Table Of ContentDigital Signal Processing
A Primer With MATLAB®
Digital Signal Processing
A Primer With MATLAB®
Samir I. Abood
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of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute
endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
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Dedicated to
My great parents, who never stop giving of themselves in countless ways,
My beloved brothers and sisters;
My dearest wife, who offered me unconditional love
with the light of hope and support;
My beloved kids: Daniah, and Mustafa, whom I can't force myself to stop loving;
To all my family, the symbol of love and giving.
Contents
Preface............................................................................................................................................ xiii
Acknowledgments............................................................................................................................xv
Author ............................................................................................................................................xvii
Chapter 1 Continuous and Discrete Signals .................................................................................1
1.1 Continuous Signals............................................................................................1
1.1.1 Generation of Continuous Signals in MATLAB .................................1
1.1.2 Operations on Signals and Sequences..................................................2
1.2 Discrete-Time Signals .......................................................................................5
1.2.1 Complex Sequences..............................................................................5
1.3 Signals and Systems ..........................................................................................6
1.4 Classification of Signals and Systems...............................................................6
1.4.1 Continuous-Time and Discrete-Time Signals ......................................6
1.4.2 Analog and Digital Signals ..................................................................6
1.4.3 Deterministic and Random Signals......................................................6
1.4.4 Periodic and Nonperiodic Signals........................................................7
1.4.5 Power and Energy Signals....................................................................7
1.4.5.1 What Is Digital Signal Processing?......................................7
1.4.5.2 Why DSP?.............................................................................7
1.4.5.3 Applications (DSP)...............................................................7
1.5 Introduction to MATLAB in DSP.....................................................................8
1.5.1 MATLAB Windows.............................................................................8
1.5.2 Basic Commands in MATLAB............................................................9
1.6 Some Fundamental Sequences........................................................................11
1.6.1 Impulse Response in MATLAB ........................................................ 11
1.6.2 Signal Duration .................................................................................. 12
1.7 Generation of Discrete Signals in MATLAB.................................................. 12
Problems.....................................................................................................................17
Chapter 2 Signals Properties....................................................................................................... 19
2.1 Periodic and Aperiodic Sequences..................................................................19
2.2 Even and Odd Parts of a Signal (Symmetric Sequences).................................20
2.3 Signal Manipulations.......................................................................................23
2.3.1 Transformations of the Independent Variable....................................23
2.3.1.1 Shifting...............................................................................23
2.3.1.2 Reversal ..............................................................................23
2.3.1.3 Time-Scaling ......................................................................23
2.3.1.4 Addition, Multiplication, and Scaling ................................24
2.3.1.5 Addition..............................................................................24
2.3.1.6 Multiplication......................................................................24
2.3.1.7 Scaling................................................................................24
2.3.1.8 Signal Decomposition.........................................................24
2.4 Discrete-Time Systems....................................................................................25
2.4.1 System Properties...............................................................................25
2.4.1.1 Memoryless System............................................................25
2.4.1.2 Additivity............................................................................26
vii
viii Contents
2.4.1.3 Homogeneity.......................................................................26
2.4.1.4 Stability...............................................................................26
2.5 Linear Time-Invariant Causal Systems (LTI).................................................27
2.5.1 Linearity.............................................................................................27
2.5.2 Time-Invariance ................................................................................. 31
2.5.3 Causality............................................................................................. 32
2.6 Definitions .......................................................................................................36
2.6.1 Continuous-Time System ...................................................................36
2.6.2 Discrete-Time System ........................................................................37
2.6.2.1 Delay Operator....................................................................38
2.6.2.2 Convolution Property..........................................................38
2.6.2.3 Impulse Function................................................................38
2.6.2.4 Impulse Response...............................................................38
2.6.2.5 Frequency Response...........................................................38
2.7 System Output .................................................................................................39
2.7.1 Causality.............................................................................................39
2.7.2 Stability..............................................................................................39
2.7.3 Invertibility.........................................................................................40
2.7.4 Memory..............................................................................................40
Problems.....................................................................................................................42
Chapter 3 Convolution ................................................................................................................ 47
3.1 Linear Convolution..........................................................................................47
3.2 Convolution Properties .................................................................................... 47
3.2.1 Commutative Property ....................................................................... 47
3.2.2 Associative Property ..........................................................................48
3.2.3 Distributive Property..........................................................................48
3.3 Types of Convolutions.....................................................................................48
3.3.1 Equations Method ..............................................................................49
3.3.1.1 Convolution of Two Sequences in MATLAB..................... 53
3.3.2 Graphical Method ..............................................................................54
3.3.3 Tabular Method .................................................................................. 55
Problems.....................................................................................................................60
Chapter 4 Difference Equations..................................................................................................65
4.1 Difference Equations and Impulse Responses................................................65
4.2 System Representation Using Its Impulse Response.......................................66
4.3 The Methods That One May Use to Solve the Difference Equations ............. 67
4.4 The Classical Approach...................................................................................68
Problems.....................................................................................................................72
Chapter 5 Discrete-Time Fourier Series (DTFS)........................................................................ 75
5.1 DTFS Coefficients of Periodic Discrete Signals ............................................. 75
5.2 Parseval’s Relation...........................................................................................77
5.3 Discreet Fourier Series....................................................................................79
Problems.....................................................................................................................83
Contents ix
Chapter 6 Discrete-Time Fourier Transform (DTFT) ................................................................85
6.1 Frequency Response........................................................................................85
6.2 DTFT for Any Discrete Signal........................................................................87
6.3 Inverse DTFT..................................................................................................88
6.4 Interconnection of Systems .............................................................................89
6.5 DTFT Properties ............................................................................................. 91
6.6 Applications of DTFT ..................................................................................... 91
6.7 LSI Systems and Difference Equations...........................................................91
6.8 Solving Difference Equations Using DTFT....................................................93
6.9 Frequency Response in MATLAB................................................................ 102
Problems...................................................................................................................106
Chapter 7 Discrete Fourier Transform (DFT) .......................................................................... 109
7.1 Method of Decimation-in-Frequency............................................................109
7.2 Method of Decimation-in-Time..................................................................... 112
7.3 Properties of Discrete Fourier Transform ..................................................... 117
7.4 Discrete Fourier Transform of a Sequence in MATLAB ............................. 122
7.5 Linear Convolution Using the DFT...............................................................123
7.6 Generation of Inverse Discrete Fourier Transform in MATLAB.................124
Problems...................................................................................................................125
Chapter 8 Fast Fourier Transform (FFT).................................................................................. 129
8.1 Fast Fourier Transform Definition................................................................. 129
8.1.1 Decimation-in-Time FFT .................................................................1 29
8.1.2 Decimation-in-Frequency FFT ........................................................1 29
8.2 Finding the FFT of Different Signals in MATLAB...................................... 134
8.3 Power Spectral Density Using Square Magnitude and Autocorrelation ....... 135
8.3.1 Equivalence of FFT and N-phase Sequence Component
Transformation.................................................................................136
Problems...................................................................................................................140
Chapter 9 Z-Transform.............................................................................................................. 143
9.1 Z-Transform Representation.......................................................................... 143
9.2 Region of Convergence (ROC)...................................................................... 144
9.3 Properties of the z-transform ......................................................................... 147
9.4 Inverse z-transform........................................................................................151
9.4.1 Partial Fraction Expansion and a Look-up Table.............................151
9.4.2 Power Series.....................................................................................154
9.4.3 Contour Integration..........................................................................154
Problems...................................................................................................................163
Chapter 10 Z-Transform Applications in DSP............................................................................165
10.1 Evaluation of LTI System Response Using Z-Transform..............................165
10.2 Digital System Implementation from Its Function........................................165
10.3 Pole-Zero Diagrams for a Function in the z-Domain....................................172
10.4 Frequency Response Using z-Transform ....................................................... 172
Problems................................................................................................................... 174
