Contents

Preface xi

Chapter 1 Introduction to Filters and Filter Design Software 1

1.1 Filter Selectivity 2

1.1.1 Lowpass Filters 3

1.1.2 Highpass Filters 4

1.1.3 Bandpass Filters 5

1.1.4 Bandstop Filters 5

1.2 Filter Approximation 6

1.3 Filter Implementation 8

1.4 WFilter - Filter Design Software 9

1.5 Conclusion 14

Chapter 2 Analog Filter Approximation Functions 15

2.1 Filter Transfer Functions 15

2.1.1 Transfer Function Characterization 16

2.1.2 Pole-Zero Plots and Transfer Functions 17

2.1.3 Normalized Transfer Functions 18

2.2 Butterworth Normalized Approximation Functions 19

2.2.1 Butterworth Magnitude Response 19

2.2.2 Butterworth Order 20

2.2.3 Butterworth Pole Locations 20

2.2.4 Butterworth Transfer Functions 21

2.3 Chebyshev Normalized Approximation Functions 27

2.3.1 Chebyshev Magnitude Response 27

2.3.2 Chebyshev Order 28

2.3.3 Chebyshev Pole Locations 28

2.3.4 Chebyshev Transfer Functions 29

2.4 Inverse Chebyshev Normalized Approximation Functions 34

viii Practical Analog and Digital Filter Design

2.4.1 Inverse Chebyshev Magnitude Response 34

2.4.2 Inverse Chebyshev Order 35

2.4.3 Inverse Chebyshev Pole-Zero Locations 35

2.4.4 Inverse Chebyshev Transfer Functions 37

2.5 Elliptic Normalized Approximation Functions 43

2.5.1 Elliptic Magnitude Response 43

2.5.2 Elliptic Order 45

2.5.3 Elliptic Pole-Zero Locations 45

2.5.4 Elliptic Transfer Functions 47

2.6 Comparison of Approximation Methods 52

2.7 Conclusion 54

Chapter 3 Analog Lowpass, Highpass, Bandpass, and Bandstop Filters 55

3.1 Unnormalized Lowpass Approximation Functions 55

3.1.1 Handling a First-Order Factor 57

3.1.2 Handling a Second-Order Factor 58

3.2 Unnormalized Highpass Approximation Functions 60

3.2.1 Handling a First-Order Factor 61

3.2.2 Handling a Second-Order Factor 62

3.3 Unnormalized Bandpass Approximation Functions 64

3.3.1 Handling a First-Order Factor 66

3.3.2 Handling a Second-Order Factor 66

3.4 Unnormalized Bandstop Approximation Functions 72

3.4.1 Handling a First-Order Factor 73

3.4.2 Handling a Second-Order Factor 73

3.5 Analog Frequency Response 76

3.5.1 Mathematics for Frequency Response Calculation 76

3.5.2 C Code for Frequency Response Calculation 80

3.6 Saving the Filter Parameters 82

3.7 Conclusion 84

Chapter 4 Analog Filter Implementation Using Active Filters 85

4.1 Implementation Procedures for Analog Filters 85

4.2 Lowpass Active Filters Using Op-amps 87

4.3 Highpass Active Filters Using Op-amps 92

4.4 Bandpass Active Filters Using Op-amps 96

4.5 Bandstop Active Filters Using Op-amps 98

4.6 Implementing Complex Zeros with Active Filters 103

4.7 Analog Filter Implementation Issues 106

4.7.1 Component Selection 106

4.7.2 Sensitivity Analysis 108

4.8 Using WFilter in Active Filter Implementation 111

4.9 Conclusion 113

Contents ix

Chapter 5 Introduction to Discrete-Time Systems 115

5.1 Analog-to-Digital Conversion 115

5.1.1 Frequency Spectrum and Sampling Rate 116

5.1.2 Quantization of Samples 118

5.1.3 A Complete Analog-to-Digital-to-Analog System 119

5.2 Linear Difference Equations and Convolution 120

5.2.1 Linear Difference Equations 121

5.2.2 Impulse Response and Convolution 124

5.3 Discrete-Time Systems and z-Transforms 126

5.4 Frequency Response of Discrete-Time Systems 130

5.5 Playing Digitized Waveforms on a Computer System 137

5.6 Conclusion 139

Chapter 6 Infinite Impulse Response Digital Filter Design 141

6.1 Impulse Response Invariant Design 142

6.2 Step Response Invariant Design 146

6.3 Bilinear Transform Design 151

6.4 C Code for IIR Frequency Response Calculation 158

6.5 Conclusion 160

Chapter 7 Finite Impulse Response Digital Filter Design 161

7.1 Using Fourier Series in Filter Design 161

7.1.1 Frequency Response and Impulse Response

Coefficients 162

7.1.2 Characteristics of FIR Filters 165

7.1.3 Ideal FIR Impulse Response Coefficients 166

7.2 Windowing Techniques to Improve Design 170

7.3 Parks-McClellan Optimization Procedure 177

7.3.1 Description of the Problem 177

7.3.2 The Remez Exchange Algorithm 179

7.3.3 Using the Parks-McClellan Algorithm 180

7.3.4 Limitations of the Parks-McClellan Algorithm 183

7.4 C Code for FIR Frequency Response Calculation 183

7.5 Conclusion 185

Chapter 8 Digital Filter Implementation Using C 187

8.1 Digital Filter Implementation Issues 187

8.1.1 Input and Output Signal Representation 188

8.1.2 Coefficient Representation 190

8.1.3 Retaining Accuracy and Stability 192

8.2 C Code for IIR Filter Implementation 194

8.3 C Code for FIR Filter Implementation 200

8.3.1 Real-Time Implementation of FIR Filters 201

x Practical Analog and Digital Filter Design

8.3.2 Nonreal-Time Implementation of FIR Filters 203

8.4 Filtering Sound Files 205

8.5 Conclusion 207

Chapter 9 Digital Filtering Using the FFT 209

9.1 The Discrete Fourier Transform (DFT) 209

9.2 The Fast Fourier Transform (FFT) 214

9.2.1 The Derivation of the FFT 215

9.2.2 The Inverse FFT 217

9.3 C Code for the FFT 218

9.4 Application of FFT to Filtering 221

9.5 Conclusion 225

Appendix A Technical References 227

Appendix B Filter Design Software and C Code 229

Appendix C Filter Design Using C 231

Appendix D C Code for Normalized Approximation Functions 233

Appendix E C Code for Unnormalized Approximation Functions 239

Appendix F C Code for Active Filter Implementation 247

Appendix G C Code for IIR Filter Design 253

Appendix H C Code for FIR Filter Design 257

Appendix I Filtering Sound Files 259

About the Author 263

Index 265