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1 Introduction 1

2 Discrete-Time Signals and Systems 8

2.0 Introduction 8

2.1 Discrete-Time Signals：Sequences 9

2.2 Discrete-Time Systems 17

2.3 Linear Time-Invariant Systems 21

2.4 Properties of Linear Time-Invariant Systems 27

2.5 Linear Constant-Coefficient Difference Equations 33

2.6 Frequency-Domain Representation of Discrete-Time Signals and Systems 39

2.7 Representation of Sequences by Fourier Transforms 45

2.8 Symmetry Properties of the Fourier Transform 52

2.9 Fourier Transform Theorems 56

2.10 Discrete-Time Random Signals 63

2.11 Summary 67

Problems 68

3 Sampling of Continuous-Time Signals 80

3.0 Introduction 80

3.1 Periodic Sampling 80

3.2 Frequency-Domain Representation of Sampling 82

3.3 Reconstruction of a Bandlimited Signal from Its Samples 87

3.4 Discrete-Time Processing of Continuous-Time Signals 91

3.5 Continuous-Time Processing of Discrete-Time Signals 99

3.6 Changing the Sampling Rate Using Discrete-Time Processing 101

3.7 Practical Considerations 112

3.8 Summary 130

Problems 131

4 The z-Transform 149

4.0 Introduction 149

4.1 The z-Transform 149

4.2 Properties of the Region of Convergence for the z-Transform 160

4.3 The Inverse z-Transform 165

4.4 z-Transform Properties 172

4.5 The Inverse z-Transform Using Contour Integration 181

4.6 The Complex Convolution Theorem 184

4.7 Parseval’s Relation 186

4.8 The Unilateral z-Transform 188

4.9 Summary 191

Problems 192

5 Transform Analysis of Linear Time-Invariant Systems 202

5.0 Introduction 202

5.1 The Frequency Response of LTI Systems 203

5.2 System Functions for Systems Characterized by Linear Constant-Coefficient Difference Equations 206

5.3 Frequency Response for Rational System Functions 213

5.4 Relationship Between Magnitude and Phase 230

5.5 Allpass Systems 234

5.6 Minimum-Phase Systems 240

5.7 Linear Systems with Generalized Linear Phase 250

5.8 Summary 270

Problems 270

6 Structures for Discrete-Time Systems 290

6.0 Introduction 290

6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations 291

6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations 297

6.3 Basic Structures for IIR Systems 300

6.4 Transposed Forms 309

6.5 Basic Network Structures for FIR Systems 313

6.6 Lattice Structures 317

6.7 Overview of Finite-Precision Numerical Effects 328

6.8 The Effects of Coefficient Quantization 335

6.9 Effects of Roundoff Noise in Digital Filters 351

6.10 Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters 373

6.11 Summary 378

Problems 379

7 Filter Design Techniques 403

7.0 Introduction 403

7.1 Design of Discrete-Time IIR Filters from Continuous-Time Filters 406

7.2 Frequency Transformations of Lowpass IIR Filters 430

7.3 Computer-Aided Design of Discrete-Time IIR Filters 438

7.4 Design of FIR Filters by Windowing 444

7.5 Examples of FIR Filter Design by the Kaiser Window Method 458

7.6 Optimum Approximations of FIR Filters 464

7.7 Examples of FIR Equiripple Approximation 481

7.8 Comments on IIR and FIR Digital Filters 488

7.9 Summary 489

Problems 490

8 The Discrete Fourier Transform 514

8.0 Introduction 514

8.1 Representation of Periodic Sequences：The Discrete Fourier Series 515

8.2 Properties of the Discrete Fourier Series 520

8.3 Summary of Properties of the DFS Representation of Periodic Sequences 525

8.4 The Fourier Transform of Periodic Signals 526

8.5 Sampling the Fourier Transform 527

8.6 Fourier Representation of Finite-Duration Sequences：The Discrete Fourier Transform 530

8.7 Properties of the Discrete Fourier Transform 535

8.8 Summary of Properties of the Discrete Fourier Transform 547

8.9 Linear Convolution Using the Discrete Fourier Transform 548

8.10 Summary 560

Problems 561

9 Computation of the Discrete Fourier Transform 581

9.0 Introduction 581

9.1 Efficient Computation of the Discrete Fourier Transform 582

9.2 The Goertzel Algorithm 585

9.3 Decimation-in-Time FFT Algorithms 587

9.4 Decimation-in-Frequency FFT Algorithms 599

9.5 Implementation of FFT Algorithms 605

9.6 FFT Algorithms for Composite N 610

9.7 Implementation of the DFT Using Convolution 622

9.8 Effects of Finite Register Length in Discrete Fourier Transform Computations 628

9.9 Summary 641

Problems 642

10 Discrete Hilbert Transforms 662

10.0 Introduction 662

10.1 Real and Imaginary Part Sufficiency of the Fourier Transform for Causal Sequences 664

10.2 Sufficiency Theorems for Finite-Length Sequences 670

10.3 Relationships Between Magnitude and Phase 674

10.4 Hilbert Transform Relations for Complex Sequences 676

10.5 Summary 689

Problems 689

11 Fourier Analysis of Signals Using the Discrete Fourier Transform 695

11.0 Introduction 695

11.1 Fourier Analysis of Signals Using the DFT 696

11.2 DFT Analysis of Sinusoidal Signals 699

11.3 The Time-Dependent Fourier Transform 713

11.4 Block Convolution Using the Time-Dependent Fourier Transform 721

11.5 Fourier Analysis of Nonstationary Signals 723

11.6 Fourier Analysis of Stationary Random Signals：The Periodogram 730

11.7 Spectrum Analysis of Random Signals Using Estimates of the Autocorrelation Sequence 742

11.8 Summary 755

Problems 756

12 Cepstrum Analysis and Homomorphic Deconvolution 768

12.0 Introduction 768

12.1 Definition of the Complex Cepstrum 769

12.2 Homomorphic Deconvolution 771

12.3 Properties of the Complex Logarithm 775

12.4 Alternative Expressions for the Complex Cepstrum 778

12.5 The Complex Cepstrum of Exponential Sequences 779

12.6 Minimum-Phase and Maximum-Phase Sequences 781

12.7 Realizations of the Characteristic System D*[·] 787

12.8 Examples of Homomorphic Filtering 797

12.9 Applications to Speech Processing 815

12.10 Summary 825

Problems 826

Appendix A Random Signals 835

A.1 Discrete-Time Random Processes 835

A.2 Averages 837

A.3 Properties of Correlation and Covariance Sequences 841

A.4 Transform Representations of Random Signals 843

Appendix B Continuous-Time Filters 845

B.1 Butterworth Lowpass Filters 845

B.2 Chebyshev Filters 847

B.3 Elliptic Filters 849

Bibliography 851

Index 869