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1 Introduction1

2 Discrete-Time Signals and Systems8

2.0 Introduction8

2.1 Discrete-Time Signals：Sequences9

2.2 Discrete-Time Systems17

2.3 Linear Time-Invariant Systems21

2.4 Properties of Linear Time-Invariant Systems27

2.5 Linear Constant-Coefficient Difference Equations33

2.6 Frequency-Domain Representation of Discrete-Time Signals and Systems39

2.7 Representation of Sequences by Fourier Transforms45

2.8 Symmetry Properties of the Fourier Transform52

2.9 Fourier Transform Theorems56

2.10 Discrete-Time Random Signals63

2.11 Summary67

Problems68

3 Sampling of Continuous-Time Signals80

3.0 Introduction80

3.1 Periodic Sampling80

3.2 Frequency-Domain Representation of Sampling82

3.3 Reconstruction of a Bandlimited Signal from Its Samples87

3.4 Discrete-Time Processing of Continuous-Time Signals91

3.5 Continuous-Time Processing of Discrete-Time Signals99

3.6 Changing the Sampling Rate Using Discrete-Time Processing101

3.7 Practical Considerations112

3.8 Summary130

Problems131

4 The z-Transform149

4.0 Introduction149

4.1 The z-Transform149

4.2 Properties of the Region of Convergence for the z-Transform160

4.3 The Inverse z-Transform165

4.4 z-Transform Properties172

4.5 The Inverse z-Transform Using Contour Integration181

4.6 The Complex Convolution Theorem184

4.7 Parseval’s Relation186

4.8 The Unilateral z-Transform188

4.9 Summary191

Problems192

5 Transform Analysis of Linear Time-Invariant Systems202

5.0 Introduction202

5.1 The Frequency Response of LTI Systems203

5.2 System Functions for Systems Characterized by Linear Constant-Coefficient Difference Equations206

5.3 Frequency Response for Rational System Functions213

5.4 Relationship Between Magnitude and Phase230

5.5 Allpass Systems234

5.6 Minimum-Phase Systems240

5.7 Linear Systems with Generalized Linear Phase250

5.8 Summary270

Problems270

6 Structures for Discrete-Time Systems290

6.0 Introduction290

6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations291

6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations297

6.3 Basic Structures for IIR Systems300

6.4 Transposed Forms309

6.5 Basic Network Structures for FIR Systems313

6.6 Lattice Structures317

6.7 Overview of Finite-Precision Numerical Effects328

6.8 The Effects of Coefficient Quantization335

6.9 Effects of Roundoff Noise in Digital Filters351

6.10 Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters373

6.11 Summary378

Problems379

7 Filter Design Techniques403

7.0 Introduction403

7.1 Design of Discrete-Time IIR Filters from Continuous-Time Filters406

7.2 Frequency Transformations of Lowpass IIR Filters430

7.3 Computer-Aided Design of Discrete-Time IIR Filters438

7.4 Design of FIR Filters by Windowing444

7.5 Examples of FIR Filter Design by the Kaiser Window Method458

7.6 Optimum Approximations of FIR Filters464

7.7 Examples of FIR Equiripple Approximation481

7.8 Comments on IIR and FIR Digital Filters488

7.9 Summary489

Problems490

8 The Discrete Fourier Transform514

8.0 Introduction514

8.1 Representation of Periodic Sequences：The Discrete Fourier Series515

8.2 Properties of the Discrete Fourier Series520

8.3 Summary of Properties of the DFS Representation of Periodic Sequences525

8.4 The Fourier Transform of Periodic Signals526

8.5 Sampling the Fourier Transform527

8.6 Fourier Representation of Finite-Duration Sequences：The Discrete Fourier Transform530

8.7 Properties of the Discrete Fourier Transform535

8.8 Summary of Properties of the Discrete Fourier Transform547

8.9 Linear Convolution Using the Discrete Fourier Transform548

8.10 Summary560

Problems561

9 Computation of the Discrete Fourier Transform581

9.0 Introduction581

9.1 Efficient Computation of the Discrete Fourier Transform582

9.2 The Goertzel Algorithm585

9.3 Decimation-in-Time FFT Algorithms587

9.4 Decimation-in-Frequency FFT Algorithms599

9.5 Implementation of FFT Algorithms605

9.6 FFT Algorithms for Composite N610

9.7 Implementation of the DFT Using Convolution622

9.8 Effects of Finite Register Length in Discrete Fourier Transform Computations628

9.9 Summary641

Problems642

10 Discrete Hilbert Transforms662

10.0 Introduction662

10.1 Real and Imaginary Part Sufficiency of the Fourier Transform for Causal Sequences664

10.2 Sufficiency Theorems for Finite-Length Sequences670

10.3 Relationships Between Magnitude and Phase674

10.4 Hilbert Transform Relations for Complex Sequences676

10.5 Summary689

Problems689

11 Fourier Analysis of Signals Using the Discrete Fourier Transform695

11.0 Introduction695

11.1 Fourier Analysis of Signals Using the DFT696

11.2 DFT Analysis of Sinusoidal Signals699

11.3 The Time-Dependent Fourier Transform713

11.4 Block Convolution Using the Time-Dependent Fourier Transform721

11.5 Fourier Analysis of Nonstationary Signals723

11.6 Fourier Analysis of Stationary Random Signals：The Periodogram730

11.7 Spectrum Analysis of Random Signals Using Estimates of the Autocorrelation Sequence742

11.8 Summary755

Problems756

12 Cepstrum Analysis and Homomorphic Deconvolution768

12.0 Introduction768

12.1 Definition of the Complex Cepstrum769

12.2 Homomorphic Deconvolution771

12.3 Properties of the Complex Logarithm775

12.4 Alternative Expressions for the Complex Cepstrum778

12.5 The Complex Cepstrum of Exponential Sequences779

12.6 Minimum-Phase and Maximum-Phase Sequences781

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

12.8 Examples of Homomorphic Filtering797

12.9 Applications to Speech Processing815

12.10 Summary825

Problems826

Appendix A Random Signals835

A.1 Discrete-Time Random Processes835

A.2 Averages837

A.3 Properties of Correlation and Covariance Sequences841

A.4 Transform Representations of Random Signals843

Appendix B Continuous-Time Filters845

B.1 Butterworth Lowpass Filters845

B.2 Chebyshev Filters847

B.3 Elliptic Filters849

Bibliography851

Index869