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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 4, APRIL 1997 273

Circuits and Systems Expositions

Elliptic Functions for Filter DesignH. J. Orchard,Life Fellow, IEEE, and Alan N. Willson, Jr.,Fellow, IEEE

Abstract—This paper offers a simple description of the ele-mentary properties of the Jacobian elliptic functions and of theLanden transformation, which connects them with the circularand hyperbolic functions and thereby provides one of the most ac-curate methods of evaluating elliptic functions. The use of ellipticfunctions in creating equal-ripple lowpass filters is explained andtheir numerical evaluation is illustrated by means of an example.A Fortran program for effecting the design is included and afaster and more accurate replacement for the Matlab programELLIPAP is given.

Index Terms—Elliptic functions, filter design.

I. INTRODUCTION

I T HAS BEEN KNOWN since the 1930s that lowpass filtertransfer functions with equal-ripple loss response in both

passband and stopband can be described exactly with Jacobianelliptic functions [1]. Such filters are variously described asCauer filters, elliptic-function filters and sometimes just aselliptic filters, though the last name is to be deprecated assuggesting that the filters are egg-shaped! They have beenwidely used in the past for all kinds of analog filters, and nowfind frequent application to IIR digital filters.

Unfortunately, very few engineers receive any instructionin the properties of elliptic functions during their education,and a perception has arisen that their numerical evaluation, asneeded in a filter design, is both complicated and difficult. Thishas motivated several writers to devise ways of designing thisclass of filter without any overt reference to elliptic functions[2]–[5]. One most ingenious approach by Darlington [6], [7]adapts the Chebyshev rational fraction into a sequence oftransformations whose end result is the transfer function ofan odd-degree elliptic-function filter.

The purpose of this tutorial paper is to give a simple descrip-tion of some of the properties of the Jacobian elliptic functionsand their relationship to the circular and hyperbolic functionsas they apply to the filter problem. No attempt is made togive a mathematically rigorous and formal development of thetheory or, necessarily, to prove any of the results described;the aim is merely to convey some appreciation of the general

Manuscript received October 11, 1995; revised August 8, 1996. This workwas supported by the National Science Foundation under Grant MIP-9632698.This paper was recommended by Associate Editor W. Mathis.

The authors are with the Electrical Engineering Department, University ofCalifornia, Los Angeles, CA 90095 USA.

Publisher Item Identifier S 1057-7122(97)02727-X.

properties of these functions. The reader is assumed to havereceived a typical engineering undergraduate education whichhas included a first course in complex variable theory. Readerswho are interested in pursuing the subject in more detail arehighly recommended to the classic book by Neville [8], thoughit must be confessed that this is not a text too well suited forcasual reference.

This discussion of the Jacobian elliptic functions leads intoa description of the Landen transformation, and hence to avery accurate method of computing the functions. With thisbackground, we show how the elliptic functions can be usedto define the desired filter response by means of a pair ofparametric equations in exactly the same way as circularfunctions are used in defining the Chebyshev filter. Finally, wedescribe the details of how the poles and zeros of the transferfunction can be calculated via the Landen transformation. Thepaper includes the listing of a Fortran program for carrying outthe design and a numerical example to illustrate the steps. Afaster and more accurate replacement for the Matlab programELLIPAP is also given.

II. PERIODIC FUNCTIONS

A. Singly Periodic Functions

Among all functions of a complex variable, elliptic functionsare distinguished by being doubly periodic. In this respect theyare similar to, though somewhat more complicated than, thevery common singly periodic elementary functions. A singlyperiodic function with a period satisfies the relationship

for all , and hence alsofor all integers .

The basic elementary function is singly periodic withan imaginary period , and this same imaginaryperiod is shared by the hyperbolic functions andwhich are merely linear combinations of and .Replacing by in the exponential function changes it to asingly periodic function with a real period , which is sharedby the circular functions and .

The complex plane of the variable of a singly periodicfunction can, by virtue of the periodicity, be regarded as beingdissected into an infinite number of congruent period strips ofwidth equal to the value of the period. Corresponding to anarbitrary point lying in any one of these strips there will be

1057–7122/97$10.00 1997 IEEE

274 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 4, APRIL 1997

(a)

(b)

Fig. 1. Period strips for (a)sin z and (b)sinh z.

a congruent point in every other strip, at , such thatthe function values at all these points are identical. The periodstrips for and are of width and of infinite length,lying side-by-side, parallel to the imaginary axis, while thosefor and are the same, except that they lie parallelto the real axis. This is shown in Fig. 1.

The functions and , by contrast, have periodsof and respectively, and so their period strips will havea width , not . If , where is the circular orhyperbolic sine or cosine, we find that an infinite strip of width

, lying inside a period strip and centered around a zero ofthe function, i.e.,half of a period strip, will be mapped, one-to-one, onto the whole of the plane, whereas if is eithertangent function, then is a one-to-one mapping of thewholeof a period strip onto the entire plane.

B. Doubly Periodic Functions (Elliptic Functions)

The existence of singly periodic functions then raises thequestion whether there are functions with two, or possiblymore, distinct periods. It is not difficult to show that: 1)one cannot have functions with three or more periods and2) if a function has two periods, the ratio of the periodscannot be real; in other words, the periods must point indifferent directions in the complex plane and not lie alongthe same straight line. Subject only to this restriction, one can

Fig. 2. The fundamental rectangle.

construct a single-valued, doubly periodic function with anytwo assigned complex numbers,and , as the periods, sothat we have for all and for allintegers and .

The double periodicity associates with each value ofan infinite two-dimensional lattice of pointsin the plane at which the function values are identical.Around these points can be visualized an infinite array ofperiod parallelograms with sides and such that, for anyvalue of , the lattice forms a set of congruent points in theparallelograms. As the lattice of points canequally well be described by or

it is clear that the periods are notquite unique and can be taken, e.g., as any two of and

. This would change the shape, though not the area, ofthe corresponding period parallelograms.

Inside each period parallelogram must possess atleast one singularity, for otherwise it would possess none atall, and hence by Liouville’s theorem would be a constant.Moreover, the integral of around the boundary of aperiod parallelogram must vanish, because the contributionsto the integral from opposite sides of the parallelogram willcancel owing to the periodicity. Hence, by the Cauchy residuetheorem, the sum of the residues at the singularities inside theparallelogram must also vanish. As a simple pole with zeroresidue is no pole at all, it follows that the simplest ellipticfunctions have either one double pole with zero residue, ortwo simple poles with equal and opposite residues.

The complexity ororder of an elliptic function is measuredby the sum of the orders of the poles contained within a periodparallelogram. We see that there are no elliptic functions oforder one, and two different varieties of order two. It canreadily be appreciated that if is an elliptic function, thenso is , with the same order and periods. It follows fromthis that an elliptic function of order two has either one doublezero or two simple zeros in each period parallelogram.

An elliptic function of order two, with one double poleof zero residue per parallelogram, was first described byWeierstrass, and bears his name. Its simplicity makes it afundamental tool in the theory of the subject, and it can alsobe used as a stepping stone for the systematic constructionof the elliptic functions of order two with two simple polesper parallelogram. The latter, when properly normalized, arethe Jacobian functions with which we are primarily concerned.

ORCHARD AND WILLSON: ELLIPTIC FUNCTIONS FOR FILTER DESIGN 275

(a) (b)

(c)

Fig. 3. Pole–zero patterns for (a)snu, (b) cnu, and (c) dnu.

For practical applications they are much more useful than theWeierstrass functions whose discovery they predate by manydecades. They also have two simple zeros per parallelogramas well as two simple poles.

III. T HE JACOBIAN FUNCTIONS

In the same way that it is convenient to have six differentcircular functions, i.e., sine, cosine, tangent, and their recip-rocals, there are twelve different Jacobian elliptic functions.Just as the sine and the tangent have periods ofand ,respectively, so the Jacobian functions also have periods thatdiffer from one group to another by a factor of two. It has longbeen the standard practice to describe the periods in terms ofquantities that are referred to as thequarterperiods, whichplay the same role for the elliptic functions as does forthe circular and hyperbolic functions.

The nature of the practical problems to which the Jacobianfunctions find application normally require one quarterperiod,denoted by , to be real and one, denoted by , to beimaginary. We shall restrict our discussion to this special, butvery important, case. The rectangle in the first quadrant of thecomplex plane of the variable that has its corners at the points

and is referred to as thefundamentalrectangle. The corners of this rectangle are denoted by letters.The corner at the origin is S (Starting point), the diagonallyopposite corner is D, the corner that coincides with the realaxis is C, and the corner that is normal to the real axis is N.

(The alliteration in this description is intended as a mnemonic!)The arrangement is shown in Fig. 2.

Every Jacobian function has a zero at one corner of thisrectangle and a pole at some other. As the zero can first beplaced at any one of the four corners, and then the pole placedat any one of the remaining three corners, this gives a total of 4

3 12 possibilities; these are the twelve different Jacobianfunctions mentioned above. Each of the functions has a two-letter name which neatly indicates its pole–zero pattern. Thefirst letter is that of the corner containing the zero, and thesecond that containing the pole. Thus the sn function, e.g., hasa zero at S and a pole at N, and so on.

The pole–zero pattern for each of the twelve functions thenextends fairly simply from the fundamental rectangle to therest of the complex plane. Both the poles and the zeros of eachfunction are arranged on the same basic geometrical lattice ofpoints, namely for all integers and , andare distinguished one from another only by the displacement

of the lattice from the origin. This displacement is equal tothat of the pole or zero of the specific function at its corner ofthe fundamental rectangle. For example, thefunction hasits pole at N, for which , so the set of poles lies at

, while the zero is at S, for which , sothe set of zeros lies at . The pole–zero patternsfor the three classic functions are shown in Fig. 3.

Around the boundary of the fundamental rectangle thereare, of course, two possible paths from the zero to the poleof each Jacobian function. The one that includes the side of


Fig. 4. Forw = snu, the four rectangles shown in theu-plane map onto thewhole of thew-plane. The number in each separate rectangle is the numberof the quadrant in thew-plane onto which it maps.

the rectangle lying on the positive real axis will always bemapped by the function onto the whole of the positive realaxis, while the other will be mapped onto the whole of eitherthe positive or negative imaginary axis. The function valueon the boundary of the fundamental rectangle always changesfrom real to imaginary, or vice versa, at a corner where thereis a pole or a zero. This means that each Jacobian functionmaps the fundamental rectangle onto either the first or fourthquadrant.

If we visualize the whole of the plane of the variablecovered by rectangles congruent to the fundamental rectangleof an elliptic function and having corners at

, as in Fig. 3, then each such rectangle will bemapped by onto one of the four quadrants of theplane.Any 2 2 block of four of these rectanglesthat has a zeroof the function f at its centerwill always be mapped onto thewhole of the plane. For example, the function has a zeroat the origin and the four rectangles that surround this zerowill be mapped onto the whole plane, as shown in Fig. 4.

The circular functions could, if desired, be arranged tohave their period specified independently as a parameter,but it is simpler for both definition and tabulation to havethe period fixed at or and to let the user adjust theperiod by a scale factor on the variable. In exactly the sameway the Jacobian functions could have both their real andimaginary quarterperiods as two independent parameters. Butthe essential properties of the functions depend only on theshapeof the fundamental rectangle, not its absolute size, andthe standard practice is to have just one parameter that specifiesthis shape, leaving the size to be adjusted automatically tosatisfy certain normalization constraints.

The Jacobian elliptic functions originated, and also derivedtheir name, from inverting an integral that was used in com-puting the arc length of an ellipse. The function ,e.g., is defined in all respects by the integral

(1)

The parameter that occurs in this integral is referred to asthe modulus(not to be confused with the other use of this

word to mean the magnitude of a complex number) and, forthe practical case we are considering, is real and satisfies

. It is often specified through amodular angle as

. Associated with is a complementary modulus. Usually, the modulus of the functions one is

dealing with is known and need not be explicitly stated foreach function as it occurs, but if it must be made clear wewrite, e.g., .

This way of defining the Jacobian functions causes not onlythe ratio of the quarterperiods to be fixed by the singleparameter , but also their absolute sizes, and thereby achievessome very simple normalizations of all twelve functions. Inparticular, the six functions ( ) that haveeither a zero or a pole at the origin are odd functions, withunit derivative there if a zero, or unit residue if a pole. Theremaining six functions ( ) are even andhave unit value at the origin. Another consequence of thisnormalization is that if represent any three ofthe Jacobian function . The threefunctions were the original ones that Jacobi obtainedin 1827 by inverting elliptic integrals, and the other nine wereintroduced in 1882 by Glaisher as reciprocals and quotientsof these first three.

If, in (1), we set and note that we getan integral expression for the quarterperiodin terms of themodulus , namely,

(2)

and if one replaces by in (2) then the resulting integralgives instead of . When , the integral in (1)simplifies to that for the arcsin function and we get

. As , we see from (2) thatwhen . Similarly, when , the integral

in (1) simplifies to that for the function and we get. In this case the integral in (1) diverges

as , and so from (2) we see that when .By replacing by in (2) we can deduce thatwhen (and ) and when (and

).The squares of the circular and hyperbolic sines and cosines

satisfy simple linear relationships, namely,

(3)

and from these all the remaining circular and hyperbolicfunctions may be found. The Jacobian functions possessproperties exactly analogous to these. They hold between anytwo of the three functions that share common poles. Thethree classic functions form such a set with poles at

and satisfy

(4)

From these, using the properties given above, one can derivethe relations between the members of the other three sets ofcopolar functions.


Finally, we consider the effect of changing the argument ofa Jacobian function from to , sometimes calledJacobi’simaginary transformation. This change amounts to rotating the

-plane around the origin by 90, and when applied to thecircular or hyperbolic functions would rotate the period stripsof one class of function into the period strips of the other. Wehave, e.g., the well-known results

(5)

For the fundamental rectangle of the Jacobian functions this90 rotation around the origin proves to be equivalent to arotation of the plane around a 45line through the origin. Itthen becomes obvious that, in the fundamental rectangle, thereal and imaginary quarterperiods, and exchange roles,as hence also do and . Apart from being interchanged inposition, the quarterperiods do not alter in size.

This rotation of the fundamental rectangle carries with it thepole and zero at two of its corners. A pole or zero at the cornerslabeled S or D will thus still be at these same labeled cornersafter the rotation, but a pole or zero at the corner labeled Cwill appear at the corner labeled N, and vice versa. Thus in thename of any Jacobian function so transformed we must change

to and to , and then change the modulus fromto .We get the following results for the three classical functions:

(6)

The results for the remaining nine functions can be found bytaking quotients and reciprocals of these three. The six oddfunctions will involve a factor , as with the function,while the six even functions will remain real, as with theand functions.

Although we began by noting that double periodicity isthe distinguishing characteristic of all elliptic functions, adetailed knowledge of precisely what periods each of thetwelve Jacobian functions possesses is of little use for ourpresent purposes, and we shall pursue the matter no furtherthan to mention, what the reader may already have deduced,that the and functions we shall later use have periodsof and .

IV. THE LANDEN TRANSFORMATION

If the modulus tends to zero, the ratio will tend toinfinity and the Jacobian functions and their period rectangleswill degenerate into the circular functions and their periodstrips respectively. Conversely if tends to unity,will tend to zero and the Jacobian functions and their periodrectangles will degenerate into the hyperbolic functions andtheir period strips. At the circular-function limit ,

, , and , whereas at the hyperbolic-function limit , , , and .

There is complete symmetry between the primed and unprimedparameters.

This exhibits the elliptic functions as occupying a contin-uous path extending from the circular functions at one endto the hyperbolic functions at the other, and withandacting as symmetric measures of position along it. At thecenter of the path the fundamental rectangle is a square with

and . The Landentransformation is a method of moving in discrete steps alongthis path, in either direction, by modifying or so that ineach step the ratio is either doubled or halved. Thefunction values before and after one such transformation arequite simply related algebraically.

Four or five successive transformations are normally suf-ficient to move from any practical value of occurring ina filter design to a point where the elliptic functions arenumerically indistinguishable from the limiting circular orhyperbolic functions. The latter can be evaluated and then,by computing the intermediate elliptic functions, one fromanother, via the algebraic relations connecting them, one canfind the wanted elliptic functions. In the filter case it is simplerto carry out this computation from the circular-function end ofthe system, even though slightly fewer transformations mightbe needed to reach the hyperbolic-function end. This is becausethe circular functions needed to start the calculation are easierto find than the hyperbolic functions.

Consider the function

(7)

in which we include as a scale factor on the variablesothat the side of the fundamental rectangle lying along the realaxis in the plane is normalized to unit length, independent ofthe value of . The first term has poles at ,while the second has poles at the zeros of thefunction inthe denominator, which are at .The residues at all these poles with respect to the variable

have the value . The factor in the numeratorof the second term is needed because the zeros of thefunction in the denominator have a derivative with respectto of magnitude . The complete function thus has polesat with residues . These are exactlythe poles of an function whose modulus is chosen so thatthe ratio of its quarterperiods is one half of .

Let such a function be , where the modulus ischosen so that the quarterperiodsand satisfy

. Using this result we see that the poles ofare at . When ,both and are equal to unity, so a factor

is therefore needed to complete the following identity:

(8)

The function with modulus is one Landen transfor-mation further from the circular function end of the paththan the function with modulus , and (8) is the algebraicrelation referred to above. The normalization achieved by thequarterperiod factor on and the change of shape arising


from the transformation cause the fundamental rectangle formodulus to occupy exactly thebottom halfof the rectanglefor modulus .

A result identical to (8) would apply also to the dc function.Both and functions have their poles and zeros interlacedalong lines parallel to the imaginary axis, i.e., to the periodstrips of the circular functions, and it is this which makesthem suitable for tracking from the circular-function end ofthe path. Next, to complement the relation in (8), we need toknow how to calculate from .

This can be found by evaluating both sides of (8) at the Dcorner of the fundamental rectangle for modulus, i.e., at thepoint . From the remarks above it will be seenthat the D corner of the rectangle for modulusis only halfway along the side from the C corner to the D corner in therectangle for modulus . The value of the function at itsD corner is equal to its modulus, whereas midway from C toD the value is equal to the square root of the modulus. Usingthis information we find that (8) reduces toat this value of .

To organize the notation in a form more suitable for calcula-tion, we indicate the moduli and quarterperiods of the ellipticfunctions that occur after successive Landen transformationstoward the circular-function end of the path by and ,where is an integer and corresponds to the wantedfunctions. In this notation the result obtained above becomes

(9)

Inverting (9) leads to a quadratic in whose twozeros are reciprocals of one another. Because theare allless than unity we must here choose the smaller one, whichcan be written in a computationally stable form as

(10)

The recurrence in (10) generates a sequence of moduli,which rapidly tends to zero with increasing, and is ter-minated when the modulus is less than , where isthe number of decimal digits used in the arithmetic. Notethat this reduction in size of the moduli isnot achieved bysubtractions, which could adversely affect numerical accuracy,but by division and squaring, and that, apart from the usualminor effects of rounding, the full -digit accuracy will bepreserved throughout the sequence.

Although only two of the twelve Jacobian functions,and, can be tracked back from the circular-function end, any

of the other eleven functions can be found from whicheverof these two happens to have been employed by using therelationships derived from (4). Fortunately, the filter problemuses primarily the or functions which require only takingreciprocals of the tracked functions. The function is thefunction shifted by a quarterperiod, i.e., . Inthe new notation both the and the functions satisfy the

rewritten version of (8), namely

(11)

As in (8), the arguments of all the elliptic functions areexpressed as a certain fraction, here denoted by, of theappropriate quarterperiods. This is particularly convenient forthe filter application because almost all of the wanted ellipticfunctions have arguments that appear naturally as prescribedfractions of the quarterperiod. The limiting value of thisargument as one approaches the circular-function end of thepath is thus , and the recurrence in (11) starts therewith . The identical recurrence for the functionwould start with .

If we transform our elliptic functions instead to thehyperbolic-function end of the path, then it is, not ,that is used for tracking the elliptic functions. Again we attachan integer subscript to the moduli and quarterperiods toindicate the values obtained after Landen transformationstoward the hyperbolic end. There should be no confusionwith the subscripts previously described as we never use bothpaths at the same time.

As in the case of the path from the circular-functionend, only two Jacobian functions can be tracked from thehyperbolic-function end. These are the and functionswhich have their poles and zeros interlaced along lines parallelto the real axis, i.e., to the period strips of the hyperbolic func-tions. The derivation of the formula for or , correspondingto (8), follows an almost identical argument with but one smalldifference. Now, the fundamental rectangle in theplane formodulus must occupy theleft-half (not the bottom half) ofthe rectangle for modulus. This requires to be chosen sothat , and a factor , in lieu of just , on thevariable on the left-hand side. The formula relating adjacentvalues of is found in the same way as for in (10) and is

(12)

This behaves with increasing exactly as did in (10), andthe sequence is terminated in the same way.

The recurrence for the function, corresponding to (11)for the function, is

(13)

The recurrence for the function differs from this only byhaving a plus sign in place of the minus sign between the twomain terms inside the square brackets.

At the circular-function end has the limiting value, and the starting circular functions thus have the simple


TABLE I

argument . But, at the hyperbolic-function end, tendsto infinity, and it is that has a unique limiting value.The argument of the starting hyperbolic functions is thereforetimes this limiting value. If we take transformations to reachnumerical equality with the hyperbolic functions we must firstcalculate using, e.g., the following approximation, derivedfrom the theory of theta functions [8], which is extremelyaccurate when is very small

(14)

Then, the starting hyperbolic function for the recurrence in(13) is

(15)

We illustrate these methods of calculation by showingthe steps in finding , first from the circular-function end and then from the hyperbolic-function end.Double precision arithmetic, equivalent to 14 decimal digits,was used throughout. Table I shows the quantities involved inthe calculation from the circular-function end. Only the firstten out of the 14 digits used are given in order to simplify thedisplay. Five transformations suffice to reduce the modulus toless than 10 , and adjacent to this smallest modulus in thetable is the value of , which is the startingcircular function. After tracking this via (11) back towe get the corresponding function value. The reciprocal ofthis is .

Table II shows the similar quantities occurring in the calcu-lation from the hyperbolic-function end. Five steps are neededhere also to reduce the complementary modulus to below10 . At this stage we use (14) to find the real quarterperiodas and this is thendivided by to give . One-third of thisgives the argument of the function that appears inthe table for , adjacent to the smallest complementarymodulus. The function is then tracked back to byusing (13) and the wanted function found via (4), rewrittenas . This gives the same value, to 14 decimaldigits, as the function calculated previously from the circular-function end.

V. THE ELLIPTIC-FUNCTION FILTER

For the first half century following the invention of electricfilters, designers used what seemed the most natural way

TABLE II

of describing their behavior, with an input/output transferfunction. But then, with the growth of system theory that usedoutput/input transfer functions, filter designers were (to putit politely) persuaded to adopt the same convention. We now,e.g., see the response of purely passive filters plotted as a gain,in decibels, withall ordinates negative! It is not our intentionhere to argue the merits of the two choices, but merely to pointout that in this paper we shall describe the behavior with theinput/output transfer function .

The lowpass filter that we wish to discuss, with equal-rippleloss response in the passband and equal-minima responsein the stopband, can be regarded as a generalization of theChebyshev filter whose loss is equal-ripple in the passbandand monotonically increasing in the stopband. To make thedescription of the former easier to follow, we analyze theChebyshev filter in a way that allows a simple step-by-stepgeneralization to the case of the elliptic-function filter.

We define the normalized Chebyshev filter in terms ofthe three complex variables , and by the parametricequations

(16a)

(16b)

where

dB (17)

is the peak-to-peak magnitude of the equal-ripple passbandloss and . One can, of course, eliminate the parameter

and end up with an equation containing a Chebyshevpolynomial, but it is more difficult to analyze in that form. Thequantity is used in place of a single parametric variableso that the factor can later reappear as a Jacobian quar-terperiod. The integer defines the degree of the polynomial

and causes the period strips of the cosine in (16a) to beone th the width of those for the cosine in (16b), so thatof the former fit exactly, side-by-side, into one of the latter.

To find the natural frequencies of the filter, which are thezeros of , we set (16a) to zero and solve to obtain thecorresponding values of . These are then substituted into(16b) to obtain the wanted values of, and hence of. Setting(16a) to zero gives

(18)

For to be imaginary, must lie on those lines in the-plane that map onto the imaginary axis of the plane of the


cosine. These are the lines parallel to the imaginary axis inthe -plane having real axis values of . Henceif

or

(19)

For these values of

(20)

Hence from (18) we can take

(21)

Using a negative sign for the right-hand side leads to exactlythe same set of zeros for . From these we have toextract the ones lying in the left half of the-plane to get thezeros of . Solving (21) for gives

(22)

Setting

(23)

we get the expression for the natural frequencies as

(24)

As is a polynomial it has an th-order pole at .To generalize the Chebyshev filter into the elliptic-function

filter it is necessary merely to change the cosine functionsappearing in (16) into their elliptic-function equivalents. Butwe find that there are two different elliptic functions, and

, that share the cosine as their limiting form when ,and the correct choice for our application becomes clear onlyby considering the path in the-plane that maps onto thepositive real -axis via (16b). The real axismaps onto the passband , while the imaginary axis

maps onto the stopband . The origin in the -plane is the point that separates the equal-ripple passband fromthe monotonically increasing stopband. The Chebyshev filteritself has no explicit transition band as such, distinct from thestopband, although the specification that it has to meet will,of course, usually include one.

The elliptic-function filter, on the other hand, does havean explicit transition band lying between the equal-ripplepassband and the equal-minima stopband, and this requiresthe path in the -plane that maps onto the positive real-axis

(a)

(b)

Fig. 5. The mappings of (a)cd(uK; k) and (b) cd(4uG; g) onto theu-plane, given that4K0=K = G0=G.

to have three distinct segments corresponding to the passband,transition band and stopband. Only the cd function can do this.The path around the fundamental rectangle, from the zero atC to the pole at D, that maps onto the positive real axis of the

function traversesthree sides of the rectangle which canthen be arranged to correspond to the three required bands.

We give first the equations that define the filter and thenexplain how they are related. They are

(25a)

(25b)

(25c)

where and are the quarterperiods associated with themodulus in (25b), while and relate similarly to modulus

in (25a). As before, the integer defines the degree of therational function , and the passband ripple is given by(17).

The mapping of the boundary of the fundamental rectanglein the -plane for (25b) onto the -axis, is shown in Fig. 5(a),which indicates the values corresponding to the corners,and the three sides belonging to the passband, transition bandand stopband. Owing to the inclusion of the quarterperiodfactors or on the variable , the normalized passband

maps from the same range, , for boththe Chebyshev filter and the elliptic-function filter. The edgeof the stopband is at , and this defines the modulus.

The modulus of the other elliptic function is chosenimplicitly by the need to meet the condition on the quarterperi-ods specified in (25c). This condition causes the fundamental


rectangle of to be of such a shape and size (inthe variable ) that of them can be fitted exactly side-by-side into that of . Fig. 5(b) shows, for the case

, how the pole–zero pattern and function values ofappear superimposed on the fundamental rectangle

of .It is this satisfaction of (25c) by the choice of, and the

resulting simple geometric relationship of the two fundamen-tal rectangles that make a rational function of

, just as is a polynomial in .Note that as tends to zero, so also does, and then (25a)and (25b) degenerate into (16a) and (16b), respectively. Thiscauses the elliptic-function filter to reduce, in the limit, to theChebyshev filter. In fact it is the transition band of the elliptic-function filter, not its stopband, that becomes, in this limit, thestopband of the Chebyshev filter.

To find the natural frequencies of we use the samemethod as for the Chebyshev filter. Setting (25a) to zero gives

(26)

For to be imaginary, must lie on those lines in the-plane that map onto the imaginary axis of thefunction.

These are the lines parallel to the imaginary axis in the-plane that pass through the poles and zeros of andhave real-axis values of Hence

or

(27)

which is precisely the value found in (19) for the Chebyshevfilter. A direct generalization of (20), with the use of (6), gives

(28)

and so from (26) we can take

(29)

which is the counterpart of (21).At this stage we transform the elliptic functions and their

parameters in (25) by a sequence of Landen transformationstoward the circular-function end of the Landen chain. Wegenerate first the descending sequence of moduliusing(10) and terminating with when the elliptic functions arenumerically indistinguishable from circular functions. Then wecreate the corresponding sequence. However, we are notgiven explicitly, only its relation to implicitly by(25c). When the latter is visualized after Landen transfor-mations, both and will have become equal atand so (25c) reduces to

(30)

Just as we can calculate from at the hyperbolic-functionend using (14), so we can use the complement of (14) to find

from at the circular-function end when is very small.This leads, after some fairly obvious elementary algebra, to

(31)

From this value of we can construct the sequencebackward to using the recurrence (9) which appears here as

(32)

For large values of , (31) may create a smaller thanthe lower (nonzero) limit of the floating-point arithmetic, thusmaking it impossible to compute the Landen chain ofvia (32). If this happens one can instead calculate , orpossibly even , from . This gives a shorter Landenchain for the than for the , reflecting the reduced sizeof compared to . At these very small values of , onecan safely replace by unity, and then (32) simplifies to

. Combining this with (31) leads to

and

(33)

Once we have , we can use the fact that the magnitude ofthe function in (25a) is equal to at the stopband lossminima to obtain the loss there with the formula

dB (34)

With the two sequences of moduli and computed,there remains the task of finding the zeros and poles of .We consider first the slightly more complicated case of thecomplex zeros. The Landen transformation was described interms of obtaining values of real elliptic functions from theirreal limiting circular-function values via the recurrence in (11).But these are all analytic functions, and so it should comeas no surprise that the recurrence also holds for complexvalues of these functions. We merely have to find the complexvalues of for the correct Chebyshev filter, as in (24),and transform their reciprocals via (11) for the functionback from modulus to modulus . The reciprocals of theend results of these transformations give the wanted naturalfrequencies.

A Chebyshev filter is defined by two parameters, the degreeand the quantity that controls the size of the passband

ripple. Clearly, the degree of the correct Chebyshev filter willbe the same as the degree of the wanted elliptic-function filter,but the magnitudes of the passband ripples of the two filtersprove to be very slightly different from one another. Thereason for this lies in (29) which shows that the value of

for the elliptic-function filter is itself equal to an ellipticfunction whose value will change, along with all the others,as the system is transformed to the circular-function end ofthe chain. We have to consider how the given value ofismodified in this process.


We begin by writing the reciprocal of (29), and attachinga subscript zero to the given value ofin preparation forgenerating a sequence of values, just as we did withand :

(35)

In the complete chain of Landen transformations, ellipticfunctions with modulus reduce to circular functions at theend where and to hyperbolic functions at the end where

. But for elliptic functions with modulus the situationis reversed; they will reduce to hyperbolic functions when

, because then . So, as we transform the mainelliptic functions of (25) to the circular-function end of thechain, we have to realize that the function appearing in(35), with modulus , will in fact be going to what, for it,will be the hyperbolic-function end.

The recurrence for tracking the function from thehyperbolic-function end is given in (13), but with thedifference that here the modulus is, not , so the rolesplayed by these two quantities are interchanged. Thus therecurrence for (written in place of the elliptic-functionthat it represents) for tracking it from to would be

(36)

But we are starting from and want , not the other wayaround, so we must invert (36) to get

(37)

where

In most cases the change in value fromto is quitesmall and the recurrence in (37) is often omitted. The size ofthe passband ripple in the Chebyshev filter is simply madeequal to that specified for the elliptic-function filter, whichthen will end up with a minutely smaller ripple size thanplanned. The precise magnitude of the ripple is normally notvery critical.

The next step is to find the natural frequencies of theChebyshev filter using (24), with the value of found from(37). The recurrence in (11) for the function relates valuesof the reciprocal of , as defined in (25b), and has to bemodified to become instead a recurrence in the reciprocal of

. This requires only a change in the sign of the lastterm. Thus if is the reciprocal of a typical complex naturalfrequency, starting with as that of the Chebyshev filter, weget the recurrence

(38)

The reciprocal of is the corresponding natural frequency, orzero, of for the elliptic-function filter.

The poles of , which are the frequencies of infiniteloss, lie on the axis at values given by

(39)

where denotes the integer part of. We compute the valuesof using (11), with the starting circularfunction value , and divide by the modulus

. When is odd there will be one pole of at infinity.A Fortran program for carrying out all the calculations

described above is given in Fig. 6. All floating-point variables,real and complex, are in double precision and have beenchosen to reflect as far as possible the variables used in thetext. One very popular, commercially available program fordesigning elliptic-function filters is available in the MatlabSignal Processing Toolbox with the title “ellipap.” Whereasthe design scheme described above takes the degree, thepassband ripple and the elliptic modulus as the startingparameters, the Matlab program uses instead, and thestopband minimum loss . One can easily calculate from

and with (17) and (35), but before the filter designcan begin one has to obtain the modulus, which defines theprincipal elliptic functions.

The Matlab program does this by finding first the twoquarterperiods and from via the arithmetic–geometricmean. An optimization program is then used to findsothat its quarterperiods and satisfy (25c). Yet anotheroptimization program step is used to find to satisfy(29). The , , and functions of arguments andmodulus are computed by the classical arithmetic–geometricmean approach, and the poles of found almost directlyfrom these using (39).

Finally, the complex zeros of are obtained throughan addition theorem for , with as in(27); this is described by Darlington [1] in his equation (65).The two optimization steps are rather time consuming and, asarranged, limit the accuracy of the Matlab-computed poles andzeros to four or five decimal digits.

The design scheme we have described, using the Landentransformation with as a given parameter, can be adaptedwith only trivial changes to accept in lieu of . As in theMatlab program one computes from and andthen forms the Landen chain for , terminating with ata value just safely above the lower limit of the floating-pointarithmetic. is found from via (31), and the Landenchain for computed backward to using (32) with

replacing . The Landen chains for both and are thusavailable with the same effort as in the version starting with

, while the rest of the design remains unchanged.This approach, with as a starting parameter, was pro-

grammed in Matlab. The program (an M-file, ellipap1.m, givenin Fig. 7) which is a direct replacement for “ellipap,” but need-ing no other subroutines, has a running time approximately 80times shorter than “ellipap” and gives poles and zeros to thefull accuracy of the floating-point arithmetic.

It is a matter of personal preference whether one usesoras one of the starting parameters. The fact that the degreemust be an integer means that the smallest value ofthat

meets the specification will normally provide some margin inperformance that should be distributed among the quantities

, , and . How best to do this requires some engineeringjudgment, and the user will need to run the program severaltimes to achieve an optimum distribution of this margin in


Fig. 6. Listing of Fortran program for designing elliptic-function filter.

the design, no matter what choice of starting parameters isadopted.

VI. NUMERICAL EXAMPLE

We choose for our numerical example a typical filter ofdegree 7, with a passband ripple of 0.1 dB and a normalizedstopband edge frequency at . This gives a modulus

, which is the same value we used inour illustration of the Landen transformation in Section IV.Table III gives first the values of , found by using (10) andwhich repeat those appearing in Table I.

From we find by using (31), with . The valuesof are then computed using (32), starting with, and aregiven in Table III next to the column of . is found fromthe prescribed passband ripple, in decibels, by solving (14),for which the straightforward formula would be

(40)

The subtraction in (40) can cause some loss of accuracy whenis small. One can avoid this by the following artifice. We

change from its given value in decibels to its equivalent innepers by multiplying by , and now let represent


Fig. 7. Matlab M-file ellipap1.m.

this quantity. The more accurate expression foris then

(41)

The subtraction in (40) occurs here inside the hyperbolic sine,and most subroutines for will return the function valueto full accuracy regardless of the size of. With this value for

one can now compute the column of in Table III using(37). As can be seen from this column, the limiting value isalready reached by . Such behavior is fairly typical of mostfilters. Only when conditions call for a relatively large value

of (e.g., low degree and small stopband loss) do we find amuch greater change in. As soon as we have the values of

and for the elliptic-function filter the minimum stopbandloss given by (34) can be found. Here it is 55.43 dB.

The next main step is the calculation of the complex naturalfrequencies of the elliptic-function filter. We find first thenatural frequencies of the Chebyshev filter of degree sevenusing the limiting value of and the formulas in (22)–(24).Table IV gives the values for those in the upper half-plane.The reciprocals of these complex zeros are then tracked backto the corresponding elliptic functions of modulus 0.8, one at a


TABLE III

TABLE IV

time, using (38) and the in Table III. All these intermediatesteps for the reciprocal of just one of the zeros, the one for

in Table IV, are shown in Table V as the complex.The reciprocal of in this column is the entry for ofthe upper half -plane complex zeros of the wanted elliptic-function filter shown in the column labeled in Table VI.The calculations for the other zeros, including the one realzero, proceed in exactly the same way.

Finally, the real values for the poles of given in(39) are calculated by making Landen transformations from

to using (11), andthen dividing by 0.8. The intermediate steps for the pole with

are shown as the quantities on the right-hand sideof Table V. It starts with and ends with

. This quantity dividedby 0.8 is in Table VI. The other poles arefound in the same way.

The reader may be interested to compare the steps inthe above calculations with those given by Gazsi [7] inhis Example 4. This is also for a filter of degree 7, with

dB and . The ultimate aimin his paper is an IIR digital filter, but he has to design ananalog filter in the domain as a first step. His calculationsare for the design method described by Darlington [6] whichis an iteration based on the Chebyshev rational fraction. Thefrequency scale is normalized to the geometric mean of thepassband and stopband edge frequencies, rather than just thepassband edge, which is nice from a theoretical point ofview, but a practical nuisance. This accounts, in part, forthe appearance of the Gazsi variables and

in place of the and used above.At first sight the Darlington iteration seems to be nothing

more than the Landen transformation in another guise. Butfurther examination shows it to be a more general approachthat is being specialized to the elliptic-function case, and thatsome, but not all, of its formulas are slightly different fromthose of the Landen transformation. The greater generality ofthe approach, however, is not used to allow any generalization

TABLE V

TABLE VI

of the elliptic-function filter. Darlington indicates that hisscheme is limited to filters of odd degree, though it can easilybe extended to the even-degree case [9]. The accuracy andthe amount of programming involved are about the same asfor the classical method using elliptic functions that has beendescribed here. The latter works equally well for any positiveinteger degree.

VII. A LTERNATIVE COMPUTATIONAL APPROACHES

Although the previous sections have described all theneeded computations in terms of Landen transformations,this tutorial paper would be incomplete without at least somemention of an alternative approach using theta functions whichhas proved appealing to several writers. The theta functionsare integral functions, i.e., analytic functions whose onlysingularities are at infinity. Polynomials are the simplestexamples of integral functions, having a finite number ofzeros. The function is a more complicated example,having an infinity of zeros, uniformly spaced along the realaxis. The theta functions are yet one step more complicatedin that they have a doubly periodic array of zeros coveringthe whole complex plane of the variable. These zeros are at

for all integers and , just as describedin Section III for the poles and zeros of the Jacobian functions.

As in that section, the parameter can take any one ofthe four values corresponding to the corners, S, C, N, and Dof the fundamental rectangle, and this leads to four distincttheta functions which we denote, following Neville [8], by

and . The function has a zero atthe origin, and the derivative there is arranged to be unity. Theother three theta functions each have a zero at one of the otherthree corners of the fundamental rectangle, as designated by itssubscript, and a constant value at the origin which is arrangedto be unity. Various normalizations of the theta functions occurin the literature, but the above, given by Neville, appears tobe the most logical. It then confers the property that, if


and represent any two of the Jacobian function

Even though the theta functions have a doubly periodicpattern of zeros, they cannot, as integral functions, be doublyperiodic. They can, however, be singly periodic, and in theclassic form are so arranged to behave along the real axis.This allows them to be represented by a Fourier series, ofsines for the one odd function , and of cosines for the otherthree which are even. The coefficients of these sines or cosinesare merely integer powers of the quantity

(42)

which connects the theta functions, slightly indirectly, to themodulus of the associated elliptic functions. Withreal and

, we have also real and . For mostpractical values of in filter design, say isless than 0.23. The power ofthat is the coefficient of thethterm is either or and this, combined with the smallvalue of , causes the Fourier series for the theta functions toconverge extremely rapidly.

We illustrate the typical form of the theta functions by givingthe series for one of them, namely,

(43)

where . If we set the elliptic-function variableto be , as in (11) of Section III, the basic argumentof the Fourier series becomes . This exhibits thesame preservation of the argument as a fixed fraction,, ofthe quarterperiod as occurs in the Landen transformation fromthe circular-function end.

A filter specification normally prescribes the modulus, andnot , so the latter must be found first from. This can beachieved via

(44a)

(44b)

which requires roughly as much computing as the Landenchain of values.

Thereafter, finding the poles of given by dc ,as in (39), involves about the same effort whether one usesthe Landen transformation or the theta functions becausethese quantities are all real. It is in the calculation of thecomplex natural frequencies that the Landen transformationshows its superiority. Using (38), with complex valuesfor the reciprocals of the natural frequencies, it can track thelatter from the Chebyshev filter to the elliptic-function filterwith complex arithmetic as easily as it can real values for thepoles. In the theta-function approach there is no correspondingpossibility yet revealed that can compute the complex naturalfrequencies directly. Instead, one has to generate a numberof real elliptic functions, including a special case for the onereal natural frequency (or its equivalent for the case of evendegree), and find the complex natural frequencies by using an

addition theorem for the cd function, similar to that given byDarlington [1].

The first main reference to the details of the theta-functionapproach can be found in a 1957 paper by Grossman [10],a colleague of Darlington at Bell Laboratories. This workdescribes only the case of odd degree filters which was allthat was thought necessary for analog filters of that time. Ithas since been generalized to cover all integer degrees, asrequired for digital filters, in a book by Antoniou [11] whichprovides a nice, easy-to-follow listing of all the calculations.

A further paper, by Amstutz [12], which discusses thecomplete design ofLC ladder filters, includes a section onelliptic functions and their calculation. Here, the author scalesthe Jacobian to give a function defined by

(45)

This makes have a constant imaginary quarterperiod ofand a real quarterperiod , in place of

and , respectively, for . The modification conferssome advantages in handling the filter approximation problem,though it is doubtful whether this is worth the complexity ofdealing with a nonstandard elliptic function.

Computation of all the real elliptic functions required forthe loss poles and for the calculation of the complex naturalfrequencies is achieved by using an infinite product for ,which is equivalent to using an infinite product for the corre-sponding theta functions when arranged to be periodic in thedirection of the imaginary axis, as in [13]. The complex naturalfrequencies are then found, as described above, via an additiontheorem. The Landen transformation is briefly mentioned, butit is not clear whether it is to be used in conjunction with, orinstead of, the infinite product for .

With present-day computers, the time taken to calculate thepoles and zeros by either the Landen transformation or any ofthese alternatives is negligible, and both can give accuracies towithin a few units in the last decimal digit of the floating-pointmantissa. Over the past 50 years one of the authors has triedout almost every approach to computing elliptic functions, andafter some initial enthusiasm for theta functions [13] finallydecided that the Landen transformation was the way to go.Accordingly, the present description of these functions waswritten so as to lead naturally into their evaluation by theLanden transformation and then their use in creating equal-ripple transfer functions. The aim has been to provide someunderstanding of the principles of the Jacobian functions ratherthan a “cookbook” approach to their calculation.

VIII. C ONCLUSIONS

The twelve Jacobian elliptic functions have been presentedas the simplest possible doubly periodic functions with aparameter, called the modulus, controlling the periodic be-havior. In practical applications the modulus is real andlies between zero and unity. When it is zero, the ellipticfunctions degenerate into one of the six singly periodic circularfunctions, while when it is unity they degenerate into oneof the six singly periodic hyperbolic functions. Many of the


properties of the Jacobian functions are simple generalizationsof well-known relations between these elementary functions.

For any value of the modulus, the Jacobian functions canbe visualized as lying somewhere along a path that has thecircular functions at one end and the hyperbolic functions atthe other. The Landen transformation allows one to move indiscrete steps along this path, and the formulas relating thevalues of the functions and their moduli before and after onesuch step are quite simple. Four or five steps are all that isneeded to transform an elliptic function of typical modulusto the point where it is numerically indistinguishable from acircular function. The elliptic function can then be computedby transforming the value of this circular function, step-by-step along this path, back to where the modulus has the desiredvalue. The program for arranging these computations is bothsimple and extremely accurate. The tools for these calculationshave been refined by mathematicians for more than 150 years,and no simpler or better way of computing the functionsrequired in the design of this class of filter is known.

REFERENCES

[1] S. Darlington, “Synthesis of reactance 4-poles which produce prescribedinsertion loss characteristics,”J. Math. Phys.,vol. 18, pp. 257–353,1939.

[2] R. N. Gadenz and G. C. Temes, “Computational algorithm for the designof elliptic filters,” Electron. Lett.,vol. 8, pp. 323–324, June 29, 1972.

[3] M. D. Lutovac and D. M. Rabrenovic, “A simplified design of someCauer filters without Jacobian elliptic functions,”IEEE Trans. CircuitsSyst., II,vol. 39, pp. 666–671, Sept. 1992.

[4] , “Algebraic design of some lower order elliptic filters,”Electron.Lett., vol. 29, pp. 192–193, Jan. 21, 1993.

[5] , “Minimum stopband attenuation of Cauer filters without ellipticfunctions and integrals,”IEEE Trans. Circuits Syst. I,vol. 40, pp.618–621, Sept. 1993.

[6] S. Darlington, “Simple algorithms for elliptic filters and generalizationsthereof,” IEEE Trans. Circuits Syst.,vol. CAS-25, pp. 975–980, Dec.1978.

[7] L. Gazsi, “Explicit formulas for lattice wave digital filters,”IEEE Trans.Circuits Syst.,vol. CAS-32, pp. 68–88, Jan. 1985.

[8] E. H. Neville, Jacobian Elliptic Functions. Oxford, England: Claren-don, 1944.

[9] D. Baez-Lopez, “Computation of even order elliptic filter functions,”Electron. Lett.,vol. 22, p. 1325, Dec. 4, 1986.

[10] A. J. Grossman, “Synthesis of Tchebycheff parameter symmetricalfilters,” Proc. IRE,vol. 45, pp. 454–473, 1957.

[11] A. Antoniou, Digital Filters: Analysis, Design and Applications,NewYork: McGraw-Hill, 1993, second ed.

[12] P. Amstutz, “Elliptic approximation and elliptic filter design on smallcomputers,”IEEE Trans. Circuits Syst.,vol. CAS-25, pp. 1001–1011,1978.

[13] H. J. Orchard, “Computations of elliptic functions of rational fractionsof a quarterperiod,”IRE Trans. Circuit Theory,vol. CT-5, pp. 352–355,1958.

H. J. Orchard (SM’65–F’68–LF’95) was born andeducated in England.

From 1942 to 1961, he was with the EngineeringDepartment of the British Post Office; between1942 and 1947 he taught at the Central TrainingSchool in Cambridge, England, and, from 1947 to1961, he worked on general problems of networkdesign in their Research Laboratories in London.In 1961, he emigrated to the U.S. and became aconsultant on network design to GTE Lenkurt Inc.,San Carlos, CA, and was head of their Networks

and Mathematics Group from 1963 to 1970. In 1970, he became a Professorwith the Electrical Engineering Department, University of California, LosAngeles, and remained in this position until his retirement in 1991 whenhe became Professor Emeritus. In 1969, he was an Associate Editor of theIEEE TRANSACTIONS ON CIRCUIT THEORY, and was also a co-winner, with G.C. Temes, of the first “Outstanding Paper Award” of the Group on CircuitTheory.

Alan N. Willson, Jr. (S’66–M’67–SM’73–F’78)was born in Baltimore, MD on October 16, 1939.He received the B.E.E. degree from the Georgia In-stitute of Technology, Atlanta, GA, in 1961, and theM.S. and Ph.D. degrees from Syracuse University,Syracuse, NY, in 1965 and 1967, respectively.

From 1961 to 1964, he was with IBM, Pough-keepsie, NY. He was an Instructor of electricalengineering at Syracuse University from 1965 to1967. From 1967 to 1973, he was a Member of theTechnical Staff at Bell Laboratories, Murray Hill,

NJ. Since 1973, he has been on the faculty of the University of California,Los Angeles, where he is now a Professor of engineering and applied sciencewith the Electrical Engineering Department. In addition, he served as theAssistant Dean for Graduate Studies, from 1977 through 1981, of the UCLASchool of Engineering and Applied Science and is currently Associate Deanof Engineering. He has been engaged in research concerning computer-aidedcircuit analysis and design, the stability of distributed circuits, properties ofnonlinear networks, theory of active circuits, digital signal processing, analogcircuit fault diagnosis, and integrated circuits for signal processing. He is theEditor of the bookNonlinear Networks: Theory and Analysis(IEEE Press,1974).

Dr. Willson is a member of Eta Kappa Nu, Sigma Xi, Tau Beta Pi, theSociety for Industrial and Applied Mathematics, and the American Society forEngineering Education. From 1977 to 1979, he served as Editor of the IEEETRANSACTIONS ONCIRCUITS AND SYSTEMS. In 1980, he was General Chairmanof the 14th Asilomar Conference on Circuits, Systems, and Computers. During1984 he served as President of the IEEE Circuits and Systems Society. Hewas the recipient of the 1978 and 1994 Guillemin–Cauer Awards of the IEEECircuits and Systems Society, the 1982 George Westinghouse Award of theAmerican Society for Engineering Education, the 1982 Distinguished FacultyAward of the UCLA Engineering Alumni Association, the 1984 Myril B. ReedBest Paper Award of the Midwest Symposium on Circuits and Systems, andthe 1985 and 1994 W.R.G. Baker Awards of the IEEE.

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