Foreword (Historical Perspective) xi
Preface xiii
Introduction xv
1 Signal Digitizing - Sampling and Coding 1
1.1 Fourier Analysis 1
1.2 Distributions 4
1.3 Some Commonly Studied Signals 6
1.4 The Norms of a Function 12
1.5 Sampling 13
1.6 Frequency Sampling 14
1.7 The Sampling Theorem 15
1.8 Sampling of Sinusoidal and Random Signals 16
1.9 Quantization 20
1.10 The Coding Dynamic Range 22
1.11 Nonlinear Coding with the 13-segment A-law 24
1.12 Optimal Coding 26
1.13 Quantity of Information and Channel Capacity 28
1.14 Binary Representations 29
2 The Discrete Fourier Transform 35
2.1 Definition and Properties of the Discrete Fourier Transform 36
2.2 Fast Fourier Transform (FFT) 38
2.3 Degradation Arising fromWordlength Limitation Effects 45
2.4 Calculation of a Spectrum Using the DFT 46
2.5 Fast Convolution 50
2.6 Calculations of a DFT Using Convolution 51
2.7 Implementation 52
3 Other Fast Algorithms for the FFT 55
3.1 Kronecker Product of Matrices 55
3.2 Factorizing the Matrix of a Decimation-in-Frequency Algorithm 56
3.3 Partial Transforms 58
3.4 Lapped Transform 66
3.5 Other Fast Algorithms 67
3.6 Binary Fourier Transform - Hadamard 71
3.7 Number-Theoretic Transforms 71
4 Time-Invariant Discrete Linear Systems 77
4.1 Definition and Properties 77
4.2 The Z-Transform 78
4.3 Energy and Power of Discrete Signals 80
4.4 Filtering of Random Signals 82
4.5 Systems Defined by Difference Equations 83
4.6 State Variable Analysis 85
5 Finite Impulse Response (FIR) Filters 89
5.1 FIR Filters 89
5.2 Practical Transfer Functions and Linear Phase Filters 91
5.3 Calculation of Coefficients by Fourier Series Expansion for Frequency Specifications 94
5.4 Calculation of Coefficients by the Least-Squares Method 97
5.5 Calculation of Coefficient by Discrete Fourier Transform 99
5.6 Calculation of Coefficients by Chebyshev Approximation 100
5.7 Relationships Between the Number of Coefficients and the Filter Characteristic 102
5.8 Raised-Cosine Transition Filter 104
5.9 Structures for Implementing FIR Filters 106
5.10 Limitation of the Number of Bits for Coefficients 107
5.11 Z-Transfer Function of an FIR Filter 109
5.12 Minimum-Phase Filters 111
5.13 Design of Filters with a Large Number of Coefficients 113
5.14 Two-Dimensional FIR Filters 114
5.15 Coefficients of Two-Dimensional FIR Filters by the Least-Squares Method 118
6 Infinite Impulse Response (IIR) Filter Sections 123
6.1 First-Order Section 123
6.2 Purely Recursive Second-Order Section 127
6.3 General Second-Order Section 134
6.4 Structures for Implementation 138
6.5 CoefficientWordlength Limitation 140
6.6 Internal DataWordlength Limitation 141
6.7 Stability and Limit Cycles 142
7 Infinite Impulse Response Filters 147
7.1 General Expressions for the Properties of IIR Filters 147
7.2 Direct Calculations of the Coefficients Using Model Functions 148
8 Digital Ladder Filters 173
8.1 Properties of Two-Port Circuits 173
8.2 Simulated Ladder Filters 176
8.3 Switched-Capacitor Filters 180
8.4 Lattice Filters 183
8.5 Comparison Elements 187
9 Complex Signals - Quadrature Filters - Interpolators 189
9.1 The Fourier Transform of a Real and Causal Set 189
9.2 Analytic Signals 192
9.3 Calculating the Coefficients of an FIR Quadrature Filter 195
9.4 Recursive 90° Phase Shifters 197
9.5 Single Side-Band Modulation 199
9.6 Minimum-Phase Filters 200
9.7 Differentiator 201
9.8 Interpolation Using FIR Filters 202
9.9 Lagrange Interpolation 203
9.10 Interpolation by Blocks - Splines 204
9.11 Interpolations and Signal Restoration 206
9.12 Conclusion 208
10 Multirate Filtering 213
10.1 Decimation and Z-Transform 213
10.2 Decomposition of a Low-Pass FIR Filter 217
10.3 Half-Band FIR Filters 220
10.4 Decomposition with Half-Band Filters 222
10.5 Digital Filtering by Polyphase Network 224
10.6 Multirate Filtering with IIR Elements 227
10.7 Filter Banks Using Polyphase Networks and DFT 227
10.8 Conclusion 229
11 QMF Filters and Wavelets 233
11.1 Decomposition into Two Sub-Bands and Reconstruction 233
11.2 QMF Filters 233
11.3 Perfect Decomposition and Reconstruction 236
11.4 Wavelets 238
11.5 Lattice Structures 242
12 Filter Banks 245
12.1 Decomposition and Reconstruction 245
12.2 Analyzing the Elements of the Polyphase Network 247
12.3 Determining the Inverse Functions 248
12.4 Banks of Pseudo-QMF Filters 249
12.5 Determining the Coefficients of the Prototype Filter 253
12.6 Realizing a Bank of Real Filters 254
13 Signal Analysis and Modeling 259
13.1 Autocorrelation and Intercorrelation 259
13.2 Correlogram Spectral Analysis 261
13.3 Single-Frequency Estimation 262
13.4 Correlation Matrix 264
13.5 Modeling 266
13.6 Linear Prediction 268
13.7 Predictor Structures 270
13.8 Multiple Sources - MIMO 273
13.9 Conclusion 275
14 Adaptive Filtering 279
14.1 Principle of Adaptive Filtering 279
14.2 Convergence Conditions 282
14.3 Time Constant 284
14.4 Residual Error 285
14.5 Complexity Parameters 286
14.6 Normalized Algorithms and Sign Algorithms 288
14.7 Adaptive FIR Filtering in Cascade Form 289
14.8 Adaptive IIR Filtering 291
14.9 Conclusion 293
15 Neural Networks 297
15.1 Classification 297
15.2 Multilayer Perceptron 299
15.3 The Backpropagation Algorithm 300
15.4 Examples of Application 303
15.5 Convolution Neural Networks 306
15.6 Recurrent/Recursive Neural Networks 307
15.7 Neural Network and Signal Processing 308
15.8 On Activation Functions 309
15.9 Conclusion 310
16 Error-Correcting Codes 313
16.1 Reed-Solomon Codes 313
16.2 Convolutional Codes 319
16.3 Conclusion 331
17 Applications 335
17.1 Frequency Detection 335
17.2 Phase-locked Loop 337
17.3 Differential Coding of Speech 338
17.4 Coding of Sound 339
17.5 Echo Cancelation 340
17.6 Television Image Processing 342
17.7 Multicarrier Transmission - OFDM 344
17.8 Mobile Radiocommunications 347
References 349
Exercises: Solutions and Hints 351
Index 363
