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Chapter 1 Discrete-Time Systems 1
1.1 Introduction 2
1.2 Discrete-Time Signals 3
1.3 Discrete-Time Systems 6
1.3.1 Linearity 6
1.3.2 Time Invariance 6
1.3.3 Causality 6
1.3.4 Stability 9
1.4 Difference Equations and Time-Domain
Response 10
1.5 Sampling of Continuous-Time Signals 14
1.5.1 Basic Principles 14
1.5.2 Sampling Theorem 15
1.6 Discrete-Time Signals and Systems with
MATLAB 22
1.7 Summary 23
Exercises 23
MATLAB Exercises 27
Chapter 2 The z and Fourier
Transforms 28
2.1 Introduction 29
2.2 Definition of the z Transform 30
2.3 Inverse z Transform 36
2.3.1 Computation Based on Residue
Theorem 37
2.3.2 Computation Based on Partial-Fraction Expansions 40
2.3.3 Computation Based on
Polynomial Division 41
2.3.4 Computation Based on Series
Expansion 43
2.4 Properties of the z Transform 43
2.4.1 Linearity 43
2.4.2 Time-Reversal 44
2.4.3 Time-Shift Theorem 44
2.4.4 Multiplication by An
Exponential 44
2.4.5 Complex Differentiation 45
2.4.6 Complex Conjugation 45
2.4.7 Real and Imaginary Sequences 46
2.4.8 Initial Value Theorem 46
2.4.9 Convolution Theorem 46
2.4.10 Product of Two Sequences 47
2.4.11 Parseval’s Theorem 48
2.4.12 Table of Basic z Transforms 48
2.5 Transfer Functions 48
2.6 Stability in the z Domain 51
2.7 Frequency Response 53
2.8 Fourier Transform 58
2.9 Properties of the Fourier Transform 61
2.9.1 Linearity 61
2.9.2 Time-Reversal 61
2.9.3 Symmetric and Antisymmetric
Sequences 62
2.9.4 Convolution Theorem 63
2.9.5 Product of Two Sequences 63
2.9.6 Parseval’s Theorem 63
2.10 Transfer Functions with MATLAB 63
2.11 Summary 65
Exercises 65
Chapter 3 Discrete Transforms 68
3.1 Introduction 69
3.2 Discrete Fourier Transform 69
3.3 Properties of the DFT 75
3.3.1 Linearity 75
3.3.2 Time-Reversal 75
3.3.3 Time-Shift Theorem 76
3.3.4 Circular Frequency-Shift Theorem (Modulation Theorem) 77
3.3.5 Circular Convolution in Time 77
3.3.6 Correlation 78
3.3.7 Real and Imaginary Sequences 78
3.3.8 Symmetric and Antisymmetric
Sequences 79
3.3.9 Parseval’s Theorem 81
3.3.10 Relationship between the
DFT and the z Transform 81
3.4 Digital Filtering Using the DFT 81
3.4.1 Linear and Circular
Convolutions 81
3.4.2 Overlap-and-Add Method 85
3.4.3 Overlap-and-Save Method 87
3.5 Fast Fourier Transform 90
3.6 Other Discrete Transforms 91
3.6.1 Discrete Cosine Transform 91
3.6.2 Discrete Hartley Transform 96
3.6.3 Hadamard Transform 97
3.6.4 Other Important Transforms 98
3.7 Signal Representations 98
3.8 Discrete Transforms with MATLAB 101
3.9 Summary 102
Exercises 102
Chapter 4 The Fast Fourier
Transform 105
4.1 Relationship of the FFT to the DFT 106
4.2 Hints on Using FFTs in Practice 107
4.2.1 Sample Fast Enough and Long Enough 107
4.2.2 Manipulating the Time Date
Prior to Transformation 108
4.2.3 Enhancing FFT Results 109
4.2.4 Interpreting FFT Results 109
4.3 Derivation of the Radix-2 FFT
Algorithm 110
4.4 FFT Input/Output Data Index Bit
Reversal 116
4.5 Radix-2 FFT Butterfly Structures 117
4.6 Alternate Single-Butterfly Structures 120
4.7 Fast Fourier Transform with
MATLAB 123
4.8 Summary 124
Exercises 124
MATLAB Exercises 125
Chapter 5 Digital Filter Structures 126
5.1 Block Diagram Representation 127
5.1.1 Basic Building Blocks 128
5.1.2 Analysis of Block Diagrams 128
5.1.3 The Delay-Free Loop Problem 129
5.1.4 Canonic and Non-Canonic Structures 130
5.2 Equivalent Structures 131
5.3 Basic FIR Digital Filter Structures 132
5.3.1 Direct-Form Structures 132
5.3.2 Cascade-Form Structures 133
5.3.3 Polyphase Realization 133
5.3.4 Linear-Phase FIR Structures 135
5.3.5 Tapped Delay Line 136
5.4 Basic IIR Digital Filter Structures 137
5.4.1 Direct-Form Structures 137
5.4.2 Cascade Realizations 139
5.4.3 Parallel Realizations 142
5.5 Realization of Basic Structures Using
MATLAB 143
5.5.1 Cascade Realization 143
5.5.2 Parallel Realization 144
5.6 Allpass Filters 146
5.6.1 Realization Based on the
Multiplier Extraction Approach 146
5.6.2 Realization Based on the
Two-Pair Extraction Approach 150
5.7 IIR Tapped Cascaded Lattice Structures 155
5.7.1 Realization of an All-Pole
IIR Transfer Function 156
5.7.2 Gray-Markel Method 156
5.7.3 Realization Using MATLAB 159
5.8 FIR Cascaded Lattice Structures 160
5.8.1 Realization of a Pair of Arbitrary
FIR Transfer Functions 160

5.8.2 Realization of a Pair of
Mirror-Image FIR Transfer
Functions 164
5.8.3 Realization of a Pair of Power-Complementary FIR
Transfer Functions 164
5.8.4 Realization of a single FIR
Transfer Function 165
5.8.5 Realization Using MATLAB 165
5.9 Summary 166
Exercises 167
MATLAB Exercises 180
Chapter 6 IIR Digital Filter Design 182
6.1 Preliminary Considerations 183
6.1.1 Digital Filter Specifications 183
6.1.2 Selection of the Filter Type 186
6.1.3 Basic Approach to IIR
Digital Filter Design 187
6.1.4 IIR Digital Filter Order
Estimation 187
6.1.5 Scaling the Digital Transfer
Function 188
6.2 Bilinear Transformation Method of
IIR Filter Design 188
6.2.1 The Bilinear Transformation 188
6.2.2 Design of Low-Order Digital
Filters 191
6.3 Design of Lowpass IIR Digital Filters 194
6.4 Design of Highpass, Bandpass, and Bandstop IIR Digital Filters 196
6.5 Spectral Transformation of IIR Filters 201
6.5.1 Lowpass-to-Lowpass Transformation 202
6.5.2 Other Transformations 205
6.5.3 Spectral Transformation Using MATLAB 206
6.6 IIR Digital Filter Design Using
MATLAB 208
6.7 Summary 211
Exercises 211
MATLAB Exercises 217
Chapter 7 FIR Digital Filter Design 219
7.1 Preliminary Considerations 220
7.1.1 Basic Approaches to FIR
Digital Filter Design 220
7.1.2 Estimation of the Filter Order 221
7.2 FIR Filter Design Based on
Windowed Fourier Series 224
7.2.1 Least Integral-Squared
Error Design of FIR Filters 224
7.2.2 Impulse Response Of
Ideal Filters 225
7.2.3 Gibbs Phenomenon 228
7.2.4 Fixed Window Function 231
7.2.5 Adjustable Window Functions 236
7.2.6 Impulse Responses of FIR
Filters with a Smooth Transition 239
7.3 Computer-Aided Design of Equiripple Linear-Phase FIR Filters 241
7.3.1 The Parks-McClellan
Algorithm 242
7.3.2 The Shpak-Antoniou
Algorithm 250
7.4 Design of Minimum-Phase FIR Filters 251
7.5 FIR Digital Filter Design Using
MATLAB 252
7.5.1 FIR Digital Filter Order
Estimation Using MATLAB 252
7.5.2 Equiripple Linear-Phase FIR
Filter Design Using MATLAB 254
7.5.3 Minimum-Phase FIR Filter
Design Using MATLAB 263
7.5.4 Window-Based FIR Filter
esign Using MATLAB 267
7.6 Summary 269
Exercises 270
MATLAB Exercises 278
Bibliography 282