Linear systems and signals

Cover......Page 1
Half title......Page 3
Series page......Page 4
Title page......Page 5
Copyright page......Page 6
CONTENTS......Page 7
PREFACE......Page 17
  B.1-1 A Historical Note......Page 21
  B.1-2 Algebra of Complex Numbers......Page 25
 B.2 SINUSOIDS......Page 36
  B.2-1 Addition of Sinusoids......Page 38
  B.3-1 Monotonic Exponentials......Page 40
  B.3-2 The Exponentially Varying Sinusoid......Page 42
 B.4 CRAMER’S RULE......Page 43
 B.5 PARTIAL FRACTION EXPANSION......Page 45
  B.5-1 Method of Clearing Fractions......Page 46
  B.5-2 The Heaviside “Cover-Up” Method......Page 47
  B.5-3 Repeated Factors of Q(x)......Page 51
  B.5-4 A Combination of Heaviside “Cover-Up” and Clearing Fractions......Page 52
  B.5-5 Improper F(x) with m = n......Page 54
  B.5-6 Modified Partial Fractions......Page 55
 B.6 VECTORS AND MATRICES......Page 56
  B.6-1 Some Definitions and Properties......Page 57
  B.6-2 Matrix Algebra......Page 58
  B.7-1 MATLAB Overview......Page 62
  B.7-2 Calculator Operations......Page 63
  B.7-3 Vector Operations......Page 65
  B.7-4 Simple Plotting......Page 66
  B.7-5 Element-by-Element Operations......Page 68
  B.7-6 Matrix Operations......Page 69
  B.7-7 Partial Fraction Expansions......Page 73
  B.8-3 Sums......Page 74
  B.8-6 Trigonometric Identities......Page 75
  B.8-7 Common Derivative Formulas......Page 76
  B.8-8 Indefinite Integrals......Page 77
 REFERENCES......Page 78
 PROBLEMS......Page 79
 1.1 SIZE OF A SIGNAL......Page 84
  1.1-2 Signal Power......Page 85
  1.2-1 Time Shifting......Page 91
  1.2-2 Time Scaling......Page 93
  1.2-3 Time Reversal......Page 96
  1.2-4 Combined Operations......Page 97
  1.3-2 Analog and Digital Signals......Page 98
  1.3-3 Periodic and Aperiodic Signals......Page 99
 1.4 SOME USEFUL SIGNAL MODELS......Page 102
  1.4-1 The Unit Step Function u(t)......Page 103
  1.4-2 The Unit Impulse Function δ(t)......Page 106
  1.4-3 The Exponential Function e^{st}......Page 109
  1.5-1 Some Properties of Even and Odd Functions......Page 112
  1.5-2 Even and Odd Components of a Signal......Page 113
 1.6 SYSTEMS......Page 115
  1.7-1 Linear and Nonlinear Systems......Page 117
  1.7-2 Time-Invariant and Time-Varying Systems......Page 122
  1.7-3 Instantaneous and Dynamic Systems......Page 123
  1.7-4 Causal and Noncausal Systems......Page 124
  1.7-5 Continuous-Time and Discrete-Time Systems......Page 127
  1.7-7 Invertible and Noninvertible Systems......Page 129
  1.7-8 Stable and Unstable Systems......Page 130
  1.8-1 Electrical Systems......Page 131
  1.8-2 Mechanical Systems......Page 134
  1.8-3 Electromechanical Systems......Page 138
 1.9 INTERNAL AND EXTERNAL DESCRIPTIONS OF A SYSTEM......Page 139
 1.10 INTERNAL DESCRIPTION: THE STATE-SPACE DESCRIPTION......Page 141
  1.11-1 Anonymous Functions......Page 146
  1.11-2 Relational Operators and the Unit Step Function......Page 148
  1.11-3 Visualizing Operations on the Independent Variable......Page 150
  1.11-4 Numerical Integration and Estimating Signal Energy......Page 151
 1.12 SUMMARY......Page 153
 REFERENCES......Page 155
 PROBLEMS......Page 156
 2.1 INTRODUCTION......Page 170
 2.2 SYSTEM RESPONSE TO INTERNAL CONDITIONS: THE ZERO-INPUT RESPONSE......Page 171
  2.2-1 Some Insights into the Zero-Input Behavior of a System......Page 181
 2.3 THE UNIT IMPULSE RESPONSE h(t)......Page 183
 2.4 SYSTEM RESPONSE TO EXTERNAL INPUT: THE ZERO-STATE RESPONSE......Page 188
  2.4-1 The Convolution Integral......Page 190
  2.4-2 Graphical Understanding of Convolution Operation......Page 198
  2.4-3 Interconnected Systems......Page 210
  2.4-4 A Very Special Function for LTIC Systems: The Everlasting Exponential e^{st}......Page 213
  2.4-5 Total Response......Page 215
  2.5-1 External (BIBO) Stability......Page 216
  2.5-2 Internal (Asymptotic) Stability......Page 218
  2.5-3 Relationship Between BIBO and Asymptotic Stability......Page 219
  2.6-1 Dependence of System Behavior on Characteristic Modes......Page 223
  2.6-2 Response Time of a System: The System Time Constant......Page 225
  2.6-3 Time Constant and Rise Time of a System......Page 226
  2.6-4 Time Constant and Filtering......Page 227
  2.6-6 Time Constant and Rate of Information Transmission......Page 229
  2.6-7 The Resonance Phenomenon......Page 230
 2.7 MATLAB: M-FILES......Page 232
  2.7-1 Script M-Files......Page 233
  2.7-2 Function M-Files......Page 234
  2.7-3 For-Loops......Page 235
  2.7-4 Graphical Understanding of Convolution......Page 237
 2.8 APPENDIX: DETERMINING THE IMPULSE RESPONSE......Page 240
 2.9 SUMMARY......Page 241
 PROBLEMS......Page 243
 3.1 INTRODUCTION......Page 257
  3.1-1 Size of a Discrete-Time Signal......Page 258
 3.2 USEFUL SIGNAL OPERATIONS......Page 260
  3.3-1 Discrete-Time Impulse Function δ[n]......Page 265
  3.3-2 Discrete-Time Unit Step Function u[n]......Page 266
  3.3-3 Discrete-Time Exponential γ^n......Page 267
  3.3-4 Discrete-Time Sinusoid cos(Omega n+θ)......Page 271
  3.3-5 Discrete-Time Complex Exponential e^{jOmega n}......Page 272
 3.4 EXAMPLES OF DISCRETE-TIME SYSTEMS......Page 273
  3.4-1 Classification of Discrete-Time Systems......Page 282
 3.5 DISCRETE-TIME SYSTEM EQUATIONS......Page 285
  3.5-1 Recursive (Iterative) Solution of Difference Equation......Page 286
 3.6 SYSTEM RESPONSE TO INTERNAL CONDITIONS: THE ZERO-INPUT RESPONSE......Page 290
 3.7 THE UNIT IMPULSE RESPONSE h[n]......Page 297
  3.7-1 The Closed-Form Solution of h[n]......Page 298
 3.8 SYSTEM RESPONSE TO EXTERNAL INPUT: THE ZERO-STATE RESPONSE......Page 300
  3.8-1 Graphical Procedure for the Convolution Sum......Page 308
  3.8-2 Interconnected Systems......Page 314
  3.8-3 Total Response......Page 317
  3.9-1 External (BIBO) Stability......Page 318
  3.9-2 Internal (Asymptotic) Stability......Page 319
  3.9-3 Relationship Between BIBO and Asymptotic Stability......Page 321
 3.10 INTUITIVE INSIGHTS INTO SYSTEM BEHAVIOR......Page 325
  3.11-1 Discrete-Time Functions and Stem Plots......Page 326
  3.11-2 System Responses Through Filtering......Page 328
  3.11-3 A Custom Filter Function......Page 330
  3.11-4 Discrete-Time Convolution......Page 331
 3.13 SUMMARY......Page 333
 PROBLEMS......Page 334
 4.1 THE LAPLACE TRANSFORM......Page 350
  4.1-1 Finding the Inverse Transform......Page 358
  4.2-1 Time Shifting......Page 369
  4.2-2 Frequency Shifting......Page 373
  4.2-3 The Time-Differentiation Property......Page 374
  4.2-4 The Time-Integration Property......Page 376
  4.2-6 Time Convolution and Frequency Convolution......Page 377
 4.3 SOLUTION OF DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS......Page 380
  4.3-1 Comments on Initial Conditions at 0^− and at 0^+......Page 383
  4.3-2 Zero-State Response......Page 386
  4.3-3 Stability......Page 391
 4.4 ANALYSIS OF ELECTRICAL NETWORKS: THE TRANSFORMED NETWORK......Page 393
  4.4-1 Analysis of Active Circuits......Page 402
 4.5 BLOCK DIAGRAMS......Page 406
 4.6 SYSTEM REALIZATION......Page 408
  4.6-1 Direct Form I Realization......Page 409
  4.6-2 Direct Form II Realization......Page 410
  4.6-3 Cascade and Parallel Realizations......Page 413
  4.6-4 Transposed Realization......Page 416
  4.6-5 Using Operational Amplifiers for System Realization......Page 419
 4.7 APPLICATION TO FEEDBACK AND CONTROLS......Page 424
  4.7-1 Analysis of a Simple Control System......Page 426
 4.8 FREQUENCY RESPONSE OF AN LTIC SYSTEM......Page 432
  4.8-1 Steady-State Response to Causal Sinusoidal Inputs......Page 438
 4.9 BODE PLOTS......Page 439
  4.9-2 Pole (or Zero) at the Origin......Page 442
  4.9-3 First-Order Pole (or Zero)......Page 444
  4.9-4 Second-Order Pole (or Zero)......Page 446
  4.9-5 The Transfer Function from the Frequency Response......Page 455
  4.10-1 Dependence of Frequency Response on Poles and Zeros of H(s)......Page 456
  4.10-2 Lowpass Filters......Page 459
  4.10-4 Notch (Bandstop) Filters......Page 461
  4.10-5 Practical Filters and Their Specifications......Page 464
 4.11 THE BILATERAL LAPLACE TRANSFORM......Page 465
  4.11-1 Properties of the Bilateral Laplace Transform......Page 471
  4.11-2 Using the Bilateral Transform for Linear System Analysis......Page 472
 4.12 MATLAB: CONTINUOUS-TIME FILTERS......Page 475
  4.12-1 Frequency Response and Polynomial Evaluation......Page 476
  4.12-2 Butterworth Filters and the Find Command......Page 479
  4.12-3 Using Cascaded Second-Order Sections for Butterworth Filter Realization......Page 481
  4.12-4 Chebyshev Filters......Page 483
 4.13 SUMMARY......Page 486
 PROBLEMS......Page 488
 5.1 THE z-TRANSFORM......Page 508
  5.1-1 Inverse Transform by Partial Fraction Expansion and Tables......Page 515
  5.1-2 Inverse z-Transform by Power Series Expansion......Page 519
  5.2-1 Time-Shifting Properties......Page 521
  5.2-2 z-Domain Scaling Property (Multiplication by γ^n)......Page 525
  5.2-4 Time-Reversal Property......Page 526
  5.2-5 Convolution Property......Page 527
 5.3 z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS......Page 530
  5.3-1 Zero-State Response of LTID Systems: The Transfer Function......Page 534
  5.3-2 Stability......Page 538
 5.4 SYSTEM REALIZATION......Page 539
 5.5 FREQUENCY RESPONSE OF DISCRETE-TIME SYSTEMS......Page 546
  5.5-1 The Periodic Nature of Frequency Response......Page 552
  5.5-2 Aliasing and Sampling Rate......Page 556
 5.6 FREQUENCY RESPONSE FROM POLE-ZERO LOCATIONS......Page 558
 5.7 DIGITAL PROCESSING OF ANALOG SIGNALS......Page 567
 5.8 THE BILATERAL z-TRANSFORM......Page 574
  5.8-1 Properties of the Bilateral z-Transform......Page 579
  5.8-2 Using the Bilateral z-Transform for Analysis of LTID Systems......Page 580
 5.9 CONNECTING THE LAPLACE AND z-TRANSFORMS......Page 583
 5.10 MATLAB: DISCRETE-TIME IIR FILTERS......Page 585
  5.10-1 Frequency Response and Pole-Zero Plots......Page 586
  5.10-2 Transformation Basics......Page 587
  5.10-3 Transformation by First-Order Backward Difference......Page 588
  5.10-4 Bilinear Transformation......Page 589
  5.10-5 Bilinear Transformation with Prewarping......Page 590
  5.10-6 Example: Butterworth Filter Transformation......Page 591
  5.10-8 Using Cascaded Second-Order Sections to Improve Design......Page 592
 5.11 SUMMARY......Page 594
 PROBLEMS......Page 595
 6.1 PERIODIC SIGNAL REPRESENTATION BY TRIGONOMETRIC FOURIER SERIES......Page 613
  6.1-1 The Fourier Spectrum......Page 618
  6.1-2 The Effect of Symmetry......Page 627
  6.1-3 Determining the Fundamental Frequency and Period......Page 629
 6.2 EXISTENCE AND CONVERGENCE OF THE FOURIER SERIES......Page 632
  6.2-1 Convergence of a Series......Page 633
  6.2-2 The Role of Amplitude and Phase Spectra in Waveshaping......Page 635
 6.3 EXPONENTIAL FOURIER SERIES......Page 641
  6.3-1 Exponential Fourier Spectra......Page 644
  6.3-2 Parseval’s Theorem......Page 652
  6.3-3 Properties of the Fourier Series......Page 655
 6.4 LTIC SYSTEM RESPONSE TO PERIODIC INPUTS......Page 657
 6.5 GENERALIZED FOURIER SERIES:SIGNALS AS VECTORS......Page 661
  6.5-1 Component of a Vector......Page 662
  6.5-2 Signal Comparison and Component of a Signal......Page 663
  6.5-3 Extension to Complex Signals......Page 665
  6.5-4 Signal Representation by an Orthogonal Signal Set......Page 667
 6.6 NUMERICAL COMPUTATION OF D_n......Page 679
  6.7-1 Periodic Functions and the Gibbs Phenomenon......Page 681
  6.7-2 Optimization and Phase Spectra......Page 684
 6.8 SUMMARY......Page 687
 REFERENCES......Page 688
 PROBLEMS......Page 689
 7.1 APERIODIC SIGNAL REPRESENTATION BY THE FOURIER INTEGRAL......Page 700
  7.1-1 Physical Appreciation of the Fourier Transform......Page 707
 7.2 TRANSFORMS OF SOME USEFUL FUNCTIONS......Page 709
  7.2-1 Connection Between the Fourier and Laplace Transforms......Page 720
 7.3 SOME PROPERTIES OF THE FOURIER TRANSFORM......Page 721
 7.4 SIGNAL TRANSMISSION THROUGH LTIC SYSTEMS......Page 741
  7.4-1 Signal Distortion During Transmission......Page 743
  7.4-2 Bandpass Systems and Group Delay......Page 746
 7.5 IDEAL AND PRACTICAL FILTERS......Page 750
 7.6 SIGNAL ENERGY......Page 753
 7.7 APPLICATION TO COMMUNICATIONS: AMPLITUDE MODULATION......Page 756
  7.7-1 Double-Sideband, Suppressed-Carrier (DSB-SC) Modulation......Page 757
  7.7-2 Amplitude Modulation (AM)......Page 762
  7.7-3 Single-Sideband Modulation (SSB)......Page 766
 7.8 DATA TRUNCATION: WINDOW FUNCTIONS......Page 769
 7.9 MATLAB: FOURIER TRANSFORM TOPICS......Page 775
  7.9-1 The Sinc Function and the Scaling Property......Page 777
  7.9-2 Parseval’s Theorem and Essential Bandwidth......Page 778
  7.9-3 Spectral Sampling......Page 779
  7.9-4 Kaiser Window Functions......Page 780
 7.10 SUMMARY......Page 782
 REFERENCES......Page 783
 PROBLEMS......Page 784
 8.1 THE SAMPLING THEOREM......Page 796
  8.1-1 Practical Sampling......Page 801
 8.2 SIGNAL RECONSTRUCTION......Page 805
  8.2-1 Practical Difficulties in Signal Reconstruction......Page 808
  8.2-2 Some Applications of the Sampling Theorem......Page 816
 8.3 ANALOG-TO-DIGITAL (A/D) CONVERSION......Page 819
 8.4 DUAL OF TIME SAMPLING: SPECTRAL SAMPLING......Page 822
 8.5 NUMERICAL COMPUTATION OF THE FOURIER TRANSFORM: THE DISCRETE FOURIER TRANSFORM......Page 825
  8.5-1 Some Properties of the DFT......Page 838
  8.5-2 Some Applications of the DFT......Page 840
 8.6 THE FAST FOURIER TRANSFORM (FFT)......Page 844
  8.7-1 Computing the Discrete Fourier Transform......Page 847
  8.7-2 Improving the Picture with Zero Padding......Page 849
  8.7-3 Quantization......Page 851
 8.8 SUMMARY......Page 854
 PROBLEMS......Page 855
 9.1 DISCRETE-TIME FOURIER SERIES (DTFS)......Page 865
  9.1-1 Periodic Signal Representation by Discrete-Time Fourier Series......Page 866
  9.1-2 Fourier Spectra of a Periodic Signal x[n]......Page 868
 9.2 APERIODIC SIGNAL REPRESENTATION BY FOURIER INTEGRAL......Page 875
  9.2-1 Nature of Fourier Spectra......Page 878
  9.2-2 Connection Between the DTFT and the z-Transform......Page 886
 9.3 PROPERTIES OF THE DTFT......Page 887
 9.4 LTI DISCRETE-TIME SYSTEM ANALYSIS BY DTFT......Page 898
  9.4-1 Distortionless Transmission......Page 900
  9.4-2 Ideal and Practical Filters......Page 902
 9.5 DTFT CONNECTION WITH THE CTFT......Page 903
  9.5-1 Use of DFT and FFT for Numerical Computation of the DTFT......Page 905
 9.6 GENERALIZATION OF THE DTFT TO THE z-TRANSFORM......Page 906
  9.7-1 Computing the Discrete-Time Fourier Series......Page 909
  9.7-2 Measuring Code Performance......Page 911
  9.7-3 FIR Filter Design by Frequency Sampling......Page 912
 REFERENCE......Page 918
 PROBLEMS......Page 919
10 STATE-SPACE ANALYSIS......Page 928
  10.1-1 Derivatives and Integrals of aMatrix......Page 929
  10.1-2 The Characteristic Equation of a Matrix: The Cayley–Hamilton Theorem......Page 930
  10.1-3 Computation of an Exponential and a Power of aMatrix......Page 932
 10.2 INTRODUCTION TO STATE SPACE......Page 933
  10.3-1 Electrical Circuits......Page 936
  10.3-2 State Equations from a Transfer Function......Page 939
 10.4 SOLUTION OF STATE EQUATIONS......Page 946
  10.4-1 Laplace Transform Solution of State Equations......Page 947
  10.4-2 Time-Domain Solution of State Equations......Page 953
 10.5 LINEAR TRANSFORMATION OF A STATE VECTOR......Page 959
  10.5-1 Diagonalization of Matrix A......Page 963
 10.6 CONTROLLABILITY AND OBSERVABILITY......Page 967
 10.7 STATE-SPACE ANALYSIS OF DISCRETE-TIME SYSTEMS......Page 973
  10.7-1 Solution in State Space......Page 975
  10.7-2 The z-Transform Solution......Page 979
  10.8-1 z-Transform Solutions to Discrete-Time, State-Space Systems......Page 981
  10.8-2 Transfer Functions from State-Space Representations......Page 984
  10.8-3 Controllability and Observability of Discrete-Time Systems......Page 985
  10.8-4 Matrix Exponentiation and the Matrix Exponential......Page 988
 10.9 SUMMARY......Page 989
 PROBLEMS......Page 990
INDEX......Page 995