DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal Processing Perspective THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE Design and Analysis of A Signal Processing Perspective Larry D. Paarmann Associate Professor of Electrical Engineering Wichita State University Wichita, Kansas KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-48012-3 Print ISBN: 0-7923-7373-1 ©2003 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©2001 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstore at: http://ebooks.kluweronline.com PREFACE nalog filters, that is continuous-time filters, or filters that can be implemented with resistors, capacitors, inductors, specialized elements or devices, etc., have enjoyed a long history of use in electrical engineering applications. In fact, it can be said without fear of contradiction that the modern technological world, as we know it, would not exist without analog filters. Even though digital filters, and digital signal processing in general, has experienced great growth and development in recent years, analog filters are an important topic. At the university where the author is an associate professor, a course in analog filters, taught at the first-year graduate / senior level, is one course in the graduate program of signal processing. Many of the concepts in analog filter theory help establish a foundation of understanding that assists in more advanced courses on digital filters, modern filters, adaptive filters, spectral estimation, etc. And, of course, analog filter concepts, and the ability to design and analyze them, is important in the additional area of analog circuit design, mixed signal circuit1 design, and in integrated circuit development. Therefore, this textbook presents analog filter theory from a signal processing perspective (i.e., stressing the signals and systems concepts), but also including analog circuit design and analysis as well. Concepts such as the relationships among the time domain, frequency domain, and s domain are stressed. Other things stressed are inherent trade-offs dictated by theory, that have nothing to do with implementation. For example, attempting to eliminate the time-domain ringing of a high-order Butterworth bandpass filter is an exercise in futility, as theory clearly reveals. Chebyshev and elliptic filters will ring even more: s domain analysis, and the equivalent time-domain analysis clearly reveal this, whereas frequency domain analysis alone may not suggest it. As an educator, and one who concentrates in the area of signal processing, the author believes these concepts to be of vital importance. Almost any book on analog filters will include signal processing / systems concepts 1 Mixed signal circuits are those that include analog signals, and analogcircuitry to process and amplify them, and also digital signals and associated digital circuitry. An example is an integrated circuit with an analog input signal and an analog output signal, but with some digital signal processing in between, which requires an A/D with some analog signal conditioning on the input,perhaps including an anti-aliasing filter, and a D/A with some signal conditioning on the output, perhaps including a reconstruction filter. as well as implementation, and the present book is no exception. Most books on analogfilterdesignbrieflypresentthe signalprocessing / systems concepts, and then concentrate on a variety of filter implementation methods. The present book reverses the emphasis, stressing signalprocessingconcepts. Thepresentbookdoes notignore implementation, as itdoes present filter implementation topics in PartII: passive filters, andoperationalamplifieractive filters. However, greateremphasisonsignal processing / systems concepts are included in Part I of the book than istypical. As suggestedabove, this emphasis makesthe bookmore appropriate as part ofa signal processing curriculum, but shouldalso be ofinterest to those in analog circuit design aswell. The intendedaudienceforthis book includesanyonewith astandardelectrical engineering background, with aB.S. degree orbeyond, oratthe senior level. The most importantbackground subjects are Laplace andFourier transformtheory, and concepts in basic systems and signals, and, of course, basic circuits as well. A background in communications systems wouldbe helpful in fully appreciating the applicationexamples given in Chapter 1, buttheseexamples are given to illustrate analogfilter applications, and acommunicationsbackgroundis notaprerequisite for understanding the developments in the book. While MATLAB2 and SPICE3 are softwarepackagesused, andfamiliarity withthem wouldbeanasset, itisassumed that the adept student can learn these software tools on his own, or with minimal guidance, if they are not already known. A brief introduction to MATLAB is given in Appendix A. Analog electrical signals are so named because they are analogous to some other signal from which they are derived: acoustic, electromagnetic, mechanical motion, etc. Analogfiltersprocessanalog signals. However, they arealsoanalogous inanother respect. The physically-constructed filter, i.e. the realized filter, responds in the time-domain and frequency-domain in a manner that is analogous to the theoretical filter, as defined by, say, as is often done, the magnitude-squared frequencyresponse. This suggestsanimportantconcept. Aparticularfilter response, as perhaps defined by the magnitude-squared frequency response, such as a Butterworth response, is mathematically defined. The realization is, at best, an approximation. Therefore, the “filter” is definedmathematically and abstractly. All realizations are approximations. A “circle” often refers to a geometrical drawing representing a circle, as a “filter” often refers to a physical realization of a filter. Hence, in this textbook, theory is stressed and presented first; implementation (a schematic drawing) follows, and, in practice, a realization (physical circuit) would oftenfollowthat. Itisafascinatingconfirmationofthevalueoftheory, that trial and error, andexperimentation, wouldnevercome up with a Butterworth filter design, but theory elegantly andclearly develops it, and states the result in a very simple form. 2MATLABisaregisteredtrademark ofTheMath Works,Inc.,andisahigh-levellanguage fortechnical computing. 3SPICE isanabbreviation forSimulationProgramwithIntegrated Circuit Emphasis,andisapowerful circuit simulationcomputingprogram. Manycommercially availablecircuitsimulationprograms arebased on SPICE. vi The term “approximation” is used in two ways in this book. In Part I it refers to a filter design H(s) that only approximates an ideal filter. As pointed out in Chapter2, the term “ideal filter” is an unfortunate choice of words, as a conventional “ideal filter” can only be conceived of as being ideal in the frequency domain. A conventional “ideal filter” has some very non-ideal characteristics, such as being non- causal, for example. Nevertheless, such “ideal filters” are often a starting point, and then classical filter designs are referred to as approximations, since their magnitude frequency response will only approximate that of the ideal response. The term “approximation” is also used in this book in the sense in which it was used two paragraphs above. A physical realization will only approximate the filter design H(s). This is because of physical limitations, such as component value tolerances, etc. So a realized filter may be thought of as doubly an approximation. The physical realization only approximates H(s), and H(s) only approximates some “ideal” response. A valuable relationship between analog filter theory and analysis and modern digital signal processing is made by the application of MATLAB to both the design and analysis of analog filters. MATLAB was used significantly in developing the material presented in this book, and throughout the textbook computer-oriented problems are assigned. The disk that accompanies this book contains MATLAB functions and m-files written specifically for this book. The MATLAB functions on the disk extend basic MATLAB capabilities in terms of the design and analysis of analog filters. The m-files are used in a number of examples in the book. They are included on the disk as an instructional aid. See Appendix B for a description of the contents of the accompanying disk. These functions and m-files are intended to be used with MATLAB, version 5, Student Edition, and are not stand-alone. Therefore, familiarity with MATLAB is essential, or the willingness to study it on one's own, for maximum benefit from the study of this book. In Chapter 1, Introduction, basic filtering concepts are presented, such as how a filter is used to estimate a signal from a noisy version of it, or to separate signals based on their frequency content. Chapter 1 also gives a number of practical examples of where a properly designed analog filter can be of significant practical use. It also gives an overview of the text, and therefore chapters of the book will only be briefly introduced here. In PART I, Approximation Design and Analysis, consisting in Chapters2 through 9, fundamental concepts and the design and analysis all of the common classical filters are theoretically presented: Butterworth, Chebyshev, elliptic and Bessel. Some filter designs, such as Gaussian and Legendre, which are not as well known, are also covered. In PART II, Implementation and Analysis, consisting of Chapters 10 and 11, implementation of a filter in a circuit schematic diagram is presented. Chapter 10 introduces passive filter design, and Chapter 11 introduces active filter design. vii Features of this book that may be of interest include the following: There are over 200 figures in the book. Many of these figures report basic characteristics of given analog filter design methods: these data graphs were obtained from MATLAB simulations. The data graphs mentioned immediately above include the magnitude frequency response, the phase response, phase delay, group delay, unit impulse response, and unit step response, for several filter orders. These data graphs are for filters with a normalized 3 dB cutoff frequency for ease of comparing different filters. Not only are all of the classical filter design methods covered (Butterworth, Chebyshev Type I, Chebyshev Type II, Bessel, and elliptic), but other methods are also included: Gaussian, Legendre, ultraspherical, Papoulis, and Halpern. There are over 100 examples in the book. There is a total of 345 homework problems in the book, appearing at the ends of the chapters. On the accompanying disk (standard 3 1/2 inch PC floppy) there is over 30 MATLAB m-files and functions written specifically for this book. The functions include filter designs for Gaussian, Legendre, ultra- spherical, Papoulis, and Halpern filters. See Appendix B for a complete list of the contents of the disk. A solutions manual, containing the solutions for selected homework problems, is available from the publisher for qualified instructors who have adopted the book for classroom use. This book has grown out of the author’s experience of teaching a course on analog filters over the past ten years. The author would like to express his appreciation to the classes of students at Wichita State University who have taken the course on analog filters with the author, have suffered through earlier manuscript versions that preceded this book, and offered comments and suggestions toward improving the final result. Being their teacher has been a rewarding experience. Larry D. Paarmann viii TABLE OF CONTENTS Page PREFACE v Chapter 1. INTRODUCTION 1 1.1 Filtering Concepts 1 1.2 Classes of Filters 4 1.3 Applications of Analog Filters 8 1.4 Historical Perspective 15 1.5 A Note on MATLAB 16 1.6 Overview of the Text 17 1.7 Chapter 1 Problems 19 PART I Approximation Design and Analysis 2. ANALOG FILTER DESIGN AND ANALYSIS CONCEPTS 23 2.1 Time, Frequency, and s Domains 24 2.2 The Paley-Wiener Theorem 34 2.3 Time-Bandwidth Products 40 2.4 Frequency Band Definitions 51 2.5 Filter Selectivity and Shaping Factor 52 2.6 Imposed Constraints 54 2.7 Analog Filter Design Theorem 58 2.8 First-Order Transfer Functions 66 2.9 Second-Order Transfer Functions 70 2.10 Transfer Functions with Orders Greater than Two 76 2.11 Minimum-Phase Transfer Functions 77 2.12 All-Pass Transfer Functions 78 2.13 Time-Domain Response 80 2.14 Phase Delay and Group Delay 81 2.15 Hilbert Transform Relations 88 2.16 Frequency Scaling 98 2.17 Chapter 2 Problems 102
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