Design and Analysis of Analog Filters: A Signal Processing Perspective

453 Pages·2001·16.05 MB·english

by  Paarmann L.D.

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
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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
TABLE OF CONTENTS (continued)
Page
Chapter
3. BUTTERWORTH FILTERS 113
3.1 Maximally-Flat Magnitude 113
3.2 Filter Selectivity and Shaping Factor 115
3.3 Determination of Order 117
3.4 Pole Locations 119
3.5 Phase Response, Phase Delay, and Group Delay 122
3.6 Time-Domain Response 125
3.7 Chapter 3 Problems 126
4. CHEBYSHEV TYPE I FILTERS 131
4.1 Equiripple Passband Magnitude 131
4.2 Filter Selectivity and Shaping Factor 136
4.3 Determination of Order 137
4.4 Chebyshev Polynomials 138
4.5 Pole Locations 140
4.6 Phase Response, Phase Delay, and Group Delay 143
4.7 Time-Domain Response 146
4.8 Comparison with Butterworth Filters 148
4.9 Chapter 4 Problems 150
5. CHEBYSHEV TYPE II FILTERS 155
5.1 Equiripple Stopband Magnitude 155
5.2 Filter Selectivity and Shaping Factor 160
5.3 Determination of Order 161
5.4 Inverse Chebyshev Polynomials 162
5.5 Location of the Poles and Zeros 163
5.6 Phase Response, Phase Delay, and Group Delay 166
5.7 Time-Domain Response 169
5.8 Comparison with Butterworth
and Chebyshev Type I Filters 170
5.9 Chapter 5 Problems 172
6. ELLIPTIC FILTERS 177
6.1 Introduction 177
6.2 Elliptic Integrals and Jacobian Elliptic Functions 182
6.3 Equiripple Passband and Stopband Magnitude 188
6.4 Computing Given N and 195
6.5 Filter Selectivity and Shaping Factor 196
6.6 Determination of Order 197
x