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1 Sampling and Reconstruction1

1.1 Introduction1

1.2 Review of Analog Signals1

1.3 Sampling Theorem4

1.3.1 Sampling Theorem6

1.3.2 Antialiasing Prefilters7

1.3.3 Hardware Limits8

1.4 Sampling of Sinusoids9

1.4.1 Analog Reconstruction and Aliasing10

1.4.2 Rotational Motion27

1.4.3 DSP Frequency Units30

1.5 Spectra of Sampled Signals30

1.5.1 Discrete-Time Fourier Transform31

1.5.2 Spectrum Replication33

1.5.3 Practical Antialiasing Prefilters38

1.6 Analog Reconstructors43

1.6.1 Ideal Reconstructors44

1.6.2 Staircase Reconstructors46

1.6.3 Anti-Image Postfilters47

1.7 Basic Components of DSP Systems54

1.8 Problems57

2 Quantization63

2.1 Quantization Process63

2.2 Oversampling and Noise Shaping67

2.3 D/A Converters73

2.4 A/D Converters77

2.5 Analog and Digital Dither86

2.6 Problems93

3 Discrete-Time Systems98

3.1 Input/Output Rules99

3.2 Linearity and Time Invariance103

3.3 Impulse Response106

3.4 FIR and IIR Filters108

3.5 Causality and Stability115

3.6 Problems120

4 FIR Filtering and Convolution124

4.1 Block Processing Methods125

4.1.1 Convolution125

4.1.2 Direct Form126

4.1.3 Convolution Table129

4.1.4 LTI Form130

4.1.5 Matrix Form132

4.1.6 Flip-and-Slide Form134

4.1.7 Transient and Steady-State Behavior135

4.1.8 Convolution of Infinite Sequences137

4.1.9 Programming Considerations142

4.1.10 Overlap-Add Block Convolution Method146

4.2 Sample Processing Methods149

4.2.1 Pure Delays150

4.2.2 FIR Filtering in Direct Form155

4.2.3 Programming Considerations163

4.2.4 Hardware Realizations and Circular Buffers165

4.3 Problems181

5 z-Transforms186

5.1 Basic Properties186

5.2 Region of Convergence189

5.3 Causality and Stability196

5.4 Frequency Spectrum200

5.5 Inverse z-Transforms205

5.6 Problems214

6 Transfer Functions217

6.1 Equivalent Descriptions of Digital Filters217

6.2 Transfer Functions217

6.3 Sinusoidal Response232

6.3.1 Steady-State Response232

6.3.2 Transient Response235

6.4 Pole/Zero Designs246

6.4.1 First-Order Filters246

6.4.2 Parametric Resonators and Equalizers248

6.4.3 Notch and Comb Filters253

6.5 Deconvolution Inverse Filters＊and Stability258

6.6 Problems263

7 Digital Filter Realizations269

7.1 Direct Form269

7.2 Canonical Form275

7.3 Cascade Form281

7.4 Cascade to Canonical288

7.5 Hardware Realizations and Circular Buffers297

7.6 Quantization Effects in Digital Filters310

7.7 Problems311

8 Signal Processing Applications321

8.1 Digital Waveform Generators321

8.1.1 Sinusoidal Generators321

8.1.2 Periodic Waveform Generators326

8.1.3 Wavetable Generators335

8.2 Digital Audio Effects355

8.2.1 Delays Echoes and Comb Filters355

8.2.2 Flanging Chorusing＊and Phasing360

8.2.3 Digital Reverberation367

8.2.4 Multitap Delays379

8.2.5 Compressors Limiters Expanders and Gates384

8.3 Noise Reduction and Signal Enhancement388

8.3.1 Noise Reduction Filters388

8.3.2 Notch and Comb Filters404

8.3.3 Line and Frame Combs for Digital TV416

8.3.4 Signal Averaging429

8.3.5 Savitzky-Golay Smoothing Filters434

8.4 Problems462

9 DFT/FFT Algorithms472

9.1 Frequency Resolution and Windowing472

9.2 DTFT Computation483

9.2.1 DTFT at a Single Frequency483

9.2.2 DTFT over Frequency Range486

9.2.3 DFT488

9.2.4 Zero Padding490

9.3 Physical versus Computational Resolution491

9.4 Matrix Form of DFT495

9.5 Modulo-N Reduction497

9.6 Inverse DFT505

9.7 Sampling of Periodic Signals and the DFT508

9.8 FFT513

9.9 Fast Convolution524

9.9.1 Circular Convolution524

9.9.2 Overlap-Add and Overlap-Save Methods530

9.10 Problems533

10 FIR Digital Filter Design541

10.1 Window Method541

10.1.1 Ideal Filters541

10.1.2 Rectangular Window544

10.1.3 Hamming Window549

10.2 Kaiser Window551

10.2.1 Kaiser Window for Filter Design551

10.2.2 Kaiser Window for Spectral Analysis565

10.3 Frequency Sampling Method567

10.4 Other FIR Design Methods568

10.5 Problems569

11 IIR Digital Filter Design573

11.1 Bilinear Transformation573

11.2 First-Order Lowpass and Highpass Filters576

11.3 Second-Order Peaking and Notching Filters583

11.4 Parametric Equalizer Filters592

11.5 Comb Filters601

11.6 Higher-Order Filters604

11.6.1 Analog Lowpass Butterworth Filters605

11.6.2 Digital Lowpass Filters611

11.6.3 Digital Highpass Filters614

11.6.4 Digital Bandpass Filters618

11.6.5 Digital Bandstop Filters623

11.6.6 Chebyshev Filter Design626

11.7 Problems640

12 Interpolation Decimation and Oversampling644

12.1 Interpolation and Oversampling644

12.2 Interpolation Filter Design650

12.2.1 Direct Form650

12.2.2 Polyphase Form652

12.2.3 Frequency Domain Characteristics657

12.2.4 Kaiser Window Designs660

12.2.5 Multistage Designs661

12.3 Linear and Hold Interpolators669

12.4 Design Examples674

12.4.1 4-fold Interpolators674

12.4.2 Multistage 4-fold Interpolators678

12.4.3 DAC Equalization683

12.4.4 Postfilter Design and Equalization687

12.4.5 Multistage Equalization691

12.5 Decimation and Oversampling699

12.6 sampling Rate Converters704

12.7 Noise Shaping Quantizers712

12.8 Problems720

13 Appendices728

A Random signals728

A.1 Autocorrelation Functions and Power Spectra728

A.2 Filtering of Random Signals732

B Random Number Generators734

B.1 Uniform and Gaussian Generators734

B.2 Low-Frequency Noise Generators740

B.3 I/f Noise Generators745

B.4 Problems749

C Complex Arithmetic in C752

D MATLAB Functions755

References773

Index790