Discrete Systems and Digital Signal Processing with MATLAB, Second Edition

by ElAli; Taan S.

• ISBN13: 9781439828182
• ISBN10: 1439828180
• Edition: 2nd
• Format: Hardcover
• Copyright: 2011-12-05
• Publisher: CRC Press

Summary

Suitable for teaching a one-semester course, this textbook covers traditional topics and includes stand-alone chapters on sampling and transformations. It explains the subject matter with an easy-to-follow mathematical development and many solved examples and contains comprehensive chapters on IIR and FIR digital filter design, state-space, DFT and FFT, and block diagrams. The author presents examples solved with MATLAB, but readers will not need to be fluent in this powerful programming language, because they are presented in a self-explanatory way.

Table of Contents

Preface	p. xv
Acknowledgments	p. xvii
Author	p. xix
Signal Representation	p. 1
Introduction	p. 1
Why Do We Discretize Continuous Systems?	p. 3
Periodic and Nonperiodic Discrete Signals	p. 3
Unit Step Discrete Signal	p. 4
Impulse Discrete Signal	p. 6
Ramp Discrete Signal	p. 6
Real Exponential Discrete Signal	p. 7
Sinusoidal Discrete Signal	p. 7
Exponentially Modulated Sinusoidal Signal	p. 11
Complex Periodic Discrete Signal	p. 11
Shifting Operation	p. 15
Representing a Discrete Signal Using Impulses	p. 16
Reflection Operation	p. 20
Time Scaling	p. 20
Amplitude Scaling	p. 20
Even and Odd Discrete Signal	p. 21
Does a Discrete Signal Have a Time Constant?	p. 24
Basic Operations on Discrete Signals	p. 25
Modulation	p. 25
Addition and Subtraction	p. 26
Scalar Multiplication	p. 26
Combined Operations	p. 26
Energy and Power Discrete Signals	p. 28
Bounded and Unbounded Discrete Signals	p. 30
Some Insights: Signals in the Real World	p. 31
Step Signal	p. 31
Impulse Signal	p. 31
Sinusoidal Signal	p. 31
Ramp Signal	p. 32
Other Signals	p. 32
End of Chapter Examples	p. 32
End of Chapter Problems	p. 53
Discrete System	p. 57
Definition of a System	p. 57
Input and Output	p. 57
Linear Discrete Systems	p. 58
Time Invariance and Discrete Signals	p. 61
Systems with Memory	p. 62
Causal Systems	p. 63
Inverse of a System	p. 64
Stable System	p. 65
Convolution	p. 66
Difference Equations of Physical Systems	p. 69
Homogeneous Difference Equation and Its Solution	p. 70
Case When Roots Are All Distinct	p. 73
Case When Two Roots Are Real and Equal	p. 73
Case When Two Roots Are Complex	p. 74
Nonhomogeneous Difference Equations and Their Solutions	p. 75
How Do We Find the Particular Solution?	p. 77
Stability of Linear Discrete Systems: The Characteristic Equation	p. 77
Stability Depending on the Values of the Poles	p. 77
Stability from the Jury Test	p. 78
Block Diagram Representation of Linear Discrete Systems	p. 80
Delay Element	p. 80
Summing/Subtracting Junction	p. 81
Multiplier	p. 81
From the Block Diagram to the Difference Equation	p. 82
From the Difference Equation to the Block Diagram: A Formal Procedure	p. 83
Impulse Response	p. 86
Correlation	p. 88
Cross-Correlation	p. 88
Auto-Correlation	p. 90
Some Insights	p. 91
How Can We Find These Eigenvalues?	p. 91
Stability and Eigenvalues	p. 92
End of Chapter Examples	p. 93
End of Chapter Problems	p. 135
Fourier Series and the Fourier Transform of Discrete Signals	p. 141
Introduction	p. 141
Review of Complex Numbers	p. 141
Definition	p. 142
Addition	p. 143
Subtraction	p. 143
Multiplication	p. 143
Division	p. 144
From Rectangular to Polar	p. 144
From Polar to Rectangular	p. 145
Fourier Series of Discrete Periodic Signals	p. 145
Discrete System with Periodic Inputs: The Steady-State Response	p. 147
General Form for yss(n)	p. 151
Frequency Response of Discrete Systems	p. 152
Properties of the Frequency Response	p. 154
Periodicity Property	p. 154
Symmetry Property	p. 155
Fourier Transform of Discrete Signals	p. 157
Convergence Conditions	p. 159
Properties of the Fourier Transform of Discrete Signals	p. 159
Periodicity Property	p. 159
Linearity Property	p. 160
Discrete-Time Shifting Property	p. 160
Frequency Shifting Property	p. 160
Reflection Property	p. 161
Convolution Property	p. 161
Parseval's Relation and Energy Calculations	p. 164
Numerical Evaluation of the Fourier Transform of Discrete Signals	p. 165
Some Insights: Why Is This Fourier Transform?	p. 170
Ease in Analysis and Design	p. 170
Sinusoidal Analysis	p. 170
End of Chapter Examples	p. 171
End of Chapter Problems	p. 185
z-Transform and Discrete Systems	p. 191
Introduction	p. 191
Bilateral z-Transform	p. 191
Unilateral z-Transform	p. 193
Convergence Considerations	p. 196
Inverse z-Transform	p. 199
Partial Fraction Expansion	p. 199
Long Division	p. 201
Properties of the z-Transform	p. 202
Linearity Property	p. 203
Shifting Property	p. 203
Multiplication by e-an	p. 205
Convolution	p. 205
Representation of Transfer Functions as Block Diagrams	p. 206
x(n), h(n), y(n), and the z-Transform	p. 208
Solving Difference Equation Using the z-Transform	p. 209
Convergence Revisited	p. 211
Final-Value Theorem	p. 214
Initial-Value Theorem	p. 215
Some Insights: Poles and Zeroes	p. 215
Poles of the System	p. 216
Zeros of the System	p. 216
Stability of the System	p. 216
End of Chapter Exercises	p. 217
End of Chapter Problems	p. 249
State-Space and Discrete Systems	p. 259
Introduction	p. 259
Review on Matrix Algebra	p. 260
Definition, General Terms, and Notations	p. 260
Identity Matrix	p. 260
Adding Two Matrices	p. 261
Subtracting Two Matrices	p. 261
Multiplying a Matrix by a Constant	p. 261
Determinant of a Two-by-Two Matrix	p. 261
Transpose of a Matrix	p. 262
Inverse of a Matrix	p. 262
Matrix Multiplication	p. 262
Eigenvalues of a Matrix	p. 263
Diagonal Form of a Matrix	p. 263
Eigenvectors of a Matrix	p. 263
General Representation of Systems in State Space	p. 264
Recursive Systems	p. 264
Nonrecursive Systems	p. 266
From the Block Diagram to State Space	p. 267
From the Transfer Function H(z) to State Space	p. 270
Solution of the State-Space Equations in the z-Domain	p. 277
General Solution of the State Equation in Real Time	p. 278
Properties of An and Its Evaluation	p. 280
Transformations for State-Space Representations	p. 283
Some Insights: Poles and Stability	p. 285
End of Chapter Examples	p. 286
End of Chapter Problems	p. 315
Block Diagrams and Review of Discrete System Representations	p. 323
Introduction	p. 323
Basic Block Diagram Components	p. 324
Ideal Delay	p. 324
Adder	p. 324
Subtractor	p. 324
Multiplier	p. 325
Block Diagrams as Interconnected Subsystems	p. 325
General Transfer Function Representation	p. 325
Parallel Representation	p. 325
Series Representation	p. 326
Basic Feedback Representation	p. 326
Controllable Canonical Form Block Diagrams with Basic Blocks	p. 327
Observable Canonical Form Block Diagrams with Basic Blocks	p. 329
Diagonal Form Block Diagrams with Basic Blocks	p. 330
Distinct Roots Case	p. 330
Repeated Roots Case	p. 332
Parallel Block Diagrams with Subsystems	p. 332
Distinct Roots Case	p. 332
Repeated Roots Case	p. 333
Series Block Diagrams with Subsystems	p. 334
Distinct Real Roots Case	p. 334
Mixed Complex and Real Roots Case	p. 335
Block Diagram Reduction Rules	p. 335
Using the Reduction Rules	p. 335
Using Mason's Rule	p. 335
End of Chapter Examples	p. 336
End of Chapter Problems	p. 359
Discrete Fourier Transform and Discrete Systems	p. 365
Introduction	p. 365
Discrete Fourier Transform and the Finite-Duration Discrete Signals	p. 366
Properties of the DFT	p. 367
How Does the Defining Equation Work?	p. 367
DFT Symmetry	p. 369
DFT Linearity	p. 371
Magnitude of the DFT	p. 371
What Does k in X(k), the DFT, Mean?	p. 372
Relation the DFT Has with the Fourier Transform of Discrete Signals, the z-Transform, and the Continuous Fourier Transform	p. 373
DFT and the Fourier Transform of x(n)	p. 373
DFT and the z-Transform of x(n)	p. 374
DFT and the Continuous Fourier Transform of x(t)	p. 374
Numerical Computation of. the DFT	p. 377
Fast Fourier Transform: A Faster Way of Computing the DFT	p. 378
Applications of the DFT	p. 380
Circular Convolution	p. 380
Linear Convolution	p. 384
Approximation to the Continuous Fourier Transform	p. 385
Approximation to the Coefficients of the Fourier Series and the Average Power of the Periodic Signal x(t)	p. 386
Total Energy in the Signal x(n) and x(f)	p. 391
Block Filtering	p. 393
Correlation	p. 393
Some Insights	p. 394
DFT Is the Same as the fft	p. 394
DFT Points Are the Samples of the Fourier Transform, of x(n)	p. 394
How Can We Be Certain That Most of the Frequency Contents of x(t) Are in the DFT?	p. 395
Is the Circular Convolution the Same as the Linear Convolution?	p. 395
Is X(w) ≅ X(K) ?	p. 395
Frequency Leakage and the DFT	p. 395
End of Chapter Exercises	p. 396
End of Chapter Problems	p. 415
Sampling and Transformations	p. 421
Need for Converting a Continuous Signal to a Discrete Signal	p. 421
From the Continuous Signal to Its Binary Code Representation	p. 422
From the Binary Code to the Continuous Signal	p. 423
Sampling Operation	p. 424
Ambiguity in Real-Time Domain	p. 424
Ambiguity in the Frequency Domain	p. 427
Sampling Theorem	p. 427
Filtering before Sampling	p. 428
Sampling and Recovery of the Continuous Signal	p. 429
How Do We Discretize the Derivative Operation?	p. 434
Discretization of the State-Space Representation	p. 438
Bilinear Transformation and the Relationship between the Laplace-Domain and the z-Domain Representations	p. 440
Other Transformation Methods	p. 445
Impulse Invariance Method	p. 446
Step Invariance Method	p. 446
Forward Difference Method	p. 446
Backward Difference Method	p. 446
Bilinear Transformation	p. 446
Some Insights	p. 449
Choice of the Sampling Interval Ts	p. 449
Effect of Choosing Ts on the Dynamics of the System	p. 449
Does Sampling Introduce Additional Zeros to the Transfer Function H(z)?	p. 450
End of Chapter Examples	p. 450
End of Chapter Problems	p. 467
Infinite Impulse Response Filter Design	p. 473
Introduction	p. 473
Design Process	p. 474
Design Based on the Impulse Invariance Method	p. 475
Design Based on the Bilinear Transform Method	p. 477
HR Filter Design Using MATLAB“	p. 480
From the Analogue Prototype to the HR Digital Filter	p. 481
Direct Design	p. 481
Some Insights	p. 482
Difficulty in Designing HR Digital Filters in the z-Domain	p. 482
Using the Impulse Invariance Method	p. 484
Choice of the Sampling Interval Ts	p. 484
End of Chapter Examples	p. 484
End of Chapter Problems	p. 515
Finite Impulse Response Digital Filters	p. 521
Introduction	p. 521
What Is an FIR Digital Filter?	p. 521
Motivating Example	p. 521
FIR Filter Design	p. 524
Stability of FIR Filters	p. 526
Linear Phase of FIR Filters	p. 527
Design Based on the Fourier Series: The Windowing Method	p. 528
Ideal Lowpass FIR Filter Design	p. 529
Other Ideal Digital FIR Filters	p. 531
Windows Used in the Design of the Digital FIR Filter	p. 532
Which Window Does Give the Optimal h(n)?	p. 534
Design of a Digital FIR Differentiator	p. 535
Design of Comb FIR Filters	p. 537
Design of a Digital Shifter: The Hilbert Transform Filter	p. 539
From IIR to FIR Digital Filters: An Approximation	p. 540
Frequency Sampling and FIR Filter Design	p. 540
FIR Digital Design Using MATLAB“	p. 541
Design Using Windows	p. 541
Design Using Least-Squared Error	p. 542
Design Using the Equiripple Linear Phase	p. 542
How to Obtain the Frequency Response	p. 542
Some Insights	p. 543
Comparison with IIR Filters	p. 543
Different Methods Used in the FIR Filter Design	p. 543
End of the Chapter Examples	p. 544
End of Chapter Problems	p. 572
Bibliography	p. 579
Index	p. 581
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