Table Of ContentHANDBOOK OF TABLES FOR ELLIPTIC-FUNCTION FILTERS
HANDBOOK OF TABLES FOR
ELLIPTIC-FUNCTION FILTERS
by
Kendall L. Su
Georgia Institute of Technology
~.
"
KLUWER ACADEMIC PUBLISHERS
Boston/Dordrecht/London
Distributors for North America:
Kluwer Academic Publishers
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Norwell, Massachusetts 02061 USA
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Library of Congress Cataloging-in-Publication Data
Su, Kendall L. (Kendall Ling-chiao), 1926-
Handbook of tables for elliptic-function filters / by Kendall L.
Suo
p. cm.
Includes bibliographical references (p. ).
ISBN-13: 978-1-4612-8829-9 e-ISBN-13:978-1-4613-1547-6
DOl: 10.1007/978-1-4613-1547-6
1. Electric filters-Design and construction-Tables. 2. Electric
network synthesis-Tables. 3. Functions, Elliptic-Tables.
1. Title.
TK7872.F5S8 1990
621.381 '5324-dc20 90-4540
CIP
Copyright © 1990 by Kluwer Academic Publishers
Softcover reprint ofthe hardcover 1st edition 1990
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photocop
ying, recording, or otherwise, without the prior written permission of the publisher,
Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell,
Massachusetts 02061.
CONTENTS
Preface Vll
Chapter 1 General Information 1
Normalization, Terminology, and Notation 2
Locations of Maximum and Minimum Magnitudes 5
Accuracy and Significant Figures 6
Chapter 2 Tables with Prescribed ap and Ws 23
Chapter 3 Tables with Prescribed a and as 131
p
Chapter 4 Interpolation and Summaries 267
In terpolation 267
a. Two-point or linear interpolation 267
b. Three-point or parabolic interpolation 268
Summaries of Filter Parameters 271
Bibliography 287
Symbol Index 289
Subject Index 291
v
PREFACE
This handbook is inspired by occasional questions from my stu
dents and coworkers as to how they can obtain easily the best network
functions from which they can complete their filter design projects to
satisfy certain criteria. They don't need any help to design the filter.
They need only the network function. It appears that this crucial
step can be a bottleneck to designers. This handbook is meant to
supply the information for those who need a quick answer to a simple
question of this kind.
There are three most useful basic standard low-pass magnitude
characteristics used in filter design. These are the Butterworth, the
Chebyshev, and the elliptic characteristics. The Butterworth charac
teristic is maximally flat at the origin. The Chebyshev characteristic
gives equal-ripple variation in the pass band. The elliptic character
istic gives equal-ripple variation in both the pass band and the stop
band.
The Butterworth and the Chebyshev characteristics are fairly
easy to use, and formulas for their parameters are widely available
and fairly easy to apply. The theory and derivation of formulas for
the elliptic characteristic, however, are much more difficult to handle
and understand. This is chiefly because their original development
made use of the Jacobian elliptic functions, which are not familiar to
most electrical engineers. Although there are several other methods
of developing this characteristic, such as the potential analogy, the
Chebyshev rational functions, and numerical techniques, most filter
designers are as unfamiliar with these methods as they are with the
elliptic functions.
Vll
Vlll PREFACE
For this reason, the elliptic magnitude characteristic remains
somewhat of a mystery to many engineers. Although it is well es
tablished that the elliptic characteristic offers the best performance
for the same filter complexity, some engineers may resort to using
sub-optimal filter functions out of expediency. This sacrifice is un
necessary and can easily be remedied if a designer has access to the
network function. In passive filters, we do pay a small price in terms
of network complexity for using elliptic filters over Butterworth· and
Chebyshev filters. In numerous other applications, such as active
filters, microwave filters, and digital filters, it is usually more advan
tageous to use elliptic filters whenever the filtering requirements are
moderately or extremely stringent.
From a practical point of view, the basic principles and mathe
matical intricacies that lead to the network functions that have the
elliptic magnitude behavior are not really important to occasional
filter designers, as long as they are able to obtain the network func
tion for a given set of specifications. This handbook is designed to
fulfill this need and enable designers to bypass the need to delve into
the relatively complex mathematical formulas and unfamiliar func
tions. It is hoped that this handbook will enable engineers to obtain
the best network functions for their tasks at hand without having to
worry about where they came from or how they were generated.
Although a few tables for elliptic-function filters do exist, they
tend to fall into two categories. Most of the more extensive ones
were originally generated primarily for passive filter synthesis. In
those tables, not only network functions but also element values are
tabulated. Because of the earlier emphasis on passive realization,
the parameters they use usually require indirect interpretations for
applications to filters other than passive ones. For example, the
transition band ratios are varied by varying one of the parameters
related to the elliptic function through the elliptic integral of the first
kind-the modular angle. Another example is that passband ripples
are not given in terms of dB. Rather, they are given in terms of the
(maximum) reflection coefficient in the pass band. These tables are
simply collections of computed network function coefficients with
out regard to the convenience of their use. To borrow a term from
PREFACE IX
computer jargon, these tables are not "user friendly." Interpolation
is almost always required since most listed parameters are not the
same as those used in the usual engineering specifications.
The other category of tabulations available is tables included in
modern filters textbooks. Their notation and terminology are more
contemporary and easier to use. These tables are typically fairly
short and are intended primarily as examples for the text narrative.
They are not sufficiently extensive to serve as stand-alone references
for designers to look up from time to time.
Another major inconvenience of existing tables is that all of them
tabulate everything for one n (order of network function) at a time.
This makes it necessary to jump from table to table when one wishes
to know which n is high enough for a given task.
This handbook lists all entries with the same parameters in the
same table as n is increased from 2 to 12. A user will not have
to leaf through several tables to locate the parameter combinations
in question. The spacing of parameters is sufficiently fine that an
engineer can usually find the functional coefficients needed in one of
the tables. If, on some rare occasions, interpolations should become
necessary, very good approximation can be obtained from entries in
these tables with relative ease.
All tables in this handbook are for elliptic filter functions classi
cally known as Case A. These functions are not suitable for passive
ladder realization without mutual inductances or ideal transformers
when n is even. The difficulty stems from two properties of the el
liptic magnitude characteristics for even n: (1) the magnitude is not
equal to zero at infinity, (2) the magnitude is not a maximum at the
origin. To circumvent property (1), a transformation can be used to
move the highest transmission zero to infinity. The resulting network
functions are known as those of Case B. To also circumvent prop
erty (2), the lowest maximum is moved to the origin. The resulting
network functions are known as those of Case C. Neither Case B
nor Case C is addressed in this handbook, both because they are of
limited importance in modern filter applications and because these
transformations always degrade the filter performance.
x PREFACE
To facilitate the look-up procedure for which this handbook is
meant to be used, the narrative has been kept to a minimum. The
reader is referred to the references listed in the Bibliography for de
tailed theory, derivations, and formulas. Chapter 2 contains tabu
lations for given pairs of passband ripple and transition band ratio.
Chapter 3 contains tabulations for given pairs of passband ripple and
stopband attenuation. Chapter 4 contains a brief discussion of two
simple schemes of interpolation that are useful when certain parame
ters do not coincide exactly with the ones tabulated. It also contains
summaries of the interrelationship among the four key parameters
extracted from tables in Chapters 2 and 3.
Kendall L. Su
HANDBOOK OF TABLES FOR ELLIPTIC-FUNCTION FILTERS
Chapter 1
GENERAL INFORMATION
One of the key steps in the design of a filter is the selection of a
network function, H{s), such that its magnitude characteristic satis
fies a certain set of specifications. For many applications in which the
bounds in both the pass band and the stop band are constant, it is
adequate to utilize one of the several standard characteristics. These
standard characteristics are usually studied in their low-pass form
and can be adapted to high-pass, band-pass, or band-elimination
applications through some simple frequency transformations.
Two standard low-pass magnitude characteristics are of the all
pole type-all their network function zeros are located at infinity.
The Butterworth characteristic is maximally-flat at the origin. The
Chebyshev characteristic has equal-ripple variation in the pass band.
Both characteristics decrease monotonically outside the pass band.
Because of their all-pole property, network functions for these char
acteristics are easy to compute and the configurations of these filters
are comparatively simple. For moderate filtering requirements, these
two types of filters are adequate.
The third standard magnitude characteristic gives equal-ripple
variation both in the pass ba.nd and in the stop band. This type of
characteristic is commonly known as the elliptic characteristic be
cause the derivation of such functions makes use of the Jacobian
elliptic functions. Filters having this characteristic are known as the
Gauer or elliptic filters. Network functions with this characteristic
have transmission zeros on the jw axis. The presence of these finite
zeros greatly enhances the performance of these filters. It is gener
ally accepted that this is the optimum low-pass characteristic when
the specifications in the pass band and in the stop band are both
constant.
1
