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1 Introduction 1

2 Discrete-Time Signals and Systems 9

2.0 Introduction 9

2.1 Discrete-Time Signals 10

2.2 Discrete-Time Systems 17

2.2.1 Memoryless Systems 18

2.2.2 Linear Systems 19

2.2.3 Time-Invariant Systems 20

2.2.4 Causality 22

2.2.5 Stability 22

2.3 LTI Systems 23

2.4 Properties of Linear Time-Invariant Systems 30

2.5 Linear Constant-Coefficient Difference Equations 35

2.6 Frequency-Domain Representation of Discrete-Time Signals and Systems 40

2.6.1 Eigenfunctions for Linear Time-Invariant Systems 40

2.6.2 Suddenly Applied Complex Exponential Inputs 46

2.7 Representation of Sequences by Fourier Transforms 48

2.8 Symmetry Properties of the Fourier Transform 54

2.9 Fourier Transform Theorems 58

2.9.1 Linearity of the Fourier Transform 59

2.9.2 Time Shifting and Frequency Shifting Theorem 59

2.9.3 Time Reversal Theorem 59

2.9.4 Differentiation in Frequency Theorem 59

2.9.5 Parseval's Theorem 60

2.9.6 The Convolution Theorem 60

2.9.7 The Modulation or Windowing Theorem 61

2.10 Discrete-Time Random Signals 64

2.11 Summary 70

Problems 70

3 The z-Transform 99

3.0 Introduction 99

3.1 z-Transform 99

3.2 Properties of the ROC for the z-Transform 110

3.3 The Inverse z-Transform 115

3.3.1 Inspection Method 116

3.3.2 Partial Fraction Expansion 116

3.3.3 Power Series Expansion 122

3.4 z-Transform Properties 124

3.4.1 Linearity 124

3.4.2 Time Shifting 125

3.4.3 Multiplication by an Exponential Sequence 126

3.4.4 Differentiation of X（z） 127

3.4.5 Conjugation of a Complex Sequence 129

3.4.6 Time Reversal 129

3.4.7 Convolution of Sequences 130

3.4.8 Summary of Some z-Transform Properties 131

3.5 z-Transforms and LTI Systems 131

3.6 The Unilateral z-Transform 135

3.7 Summary 137

Problems 138

4 Sampling of Continuous-Time Signals 153

4.0 Introduction 153

4.1 Periodic Sampling 153

4.2 Frequency-Domain Representation of Sampling 156

4.3 Reconstruction of a Bandlimited Signal from Its Samples 163

4.4 Discrete-Time Processing of Continuous-Time Signals 167

4.4.1 Discrete-Time LTI Processing of Continuous-Time Signals 168

4.4.2 Impulse Invariance 173

4.5 Continuous-Time Processing of Discrete-Time Signals 175

4.6 Changing the Sampling Rate Using Discrete-Time Processing 179

4.6.1 Sampling Rate Reduction by an Integer Factor 180

4.6.2 Increasing the Sampling Rate by an Integer Factor 184

4.6.3 Simple and Practical Interpolation Filters 187

4.6.4 Changing the Sampling Rate by a Noninteger Factor 190

4.7 Multirate Signal Processing 194

4.7.1 Interchange of Filtering with Compressor/Expander 194

4.7.2 Multistage Decimation and Interpolation 195

4.7.3 Polyphase Decompositions 197

4.7.4 Polyphase Implementation of Decimation Filters 199

4.7.5 Polyphase Implementation of Interpolation Filters 200

4.7.6 Multirate Filter Banks 201

4.8 Digital Processing of Analog Signals 205

4.8.1 Prefiltering to Avoid Aliasing 206

4.8.2 A/D Conversion 209

4.8.3 Analysis of Quantization Errors 214

4.8.4 D/A Conversion 221

4.9 Oversampling and Noise Shaping in A/D and D/A Conversion 224

4.9.1 Oversampled A/D Conversion with Direct Quantization 225

4.9.2 Oversampled A/D Conversion with Noise Shaping 229

4.9.3 Oversampling and Noise Shaping in D/A Conversion 234

4.10 Summary 236

Problems 237

5 Transform Analysis of Linear Time-Invariant Systems 274

5.0 Introduction 274

5.1 The Frequency Response of LTI Systems 275

5.1.1 Frequency Response Phase and Group Delay 275

5.1.2 Illustration of Effects of Group Delay and Attenuation 278

5.2 System Functions—Linear Constant-Coefficient Difference Equations 283

5.2.1 Stability and Causality 285

5.2.2 Inverse Systems 286

5.2.3 Impulse Response for Rational System Functions 288

5.3 Frequency Response for Rational System Functions 290

5.3.1 Frequency Responseof 1st-Order Systems 292

5.3.2 Examples with Multiple Poles and Zeros 296

5.4 Relationship between Magnitude and Phase 301

5.5 All-Pass Systems 305

5.6 Minimum-Phase Systems 311

5.6.1 Minimum-Phase and All-Pass Decomposition 311

5.6.2 Frequency-Response Compensation of Non-Minimum-Phase Systems 313

5.6.3 Properties of Minimum-Phase Systems 318

5.7 Linear Systems with Generalized Linear Phase 322

5.7.1 Systems with Linear Phase 322

5.7.2 Generalized Linear Phase 326

5.7.3 Causal Generalized Linear-Phase Systems 328

5.7.4 Relation of FIR Linear-Phase Systems to Minimum-Phase Systems 338

5.8 Summary 340

Prohlems 341

6 Structures for Discrete-Time Systems 374

6.0 Introduction 374

6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations 375

6.2 Signal Flow Graph Representation 382

6.3 Basic Structures for IIR Systems 388

6.3.1 Direct Forms 388

6.3.2 Cascade Form 390

6.3.3 Parallel Form 393

6.3.4 Feedback in IIR Systems 395

6.4 Transposed Forms 397

6.5 Basic Network Structures for FIR Systems 401

6.5.1 Direct Form 401

6.5.2 Cascade Form 402

6.5.3 Structures for Linear-Phase FIR Systems 403

6.6 Lattice Filters 405

6.6.1 FIR Lattice Filters 406

6.6.2 A11-Pole Lattice Structure 412

6.6.3 Generalization of Lattice Systems 415

6.7 Overview of Finite-Precision Numerical Effects 415

6.7.1 Number Representations 415

6.7.2 Quantization in Implementing Systems 419

6.8 The Effects of Coefficient Quantization 421

6.8.1 Effects of Coefficient Quantization in IIR Systems 422

6.8.2 Example of Coefficient Quantization in an Elliptic Filter 423

6.8.3 P0lesof Quantized 2nd-Order Sections 427

6.8.4 Effects of Coefficient Quantization in FIR Systems 429

6.8.5 Example of Quantization of an Optimum FIR Filter 431

6.8.6 Maintaining Linear Phase 434

6.9 Effects of Round-off Noise in Digital Filters 436

6.9.1 Analysis of the Direct Form IIR Structures 436

6.9.2 Scaling in Fixed-Point Implementations of IIR Systems 445

6.9.3 Example of Analysis of a Cascade IIR Structure 448

6.9.4 Analysis of Direct-Form FIR Systems 453

6.9.5 Floating-Point Realizations of Discrete-Time Systems 458

6.10 Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters 459

6.10.1 Limit Cycles Owing to Round-off and Truncation 459

6.10.2 Limit Cycles Owing to Overflow 462

6.10.3 Avoiding Limit Cycles 463

6.11 Summary 463

Problems 464

7 Filter Design Techniques 493

7.0 Introduction 493

7.1 Filter Specifications 494

7.2 Design of Discrete-Time IIR Filters from Continuous-Time Filters 496

7.2.1 Filter Design by Impulse Invariance 497

7.2.2 Bilinear Transformation 504

7.3 Discrete-Time Butterworth,Chebyshev and Elliptic Filters 508

7.3.1 Examples of IIR Filter Design 509

7.4 Frequency Transformations of Lowpass IIR Filters 526

7.5 Design ofFIR Filters by Windowing 533

7.5.1 Properties of Commonly Used Windows 535

7.5.2 Incorporation of Generalized Linear Phase 538

7.5.3 The Kaiser Window Filter Design Method 541

7.6 Examples of FIR Filter Design by the Kaiser Window Method 545

7.6.1 Lowpass Filter 545

7.6.2 Highpass Filter 547

7.6.3 Discrete-Time Differentiators 550

7.7 Optimum Approximations ofFIR Filters 554

7.7.1 Optimal Type Ⅰ Lowpass Filters 559

7.7.2 Optimal Type Ⅱ Lowpass Filters 565

7.7.3 The Parks-McClellan Algorithm 566

7.7.4 Characteristics of Optimum FIR Filters 568

7.8 Examples of FIR Equiripple Approximation 570

7.8.1 Lowpass Filter 570

7.8.2 Compensation for Zero-Order Hold 571

7.8.3 Bandpass Filter 576

7.9 Comments on IIR and FIR Discrete-Time Filters 578

7.10 Design of an Upsampling Filter 579

7.11 Summary 582

Problems 582

8 The Discrete Fourier Transform 623

8.0 Introduction 623

8.1 Representation of Periodic Sequences:The Discrete Fourier Series 624

8.2 Properties ofthe DFS 628

8.2.1 Linearity 629

8.2.2 Shift of a Sequence 629

8.2.3 Duality 629

8.2.4 Symmetry Properties 630

8.2.5 Periodic Convolution 630

8.2.6 Summary of Properties of the DFS Representation of Periodic Sequences 633

8.3 The Fourier Transform of Periodic Signals 633

8.4 Sampling the Fourier Transform 638

8.5 Fourier Representation of Finite-Duration Sequences 642

8.6 Properties ofthe DFT 647

8.6.1 Linearity 647

8.6.2 Circular Shift of a Sequence 648

8.6.3 Duality 650

8.6.4 Symmetry Properties 653

8.6.5 Circular Convolution 654

8.6.6 Summary of Properties of the DFT 659

8.7 Linear Convolution Using the DFT 660

8.7.1 Linear Convolution of Two Finite-Length Sequences 661

8.7.2 Circular Convolution as Linear Convolution with Aliasing 661

8.7.3 Implementing Linear Time-Invariant Systems Using the DFT 667

8.8 The Discrete Cosine Transform（DCT） 673

8.8.1 Definitions ofthe DCT 673

8.8.2 Definition of the DCT-1 and DCT-2 675

8.8.3 Relationship between the DFT and the DCT-1 676

8.8.4 Relationship between the DFT and the DCT-2 678

8.8.5 Energy Compaction Property of the DCT-2 679

8.8.6 Applications of the DCT 682

8.9 Summary 683

Problems 684

9 Computation of the Discrete Fourier Transform 716

9.0 Introduction 716

9.1 Direct Computation of the Discrete Fourier Transform 718

9.1.1 Direct Evaluation of the Definition of the DFT 718

9.1.2 The Goertzel Algorithm 719

9.1.3 Exploiting both Symmetry and Periodicity 722

9.2 Decimation-in-Time FFr Algorithms 723

9.2.1 Generalization and Programming the FFT 731

9.2.2 In-Place Computations 731

9.2.3 Alternative Forms 734

9.3 Decimation-in-Frequency FFT Algorithms 737

9.3.1 In-Place Computation 741

9.3.2 Alternative Forms 741

9.4 Practical Considerations 743

9.4.1 Indexing 743

9.4.2 Coefficients 745

9.5 More General FFT Algorithms 745

9.5.1 Algorithms for Composite Values of N 746

9.5.2 Optimized FFr Algorithms 748

9.6 Implementation of the DFT Using Convolution 748

9.6.1 Overview of the Winograd Fourier Transform Algorithm 749

9.6.2 The Chirp Transform Algorithm 749

9.7 Effects of Finite Register Length 754

9.8 Summary 762

Problems 763

10 Fourier Analysis of Signals Using the Discrete Fourier Transform 792

10.0 Introduction 792

10.1 Fourier Analysis of Signals Using the DFT 793

10.2 DFT Analysis of Sinusoidal Signals 797

10.2.1 The Effect of Windowing 797

10.2.2 Properties of the Windows 800

10.2.3 The Effect of Spectral Sampling 801

10.3 The Time-Dependent Fourier Transform 811

10.3.1 Invertibility ofX[n,） 815

10.3.2 Filter Bank Interpretation of X[n,） 816

10.3.3 The Effect of the Window 817

10.3.4 Sampling in Time and Frequency 819

10.3.5 The Overlap－Add Method of Reconstruction 822

10.3.6 Signal Processing Based on the Time-Dependent Fourier Transform 825

10.3.7 Filter Bank Interpretation of the Time-Dependent Fourier Transform 826

10.4 Examples of Fourier Analysis of Nonstationary Signals 829

10.4.1 Time-Dependent Fourier Analysis of Speech Signals 830

10.4.2 Time-Dependent Fourier Analysis of Radar Signals 834

10.5 Fourier Analysis of Stationary Random Signals:the Periodogram 836

10.5.1 The Periodogram 837

10.5.2 Properties of the Periodogram 839

10.5.3 Periodogram Averaging 843

10.5.4 Computation of Average Periodograms Using the DFT 845

10.5.5 An Example of Periodogram Analysis 845

10.6 Spectrum Analysis of Random Signals 849

10.6.1 Computing Correlation and Power Spectrum Estimates Using the DFT 853

10.6.2 Estimating the Power Spectrum of Quantization Noise 855

10.6.3 Estimating the Powet Spectrum of Speech 860

10.7 Summary 862

Problems 864

11 Parametric Signal Modeling 890

11.0 Introduction 890

11.1 All-Pole Modeling of Signals 891

11.1.1 Least-Squares Approximation 892

11.1.2 Least-Squares Inverse Model 892

11.1.3 Linear Prediction Formulation of All-Pole Modeling 895

11.2 Deterministic and Random Signal Models 896

11.2.1 All-Pole Modeling of Finite-Energy Deterministic Signals 896

11.2.2 Modeling of Random Signals 897

11.23 Minimum Mean-Squared Error 898

11.2.4 Autocorrelation Matching Property 898

11.2.5 Determination of the Gain Parameter G 899

11.3 Estimation of the Correlation Functions 900

11.3.1 The Autocorrelation Method 900

11.3.2 The Covariance Method 903

11.3.3 Comparison of Methods 904

11.4 Model Order 905

11.5 All-Pole Spectrum Analysis 907

11.5.1 All-Pole Analysis of Speech Signals 908

11.5.2 Pole Locations 911

11.5.3 All-Pole Modeling of Sinusoidal Signals 913

11.6 Solution of the Autocorrelation Normal Equations 915

11.6.1 The Levinson－Durbin Recursion 916

11.6.2 Derivation of the Levinson－Durbin Algorithm 917

11.7 Lattice Filters 920

11.7.1 Prediction Error Lattice Network 921

11.7.2 All-Pole Model Lattice Network 923

11.7.3 Direct Computation of the k-Parameters 925

11.8 Summary 926

Problems 927

12 Discrete Hilbert Transforms 942

12.0 Introduction 942

12.1 Real-and Imaginary-Part Sufficiency of the Fourier Transform 944

12.2 Sufficiency Theorems for Finite-Length Sequences 949

12.3 Relationships Between Magnitude and Phase 955

12.4 Hilbert Transform Relations for Complex Sequences 956

12.4.1 Design of Hilbert Transformers 960

12.4.2 Representation of Bandpass Signals 963

12.4.3 Bandpass Sampling 966

12.5 Summary 969

Problems 969

13 Cepstrum Analysis and Homomorphic Deconvolution 980

13.0 Introduction 980

13.1 Definition of the Cepstrum 981

13.2 Definition of the Complex Cepstrum 982

13.3 Properties of the Complex Logarithm 984

13.4 Alternative Expressions for the Complex Cepstrum 985

13.5 Properties of the Complex Cepstrum 986

13.5.1 Exponential Sequences 986

13.5.2 Minimum-Phase and Maximum-Phase Sequences 989

13.5.3 Relationship Between the Real Cepstrum and the Complex Cepstrum 990

13.6 Computation ofthe Complex Cepstrum 992

13.6.1 Phase Unwrapping 993

13.6.2 Computation of the Complex Cepstrum Using the Logarithmic Derivative 996

13.6.3 Minimum-Phase Realizations for Minimum-Phase Sequences 998

13.6.4 Recursive Computation of the Complex Cepstrum for Minimum-and Maximum-Phase Sequences 998

13.6.5 The Use of Exponential Weighting 1000

13.7 Computation of the Complex Cepstrum Using Polynomial Roots 1001

13.8 Deconvolution Using the Complex Cepstrum 1002

13.8.1 Minimum-Phase/Allpass Homomorphic Deconvolution 1003

13.8.2 Minimum-Phase/Maximum-Phase Homomorphic Deconvolution 1004

13.9 The Complex Cepstrum for a Simple Multipath Model 1006

13.9.1 Computation of the Complex Cepstrum bv z-Transform Analysis 1009

13.9.2 Computation of the Cepstrum Using the DFT 1013

13.9.3 Homomorphic Deconvolution for the Multipath Model 1016

13.9.4 Minimum-Phase Decomposition 1017

13.9.5 Generalizations 1024

13.10 Applications to Speech Processing 1024

13.10.1 The Speech Model 1024

13.10.2 Example of Homomorphic Deconvolution of Speech 1028

13.10.3 Estimating the Parameters of the Speech Model 1030

13.10.4 Applications 1032

13.11 Summary 1032

Problems 1034

A Random Signals 1043

B Continuous-Time Filters 1056

C Answers to Selected Basic Problems 1061

Bibliography 1082

Index 1091