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Introduction 1

1 Discrete-time signals and systems 5

1.1 Introduction 5

1.2 Discrete-time signals 6

1.3 Discrete-time systems 10

1.3.1 Linearity 10

1.3.2 Time invariance 11

1.3.3 Causality 11

1.3.4 Impulse response and convolution sums 14

1.3.5 Stability 16

1.4 Difference equations and time-domain response 17

1.4.1 Recursive×nonrecursive systems 21

1.5 Solving difference equations 22

1.5.1 Computing impulse responses 31

1.6 Sampling of continuous-time signals 33

1.6.1 Basic principles 34

1.6.2 Sampling theorem 34

1.7 Random signals 53

1.7.1 Random variable 54

1.7.2 Random processes 58

1.7.3 Filtering a random signal 60

1.8 Do-it-yourself:discrete-time signals and systems 62

1.9 Discrete-time signals and systems with MATLAB 67

1.10 Summary 68

1.11 Exercises 68

2 The z and Fourier transforms 75

2.1 Introduction 75

2.2 Definition of the z transform 76

2.3 Inverse z transform 83

2.3.1 Computation based on residue theorem 84

2.3.2 Computation based on partial-fraction expansions 87

2.3.4 Computation based on series expansion 92

2.4 Properties of the z transform 94

2.4.1 Linearity 94

2.4.2 Time reversal 94

2.4.3 Time-shift theorem 95

2.4.4 Multiplication by an exponential 95

2.4.5 Complex differentiation 95

2.4.6 Complex conjugation 96

2.4.7 Real and imaginary sequences 97

2.4.8 Initial-value theorem 97

2.4.9 Convolution theorem 98

2.4.10 Product of two sequences 98

2.4.11 Parseval's theorem 100

2.4.12 Table of basic z transforms 101

2.5 Transfer functions 104

2.6 Stability in the z domain 106

2.7 Frequency response 109

2.8 Fourier transform 115

2.9 Properties of the Fourier transform 120

2.9.1 Linearity 120

2.9.2 Time reversal 120

2.9.3 Time-shift theorem 120

2.9.4 Multiplication by a complex exponential(frequency shift,modulation) 120

2.9.5 Complex differentiation 120

2.9.6 Complex conjugation 121

2.9.7 Real and imaginary sequences 121

2.9.8 Symmetric and antisymmetric sequences 122

2.9.9 Convolution theorem 123

2.9.10 Product of two sequences 123

2.9.11 Parseval's theorem 123

2.10 Fourier transform for periodic sequences 123

2.11 Random signals in the transform domain 125

2.11.1 Power spectral density 125

2.11.2 Whitenoise 128

2.12 Do-it-yourself:the z and Fourier transforms 129

2.13 The z and Fourier transforms with MATLAB 135

2.14 Summary 137

2.15 Exercises 137

3 Discrete transforms 143

3.1 Introduction 143

3.2 Discrete Fourier transform 144

3.3 Properties of the DFT 153

3.3.1 Linearity 153

3.3.2 Time reversal 153

3.3.3 Time-shift theorem 153

3.3.4 Circular frequency-shift theorem(modulation theorem) 156

3.3.5 Circular convolution in time 157

3.3.6 Correlation 158

3.3.7 Complex conjugation 159

3.3.8 Real and imaginary sequences 159

3.3.9 Symmetric and antisymmetric sequences 160

3.3.10 Parseval's theorem 162

3.3.11 Relationship between the DFT and the z transform 163

3.4 Digital filtering using the DFT 164

3.4.1 Linear and circular convolutions 164

3.4.2 Overlap-and-add method 168

3.4.3 Overlap-and-save method 171

3.5 Fast Fourier transform 175

3.5.1 Radix-2 algorithm with decimation in time 176

3.5.2 Decimation in frequency 184

3.5.3 Radix-4 algorithm 187

3.5.4 Algorithms for arbitrary values of N 192

3.5.5 Alternative techniques for determining the DFT 193

3.6 Other discrete transforms 194

3.6.1 Discrete transforms and Parseval's theorem 195

3.6.2 Discrete transforms and orthogonality 196

3.6.3 Discrete cosine transform 199

3.6.4 A family of sine and cosine transforms 203

3.6.5 Discrete Hartley transform 205

3.6.6 Hadamard transform 206

3.6.7 Other important transforms 207

3.7 Signal representations 208

3.7.1 Laplace transform 208

3.7.2 The z transform 208

3.7.3 Fourier transform(continuous time) 209

3.7.4 Fourier transform(discrete time) 209

3.7.5 Fourier series 210

3.7.6 Discrete Fourier transform 210

3.8 Do-it-yourself:discrete transforms 211

3.9 Discrete transforms with MATLAB 215

3.10 Summary 216

3.11 Exercises 217

4 Digital filters 222

4.1 Introduction 222

4.2 Basic structures of nonrecursive digital filters 222

4.2.1 Direct form 223

4.2.2 Cascade form 224

4.2.3 Linear-phase forms 225

4.3 Basic structures of recursive digital filters 232

4.3.1 Direct forms 232

4.3.2 Cascade form 236

4.3.3 Parallel form 237

4.4 Digital network analysis 241

4.5 State-space description 244

4.6 Basic properties of digital networks 246

4.6.1 Tellegen's theorem 246

4.6.2 Reciprocity 248

4.6.3 Interreciprocity 249

4.6.4 Transposition 249

4.6.5 Sensitivity 250

4.7 Useful building blocks 257

4.7.1 Second-order building blocks 257

4.7.2 Digital oscillators 260

4.7.3 Comb filter 261

4.8 Do-it-yourself:digital filters 263

4.9 Digital filter forms with MATLAB 266

4.10 Summary 270

4.11 Exercises 270

5 FIR filter approximations 277

5.1 Introduction 277

5.2 Ideal characteristics of standard filters 277

5.2.1 Lowpass,highpass,bandpass,and bandstop filters 278

5.2.2 Differentiators 280

5.2.3 Hilbert transformers 281

5.2.4 Summary 283

5.3 FIR filter approximation by frequency sampling 283

5.4 FIR filter approximation with window functions 291

5.4.1 Rectangular window 294

5.4.2 Triangular windows 295

5.4.3 Hamming and Hann windows 296

5.4.4 Blackman window 297

5.4.5 Kaiser window 299

5.4.6 Dolph-Chebyshev window 306

5.5 Maximally flat FIR filter approximation 309

5.6 FIR filter approximation by optimization 313

5.6.1 Weighted least-squares method 317

5.6.2 Chebyshev method 321

5.6.3 WLS-Chebyshev method 327

5.7 Do-it-yourself:FIR filter approximations 333

5.8 FIR filter approximation with MATLAB 336

5.9 Summary 342

5.10 Exercises 343

6 IIR filter approximations 349

6.1 Introduction 349

6.2 Analog filter approximations 350

6.2.1 Analog filter specification 350

6.2.2 Butterworth approximation 351

6.2.3 Chebyshev approximation 353

6.2.4 Elliptic approximation 356

6.2.5 Frequency transformations 359

6.3 Continuous-time to discrete-time transformations 368

6.3.1 Impulse-invafiance method 368

6.3.2 Bilinear transformation method 372

6.4 Frequency transformation in the discrete-time domain 378

6.4.1 Lowpass-to-lowpass ffansformation 379

6.4.2 Lowpass-to-highpass transformation 380

6.4.3 Lowpass-to-bandpass transformation 380

6.4.4 Lowpass-to-bandstop transformation 381

6.4.5 Variable-cutoff filter design 381

6.5 Magnitude and phase approximation 382

6.5.1 Basic principles 382

6.5.2 Multivariable function minimization method 387

6.5.3 Alternative methods 389

6.6 Time-domain approximation 391

6.6.1 Approximate approach 393

6.7 Do-it-yourself:IIR filter approximations 394

6.8 IIR filter approximation with MATLAB 399

6.9 Summary 403

6.10 Exercises 404

7 Spectral estimation 409

7.1 Introduction 409

7.2 Estimation theory 410

7.3 Nonparametric spectral estimation 411

7.3.1 Periodogram 411

7.3.2 Periodogrum variations 413

7.3.3 Minimum-variance spectral estimator 416

7.4 Modeling theory 419

7.4.1 Rational transfer-function models 419

7.4.2 Yule-Walker equations 423

7.5 Parametric spectral estimation 426

7.5.1 Linear prediction 426

7.5.2 Covariance method 430

7.5.3 Autocorrelation method 431

7.5.4 Levinson-Durbin algorithm 432

7.5.5 Burg's method 434

7.5.6 Relationship of the Levinson-Durbin algorithm to a lattice structure 438

7.6 Wiener filter 438

7.7 Other methods for spectral estimation 441

7.8 Do-it-yourself:spectral estimation 442

7.9 Spectral estimation with MATLAB 449

7.10 Summary 450

7.11 Exercises 451

8 Multirate systems 455

8.1 Introduction 455

8.2 Basic principles 455

8.3 Decimation 456

8.4 Interpolation 462

8.4.1 Examples of interpolators 464

8.5 Rational sampling-rate changes 465

8.6 Inverse operations 466

8.7 Noble identities 467

8.8 Polyphase decompositions 469

8.9 Commutator models 471

8.10 Decimation and interpolation for efficient filter implementation 474

8.10.1 Narrowband FIR filters 474

8.10.2 Wideband FIR filters with narrow transition bands 477

8.11 Overlapped block filtering 479

8.11.1 Nonoverlapped case 480

8.11.2 Overlapped input and output 483

8.11.3 Fast convolution structure Ⅰ 487

8.11.4 Fast convolution structure Ⅱ 487

8.12 Random signals in multirate systems 490

8.12.1 Interpolated random signals 491

8.12.2 Decimated random signals 492

8.13 Do-it-yourself:multirate systems 493

8.14 Multirate systems with MATLAB 495

8.15 Summary 497

8.16 Exercises 498

9 Filter banks 503

9.1 Introduction 503

9.2 Filter banks 503

9.2.1 Decimation of a bandpass signal 504

9.2.2 Inverse decimation of a bandpass signal 505

9.2.3 Critically decimated M-band filter banks 506

9.3 Perfect reconstruction 507

9.3.1 M-band filter banks in terms of polyphase components 507

9.3.2 Perfect reconstruction M-band filter banks 509

9.4 Analysis of M-band filter banks 517

9.4.1 Modulation matrix representation 518

9.4.2 Time-domain analysis 520

9.4.3 Orthogonality and biorthogonality in filter banks 529

9.4.4 Transmultiplexers 534

9.5 General two-band perfect reconstruction filter banks 535

9.6 QMF filterbanks 540

9.7 CQF filterbanks 543

9.8 Block transforns 548

9.9 Cosine-modulated filter banks 554

9.9.1 The optimization problem in the design of cosine-modulated filter banks 559

9.10 Lapped transforms 563

9.10.1 Fast algorithms and biorthogonal LOT 573

9.10.2 Generalized LOT 576

9.11 Do-it-yourself:filter banks 581

9.12 Filter banks with MATLAB 594

9.13 Summary 594

9.14 Exercises 595

10 Wavelet transforms 599

10.1 Introduction 599

10.2 Wavelet transforms 599

10.2.1 Hierarchical filter banks 599

10.2.2 Wavelets 601

10.2.3 Scaling functions 605

10.3 Relation between x(t)and x(n) 606

10.4 Wavelet transforms and time-frequency analysis 607

10.4.1 The short-time Fourier transform 607

10.4.2 The continuous-time wavelet transform 612

10.4.3 Sampling the continuous-time wavelet transform:the discrete wavelet transform 614

10.5 Multiresolution representation 617

10.5.1 Biorthogonal multiresolution representation 620

10.6 Wavelet transforms and filter banks 623

10.6.1 Relations between the filter coefficients 629

10.7 Regularity 633

10.7.1 Additional constraints imposed on the filter banks due to the regularity condition 634

10.7.2 Apractical estimate of regularity 635

10.7.3 Number of vanishing moments 636

10.8 Examples of wavelets 638

10.9 Wavelet transforms of images 641

10.10 Wavelet transforms of finite-length signals 646

10.10.1 Periodic signal extension 646

10.10.2 Symmetric signal extensions 648

10.11 Do-it-yourself:wavelet transforms 653

10.12 Wavelets with MATLAB 659

10.13 Summary 664

10.14 Exercises 665

11 Finite-precision digital signal processing 668

11.1 Introduction 668

11.2 Binary number representation 670

11.2.1 Fixed-point representations 670

11.2.2 Signed power-of-two representation 672

11.2.3 Floating-point representation 673

11.3 Basic elements 674

11.3.1 Properties of the two's-complement representation 674

11.3.2 Serial adder 674

11.3.3 Serial multiplier 676

11.3.4 Parallel adder 684

11.3.5 Parallel multiplier 684

11.4 Distributed arithmetic implementation 685

11.5 Product quantization 691

11.6 Signal scaling 697

11.7 Coefficient quantization 706

11.7.1 Deterministic sensitivity criterion 708

11.7.2 Statistical forecast of the wordlength 711

11.8 Limit cycles 715

11.8.1 Granular limit cycles 715

11.8.2 Overflow limit cycles 717

11.8.3 Elimination of zero-input limit cycles 719

11.8.4 Elimination of constant-input limit cycles 725

11.8.5 Forced-response stability ofdigital filters with nonlinearities due to overflow 729

11.9 Do-it-yourself:finite-precision digital signal processing 732

11.10 Finite-precision digital signal processing with MATLAB 735

11.11 Summary 735

11.12 Exercises 736

12 Efficient FIR structures 740

12.1 Introduction 740

12.2 Latticeform 740

12.2.1 Filter banks using the lattice form 742

12.3 Polyphase form 749

12.4 Frequency-domain form 750

12.5 Recursive running sum form 750

12.6 Modified-sinc flter 752

12.7 Realizations with reduced number of arithmetic operations 753

12.7.1 Prefilter approach 753

12.7.2 Interpolation approach 756

12.7.3 Frequency-response masking approach 760

12.7.4 Quadrature approach 771

12.8 Do-it-yourself:efficient FIR structures 776

12.9 Efficient FIR structures with MATLAB 781

12.10 Summary 782

12.11 Exercises 782

13 Efficient IIR structures 787

13.1 Introduction 787

13.2 IIR parallel and cascade filters 787

13.2.1 Parallel form 788

13.2.2 Cascade form 790

13.2.3 Error spectrum shaping 795

13.2.4 Closed-form scaling 797

13.3 State-space sections 800

13.3.1 Optimal state-space sections 801

13.3.2 State-space sections without limit cycles 806

13.4 Lattice filters 815

13.5 Doubly complementary filters 822

13.5.1 QMF filter bank implementation 826

13.6 Wave filters 828

13.6.1 Motivation 829

13.6.2 Wave elements 832

13.6.3 Lattice wave digital filters 848

13.7 Do-it-yourself:efficient IIR structures 855

13.8 Efficient IIR structures with MATLAB 857

13.9 Summary 857

13.10 Exercises 858

References 863

Index 877