Preface, x
Acknowledgments, xii
List of abbreviations, xiii
List of principal symbols, xvi
Introduction, xxi
1 Signal representation and system response, 1
1.1 Introduction, 1
1.2 Signal classification, 3
1.2.1 Energy signals, 3
1.2.2 Power signals, 4
1.3 Fourier series, 5
1.4 The Fourier transform, 13
1.5 Laplace transform, 17
1.6 Transform analysis of linear systems, 21
1.6.1 Superposition, 21
1.6.2 Linear ordinary differential equations, 22
1.6.3 Response of linear system to a periodic input, 23
1.6.4 General approach, 25
1.7 Transfer function, 25
1.7.1 Method (i), 28
1.7.2 Method (ii), 28
1.8 Summary, 30
1.9 Problems, 30
2 Time domain description and convolution, 33
2.1 Introduction, 33
2.2 The impulse response, 33
2.2.1 The impulse, 34
2.2.2 Signal representation, 35
2.2.3 System response to an impulse, 36
2.3 Convolution, 39
2.4 Properties of convolution, 46
2.4.1 Time delay, 47
2.5 Summary, 52
2.6 Problems, 54
3 Transfer function and system characterisation, 56
3.1 Introduction, 56
3.2 Transfer function, poles and zeros, 56
3.3 Transfer function and frequency response, 58
3.3.1 Frequency response from pole/zero diagram, 59
3.3.2 Fourier transform of periodic signals, 61
3.3.3 Measurement of frequency response, 64
3.3.4 Bode plots, 65
3.3.5 Fourier and Laplace, 69
3.4 Transfer function and impulse response, 71
3.5 Time domain response of first and second order systems, 73
3.5.1 First order systems, 74
3.5.2 Second order systems, 75
3.6 Rise time and bandwidth, 78
3.7 Summary, 81
3.8 Problems, 82
4 Sampled data systems and the z-transform, 85
4.1 Introduction, 85
4.2 Sampled data systems and aliasing, 88
4.2.1 Sampling theorem - the Nyquist criterion, 94
4.2.2 Practical sampled data systems, 95
4.3 The z-transform,98
4.3.1 The inverse z-transform, 103
4.3.2 Delay theorem, 105
4.4 Digital filters and discrete convolution, 106
4.4.1 Discrete convolution, 109
4.5 Poles and stability, 112
4.6 Frequency response of a digital filter, 113
4.7 Example of a complete system, 116
4.8 Summary, 122
4.9 Problems, 123
5 Infinite impulse response digital filters, 126
5.1 Introduction, 126
5.2 Analogue prototype filters, 127
5.2.1 Introduction, 127
5.2.2 Butterworth prototype polynomials, 128
5.2.3 Chebyshev prototype polynomials, 130
5.3 Digital filter structure, 132
5.3.1 Introduction, 132
5.3.2 The canonical form, 133
5.3.3 Parallel and cascade realisations, 136
5.4 Filter design methods, 137
5.4.1 Introduction, 137
5.4.2 The bilinear z-transform, 138
5.4.3 Filter transformation, 143
5.5 Finite precision effects, 145
5.5.1 Filter coefficient quantisation errors, 145
5.5.2 Limit cycles, 147
5.5.3 IIR filter hardware, 147
5.6 Summary, 148
5.7 Problems, 148
6 Finite impulse response digital filters, 150
6.1 Introduction, 150
6.2 Finite theory and frequency response, 150
6.2.1 Transfer function, 150
6.2.2 Frequency response, 151
6.2.3 Phase response, 152
6.3 Linear phase filters, 152
6.3.1 Principles, 152
6.3.2 Linear and nonlinear phase filters, 155
6.4 Linear phase filter design, 156
6.4.1 Fourier series method, 156
6.4.2 Window effects, 159
6.4.3 Design summary, 163
6.4.4 Design optimisation techniques, 165
6.5 Finite precision effects, 168
6.5.1 Noise reduction through the filter, 168
6.5.2 Filter coefficient quantisation errors, 169
6.5.3 FIR filter hardware, 172
6.6 FIR filter applications, 172
6.6.1 Matched filter detector, 172
6.6.2 Matched filter applications, 173
6.6.3 Other receiver designs, 174
6.7 Summary, 174
6.8 Problems, 175
7 Random signal analysis, 176
7.1 Introduction, 176
7.2 Random process, 176
7.3 Averages and spectral representations, 181
7.3.1 Autocorrelation and autocovariance, 183
7.3.2 Correlation and dependence, 185
7.3.3 Power spectral density, 185
7.3.4 Alternative representations of a random process, 187
7.4 Random signal and discrete linear systems, 188
7.4.1 Cross-correlation between the input and output of a filter, 191
7.5 Spectral factorisation, inverse and whitening filters, 192
7.5.1 Inverse filters, 193
7.5.2 Noise whitening, 195
7.5.3 Cross-correlation between two filter outputs, 196
7.6 Filter noise calculations, 197
7.6.1 Quantisation noise, 199
7.6.2 Dynamic range, 201
7.7 Summary, 203
7.8 Problems, 204
8 Adaptive filters, 206
8.1 Introduction, 206
8.2 Wiener filters, 207
8.2.1 Wiener FIR filter, 208
8.2.2 Application to channel estimation, 211
8.3 Algorithms for adaptive filtering, 215
8.3.1 Recursive least squares, 217
8.3.2 Stochastic gradient methods, 219
8.3.3 A comparison of algorithms, 227
8.4 Applications, 230
8.4.1 Adaptive line enhancement, 231
8.4.2 Adaptive tone suppression, 233
8.4.3 Noise whitening, 233
8.4.4 Echo cancellaion, 234
8.4.5 Channel equalisation, 235
8.5 Summary, 238
8.6 Problems, 238
9 The Fourier transform and spectral analysis, 240
9.1 Development of the discrete Fourier transform, 240
9.1.1 The continuous Fourier transform, 240
9.1.2 Fourier tranform of a finite length data record, 241
9.1.3 Definition of the DFT, 243
9.1.4 Properties of the DFT, 245
9.2 Computation of the discrete Fourier transform, 246
9.2.1 DFT matrix coefficient values, 246
9.2.2 Matrix formulation of the DFT, 246
9.2.3 Analogies for the DFT, 251
9.3 Resolution and window responses, 253
9.3.1 Resolution, 253
9.3.2 Leakage effects, 253
9.3.3 The rectangular window, 254
9.3.4 The Hanning window, 257
9.3.5 The Hamming window, 259
9.3.6 The Dolph-Chebychev window, 259
9.3.7 Window comparisons, 261
9.4 Fundamentals of spectral analysis, 262
9.5 Classical spectral analysis, 264
9.6 Modern spectral analysis, 267
9.6.1 Introduction to parametric techniques, 267
9.6.2 Autoregressive spectrum analysis, 268
9.7 Comparison of spectral analysis techniques, 272
9.8 Application of AR techniques in speech coders, 273
9.9 Summary, 276
9.10 Problems, 276
10 The fast Fourier transform, 278
10.1 Introduction, 278
10.2 Partitioning of the DFT into two half-size matrices, 278
10.3 Radix-2 FFT, 284
10.3.1 Decimation-in-time algorithm, 284
10.3.2 Decimation-in-frequency algorithm, 288
10.4 Implementation considerations, 288
10.4.1 Complex multiplier hardware, 288
10.4.2 Alternative radix arithmetic approaches, 289
10.4.3 Real valued data, 290
10.4.4 Inverse transforms, 292
10.5 Applications, 295
10.6 Summary, 297
10.7 Problems, 298
11 Multirate signal processing, 299
11.1 Introduction, 299
11.2 Decimation, interpolation, imaging and aliasing, 300
11.2.1 Decimation, 303
11.2.2 Bandpass sampling, 306
11.2.3 Interpolation, 310
11.3 Applications of multirate systems, 311
11.3.1 Transmultiplexers, 311
11.3.2 Analysis and synthesis filterbanks, 311
11.3.3 Filterbank design approaches, 314
11.4 Audio and speech processing, 319
11.4.1 Speech and image signal coding, 321
11.5 Summary, 325
11.6 Problems, 325
Solutions to self-assessment questions, 327
Bibliography, 343
Appendix A Matrix theory revision, 349
Appendix B Signal transforms, 352
Index, 354