Signals and Systems Using MATLAB

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1 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

2 PREFACE ACKNOWLEDGMENTS xv xxiii. ; lt. 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 Tin ' Cont" -time 5ig: : a.is and " : v,1 3." 1 Continuous-time Signals 65 Introduction 65 Classification of Time-Dependent Signals 66

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

3 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 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

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

4 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 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 from Here? 317 Problems 318 Basic Problems 318 Problems Using MATLAB 327

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

5 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

6 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 " 3 r~ : ; 't of 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

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?

7 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

8 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 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

9 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 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 TEC Website for this Book: Companion Website for this Book:

System analysis and signal processing

System analysis and signal processing with emphasis on the use of MATLAB PHILIP DENBIGH University of Sussex ADDISON-WESLEY Harlow, England Reading, Massachusetts Menlow Park, California New York Don Mills,