is Next? 481 Problems 482

Basic Problems 482 Problems Using MATLAB 485

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D - • ;.' Styiiaj.s and Systems

CHAPTERS Sampling Theory 493 Introduction 493 Uniform Sampling 494

Pulse Amplitude Modulation 494 Ideal Impulse Sampling 495 Reconstruction of the Original

Continuous-time Signal 503 Signal Reconstruction from Sine Interpolation 507 The Nyquist-Shannon Sampling Theorem 508 Sampling Simulations with MATLAB 509 Sampling Modulated Signals 513

Practical Aspects of Sampling 515 Sample-and-Hold Sampling 515 Quantization and Coding 517

C o n t e n t s

Sampling, Quantizing, and Coding with MATLAB 520 Application to Digital Communications 523

Pulse Code Modulation 524 Time-Division Multiplexing 527

What Have We Accomplished? Where Do We Go from Here? 528

Problems 528 Basic Problems 528 Problems Using MATLAB 532

Discrete-time Signals and Systems 535 Introduction 535 Discrete-time Signals 536

Periodic and Aperiodic Signals 538 Finite Energy and Finite Power

Discrete-time Signals 542 Even and Odd Signals 545 Basic Discrete-time Signals 549

Discrete-time Systems 560 Recursive and Non-Recursive Discrete-time

Systems 562 Dynamic Discrete-time Systems Represented by

Difference Equations 567 The Convolution Sum 568 Linear and Non-linear Filtering with MATLAB 576 Causality and Stability of Discrete-time Systems. 579

What Have We Accomplished? Where Do We Go from Here? 584

Problems 584 Basic Problems 584 Problems Using MATLAB 595

The Z-transform 603 Introduction 603 Laplace Transform of Sampled Signals 604 Two-Sided Z-transform 607

Region of Convergence 608 One-sided Z-transform 614

Signal Behavior and Poles 614 Computing Z-transforms with Symbolic MATLAB 618 Convolution Sum and Transfer Function 620 Interconnection of Discrete-time Systems 629 Initial and Final Value Properties 630

One-sided Z-transform Inverse 632 Long-division Method 633 Partial Fraction Expansion 634 Inverse Z-transform with MATLAB 637 Solution of Difference Equations 642 Inverse of Two-sided Z-transforms 652

State-Variable Representation 655 What Have We Accomplished? Where Do We

Go from Here? 668 Problems 669

Basic Problems 669 Problems Using MATLAB 679

Fourier Analysis of Discrete-time Signals and Systems 683 Introduction 683 The Discrete-Time Fourier Transform (DTFT) 684

Sampling, Z-transform, Eigenfunctions, and the DTFT 685

Duality in Time and in Frequency 687 Computation of the DTFT Using MATLAB 689 Time and Frequency Supports 692 Energy/Power of Aperiodic Discrete-time Signals 699 Time and Frequency Shifts 701 Symmetry 702 Convolution Sum 708

Fourier Series of Discrete-time Periodic Signals 710 Circular Representation of Discrete-time

Periodic Signals 711 Complex Exponential Discrete Fourier Series 712 Connection with the Z-transform 714 DTFT of Periodic Signals 715 Response of LTI Systems to Periodic Signals 718 Circular Shifting and Periodic Convolution 719

The Discrete Fourier Transform (DFT) 727 DFT of Periodic Discrete-time Signals 727 DFT of Aperiodic Discrete-time Signals 728 Computation of the DFT via the FFT 730 Linear and Circular Convolution 735 The Fast Fourier Transform Algorithm 740 Computation of the Inverse DFT 746

What Have We Accomplished? Where Do We Go from Here? 749

Problems 751 Basic Problems 751 Problems Using MATLAB 760

Introduction to the Design of Discrete Filters 769 Introduction 769 Frequency Selective Discrete Filters 771

Phase Distortion 771 IIR and FIR Discrete Filters 774

Filter Specifications 778 Frequency Specifications 778 Time-Domain Specifications 783

IIR Filter Design 783 Transformation Design of IIR Discrete Filters 784 Design of Butterworth Low-pass Discrete Filters 788 Design of Chebyshev Low-pass Discrete Filters 796 Rational Frequency Transformations 802 General IIR Filter Design with MATLAB 806

FIR Filter Design 809 Window Design Method 810 Window Functions 812 Linear Phase and Symmetry of the

Impulse Response 813 Realization of Discrete Filters 820

Realization of IIR Filters 821 Realization of FIR Filters 828

What Have We Accomplished? Where Do We Go from Here? 829

Problems 830 Basic Problems 830 Problems Using MATLAB 836

Useful Formulas 845 Trigonometric Relations 845 Hyperbolic Trigonometry Relations 846 Calculus 846

BIBLIOGRAPHY 847

INDEX 851

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