MULTIRATE DIGITAL SIGNAL PROCESSING

Transcription

1 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

2 Contents PREFACE ACKNOWLEDGMENTS 1 INTRODUCTION 1.0 Basic Considerations Sampling Rate Conversion Examples of Multirate Digital Systems Sampling Rate Conversion in Digital Audio Systems Conversion Between Delta Modulation and PCM Signal Coding Formats Digital Time Division Multiplexing (TDM) to Frequency Division Multiplexing (FDM) Translation Sub-Band Coding of Speech Signals Short-Time Spectral Analysis and Synthesis Scope of the Book 9 References 10 2 BASIC PRINCIPLES OF SAMPLING AND SAMPLING RATE CONVERSION 2.0 Introduction 13

3 2.1 Uniform Sampling and the Sampling Theorem Uniform Sampling Viewed as a Modulation Process Spectral Interpretations of Sampling The Sampling Theorem Reconstruction of an Analog Signal from Its Samples Summary of the Implications of the Sampling Theorem Sampling Rate Conversion - An Analog Interpretation Sampling Rate Conversion - A Direct Digital Approach Relationship to Time-Varying Systems Sampling Rate Reduction - Decimation by an Integer Factor M Sampling Rate Increase - Interpolation by an Integer Factor L Sampling Rate Conversion by a Rational Factor M/L Decimation and Interpolation of Bandpass Signals The Sampling Theorem Applied to Bandpass Signals Integer-Band Decimation and Interpolation Quadrature Modulation of Bandpass Signals Single-Sideband Modulation Discussion Summary 57 References 57 STRUCTURES AND NETWORK THEORY FOR MULTIRATE DIGITAL SYSTEMS 3.0 Introduction Signal-Flow Graph Representation of Digital Systems Signal-Flow Graphs: Basic Principles Commutation of Branch Operations and Circuit Identities Transposition and Duality for Multirate Systems A Review of Structures for Linear Time-Invariant Filters FIR Direct-Form Structure Transposed Direct-Form Structure for FIR Filters Structures for HR Digital Filters Structures for FIR Decimators and Interpolators Direct and Transposed Direct-Form FIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate Polyphase FIR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate 79

4 Contents ix Polyphase Structures Based on Clockwise Commutator Models FIR Structures with Time-Varying Coefficients for Interpolation/Decimation by a Factor of L/M HR Structures for Decimators and Interpolators with Integer Changes in Sampling Rate Polyphase Structures for HR Decimators and Interpolators Direct-Form Structures and Structures with Time-Varying Coefficients for HR Decimators and Interpolators Comparison of Structures for Decimation and Interpolation Advanced Network Concepts of Linear Multirate and Time-Varying Structures System Representation of Linear Time-Varying and Multirate Networks Cascading Networks and Commutation of Network Elements Network Duality Network Transposition and Tellegen's Theorem Transposition of Complex Networks Summary 124 References DESIGN OF DIGITAL FILTERS FOR DECIMATION AND INTERPOLATION Introduction Digital Filter Design Fundamentals Basic Considerations and Properties Advantages and Disadvantages of FIR and HR Filters for Interpolation and Decimation Filter Specifications for Sampling Rate Changing System The Prototype Filter and Its Polyphase Representation Ideal Frequency Domain Characteristics for Interpolation Filters Ideal Frequency Domain Characteristics for Decimation Filters Time Domain Properties of Ideal Interpolation Filters Time Domain Properties of Ideal Decimation Filters Filter Design Procedures for FIR Decimators and Interpolators 143

5 4.3.1 FIR Filters Based on Window Designs Equiripple (Optimal) FIR Designs The Effects of the <t> Bands for Equiripple Designs Equiripple FIR Filters for Interpolation with Time Domain Constraints Half-Band FIR Filters - A Special Case of FIR Designs for Conversion by Factors of Minimum Mean-Square-Error Design of FIR Interpolators - Deterministic Signals Solution of the Matrix Equation Properties of the Minimum Mean-Square-Error Interpolators Design of FIR Interpolators with Minimax Error in the Frequency Domain Design of FIR Interpolators with Minimax Error in the Time Domain Linear Interpolation Lagrange Interpolators Discussion Filter Design Procedures for HR Decimators and Interpolators Ideal Characteristics and Practical Realizations of HR Decimators and Interpolators Conventional HR Filter Designs Special HR Designs Based on the Transformation of Conventional Designs A Direct Design Procedure for Equiripple HR Filters for Decimation and Interpolation Comparisons of HR and FIR Designs of Interpolators and Decimators 188 References 190 MULTISTAGE IMPLEMENTATIONS OF SAMPLING RATE CONVERSION 5.0 Introduction Computational Efficiency of a 2-Stage Structure - A Design Example Terminology and Filter Requirements for Multistage Designs Overall Filter Requirements Lowpass Filter Requirements for Individual Stages Filter Specifications for Individual Stages which Include "Don't-Care" Bands Passband and Stopband Tolerance Requirements 205

6 Contents xi Design Considerations Multistage FIR Designs Based on an Optimization Procedure Analytic Expressions for the Required Filter Order for Each Stage of a Multistage Design Design Criteria Based on Multiplication Rate Design Criteria Based on Storage Requirements Design Curves Based on Computer-Aided Optimization Application of the Design Curves and Practical Considerations Multistage Structures Based on Half-Band FIR Filters Half-Band Designs with No Aliasing in the Final Transition Band Half-Band Designs for Power-of-2 Conversion Ratios and Aliasing in the Final Transition Band Multistage FIR Designs Based on a Specific Family of Half-Band Filter Designs and Comb Filters Comb Filter Characteristics A Design Procedure Using a Specific Class of Filters Multistage Decimators and Interpolators Based on HR Filter Designs Considerations in the Implementation of Multistage Decimators and Interpolators Summary 249 References MULTIRATE IMPLEMENTATIONS OF BASIC SIGNAL PROCESSING OPERATIONS Introduction Multirate Implementation of Lowpass Filters Design Characteristics of the Lowpass Filters Multistage Implementations of the Lowpass Filter Structure Some Comments on the Resulting Lowpass Filters Design Example Comparing Direct and Multistage Implementations of a Lowpass Filter Multirate Implementation of a Bandpass Filter Pseudo Integer-Band, Multirate Bandpass Filter Implementations Alternative Multirate Implementations of Bandpass Filters Multirate Implementation of Narrow-Band Highpass and Bandstop Filters 270

7 xii Contents 6.3 Design of Fractional Sample Phase Shifters Based on Multirate Concepts Design of Phase Shifter Networks with Fixed Phase Offsets Multirate Implementation of a Hubert Transformer Narrow-Band, High-Resolution Spectral Analysis Using Multirate Techniques Sampling Rate Conversion Between Systems with Incommensurate Sampling Rates Summary 286 References MULTIRATE TECHNIQUES IN FILTER BANKS AND SPECTRUM ANALYZERS AND SYNTHESIZERS Introduction General Issues and Definitions Uniform DFT Filter Banks and Short-Time Fourier Analyzers and Synthesizers Filter Bank Interpretation Based on the Complex Modulator Complex Bandpass Filter Interpretation Polyphase Structures for Efficient Realization of Critically Sampled DFT Filter Banks Polyphase Filter Bank Structures for K=MI Weighted Overlap-Add Structures for Efficient Realization of DFT Filter Banks A Simplified Weighted Overlap-Add Structure for Windows Shorter than the Transform Size Comparison of the Polyphase Structure and the Weighted Overlap-Add Structure Filter Design Criteria for DFT Filter Banks Aliasing and Imaging in the Frequency Domain Filter-Bank-Design by Frequency Domain Specification The Filter-Bank-Sum Method Aliasing and Imaging in the Time Domain Filter Bank Design by Time Domain Specification The Overlap-Add Method Relationship of Time and Frequency Domain Specifications Effects of Multiplicative Modifications in the DFT Filter Bank and Methods of Fast Convolution The General Model for Multiplicative Modifications Modifications in the Filter-Bank-Sum Method Modifications in the Overlap-Add Method 352

8 Contents Time-Invariant Modifications and Methods of Fast Convolution Other Forms of Filter Bank Modifications and Systems Generalized Forms of the DFT Filter Bank The Generalized DFT (GDFT) The GDFT Filter Bank Polyphase Structure for the GDFT Filter Bank Weighted Overlap-Add Structure for the GDFT Filter Bank Filter Design Criteria for the GDFT Filter Bank Uniform Single-Sideband (SSB) Filter Banks Realization of SSB Filter Banks from Quadrature Modulation Designs Critically Sampled SSB Filter Banks with k 0-1/ SSB Filter Banks Based on Jt 0 = 1/2 Designs Filter Banks Based on Cascaded Realizations and Tree Structures Quadrature Mirror Filter (QMF) Bank Design Finite Impulse Response (FIR) Designs for QMF Filter Banks Polyphase Realization of Quadrature Mirror Filter Banks Equivalent Parallel Realizations of Cascaded Tree Structures Summary 395 References 396 Appendix INDEX