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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-28, NO. 12, DECEMBER 1981 1125

Broadbanding: Ga in Equalization Directly F rom Data

J. W ILLIAM HELTON, MEMBER, IEEE

Ahstrucr -This paper begins by presenting a powerful method which is easy to apply to many broad-band circuit design problems. Anyone with a broad-band design problem, which in the narrow-band case amounts to finding a point inside of a certain circle (say on the Smith chart), might find the method here very useful (Section I).

Gain equalization problems fall into this category and the main subject of this paper is a conceptually appealing, highly practical, and very flexible theory of gain equalization. The clever matching theory developed by Fano and Youla in principle handles passive one-ports well except for some difficulty in computing gain-bandwidth limitations. It converts the main problem into computing solutions to a system of nonlinear equations which are in practice so formidable that typical text book treatments [8], [9] never address the issue of solving them systematically. Also classical theory requires the load and gains to be specified as rational functions. Our theory does a good job on gain-bandwidth limitations, reduces all problems to ones of finding eigenvalues and eigenvectors of a given matrix, and only requires the load and gains be specified as data on a frequency band. Our theory is highly effective for multiports and so settles the old impedance matching problem for passive multiport circuits. The concrete results which we present here are:

(1) Two numerically efficient ways to determine theoretical gain bandwidth limitations for one-ports and n-ports;

(2) For one-ports a quick way to compute the frequency response function for the optimal coupling circuit directly from the answer obtained in (1).

The recent advance of broad-band microwave technology has produced a need for more general and more flexible theories of gain equalization. The type of theory called for is based on measured data and avoids rational functions and spectral factorizations until late in the design process. One typically specifies a desired gain profile G(jo) and then wants to find the largest multiple KG of it which is realizable. The procedure described herein is well suited to these needs since is requires only measured data and since determination of K is automatic. A very different method for broadbanding which fills these needs was developed by Carlin [lo]. It is a clever approach with quite a few compromises. One possible use of the lengthier rigorous procedure here would be to check the accuracy of Carlin’s method.

In addition to quantitative results. we present some (much more easily learned) qualitative properties which every circuit (passive or active) de- signed to optimize gain possesses. They might be of considerable practical use in that any designer can learn them instantly and thereby obtain a certain (small) amount of general orientation very cheaply. The section on qualitative results, Section III, can be read independently of the rest of the paper (except for Fig. 1.1 and environs) and that might be best for some practical designers who have little taste for theory.

Also in this paper we describe a certain viewpoint to the matching problem itself. From this perspective the matching problem is an elegant mathematical problem which fits solidly into a long line of classical mathematics. The classical mathematics underlies the computation of “pre-

Manuscript received September 6, 1979; revised May 5, 1981 and July 6, 1981. This work was supported in part by the National Science Foundation.

The author is with the University of California, San Diego, La Jolla, CA 92093.

diction error” in Wiener’s prediction theory. Our contention is that the matching problem is a very natural nonlinear analog of the classical prediction error problem. This formulation might broaden the appeal of impedance matching theory since it is easy to remember and is intriguing to the many systems theorists who are schooled in linear prediction theory (Section IV).

I. AN APPROACH TO BROADBANDING

Quite a few design techniques for narrow-band circuits depend heavily on the construction of disks in the complex plane and the selection of points inside them which satisfy certain properties. Broad-band problems require one to work with (frequency response) functions and not just single complex numbers. Thus one needs to work with “disks” in function space; so we shall soon formalize an appropriate notion of a “disk” in a function space. The big difficulty with broad-band design (which distinguishes it from the narrow-band case) is that a given “disk” (dictated by your design objectives) may contain no physically realizable (no poles in R.H.P.) frequency response function. The purpose of this section is to present a practical test which tells you if a given disk contains a physical response function f and if So how to compute f. After presenting the test we demonstrate its use on a gain equalization problem.

A. The General Method

We shall phrase the theory in terms of functions f(e”) defined on the unit circle rather than in terms of f(jw) defined on thejo axis. Obviously the two formulations. are equivalent and one can use a bilinear transformation to convert one to another. G iven a bounded function c(ej*) and a strictly positive function r(e@) define the “disk” A’c in function space with center c and radius r to be all functions h(ej’) which satisfy

for all 8. One is very much justified in thinking of this as a disk in the same spirit as one would think of a disk in the complex plane. We shall let H” denote all functions k( e@) with a bounded analytic continuation k(p) onto the unit disk lpi< 1. The test we shall present settles the question:

G iven c and r does there exist an Hm function h in the disk 11’,? If so how do you construct h?

There is an exact generalization of all this to matrix functions which applies to mu ltiports. Let C(ej’), P(eje), and R(ej’) be bounded nXn matrix valued functions with

0098-4094/81/1200-1125$00.75 01981 IEEE

1126 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-28, NO. 12, DECEMBER 1981

P* and R* strictly positive definite. Define APdR to be the set of H(ej”) which satisfy

(H-C)P*(H-C)*GR*.

Then one can ask: Does a particular disk APdR contain an HE” function and if so how do you build it? Here Hgn denotes matrix valued functions with Hm entries.

There are various function spaces one could use. The one we concentrate on in Section I is %L$“;,) the space of n X n matrix valued rational functions, which have no poles on the circle {e je* all t9}. Also we shall specialize to real . functions, namely, functions which though defined on the circle have rational continuations which are real on the real axis. Denote the real rational functions in Lm and Hm by RAL” and R%H”O, respectively. Our solution to the question is formulated in terms of two lemmas. At first reading it is not necessary to master the details of these lemmas but just to observe that they completely answer the question.

Lemma I: Suppose C(eie), P(eje)*, R(eie)* are rational real n X n matrix functions and that P* and R* are strictly positive definite, Then there is a rational real H,$” function H in the disk A2 R if and only if the largest eigenvalue 70’ of

(x$&-2] -9c~)w = 2TRZW

is less than or equal to one. Here X, and 9F stand for the Hankel and Toeplitz matrices’ with generating function F. Moreover, for any 7 2 r0 there is a unique real rational H, which satisfies

(C(ej”)-H,(e-@))P(eje)*

(c(eje)--H,(eje))*=T2R(eje)2 for all 8.

The proof of the lemma is constructive; since several proofs are possible there are in principle several construc- tions embedded in the mathematics literature. For scalar functions and ~=rc there is an extremely polished one which can be used to construct HT, from Lemma I.

‘The definition of H@el and Toeplitz matrices goes as follows. Given f(ej’) on O<B<2n let fE denote its ~th Fourier coefficient. Define the Hankel matrix (respectively, the Toeplitz matrix) for f to be the infinite matrix

i-, L-2 i-3 ‘3c,= f-2 j-3 ..’

1 I j-, ...

respectively,

L-2 i-1 i, .

\ I Forf in L*

“6 each]. is of course an n X n matrix. Denote the tram

matrix M 9 ose of a

y M’, the conjugate transpose by M*. For pracuc use one would truncate the infinite Hankel and Toeplitz matrices to finite ones. This approximation gets good quickly provided the Fourier coefficients of f converge quickly to zero.

Lemma II: For Lemma I with scalar valued functions and P= 1 here is a recipe from computing H7,.

(1) Save the largest eigenvalue ~0’ and corresponding eigenvector w from Lemma I..

(2) Compute the minimal phase spectral factor (Y of r*, that is, ] o( eje)] * = r( ej’)*, with a a function in Hm with no zeros.

(3) (a) Compute the vector x=Tzw. (b) Check whether [ Xc,,JX= x. (c) If not, then

satisfies

(4 Set G h, f2, . * - ) equal to the vector x if (b) holds and equal toy if (c) holds.

(4) Compute the function f(de)=fo+f,eje+f2dze+ . . . .

(5) Compute the function

Mo(eie)= c( eie)

[ 1 TOa(eie) _

where [g] _ stands for Co z gpke-jke.

k=l

(6) The answer is

It belongs to H” and satisfies ] c - HTo ] * E $r* provided c and r are rational.

Since the lemmas involve nothing more than Fourier transforming and solving the standard generalized eigenvalue problem, both very standard computer procedures for which many packages exist, this would be quite a straightforward test to implement.

Later we shall use the lemmas to perform a very thorough analysis of the gain equalization problem, but first we give a more informal discussion of why we think this is a highly useful tool in broad-band circuit design. Frequently one can think of the set R%H” of real rational I-i” functions as corresponding to the set of all physically realizable circuits. Frequently the major concern is some performance criterion p(H) on the physically realizable circuits. The main problem is to find a physical H so that p(H) is less than a given value a. In other words the issue is whether or not the set

c,A {HELm: p(H)<a}

contains an H in R’?%Hm. Our main point is that whenever f?a is a disk then the lemmas answer the question and also construct a solution.

The extent to which this scheme is applicable depends largely on how many performance criteria give rise to sublevel sets l?a which are disks. Many which occur in classical circuit theory do. For example, the gain in each of the gain optimization problems in this paper is a function p

HELTON: BROADBANDING 1127

I I “cl I---

,------- l : r, # I

l I Q ;r

------ u

lOSSleSS coupling circuit

El S

I load

Fig. I.

Xl-port

on R9LHz” whose sublevel sets are disks. That is how we solve the problems. An easy rule of thumb to use for the scalar (one-port) case is the obvious fact:

Lemma III: If p(H)=SUpef(l L(H)(e”)I) where L is some linear fractional map, that is,

L(H)(ej’)= k(eje)H(eje)+l(eje)

m(eje)H(eje)+n(eje)

and f is a monotone function, then the sublevel sets lZ= for p are disks.

There are certainly many other p’s which yield disks, but linear fractional ones contain many performance criteria which occur in classical network theory. The underlying reason is that circuit connection is a linear fractional operation, objects of concern (such as voltage/voltage transfer ratios, voltage/current transfer ratios, power transfer ratios, tan*(phase)) are ratios of sums in many situations, and even the conversion laws for various for- ma lisms (e.g. impedance, scattering, hybrid, or admittance) are linear fractional.

B. A Gain Equalization Example

We turn now to gain equalization, the ma in topic of the paper. It provides a splendid example of the method just described. Since the ma in objective of the paper is to set down a thorough theory of gain equalization we shall begin by describing precisely the very general gain equalization problem we shall ultimately solve. Then we take the simplest case and show how to solve it using our method.

The gain equalization problem, best described by F ig. 1, is: G iven S(jo), the scattering matrix, for a passive n-port and given a function G(jw) on a frequency band $, does there exist a lossless coupling circuit U so that the transducer power gain of the circuit in F ig. 1 over the frequency band $ equals G(jw)? If so, find the frequency response function for a U which realizes this gain.

Here the sources are ideal voltage sources uj each in series with a unit resistor. The maximum power available from each source when operating at voltage vj is I vj ] */4. So we call (I 0, I2 -t * . .. + I v,, I *)/4 the total available source power corresponding to source voltage 3= (u ,, v2. . . u,) and denote it by PA(C). For each source voltage v’ let PL(u’; jo) denote the power dissipated by the load S(jw) in F ig. 1. The transducer power gain G ;( jw) in mode v’ is defined to be PL( u’; jw)/Pa( 6’). The worst gain at frequency w is

G(jw)= infGdjw) G

and this is what we call the transducer power gain.

Consider the case where S is a one-port. It is easy to derive (see the discussion in Section V) that the transducer power gain at frequency w in F ig. 1 equals

G(jw)=l-[S(H(jo),S(jw))]* (1.1)

where S(h, k) is defined to equal

6(h,k)= +q$ I I and H is the lower diagonal term of the scattering matrix (normalized to unit resistors) for the lossless coupling circuit U. That is,

By Darlington’s construction any H( jw) in R%Hm with ]I H ]I ~1 can be embedded in some lossless U as its 22 entry. Since the circuit in F ig. 1 is entirely passive the transducer power gain G is always between 0 and 1. One can check that any H which satisfies equality (1.1) with O<G(jw)< 1 automatically satisfies II H II& 1. Thus a U exists delivering at least gain G( jw) for all .w if and only if there is a bounded real H satisfying (1.1)

The linear fractional form of (1.1) makes us suspect that the set of functions H satisfying (1.1) with an inequality < is a disk, and indeed a simple algebra calculation reveals that it equals A: where

and

c(jw)= G(P)S(jo) l-]s(jo)]*(l-G(jw)

r( jW)*=(l-G( jw)) 1-pQ.d)~2 * 1 l-]S(jw)]*(l-G( jw)) .

0.3) Thus use a bilinear map to transform c and r to functions of eje and apply Lemma I to determine if G is realizable for S; the first ma in result in’this paper on gain-bandwidth lim itations (Section II, Test I) amounts to the explicit statement one gets (in a more general situation). If G is realizable then Lemma II tells you how to find an H which can be used in the Darlington construction to find U for a coupling circuit with transducer power gain identically equal to G . It should be clear to the reader at this point that the method gives a complete solution to the problem. This solution is stated very formally as Test I of Section II and there its practical consequences for broadbanding are discussed.

It m ight be worth emphasizing the well-known fact that there are gain problems for amp lifiers which reduce directly to passive problems. In particular suppose one is given a unilateral amp lifying device and wants to build a passive equalizer heaving no feedback or feedforward. The formula for the gain of this amp lifier is a product of two terms like (1.2) (see [19, fig. 1001). As a consequence the mathematical problem of designing the amp lifier to have a prescribed gain is equivalent to the one we just solved.


Also we mention that another performance criterion in the design of such an amplifier which yields disks is the “noise figure.” See [19, ch. IV, pp. 26-27, fig. 1091. Sub- level sets of the noise figure are disks. Thus the methods here apply directly to the design of an amplifier having “noise figure” less than a prescribed amount at each frequency. Actually, one sees from the form of the noise figure equations that mathematics behind the other two theories of broad-band gain equilization (Fano-Youla, Carlin) could be used to attack this problem. However, the possibly far-reaching observation that the whole theory -of broad-band gain equalization can be brought to bear on noise figure minimization appears to be new and is largely a result of the conceptual simplicity of our approach.

That completes the presentation of our basic broadbanding method. More appears in Section IV and Section V, but the rest of the paper is directed toward a study of broad-band gain equalization. The paper, in fact, sum- marizes a five-year mathematical study of gain equalization [4], [5], [ 12]-[ 171. These can be consulted for greater detail.

II. GAIN-BANDWIDTH LIMITATIONS AND REALIZING THE EQUALIZERS

A. One-Ports

We begin with the one-port case and describe it exten- sively. The problem is to determine which gains on a certain frequency band !J are realizable for a given load S. This problem effectively breaks into two different problems depending on what information about the reflection coefficient S of the load is given:

(I) the values of S(jw) are known to within a tolerance e forjw in some band $+ which contains 4, or

(II) the function S is given exactly as a rational function and we know its values exactly everywhere in the complex plane.

Problem I corresponds to the usual industrial circum- stance where S is determined as data taken from a spec- trum analyzer. So our ultimate goal even in presenting results on Problem II will be to address Problem I. Before stating results we will discuss what one can expect on naive grounds about Problems I and II.

Our first point is that the relation between Problems I and II is surprisingly tenuous. While the data S( jw) on $ + in theory has a unique analytic continuation to the whole R.H.P., in practice the values of S at any reasonable distance from $ + are VERY BADLY determined. The extent to which this is true is appalling and easily demon- strated by the following example. Suppose our measured data on $ + is identically 0 and that we trust these measure- ments to within E. What can we say about the values of a bounded real function S which is consistent with these data, that is (S(~W)]<E for jwE$ +? The answer on thejw axis away from $ + is that we can say nothing. After all let W be any positive function on the jw axis with values less thaneon!!f+, less than 1 on the whole axis and uniformly bounded about 0. Then any spectral factorization S of W,

’ that is SS = W is consistent with the measurement 0. How-

ever, the absolute values IS( away from $ + are arbitrary! The values of such S(p) for p inside the R.H.P. aren’t completely arbitrary (one can use the maximum principle a la Phragmen-Lindelof to get estimates) but they aren’t at all near 0 unless p is near 4 + . 1

The second point is that problem I is well posed and the answer is not affected by the fact that S off of $ + is hard to determine. To see this suppose S, and S, are the. scattering functions for two load circuits which satisfy

and ]S,(jw)]<r<l and ]S,(jo)]<r<l

for jw in $. A coupling circuit U, with transducer power gain G for S, when coupled to S, has gain uniformly within

E IS,-S2l < 1 - lrl* 1 - lrl*

of G on g. In other words for data bounded well below 1 the gains obtainable from S, are within

cz 1 - lrl*

of those obtainable from any S, which is within c of S,. The closer the data is to having absolute value 1 the less precision there is in the problem. However, for fixed r a small measurement error e leads to a small uncertainty in obtainable gains.

Thus one would hope that there is a method for broadbanding which does not require analytic continuation of S off 4+. On the other hand, continuing S to the jo axis is not inherently bad since mistakes in the continuation off 9 + don’t have much effect on the answer, but certainly one should avoid spending much computer time calculating properties of the continuing function which have nothing intrinsic to do with the equalization problem (such as the transmission zeros of S which dominate classical equalization theory). In this section we present two basically different tests to determine if a gain is realizable. The first does not use an analytic continuation of S; the second does. The two tests have complementary strong points. In the section on proofs we give a general theorem which contains each as a special case.

We shall now state the method for determining when a gain is realizable for S which was derived in Section I. Actually we give a refinement of it. Throughout the paper we shall be interested only in the case where ]S( jo)] does not take the value 1 on $l+. Although the theory could be extended to handle this case, it would complicate the exposistion greatly, and the author has never seen an industrial case where this occurs. The first step is to state definitions.

Our theory heavily involves matrices of the form TkTk-X,Xf*

so we denote this matrix by A( f, k). Since the author’s mathematical theorems are proved for functions analytic on the disk rather than on the R.H.P., in the tests pre- sented here the first step will always be to use the Cayley

HELTON: BROADBANDING

transform p +p- l/p+ 1 to convert data on the jw axis to data on the circle. (This step is avoidable since after some computation one could work directly on the jw axis and a theoretically inclined reader could carry out the transformation.) So we define the map w: unit circle+jw axis by

w(ejO)= 5. The basic test is: We are given functions S and G on $

satisfying ]S( jo)] <r< 1 and O<G( jw)<l on 4. Then G will be called an approximately realizable gain for S provided there is a sequence U, of lossless coupling circuits so that the resulting transducer power gains G , converge* to G .

CONSTRUCT:

1) e(ejs)=S(w(ejs)) on gwgw-‘(g).

on $,,

off gw.

TEST I. Set

f= - l-,u;;l-g) andir=(l-d( ,-~~~‘~,,)*.

Note fro and k= 1 off of $,,. Then G is an approximately realizable gain for the load S (if) and only if A( f, k) is a (strictly) positive definite matrix. Also if $ is the jw axis and u, G are rational then the adjective approximately can be removed.

Comments. 1) The test is very practical. One smooths f and k a little

at the ends of $l before (fast) Fourier transforming (or windows the Fourier transform). On truncates the infinite matrices to decent size finite matrices. One forms the matrix A( f, k) from these truncations:

If A( f, k) is positive definite, then G is a realizable gain for S.

If A( f, k) is not positive definite, then G is not a realizable gain for S. 2) Determining if a matrix is positive definite amounts

to showing that its smallest eigenvalue is positive. One could use a standard package for doing this such as Eispack. A classic book on the subject is [23].

3) When S and G are rational functions this indeed is the gain-bandwidth test derived in Section I-B. To see this note that f and k in Test I are just the functions c and r* of (1.3) transformed by the mapp ing w:

f(eje)=c(w(eje)) and k(ej’)=r(w(eis))*.

Application of Lemma I gives Test I. Further discussion appears in Section IV.

Wh ile Test I completely solves the gain-bandwidth lim i- tations problem in a very simple way it has one drawback: the gain G enters the formula nonlinearly. A common

*Here convergence means G, is a sequence of functions which con- verges uniformly to G on each closed set in the interior of 5.

1129

industrial problem is to realize the biggest gain of the form , KG where G is given and K is a constant to be found. Certainly one could use Test I; just substitute KG for G and perform Test I on a grid of K ‘s. However, there is another test in which the gain (and consequently K) appear linearly. This second test is

CONSTRUCT: 1) and 2) as before. 3) A function a0 which is analytic and uniformly less

than 1 on the disk which approximates (I closely on $,,. 4)Set 6=(1-]a,]*)-‘. TEST II. G is an approximately realizable gain for the load S (if

and) only if

A(&~J*)-$,,

is (strictly) positive definite. The accuracy of the test naturally depends on the accuracy of the approximation ua to 6.

Comments. 1) The gain appears linearly in the key matrix, so sweep-

ing K in KG is particularly easy. In fact the key matrix has the form

A-KB

and we must determine for which K it is positive definite. Now the transition from being positive definite to not being positive definite occurs at the lowest K~ for which there exists an x#O satisfying

Ax=K~Bx.

Since B is positive definite (one m ight want to add a small (~1 to B to improve regularity) this is the standard generalized eigenvalue problem. Since we want to find the lowest general ized eigenvalue for A, Eispack or [23] is again appropriate.

2) To implement Test II one proceeds pretty much as before (see Test I, Comment 1), but now there are two additional points to consider. The first is that one must construct an analytic approximation a= to u on 4, or 4: > $,,. This is a very standard approximation problem and since many people are far more knowledgeable about it than I it would not be labored here. The way it is done in many industries is simply to m inimize

2 F(a,;..,a,)=

jl s+ u(eje)- 5 a,eine de w n=l

over sequences (a,, . . . , a,,,,). This is fast and in practice the approximation

u,(eje)k 5 anejne ??=I

obtained is very accurate on g,,,. To apply Test II we need a function a, with ]a,] uniformly less than 1. Thus it m ight be advisable to m inimize F with an added relaxation term which punishes large a,‘~; say m inimize


3) This brings us to the second point. Suppose ]a,] is near 1 at some point e jeo. By assumption e jeo is off .$t, but still this gives a problem because

a=1 1 - ]u]2

will have a sharp spike at 0,. This makes no difference theoretically but in practice causes numerical difficulties. The overriding factor determining how rapidly finite truncations of infinite Toeplitz and Hankel matrices $If and X, converge to them is obviously how rapidly the Fourier coefficients off converge to 0. For the Toeplitz and Hankel matrices in Test II the crucial functions are 58 and S* (we assume a practitioner would automatically window the coefficients of gS*). The function typically will be well behaved and 6 will be fine except for the spikes. Thus finite truncations converge rapidly if and only if 6 does not have very sharp spikes. Note this problem did not arise at all in Test I and it is the price we pay for getting the gain to appear linearly.

After some thought one is inclined to improve a bad situation by taking the spike in a function 6 and just lopping off the top smoothly. Fortunately, this works very well: Clearly it makes the Fourier coefficients of 58’ and S* decrease more quickly and we shall do a rigorous estimate which shows that a slight modification of the procedure usually introduces little error. To describe the lopping procedure and the error introduced precisely define +. to be a smooth function which is identically one outside of [do--c, e,+c] and which dips modestly below 1 near e,; set

a,= 1 1 -&$J]’ .

Clearly, judicious choice of +. smoothes the spike in 6,. Our recommendation would be to use the matrix

(2-l)

instead of the matrix in Test I. One can prove (see [18]) that the error introduced is less than would be introduced by any error in measurement of data on g which is less than

1 1 /

eO+c -- r d e,-c In+,.

Here d is the distance from the points ej(‘okr) to gw. For example, suppose (p. is about 0.9 on [0,-e, e,+e]. Then

1 a,<- l-0.81 -’ and so its spike is much dulled, but the error introduced corresponds to a data error of less than

; f2r(0.105)-0.06;.

Several spikes might have to be smoothed; each time it is done an error of this magnitude might enter. However, the conclusion is: Spikes well away from 4 do not have much influence.

4) The proof of Test 2 requires a substantial addition to Lemma I, see Section IV.

B. Limitations for Multiports

Both Test I and Test II generalize directly to multiports. We shall give the generalization of Test II first. It is

CONSTRUCT: I), 2) and 3) as before. 4) Set 6 =(1- a,*~,)-’ and &=(l- ~,a,*)-‘. TEST III. If the matrix

is strictly positive definite, then there is a lossless equalizer U so that the transducer power gain G;( jw) is “G( jw) on $ for all modes v’. Conversely, if %(a,, g) is not positive definite, then no such equalizing circuits exist. Moreover, when $ equals the whole jw axis and u and g are rational, then X(0, g)aO implies there is a lossless rational u so that the gain G; in any mode precisely equals G. The accuracy of the test depends on the accuracy of the approximation a, to u.

The generalization of Test I to multiports is

TEST IK To determine if the function G is a realizable transducer power gain for S construct a matrix

X,h SF - 3Cgo*[l-~a*(,-g)l(~,-~~*)~~~*(,-~)-~~)-’ *~;o’[~-oo*(l-g),-’ + ~,-~)[,-o’o~[oo*(,-~)-I)~

(2.4 Then Test III holds with X, replacing X.

Test IV will be proved in Section V when g is the whole jo axis and the idea behind Test III should be clear from Section V.

Comments. The comments which follow Tests I and II for one ports

apply readily here.

C. Finding an Optimal Coupling Circuit For one-ports Lemma II together with the Darlington

construction immediately tells one the frequency response function of an equalizing circuit U which delivers a realizable gain. The test builds directly on Test I or Test II (whichever you are using). Consider, for example, Test II; in particular consider the gain sweep problem discussed in Comment 1) after Test II. We shall assume that you have applied Test II, have the solution in hand .and we shall describe how you build the optimal coupling circuit. Recall that the issue in the test was to find the smallest ~~ for which

is positive definite. The fastest numerical procedure for doing this would be to use a standard package which minimizes the Rayleigh quotient for a finite truncation of the Aw = KBW problem. Such a method gives not only ~~ but also the corresponding eigenvector w. So the first step in COnStrUCting our Coupling circuit is to Compute ~~ and w in Test II. Then use Lemma II.. The resulting algorithm appears in [ 181.


Similar algorithms follow from routine mathematical work based on [2] or [5].

D. Comparison with Existing Theory for One-Ports

The ma in existing theory for gain equalization was developed mostly by Fano and Youla. It is impressive work which is described thoroughly in [8] and [21]. The ma in bottleneck in the theory is that one must solve a system of simultaneous equations (see [21, ch. 8, sec. 61) of the following form.

G iven complex numbers pk, pk, for k = 1,. . . , n; find complex numbers a, with Rear, ~0 and integers nk and z = * 1 satisfying for all k

lnll-G( jw)ldw. (2.3)

The standard prescription (see [8, ch. 4, sec. 71 [21, ch. 8, sec. 5c]) is (or is equivalent to): G iven S computep, and Pk from an explicit recipe. Then G is realizable for S if and only if this system has a solution n,;**,n,,a,,a,, . .. . This solution can be used to write down U.

This, of course, is formidably nonlinear in the a’s (and this is the simplest case-more complicated cases add polynomial nonlinearities in a). Wh ile [21] indicates that the gain equalization problem for one-ports is solved neither [21] nor [S] offer any general advice on solving these equations. They do work many nice examples, but these give the impression that when the load has more than three reactive elements computations get out of hand.

Tests I and II of this paper completely avoid the nonlinear system (2.3). Instead of solving (2.3) one must solve an eigenvalue problem. Wh ile eigenvalue problems are indeed nonlinear they are the world’s most standard nonlinear problems and so are readily solved. We also recall that the ma in disadvantage of Test I is that the gain appears nonlinearly. This is also a disadvantage of (2.3).

The fact that one could use a matrix test rather than (2.3) to determine gain realizability for many one-ports was surely known to Youla. His paper with Saito [24] applied the mathematical theory of interpolation to design of ladder filters, and a remark in the paper indicated it could be used in broad-band matching. Unfortunately, this fine paper received little attention. Chen used it in [8] to obtain a special case (Theorem 4.2) of our EXTREMAL principle to be mentioned, but he did not apply it to actually solving the gain-bandwidth or equalization problem. In 1976 the author began using very general interpolation theory on broad-band matching of mu ltiports and subsequently dis- covered [24]. Results are announced in [16]. As one has seen the results here bypass interpolation theory and use a type of approximation theory which ( though related) applies much more directly.

O ther advantages of Tests I and II center around not having to computep, and pk. When S is given as a rational function these are in principle straightforward to compute.

The reader has already seen plenty of discussion about an S for which less is known. Consequently we will not further belabor the advantages of not computing (the transmission zeros pk) and pk.

III. QUALITATIVE PROPERTIES OF ACTIVE AND PASSIVE CIRCUITS

The purpose of this section is to list some simple, easy to learn, qualitative properties of gain equalization (as op- posed to Section II which was entirely quantitative). It can be read independently of the rest of the paper except that F ig. 1 and the surrounding commentary are required. Wh ile the rest of the paper treats only passive circuits, here we also describe some basic though surprising behavior of (broad-band) amp lifiers. Hopefully, this section will be interesting to design engineers; despite the fact that most industrial problems have many special constraints, know- ing what is possible in ideal circumstances frequently gives one a feel for the lay of the land.

The section splits into two parts: one on passive circuits and one on amp lifiers.

A. Passive Circuits

The situation is the one described in F ig. 1: G iven the load S, we want to find a lossless equalizer U with prescribed transducer power gain (TPG) G(o). What can be said about-the realizable gains? For one-ports with a fixed rational strictly passive load S the following two things are true:

MONOTONOCITY PRINCIPLE: Suppose G is a rational function 0 < G < 1. IF some realizable TPG is greater than or equal to G for all frequencies, THEN G itself is a realizable TPG.

UNIQUE EXTREMAL PRINCIPLE: F ix rational G 2 0. Let ~~ be the largest number K for which KG is a realizable TPG. THEN no realizable TPG is bigger than KEG at all frequencies. For example, if one chooses U, to maximize max,inf,G,(w), then the.resulting gain GuO is constant.

The MONOTONOCITY PRINCIPLE holds in a strong form for n-ports (see [12], [15], [16]). The proof is so immediate from Test II or Test III that we om it it. The UNIQUE EXTREMAL PRINCIPLE holds in a much weaker form for n-ports. The UNIQUE EXTREMAL PRINCIPLE for one-ports with transmission zeros of order 1 and G = 1 is essentially due to Youla (c.f., [8, th. 4.21).

B. Amplifiers

For amp lifiers the situation we analyze is this. We are given a linear frequency dependent amp lifying device (or a small signal mode l of a nonlinear device) and want to add passive circuitry so that the result is a good amp lifier. We shall always assume that the load on the amp lifier is a resistor and that the internal impedance of the source is frequency independent. We focus on designing passive equalizing circuitry to maximize or to prescribe the gain of the amp lifier over all frequencies.

For a one-port reflection-type amp lifier (based on an


amplifying device which amplifies at each frequency), the following can be derived from the main theorem of [4] and an approximation argument.

SPIKED GAIN PRINCIPLE: Suppose the (small signal) gain function G(w) for a particular stable amplifier can not be significantly increased (G’(w)>G(w) all w) by any modification of the passive equalizing circuitry in that amplifier which leaves the amplifier stable. THEN there are certain frequencies w,, . . *, wk at which G is very, very large (has spikes). Such spikes wi,. . . , wk always exist unless all amplifying devices in the amplifier are frequency independent. For example, a stable amplifier for which inf, G(w) is near maximum has gain which is nearly constant except for a finite number of spikes.

Intuitively the principle is a natural consequence of stability being a major constraint. Namely, in the optimum design the amplifier is just about-to go unstable; a pole when crossing from left to right half plane produces a spike. This suggests one might base an approach to amplifier design on identifying the location of the spikes and keeping them away from the operating band. Note that stability is indeed what forces the spikes, since MONO- TONOCITY and UNIQUE EXTREMALITY hold for amplifiers not subject to the stability constraint; for reflection type, Section V [12], for transistor type [5].

We conclude with a different type of fact about amplifiers.3

DISSIPATION PRINCIPLE: If any (stable) amplifier which is built with passive equalizing circuitry has a certain gain function G, then there is an (stable) amplifier built with the same amplifying device and of the same type, but built with lossless circuitry g which has gain greater than or equal to G. In other words, you cannot get better gain with passive circuitry than lossless circuitry.

IV. THEORY AND PERSPECTIVE

This section gives the main ideas behind our results and extensions of them. Also in the course of our first two subsections one sees that broad-band matching fits nicely into a line of classical mathematics. The basic issue in this mathematical development is the following question:

QUESTION: Given a function fEL” what is the distance from f to H”? Naturally, the answer depends on what we mean by distance (which metric do we take on L”). The earliest case studied took the distance between f, g in L” to be II f-gll Lzc,,,de) where w is a given weight function. Problem: Given f, find

inf II f-gll Lz. gEHm

The distance question was answered by Szego and his analysis was later used by Wiener as the basis for his

3While this principle is very plausible it is nontrivial to prove. For one-port reflection type amplifiers it is an application of the fundamental mathematical theorem that any analytic mapping of the disk into itself decreases the “ Poincar6 ” distance between points. For multiport amplifiers a significant generalization of this fact [6] is necessary [5].

theory of prediction. The minimum distance itself has the physical interpretation of “prediction error.”

The QUESTION when one uses the L* metric (instead of L*( wde)) also is classical. Standard results on this problem are described in subsection (a) and used to prove Lemmas I and II. Subsection (b) discusses the most ele- mentary “non-Euclidean” metric P( , ) on L* and shows that the QUESTION for the metric P( , ) corresponds to the physical study of broad-band matching.

A. The Basic Lemmas, the Tests and Sup Norm Approxima- tion

Take the distance between f, g in L”( M,,) to be II f-gll LmcMn)= sup II f(ej’)-g(eje)ll M,. e

Here M,, denotes the n Xn matrices and Lm( M,) denotes the bounded matrix valued functions on the circle.

The QUESTION what is the distance from a given f in Lm(M,) to H”(M,) was settled by Nehari for n= 1 and various ingredients of the general solution were developed by many others (see [17] for a list). The answer is

Theorem. (c.f., [2]) If F is in Lz, then inf IIF-MMl,~,=III~,1t1.

WEHE”

If F is continuous (Hintzman [20]), then the L$” distance to the continuous functions in Hz” also equals 111 X,111. If F is rational the closest function MO in Hz is unique and rational. Moreover, if y(e”) is any rationd function satisfying 111 X,1]] *<y(eje)* for all 8, then there is a rational M in Hz which satisfies n

II F(eje)-M(eje)ll~n=y(eje)2.

Here 11) 3c, 111 stands for the norm of the matrix X, acting as an operator on the Hilbert space I,& of square summable 1z” valued sequences.

Lemma I follows from this. To prove Lemma I let a, and a2 be minimal phase

(matrix) spectral factors P*=a,a; R*= a2az.

Then H is in A2 R if and only if

lla;‘Ca,--Kll,, <l Mn for K= a; ‘Ha,. Such an H in H” exists if and only if a K in H” exists which by the Lm distance theorem is equivalent to lll~,l~c,, 111~ 1. The norm inequality can be rewrit- ten as

~a~bx,(3C,~bq)*~I

which some simple tricks with Toeplitz and Hankel operators convert to

X,( ET,-,)pJcr; d TR2 .

This is the inequality in-Lemma I. When C, P, R are real functions a real solution exists by symmetry. The finer structure in Lemma I follows from that in the theorem.

Test I of Section II looks close to being just the applica- . tion of Lemma I to the gain equalization problem which


we did in Section I. It differs because in Section I all functions are rational and Test I involves step functions because of the need to restrict attention to an arc $,, on the circle. To prove Test I requires the full Nehari theorem for F in L* and not just the theorem for continuous F. This type of procedure is the one we always use to cut down from the full circle to an interval.

F ig. 1; if S is a passive one-port then their cascade has response function %JS).

The transducer power gain in our basic circuit (Fig. 1) for one-ports is

Lemma II follows directly from the algorithm given in [20] for actually computing the closest M ,, to a given rational F. See [ 181 for details.

For a fixed U the lowest gain is at the frequency where the m ismatch is biggest. Since arctanh is a monotone function maximizing (or m inimizing) the m ismatch 6 corresponds to maximizing (or m inimizing) the Poincare ‘metric p. Thus

B. The Poincart? Distance to Hm and Gain Equalization m inG(jw)=l-[tanhP(0,~~(S))]2. The gain-bandwidth lim itations problem involves the

distance to H” but with yet another metric, name ly, a The broad-band impedance matching problem: Maximize

Poincare type metric. Let us discuss for a moment the this over all lossless U, is equivalent to finding

classical Poincare metric. It was introduced near the turn m= m inP(O,%~(S)). of the century and is one of the most basic metrics in mathematics besides the Euclidean metric. It is defined on the open unit disk {z: IL] < l} by

Sl -s2 p(s,,s,)=arctanh ~ I I I-s,s, *

The metric p is invariant under linear fractional maps of the disk onto itself and it is used by mathematicians whenever studying such maps. If geodesics with respect to p are thought of as lines, then they satisfy all of Euclidean axioms except for the parallel postulate, thus yielding a “non-Eucl idean” geometry. This “Poincare disk” as it is called is the paradigm for one of the two basic types of planar non-Eucl idean geometries. It is amusing to note that a picture of it appear ing in most mathematics books looks much like an electrical engineers’ Smith chart.

The function space analog of the unit disk is the unit bal l%3L”ofL”definedby’33Lm={f~L”: IIfIILm<l}.A natural “Poincare metric” on %L” is

P(f, s>= sypP( f(eie), g(@)).

Again the QUESTION is: G iven f~ ‘%Lm, find the Poin- care. distance of f to %Hm = H* f~ %L”. This metric is invariant under maps ‘3”: %Lm -+ %Lm defined by

%JS)=A+BS(I-DDS)-‘C (4.1) where

is the scattering matrix for any lossless two-port with IIAIl,m<l.

The basic reason the Poincare metric arises so naturally in electronics is that the ratio of delivered to available

- power in F ig. 1 can be easily computed to be

”

We shall now show that this is equivalent to the “Poincart distance” problem. F irst use the fact that ‘3u and complex conjugation are P-isometries to get

m= m inP( F;‘(O) , S). u

Darlington’s theorem tells us that as U sweeps through all rational lossless functions %JO) sweeps through the rational functions in %Hm. A simple variant on Darlington’s theorem says that F;‘(O) does also. Thus

The rational H’s in %H* are dense in %CHm, the con- t inuous functions in ‘%Hm. It turns out that for rational 9 the m inimum over all of $?JH~ is always achieved with a rational H so we haven’t been very careful in distinguishing ‘%H” from %CHoo. We have obtained that the matching problem is the Poincart! distance problem for functions S in ‘%H” as desired. Note that whether we used 6 or p in the argument makes no difference; arcth was only applied to 6 to give the mathematically familiar object p.

The problem of finding the scattering matrix for an optimal matching circuit U can be reduced using standard techniques of Fano-Youla theory to actually finding a g, in ‘%H” ‘PoincarC’ closest to K

The basic problem of achieving a prescribed gain G is attacked in the same way but uses a weighted supremum

PW(f? g)=supw(jw)p(f(jw),g(jw)). w

Take the weight to be 1 /arctanh( 1 - G)‘12. Then the argument in the preceding paragraph tells us that a gain greater than G is realizable for S if and only if there is a U so that

G= power to load available source power =1--6(H,,?)2 if and only if for some rational H in 6i3Hm

where 6 is given by (1.2). That is, 6 is just the business part laP,(H,S) (4.2a)

of the Poincare metric and we call it the power m ismatch or more simply between source and load. Here S denotes the scattering parameter for the internal impedance of the source. Also

1-G(j~+S(H(jo),S(jw))~ (4.2b)

the maps ‘%u correspond to lossless cascade corrections in which is inequality (1.1). The study of optimizing and


achieving gains on a band !J goes in exactly the same way except one takes supjUE9 instead of supjo over all w.

We mention that there is a natural non-Euclidean geometry on %L”, P( , ). Border’s thesis uses the geodesics to study impedance matching generally; specializing his general theory has several specific successes.

C. A General Gain Equalization Theory

There is a big family of equivalent tests and solutions to the gain equalization problem which can be gotten by combining the lemmas of Section I with what could intuitively be thought of as a coordinate change in function space. A linear fractional map with functions as coefficients will transform one disk in function space to another. One could work on either disk and by doing this properly get different forms of the solution to a problem.

Here we shall develop the special case of linear fractional coordinate changes appropriate for the gain equalization problem. Let g be any point-wise Poincare isometry on %Lm, namely, a map of the form (4.1). Then H is in the disk (4.2b)= (1.1) if and only if

I-G2S(T(H),Q(~))2. (4.3)

In other words, (1.1) is satisfied if and only if there is a function K in

Range%&{%(H): HE’%Hm}

which lies in the disk

We have a variation on the main problem in 1. Instead of, determining if H intersects a certain disk we must determine if Range ‘9intersects a certain disk. Fortunately for ‘%with rational coefficients one can actually compute Range 9 and fortunately it has a good form for solving the problem. Thus one gets a test based on 9(g) rather than g

Let us be more specific. Suppose d is rational with Ild II ,,<l. Define?&-dby

GA-d(H)=-d+(l-Id12)H(1-dH)-’ .

on H in %L”. It is a (point-wise) Poincare isometry and ‘$ - d< 2) = 0. By the formula (1.3), we see that H satisfies (4.3) if and only if

Icd-s-d(H)I<rd

where

GF cd= I-(l-G)IF12

rj=(l-G) 1-lF12 2 I l-IF12(1-G) ’

That is, F=FA--d(S). (4.4)

where K= H(l -dH)-‘. If K satisfies (4.3), then K automatically has the form K=H(l -dH)-’ for some H in

%Hm. A key question is: which K’s have this form? This question is answered4 (in great generality) by Lemma 4.3 [12]: For dE%H”

precisely those K ‘s which are in Hm

come from an H in 93H”. (A similar fact plays a key role in the Fano-Youla theory and is the main vestige of that theory here.)

Our problem is of the same type as the one in Section I; only the values of c and r are different from those in (1.3). Thus the argument in Section I which uses Lemmas I and II gives the following theorem when G, S, and d are rational. (You can also remove the word approximate and the requirement of strict inequality.)

Theorem 4. I. Suppose’ d is in 9 H”. Set

fd= c,+d

1-P12 Then G is an approximately realizable gain for S (if and) only if the matrix A( fd, kd) is (strictly) positive definite.

Clearly Test I is a corollary of Theorem 4.1 gotten by taking d=O. Test II follows immediately by setting d=S; note O=F=c,. The point behind Test II is that we have chosen an 9 which maps ,? to 0 and when S=O the gain enters formulas (4.3) linearly. Thus we get a test in which the gain appears linearly. Recall there were some numerical problems with Test II near a point z, where IS( = 1. Basically, this is because the map $& _ s is singular at zI.

So an obvious ploy is to take d close to S with II d II Lm =C 1. For example, in the notation of Comment 3 after Test II (which is on the circle not the R.H.P.) one could take d=cua. Then there is no loss of accuracy, no intrinsic numerical problem, and since c,, is small the gain almost appears linearly in Theorem 4.1. Another possibility would be to take d=&u = la12a and apply the theorem of footnote 5 for rational d. Here one would select an explicit $$J to be rational of low order so its poles would be known and not have to be determined numerically. Other choices of d would accomplish other purposes; in short there is enor- mous flexibility one could use to advantage.

4The proof is so simple we give the basic idea here. Our proof follows the lines of Theorem 4,lB [ 121 which is done rigorously. If HE H” and IIHII,,Sl, then ]]d(e~@)H(eJe)IILm<l and so 1-h has winding number 0. That is, 1 -ds has no zeros in the unit disk and (1 -dH)- ’ is in Hm. Consequently K= H(l -dH)-’ is in Mm. If K satisfies the estimate (4.3), then H=K(l-dK)-’ satisfies ]H(e”)]Gl. If in addition KEHm, then the identity 1-t dK= ( 1 - dH) - ’ implies that ( 1 -dH) - ’ E Hm. Since (I- dH) - ’ has winding number 0, its inverse is in Hm. Thus (1 + dK) - ’ and consequently H lies in Hm.

‘The more genera1 theorem (obtained from Lemma 4.3 [l2]) which holds for any rational d having ! poles inside the unit disk goes as follows. Factor d as d= rt-’ where r and o are in Hm and coprime and 1 o( ej’)] = 1. Define d, to be a rational function in Hm with the property that (d, - d-‘)1/02 is analytic near the zeros of c (that is, near the poles of d in the disk). Then

Theorem 4.2 Suppose d is rational. Set

c,+d+(l-]d2)d, fd= (1 $2,“2 kd=sme.

Then G is a realizable gain for S (if and) only if the matrix A( fd, kd) has at most I (nonpositive) negative eigenvalues.


D. Multiports

Let $I&, denote {ME M ,,: II M II ,,,,< l} the unit ball of n X n matrices. The scattering matrix of any lossy n port is a function with values S( jw) in C&%4,. As we .have seen for one-ports the Poincare metric (power m ismatch on %U,) played a ma jor role. The Poincare metric is invariant under bi-holomorphic mapp ing (lossless cascade) and it is essentially the unique metric with that property (any other invariant metric is a function of it). In higher dimensions the situation is different because there are infinitely many invariant metrics.

Many such metrics can be built from the eigenvalues of the following “cross ratio” of matrices S, HE J4,

&(S,H)=(l-SS*) -“*(23-S)(S*H-1)-I

*(s*-H*)(l-SH*)-‘(I-SS*)“*.

For S, H in ‘3&U,, this has positive definite matrices for values and its key property is that its eigenvalues are invariant under the transformations F,, of (4.1) where U is now a unitary 2n X2n matrix. That is,

e.v. &(9JS),CFu(H))= e.v. &(S,H)=e.v. &(H,S).

The first cross ratio of this type is due to Siegel [22]; it was significantly general ized and introduced to engineering by Youla.

Now let’s look at the gain equalization problem. In F ig. 1 for each voltage v’ there is an x;with IIx;ll = 1 so that

G ,-(jo)=l-Il~~~j,,(S(jw))x~ll~.

The worst mode u’ gives a gain of

G(j~)=l-ll~~(j,,(S(jW))II~,

=l-llF(o,~~,j,,(S(~w)))llM,.

Now we are in a position to proceed exactly as in the one-port case. We get that a transducer power gain of at least G is realizable for S if and only if there is an H in % fG” which satisfies

l-G(jw)W(S(~o)*,H(jo))ll,~ (4.4) as a generalization of (4.2b).

To see if such an H exists we proceed exactly as before. “Poincare circles” in CB3, are all Euclidean circles and one simply must compute which one. Executing this gives Test IV (see [18]); a change of coordinates gives Test III.

V. CONNECTION WITH CLASSICAL TRANSMISSION ZERO THEORY

This section concerns Test II and gives a more flexible way to view it. This is of theoretical interest since it shows how the transmission zeros of S which dominate classical equalization theory are naturally related to Test II. The transmission zeros of S are defined to be the poles of S-1 -]s]*)- ’ in the R.H.P. Transmission zeros have no intrinsic relationship to equalization theory. For example, they have no connection at all with Test I and with the general equalization Theorem 5.1 for any d other than d=S. The case d=S amounts to Test II. Thus the author

considers it something of a historical accident that trans- m ission zeros arose at all in equalization theory. Their role is to give a basis in which the key matrix A of Test II is finite dimensional. We begin our presentation by analyzing Hankel matrices.

There is another way to look at Hankel matrices. Since the Fourier transform can be used to identify I* with L*( II), the square integrable functions on the circle II, the matrix X, can be identified with an operator X, on L*(H). To describe this operator define H* (resp. a*) to be those functions in L* whose negative (resp. nonnegative) Fourier coefficients vanish, define Ps to be the orthogonal projection onto the subspace 5 of L*, and define Gsn, to be the operator on L*(n) .which mu ltiplies each function g in L* by the function F, that is

[%,g](eje)=F(e-@)g(e@).

Then X, can be identified with x,=P;mMPH2

and CJ,,., with TM= P,i%,f$*.

Now H* can be thought of as those functions in L*(n) with analytic continuations onto the disk and w* those which continue onto the exterior. So if M is uniformly bounded and ‘analytic on the disk, then X,=0 and so has no range. Let us consider only scalar valued M and L*(n) functions for this discussion. One can compute that for

and ]z]<l

b,(ej’)= --+p

[xbzg](eje)=& ,-,ie for any g in H*. Thus the range of XbZ is one dimensional and

1 z-eie

is a basis for it. From linearity 1

zI- eie

is a basis for range of X, whenever M is analytic on the disk except for simple poles at the z,. Higher order poles behave similarly.

Now we convert Test II into a finite dimensional test. Throughout this section we shall assume that S is given as a rational function II S I] L, < 1, and that Cl is the whole jw axis. Also assume S has order 1 transmission zeros only denote them byp,; . ., p,, . The function u = a0 used in our test is a linear fractional transform of S and so it is rational; moreover, the points

1 -PI zl= I+p,

are the poles of


.

(l-aa)%=&?

in the disk. The crux of Test II was to determine if the infinite matrix

A&,(1-g)s*)

is positive definite. This is the matrix in the e -j”’ basis for the operator on p* we shall denote by

For 1 -g uniformly bigger than 0 the operator A’ is positive definite if and only if the matrix A” defined by

is positive definite; here (Y is the spectral factor ]a]*=(1 - g)6* satisfying (Y and (Y-’ E H”. This is because

Clearly, A” is positive definite if and only if ( A”f, f )a0 for f in the range of XGou- ,. The poles of &!(u -’ inside the disk are precisely at the transmission zeros z, of u, so the range of XGoaml, equals the span of the bz,. Thus A” is positive definite if and only if the finite dimensional matrix with entries

(A”bZ,~ bJ (5-l) is positive definite.

The point is that if we express the operator A” as a matrix with respect to the basis e -9 for g*, then we get an infinite matrix equivalent to the one used in Test II. If we use the “transmission zero basis” above we get the finite matrix above. We emphasize that actually S need not be given as a rational function-only the transmission zeros are needed to make this work. Consequently, if a designer has measured data and wants to ‘fit it with an S having prescribed transmission zeros he might well prefer to use the bz, basis. One possibility would be to use some of each basis if we favored some transmission zeros and not the others. The author has not investigated this or any -other basis one might try.

The crucial matrix in (5.1) can be computed exp1icit.y using standard operator theory methods as in Section V [ 151. On obtains that the gain-bandwidth Test II in terms of transmission zeros is:

1) Compute the z, from S. 2) Expand the function asa-’ outside the disk as

where A is analytic outside the disk and 3 means c= l/W. 3) Find

as above. 4) See if the matrix with these entries is positive definite.

Several matrix tests appear in [ 131. Note that G appears nonlinearlv. So in obtaining finite dimensional matrices we

[lOI

ti:;

(131

1141

[I51

have sacrificed linear dependence on gain. A procedure similar to all this allows one to convert Test I or any test coming from Theorem 4.1 to one involving only finite dimensional matrices provided S and G are rational. How- ever, the transmission zeros of S have nothing to do with the process. In Test I what matters is the poles zI of

inside the disk. Find the z,‘s and use the bz, as before. For Theorem 4.1 what matters are the poles of fd.

Hopefully, enough has been explained here so that an interested reader could explore the relative merits of using different bases in the problem. The authors’ strong prefer- ence is for the simple e jne basis. He suspects that the most serious use for others will come when S has a transmission zero &, on g. That has been ruled out in this paper by hypotheses. To treat it in Test II one would probably want to include a b, in the basis and take the limit of the resulting matrix as z approaches &,.

ACKNOWLEDGMENT

The author wishes to thank H. Carlin, W. Ku, and R. Tucker of Cornell University; A. Podel, B. Kendall, and D. Mellor of Hewlitt Packard Corporation and W. Peterson of Varian Associated for providing valuable perspectives on exactly what is and is not important in gain equalization. In particular their emphasis on using no information be- yond measured data had great influence on this paper. Also many conversations with the mathematician J. Ball were extremely valuable.

[II

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141

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PI

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acting in H” (broadband matching),” in Topic in Functional Anal- vsis. Advances in Mathematics Sunnlementarv Studies, Vol. 3, 1978 ‘pp. ‘129-157.

&.

J. Will iam Helton, “Operator theory and broadband matching,” Fourteenth Allerton Conference on Circuits and Systems, 1976, pp. 91-98.

“Non Euclidean functional analysis and electronics,” (in preparation).

An expanded version of this paper which is not intended for publiration, but which is available upon request. Hewlitt Packard, Application Notes 154, S-parameter design, April 1972 W . Hintzman, “Best uniform approximation via annihilating mea- sures,” Bull. AMS, Vol. 76, No. 5 (September 1970), pp. 1062-66. E. Kuh and R. A. Rohrer, Theory of Linear Active Networks, Holden-Day, San Francisco, 1967. C. L. Siegel, Symplectic geometry, Academic Press, New York, 1964. Wilkenson, The Algebraic Eigenvalue Problem, Oxford Umversity Press, London, 1965. D. C. Youla and M. Saito, “Interpolation with positive real functions,” J. of Franklin Institute, vol. 284, no. 2 (1967), pp. 77-108.

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J. Will iam Helton (M’73) was born in Jackson- ville. TX. on November 21. 1944. He received the bachelors degree in mathematics from the Uni- versity of Texas, Austin, in 1964, the masters degree and the Ph.D. degree in mathematics from Stanford University, Stanford, CA, in 1966 and 1968, respectively.

From 1968 to 1973 he was at State University of New York, Stony Brook, as an Assistant and then Associate Professor, In 1974 he visited Uni- versity of California; Los Angeles, for six months

and subsequently moved to University of California, San Diego, as an Associate Professor. Currently, he is a Professor of mathematics at UCSD. He is an editor of the Integral Equations and Operator Theory and the Journal of Operator Theory. In January of 1980 he delivered one of the principle addresses to the American Mathematical Society. His interests are: circuit theory, distributed systems, and aspects of the theory of operators on Hilbert space which come from circuits, systems, differential equations, and integral equations.

Synthesis o f Lossy Lumped-Distributed Cascade Networks

ALOIS J. RIEDERER, MEMBER, IEEE, AND LOUIS WEINBERG, FELLOW, IEEE

Abstruct-The synthesis of lossy lumped-distributed networks is important for many applications; for example, in the analysis of large systems such as the interconnections of the circuits on an LSI or VLSI silicon chip, such networks have been used as models, and a solution of the synthesis problem will thus aid in the design of these chips. In this paper single- variable realizability conditions and synthesis procedures are established for the class of lossy and/or lossless lumped-distributed cascade networks described by an input-impedance expression of the form

i=o with ai( b,(s) real polynomials in s.

The cascade networks consist of commensurate, uniform, lossless transmission lines interconnected by passive, lumped (lossless and/or lossy) two-ports and terminated in a passive load .which can be prescribed as part of the specifications.

Moreover, the results of this paper are also applicable to lumped- distributed cascade networks which contain noncommensurate, tapered and/or lossy transmission lines (e.g., RC lines, distortionless lines) and to

Manuscript received March 3 1, 1980; revised April 22, 198 1 and June 3, 1981. This work was supported in part by a Trenton State College Research Grant.

A. J. Riederer is with the Department of Engineering Technology, Trenton State College, Trenton, NJ 08625.

L. Weinberg is with the Department of Electrical Engineering, City College, and the Department of Mathematics, Graduate Center, City University of New York, NY 1003 1.

nonelectrical systems which can be modeled as distributed or lumped- distributed cascades of types similar to the ones described above (e.g., acoustic filters).

I. INTRODUCTION

I N THE analysis of large systems such as the interconnections of the circuits on an LSI or VLSI silicon chip,

lossy m ixed lumped-distributed networks have been used as mode ls. In this paper we solve the problem of the synthesis of such networks, name ly, networks connected in cascade. This synthesis is the latest step in a natural evolution of synthesis of lumped and distributed networks, whose development we trace briefly below.

As is well known, the synthesis of filters containing lumped elements has raised network theory, both analysis and synthesis, to a sophisticated science [l], [2]. This filter theory has been well known for a number of decades. Except for some unsolved problems involving networks without transformers, a fairly complete theory is available. For driving-point functions Brune’s theorem provides a complete characterization, name ly, the necessary and suffi- cient condition for the realization of a driving-point function by a network containing lumped elements is that the function be rational and positive real (pr). Darlington’s theory then solved the filter problem by relating the driving-point function to the transmission function of a

0098-4094/81/1200-1137$00.75 01981 IEEE

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