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