كتاب Discrete Systems and Digital Signal Processing with MATLAB
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منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب Discrete Systems and Digital Signal Processing with MATLAB

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كتاب Discrete Systems and Digital Signal Processing with MATLAB Empty
مُساهمةموضوع: كتاب Discrete Systems and Digital Signal Processing with MATLAB   كتاب Discrete Systems and Digital Signal Processing with MATLAB Emptyالجمعة 14 أكتوبر 2022, 1:58 am

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Discrete Systems and Digital Signal Processing with MATLAB
Taan S. ElAli

كتاب Discrete Systems and Digital Signal Processing with MATLAB D_s_a_10
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Table of Contents
1 Signal Representation 1
1.1 Introduction 1
1.2 Why Do We Discretize Continuous Systems? 2
1.3 Periodic and Nonperiodic Discrete Signals .3
1.4 The Unit Step Discrete Signal 4
1.5 The Impulse Discrete Signal 6
1.6 The Ramp Discrete Signal 6
1.7 The Real Exponential Discrete Signal .7
1.8 The Sinusoidal Discrete Signal 7
1.9 The Exponentially Modulated Sinusoidal Signal . 11
1.10 The Complex Periodic Discrete Signal 11
1.11 The Shifting Operation 15
1.12 Representing a Discrete Signal Using Impulses 16
1.13 The Reflection Operation .18
1.14 Time Scaling .19
1.15 Amplitude Scaling 20
1.16 Even and Odd Discrete Signal .21
1.17 Does a Discrete Signal Have a Time Constant? 23
1.18 Basic Operations on Discrete Signals 25
1.18.1 Modulation .25
1.18.2 Addition and Subtraction .25
1.18.3 Scalar Multiplication .25
1.18.4 Combined Operations .26
1.19 Energy and Power Discrete Signals .28
1.20 Bounded and Unbounded Discrete Signals .30
1.21 Some Insights: Signals in the Real World .30
1.21.1 The Step Signal 31
1.21.2 The Impulse Signal 31
1.21.3 The Sinusoidal Signal .31
1.21.4 The Ramp Signal 31
1.21.5 Other Signals 32
1.22 End of Chapter Examples .32
1.23 End of Chapter Problems 50
2 The Discrete System . 55
2.1 Definition of a System .55
2.2 Input and Output .55
2.3 Linear Discrete Systems 56
2.4 Time Invariance and Discrete Signals 582.5 Systems with Memory 60
2.6 Causal Systems .61
2.7 The Inverse of a System 62
2.8 Stable System 63
2.9 Convolution 64
2.10 Difference Equations of Physical Systems 68
2.11 The Homogeneous Difference Equation and Its Solution .69
2.11.1 Case When Roots Are All Distinct .71
2.11.2 Case When Two Roots Are Real and Equal 72
2.11.3 Case When Two Roots Are Complex .72
2.12 Nonhomogeneous Difference Equations and their
Solutions .73
2.12.1 How Do We Find the Particular Solution? .75
2.13 The Stability of Linear Discrete Systems:
The Characteristic Equation 75
2.13.1 Stability Depending On the Values of the Poles 75
2.13.2 Stability from the Jury Test 76
2.14 Block Diagram Representation of Linear Discrete
Systems .78
2.14.1 The Delay Element 79
2.14.2 The Summing/Subtracting Junction 79
2.14.3 The Multiplier 79
2.15 From the Block Diagram to the Difference Equation .81
2.16 From the Difference Equation to the Block Diagram:
A Formal Procedure .82
2.17 The Impulse Response .85
2.18 Correlation .87
2.18.1 Cross-Correlation .87
2.18.2 Auto-Correlation 89
2.19 Some Insights 90
2.19.1 How Can We Find These Eigenvalues? .90
2.19.2 Stability and Eigenvalues .91
2.20 End of Chapter Examples .91
2.21 End of Chapter Problems .134
3 The Fourier Series and the Fourier Transform of Discrete
Signals 143
3.1 Introduction 143
3.2 Review of Complex Numbers .143
3.2.1 Definition 145
3.2.2 Addition .145
3.2.3 Subtraction .145
3.2.4 Multiplication 145
3.2.5 Division 146
3.2.6 From Rectangular to Polar 146
3.2.7 From Polar to Rectangular 1463.3 The Fourier Series of Discrete Periodic Signals 147
3.4 The Discrete System with Periodic Inputs: The Steady-State
Response 150
3.4.1 The General Form for yss(n) .153
3.5 The Frequency Response of Discrete Systems 154
3.5.1 Properties of the Frequency Response 157
3.5.1.1 The Periodicity Property 157
3.5.1.2 The Symmetry Property .157
3.6 The Fourier Transform of Discrete Signals 159
3.7 Convergence Conditions .161
3.8 Properties of the Fourier Transform of Discrete Signals .162
3.8.1 The Periodicity Property .162
3.8.2 The Linearity Property .162
3.8.3 The Discrete-Time Shifting Property .163
3.8.4 The Frequency Shifting Property .163
3.8.5 The Reflection Property .163
3.8.6 The Convolution Property 164
3.9 Parseval’s Relation and Energy Calculations 167
3.10 Numerical Evaluation of the Fourier Transform of
Discrete Signals 168
3.11 Some Insights: Why Is This Fourier Transform? 172
3.11.1 The Ease in Analysis and Design .172
3.11.2 Sinusoidal Analysis 173
3.12 End of Chapter Examples .173
3.13 End of Chapter Problems .189
4 The z-Transform and Discrete Systems . 195
4.1 Introduction 195
4.2 The Bilateral z-Transform .195
4.3 The Unilateral z-Transform 197
4.4 Convergence Considerations .200
4.5 The Inverse z-Transform .203
4.5.1 Partial Fraction Expansion 203
4.5.2 Long Division 206
4.6 Properties of the z-Transform 207
4.6.1 Linearity Property .207
4.6.2 Shifting Property .207
4.6.3 Multiplication by e-an 209
4.6.4 Convolution .210
4.7 Representation of Transfer Functions as Block Diagrams 210
4.8 x(n), h(n), y(n), and the z-Transform .212
4.9 Solving Difference Equation using the z-Transform 214
4.10 Convergence Revisited 216
4.11 The Final Value Theorem 219
4.12 The Initial-Value Theorem 2194.13 Some Insights: Poles and Zeroes .220
4.13.1 The Poles of the System .220
4.13.2 The Zeros of the System 221
4.13.3 The Stability of the System .221
4.14 End of Chapter Exercises 221
4.15 End of Chapter Problems .255
5 State-Space and Discrete Systems 265
5.1 Introduction 265
5.2 A Review on Matrix Algebra .266
5.2.1 Definition, General Terms and Notations .266
5.2.2 The Identity Matrix 266
5.2.3 Adding Two Matrices 267
5.2.4 Subtracting Two Matrices 267
5.2.5 Multiplying A Matrix by a Constant .267
5.2.6 Determinant of a Two-by-Two Matrix 268
5.2.7 Transpose of A Matrix 268
5.2.8 Inverse of A Matrix .268
5.2.9 Matrix Multiplication .269
5.2.10 Eigenvalues of a Matrix .269
5.2.11 Diagonal Form of a Matrix .269
5.2.12 Eigenvectors of a Matrix 269
5.3 General Representation of Systems in State-Space 270
5.3.1 Recursive Systems 270
5.3.2 Nonrecursive Systems 272
5.3.3 From the Block Diagram to State-Space .273
5.3.4 From the Transfer Function H(z) to State-Space 276
5.4 Solution of the State-Space Equations in the z-Domain 283
5.5 General Solution of the State Equation in Real-Time 284
5.6 Properties of An and Its Evaluation 285
5.7 Transformations for State-Space Representations 289
5.8 Some Insights: Poles and Stability 291
5.9 End of Chapter Examples .292
5.10 End of Chapter Problems .322
6 Modeling and Representation of Discrete Linear Systems . 329
6.1 Introduction 329
6.2 Five Ways of Representing Discrete Linear Systems .330
6.2.1 From the Difference Equation to the Other Four
Representations .330
6.2.1.1 The Difference Equation Representation .330
6.2.1.2 The Impulse Response Representation 331
6.2.1.3 The z-Transform Representation .332
6.2.1.4 The State-Space Representation 333
6.2.1.5 The Block Diagram Representation 3346.2.2 From the Impulse Response to the Other
Four Representations 335
6.2.2.1 The Impulse Response Representation 335
6.2.2.2 The Transfer Function Representation .335
6.2.2.3 The Different Equation Representation .336
6.2.2.4 The State-Space Representation 336
6.2.2.5 The Block Diagram Representation 337
6.2.3 From the Transfer Function to the Other
Four Representations 337
6.2.3.1 The Transfer Function Representation .337
6.2.3.2 The Impulse Response Representation 338
6.2.3.3 The Difference Equation Representation .338
6.2.3.4 The State-Space Representation 339
6.2.3.5 The Block Diagram Representation 339
6.2.4 From the State-Space to the Other Four Representations .340
6.2.4.1 The State-Space Representation 340
6.2.4.2 The Transfer Function Representation .340
6.2.4.3 The Impulse Response Representation 341
6.2.4.4 The Difference Equation Representation .341
6.2.4.5 The Block Diagram Representation 342
6.2.5 From the Block Diagram to the Other Four
Representations .343
6.2.5.1 The State-Space Representation 343
6.2.5.2 The Transfer Function Representation .344
6.2.5.3 The Impulse Response Representation 345
6.2.5.4 The Difference Equation Representation .345
6.3 Some Insights: The Poles Considering Different Outputs within
the Same System 346
6.4 End of Chapter Exercises 346
6.5 End of Chapter Problems .361
7 The Discrete Fourier Transform and Discrete Systems . 365
7.1 Introduction 365
7.2 The Discrete Fourier Transform and the Finite-Duration
Discrete Signals 366
7.3 Properties of the Discrete Fourier Transform 367
7.3.1 How Does the Defining Equation Work? .367
7.3.2 The DFT Symmetry 368
7.3.3 The DFT Linearity 370
7.3.4 The Magnitude of the DFT .371
7.3.5 What Does k in X(k), the DFT, Mean? 372
7.4 The Relation the DFT Has with the Fourier Transform of
Discrete Signals, the z-Transform and the Continuous
Fourier Transform 3737.4.1 The DFT and the Fourier Transform of x(n) 373
7.4.2 The DFT and the z-Transform of x(n) .374
7.4.3 The DFT and the Continuous Fourier Transform of x(t) .376
7.5 Numerical Computation of the DFT 377
7.6 The Fast Fourier Transform: A Faster Way of
Computing the DFT .378
7.7 Applications of the DFT .380
7.7.1 Circular Convolution .380
7.7.2 Linear Convolution 384
7.7.3 Approximation to the Continuous Fourier Transform .385
7.7.4 Approximation to the Coefficients of the Fourier Series
and the Average Power of the Periodic Signal x(t) .387
7.7.5 Total Energy in the Signal x(n) and x(t) 391
7.7.6 Block Filtering .393
7.7.7 Correlation .395
7.8 Some Insights 395
7.8.1 The DFT Is the Same as the fft .395
7.8.2 The DFT Points Are the Samples of the Fourier
Transform of x(n) 395
7.8.3 How Can We Be Certain That Most of the Frequency
Contents of x(t) Are in the DFT? 395
7.8.4 Is the Circular Convolution the Same as the
Linear Convolution? .396
7.8.5 Is ? .396
7.8.6 Frequency Leakage and the DFT .396
7.9 End of Chapter Exercises 396
7.10 End of Chapter Problems .415
8 Analogue Filter Design 421
8.1 Introduction 421
8.2 Analogue Filter Specifications .422
8.3 Butterworth Filter Approximation 425
8.4 Chebyshev Filters .428
8.4.1 Type I Chebyshev Approximation .428
8.4.2 Inverse Chebyshev Filter (Type II Chebyshev Filters) .431
8.5 Elliptic Filter Approximation .433
8.6 Bessel Filters 434
8.7 Analogue Frequency Transformation .437
8.8 Analogue Filter Design using MATLAB 438
8.8.1 Order Estimation Functions 439
8.8.2 Analogue Prototype Design Functions .440
8.8.3 Complete Classical IIR Filter Design .440
8.8.4 Analogue Frequency Transformation 442
8.9 How Do We Find the Cut-Off Frequency Analytically? .443
8.10 Limitations 447
X w X k ( ) ( ) ≅8.11 Comparison between Analogue Filter Types 447
8.12 Some Insights: Filters with High Gain vs. Filters with Low Gain
and the Relation between the Time Constant and the Cut-Off
Frequency for First-Order Circuits and the Series RLC Circuit .448
8.13 End of Chapter Examples .449
8.14 End of Chapter Problems .479
9 Transformations between Continuous and Discrete
Representations . 487
9.1 The Need for Converting Continuous Signal to a Discrete
Signal 487
9.2 From the Continuous Signal to Its Binary Code Representation 488
9.3 From the Binary Code to the Continuous Signal .490
9.4 The Sampling Operation .490
9.4.1 Ambiguity in Real-Time Domain .490
9.4.2 Ambiguity in the Frequency Domain .492
9.4.3 The Sampling Theorem 493
9.4.4 Filtering before Sampling 494
9.4.5 Sampling and Recovery of the Continuous Signal .496
9.5 How Do We Discretize the Derivative Operation? 500
9.6 Discretization of the State-Space Representation .504
9.7 The Bilinear Transformation and the Relationship between the
Laplace-Domain and the z-Domain Representations 506
9.8 Other Transformation Methods .512
9.8.1 Impulse Invariance Method 512
9.8.2 The Step Invariance Method .512
9.8.3 The Forward Difference Method 512
9.8.4 The Backward Difference Method .512
9.8.5 The Bilinear Transformation .512
9.9 Some Insights 515
9.9.1 The Choice of the Sampling Interval Ts 515
9.9.2 The Effect of Choosing Ts on the Dynamics
of the System .515
9.9.3 Does Sampling Introduce Additional Zeros for
the Transfer Function H(z)? .516
9.10 End of Chapter Examples .517
9.11 End of Chapter Problems .534
10 Infinite Impulse Response (IIR) Filter Design . 541
10.1 Introduction 541
10.2 The Design Process 542
10.2.1 Design Based on the Impulse Invariance Method 542
10.2.2 Design Based on the Bilinear Transform Method .545
10.3 IIR Filter Design Using MATLAB .54810.3.1 From the Analogue Prototype to the IIR Digital Filter 548
10.3.2 Direct Design .548
10.4 Some Insights 550
10.4.1 The Difficulty in Designing IIR Digital Filters in the
z-Domain 550
10.4.2 Using the Impulse Invariance Method .552
10.4.3 The Choice of the Sampling Interval Ts 552
10.5 End of Chapter Examples .552
10.6 End of Chapter Problems .584
11 Finite Impulse Response (FIR) Digital Filters 591
11.1 Introduction 591
11.1.1 What Is an FIR Digital Filter? .591
11.1.2 A Motivating Example .591
11.2 FIR Filter Design 594
11.2.1 Stability of FIR Filters 596
11.2.2 Linear Phase of FIR Filters 597
11.3 Design Based on the Fourier Series: The Windowing Method 598
11.3.1 Ideal Lowpass FIR Filter Design 599
11.3.2 Other Ideal Digital FIR Filters 601
11.3.3 Windows Used in the Design of the Digital FIR Filter 602
11.3.4 Which Window Gives the Optimal h(n)? .604
11.3.5 Design of a Digital FIR Differentiator .605
11.3.6 Design of Comb FIR Filters .607
11.3.7 Design of a Digital Shifter: The Hilbert Transform Filter 609
11.4 From IIR to FIR Digital Filters: An Approximation .610
11.5 Frequency Sampling and FIR Filter Design 610
11.6 FIR Digital Design Using MATLAB . 611
11.6.1 Design Using Windows . 611
11.6.2 Design Using Least-Squared Error 612
11.6.3 Design Using the Equiripple Linear Phase 612
11.6.4 How to Obtain the Frequency Response 612
11.7 Some Insights 613
11.7.1 Comparison with IIR Filters .613
11.7.2 The Different Methods Used in the FIR Filter Design .613
11.8 End of the Chapter Examples 614
11.9 End of Chapter Problems .644
References 649
Index . 651
Index
A
Active circuit elements, 422
Active filters, 422
AC voltage source, 31
A/D, see Analogue-to-digital conversion
Algebra, easy-to-manipulate, 165
Algebraic equations, 115
Aliasing, 529, 531, 532, 534
prevention of, 617
sampling without, 493
Amplitude scaling, 20
Analogue-to-digital (A/D) conversion, 488
Analogue filter design, 421–485
analogue filter design using MATLAB, 438–442
analogue frequency transformation, 442
analogue prototype design functions, 440
complete classical IIR filter design,
440–442
order estimation functions, 439
analogue filter specifications, 422–425
analogue frequency transformation, 437–438
Bessel filters, 434–437
Butterworth filter approximation, 425–428
Chebyshev filters, 428–433
inverse Chebyshev filter, 431–433
Type I Chebyshev approximation,
428–431
comparison between analogue filter types,
447–448
cut-off frequency, 443–447
elliptic filter approximation, 433–434
examples, 449–478
insights, 448–449
limitations, 447
problems, 479–485
Analogue frequency, 159, 437
Analogue prototype
functions, 440, 548
IIR digital filter and, 548, 559, 565, 573
Analogue transformation functions, 549
Anti-aircraft gun, 31
Approximation methods
Bessel filters, 434
Butterworth, 425
Chebyshev Type I, 428
Chebyshev Type II, 428, 431
elliptic, 433
equiripple behavior of, 428
monotonic behavior of, 428
Auto-correlation, 87
definition of, 89, 394
important application of, 395
Auxiliary equation, system, 220
B
Backward difference transformation, 512, 514
Band-limited signal, 492, 493
Bandpass filter(s), 439
approximation
Fourier series method, 633
frequency sampling, 633
bode plot, 470
fifth-order, 468
fourth-order, 567
ideal, 601
magnitude response for, 601
transfer functions of, 570
transforming lowpass filter to, 542
Bandstop cut-off frequencies, 574
Bandstop filter (BSF), 423, 439
center frequency, 475
ideal, 601
magnitude response, 471, 601
sixth-order, 574
transforming lowpass filter to, 542
Bessel filter(s), 434
design functions, 441
frequency response of, 436
phase response, 436
transfer function, 435
BIBO system, see Bounded-input bounded-output
system
Bilateral z-transform, 195
Bilinear transformation, 506, 512, 514
coefficients, 569652 Discrete Systems and Digital Signal Processing with MATLAB
filter design using, 545
impulse invariance transformation vs., 553
magnitude response, 554, 560
pole obtained using, 547
Binary code, 488, 490
Blackman window, 603, 616
Block diagram, 80, 81, 83, 298
difference equation and, 81, 82
drawing of from state and output
equation, 342
representation, 112, 113, 115, 210,
334, 335, 337
linear discrete systems, 78
MATLAB scripts, 359
states matrices, 353
state-space representation, 336
Block filtering, 393, 413
Bounded-input bounded-output (BIBO)
system, 63
BPF, see Passband filter
Broadcast signals, 329
BSF, see Bandstop filter
Butterworth filter(s), 423, 460
approximation, 425
characteristics, 453
cut-off frequency, 440, 553
design of, 577, 582
magnitude response of, 454, 463, 551
monotonicity in passband, 446
nonlinear phase, 613
pole zero plot of, 465
C
Capacitor, charging and discharging of, 1
CAT scan operation, 487
Causal filters, difference equation, 594
Causal systems, definition of, 61
Center frequency, bandstop filter, 475
Characteristic equation, 76, 102, 104, 110
characteristic root of, 123
coefficients, 77
difference equation representation, 330
roots of, 75
Chebyshev filter(s), 428, 462
cut-off frequency for, 430
design functions, 441
inverse, 431
magnitude response of, 432, 463, 464
nonlinear phase, 613
pole locations, 433
pole zero plot of, 466
transfer function, 429, 431
Chebyshev Type II approximation, MATLAB
function, 469
Circle of unity magnitude radius, 200
Circular convolution, 372, 380, 383
definition of, 382
equation, 385
linear convolution and, 396
Comb filter(s)
design of, 607
transfer function of, 624
Command line prompt, 450
Communication channel interference, 329
Complex number(s)
addition, 145
complex conjugate terms and, 230
definition of, 145
division, 146
multiplication, 145
polar to rectangular, 146
rectangular to polar, 146
review of, 143
subtraction, 145
z-transform, 200
Conditional statements, building of using
analogue circuits, 488
Constant coefficients realization, 595
Continuous filter, cut-off frequency of, 509
Continuous Fourier transform, 528
approximated, 376, 385
discrete Fourier transform and, 375
Continuous frequency, 365
Continuous radian frequency, 10
Continuous signal(s), 1, 2, 489
analogue frequency of, 159
binary code representation, 488
discretized, 24
Fourier series approximation, 387
frequency domain, 496
MATLAB simulated, 528
process of discretizing, 488
radar station, 487
sampling and recovery of, 496
Continuous system
differential equation, 522
impulse response, 527
input-output relationship, 507
oscillatory plot of, 523
partial fraction expansion, 525
plots, 519
state-space, 524
transfer functions, 527
Continuous value, 503
Continuous wave, example of, 1
Control systems, 488
ConvergenceIndex 653
conditions, 161
z-transform, 200
Convolution, 64
circular, 372, 380, 383
definition of, 382
equation, 385
linear convolution and, 396
equation, 66, 67, 104, 210
frequency response, 600
linear, 381, 412
equation, 382
output using, 413
property, 224
Fourier transform, 164
z-transform, 254
real-time, 225
result, 96, 381
sum(s), 66, 68, 173
equation, 593
evaluation of output using, 335
z-transform, 210
Correlation, 87
radar signal, 394
signals, discrete Fourier transform, 368
Cross-correlation, 87, 407, 408, 409
equation, 88, 394
important application of, 132
Cross multiplication, 348
Cubic-spline interpolation, 533
Current, continuous signal for, 10
Cut-off frequency(ies), 498, 543
analogue filter, 546
bandstop, 574
Butterworth filter, 440, 553
calculation of, 443
continuous filter, 509
digital filter, 546
first-order circuits, 448
highpass filter, 441, 445, 559, 620
lowpass filter, 445, 640
normalized, 557
use of MATLAB to estimate, 451
D
D/A, see Digital-to-analogue conversion
Data values, distorted, 329
Decimation in time, 379
Defining equation, discrete Fourier transform, 367
Delay elements, 79, 212
Delta signal, 331
Dense spectrum, 412
Derivative operation, discretizing of, 500
DFT, see Discrete Fourier transform
Difference equation, 174
block diagram and, 81, 82
causal, 232, 594
change of coefficients in, 614
delta signal as input, 331
digital FIR filter, 610
equivalent, 502
first-order, 155
general, 106, 123, 155
homogeneous, 69, 70, 101
impulse response, 85
inverse z-transform and, 608
linear, 330
model plots, 355
nonhomogeneous, 73, 86
nonrecursive filters, 594
Nth order, 236
physical systems, 68
representation, 126, 330, 336, 338, 341, 358
cross multiplication, 348
impulse response, 351
solving of using z-transform, 214
state-space representation, 333
step response, 350
systems represented as, 82, 112
unity coefficient, 348
z-transformed, 241, 332
Differentiation property, 223
Differentiator
design of using MATLAB, 637
FIR digital, 612
system, input to, 605
transfer function, 606
Digital-to-analogue (D/A) conversion, 488
Digital filter, 393
analogue to, 549
IIR to FIR, 610
transfer function, 509
Digital frequency, 160, 365, 496
maximum, 630
normalized, 634
Digital passband, normalized, 571
Digital processor, 487
Digital shifter, design of, 609
Discrete Fourier transform (DFT), 366, 507,
508, 604, 628
approximation using, 400
block filtering with, 412
continuous system, 506
correlation signals for, 368
equation, 366, 377
exact approximation using, 401
frequency index for, 366
inverse, 378
properties of, 372654 Discrete Systems and Digital Signal Processing with MATLAB
with zero padding, 414
Discrete Fourier transform and discrete systems,
365–419
applications of DFT, 380–395
approximation to coefficients of Fourier
series, 387–391
approximation to continuous Fourier
transform, 385–387
block filtering, 393–394
circular convolution, 380–384
correlation, 394–395
linear convolution, 384–385
total energy in signal, 391–393
discrete Fourier transform and finite-duration
discrete signals, 366–367
exercises, 396–415
fast Fourier transform, 378–380
insights, 395–396
circular convolution and linear
convolution, 396
DFT points are samples of Fourier
transform of x(n), 395
DFT same as fft, 395
frequency contents of x(t) in DFT,
395–396
frequency leakage and DFT, 396
|X(w)|, 396
numerical computation of DFT, 377–378
problems, 415–419
properties of discrete Fourier transform,
367–373
defining equation, 367–368
DFT linearity, 370–371
DFT symmetry, 368–370
magnitude of DFT, 371
meaning of DFT, 372–373
relation of DFT with Fourier transform of
discrete signals, 373–377
DFT and continuous Fourier transform of
x(t), 375–377
DFT and Fourier transform of x(n),
373–374
DFT and z-transform of x(n), 374–375
Discrete linear systems, ways of representing,
330–333
block diagram representation, 334
difference equation representation, 330–331
impulse response representation, 331–332
state-space representation, 333–334
z-transform representation, 332–333
Discrete matrix, 506
Discrete periodic signals, Fourier series of, 147
Discrete signal(s), 489
average power in, 28
basic operations, 25–28
addition and subtraction, 25
combined operations, 26–28
modulation, 25
scalar multiplication, 25
bounded, 30
complex periodic, 10
conversion of continuous signal to, 487
decaying exponential, 8
decaying sinusoidal, 12
digitized, 3
even, 21
example of, 2
finite-duration, 366
Fourier transform of, 159, 373, 395
growing exponential, 8, 10
impulse, 6
odd, 21
Parseval’s relation for, 167
periodic, 3, 4
plot of, 143
ramp, 6, 7
real exponential, 7
representation of, 16, 21
shifted, 15
sinusoidal, 7, 9
time constant, 23, 24
time invariance and, 58
time-scaled, 19
total energy in, 28
unbounded, 30
unit step, 4, 5, 198
z-transform of, 199
Discrete system, 55–141
block diagram representation of linear
discrete systems, 78–81
delay element, 79
multiplier, 79–81
summing/subtracting junction, 79
causal systems, 61–62
convolution, 64–68
correlation, 87–89
autocorrelation, 89
cross-correlation, 87–89
definition of system, 55
difference equations of physical systems, 68
difference equations representing, 350–351
examples, 91–134
frequency response of, 154
from block diagram to difference equation,
81–82
from difference equation to block diagram,
82–85
homogeneous difference equation and
solution, 69–73
case when roots are all different, 71Index 655
case when two roots are complex, 72–73
case when two roots are real and equal, 72
impulse response, 85–87, 151
input and output, 55–56
insights, 90–91
eigenvalues, 90–91
stability and eigenvalues, 91
inverse of system, 62–63
linear discrete systems, 56–58
nonhomogeneous difference equations and
solution, 73–75
with periodic inputs, 150
problems, 134–141
stability of linear discrete systems, 75–78
stability depending on values of poles,
75–76
stability from jury test, 76–78
stable system, 63–64
systems with memory, 60–61
time invariance and discrete signals, 58–60
Discrete time
domain, 254
shifting property, Fourier transform, 163
Discrete value, 503
Discretization
formula, 502
interval, 506
methods of, 521
state-space representation, 504
Discrimination parameter, 425, 434
Dynamics matrix, 291
E
Edge frequencies, 635, 641
Eigenvalues, 291, 301
Electrical switch, 31
Electromagnetic signal, 1
Element-by-element multiplication, 28, 41
Elevator system, 55
Elliptic filter(s), 462
approximation, 433
design, 441, 458, 582
magnitude-squared response, 434, 459
nonlinear phase, 613
pole zero plot of, 467
ripples, 562
roll-off characteristics, 448
transfer function of, 433
Ending index, 98
Energy
calculations, 167
discrete signal, 28
finite, 29, 30
signals, cross-correlation equations for, 88
spectrum density, signal, 395
total, 391, 392, 393
use of MATLAB to find total, 34
use of Parseval’s theorem to find, 168
Equation(s)
algebraic, 91, 115
analogue filter, 422
auxiliary, 91, 220
characteristic, 69, 76, 102, 104, 110
characteristic root of, 123
coefficients in, 77
difference equation representation, 330
roots of, 75
system, 220
Chebyshev filter, 431
circular convolution, 385
convolution, 66, 67, 104, 210, 593
cross-correlation, 88, 394
defining, discrete Fourier transform, 367
difference, 174
block diagram and, 81, 82
causal filters, 594
change of coefficients in, 614
delta signal as input, 331
digital FIR filter, 610
equivalent, 502
first-order, 155
general, 106, 123, 155
homogeneous, 69, 70, 101
impulse response, 85
inverse z-transform and, 608
linear, 330
model, 355
nonhomogeneous, 73, 86
nonrecursive filters, 594
Nth order, 236
physical systems, 68
representation, 126, 330, 336, 338, 341,
345, 348, 358
solving of using z-transform, 214
state-space representation, 333
step response, 350
systems represented as, 82, 112
unity coefficient, 348
z-transformed, 241, 332
discrete Fourier transform, 366, 377
Fourier transform, 164
inverse transform on, 233
linear convolution, 382
matrix state, transfer function calculated
from, 340
output, 300, 306, 313
in matrix form, 319
z-domain, 314656 Discrete Systems and Digital Signal Processing with MATLAB
simultaneous algebraic, 111
state, 298, 300, 306, 313
discrete state-space approximation, 505
in matrix form, 319, 337
obtaining of by inspection, 339
-space matrix, 299
summation, 604
z-transform, 196, 198, 210
Equiripple linear phase, FIR filter design using,
612
Exponential signal, MATLAB script to simulate, 37
F
Fast Fourier transform (FFT), 149,378
development of, 379
implementation of in MATLAB, 380
FFT, see Fast Fourier transform
Filter
active, 422
analogue
prototype functions, 440
specifications, 425
zero-pole plot, 423
average, frequency response, 593
bandpass, 439
bode plot, 470
fifth-order, 468
fourth-order, 567
ideal, 601
magnitude response for, 601
transfer functions of, 570
bandstop, 423, 439
center frequency, 475
ideal, 601
magnitude response, 471, 601
sixth-order, 574
transformed, 471, 542
Bessel, 434
design functions, 441
frequency response of, 436
group delay, 435, 436
phase response, 436
transfer function, 435
Butterworth, 423, 460
approximation, 425
characteristics, 453
cut-off frequency, 440, 553
design of, 577, 582
magnitude response of, 454, 463, 551
monotonicity in passband, 446
nonlinear phase, 613
pole zero plot of, 465
causal, difference equation, 594
Chebyshev, 428, 462
cut-off frequency for, 430
design functions, 441
equation, 431
inverse, 431
magnitude response of, 432, 463, 464
nonlinear phase, 613
pole locations, 433
pole zero plot of, 466
circuit elements, 422
coefficients, 619
comb
digital, 607
transfer function of, 624
continuous, cut-off frequency of, 509
design
direct, 549, 550, 555, 575, 576
functions, 441
IIR, 440, 610
indirect method, 575
limitations in, 447
transfer function, 509
transformation from analogue to, 549
use of windows in, 602
elements used in building of, 422
elliptic, 462
approximation, 433
design, 441, 458, 582
magnitude response of, 434, 459, 464
nonlinear phase, 613
pole zero plot of, 467
ripples, 562
roll-off characteristics, 448
transfer function of, 433
function, initial conditions for, 236
group delay of, 423
high gain, 448
highpass, 423
analogue, cut-off frequency, 441
cut-off frequency, 559, 620
ideal, 601
magnitude response for, 583, 601
Hilbert transform, 609, 612, 639
impulse response, 593
linear phase, 422
low gain, 448
lowpass, 421, 423, 424
analogue Bessel, 435
cutoff frequency of, 445, 640
ideal, 498, 599
impulse response, 499
limitations in design of, 446
maximum gain of unity, 472
peak passband ripple, 449
specifications, 424Index 657
transforming, 542
use of MATLAB to design, 634
magnitude response, 431, 469
noncausal, 608
nonrecursive, difference equation, 594
order
estimated, 433
required, 430
output, delay in, 597
parameters specifying, 430
passive, 422
phase shift of, 423
problems of designing, 422
prototype, 438
sixth-order, 554
specifications, 439, 446, 472
types, comparison between analogue, 446
Final value theorem, 219
Finite duration signals, 87, 96
Finite impulse response (FIR) digital filters,
591–648
definition, 591
design based on Fourier series, 598–609
design of comb FIR filters, 607–608
design of digital FIR differentiator,
605–607
design of digital shifter, 609
ideal lowpass FIR filter design, 599–601
other ideal digital FIR filters, 601–602
window giving optimal h(n), 604–605
windows used in design of digital FIR
filter, 602–604
examples, 614–644
FIR digital design using MATLAB, 611–613
design using equiripple linear phase, 612
design using least-squared error, 612
design using windows, 611–612
obtaining frequency response, 612–613
FIR filter design, 594–598
linear phase of FIR filters, 597–598
stability of FIR filters, 596–597
frequency sampling and FIR filter design,
610–611
from IIR to FIR digital filters, 610
insights, 613–614
comparison with IIR filters, 613
different method used in FIR filter design,
613–614
motivating example, 591–594
problems, 644–648
FIR digital filters, see Finite impulse response
digital filters
First-order circuits, cut-off frequency for, 448
First-order difference equation, 155
First-order systems, output for, 90
Forward difference transformation, 512, 514
Fourier, Joseph, 143
Fourier series
approximation, 387
coefficients, 148, 387, 389
approximation to, 391
finding of using MATLAB, 390
filter design based on, 598
magnitude coefficients, 149
Fourier series and Fourier transform of discrete
signals, 143–194
convergence conditions, 161
discrete system with periodic inputs, 150–154
examples, 173–188
Fourier series of discrete periodic signals,
147–149
Fourier transform of discrete signals, 159–161
frequency response of discrete systems,
154–159
periodicity property, 157
symmetry property, 157–159
insights, 172–173
ease in analysis and design, 172–173
sinusoidal analysis, 173
numerical evaluation of Fourier transform of
discrete signals, 168–172
Parseval’s relation and energy calculations,
167–168
problems, 189–194
properties of Fourier transform of discrete
signals, 162–167
convolution property, 164–167
discrete-time shifting property, 163
frequency shifting property, 163
linearity property, 162
periodicity property, 162
reflection property, 163–164
review of complex numbers, 143–147
addition, 145
definition, 145
division, 146
from polar to rectangular, 146–147
from rectangular to polar, 146
multiplication, 145–146
subtraction, 145
Fourier transform
approximation to magnitude of, 397
calculation of, 533
continuous, 375, 376, 528
discrete, 159, 366, 373
ease in analysis and design, 172
fast, 378
development of, 379
implementation of in MATLAB, 380
MATLAB simulated, 529658 Discrete Systems and Digital Signal Processing with MATLAB
pairs, 166
properties, 166
sampled signal, 507
Frequency(ies)
analogue, 159, 507
axis, 400
center, bandstop filter, 475
components
signal, 149, 592
use of DFT to find, 390
continuous, 365
cut-off, 498, 543
analogue filter, 546
bandstop, 574
Butterworth filter, 440, 553
calculation of, 443
continuous filter, 509
digital filter, 546
highpass filter, 445, 559, 620
lowpass filter, 445, 640
normalized, 557
digital, 160, 365, 496, 511
domain(s)
ambiguity in, 492
continuous signal, 496
convolution in, 164, 165
discrete signal in, 144
representation, 195, 497
sampling operation in, 490
signal in, 365
edge, 635, 641
index, 366, 372
input signal, 572
leakage, 396
noise at high, 621
output, 173
pairs, 638
passband edge, 433
passing, 158
points, 637
radian, 497, 498, 624
rejecting, 158
relationship between analogue and
digital, 542
resolution, 395, 407, 415
response, 170, 171, 172
Bessel filter, 436
comparison of, 595
computations, 451, 452
convolution between, 600
defined, 155
discrete systems, 154
equation used to find, 155
function, 157, 158
general, 156
how to obtain, 612
magnitude of, 178, 186
MATLAB, 175, 594, 623
periodicity property, 157
properties of, 157
simplified, 608
symmetry property, 157
sampling, 372, 491, 497, 632
bandpass filter approximation using, 633
FIR filter design and, 610–611
shifting property, Fourier transform, 163
sinusoidal wave of, 492
spacing, 170
spectrum, input, 403
transformation(s), 541, 543
analogue, 437
functions, 442
lowpass filter, 542
value, 154
Fundamental period, 13, 14, 15
G
Gamma function, 435
Geometric series sum, 67, 105, 148, 160, 200
Grid interval, 529
Group delay, Bessel filter, 435, 436
H
Hamming window, 396, 603, 616, 630
Hanning window, 396, 603
approximation using, 402
magnitude frequency response plot of, 616
Highpass filter (HPF), 423
cut-off frequency, 559, 620
ideal, 601
magnitude response for, 583, 601
Hilbert transform filter, 609, 612, 639
Homogeneous solution, 68
complex roots in, 72
distinct roots in, 71
equal roots in, 72
HPF, see Highpass filter
I
Identity matrix, 289
IIR filter design, see Infinite impulse response
filter design
Impulse(s)Index 659
discrete signal, 6
representing discrete signal using, 16
shifted, 16
sum of, 18
Impulse invariance
filter design using, 542
transformation, 512, 513, 515
bilinear transformation vs., 553
IIR digital filters based on, 545
magnitude response, 554, 560
numerator and denominator
coefficients, 555
Impulse response, 85, 100, 108, 114, 241,
247, 593
calculation of, 119
causal, 216
continuous system, 527
convolution between step input signal and,
341, 345
defined, 174, 405
difference equation representation, 351
discrete system, 124, 177
final, 123
linear system, 403
lowpass filter, 499
model, 350, 355
plotting of actual, 134
representation, 112, 331, 338, 341, 348, 358
samples, FIR filter, 610
solution for, 105
state-space representation, 355–356
use of MATLAB to find, 105, 113, 116, 119,
120, 351
use of z-transform to find, 215
Impulse signal, 31, 35, 111
discrete signals using, 18
Fourier transform of, 161
MATLAB-generated, 37
response to, 65
shifted, 65
Infinite impulse response (IIR) filter design,
541–589
design process, 542–548
bilinear transform method, 545–548
impulse invariance method, 542–545
examples, 552–584
IIR filter design using MATLAB, 548–550
direct design, 549–550
from analogue prototype to IIR digital
filter, 548–549
insights, 550–552
choice of sampling interval Ts, 552
difficulty in designing IIR digital filters in
z-domain, 550–552
using impulse invariance method, 552
problems, 584–589
Infinite series, 161
Initial condition vector, 237, 239, 311
Initial index, 41
Initial-value theorem, 219, 220, 248
Input
divided, 178
frequency, 154
frequency response at, 186
magnitude of, 182
spectrum of, 403
magnitude of, 154, 173
matrix, 504
noise, auto-correlation of, 132
–output relation, 56, 60, 68, 508
particular solutions for selected, 75
phase shift of, 178
signal(s), 55, 57
change of magnitude of, 393
frequencies, 172, 572
step, 240
vector, 291, 504
Inspection
obtaining state equations by, 339
state-space representation by, 317
Interchange of summation property, 164
Inverse Chebyshev filter, 431
Inverse transform, 232, 298
calculation of, 404
use of MATLAB to find, 315
Inverse z-transform, 203, 335
difference equation and, 608
long division, 206
partial fraction expansion, 203
J
Jacobian elliptic function, 433
Jury test, 76, 78
K
Kaiser window, 603, 604
L
Laplace domain, 472, 506, 510
Laplace transform, 423, 519
inverse, 524
state vector, 520
transition matrix, 520660 Discrete Systems and Digital Signal Processing with MATLAB
Lindex, 42, 43
Linear convolution, 381, 412
equation, 382
output using, 413
Linear discrete systems, 56
block diagram representation of, 78
stability of, 75
Linearity property
discrete Fourier transform, 371
Fourier transform, 162
z-transform, 207, 226
Linear system(s)
impulse response, 403
representation, 56
types of, 365
Linear time-invariant (LTI) model, 438
Linear time-invariant system(s), 90
analysis and design of, 329
multiple-inputs multiple-outputs in, 346
output of, 173
transfer function of, 220, 329
Linear time-variant system, 153
Load resistor, 31
Long division, 206, 216, 227, 610, 627
Lowpass filter (LPF), 421, 423, 424
analogue Bessel, 435
cut-off frequency for, 445, 640
ideal, 498, 599
impulse response, 499
limitations in design of, 446
maximum gain of unity, 472
peak passband ripple, 449
prototype, 438
specifications, 424
transforming, 542
use of MATLAB to design, 634
LPF, see Lowpass filter
LTI model, see Linear time-invariant model
M
Magnitude
plot, 158, 159, 160
requirements, design by satisfying, 541
response, 158, 427, 429
Butterworth filter, 454, 551
characteristic, 434
Chebyshev filter, 432
elliptic filter, 434, 459
highpass filter, 583
MATLAB
analogue filter design using, 438
approximation to continuous Fourier
transform, 385
calculation of average power using, 33
complex poles, 517
control toolbox, 469
cut-off frequencies, 451
data entered as row vectors in, 171
default scaling, 612
design of ideal differentiator using, 637
DFT equation implemented on digital
computer using, 377
digital frequencies using, 569, 576
direct design method, 557, 563
filter function, 124
finding of Fourier series coefficients
using, 390
FIR digital design using, 611
frequency response, 623
function, 225, 227
Chebyshev Type II approximation, 469
direct design, 550
general form of, 42
roots, 76
use of to find step response, 347, 348
zeros, 35
-generated exponential decaying signal, 38
-generated impulse signal, 37
-generated sinusoidal signal, 39
-generated step signal, 35, 36
IIR filter design using, 548
implementation of fast Fourier transform
in, 380
impulse invariance transformation, 544
impulse response, 105, 113, 116, 119, 120
inverse transform, 315
limitations in filter design using, 447
lowpass filter design, 634
magnitude of frequency response, 175
partial fraction expansion using, 303
phase response displayed by, 450
random signal generation, 409
reconstruction implementations, 533
recursion, 303, 308
script(s), 98, 180, 182
block diagram representation, 359
discrete input signal, 580
exponential signal, 37
frequency responses, 594
Hanning windows, 402
IIR digital bandpass filters, 571–572
IIR digital bandstop filters, 577
inverse discrete Fourier transform, 378
response plotting due to step input, 240
state-space matrix system, 347
transfer function representation, 357
shifting function, 49
signal processing toolbox, 438, 439Index 661
simulated continuous signal, 528
simulated Fourier transform, 529
spectral energy estimate, 410
statement, 561
step response, 311, 320
system identification toolbox, 438
system stability, 311
time-shifting property of Fourier
transform, 179
total energy, 392, 393
transfer function, 301
transformation, 565
transition matrix entries, 304
windows implemented in, 396
Matrix state equations, transfer function
calculated from, 340
Mechanical systems, modeling of, 329
Medical imaging, 487
Memory, systems with, 60, 61
Mixed system, 488
Model(s)
difference equation, 355, 361
impulse response, 350, 355, 361
linear time-invariant, 438
state-space, 346, 355, 361
transfer function, 350
Modeling and representation of discrete linear
systems, 329–364
exercises, 346–361
poles considering different outputs within
same system, 346
problems, 361–364
ways of representing discrete linear systems,
330–346
block diagram to other representations,
343–346
difference equation to other
representations, 330–334
impulse response to other representations,
335–337
state-space to other representations,
340–342
transfer function to other representations,
337–340
Modulation, 25
Multiplication, term-by-term, 384
Multiplier element, 79
N
Noise components, preventing amplification
of, 621
Noncausal filter, 608
Noncausal system, 62
Nonhomogeneous difference equations, 73
Nonlinear system, 57, 58
Nonrecursive system, 591, 614
Nyquist criteria, 400
Nyquist rate, 392, 401, 403, 529, 533, 607
O
Order estimation functions, 439
Orthogonality condition, 148
Output
continuous time system, 506
discrete Fourier transform of, 506
equation, 300, 306, 313
in matrix form, 319
z-domain, 314
frequency, 173
initial condition for, 501
matrix, 504
one-dimensional, 303
values, use of z-transform method to find, 303
vector, 334
voltage, at low frequencies, 474
P
Parseval’s Theorem, 167, 605
Partial fraction expansion, 231, 242, 253, 303
analytical solutions, 321
continuous system, 524
impulse response representation, 338
inverse z-transform, 203
transfer function representation, 228, 318, 357
z-transform representation, 332
Particular solution, 68
Passband
bandwidth, 446
edge frequency, 433
filter (BPF), 423
ripple, 425, 429
Passive circuit elements, 422, 475
Passive filters, 422
Periodicity property
Fourier transform, 162
frequency response, 157
Periodic signal
approximated, 148
average power in, 389
Phase
characteristic, nonlinear, 434
shift, 153
Physical systems, difference equations of, 68
Polar form, complex number written in, 146, 200662 Discrete Systems and Digital Signal Processing with MATLAB
Pole(s)
locations, Chebyshev filter, 433
phase of, 247
system, 220, 252
transient shape and, 346
Power
average, periodic signal, 389
calculation of using MATLAB, 33
signals, 30, 89
Proportional-integral-derivative control, design
of, 605
Prototype filters, 438
Pulse
definition, 405
signal, 16
Q
Quality factor, 446, 448
R
Radar
antenna, 31
signal correlation, 394
station, continuous signal, 487
Radian frequency, 153, 497, 498, 624
Ramp signal, 6, 7, 31
Random signal
dc component, 577
use of MATLAB to generate, 409
Rational number, 14, 15
Real exponential discrete signal, 7
Real number, 14, 23
Real-time domain, ambiguity in, 490
Record length, 407, 415
Rectangular window, 603, 630
filter design using, 618
magnitude frequency response plot of, 616
Recursive system, 614
Reflected signal, 18, 20
Reflection property, Fourier transform, 163
Region of convergence (ROC), 201, 217, 218, 253
Rindex, 42, 43
RLC series circuit, 448
ROC, see Region of convergence
Roots
magnitude of, 314
symmetry in, 552
S
Sample-and-hold process, 489
Sampling
filtering before, 494
frequency, 372, 491, 497, 531, 565,
610–611, 632
ideal, 498
interval, 10, 55, 149, 196, 388, 528, 577
changing of, 521–522
choice of, 515, 552
minimum, 572
operation, 490
period, 2, 147–148, 542, 557, 606
rate, Nyquist, 392
theorem, 493, 495
time, transition matrix at, 506
uniform, 501
Scaling factor, 25, 530
s-domain transfer function, 547
Second-order system, 221
initial conditions, 502
output for, 90
Selectivity parameter, 434
Series connection, 317
Shifting property, z-transform, 207
Shift operation, importance of, 15–16
Signal(s)
average value of, 389
band-limited, 492, 493
broadcast, 329
continuous, 1, 2, 489
analogue frequency of, 159
binary code representation, 488
discretized, 24
exponential, 23, 24
Fourier series approximation, 387
frequency domain, 496
MATLAB simulated, 528
process of discretizing, 488
processing of, 2
radar station, 487
sampled, 3
sampling and recovery of, 496
correlation, discrete Fourier transform, 368
delta, 331
discrete, 489
average power in, 28
basic operations, 25–28
bounded, 30
complex periodic, 10
conversion of continuous signal to, 487
decaying exponential, 8
decaying sinusoidal, 12
digitized, 3Index 663
even, 21
example of, 2
finite-duration, 366
Fourier transform and, 159, 373, 395
growing exponential, 8, 10
impulse, 6
odd, 21
Parseval’s relation for, 167
periodic, Fourier series of, 147
plot of, 143
ramp, 6, 7
real exponential, 7
representation of, 16, 21
shifted, 15
sinusoidal, 7, 9, 144
time constant, 23, 24
time invariance and, 58
time-scaled, 19
total energy in, 28
unbounded, 30
unit step, 4, 5, 198
z-transform of, 199
electromagnetic, 1
energy, 167
cross-correlation equations for, 88
spectrum density of, 395
exponential
decaying, MATLAB-generated, 38
MATLAB script to simulate, 37
finite duration, 87, 96, 374
Fourier series coefficients, 389
Fourier transform of, 365
frequency components in, 149, 592
frequency domain, 365
impulse, 31, 35, 111
discrete signals using, 18
Fourier transform of, 161
MATLAB-generated, 37
response to, 65
shifted, 65
input, 55, 57
change of magnitude of, 393
frequencies, 172, 572
MATLAB script, 580
noise associated with, 494
noncausal, 217
periodic, 39, 147
approximated, 148
average power in, 389
power, 30, 89
processing of in real-life situations, 487
processing toolbox, MATLAB, 438, 439
pulse, 16
radar, correlation, 394
ramp, 31
random
dc component, 577
use of MATLAB to generate, 409
reconstructed, 531
reflected, 18, 20, 163
representation of in real life, 31
sampled, 405, 507
sampling interval for, 196
scaled, 28
shifted step, 16
sinusoidal, 14, 31, 173
decaying, 12
exponentially modulated, 10
growing, 12
irregularly decaying modulated, 12
irregularly growing modulated, 12
MATLAB-generated, 39
MATLAB script to simulate, 37
model, 329
step, 31, 161
input, convolution between impulse
response and, 341, 345
MATLAB-generated, 35, 36
total energy in, 391, 392, 393
z-transform of, 195, 221–222
Signal representation, 1–53
amplitude scaling, 20–21
basic operations on discrete signals, 25–28
addition and subtraction, 25
combined operations, 26–28
modulation, 25
scalar multiplication, 25
bounded and unbounded discrete signals, 30
complex periodic discrete signal, 11–15
discrete signal time constant, 23–25
energy and power discrete signals, 28–30
even and off discrete signal, 21–23
examples, 32–50
exponentially modulated sinusoidal signal, 11
impulse discrete signal, 6
periodic and nonperiodic discrete signals, 3–4
problems, 50–53
ramp discrete signal, 6
real exponential discrete signal, 7
reason for discretizing continuous systems,
2–3
reflection operation, 18
representing discrete signal using impulses,
16–18
shifting operation, 15–16
signals in real world, 30–32
impulse signal, 31
other signals, 32
ramp signal, 31–32
sinusoidal signal, 31664 Discrete Systems and Digital Signal Processing with MATLAB
step signal, 31
sinusoidal discrete signal, 7–11
time scaling, 19–20
unit step discrete signal, 4–5
Sindex, 42
Sinusoidal discrete signal, 7, 9
Sinusoidal input signal, 173
Sinusoidal response, 108
Sinusoidal signal, 14, 31
decaying, 12
exponentially modulated, 10
growing, 12
irregularly decaying modulated, 12
irregularly growing modulated, 12
MATLAB-generated, 39
MATLAB script to simulate, 37
model, 329
Sixth-order filter, 554
Spectral energy estimate, calculation of, 410
Square-magnitude response expression, 431
Stable system, 63
Starting index, 41, 98
State equation, 298, 300, 306, 313
discrete state-space approximation, 505
in matrix form, 319
obtaining of by inspection, 339
State matrix equations, 337
State-space and discrete systems, 265–328
examples, 292–322
general representation of systems in statespace, 270–283
block diagram to state-space, 273–275
nonrecursive systems, 272–273
recursive systems, 270–272
transfer function H(z) to state-space,
276–283
general solution of state equation in real-time,
284–285
poles and stability, 291–292
problems, 322–328
properties of A″ and evaluation, 285–289
review on matrix algebra, 266–270
adding two matrices, 267
definition, general terms, and
notations, 266
determinant of two-by-two matrix, 268
diagonal form of matrix, 269
eigenvalues of matrix, 269
eigenvectors of matrix, 269–270
identity matrix, 266–267
inverse of matrix, 268
matrix multiplication, 269
multiplying matrix by constant, 267
subtracting two matrices, 267
transpose of matrix, 268
solution of state-space equations in z-domain,
283–284
transformations for state-space
representations, 289–291
State-space matrix system, MATLAB script, 347
State-space model, 346, 355
State-space representation, 310, 343
discretization of, 504
impulse response, 355–356
transformations for, 289
State-space system matrices, 352
State values, use of z-transform method
to find, 303
State vector, 289, 306, 504
Laplace transform, 520
solution, 285, 504, 505
terms of, 302
Steady-state response, 150, 153, 154, 185
Step input, 240
response plot due to, 240
signal, convolution between impulse response
and, 341, 345
Step invariance transformation, 512, 513
Step response, 108, 241, 242, 312
difference equation, 350
impulse response model, 350
state-space model, 350
transfer function model, 350
use of MATLAB to find, 311, 320, 347, 348
Step signal, 31, 35, 36, 161
Stopband
attenuation, minimum, 425
ripple, maximum, 424
specifications, 430
Summation equation, 604
Summing junction, 79, 81–82
Superposition, 178
principle, 151
response due to inputs using, 316
Symmetry property, frequency response, 157
System
auxiliary equation of, 220
bounded-input bounded-output, 63
causal, 61, 608
characteristic equation, 69, 70, 123, 220
continuous, 506
differential equation, 522
impulse response, 527
input-output relationship, 507
oscillatory plot of, 523
partial fraction expansion, 525
plots, 519
state-space, 524
transfer functions, 527
control, 488Index 665
definition of, 55
discrete
difference equations representing,
350–351
Fourier transform, input to, 371
with periodic inputs, 150
eigenvalues, 91, 221, 290, 301
first-order, 80, 90
frequency response for, 156, 184
function, numerator and denominator
coefficients of, 438
highpass, 478
identification toolbox, MATLAB, 438
impulse response for, 67, 85, 155
initial condition vector, 311
inverse of, 62
linear, 75, 78, 173, 365
linear time-variant, 153
analysis and design of, 329
multiple-inputs multiple-outputs in, 346
output of, 173
system, 90, 329
transfer function of, 220
lowpass, 478
magnitude of, 154
matrix, 301, 333, 352
mechanical modeling of, 329
with memory, 60, 61
mixed, 488
noncausal, 62, 218, 235
nonlinear, 57, 58
nonrecursive, 591, 614
output, 130, 150, 167, 333
physical, difference equations of, 68
poles of, 220, 346
recursive, 614
second-order, 221
initial conditions, 502
output for, 90
stability, 90, 91, 221, 292
poles and, 252
use of MATLAB to check, 311
stable, 63, 75, 101, 236, 354
state-space, see State-space and discrete
systems
steady-state output in, 153, 595
steady-state response of, 220
third-order, output for, 90
time invariant, 59, 65, 66
time variant, 60
transfer function, 212, 230, 290
unknown parameters of, 329
unstable, 78, 235, 292, 314
zeroes of, 221
zero input, 286
T
Thermal interferences, 3
Third-order systems, output for, 90
Third-order transfer function, 210
Time
ambiguity in, 492, 494
constant, discrete, 23, 24, 25
invariance, discrete signals and, 58
invariant system, 59, 65, 66
-scaled discrete signal, 19
scaling, 19, 41
variant system, 60
Time domain, 251
characteristics, 448
no ambiguity in, 495
Total energy
signal, 391, 392, 393
use of MATLAB to calculate, 392, 393
Transfer function(s), 218, 298
additional zeroes introduced for, 516
analogue filter design, 421
Bessel filter, 435
calculation of, 310, 340
Chebyshev filter, 429
comb filter, 624
continuous, 506, 515, 517
denominator coefficient of, 329
differentiator, 606
digital filter, 509
discrete, 518, 519
elliptic filter, 433
IIR filter, 554, 561, 568, 570
magnitude, calculation of, 445
model, 350
numerator coefficient of, 329
partial fraction expansion and, 318
phase angle of, 422
representation, 332, 335, 344, 347
as block diagrams, 210
MATLAB script, 357
partial fraction expansion, 357
roots of denominator in, 332
s-domain, 547
third-order, 210
use of MATLAB to find, 301
z-domain, 216, 230
Transformation, see also specific types
functions, 442
matrix, 289, 310
methods, 512
backward difference, 512, 514
bilinear, 512, 514
forward difference, 512, 514
impulse invariance, 512, 513, 515666 Discrete Systems and Digital Signal Processing with MATLAB
step invariance, 512, 513
Transformations between continuous and discrete
representations, 487–539
bilinear transformation and relationship
between Laplace-domain and zdomain representations, 506–511
discretization of derivative operation,
500–503
discretization of state-space representation,
504–506
examples, 517–534
from binary code to continuous signal, 490
from continuous signal to binary code
representation, 488–490
insights, 515–516
choice of sampling interval Ts, 515
effect of choosing Ts on dynamics of
system, 515–516
introduction of additional zeroes for
transfer function H(z), 516
need for converting continuous signal to
discrete signal, 487–488
other transformation methods, 5515
backward difference method, 512
bilinear transformation, 512–515
forward difference method, 512
impulse invariance method, 512
step invariance method, 512
problems, 534–539
sampling operation, 490–500
ambiguity in frequency domain, 492–493
ambiguity in real-time domain, 490–492
filtering before sampling, 494–496
sampling and recovery of continuous
signal, 496–500
sampling theorem, 493
Transition matrix, 285
continuous system, 525
Laplace transform, 520
sampling time, 505
use of MATLAB to verify entries in, 304
Transmission matrix, 504
Trigonometric identities, 111
U
Unique set, 148
Unit circle, 248, 253, 301
Unit step discrete signal, 4, 5, 198
V
Variable coefficients realization, 596
Voltage
divider, 472
sources, AC, 31
value, 1
W
Warping, 545, 546
Waveform generation, 487
Window(s)
Blackman, 603, 616
default, 634
definition, 600
FIR filter design using, 611
Hamming, 396, 603, 616, 630
Hanning, 396, 603
magnitude frequency response plot of, 616
MATLAB script, 402
Kaiser, 603, 604
rectangular, 603, 630
filter design using, 618
magnitude frequency response plot of, 616
use of in filter design, 602
Z z
-domain
design of IIR filters in, 550
input in, 213
multiplication in, 225
output in, 213, 314
representations, bilinear transformation and,
506
state vector in, 333
Zero input, response to system with, 286
Zero-order hold method, 532
Zero padding, 414, 415
Zero-pole-gain form, 441, 446
z-transform and discrete systems, 195–263
bilateral z-transform, 195–197
convergence, 200–203, 216–218
exercises, 221–255
final value theorem, 219
initial-value theorem, 219–220
inverse z-transform, 203–207
long division, 206–207
partial fraction expansion, 203–206
poles and zeroes, 220–221
poles of system 220
stability of system, 221
zeroes of system, 221
problems, 255–263
properties of z-transform, 207–210Index 667
convolution, 210
linearity property, 207
multiplication by e-an, 209
shifting property, 207–209
representation of transfer functions as block
diagrams, 210–212
solving difference equation using z-transform,
214–216
unilateral z-transform, 197–200
x(n), h(n), y(n), and z-transform, 212–214

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