Filter Design With Time Domain Mask Constraints: Theory and Applications

Transcription

1 Filter Design With Time Domain Mask Constraints: Theory and Applications

2 Applied Optimization Volume 56 Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald Hearn University of Florida, U.S.A. The titles published in this series are listed at the end of this volume.

3 Filter Design With Time Domain Mask Constraints: Theory and Applications by Ba-Ngu Vo The University of Melbourne, Australia Antonio Cantoni The University of Western Australia, Australia and KokLay Teo The Hong Kong Polytechnic University, Hong Kong SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All AII Rights Reserved 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1 st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

5 v CONTENTS List of Figures... XI List of Tables xv Preface xvii CHAPTER 1 INTRODUCTION Applications... 2 Pulse compression TV waveform equalization... 5 Data channel equalization Design to meet standards Deconvolution Envelope Constrained Filtering Historical Notes Road Map CHAPTER 2 FILTERING WITH CONVEX RESPONSE CONSTRAINTS Analog Filtering with Convexly Constrained Responses The cost functional The feasible region Filter Design with Envelope Constraints Convex Programming Finite Dimensional Analog CCR Filters Discrete-time CCR Filter Design Continuous-time CCR Filtering via DSP Approach Problem formulation for hybrid filter Feasible region... 42

6 vi 2.7 Finite Dimensional Hybrid Filter for CCR Filtering FIR digital processor Appendix CHAPTER 3 ANALYSIS AND PROBLEM CHARACTERIZATION Duality of Quadratic Program Dual Problems for EC Filtering Unconstrained dual problem Dual Problem for Finite Dimensional Filter Semi-Infinite Programming Optimality conditions Transformation technique via dual parametrization Getting the primal solution Linearly Constrained Quadratic SIP and EC Filters Finite dimensional dual problem with mxn support points Finit~ dimensional dual problem with n support points Finite-dimensional EC filters Appendix CHAPTER 4 DISCRETE-TIME EC FILTERING ALGORITHMS Discrete-time EC Filtering Problem QP via Active Set Strategy QP with linear inequality constraints QP with inequality constraints Iterative Algorithm via The Primal-Dual Method Non-smooth dual problem... \ Steepest ascent with directional differentials Iterative Algorithm using Augmented Cost Approximation results

7 vii Update equations Tapped Delay Line FIR Filters Discrete-time Laguerre Networks Appendix A Appendix B CHAPTER 5 NUMERICAL METHODS FOR CONTINUOUS TIME EC FIL TERING Continuous-time EC Filtering Analog filters Hybrid filters Non-iterative Method Primal Dual Method Discretization of dual problem for analog filters Discretization of dual problem for hybrid filters..., 170 Finite filter structures Steepest ascent algorithm Penalty Approach for Semi-Infinite Programming Approximations by conventional constrained problems Approximations by unconstrained problems Approximating convex problems Affine functional inequality constrained problems SIP with quadratic cost and affine constraints Application to finite dimensional EC filters Laguerre Networks in Continuous-time EC Filtering Application to channel equalization Hybrid filter with FIR Digital Components Linear interpolator..., 195 Butterworth and Bessel post-filtering

8 viii Chebyshev and elliptic post-filtering Appendix CHAPTER 6 ROBUST ENVELOPE CONSTRAINED FILTERING Constraint Function Transformation to Smooth Problem Problem conversion Application to Analog ECUI Filtering Problem Finite dimensional filter for analog problem Example using Walsh functions Approximations for finite dimensional filter Discrete-time Hybrid ECUI Filtering Approximations for hybrid filters Finite dimensional hybrid filters Constraint Robustness Characterization of filter structure Finite dimensional filter Numerical example with Laguerre filter EC Filtering with Uncertain Implementation Examples with finite dimensional filters Appendix APPENDIX A MATHEMA TICAL BACKGROUND A.l Topological Space A.2 Metric Spaces A.3 Vector Spaces A.4 Normed Spaces A.5 Inner Product Spaces

9 IX A.6 Linear Operators A.7 Linear Functionals and Dual Spaces A.8 Measures and Integration APPENDIX B OPTIMIZATION THEORY B.l Projection Theorem B.2 Hahn-Banach Theorem B.3 Positive Cones and Convex Mappings B.4 Gateaux and Frechet Differentials B.5 Lagrange Multipliers References Index

10 xi List of Figures Figure Pulse Compression... 3 Figure Pulse shape constraints for radar/sonar problem... 4 Figure K-rating mask for equalization of TV channel Figure Model of a data channel Figure Mask for constraints at sampling instances... 8 Figure Mask for handling timing jitter... 8 Figure Impulse response of coaxial cable for various lengths Figure DSX3 pulse template, coaxial cable response and filter output Figure Pre- shaping of pulse Figure ANSI T1.403 for T Mb/s Figure Receiver model and output mask Figure Magnitude response constraint Figure Antenna receiver Figure EC filtering with uncertain input Figure Parallel filter structure Figure Transversal filter structure Figure Configuration for digital processing of continuous-time signal Figure Configuration of optimal analog EC filter Figure Configuration of optimal hybrid EC filter Figure Configuration for an adaptive EC filter Figure Penalty allocator Figure Flow chart for line search Figure A tapped delay line FIR filter Figure Optimum EC filter output for a 13-bit Barker-coded input Figure Augmented cost and noise gain of Barker coded example Figure Optimum EC filter output for a rectangular input Figure Augmented cost and noise gain of rectangular input example Figure Optimum and sub-optimum EC filter and their outputs Figure Augmented cost and noise gain of DSX-3 example Figure Discrete-time Laguerre network Figure Signals and output mask for Laguerre and FIR EC filter Figure Bound on slopes

11 XlI Figure Lower bound Figure B lock diagram of a Laguerre network Figure Augmented cost function for coaxial cable example Figure Magnitude responses of equalizer, unequalized and equalized cable Figure DSX3 pulse template, coaxial cable response and filter output Figure Augmented cost function for fourth order circuit example Figure DSX3 pulse template, filter input and output Figure Magnitude responses of equalizer, unequalized and equalized circuit Figure Rectangular pulse response of post-filter Figure FIR-linear-interpolator filter output Figure FIR-Butterworth and FIR-Bessel filter outputs Figure Magnitude responses of 5th order Butterworth, Bessel and Linear Interpolator Figure Magnitude responses of unequalized and equalized cable Figure FIR-Chebyshev-l and FIR-elliptic filter outputs Figure Magnitude responses of 5th order Chebyshev-l and elliptic filters.200 Figure Magnitude responses of unequalized and equalized cable.. : Figure Magnitude responses of elliptic and Chebyshev-2 filters Figure FIR-Cbebysbev-2 and FIR-elliptic filter output Figure Magnitude responses of unequalized and equalized cable Figure Illustration for lemma Figure bit Barker-coded signal with input mask Figure Filter outputs Figure Impulse response of optimal ECUI filter Figure DSX3 example Figure Impulse response of optimal ECUI filter Figure Response of filter to nominal Barker Coded input Figure Response of filter to perturbed inputs Figure Filter's response to nominal input Figure Filter's response to perturbed inputs Figure Weighted constraint robustness margin Figure Laguerre filter - EC approach Figure Magnitude response EC approach Figure Laguerre filter - EC approach with constraint robustness

12 xiii Figure Magnitude response with constraint robustness Figure Response of perturbed filter Figure Response of perturbed robust filter Figure Signal Energy = Figure Signal Energy = 6.25, - 9% increase Figure Magnitude response of perturbations Figure Magnitude response of perturbations of robust filter

13 xv List of Tables Table Simulation results for 13-bit Barker-coded signal example Table results for rectangular pulse example Table Simulation results for DSX-3 pulse template Table Laguerre and FIR filters Table Performance of Laguerre filters for various dominant pole values Table Simulation data for linear interpolator, Butterworth and Bessel postfilters Table Simulation data for Chebyshev type 1 and elliptic post-filters

14 Preface In the signal processing literature, filter design is a well-established area and there has been a considerable amount of work devoted to it. Whether it's classical or at the leading edge of research, the literature primarily concentrates on frequency domain constraints (such as passband ripples, stopband attenuation). However, very little attention has been directed towards filter design with time domain constraints in the form of envelopes or masks, which have become increasingly important in the performance specification of modern telecommunication systems. Time domain filter design problems are commonly known in the literature as time-domain synthesis problems and deal with finding a network to give a prescribed response for a given excitation, both of which are specified as functions of time. This class of problems arises in pulse-shaping circuits, pulse transmission systems, delay network, transmission channel equalization, video distribution systems, and the like. In time domain synthesis problems, often the performance criterion is the mean square error between the filter output and some desired signal. In many practical signal processing problems, this soft least-square approach is artificial. The approach may also yield unsatisfactory results because large narrow excursions from the desired shape occur, and the norm of the filter can be large. In addition, the choice of an appropriate weighting function is not obvious. Moreover, the solution can be sensitive to the detailed structure of the desired pulse, and it is usually not obvious how the desired pulse should be chosen so as to obtain the best possible solution. The distinctive feature of the hard envelope-constrained filter formulation is that the output waveform is specified to lie within an envelope defined by a set of inequality constraints, rather than attempting to match it with a specific desired pulse. Therefore, in the hard envelope-constrained filter formulation, we deal with a whole set of allowable outputs, and the objective is to seek the one for which the corresponding filter results in minimum enhancement of filter input noise. The hard envelope-constrained filter formulation is more relevant than the soft least-square approach in a number of xvii

15 xviii signal processing fields. For example, in TV channel equalization, it is required that the shaped signal simply fits into a prescribed envelope called a K-mask. In digital transmission, the performance of a digital link is often specified in terms of a mask applied to the received test signal. The envelope-constrained approach is directly applicable for shaping test signals into given masks. For pulse compression applications in sonar and radar detection, envelope-constrained filters can also be used to suppress sidelobes. This monograph has evolved from work on envelope constrained filter design problems that has origins dating back to the late 60's. Over the years, there have been numerous fundamental contributions reported in the literature. In this monograph, our modest intention is to present a unified formulation that covers digital, analog and hybrid (consisting of both analog and digital parts) filters. This formulation considers not only envelope-constrained filters but also problems with more general response constraints and wider range of cost functions. Motivated by practical realization issues, finite dimensional approximations to this general problem are investigated and a number of realizable filter structures are suggested. Furthermore, the important issue of designing envelope constrained filters that are robust to implementation errors are addressed. A class of iterative algorithms is developed for solving this envelope-constrained filtering problem with finite filter structures. Convergence properties of these algorithms are addressed. In continuous-time, this technique yields optimum EC filters with analog or hybrid realizations. This is achieved without discretizing the constraints of the filter. Practical real examples are included to illustrate the effectiveness of each of the algorithms developed in this monograph. Most of the material contained in this monograph is based on the research carried out by the authors and their collaborators during the last several years. This monograph can also be viewed as an application of constrained optimization theory to a class of engineering problems in signal processing. It takes the reader through the three phases of problem solving: problem formulation, characterization of solution, and numerical techniques for solving the problem. In the formulation phase (Chapter 2) we see how a raw engineering problem is formulated by taking into account real world constraints. We also see that the first itera-

16 xix tion of the formulation yields a solution, which is not directly usable. The next iteration in the formulation process is to improve and refine the formulation and the mathematical model. The solution characterization phase (Chapter 3) marches through the mathematical analysis of optimization theory to derive the forms that the solutions take on. Interpretations of what these results mean in signal processing terms are then made. Numerical algorithms for solving these problems are perhaps the most important phase in engineering, because if we can't find the solution to the problem, the other two phases are of little use to the designer. This phase is covered in Chapters 4,5 and 6. Chapters 4 and 5 present algorithms for discrete-time and continuous-time problems respectively. We look at the discrete-time problem first because it is simpler. This also makes it easier to grasp the continuous-time problem. Chapter 6 considers the robustness problem. This monograph is intended for both engineers and applied mathematicians. The authors believe that both engineers and applied mathematicians can make further contributions to the subject. With the background acquired from this monograph, engineers will learn the mathematical rigor in problem formulation and the development of solution methods. On the other hand, applied mathematicians will find this monograph as a stepping stone towards a research area in signal processing. Moreover, the authors also believe that the monograph will be useful as a reference to practicing engineers and scientists as well as mathematicians. The background required for understanding the mathematical formulation, the algorithms and the application of these algorithms to solve practical problems is advanced calculus. However, to analyze the convergence properties of these algorithms, some results in real and functional analysis and optimization theory are required. For the convenience of the reader, essential mathematical concepts and facts in real and functional analysis are stated without proof in Appendix A, while optimization theory, in particular, convex mathematical programming, are reviewed in Appendix B. To ensure readability of the monograph amongst both engineers and applied mathematicians, the proofs are given in the appendix of each chapter so as not to disrupt the

17 xx continuity of flow. Those who are interested in the mathematical details can take excursions to the appendices. The authors believe that in this way both classes of reader can gain a better appreciation of the subject. It is a pleasure to express our gratitude to R. Evans, W. X. Zheng, G. Lin, Z. Zang, C. H. Tseng, and H. H. Dam. They have made a significant contribution to the material presented in this monograph through collaborative research projects. We wish to thank Australian Telecommunications Research Institute of Curtin University of Technology and the Co-operative Research Centre for Broadband Telecommunications and Networking for the stimulating research environment and we also appreciate the financial supports provided by the Australian Research Council, the Department of Electrical and Electronic Engineering of the University of Western Australia, and the Department of Electrical and Electronic Engineering of the University of Melbourne. Also the financial supports of the Research Committee, the Centre for Multimedia Signal Processing of the Department of Electronics and Information Engineering, the Department of Applied Mathematics, all with The Hong Kong Polytechnic University, and the Research Grant Committee of Hong Kong are gratefully acknowledged. Furthermore, we wish to express our appreciation to John R. Martindale, Senior Publishing Editor of Kluwer Academic Publishers, for his encouragement, enthusiasm and collaboration. Our sincere thanks also go to our families for their support, patience and understanding. B.Vo A. Cantoni K. L. Teo February 2001