AT&T MULTIRATE DIGITAL SIGNAL PROCESSING RONALD E. CROCHIERE LAWRENCE R. RABINER Acoustics Research Department Bell Laboratories Murray Hill, New Jersey Prentice-Hall, Inc., Upper Saddle River, New Jersey 07458
Contents PREFACE ACKNOWLEDGMENTS 1 INTRODUCTION 1.0 Basic Considerations 1 1.1 Sampling Rate Conversion 3 1.2 Examples of Multirate Digital Systems 4 1.2.1 Sampling Rate Conversion in Digital Audio Systems 4 1.2.2 Conversion Between Delta Modulation and PCM Signal Coding Formats 5 1.2.3 Digital Time Division Multiplexing (TDM) to Frequency Division Multiplexing (FDM) Translation 5 1.2.4 Sub-Band Coding of Speech Signals 6 1.2.5 Short-Time Spectral Analysis and Synthesis 8 1.3 Scope of the Book 9 References 10 2 BASIC PRINCIPLES OF SAMPLING AND SAMPLING RATE CONVERSION 2.0 Introduction 13
2.1 Uniform Sampling and the Sampling Theorem 13 2.1.1 Uniform Sampling Viewed as a Modulation Process 13 2.1.2 Spectral Interpretations of Sampling 15 2.1.3 The Sampling Theorem 18 2.1.4 Reconstruction of an Analog Signal from Its Samples 20 2.1.5 Summary of the Implications of the Sampling Theorem 21 2.2 Sampling Rate Conversion - An Analog Interpretation 22 2.3 Sampling Rate Conversion - A Direct Digital Approach 29 2.3.1 Relationship to Time-Varying Systems 29 2.3.2 Sampling Rate Reduction - Decimation by an Integer Factor M 31 2.3.3 Sampling Rate Increase - Interpolation by an Integer Factor L 35 2.3.4 Sampling Rate Conversion by a Rational Factor M/L 39 2.4 Decimation and Interpolation of Bandpass Signals 42 2.4.1 The Sampling Theorem Applied to Bandpass Signals 42 2.4.2 Integer-Band Decimation and Interpolation 43 2.4.3 Quadrature Modulation of Bandpass Signals 48 2.4.4 Single-Sideband Modulation 52 2.4.5 Discussion 56 2.5 Summary 57 References 57 STRUCTURES AND NETWORK THEORY FOR MULTIRATE DIGITAL SYSTEMS 3.0 Introduction 59 3.1 Signal-Flow Graph Representation of Digital Systems 60 3.1.1 Signal-Flow Graphs: Basic Principles 61 3.1.2 Commutation of Branch Operations and Circuit Identities 63 3.1.3 Transposition and Duality for Multirate Systems 68 3.2 A Review of Structures for Linear Time-Invariant Filters 70 3.2.1 FIR Direct-Form Structure 71 3.2.2 Transposed Direct-Form Structure for FIR Filters 72 3.2.3 Structures for HR Digital Filters 72 3.3 Structures for FIR Decimators and Interpolators 76 3.3.1 Direct and Transposed Direct-Form FIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate 76 3.3.2 Polyphase FIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate 79
Contents ix 3.3.3 Polyphase Structures Based on Clockwise Commutator Models 86 3.3.4 FIR Structures with Time-Varying Coefficients for Interpolation/Decimation by a Factor of L/M 88 3.4 HR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate 91 3.4.1 Polyphase Structures for HR Decimators and Interpolators 93 3.4.2 Direct-Form Structures and Structures with Time-Varying Coefficients for HR Decimators and Interpolators 98 3.4.3 Comparison of Structures for Decimation and Interpolation 99 3.5 Advanced Network Concepts of Linear Multirate and Time-Varying Structures 100 3.5.1 System Representation of Linear Time-Varying and Multirate Networks 101 3.5.2 Cascading Networks and Commutation of Network Elements 108 3.5.3 Network Duality 112 3.5.4 Network Transposition and Tellegen's Theorem 116 3.5.5 Transposition of Complex Networks 121 3.6 Summary 124 References 124 4 DESIGN OF DIGITAL FILTERS FOR DECIMATION AND INTERPOLATION 127 4.0 Introduction 127 4.1 Digital Filter Design Fundamentals 128 4.1.1 Basic Considerations and Properties 128 4.1.2 Advantages and Disadvantages of FIR and HR Filters for Interpolation and Decimation 130 4.2 Filter Specifications for Sampling Rate Changing System 132 4.2.1 The Prototype Filter and Its Polyphase Representation 132 4.2.2 Ideal Frequency Domain Characteristics for Interpolation Filters 136 4.2.3 Ideal Frequency Domain Characteristics for Decimation Filters 139 4.2.4 Time Domain Properties of Ideal Interpolation Filters 140 4.2.5 Time Domain Properties of Ideal Decimation Filters 142 4.3 Filter Design Procedures for FIR Decimators and Interpolators 143
4.3.1 FIR Filters Based on Window Designs 143 4.3.2 Equiripple (Optimal) FIR Designs 146 4.3.3 The Effects of the <t> Bands for Equiripple Designs 150 4.3.4 Equiripple FIR Filters for Interpolation with Time Domain Constraints 154 4.3.5 Half-Band FIR Filters - A Special Case of FIR Designs for Conversion by Factors of 2 155 4.3.6 Minimum Mean-Square-Error Design of FIR Interpolators - Deterministic Signals 157 4.3.7 Solution of the Matrix Equation 163 4.3.8 Properties of the Minimum Mean-Square-Error Interpolators 165 4.3.9 Design of FIR Interpolators with Minimax Error in the Frequency Domain 167 4.3.10 Design of FIR Interpolators with Minimax Error in the Time Domain 172 4.3.11 Linear Interpolation 175 4.3.12 Lagrange Interpolators 177 4.3.13 Discussion 180 4.4 Filter Design Procedures for HR Decimators and Interpolators 4.4.1 Ideal Characteristics and Practical Realizations of HR Decimators and Interpolators 181 4.4.2 Conventional HR Filter Designs 183 4.4.3 Special HR Designs Based on the Transformation of Conventional Designs 185 4.4.4 A Direct Design Procedure for Equiripple HR Filters for Decimation and Interpolation 186 4.5 Comparisons of HR and FIR Designs of Interpolators and Decimators 188 References 190 MULTISTAGE IMPLEMENTATIONS OF SAMPLING RATE CONVERSION 5.0 Introduction 193 5.1 Computational Efficiency of a 2-Stage Structure - A Design Example 196 5.2 Terminology and Filter Requirements for Multistage Designs 5.2.1 Overall Filter Requirements 199 5.2.2 Lowpass Filter Requirements for Individual Stages 202 5.2.3 Filter Specifications for Individual Stages which Include "Don't-Care" Bands 204 5.2.4 Passband and Stopband Tolerance Requirements 205
Contents xi 5.2.5 Design Considerations 206 5.3 Multistage FIR Designs Based on an Optimization Procedure 207 5.3.1 Analytic Expressions for the Required Filter Order for Each Stage of a Multistage Design 208 5.3.2 Design Criteria Based on Multiplication Rate 209 5.3.3 Design Criteria Based on Storage Requirements 210 5.3.4 Design Curves Based on Computer-Aided Optimization 211 5.3.5 Application of the Design Curves and Practical Considerations 216 5.4 Multistage Structures Based on Half-Band FIR Filters 218 5.4.1 Half-Band Designs with No Aliasing in the Final Transition Band 220 5.4.2 Half-Band Designs for Power-of-2 Conversion Ratios and Aliasing in the Final Transition Band 222 5.5 Multistage FIR Designs Based on a Specific Family of Half-Band Filter Designs and Comb Filters 227 5.5.1 Comb Filter Characteristics 227 5.5.2 A Design Procedure Using a Specific Class of Filters 231 5.6 Multistage Decimators and Interpolators Based on HR Filter Designs 235 5.7 Considerations in the Implementation of Multistage Decimators and Interpolators 244 5.8 Summary 249 References 249 6 MULTIRATE IMPLEMENTATIONS OF BASIC SIGNAL PROCESSING OPERATIONS 251 6.0 Introduction 251 6.1 Multirate Implementation of Lowpass Filters 252 6.1.1 Design Characteristics of the Lowpass Filters 256 6.1.2 Multistage Implementations of the Lowpass Filter Structure 258 6.1.3 Some Comments on the Resulting Lowpass Filters 260 6.1.4 Design Example Comparing Direct and Multistage Implementations of a Lowpass Filter 261 6.2 Multirate Implementation of a Bandpass Filter 263 6.2.1 Pseudo Integer-Band, Multirate Bandpass Filter Implementations 265 6.2.2 Alternative Multirate Implementations of Bandpass Filters 267 6.2.3 Multirate Implementation of Narrow-Band Highpass and Bandstop Filters 270
xii Contents 6.3 Design of Fractional Sample Phase Shifters Based on Multirate Concepts 271 6.3.1 Design of Phase Shifter Networks with Fixed Phase Offsets 274 6.4 Multirate Implementation of a Hubert Transformer 275 6.5 Narrow-Band, High-Resolution Spectral Analysis Using Multirate Techniques 280 6.6 Sampling Rate Conversion Between Systems with Incommensurate Sampling Rates 283 6.7 Summary 286 References 287 7 MULTIRATE TECHNIQUES IN FILTER BANKS AND SPECTRUM ANALYZERS AND SYNTHESIZERS 289 7.0 Introduction 289 7.1 General Issues and Definitions 290 7.2 Uniform DFT Filter Banks and Short-Time Fourier Analyzers and Synthesizers 296 7.2.1 Filter Bank Interpretation Based on the Complex Modulator 297 7.2.2 Complex Bandpass Filter Interpretation 300 7.2.3 Polyphase Structures for Efficient Realization of Critically Sampled DFT Filter Banks 303 7.2.4 Polyphase Filter Bank Structures for K=MI 311 7.2.5 Weighted Overlap-Add Structures for Efficient Realization of DFT Filter Banks 313 7.2.6 A Simplified Weighted Overlap-Add Structure for Windows Shorter than the Transform Size 324 7.2.7 Comparison of the Polyphase Structure and the Weighted Overlap-Add Structure 325 7.3 Filter Design Criteria for DFT Filter Banks 326 7.3.1 Aliasing and Imaging in the Frequency Domain 327 7.3.2 Filter-Bank-Design by Frequency Domain Specification The Filter-Bank-Sum Method 332 7.3.3 Aliasing and Imaging in the Time Domain 335 7.3.4 Filter Bank Design by Time Domain Specification The Overlap-Add Method 339 7.3.5 Relationship of Time and Frequency Domain Specifications 341 7.4 Effects of Multiplicative Modifications in the DFT Filter Bank and Methods of Fast Convolution 346 7.4.1 The General Model for Multiplicative Modifications 346 7.4.2 Modifications in the Filter-Bank-Sum Method 351 7.4.3 Modifications in the Overlap-Add Method 352
Contents 7.4.4 Time-Invariant Modifications and Methods of Fast Convolution 352 7.4.5 Other Forms of Filter Bank Modifications and Systems 355 7.5 Generalized Forms of the DFT Filter Bank 356 7.5.1 The Generalized DFT (GDFT) 356 7.5.2 The GDFT Filter Bank 358 7.5.3 Polyphase Structure for the GDFT Filter Bank 360 7.5.4 Weighted Overlap-Add Structure for the GDFT Filter Bank 362 7.5.5 Filter Design Criteria for the GDFT Filter Bank 365 7.6 Uniform Single-Sideband (SSB) Filter Banks 366 7.6.1 Realization of SSB Filter Banks from Quadrature Modulation Designs 367 7.6.2 Critically Sampled SSB Filter Banks with k 0-1/4 371 7.6.3 SSB Filter Banks Based on Jt 0 = 1/2 Designs 373 7.7 Filter Banks Based on Cascaded Realizations and Tree Structures 376 7.7.1 Quadrature Mirror Filter (QMF) Bank Design 378 7.7.2 Finite Impulse Response (FIR) Designs for QMF Filter Banks 382 7.7.3 Polyphase Realization of Quadrature Mirror Filter Banks 387 7.7.4 Equivalent Parallel Realizations of Cascaded Tree Structures 392 7.8 Summary 395 References 396 Appendix 7.1 401 INDEX