Multirate Digital Signal Processing

Contents......Page 7
Preface......Page 15
Acknowledgments......Page 19
1.0 Basic Considerations......Page 21
1.1 Sampling Rate Conversion......Page 23
1.2.1 Sampling Rate Conversion in Digital Audio Systems......Page 24
1.2.3 Digital Time Division MUltiplexing (TDM) to Frequency Division Multiplexing (FDM) Translation......Page 25
1.2.4 Sub-Band Coding of Speech Signals......Page 26
1.2.5 Short-Time Spectral Analysis- and Synthesis......Page 28
1.3 Scope of the Book......Page 29
References......Page 30
2.1.1 Uniform Sampling Viewed as a Modulation Process......Page 33
2.1.2 Spectral Interpretations of Sampling......Page 35
2.1.3 The Sampling Theorem......Page 38
2.1.4 Reconstruction of an Analog Signal from Its Samples......Page 40
2.1.5 Summary of the Implications of the Sampling Theorem......Page 41
2.2 Sampling Rate Conversion - An Analog Interpretation......Page 42
2.3.1 Relationship to Time-Varying Systems......Page 49
2.3.2 Sampling Rate Reduction - Decimation by an Integer Factor M......Page 51
2.3.3 Sampling Rate Increase - Interpolation by an Integer Factor L......Page 55
2.3.4 Sampling Rate Conversion by a Rational Factor M /L......Page 59
2.4.1 The Sampling Theorem Applied to Bandpass Signals......Page 62
2.4.2 Integer-Band Decimation and Interpolation......Page 63
2.4.3 Quadrature Modulation of Bandpass Signals......Page 68
2.4.4 Single-Sideband Modulation......Page 72
2.4.5 Discussion......Page 76
References......Page 77
3.0 Introduction......Page 79
3.1 Signal-Flow Graph Representation of Digital Systems......Page 80
3.1.1 Signal-Flow Graphs: Basic Principles......Page 81
3.1.2 Commutation of Branch Operations and Circuit Identities......Page 83
3.1.3 Transposition and Duality for Multirate Systems......Page 88
3.2 A Review of Structures for Linear Time-Invariant Filters......Page 90
3.2.1 FIR Direct-Form Structure......Page 91
3.2.3 Structures for IIR Digital Filters......Page 92
3.3.1 Direct and Transposed Direct-Form FIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate......Page 96
3.3.2 Polyphase FIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate......Page 99
3.3.3 Polyphase Structures Based on Clockwise Commutator Models......Page 106
3.3.4 FIR Structures with Time-Varying Coefficients for Interpolation/Decimation by a Factor of LIM......Page 108
3.4 IIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate......Page 111
3.4.1 Polyphase Structures for IIR Decimators and Interpolators......Page 113
3.4.2 Direct-Form Structures and Structures with Time-Varying Coefficients for IIR Decimators and Interpolators......Page 118
3.4.3 Comparison of Structures for Decimation and Interpolation......Page 119
3.5 Advanced Network Concepts of Linear Multirate and Time-Varying Structures......Page 120
3.5.1 System Representation of Linear Time-Varying and Multirate Networks......Page 121
3.5.2 Cascading Networks and Commutation of Network Elements......Page 128
3.5.3 Network Duality......Page 132
3.5.4 Network Transposition and Tellegen's Theorem......Page 136
3.5.5 Transposition of Complex Networks......Page 141
References......Page 144
4.0 Introduction......Page 147
4.1.1 Basic Considerations and Properties......Page 148
4.1.2 Advantages and Disadvantages of FIR and IIR Filters for Interpolation and Decimation......Page 150
4.2.1 The Prototype Filter and Its Polyphase Representation......Page 152
4.2.2 Ideal Frequency Domain Characteristics for Interpolation Filters......Page 156
4.2.3 Ideal Frequency Domain Characteristics for Decimation Filters......Page 159
4.2.4 Time Domain Properties of Ideal Interpolation Filters......Page 160
4.2.5 Time Domain Properties of Ideal Decimation Filters......Page 162
4.3.1 FIR Filters Based on Window Designs......Page 163
4.3.2 Equiripple (Optimal) FIR Designs......Page 166
4.3.3 The Effects of the f/J Bands for Equiripple Designs......Page 170
4.3.4 Equiripple FIR Filters for Interpolation with Time Domain Constraints......Page 174
4.3.5 Half-Band FIR Filters - A Special Case of FIR Designs for Conversion by Factors of 2......Page 175
4.3.6 Minimum Mean-Square-Error Design of FIR Interpolators - Deterministic Signals......Page 177
4.3.7 Solution of the Matrix Equation......Page 183
4.3.8 Properties of the Minimum Mean-Square-Error Interpolators......Page 185
4.3.9 Design of FIR Interpolators with Minimax Error in the Frequency Domain......Page 187
4.3.10 Design of FIR Interpolators with Minimax Error in the Time Domain......Page 192
4.3.11 Linear Interpolation......Page 195
4.3.12 Lagrange Interpolators......Page 197
4.3.13 Discussion......Page 200
4.4.1 Ideal Characteristics and Practical Realizations of IIR Decimators and Interpolators......Page 201
4.4.2 Conventional IIR Filter Designs......Page 203
4.4.3 Special IIR Designs Based on the Transformation of Conventional Designs......Page 205
4.4.4 A Direct Design Procedure for Equiripple IIR Filters for Decimation and Interpolation......Page 206
4.5 Comparisons of IIR and FIR Designs of Interpolators and Decimators......Page 208
References......Page 210
5.0 Introduction......Page 213
5.1 Computational Efficiency of a 2-Stage Structure - A Design Example......Page 216
5.2.1 Overall Filter Requirements......Page 219
5.2.2 Lowpass Filter Requirements for Individual Stages......Page 222
5.2.3 Filter Specifications for Individual Stages which Include "Don't-Care" Bands......Page 224
5.2.4 Passband and Stopband Tolerance Requirements......Page 225
5.2.5 Design Considerations......Page 226
5.3 Multistage FIR Designs Based on an Optimization Procedure......Page 227
5.3.1 Analytic Expressions for the Required Filter Order for Each Stage of a Multistage Design......Page 228
5.3.2 Design Criteria Based on Multiplication Rate......Page 229
5.3.3 Design Criteria Based on Storage Requirements......Page 230
5.3.4 Design Curves Based on Computer-Aided Optimization......Page 231
5.3.5 Application of the Design Curves and Practical Considerations......Page 236
5.4 Multistage Structures Based on Half-Band FIR Filters......Page 238
5.4.1 Half-Band Designs with No Aliasing in the Final Transition Band......Page 240
5.4.2 Half-Band Designs for Power-of-2 Conversion Ratios and Aliasing in the Final Transition Band......Page 242
5.5.1 Comb Filter Characteristics......Page 247
5.5.2 A Design Procedure Using a Specific Class of Filters......Page 251
5.6 Multistage Decimators and Interpolators Based on IIR Filter Designs......Page 255
5.7 Considerations in the Implementation of Multistage Decimators and Interpolators......Page 264
References......Page 269
6.0 Introduction......Page 271
6.1 Multira~ Implementation of Lowpass Filters......Page 272
6.1.1 Design Characteristics of the Lowpass Filters......Page 276
6.1.2 Multistage Implementations of the Lowpass Filter Structure......Page 278
6.1.3 Some Comments on the Resulting Lowpass Filters......Page 280
6.1.4 Design Example Comparing Direct and Multistage Implementations of a Lowpass Filter......Page 281
6.2 Multirate Implementation of a Bandpass Filter......Page 283
6.2.1 Pseudo Integer-Band, Multirate Bandpass Filter Implementations......Page 285
6.2.2 Alternative Multirate Implementations of Bandpass Filters......Page 287
6.2.3 Multirate Implementation of Narrow-Band Highpass and Bandstop Filters......Page 290
6.3 Design of Fractional Sample Phase Shifters Based on Multirate Concepts......Page 291
6.3.1 Design of Phase Shifter Networks with Fixed Phase Offsets......Page 294
6.4 Multirate Implementation of a Hilbert Transformer......Page 295
6.5 Narrow-Band, High-Resolution Spectral Analysis Using Multirate Techniques......Page 300
6.6 Sampling Rate Conversion Between Systems with Incommensurate Sampling Rates......Page 303
6.7 Summary......Page 306
References......Page 307
7.0 Introduction......Page 309
7.1 General Issues and Definitions......Page 310
7.2 Uniform DFT Filter Banks and Short-Time Fourier Analyzers and Synthesizers......Page 316
7.2.1 Filter Bank Interpretation Based on the Complex Modulator......Page 317
7.2.2 Complex Bandpass Filter Interpretation......Page 320
7.2.3 Polyphase Structures for Efficient Realization of Critically SamPied DFT Filter Banks......Page 323
7.2.4 Polyphase Filter Bank Structures for K-MI......Page 331
7.2.5 Weighted Overlap-Add Structures for Efficient Realization of DFT Filter Banks......Page 333
7.2.6 A Simplified Weighted Overlap-Add Structure for Windows Shorter than the Transform Size......Page 344
7.2.7 Comparison of the Polyphase Structure and the Weighted Overlap-Add Structure......Page 345
7.3 Filter Design Criteria for DFT Filter Banks......Page 346
7.3.1 Aliasing and Imaging in the Frequency Domain......Page 347
7.3.2 Filter-Bank-Design by Frequency Domain Specification The Filter-Bank-Sum Method......Page 352
7.3.3 Aliasing and Imaging in the Time Domain......Page 355
7.3.4 Filter Bank Design by Time Domain Specification The Overlap-Add Method......Page 359
7.3.5 Relationship of Time and Frequency Domain Specifications......Page 361
7.4.1 The General Model for Multiplicative Modifications......Page 366
7.4.2 Modifications in the Filter-Bank-Sum Method......Page 371
7.4.4 Time-Invariant Modifications and Methods of Fast Convolution......Page 372
7.4.5 Other Forms of Filter Bank Modifications and Systems......Page 375
7.5.1 The Generalized DFT (GDFT)......Page 376
7.5.2 The GDFT Filter Bank......Page 378
7.5.3 Polyphase Structure for the GDFT Filter Bank......Page 380
7.5.4 Weighted Overlap-Add Structure for the GDFT Filter Bank......Page 382
7.5.5 Filter Design Criteria for the GDFT Filter Bank......Page 385
7.6 Uniform Single-Sideband (SSB) Filter Banks......Page 386
7.6.1 Realization of SSB Filter Banks from Quadrature Modulation Designs......Page 387
7.6.2 Critically Sampled SSB Filter Banks with ko = 1/4......Page 391
7.6.3 SSB Filter Banks Based on ko = 1/2 Designs......Page 393
7.7 Filter Banks Based on Cascaded Realizations and Tree Structures......Page 396
7.7.1 Quadrature Mirror Filter (QMF) Bank Design......Page 398
7.7.2 Finite Impulse Response (FIR) Designs for QMF Filter Banks......Page 402
7.7.3 PolYllhase Realization of Quadrature Mirror Filter Banks......Page 407
7.7.4 Equivalent Parallel Realizations of Cascaded Tree Structures......Page 412
7.8 Summary......Page 415
References......Page 416
Appendix 7.1......Page 421
INDEX......Page 425