Signals and Systems Using MATLAB

Signals and Systems Using MATLAB
اسم المؤلف
Luis F. Chaparro
التاريخ
25 يوليو 2022
المشاهدات
61
التقييم
(لا توجد تقييمات)
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Signals and Systems Using MATLAB
Third Edition
Luis F. Chaparro
Department of Electrical and Computer Engineering
University of Pittsburgh
Pittsburgh, PA, USA
Aydin Akan
Department of Biomedical Engineering
Izmir Katip Celebi University
Izmir, Turkey
CHAPTER
FROM THE GROUND UP!
CONTENTS
0.1 Introduction 4
0.2 Examples of Signal Processing Applications . 4
0.2.1 Compact-Disc (CD) Player . 5
0.2.2 Software-Defined Radio and Cognitive Radio . 6
0.2.3 Computer-Control Systems . 7
0.3 Implementation of Digital Signal Processing Algorithms . 8
0.3.1 Microprocessors and Micro-Controllers 9
0.3.2 Digital Signal Processors . 9
0.3.3 Field Programmable Gate Arrays 9
0.4 Continuous or Discrete? 10
0.4.1 Continuous and Discrete Representations . 11
0.4.2 Derivatives and Finite Differences 12
0.4.3 Integrals and Summations . 14
0.4.4 Differential and Difference Equations 16
How to Solve Ordinary Differential Equations . 18
0.5 Complex or Real? . 20
0.5.1 Complex Numbers and Vectors 21
Euler’s Identity 24
0.5.2 Functions of a Complex Variable 27
0.5.3 Phasors and Sinusoidal Steady State 28
0.5.4 The Phasor Connection . 30
0.6 Soft Introduction to MATLAB 32
0.6.1 Numerical Computations . 33
MATLAB as a Calculator 33
MATLAB as a Signal Generator 37
Saving and Loading Data . 40
0.6.2 Symbolic Computations 41
Derivatives and Differences 43
The Sinc Function and Integration 44
Chebyshev Polynomials and Lissajous Figures . 46
Ramp, Unit-Step and Impulse Responses 49
0.7 Problems . 50
0.7.1 Basic Problems 50
0.7.2 Problems Using MATLAB
CHAPTER
CONTINUOUS-TIME SIGNALS
CONTENTS
1.1 Introduction 59
1.2 Classification of Time-Dependent Signals 60
1.3 Continuous-Time Signals 61
1.3.1 Addition, Constant Multiplication, Time Shifting and Reflection 65
1.3.2 Even and Odd Signals 68
1.3.3 Periodic and Aperiodic Signals 70
1.3.4 Finite-Energy and Finite-Power Signals 72
Superposition of Power 76
1.4 Representation of Continuous-Time Signals Using Basic Signals 77
1.4.1 Complex Exponentials 78
Sinusoids . 79
1.4.2 Unit Step, Unit Impulse and Ramp Signals . 80
The Ramp Signal . 82
Signal Generation With MATLAB 83
1.4.3 Generic Representation of Signals 91
1.5 Time Scaling, Modulation, Windowing and Integration . 93
1.6 Special Signals—the Sampling and the Sinc Signals 102
1.7 What Have We Accomplished? Where Do We Go From Here? . 104
1.8 Problems . 105
1.8.1 Basic Problems 105
1.8.2 Problems Using MATLAB .
CHAPTER
CONTINUOUS-TIME SYSTEMS
CONTENTS
2.1 Introduction 115
2.2 System Concept and Classification 116
2.3 Linear Time-Invariant (LTI) Continuous-Time Systems 117
2.3.1 Linearity 118
The Op-Amp 122
2.3.2 Time-Invariance 123
AM Communication System 124
FM Communication System 125
Vocal System 125
2.3.3 Representation of Systems by Ordinary Differential Equations . 128
2.4 The Convolution Integral . 135
2.4.1 Impulse Response and Convolution Integral 137
2.4.2 Interconnection of Systems—Block Diagrams 144
2.5 Causality 148
2.5.1 Graphical Computation of Convolution Integral . 150
2.6 Bounded Input–Bounded Output Stability . 154
2.7 What Have We Accomplished? Where Do We Go From Here? . 157
2.8 Problems . 158
2.8.1 Basic Problems 158
2.8.2 Problems Using MATLAB .
CHAPTER
THE LAPLACE TRANSFORM
CONTENTS
3.1 Introduction 167
3.2 The Two-Sided Laplace Transform . 169
3.2.1 Eigenfunctions of LTI Systems 169
3.2.2 Region of Convergence . 173
Poles and Zeros and the Region of Convergence . 174
Poles and Region of Convergence . 175
3.3 The One-Sided Laplace Transform . 178
3.4 Properties of the One-Sided Laplace Transform . 186
3.4.1 Linearity 187
3.4.2 Differentiation . 190
3.4.3 Integration . 193
3.4.4 Time-Shifting 194
3.4.5 Duality 196
3.4.6 Convolution Integral 199
3.5 Inverse Laplace Transform 200
3.5.1 Inverse of One-Sided Laplace Transforms . 201
Simple Real Poles . 203
Simple Complex Conjugate Poles 204
Double Real Poles . 208
3.5.2 Inverse of Functions Containing e−ρs Terms . 212
3.6 The Transfer Function of LTI Systems . 213
3.7 Analysis of LTI Systems Represented by Differential Equations . 220
Zero-State and Zero-Input Responses 222
Transient and Steady-State Responses . 223
3.8 Inverse of Two-Sided Laplace Transforms 227
3.9 What Have We Accomplished? Where Do We Go From Here? . 230
3.10 Problems . 230
3.10.1 Basic Problems 230
3.10.2 Problem Using MATLAB
CHAPTER
FREQUENCY ANALYSIS:
THE FOURIER SERIES
CONTENTS
4.1 Introduction 241
4.2 Eigenfunctions Revisited . 242
4.3 Complex Exponential Fourier Series . 249
4.3.1 Line Spectrum—Power Distribution Over Frequency 252
Parseval’s Power Relation 252
Symmetry of Line Spectra 253
4.3.2 Trigonometric Fourier Series . 254
4.3.3 Fourier Series and Laplace Transform 258
4.3.4 Reflection and Even and Odd Periodic Signals . 260
4.3.5 Convergence of the Fourier Series 274
4.3.6 Time and Frequency Shifting 278
4.4 Response of LTI Systems to Periodic Signals 281
4.4.1 Filtering of Periodic Signals . 284
4.5 Operations Using Fourier Series 287
4.5.1 Sum of Periodic Signals 287
4.5.2 Multiplication of Periodic Signals 288
4.5.3 Derivatives and Integrals of Periodic Signals . 290
4.5.4 Amplitude and Time Scaling of Periodic Signals . 292
4.6 What Have We Accomplished? Where Do We Go From Here? . 296
4.7 Problems . 297
4.7.1 Basic Problems 297
4.7.2 Problems Using MATLAB .
CHAPTER
FREQUENCY ANALYSIS:
THE FOURIER TRANSFORM
CONTENTS
5.1 Introduction 305
5.2 From the Fourier Series to the Fourier Transform . 306
5.3 Existence of the Fourier Transform 308
5.4 Fourier Transforms From the Laplace Transform . 309
5.5 Linearity, Inverse Proportionality and Duality 310
5.5.1 Linearity 310
5.5.2 Inverse Proportionality of Time and Frequency . 311
5.5.3 Duality 317
5.6 Spectral Representation . 319
5.6.1 Signal Modulation 319
Why Amplitude Modulation? 322
5.6.2 Fourier Transform of Periodic Signals 323
5.6.3 Parseval’s Energy Relation . 325
5.6.4 Symmetry of Spectral Representations 327
5.7 Convolution and Filtering 332
5.7.1 Basics of Filtering 334
5.7.2 Ideal Filters 336
5.7.3 Frequency Response From Poles and Zeros 341
5.7.4 The Spectrum Analyzer 346
5.8 Additional Properties . 348
5.8.1 Time Shifting 348
5.8.2 Differentiation and Integration 349
5.9 What Have We Accomplished? What Is Next? . 352
5.10 Problems . 353
5.10.1 Basic Problems 353
5.10.2 Problems Using MATLAB
CHAPTER
APPLICATION OF LAPLACE
ANALYSIS TO CONTROL
CONTENTS
6.1 Introduction 363
6.2 System Connections and Block Diagrams . 364
6.3 Application to Classical Control 368
6.3.1 Stability and Stabilization . 373
6.3.2 Transient Analysis of First- and Second-Order Control Systems 375
6.4 State-Variable Representation of LTI Systems . 382
6.4.1 Canonical Realizations . 389
6.4.2 Complete Solution From State and Output Equations . 396
Exponential Matrix Solution 396
Cramer’s Rule Solution 397
6.4.3 External and Internal Representation of Systems 399
6.5 What Have We Accomplished? What Is Next? . 402
6.6 Problems . 403
6.6.1 Basic Problems 403
6.6.2 Problems Using MATLAB .
CHAPTER
FOURIER ANALYSIS IN
COMMUNICATIONS AND FILTERING
CONTENTS
7.1 Introduction 409
7.2 Application to Communications . 410
7.2.1 AM Suppressed Carrier (AM-SC) 411
7.2.2 Commercial AM 413
7.2.3 AM Single Sideband 415
7.2.4 Quadrature AM and Frequency Division Multiplexing 416
Quadrature Amplitude Modulation (QAM) 416
Frequency Division Multiplexing (FDM) . 417
7.2.5 Angle Modulation 418
7.3 Analog Filtering . 421
7.3.1 Filtering Basics 423
Magnitude Squared Function 423
Filter Specifications 424
7.3.2 Butterworth Low-Pass Filter Design 425
Factorization . 426
Filter Design . 427
7.3.3 Chebyshev Low-Pass Filter Design 429
Filter Design . 430
Factorization . 431
7.3.4 Frequency Transformations 434
7.3.5 Filter Design With MATLAB 436
Low-Pass Filter Design 436
General Filter Design 439
7.4 What Have We Accomplished? What Is Next? . 440
7.5 Problems . 441
7.5.1 Basic Problems 441
7.5.2 Problems Using MATLAB
CHAPTER
SAMPLING THEORY
CONTENTS
8.1 Introduction 449
8.2 Uniform Sampling . 450
8.2.1 Pulse Amplitude Modulation 450
8.2.2 Ideal Impulse Sampling 451
8.2.3 Reconstruction of the Original Continuous-Time Signal . 459
8.2.4 Signal Reconstruction From Sinc Interpolation . 462
8.2.5 The Nyquist–Shannon Sampling Theorem . 463
8.2.6 Sampling Simulations With MATLAB . 464
8.2.7 Sampling Modulated Signals 468
8.3 Practical Aspects of Sampling 469
8.3.1 Sample-and-Hold Sampling . 470
8.3.2 Quantization and Coding . 472
8.3.3 Sampling, Quantizing and Coding With MATLAB . 474
8.4 Application to Digital Communications 477
8.4.1 Pulse Code Modulation . 478
Baseband and Band-Pass Communication Systems . 480
8.4.2 Time-Division Multiplexing . 481
8.5 What Have We Accomplished? Where Do We Go From Here? . 482
8.6 Problems . 482
8.6.1 Basic Problems 482
8.6.2 Problems Using MATLAB .
CHAPTER
DISCRETE-TIME SIGNALS
AND SYSTEMS
CONTENTS
9.1 Introduction 487
9.2 Discrete-Time Signals 488
9.2.1 Periodic and Aperiodic Discrete-Time Signals 490
9.2.2 Finite-Energy and Finite-Power Discrete-Time Signals 494
9.2.3 Even and Odd Discrete-Time Signals . 497
9.2.4 Basic Discrete-Time Signals . 500
Discrete-Time Complex Exponential 500
Discrete-Time Sinusoids 503
Discrete-Time Unit-Step and Unit-Sample Signals 506
Generic Representation of Discrete-Time Signals 507
9.3 Discrete-Time Systems . 511
9.3.1 Recursive and Non-recursive Discrete-Time Systems 514
9.3.2 Dynamic Discrete-Time Systems Represented by Difference Equations 518
9.3.3 The Convolution Sum . 519
9.3.4 Linear and Non-linear Filtering With MATLAB 525
Linear Filtering 526
Non-linear Filtering 526
9.3.5 Causality and Stability of Discrete-Time Systems 529
Causality . 529
Bounded Input–Bounded Output (BIBO) Stability . 531
9.4 Two-Dimensional Discrete Signals and Systems . 532
9.4.1 Two-Dimensional Discrete Signals 532
9.4.2 Two-Dimensional Discrete Systems . 537
9.5 What Have We Accomplished? Where Do We Go From Here? . 543
9.6 Problems . 543
9.6.1 Basic Problems 543
9.6.2 Problems Using MATLAB .
CHAPTER
THE Z-TRANSFORM
CONTENTS
10.1 Introduction 559
10.2 Laplace Transform of Sampled Signals . 560
10.3 Two-Sided Z-transform . 563
10.3.1 Region of Convergence . 564
ROC of Finite-Support Signals 565
ROC of Infinite-Support Signals . 567
10.4 One-Sided Z-transform . 569
10.4.1 Signal Behavior and Poles . 569
10.4.2 Computing Z-transforms With Symbolic MATLAB 573
10.4.3 Convolution Sum and Transfer Function . 574
10.4.4 Interconnection of Discrete-Time Systems 583
10.4.5 Initial- and Final-Value Properties 583
10.5 One-Sided Z-transform Inverse . 586
10.5.1 Long-Division Method 586
10.5.2 Partial Fraction Expansion . 587
10.5.3 Inverse Z-transform With MATLAB . 591
Partial Fraction Expansion . 591
A. Simple Poles 592
B. Multiple Poles 593
10.5.4 Solution of Difference Equations . 595
Approximate Solution of Ordinary Differential Equations 602
10.5.5 Inverse of Two-Sided Z-transforms . 604
10.6 State Variable Representation 606
Solution of the State and Output Equations 610
Canonical Realizations . 614
10.7 Two-Dimensional Z-transform . 618
10.8 What Have We Accomplished? Where Do We Go From Here? . 627
10.9 Problems . 627
10.9.1 Basic Problems 627
10.9.2 Problems Using MATLAB
CHAPTER
DISCRETE FOURIER ANALYSIS
CONTENTS
11.1 Introduction 638
11.2 The Discrete-Time Fourier Transform (DTFT) . 639
11.2.1 Sampling, Z-transform, Eigenfunctions and the DTFT 639
11.2.2 Duality in Time and in Frequency 641
11.2.3 Computation of the DTFT Using MATLAB . 644
11.2.4 Time and Frequency Supports 647
11.2.5 Decimation and Interpolation . 648
11.2.6 Energy/Power of Aperiodic Discrete-Time Signals 654
11.2.7 Time and Frequency Shifts 655
11.2.8 Symmetry 656
Computation of the Phase Spectrum Using MATLAB 659
11.2.9 Convolution Sum 662
11.3 Fourier Series of Discrete-Time Periodic Signals . 663
Circular Representation of Discrete-Time Periodic Signals 665
11.3.1 Complex Exponential Discrete Fourier Series . 665
11.3.2 Connection With the Z-transform . 668
11.3.3 DTFT of Periodic Signals 669
Computation of the Fourier Series Using MATLAB . 670
11.3.4 Response of LTI Systems to Periodic Signals . 671
11.3.5 Circular Shifting and Periodic Convolution . 673
11.4 The Discrete Fourier Transform (DFT) . 680
11.4.1 DFT of Periodic Discrete-Time Signals . 680
11.4.2 DFT of Aperiodic Discrete-Time Signals . 681
11.4.3 Computation of the DFT via the FFT . 683
11.4.4 Linear and Circular Convolution 688
11.4.5 The Fast Fourier Transform Algorithm . 693
Radix-2 FFT Decimation-in-Time Algorithm 695
11.4.6 Computation of the Inverse DFT 698
11.5 Two-Dimensional Discrete Transforms 701
11.6 What Have We Accomplished? Where Do We Go From Here? . 708
11.7 Problems . 708
11.7.1 Basic Problems 708
11.7.2 Problems Using MATLAB .
CHAPTER
INTRODUCTION TO THE
DESIGN OF DISCRETE FILTERS
CONTENTS
12.1 Introduction 722
12.2 Frequency Selective Discrete Filters 723
12.2.1 Phase Distortion . 724
Group Delay 725
12.2.2 IIR and FIR Discrete Filters . 726
12.3 Filter Specifications 731
12.3.1 Frequency Specifications 731
Loss Function 732
Magnitude Normalization 733
Frequency Scales . 734
12.3.2 Time Domain Specifications . 735
12.4 IIR Filter Design . 736
12.4.1 Transformation Design of IIR Discrete Filters . 736
The Bilinear Transformation . 736
Frequency Warping . 738
12.4.2 Design of Butterworth Low-Pass Discrete Filters . 740
12.4.3 Design of Chebyshev Low-Pass Discrete Filters 748
12.4.4 Rational Frequency Transformations . 753
Low-Pass to Low-Pass Transformation 753
Low-Pass to High-Pass Transformation . 755
Low-Pass to Band-Pass and Low-Pass to Band-Stop Transformations . 756
12.4.5 General IIR Filter Design With MATLAB . 757
12.5 FIR Filter Design 760
12.5.1 Window Design Method 761
12.5.2 Window Functions 763
12.5.3 Linear Phase and Symmetry of the Impulse Response 765
12.6 Realization of Discrete Filters 771
12.6.1 Realization of IIR Filters . 772
Direct Form Realization 772
Cascade Realization 775
Parallel Realization 777
12.6.2 Realization of FIR Filters 779
12.7 Two-Dimensional Filtering of Images 780
12.7.1 Spatial Filtering . 781
12.7.2 Frequency Domain Filtering . 785
12.8 What Have We Accomplished? Where Do We Go From Here? . 793
12.9 Problems .
Index
A
Acoustic signal, 5, 61, 93, 124, 553, 751
Adder circuit, 148
Adders, 65, 123, 147, 367, 384, 403, 607, 771
Aliasing, 466, 650
All-pass filter, 237, 344, 374, 650, 672, 714, 724, 753,
791
ideal, 337
Alternating current (AC), 124
Amplitude
analog, 60
maximum, 293
Amplitude modulation (AM), 94
demodulation, 411
envelope detector, 413
suppressed carrier, 411
tunable bandpass filter, 411
Analog communication systems, 410
AM, 410
angle modulation, 418
channel noise, 412
FM, 418
frequency division multiplexing (FDM), 416
quadrature amplitude modulation (QAM), 416
single sideband AM, 415
Analog control systems, 368
actuator, 370
cruise control, 371
feedback, 368
open- and closed-loop, 368
PI controller, 371
positive- and negative-feedback, 367
proportional (P) controller, 375
stability and stabilization, 373
transducer, 370
Analog filtering, 421
basics, 423
Butterworth low-pass filter, 427
Chebyshev low-pass filter, 429
Chebyshev polynomials, 430
eigenfunction property, 423
factorization, 424
frequency transformations, 434
loss function, 424
low-pass specifications, 424
magnitude and frequency normalization, 426
magnitude squared function, 423
Analog LTI systems, 117
BIBO stability, 148, 154, 531, 641, 727
causality, 148, 149
convolution integral, 137
eigenfunction property, 242
frequency response, 242, 333
impulse response, 137
impulse response, transfer function and frequency
response, 333
passivity, 155
transient analysis, 375
zero-input response, 128
zero-state response, 128
Analog systems
dc source, 335
windowing, 335
Analog-to-digital converter
A/D converter, 470
Antenna, 6, 124, 322
Antialiasing filter, 460
Application-specific integrated circuit (ASIC), 4
Armstrong, E., 418
Attenuation, 113, 233, 338, 444, 627, 729
B
Basic analog signal operations
adding, 65
advancing and delaying, 65
constant multiplying, 65
modulation, 93
reflection, 65
time scaling, 93
Basic analog signals
complex exponentials, 78
ramp, 82
sinusoids, 79
triangular pulse, 88
809810 Index
unit-impulse, 80
unit-step, 81
Basic discrete-time signals
complex exponentials, 500
sinusoids, 503
unit-step and unit-sample signals, 506
Bilinear transformation, 738
warping, 738
Biological systems, 4, 489
C
Capability of storing energy, 116, 127, 383
Capacitance, 116, 159
Capacitors, 8, 17, 30, 115, 125, 155, 423
Carrier, 124, 410
Cascade connections, 144, 365, 410, 521
Chemical plant, 363
Chirp signal, 43, 114, 715
Circuit
open, 17, 163, 340
short, 163, 340
Circuit theory, 72, 332, 412
Circular shifting, 673
Classical control, 158, 368, 402
Clock frequencies, 9
Closed-loop transfer function, 364
Comb filters, 730, 799
Complex conjugate poles, 188, 205, 237, 776
Complex variables, 20
function, 27
Continuous-time LTI systems
complete response, 223
eigenfunction property, 169
represented by ordinary differential equations, 220
steady-state response, 223
transient response, 223
unit-step response, 225
zero-input response, 222
zero-state, zero-input responses, 222
zero-state response, 222
Continuous-time signals
absolutely integrable, 73
advanced, 66
aperiodic, 60, 70
basic signals, 77
causal sinusoid, 74
complex exponentials, 77
definition, 62
delayed, 66
even, 60, 66, 68
even and odd decomposition, 69
finite energy, 60, 72
finite power, 60, 72, 73
finite support, 60
full-wave rectified, 88
odd, 60, 66, 68
periodic, 60, 70
real and imaginary parts, 78
shifting and reflecting, 67
sinusoids, 64
speech, 62
square integrable, 73
train of rectangular pulses, 89
windowing, 65
Conversion, 5, 23, 61, 360, 474
Convolution integral
commutative property, 145
distributive property, 145
Fourier, 332
graphical computation, 150
Laplace, 213
Convolution sum, 519, 574
commutative property, 521
deconvolution, 521
graphical, 576
length, 575
noncausal signals, 580
Cooley, J., 694
Cramer’s rule, 396, 611
Cut-off frequencies, 335
D
D/A converter, 62
Damped sinusoid, 78, 95, 501
Damping factor, 20, 168, 189, 560
Dc gain, 299, 425, 629, 746, 794
Dc loss, 425, 433, 733, 747, 799
Dc source, 20, 50, 267, 294, 335, 359, 546
De Moivre, A., 563
Differentiation, 12, 81, 190, 349
Digital communications, 477
PCM, 477
time-division multiplexing, 481
Digital signal processing
FFT, 693
inverse FFT, 698Index 811
Digital-to-analog converter
D/A converter, 62, 470
Dirac, P., 178
Discrete Cosine Transform (DCT)
1D-DCT, 704
2D-DCT, 706
transform coding, 704
Discrete filtering
analog signals, 723
averaging filter, 526
bilinear transformation, 736
Butterworth LPF, 740
Chebyshev LPF, 748
eigenfunction property of LTI systems, 723
FIR realizations, 779
FIR window design, 761
frequency scales, 734
frequency selective filters, 723
frequency specifications, 731
group delay, 725
IIR and FIR, 726
IIR realizations, 772
linear phase, 724
loss function, 732
median filter, 527
rational frequency transformations, 753
realization, 771
time specifications, 735
windows for FIR design, 763
Discrete filters
FIR, 582
FIR filters and convolution sum, 575
IIR, 582
Discrete Fourier series, 663
circular representation, 665
circular shifting, 673
complex exponential, 665
periodic convolution, 673
Z-transform, 668
Discrete Fourier Transform (DFT), 680
Fast Fourier Transform (FFT), 683
linear and circular convolution, 688
Discrete LTI systems
autoregressive (AR), 515
autoregressive moving average (ARMA), 517
BIBO stability, 529, 581
causality, 529
convolution sum, 519
difference equation representation, 514
moving average (MA), 514
recursive (IIR) and non-recursive (FIR), 514
response to periodic signals, 671
time-invariance, 512
Discrete-Time Fourier Transform (DTFT), 639
convergence, 639
convolution sum, 662
decimation, 648
eigenfunctions, 640
interpolation, 648
Parseval’s theorem, 654
periodic signals, 669
sampled signal, 640
symmetry, 656
time and frequency shifts, 655
time–frequency duality, 641
time–frequency supports, 647
Z-transform, 640
Discrete-time signals
absolutely summable, 494
definition, 488
discrete frequency, 500
energy and power spectrum, 654
Fibonacci sequence, 489
finite energy, 494
finite power, 494
generic representation, 507
inherently discrete-time, 489
periodic, fundamental period, 490
sample index, 489
sinusoid, 489
square summable, 494
square summable, finite energy, 494
Discrete-time sinusoids, 503, 544, 656
periodic, 490
Discretization, 11, 62, 560, 644
Disturbance, 7, 368, 691
Doppler frequency shift, 113, 444
Double poles, 189, 572
Double real poles, 208, 379
Down-sampling, 500, 552, 649, 715, 800
E
Electrical circuit, 8
Electronic noise, 7, 17
Elliptic design, 439
Elliptic filters, 445, 798812 Index
Energy
conservation property, 326
consumption, 7
continuous-time signals, 73
discrete-time signals, 494
spectrum, 326, 654
Euler, L., 23
Euler’s identity, 23, 78
F
Fast Fourier Transform (FFT), 693
inverse, 698
Fessenden, R., 124
Filtering, 245
analog basics, 334
antialiasing, 455
RC high-pass filter, 343
RC low-pass filter, 342
Filters
band-stop, 340, 435, 753, 781
bandpass, 340, 412, 440
comb, 730
denormalized, 426, 427
frequency discriminating, 334
gaussian, 720, 782
ideal, 299, 336
linear, 525, 785
magnitude-normalized, 425
median, 527
non-linear, 525
Finite calculus, 10
finite-difference, 12
summations, 14
Fourier, J.B., 249
Fourier series
absolute uniform convergence, 274
analog Parseval’s power relation, 252
coefficients, 250
coefficients from Laplace, 258
complex exponential, 249
convergence, 274
dc component, 255
even and odd signals, 260
Fourier basis, 251
fundamental frequency, 250
fundamental period, 250
Gibb’s phenomenon, 274
harmonics, 255
line spectrum, 253
mean-square approximation, 275
normality, 251
orthogonality, 251
periodic signals, 243
product of periodic signals, 288
spectrum analyzer, 252
sum of periodic signals, 287
time and frequency shifting, 278
time reversal, 260
trigonometric, 254
Fourier transform
amplitude modulation, 320
analog spectrum and line spectrum, 323
convolution integral, 332
differentiation and integration, 349
direct and inverse, 307
duality, 317
frequency shifting, 319
Laplace ROC, 309
linearity, 310
periodic analog signals, 323
shifting in time, 348
symmetry, 327
Frequencies
carrier, 412
instantaneous, 114, 419
normalized, 423
passband, 440
stopband, 445, 731
Frequency modulation (FM)
definition, 94
Frequency response
poles and zeros, 341
G
Gauss, C., 15, 701
Gaussian filter, 720, 782
Gaussian noise, 526, 659
Gibbs phenomenon
filtering, 339
Global positioning system (GPS), 4
H
Harmonic frequencies, 253, 285, 323, 671
Heaviside, O., 169
Hertz, H., 28
Human visual system, 781Index 813
I
Ideal filters
band-pass, 336
band-pass and high-pass, 336
high-pass, 336
linear phase, 336
low-pass, 336
zero-phase, 338
Image processing
2D filtering, 781
bank of filters, 790
binarization, 704
data compression, 704, 790
edge detection, 782
non-linear filtering, 785
Image processing, edge detection, 556
Images
digital, 534
discretized, 534
Inductors, 8, 30, 115, 125, 155, 423
Instability, 375
Integration, 14, 93, 100, 120, 193, 258, 272, 349, 361,
638
Inverse Laplace
partial fraction expansion, 201
two-sided, 227
with exponentials, 212
Inverse proportionality, 310
Inverse Z-transform
inspection, 586
long-division method, 586
partial fraction expansion, 587
positive powers of z, 590
proper rational functions, 587
rational functions, 587
J
Jury, E.I., 563
K
Kalman filtering, 563
L
Laplace, P.S., 169
Laplace transform
convolution integral, 199
derivative, 190
integration, 193
inverse, 171, 200
linearity, 187
poles and zeros, 174
proper rational, 201
region of convergence (ROC), 174
sampled signals, 560
two-sided, 171
L’Hôpital’s rule, 264
M
Marconi, G., 124
MATLAB
analog Butterworth and Chebyshev filter design, 434
analog Butterworth filtering, 433
control toolbox, 379
decimation and interpolation, 650
DFT and FFT, 644
discrete filter design, 749
DTFT computation, 644
FFT computation, 700
filter design, 436
Fourier series computation, 670
general discrete filter design, 757
numerical computations, 33
phase computation, 659
phase unwrapping, 659
spectrogram, 421
symbolic computations, 43
window FIR filter design, 771
Maximum frequency, 454
Message, 124, 161, 172, 320, 410, 468, 477
analog, 450, 477
digital, 450, 477
N
Noise, 7, 17, 165, 334, 526, 659, 781
Noisy signal, 13, 481, 527, 630
Non-linear filtering
median filter, 527
Nyquist
sampling rate, 464
sampling theorem, 463
Nyquist, H., 464
P
Parallel connections, 144, 364, 410
Phase modulation (PM)
definition, 94
Phasor connection, 30814 Index
Phasors
sinusoidal steady state, 244
Plant, 7, 363, 585
Pole–zero cancellation, 186, 223, 374, 399, 761
Power, 72, 73
average, 73
discrete-time signals, 494
instantaneous, 72
Power spectral density, 183
Pulse
amplitude modulation (PAM), 450
code modulation (PCM), 477
Q
Quantization
error, 473
four-level quantizer, 472
step, 473
R
Radio spectrum, 6, 416, 450, 477
Real-time processing, 65
Recursive system, 514, 540
Reference signal, 7, 239, 368, 407, 585
Resistance, 110, 116, 159
Resistors, 8, 17, 72, 107, 115, 120, 147, 155, 246, 340,
403, 423
S
Sampler
time-varying system, 453
Sampling
antialiasing filter, 455, 460
band-limited signal, 454
Discrete-Time Fourier Transform, 453
frequency aliasing, 455
holder for D/A, 471
ideal impulse, 451
Nyquist rate, 464
Nyquist–Shannon theorem, 463
Parseval’s application, 458
practical aspects, 469
quantization, 472
quantization error, 472, 473
quantizer, 472
sample-and-hold system, 470
sampling frequency, 454
sampling function, 451
signal reconstruction, 459
sinc interpolation, 462
time-division multiplexing, 481
uniform sampling, 450
Shannon, C., 464
Signal processing applications
cognitive radio, 6
compact-disc (CD) player, 5
computer-control systems, 7
software-defined radio, 6
Signal processing system, 8
analog-to-digital converter, 8
digital signal processors (DSPs), 4, 8
field-programmable gate arrays (FPGAs), 4, 8
microprocessors and micro-controllers, 8
real-time implementation, 8
Signals
absolutely summable, 494
analog, 60
band-limited, 454
causal, anticausal, noncausal, 173
causal discrete sinusoid, 494
continuous-time, 60
delayed, 66
deterministic, 60
digital, 60
discrete finite energy, 494
discrete sinusoid, 489
discrete-time, 60
finite energy discrete signal, 494
full-wave rectified, 267
modulation, 124
piecewise smooth, 274
radiated with antenna, 322
random, 60
sampled continuous-time, 489
sinc, 264
smooth, 274
square summable, 494
Sobel filters, 557, 782
Spectrum, 327, 411, 446, 465, 640, 652, 724, 781
Steinmetz, C., 28
Superposition, 76, 118, 135, 203, 244, 333, 417, 453,
512, 664, 682
Support
finite, 64, 93, 151, 179, 311, 321, 454, 479, 534,
565, 630, 644, 702
first-quadrant, 624Index 815
infinite, 70, 173, 250, 308, 336, 523, 567, 664
one-point, 311
Switch, 126, 162, 450, 471, 481
System
analog, 117
continuous-time, 117
definition, 116
digital, 117
discrete-time, 117
hybrid, 117
System connections, 364
Systems
all-pass, 248
amplitude modulation (AM), 124
analog averager, 139
discrete, 482, 560, 638, 693
echo, 155, 161
frequency modulation (FM), 125
ideal communication system, 247
linearity, 118
multi-path channel, 172
negative-feedback, 367, 403
non-linear, 123
positive-feedback, 367
sample-and-hold sampling, 470
single-input single-output, 117
time-invariance, 123
time-varying, 123
unstable, 118, 154, 236, 373
vocal system, 125
zero-order-hold, 470
T
Tesla, N., 124
Theorem, initial-value, 202
Transfer function
BIBO, 630
constant-numerator, 609, 772
discrete, 737
feedback, 403
feedforward, 376, 403
normalized, 441
one-dimensional, 626
only-zeros, 391
open-loop, 364
rational, 341, 439, 727
representation, 383, 560
two-dimensional, 626
Transient responses, 223, 600
Transmitter, 112, 124, 172, 410, 478
Triangular signal, 88, 142, 271, 292, 300
Tsypkin, Y., 563
Tukey, J., 694
Two-Dimensional Discrete Fourier Transform, 701
2D-FFT, 702
aperiodic signals, 702
Two-dimensional discrete signals, 532
impulse, 533
periodic, 534
rectangular and block periodic, 536
separable, 533
unit-ramp, 533
unit-step, 533
Two-dimensional discrete systems, 532, 537
convolution, 537
FIR, 540
IIR, 540
impulse response, 537
LSI, 537
LSI BIBO, 540
LSI causal, 542
separable, 538
Two-dimensional filtering, 781
1D vs. 2D, 781
circular filters, 788
frequency domain, 786
separable, 787
spatial, 781
Two-dimensional Z-transform, 618
convolution sum, 622
inverse, 621
stability testing, 623
U
Unit-step function, 99
V
Voice signal, 60, 124, 724
Voltage, 8, 18
ac, 239, 299
dc, 299
Voltage follower, 147, 366, 403
circuit, 147, 403
Voltage source, 17, 127, 148, 154, 192, 216, 246, 339,
376, 441
Voltage–current relation, 120816 Index
W
Walsh functions, 303
Windowed cosine, 480
Wireless communications, 112, 124, 172, 477
Z
Z-transform
connection with sampling, 563
damping radius, 562
discrete frequency, 562
inverse, 586
linearity, 569
one-sided, 569
poles and zeros, 587
ROC and uniqueness, 568
s-plane and Z-plane, 562
sampled signals, 561
significance of poles, 573
solution of difference equations, 596, 602
time-shifting, 595
transfer function, 574
two-sided transform, 563
Zadeh, L., 563
Zero-boundary conditions, 537

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