From the Inside Flap
Analog and digital filter design is of great importance throughout engineering, applied mathematics, and computer science. Filters are the staple for designers in the controls, signal processing, and communications fields. They are commonly used in a wide variety of systems, such as chemical processing plants, instrumentation, suspension systems, modems, and digital cellular phones.
When a designer uses conventional techniques and software to design a filter, the designer receives only one possible filter that meets a set of specifications, yet an infinite number of designs may exist. This book develops alternative techniques and software to produce a comprehensive set of designs that meet the specification and represent the infinite design space. Included in the set of designs are filters that have minimal order, minimal quality factors, minimal complexity, minimal sensitivity to pole-zero locations, minimal deviation from a specified group delay, approximate linear phase, and minimized peak overshoot. For digital filters, the design space also includes filters with power-of-two coefficients. These alternative filter designs are crucial when evaluating filters for synthesis in analog circuits, digital hardware, or software.
This book overcomes the gap between filter theory and practice, and it presents new algorithms and designs developed over the last five years. The book includes ready-to-use filter design algorithms and implementations of the algorithms in both MATLAB and Mathematica. In order to make the material accessible to both the practitioner and the researcher, we have divided the book into two parts. Part I reviews conventional filter design techniques, presents several new ready-to-use algorithms, and discusses many case studies. The case studies present filters that cannot be designed with conventional techniques but can be designed with advanced methods. Part II discusses the theory underlying the new advanced design methods. The book also contains appendices to show examples of using advanced filter design software in MATLAB and Mathematica, and it includes filter design problems for the reader to solve.
In designing analog and digital IIR filters, one generally relies on canned software routines or mechanical procedures that rely on extensive tables. The primary reason for this "black box" approach is that the approximation theory that underlies filter design includes complex mathematics. Unfortunately, the conventional approach returns only one design, and it hides a wealth of alternative filter designs that are more robust when implemented in analog circuits, digital hardware, and software.
In this book, we provide advanced techniques to return multiple designs that meet the user specification. The key observations underlying our advanced filter design are as follows:
Many designs satisfy the same user specification. Butterworth and Chebyshev IIR filters are special cases of elliptic IIR filters. Minimum-order filters may not be as efficient to implement as some higher-order filters.
Our approach is to search for a variety of design specifications that satisfies both the user specification and the limitations on the target implementation technology. Our algorithms return the following designs:
Minimal filter order Maximum stopband loss margin Maximum passband loss margin Minimum transition (conventional design) Maximum transition Minimal maximum quality factor Minimal implementation cost Minimal deviation from specified group delay Minimal deviation from linear phase Elliptic IIR filters with power-of-two coefficients Zero-phase elliptic IIR filters Multiplierless elliptic half-band IIR filters Multiplierless Hilbert transformers Robust low-sensitivity sharp cutoff SC filters
For example, we can design selective elliptic IIR filters for microcontrollers and for other architectures that have fixed-point arithmetic and no hardware multiplier.
The theory underlying our advanced techniques is rooted in Jacobi elliptic functions which we use to approximate the magnitude frequency response of the filter. Jacobi elliptic functions are very complex transcendental functions. For many filter orders, however, we derive closed-form solutions to design elliptic filters that only use polynomials, square roots, and logarithms. This breakthrough allows us to derive precise relationships between the user specification, implementation constraints, and the pole-zero locations of the filter. Thus, we have transformed the design space for HR filters from elliptic function approximation theory into polynomial theory, which can be understood by designers with a knowledge of algebra. In addition, final expressions are simple. Most of elliptic filters can be accurately designed ten to hundred times faster than using the classic approach.
The elliptic approximation is the most frequently used function in the design of IIR filters. In the latter part of this book, we explore many of the properties of elliptic functions such as its nesting property. These properties enable us to automate the design of filters using symbolic algebra. Transfer function poles and zeros are obtained by means of simple formulas, thereby freeing the designer from having to rely on extensive tables or canned computer programs. Symbolic design makes it possible to eliminate redundant variables, decrease the filter order, and simplify and approximate the underlying complex relations prior to the final numeric calculations.
The primary benefit of this book is convenient access to the latest advances in algorithms and software for analog and digital IIR filter design. These advanced techniques can design many types of filters that conventional techniques cannot design. A secondary benefit is a large collection of case studies for filter designs that require advanced techniques. Another benefit is a unique treatment of elliptic function filters.
The book is divided into 13 chapters.
Chapter 1 presents an overview of basic classes of continuous-time and discrete-time signals. We discuss mathematical representations of signals, and introduce the two computer environments, MATLAB and Mathematica, which we use to analyze and process signals.
In Chapter 2, we introduce fundamentals of linear system theory and define basic system properties. We present basic definitions and background mathematics that are used in this book. Since many readers are already familiar with this material, our aim is to be logically consistent rather than mathematically rigorous.
In Chapter 3, we review the definition and the salient properties of the most important transforms required by the filter design studied in this book. We focus on the phasor transformation, Fourier series and harmonic analysis, Fourier transformation, Laplace transformation, discrete Fourier transform, and z-transform. Step-by-step procedures for analyzing LTI systems in the transform domain are given.
Chapter 4 is intended to review the basics of classic analog filter design. Classification, salient properties and sensitivity of transfer functions are given. The most important analog filter realizations are presented. A detailed case study is given for realization of various transfer functions.
Chapter 5 reviews basic definitions of analog filter design. It introduces straight-forward procedures to map the filter specification into a design space. We search this design space for the optimum solution according to given criteria. We conclude this chapter by an application example in which we design a robust selective analog filter based on commercially available integrated circuits.
In Chapter 6, we present (1) case studies of optimal analog filters that cannot be designed with classic techniques, and (2) the formal, mathematical framework that underlies their solutions. We present detailed step-by-step analog filter design algorithms.
Chapter 7 presents an extensible framework for designing analog filters that exhibit several desired behavioral properties after being realized in circuits. In the framework, we model the constrained nonlinear optimization problem as a sequential quadratic programming problem. We derive the differentiable constraints and a weighted differentiable objective function for simultaneously optimizing the behavioral properties of magnitude response, phase response, and peak overshoot and the implementation property of quality factors.
Chapter 8 is intended to review the basics of classic digital IIR filter design. Classification, salient properties and sensitivity of transfer functions in the z-domain are given. The most important digital filter realizations are presented. For each realization we provide complete design equations and procedures that make the design easily applicable to a broad variety of digital filter design problems.
Chapter 9 reviews basic definitions of digital IIR filter design. It introduces straightforward procedures to map the filter specification into a design space. We search this design space for the optimum solution according to given criteria. We conclude this chapter with several important application examples in which we design low-sensitivity selective multiplierless IIR filters, power-of-two IIR filters, half-band IIR filters, 1/3band filters, narrow-band IIR filters, Hilbert transformers, and zero-phase IIR filters. Each example design is followed by a comprehensive step-by-step procedure for computing the filter coefficients.
In Chapter 10, we present (1) case studies of optimal digital filters that cannot be; designed with classic techniques, and (2) the formal, mathematical framework that underlies their solutions. We present detailed step-by-step digital filter design algorithms.
Chapter 11 presents an extensible framework for the simultaneous constrained) optimization of multiple properties of digital IIR filters. The framework optimizes the; pole-zero locations for behavioral properties of magnitude and phase response, and the; implementation property of quality factors, subject to constraints on the same properties. We formulate the constrained nonlinear optimization problem as a sequential quadratic programming problem.
Chapter 12 introduces the basic Jacobi elliptic functions and reviews the most important relations between them. Several related theorems not found in standard textbooks are presented. Various useful approximation formulas are offered to facilitate the derivation of elliptic rational functions. A nesting property of the Jacobi elliptic functions is derived. In this chapter we present a novel approach to the design of elliptic filters in which we use exact closed-form expressions based on the nesting property.
In Chapter 13, we introduce the elliptic rational function as a natural generalization of the Chebyshev polynomial and we bypass mathematical theory of special functions required in the previous chapter. We prefer to give a reader an intuitive feel of the basic properties of the elliptic rational function. Our goal is to build the known edge of the elliptic rational function using simple algebraic manipulations, even without mentioning the Jacobi elliptic functions.
Problems are included at the ends of chapters. The majority provide important practice with the concepts and techniques presented. Almost all the problems arc suitable for solution using Mathematica and MATLAB.
Our overall approach to the topic has been guided by the fact that with the recent and anticipated developments in the technologies for filter design and implementation, the importance of having equal familiarity with computer-aided techniques suitable for analyzing and designing both continuous-time and discrete-time filters has increased dramatically.
We seek to leave with each reader (student, instructor, researcher, or practicing engineer) a set of software tools Mathematica notebooks and MATLAB scripts useful for solving filter design problems of practical importance.
A notable feature of the book is a detailed step-by-step exposure to the filter analysis by transform method, or in the time domain, exemplified by self-contained Mathematica notebooks. Students can use these notebooks to (a) automate symbolic filter analysis and design in software, (b) derive closed-form expressions for, say, transfer functions, and (c) gain insight into the relevant filter design parameters and coefficients.
This book was designed for educators who wish to integrate their curriculum with computer-based learning tools. Our goal is to provide an effective and efficient environment for students to learn the theory and problem-solving skills for contemporary filter design. To accomplish this we have used a computer-biased approach in which computer solutions and theory are viewed as mutually reinforcing rather than an either-or proposal.
We believe that students learn most effectively by solving problems following a worked-out problem as a model. Software scripts for running electronic examples (of worked-out problems) can capture the essence of a key concept, and encourage active participation in learning.
Filter analysis and design is a foundation subject for many students, due to its direct engineering applications, especially in electrical engineering. The concepts which it embodies, and the analytical techniques which it employs, are valid far outside the boundaries of electrical engineering.
The subject of filter design is an extraordinarily rich one, and a variety of approaches can be taken in structuring an introductory or advanced filter design course. This text provides a broad treatment of filter design and analysis, and contains sufficient material for a one-semester or two-semester course on the subject. Students using this book are assumed to have a basic background in calculus, complex numbers, and differential equations.
A typical one-semester introductory filter design course at a sophomore-junior level using this book could comprise the following: (a) Chapters 1-3, (b) Chapter 4, (c) a choice from Chapter 5 with the emphasis placed on specification and approximation problem, (d) Chapter 8, (e) a choice from Chapter 9 (digital specification and approximation problem). Combine the text with the MATLAS Signal Processing Toolbox and the Mathematica Signals and Systems Pack to illustrate the classic filter design procedures. Proceed lightly through our Mathematica Example Notebooks and our MATLAB filter design toolbox.
A one-semester introductory analog filter design course at a sophomore-junior level using this book could comprise the following: (a) Chapters 1-3, (b) Chapters 4, 5 and 7, (c) utilize the MATLAB Signal Processing Toolbox, and the Mathematica Signals and Systems Pack, to illustrate the classic analog filter design, (d) proceed through our Mathematica Example Notebooks, and our MATLAB analog filter design toolbox.
A one-semester introductory digital filter design course at a sophomore-junior level using this book could comprise the following: (a) Chapters 1-3, (b) Chapters 8, 9 and 11, (c) utilize the MATLAB Signal Processing Toolbox, and the Mathematica Signals and Systems Pack, to illustrate the classic digital filter design, (d) proceed through our Mathematica Example Notebooks, and our MATLAB digital filter design toolbox.
In addition to these course formats this book can be used as the basic text for a thorough, two-semester sequence on advanced filter design. The portions of the book not used in an introductory course on filter design, together with other sources, can form the basis for a senior elective course. Alternatively, for a two-semester course, we suggest coverage of the first 11 chapters, proceeding lightly through Chapters 12 and 13, and covering thoroughly Chapters 5 and 9 because they introduce the design space concept in filter design.
The book can serve as a text for a sequence of two one-semester courses on analog and digital filters for senior undergraduate or first-year graduate students. Such a course could comprise the following: (a) review of Chapters 1-3, (b) Chapters 4, 5, (c) brief discussion of analog-filter design algorithms (Chapter 6) with the emphasis placed on application rather than derivation, (d) Chapter 7 in full depth, (e) Chapters 8, 9, (f) brief discussion of digital-filter design algorithms (Chapter 10) with the emphasis placed on application rather than derivation, (g) Chapter 11 in full depth, (h) a choice from Chapters 12 and 13, depending on the course orientation desired.
The book's structure allows students who are interested in only analog filters to skip chapters on digital filters, without loss of continuity, and vice versa. It should be pointed out that not all sections in every chapter are covered in class. Also, various topics can be omitted at the discretion of the instructor. Depending upon the background the students can utilize chapters 1, 2, and 3 to review and expand their knowledge of linear system theory for continuous-time and discrete-time systems.
Advanced postgraduate courses (masters's programs and Ph.D. programs) can also be prepared from Chapters 12 and 13.
Selected topics chosen from the book chapters can be used in Electronics and Electric Circuit Theory courses, too.
We have included a collection of more that 70 Mathematica notebooks, numerous MATLAB scripts, and many end-of-chapter problems and exercises. This variety and quantity will hopefully provide instructors with considerable flexibility in putting together homework sets that are tailored to the specific needs of their students. Many filter realizations, both analog and digital, presented throughout the book can help lecturers organize versatile homeworks, projects and tests for the students. In addition, the filter design algorithms (Chapters 6 and 10) can be directly programmed in any language or environment such as Visual BASIC, Visual C, Maple, DERIVE, or MathCAD.
We thank Professor Ljiljana Milić for her valuable comments which have improved the book. We would like to thank Professor Marija Hribšek, Professor Antonije Djordjević and Professor Veljko Milutinović, for their encouragement throughout the period in which the book was written. The first author is grateful to general manager Siniša Davitkov for providing him with the opportunity to work on filter design. We would like to thank Professor George S. Moschytz for his encouragement and support while working on the writing this book.
The continuing encouragement, patience, technical support, and enthusiasm provided by Prentice Hall, and in particular by Alice Dworkin have been important in bringing this project to fruition.
We also want to thank the very thoughtful and careful reviewers, Professor Igor Tsukerman, The University of Akron, and Professor Michael J. Werter, The University of California at Los Angeles, for their useful comments.
Miroslav D. Lutovac is a chief scientist at the Institute for Research and Development in Telecommunications and Electronics (IRITEL) and is an Associate Professor in the School of Electrical and Computer Engineering, both of which are located at the University of Belgrade in Belgrade, Yugoslavia. His research interests include theory and implementation of active, passive, and digital networks and systems, filter approximation, symbolic analysis and synthesis of digital filters, and multiplierless digital IIR filter design. He has published over 100 papers in these fields. He received his B.Sc. (1981), M.Sc. (1985), and D.Sc. (1991) degrees in Electrical Engineering from the University of Belgrade in Belgrade, Yugoslavia. He has managed several national projects on multichip module design and voice delta coders. He teaches courses in electronics, computer-aided design, digital signal processing, and filter analysis and design.
Dejan V. Tošić is an Associate Professor in the School of Electrical and Computer Engineering at the University of Belgrade in Belgrade, Yugoslavia. His research interests include circuit theory and analysis, filter design and synthesis, neural networks, microwave circuits, and computer-aided design. He has published over 100 papers in these fields. He is currently concentrating his research efforts on creating a general framework for the symbolic analysis of linear circuits and systems, which is suitable for research, industrial, and educational applications. Using this framework, he is developing design automation tools for optimizing the design and synthesis of analog and digital filters. He received his B.Sc. (1980), M.Sc. (1986), and D.Sc. (1996) degrees in Electrical Engineering from the University of Belgrade in Belgrade, Yugoslavia. In 1992, he won the Teacher of the Year Award from the School of Electrical and Computer Engineering at the University of Belgrade. He teaches classes in circuit theory, microwave engineering, and digital image processing.
Brian L. Evans is an Associate Professor in the Department of Electrical and Computer Engineering at The University of Texas at Austin, and is the Director of the Embedded Signal Processing Laboratory, which is part of the Center for Telecommunications and Signal Processing and the Center for Vision and Image Sciences. His research interests include real-time embedded systems; signal, image and video processing systems; system-level design; electronic design automation; symbolic computation; and filter design. Dr. Evans has published over 75 refereed conference and journal papers in these fields. He developed and currently teaches Multidimensional Digital Signal Processing, Embedded Software Systems, Real-Time Digital Signal Processing Laboratory, and Linear Systems and Signals. His B.S.E.E.C.S. (1987) degree is from the Rose-Hulman Institute of Technology, and his M.S.E.E. (1988) and Ph.D.E.E. (1993) degrees are from the Georgia Institute of Technology. From 1993 to 1996, he was a postdoctoral researcher at the University of California at Berkeley with the Ptolemy Project. Ptolemy is a research project and software environment focused on design methodology for signal processing, communications, and controls systems. In addition to Ptolemy, he has played a key role in the development and release of six other computer-aided design frameworks, including the Signals and Systems Pack for Mathematica, which has been on the market since the Fall of 1995. He is an Associate Editor of the IEEE Transactions on Image Processing, a member of the Design and Implementation of Signal Processing System Systems Technical Committee of the IEEE Signal Processing Society, and a Senior Member of the IEEE. He is a recipient of a 1997 National Science Foundation CAREER Award.
Miroslav D. Lutovac
Dejan V. Tošić
Brian L. Evans
From the Back Cover
A complete up-to-date reference for advanced analog and digital IIR filter design rooted in elliptic functions. Revolutionary in approach, this book opens up completely new vistas in basic analog and digital IIR filter design--regardless of the technology. By introducing exceptionally elegant and creative mathematical stratagems (e.g., accurate replacement of Jacobi elliptic functions by functions comprising polynomials, square roots, and logarithms), optimization routines carried out with symbolic analysis by Mathematica, and the advance filter design software of MATLAB, it shows readers how to design many types of filters that cannot be designed using conventional techniques. The filter design algorithms can be directly programed in any language or environment such as Visual BASIC, Visual C, Maple, DERIVE, or MathCAD. Signals; Systems; Transforms; Classical Analog Filter Design; Advanced Analog Filter Design Case Studies; Advanced Analog Filter Design Algorithms; Multi-criteria Optimization of Analog Filter Designs; Classical Digital Filter Design; Advanced Digital Filter Design Case Studies; Advanced Digital Filter Design Algorithms; Multi-criteria Optimization of Digital Filter Designs; Elliptic Functions; Elliptic Rational Function.
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