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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 41

Generalized Darlington SynthesisPatrick Dewilde,Fellow, IEEE

(Invited Paper)

In honor to the great scientist and engineer Sidney Darlington

Abstract—The existence of a “Darlington embedding” hasbeen the topic of vigorous debate since the time of Darlington’soriginal attempts at synthesizing a lossy input impedance througha lossless cascade of sections terminated in a unit resistor. Thispaper gives a survey of present insights in that existential ques-tion. In the first part it considers the multiport, time invariantcase, and it gives the necessary and sufficient conditions forthe existence of the Darlington embedding, namely that thematrix transfer scattering function considered must satisfy aspecial property of analyticity known as “pseudomeromorphiccontinuability” (of course aside from the contractivity conditionwhich ensures lossiness). As a result, it is reasonably easy toconstruct passive impedances or scattering functions which donot possess a Darlington embedding, but they will not be rational,i.e. they will have infinite dimensional state spaces. The situationchanges dramatically when time-varying systems are concerned.In this case also Darlington synthesis is possible and attractive,but the anomalous case where no synthesis is possible alreadyoccurs for systems with finite dimensional state spaces. We giveprecise conditions for the existence of the Darlington synthesisfor the time-varying case as well. It turns out that the mainworkhorse in modern Darlington theory is the geometry of theso called Hankel map of the scattering transfer function to beembedded. This fact makes Darlington theory of considerablylarger importance for the understanding of systems and theirproperties than the original synthesis question would seem toinfer. Although the paper is entirely devoted to the theoreticalquestion of existence of the Darlington embedding and its systemtheoretic implications, it does introduce the main algorithmused for practical Darlington synthesis, namely the ‘square rootalgorithm’ for external or inner-outer factorization, and discussessome of its implications in the final section.

I. INTRODUCTION

T RADITIONAL Darlington synthesis [1] is concernedwith the realization of a rational and ‘bounded real’

transfer function with bound 1 as a partial transferoperator of a ‘lossless’ transfer matrix , which then hasthe form

(1)

and in which the Smith–McMillan degree of is equal tothat of (it is equal to the dimension of the state space ofa minimal system theoretical realization for). In addition tothe embedding, a cascade realization of, which can alwaysbe performed in the rational case, results in an attractive

Manuscript received May 1, 1998; revised August 1, 1998.The author is with DIMES, Delft University of Technology, Delft, the

Netherlands.Publisher Item Identifier S 1057-7122(99)00540-1.

Fig. 1. The Darlington problem consists in the first place in embedding acontractive andcausal transfer functionS in a unitary and causal�. For adiscussion of the meaning of the symbols used, see Section II.

realization of as well, especially from the point of viewof selective behavior, i.e., bandpass and bandstop properties.Fig. 1 shows the Darlington set up. Signals on input and outputwires carry the interpretation of incident and reflected wavesin which the energy is measured as the norm squared quantityintegrated over time.

In this paper we consider two important generalizationsof the Darlington theory, one in the transform domain forsystems with nonrational transfer functions and the other inthe discrete time domain, for time-varying systems. We shalldiscover a striking parallel between these two cases, but somemajor and interesting differences as well. Generalizations ofthe Darlington theory to nonrational functions was a populartopic in the early seventies, and gave rise to contradictoryresults. For example, it has been stated that any contractiveand causal transfer function has a Darlington synthesis

. If is required to be unitary on the imaginary axis, the statement is certainly not true. Necessary and suf-

ficient embedding conditions were discovered independentlyby the author [2] and D. S. Arov [3]. On the other hand,if the requirement is simply that be embedded in anisometric with a finite number of ports, then necessaryand sufficient conditions are also known, and closely relatedto the classical Szego condition for the existence of a ‘spectralfactorization’, a particularly nice exposition of which is givenin [4].

In recent times, the author and his collaborators have spentquite some effort in extending the Darlington theory to thediscrete time, time-varying case, say a case that covers andextends classical linear algebra. It turns out that embeddinganomalies which in the time-invariant case only occur forsystems with infinite dimensional state space, already takeplace in time-varying systems with very low state dimensions.We shall report on these results further on in this paper.

1057–7122/99$10.00 1999 IEEE

42 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 1, JANUARY 1999

It turns out that the state-space formalism is the most effec-tive vehicle to discuss Darlington synthesis. Darlington himselfwas working strictly with rational functions in the transformdomain. Anderson and Vongpanitlerd [5] have demonstratedconvincingly the power of the state space approach in theirlandmark book. More recent work on time-varying systemsshows that Darlington theory does not need an analyticalframework, it suffices with a purely algebraic theory. Darling-ton was working strictly with a scalar function , which hewas embedding in a 2 2 matrix function of the variable.Darlington’s work was duly generalized by Belevitch [6], [7]and put in a practical context by Neirynck e.a. [8] leading tothe effective and popular ‘parametric methods’ to synthesizelossless selective cascade filters. Subsequent to this work therewere attempts to generalize it to multiple input multiple outputsystems ([7] and [9]) but these were not really successful, dueto the lack of a good degree reduction criterion for multiportfactorization. Such a criterion was first presented in [10] andlead to a new and effective method for multiport Darlingtonsynthesis [11]. It turns out that the latter method actuallygeneralizes to the nonrational case. We shall give a surveyof these results later on.

For the sake of uniform treatment, and ease of relatingto analytic properties, we shall put the theory in the-domain, where the symbol can be interpreted as a causalshift operator (for mathematical ease we useinstead of the“ ” that is more common in the engineering literature),or just as , where is the variable in the Laplacetransform of continuous time functions. The theory remainsin this way applicable to the ‘’-domain through the bilineartransformation, whose inverse is , and which mapsthe unit circle of the complex-plane on the imaginary axisof the -plane and the open unit disc of the-plane ontothe open right half plane of the-plane (it is also the so-called Smith transform). The reason why it does not mattertoo much in which variable we work is precisely the algebraicnature of the Darlington theory, and the the fact that thebilinear transformation conserves the degree of polynomialsafter transformation.

II. THE BASIC DARLINGTON SYNTHESIS PROBLEM

We now work toward a general formulation of the basicDarlington synthesis problem in the-plane context. Thereis some ambiguity in the literature concerning what is really‘Darlington synthesis’, which we will explain further on. Westart out considering an transfer function which canbe thought of as representing a causal, discrete time system,although that interpretation could be a little strenuous whenit concerns a continuous-time signal—but it does exist. If thesystem represented by is stable in some sense (we alreadyassumed it to be causal), then will be analytic insidethe unit disc and have some boundedness associated with it.There are several contexts appropriate for studying stability,we choose one which is natural to Darlington theory as itis concerned with propagation of energy, the Hilbert spacecontext. The impulse response corresponding todefines bounded map of the convolution type between two

spaces of time series whose energy is bounded. Such spacesare called ‘of type’. As input space we take time series ofthe type where each belongsto a (possibly complex) vector space of the type , in which

indicates the complex numbers. (As a rule, we take vectorsin the ‘row convention’, i.e. we write the multiplication ofa vector with a matrix as . When we handle infiniteseries or infinite matrices, we single out the zero-th element bysurrounding it with a box, for orientation purposes.) We writethe Euclidean norm of a vector as and it equals

in which is the -th (scalar) component of . A timeseries which is bounded in energy then hasthe norm

The space of such bounded time series is traditionally called. A simple matrix which maps a vector to a

vector will have an ‘operator’ norm given by

This norm isnot a Euclidean norm on the space ofentries of , such a norm would be called a ‘Frobenius norm’:

A bounded input—bounded output (BIBO) transfer operatorbetween an input space and an output space will have theproperty that its Fourier transfer function is uniformlybounded on the open unit disc of thecomplex plane in the sense that the collection of Euclideanmatrix norms has a finite upper bound when .It is known in complex function theory (for a good textbook,see [12]) that such functions have radial limitsto values on the unit circle ofthe complex plane, whose norms are also uniformly boundedwith the same bound. The space of such uniformly boundedmatrix-functions on the unit disc is traditionally called theHardy space . It is actually the subspace of measurableand uniformly bounded functions on the unit circle ,which have a uniformly bounded analytic continuation to theopen unit disc . It turns out that the bound, which is denoted

or equivalently , is actually a least upper boundon the energy magnification between the input and the outputof [12], [13]. Hence, will be BIBO stable for the energynorm on the input and output spaces iff . If,

DEWILDE: GENERALIZED DARLINGTON SYNTHESIS 43

in addition, the bound is actually equal or less than 1, thenthe system with transfer function will not amplify theinput energy, it will have a passive input/output behavior,and should be realizable by a physically passive system. Wesay that in that case is contractive (in the engineeringliterature, the term “bounded real’ is also used, but we preferthe mathematical term because of the ambiguous connotationof the former). Such an is characterized by the properties:‘ is analytic in and ’.In particular we have that is a causal transfer functionand on the unit circle (in which indicates thepositive square root of ). Note that the conditions ‘analytic in the open unit discand ’ are notsufficient to guarantee that is in (i.e. -BIBOstable)—a counterexample is given by ,which is analytic but unbounded in , and of unit modulusa.e. on the unit circle.

We need a few additional notions from matrix or operatortheory. The hermitian transform of a matrix is the matrix

whose elements are defined by in which thebar indicates complex conjugation. We say thatis hermitian,if it is square and equals its hermitian transform, . Wesay that is positive definite if it is hermitian and for all(row-)vectors of the right dimension we have .

is strictly positive definite if . Anmatrix actually defines a linear map from a (complex)

-dimensional vector space to a (complex)-dimensionalvector space through the assignment . Weshall assume that on an-dimensional vector space an innerproduct is defined as . The hermitian conjugatethen defines what is called an adjoint map through the rule

.Since a transfer function also maps an input space, now of

type , to an output space, now of type, we may extend thedefinitions to the natural in-product in -type spaces. In the‘time domain’ we have here an abstract mapgiven by convolution, while in the transfer domain the map isgiven by -transform multiplication: . Therelevant in-product in the input space (and likewise in theoutput space) is expressed in the time domain by

and, via Parseval’s theorem, in the transform domain by

The adjoint of a transfer operator will then be definedby the rule: . If is thetransfer function that corresponds to and the arethe matrix values that takes on the unit circle, then thetransfer function corresponding to its adjoint takes the values

on the unit circle. The correspondingtransfer function (i.e. function of) is then the so-calledpara-Hermitian conjugate , which is anticausalwhen is causal. It turns out that this transfer functiondefines an output-input map that produces an energy in-productrelated to the original by adjunction.

If corresponds to a passive system, then it is contrac-tive, i.e. . This will be the case iff

. (Notice that in generalexcept on the unit circle —we must carefullydistinguish the upper star, which indicates pointwise Hermitianconjugation, from the lower star, which denotes the analyticalcontinuation of the hermitian conjugate on the unit circle.) Weshall say that is “isometric” ifor, equivalently, . In that case, the energyin the output is in all circumstances equal to theenergy in the input , and the system producing it must thenbe physically lossless. A transfer function characterizationof this condition is obtained by analytic extension from theunit circle as for whatever region in thecomplex plane wherein the extension is well-defined. We saythat is ‘unitary’ if, in addition to , also

. This will be the case when is isometricand square. If is at the same time causal and unitary(i.e. it is in and it is unitary), then it is calledinner in themathematical literature, a term that we shall adopt. The term‘Darlington synthesis’ of can then just as well be termed‘inner embedding’ of . Given a contractive, itis not at all obvious that there exists an suchthat the combined is isometric. Nor is it obviousthat has an embedding into a unitary . In the nextsection we shall summarize the results known on these matters.

As the reader will have noticed, we avoid a physicalterminology to indicate mathematical properties. The reasonshould be clear: there are many ways to describe a givenphysical situation mathematically. In our treatment, we use a‘scattering’ framework throughout. This means that the signalscarry energy in the form of a quadratic norm. That is, forexample, the case when the input and output quantities ofa circuit are incident or reflected waves. If, at a given port,the input quantity would be a voltage and the outputquantity a current , then the corresponding power transportwould be measured by the real part of the mixed voltage-current product, . Given a normalizing resistancevalue we can transform and to the wave quantities

and the power transportbecomes , the difference between the powercarried by an “incident wave” taken as input signal and thepower carried by the “reflected wave”taken as output signal[14]. These power quantities are now strictly represented byquadratic norms (the energy then becomes the integral overtime). It is known in filter theory that this is indeed a pertinentway of representing selective filters connected between lossysources and loads [15].

Before proceeding to our main results, we need a fewmore notions from analytical function theory. Any (scalar)function in can be factorized as inwhich is inner (i.e. a causal transfer function of constantmodulus 1 or pure phase function) and which has theproperty of beingouter. Although outerness has some niceanalytical characterizations, which require some more theorythan needed for this paper, we shall suffice here by stating thatit is characterized by the property that it has an approximate


causal inverse, i.e. that there exists a sequence of causalbounded functions so that in thequadratic sense on the unit circle. For all engineering purposes,

has a causal inverse, it is a “minimal phase function”.The property extends to matrix functions: if ,then in which is inner, andouter—see the discussion further on.

III. I SOMETRIC AND UNITARY EMBEDDING—THE LTI CASE

In the next three sections we give a number of resultson isometric and unitary embedding for the LTI case. Westart with some general considerations and then focus onsystems with a finite dimensional state space, those are systemsrepresented by a rational transfer function.

Suppose that an causal and contractive transferfunction is given (we may assume to be square,if that would not be the case the addition of extra zerorows or columns would be adequate). Then the matrixfunction is positive definiteon the unit circle of the complex plane. We search foran causal transfer function such that

—a spectral factorizationof. A first, fundamental, but

not constructive, result was given by Masani and Wiener[16]—for a very nice treatment of the Masani Wiener theory,see Helson [4]. It states:

Theorem 1 (Masani–Wiener):Let the positive definitematrix function be invertible almost anywhere on

the unit circle of the complex plane. Then is factorableas if and only if

(2)

There is a (fairly straightforward) extension of the theorem tothe case where is not invertible a.e., but here we stick to thegeneric case. The important point isthat the factorization doesnot necessarily exist. The existence condition (2) for the matrixfunction case is an extension of the famous Szego conditionfor the scalar case [17], and it states, loosely speaking, thatthe spectrum cannot be zero on more than a verythin set. For example, if has the behavior of an ideallow pass filter, then there would be an interval in which it isidentically zero, and the condition would certainly be violated.The integral in (2) is known as theentropyof the process forwhich is the spectral density, and the property that itis larger than corresponds to a stochastic property knownas ‘indeterminatedness’ of the process. Be that as it may, if

is rational, then the Szego condition will certainly besatisfied and the desired will also exist. The point to beconsidered is the construction of so that it has at mostthe same Smith–McMillan degree as . In the section ontime-varying systems we shall given such a construction basedon a state space representation for, a construction, which isequally valid for the LTI case as a special case.

Of all the of minimal degree, which satisfy, there is one of special interest,

namely the one which isouter or minimal phase. This is the

one that we would normally construct if we use the algorithmspresented later (actually it does not matter which minimal oneis chosen for the embedding construction to follow next). Letus assume that this is the selected, but at this point we donot yet assume rationality. Consider next the transfer function

(3)

which is isometric on . As the following endeavor, we wishto find causal transfer functions and so that theoverall transfer function

(4)

is unitary on .It has been claimed in the literature that this can always be

done [18]. This is not true. One obvious property of isthat its elements have a special kind of analytic extension tooutside the closed unit disc. Expressed in terms of the complexvariable , the unitarity of (3) says that

(5)

As an analytic function in the open unit disc isuniquely defined by its boundary values , and it willhave only discrete zeros in , as points where itsrank drops from (this uniticity is a result of complexfunction analysis applied to , see [12].) Since for

, and from (3) one sees that there will exista continuation for to the whole region outside the closedunit disc (the region ) as ,which certainly exists as a matrix of meromorphic entries in.We see that if there exists a unitary embedding for , thenalso the entries of must have a meromorphic continuationto the complete region . Such functions are calledpseudo-meromorphically continuable, and they form a class oftransfer functions with special analyticity properties, namelythe pseudo-meromorphic extension is unique and it has radiallimits a.e. to the unit circle from points in . Surely, rationaltransfer functions are such, but it is easy to cook up functionswhich do not satisfy the condition and necessarily will notbe rational. An interesting example iswhich is obviously contractive in (its maximum modulus inthat region is ), it is analytically continuable everywhereacross the unit circle, but not to the whole of, since thesquare root will have an essential singularity at the point

from which a branch cut will extend to . It followsthat this hasno unitary embedding, there isno Darlingtonsynthesis for this !

The previous reasoning, convincing as it is, does not yetgive a system or circuit theoretical clue as to why the em-bedding should not exist when the transfer function has nopseudomeromorphic continuation. We explore this point in thenext section.

IV. REMEMBRANCE OF THINGS PAST

In [2], [19] the author showed that the existence of aDarlington synthesis for a contractive scattering matrix

is closely related to a basic system theoretical property


of . The property is of a geometric nature and carries overto situations where there is no transform available anymore,such as time-varying systems. In general terms (we shall makethings more precise soon), the system’s geometry viewed froman input–output point of view is characterized by four relatedspaces and a system induced map between them which iscalled the”Hankel map”. Let be the space of input signalsand the space of output signals (in our framework, bothspaces will consist of vector time series which are bounded inenergy, they are Hilbert spaces). Classical dynamic SystemTheory (for an original description see [20]) decomposeseach space in a strict past and a future: (indicates decomposition in orthogonal subspaces—andare Hilbert spaces of the -type). For example, with

we haveand . Let indicate the projection ofa signal on his future part (i.e. ), then theHankelmap maps by definition to through the recipe:

(6)

It connects the (strict) past of the system to its future. Thisconnection happens concretely through the state of the system,in the situation whereby nonzero input is only provided inthe past (i.e. the input signal is zero from on): fromits past, the system gleans information which it subsumesin its state, and then it uses this state to generate its future,under the assumption that future inputs are zero (in case thesystem is not linear a more sophisticated definition must begiven, see [20], but we stick to the linear case here). Fig. 2illustrates the situation. We can expect that the Hankel mapyields important information on the structure of the system.Its kernel , consisting of the signalsconsists of all the (strict) past input signals whose informationthe system does not retain: its future response to such a signalis zero. The orthogonal complement to in the past inputspace we call , and it is a space characteristicfor the information that the systemdoes retain. It is alsothe range of . We call it the natural input state spaceor reachability spaceof the system. At the output side thereare the duals of these spaces. The range of the Hankel map,

is the space of natural responses, those arethe responses of the system when the input is kept equal tozero. It is anatural output state spaceor observability spacefor the system. Its orthogonal complement isactually the kernel of the adjoint map , i.e. the set of outputs

. All the spaces defined in this section have ofcourse Fourier transforms, and their orthogonality relationshipsare preserved thanks to the Parseval theorem. We denote theseFourier transforms by a , e.g. is the Fourier transformof (for readers familiar with Fourier theory we mention that aspace like transforms to the celebrated Hardy space

of the unit disc , so that and are subspaces of it.then transforms to the orthogonal complement

of in ).The nullspace has the property that it is ‘shift invariant’

for the backward shift: , i.e. if some inputin the strict past generates the zero state, then so does

Fig. 2. The Hankel operator maps inputs in the strict past to outputs in thefuture.

. Such shift invariant spaces have remarkable math-ematical properties described by the famous Beurling–Laxtheorem [12], [4], [13] which we now introduce. It states thatthere exists a transfer function so that isisometric i.e. and all inputs inare right multiples of in the sense that forall with an auxiliary -dimensional space of the

type, (in the next section we shall meet rationalexamples which should make the present, general discussionless abstract). If we look at inputs in , then we see that foreach point of the unit circle, they form a -dimensionalsubspace of the (pointwise) -dimensional input space (inanalytical function theory it is proven that the dimensionis constant a.e.). Because is co-isometric, we necessarilyhave . If then the nullspace is defectivein dimension, and the corresponding input state space (theorthogonal complement of ) will have to be very large, itcannot be a finite dimensional space since it must contain aspace isomorphic to . For example, if

, the scalar case, thenis either or 1. In the first case, thenullspace is trivial and the natural input state space

becomes the entire input space, the system ‘remembers’its complete input space, or, to put it differently, the state ofthe system is but a tampered version of the complete input.If, on the other hand, , thenin which is a scalar function of unit modulus. Thenatural input space becomes , which willbe a finite dimensional space whenis rational, and evenotherwise it is still a very ‘small’ space, the system forgetsalmost everything from its past.

We say that a system isroomy [2], [19], iff , i.e. iffthe system has a pointwise fully dimensional nullspace.isin that case not only co-isometric, but unitary as well. Thenwe have that in which is a causal transferfunction, and we obtain theleft external factorization

(7)

which displays the causal as a ratio of two anticausaltransfer functions. Hence , which is analytic in the openunit disc, has an extension (in the sense of radial pointwiselimits) to a function which is meromorphic outside the unit


disc, i.e. it is pseudo-meromorphically continuable (as statedbefore, such an extension is necessarily unique). The externalfactorization (7) is actually a “left coprime factorization”because it is minimal in the sense that every other suchanticausal factorization will involve left multiples of and

(this fact needs some proof, see e.g. [19]). Hence, thecoprime factorization involves automatically the existence ofa pseudomeromorphic continuation for as explainedbefore. The converse is true as well: if possesses apseudomeromorphic continuation, then it also possesses acoprime factorization of the type (7). This fact can fairlyeasily be proven through the observation that if ispseudomeromorphically continuable, then its nullspacenecessarily has full dimension, since it will contain at leastthe space of all -dimensional vectors with entries in thestrict past that belong to the intersection of all the scalar inputnullspaces of the entries .

A dual theory, now on the output nullspace, is of coursealso possible. In this case the Beurling–Lax theorem states thatthere exists an integer and a causal and isometricfunction such that for all belongingto an auxiliary input space of the type . If thesystem is roomy then it will be meromorphically continuable,and it will follow that . In that case we have that

for a causal and a unitary ,and aright coprime external factorization

(8)

follows, again exhibiting the pseudomeromorphic continua-tion. It is not too hard to prove that ,a quantity which can be used as a kind of ‘generalized degree’[19].

V. GENERALIZED LTI DARLINGTON SYNTHESIS

We are now ready to perform the general linear timeinvariant (LTI) Darlington synthesis in abstract terms. Themethod that we shall use can, however, easily be convertedinto a constructive or algorithmic technique, at least for therational case, see the later sections of this paper. Let us assumethat we are given an isometric and roomy . It can beobtained as , result of a spectral factorization, orelse directly be given as an isometric transfer function.Because of the roominess assumption, will have a leftcoprime factorization:

(9)

in which decomposes as in accordancewith . We see that and can sharethe same factor, it is in fact not too hard to prove thatif is roomy with left coprime unitary factor ,then the minimal outer spectral factor has an externalfactorization with left factor as well. We formulate thegeneralized Darlington synthesis as a theorem.

Theorem 2: Let be a causal and isometric transferfunction. If is roomy, then the right coprime factoriza-

tion

(10)

is such that is constant and can be chosen andis a (minimal) unitary embedding for (and also for

which is part of ). Conversely, suppose thator has a minimal unitary embedding, then it is roomy,and the Darlington embedding is actually a right external innerfactor for .

Proof—Sketch:The proof makes use of a special propertyof coprime factorizations borrowed from standard Euclideanfactorization theory and adapted to the case. In thestandard theory we have that two polynomials and

are coprime iff there exist polynomials andsuch that the Bezout equation issatisfied. The property extends to polynomial matrix functions,and with some effort also to our present case, where someadded difficulties may occur because of the possible presenceof zeros on the unit circle which actually do not participate inthe division theory. The precise statement is [19]:two causaland bounded transfer functions and are left coprime(in the sense) iff there exist sequences andof causal and bounded transfer functions such that

for the entrywise quadratic norm on the unit circle.(The difference with the standard case is that the anddo not necessarily have a bounded limit themselves.) Two

additional mathematical remarks are in order: 1) the limitingprocedure given has as a consequence uniform convergenceon compact subsets of the open unit disc and in particularpointwise convergence and 2) the property actually expressesthe ‘right outerness’ of ). If we apply this principleon the coprime right external factors for wededuce the existence of series of causal and bounded transfermatrices and which are such that

Premultiplying with and using the fact thatwe find

This expression asserts the existence of the limit in quadraticnorm on the entries of from a series of elements whichare all causal. Complex function theory then produces theresult that must be causal as well. Since it is actually bydefinition anticausal, it can only be constant since this is theonly type of transfer function that can be causal and anticausalat the same time. Hence, if the isometric has an externalfactorization, which happens iff it is pseudomeromorphicallycontinuable, then it will have a Darlington embedding. Theconverse was already evident from the discussion in the


previous section in which we showed that an embeddedis necessarily pseudomeromorphically continuable.

A final remark for this section will provide the link withthe time-varying theory to be treated furtheron. It seemsthat pseudomeromorphic continuability is the central propertyof transfer functions which allows for Darlington synthesis,but there is a time-domain criterion that is equivalent andmore general. From the Darlington construction by externalfactorization given in Theorem 2, it follows that willhave a Darlington synthesis iff its corresponding has thefull range property, i.e. its constituent functions span the spaceIC at each point of the unit circle (a.e.). In the timedomain, however, this property expresses an invariance whichis of crucial importance, namely that does not contain asubspace that is doubly invariant, i.e. for whichand . It is this property that will determine thepossibility for Darlington synthesis of time-varying systemsfor which there is no useful Laplace or-transform.

VI. THE TIME-VARYING CASE

Now we consider a system , which is time-varying,and given by a contractive transfer map (the distinctionbetween and is not relevant anymore). The Darlingtonsynthesis problem will consist in the solution of the questionwhether can be embedded in a causal unitary transferfunction

(11)

The input and output spaces are, as before, spaces of quadrat-ically summable series, but because of the time variance wenow have more freedom in allowing time variations from onetime point to another. For example, we allow time varyingdimensions for the vectors in the input and output spaces. Wesuppose that at a pointin time the input vector belongs toa vector space , dependent on, and likewise, belongs toa . has dimension while has dimension . Thesequence of dimensions of the input sequence is then givenby . We assume that each inputsequence considered is bounded in quadratic norm:

We do allow that any number of spaces is actually empty,i.e. has dimension zero. Such a space then simply correspondsto a ‘place holder’ for which no value is available, i.e. ata certain time point there is no input data. In this way wecan easily embed finite matrix theory in our framework: wetake inputs as non existent up to and from some positivetime point e.g. For higher precision of notation we shallhenceforth denote the base input space as—in the sequelwe shall have to extend it. The transfer operatormaps aninput to an output and can hence be represented by

Fig. 3. The time-varying state space realization represents the calculation atat given pointt in the time sequence.

a (doubly infinite) matrix

...

...

in which each is a block matrix of dimensions .Causality is now expressed by a (block) upper triangularproperty: when .

Connected to an upper transfer operator, there is a statespace realization possible, based on the assumption that ateach point in time the system uses a state that it hasremembered from its past history, inputs a vector, computesa new state and an output , and then moves on to thenext time point. Because of the linearity of the computation,the local state space representation takes the form

(12)

in which the ‘realization’ is a collectionof linear operators (acting possibly on an infinite dimen-sional state space). The (time) series of statesthen belongs to a not yet defined series of spaces

—see Fig. 3.On the various spaces of time sequences we can define

standard operators as is also done in the time invariant case,but now we have to be a little more careful because ofthe changing dimensions, and the fact that operators whichcommute in the time invariant case may not commute anymorenow. First there is the nondynamical operator of performingan instantaneous transformation on each entry of a time series:

which corresponds to a diagonal operator

will be bounded iff there is a uniform bound on its entries:, and it will be boundedly invertible iff

each is square invertible, and there is a uniform boundon the collection of inverses . Next there


is the causal shift operator. will then be the anticausal shift. If

we collect the states of a computation in a series, then we canexpress the state space realization in terms of the four diagonaloperators , etc...,a

(13)

and we find, at least formally, the following expression for thecorresponding transfer operator:

the expression will make sense if a precise meaning to theoperator can be given (depending on the context,see further). An important remark at this point is that thecausal shift does notcommute with the diagonal operators

except when is Toeplitz, i.e. when all its entrieson the main diagonal are equal (and then necessarily squareof equal dimensions). Somewhat less is true, there is a kind ofpseudocommutation, and we have to introduce a new notationfor this:

(14)

in which is by definition adiagonal shiftof the matrixin the South–East direction. We can apply further shifts,

and, with some abuse of notation because the dimensions ofthe shift operator vary along the main diagonal, we writeapplications of the diagonal shift as

is a diagonal operator equal to a version of, shiftednotches in the South–East direction. The causal operator

can be represented by a collection of diagonal operators:

in which . It turnsout that most of the time varying theory parallels the time in-variant theory when one treats diagonals as ordinary ‘scalars’,respecting noncommutativities.

It is to be expected that the time-varying equivalent ofthe ‘Hankel operator’ will play a central role in a Darlingtontheory for time varying transfer operators. As we saw in thetime invariant theory, the Hankel operator connects the ‘strictpast’ of the system to its future, and therefore characterizesthe structure of its state. However, a time-varying system mayhave a different state structure at each of its time points,so we have to define a Hankel operator at each point.

—the Hankel map at point—will be defined by the mapand it will be

given by the matrix representation (dropping unneeded zeroentries):

......

... ...

(15)

The collection of the ’s actually form the overall Hankeloperator for the system under consideration. We build asystem’s map for which this collection is a natural operator.The trick is to consider not only one input time series at agiven time point , but to consider an appropriate sequencefor each and every time point. Stacking these input (and theresulting output) time series as one input object produces thedesired result. In this way, the input space becomes a spaceof objects of the type

......

......

...... ..

.

... ...

......

......

. . .(16)

In addition, we require that the complete scheme be boundedin energy, hence

The space so obtained we call , it is again a (somewhatcomplicated) Hilbert space of quadratically summable timeseries. Since each row in this scheme corresponds to a simpletime series for time point, we have that can be decomposedin a strict past and a strict future so thatwhen , and otherwise zero, while when

and otherwise zero. The space of the (i.e. the‘future’) we shall now call , while the space of the(i.e. the ‘strict past’) we shall call . We shall denotethe projection of on by and the projection onthe orthogonal complement (these are genuineorthogonal projections on the extended input and output spacesand ).

The action of a causal operatoron such a scheme is easyto tally, we just have to stack inputs and outputs:

......

......

... ...

. . .. . .


Fig. 4. The time-varying Hankel operator consists in a collection of matrices each of which maps an input in the strict past to an output in thefuture relative to a pointt.

We are now able to define the global Hankel operator foras

(17)

It consists in applying to elements of which are restrictedto the strict past, and looking at the effect only from timeon, for each , in one global operator. Since maps a twodimensional scheme to another two dimensional scheme, it isactually a tensor. At each time point specializes to a‘snapshot’ which is obtained by looking at its effect on the-th row of the input and looking at how it produces the-

th row of the output—this turns out to be the local Hankelmatrix . A graphical representation of the Hankel operatoris shown in Fig. 4.

Just as in the LTI case, the Hankel operator will generate thesystem’s state geometry. The following spaces are important:

• Natural Input State Space• Natural Output State Space• Input Nullspace• Output Nullspace

The crucial property of these nullspaces are their invariances.We see that is invariant for the left application of a diagonal

and of the anticausal shift :

(18)

Dually, is invariant for the application of a diagonaland the causal shift:

(19)

We call a -invariant space asliced space, because ifbelongs to it, then any row of with all other rows put to zeroalso belongs to it. A sliced space has a basis which consistsof time slices: for each time index there is a basis for thecorresponding row. A generalized Beurling–Lax theorem inthis setting is as follows:

Theorem 3 (Generalized Beurling–Lax):Any left DZ-invariant subspace of has the form

(20)

for some causal and isometric operator(i.e. ).Sketch of Proof:One considers the (so called wandering)

subspace and realizes that it is a slicedspace as well. An orthonormal sliced basis forproducesthe operator as the collection of these base vectors itself.This construction is identical to the construction used in theclassical case. For details see [21].

The (generalized) Beurling–Lax theorem leads, just as in theclassical case, to the construction of external factorizations forthe operator and to Darlington synthesis, and will providethe conditions under which it is possible. The time-varyingcase is considerably more involved than the LTI case, andalthough we are able to formulate in abstract form necessaryand sufficient conditions for the Darlington embedding toexist (Theorem 6), to treat the embedding concretely let usassume at this point that we dispose of a concrete stable statespace description for , i.e. we assume that (1) is locallyfinite, that is, it has a realization based on a collectionof finite dimensional local state spaces for each time pointand which is given, as explained above, by diagonal operators

consisting of diagonals of finite matrices and (2)the realization is such that the spectral radius of theoperator is strictly less than one. The spectral radius isdefined as

(21)

We denote this positive number as . It is actuallyequal to

which follows from working out the power product .When we say that the system has auniformly


exponentially stable or ue-stable realization. Whenthen left and right external factorizations for are easy toproduce. They are based on the construction of so callednormal formsfor the realization . We say that therealization is inoutput normal formif is isometric, i.e.if , i.e. for all . Dually,it will be in input normal form, if is isometric, i.e.

. Given a ue-stable realization, both the outputand input normal forms can easily be constructed througha state transformation. Such a state transformation takes thecharacter of a bounded and boundedly invertible diagonal map

which transforms the state to so that for each. If one applies this transformation on the state,

then one obtains a new, equivalent realization:

(22)

in other words, the realization transforms to thenew equivalent realization(in which represents the diagonal shift upward in theNorth-West direction, of course). To obtain, say, the outputnormal form, we must find so thatis isometric. Writing , this amounts to solving theequation:

(23)

or, in component form,

(24)

These recursive equations are called ‘Lyapunov-Stein’ equa-tions. When is ue-stable, then they have a unique solution,which with, for

and , is found by solving (23) as a fixed pointequation:

(25)

If we define asthe observability operator, then we see that is actuallya generalized Gramian . To find the needed forthe output normal form, we next have to factor ,which of course should be possible, since all the individualmatrices are positive semi-definite and can be factored as

.So far so good, but to achieve the output normal form,

we need the bounded invertibility of . This means that theindividual must be invertible, and that their inverses musthave a uniform upper bound. Alternatively, must have adefinite lower bound, i.e. there must exist an such that

. In other words, the system must be what we calluniformly observable. It should be clear that this is not alwaysthe case, and we obtain the result:a locally finite system whichis ue-stable will have a ue-stable output normal form if it isuniformly observable. (There is a somewhat more involvedconverse to this property, which we skip for technical reasons.)

When a ue-stable system is uniformly observable, then itwill have a right external factorization. The construction ofthe factorization is based on the output normal form. We sawin the preceeding paragraph that the system will have such aue-stable form if it is uniformly observable. Suppose now that

is in output normal form, that is. This means that each is actually isometric. Let us

now define two new diagonal operators so that therealization

(26)

is unitary, and let be the corresponding operator,

To assess the properties of we need the following lemma,which plays a central role in the subsequent discussion onDarlington realizations:

Lemma 1: Suppose that isa transfer operator which is ue-stable and for which therealization operator

is unitary. Then is unitary as well, i.e. and.

Sketch of Proof:The lemma can fairly easily be checkedby direct calculation, the crucial point in the verification is theexistence and boundedness of the inverse , whichis assured by the ue-stable assumption.

Hence , as defined, will be unitary if is ue-stable.It turns out to be the right inner factor for the right ex-ternal, coprime factorization of . This we see simply bycomputing . During the computation one needsthe (generalized) partial fraction decomposition of the product

, which, because ofis easily seen to equal. Hence (we write ‘ ’ for )

The last term in this expression is zero, because of the unitarityof the realization for . Hence, a realization for is givenby

Together these relations produce

or, if the original realization for would not have been inoutput normal form (and further explicited per time point):

(27)

in which now and . Ateach time point , this equation has the form of the famous


Jacobi - factorization of numerical linear algebra. If weassume that at time pointwe know from the previouscalculation, then the matrix

is known, and is transformed by the unitary (-type) matrix

to upper-triangular form (at least to block-upper form, butnothing prevents a further transformation to fully upper). Thetransformation produces which can then be used in thecalculation at the next time point , etc recursively.

We have obtained the following result:Theorem 4: Suppose that is a ue-stable, locally finite

transfer function which is uniformly observable, then it hasa right coprime external factorization given by

in which is inner. and can be found by the squareroot algorithm of (27).

A kind of converse of the theorem is fairly easily demon-strated by construction of ‘counterexamples’, namely by con-structing an with a ue-stable realization which is notuniformly observable. Such an will normally not have anexternal factorization. The author does not know whetherthe case where has a ue-stable realization which is notuniformly observable and for which a Darlington embeddingexists actually do exist (he thinks they do), but normally thesingularity of the Gramian will give rise to the existence ofwhat we shall call a defect space, and we shall see in Theorem5 that then a Darlington embedding does not exist.

Of course, there is a dual to the previous theorem, based onthe input normal form. For that we need the notion of ‘uniformreachability’ which will involve the reachability operatorand the reachability Gramian . We shall say thatthe realization is uniformly reachable if there exists ansuchthat . We obtain:

Theorem 5: Suppose that is a ue-stable, locally finitetransfer function which is uniformly reachable, then it has aleft coprime factorization given by

(28)

in which is inner. and can be found by an appropriatesquare root algorithm.

In contrast to what happens in the LTI case, it is possiblethat a ue-stable time-varying system possesses a left externalfactorization but not a right one and vice versa. This willbe the case,in general, when the system has a uniformlyreachable ue-realization which is not uniformly observable orvice-versa, and this can already easily happen with systemsthat are locally finite. In the time invariant case all systemswith finite dimensional state spaces have right and left coprimefactorizations. In contrast to the claim that ‘time-varyingsystem theory is but a small extension of time invariant theory’we see that the two types of systems have very differentproperties indeed!

We terminate this section by showing how Darlingtonsynthesis works for isometric systems, the more general (con-tractive) case is treated in the next section. Suppose thatisisometric, i.e. and that it is locally finite, i.e. thatit has a finite dimensional state space realization at each timepoint. We proceed by constructing a particular realizationfor , which is itself isometric, and by trying to embed it ina larger realization which is unitary and hence a candidate foran inner transfer function that embeds as

(29)

As we saw before in the LTI case, the properties of theoutput state space of will be crucial to the Darlingtonembedding theory and we may encounter some possibly verystrange behavior here. An important property of an isometricsystem like is the fact that its output nullspace actuallycontains the space (this is the space of outputs tocausal inputs under the action of the isometric operator).Hence, . But may be larger. Itwill also contain the kernel , by definition, andclearly . The ‘algebraic’ (i.e.unqualified) nullspace of will not only contain

, which is a left -invariant space, but also thelatter’s left shifts , and hence theirclosed union, which we denote as

The space turns out to be the characteristic outputnullspace of the complementary entry

for an appropriate sequence of input spacescom-plementary to . There is more the matter, however. The‘algebraic nullspace’ may be larger than . Wedenote that remaining space as

(30)

and will call it the right defect space of . The possibleexistence of this space has sometimes been overlooked in theliterature, but its properties appear to be crucial to the existenceof the Darlington embedding.

The following theorem (which does not assume local finite-ness) gives, in addition to the essential properties needed, theDarlington theory for isometric systems in abstract form.

Theorem 6: Let be a causal isometric transfer operatorand let be its natural output state space and itsright defect space. The output space then decomposes as

(31)

There exists an isometric operator with the same outputstate space and defect space and such that, forsome input space sequence

(32)

The operator


will then be isometric as well, and such that. will be unitary (hence inner) iff the defect space

.Sketch of Proof:The theorem is a direct consequence of

the generalized Beurling–Lax representation theorem appliedto the left DZ-invariant subspace :

. Furthermore, it is known fromfunctional analysis that an isometric operator will actually beunitary iff it has full range, in the present case iff .For a complete, formal proof we refer to the book [21].

In the case of ues locally finite systems, the theorem leadsto the following construction for the embedding. First, an iso-metric realization for can be obtained from an orthonormalbasis for the natural output state space . Let, at timepoint be an orthonormal basis for the range of theHankel matrix , which is assumed to be of finite dimension

. Assembling all these bases as slices of an overall operator, we see that has the form of an upper operator. The

next step is the assertion that there exist diagonal operatorsand so that can be represented as

This fact is basic to state space realization theory and followsdirectly from the shift-invariance property of whichstates that . Hence, given the choice ofbasis, we may recover corresponding diagonal operatorsand

by the recipe: and , inwhich we have used the orthonormality . Thecorresponding will automatically be isometric, but isnot necessarily ue-stable. An isometric realization foris nextfound by determining and from the relation

, specifically, . Thepoint is that, in contrast to traditional realization theory, wemay not be able to expressdirectly asbecause the inverse of may not exist.

Since is a DZ-invariant subspace, we havefrom the generalized Beurling–Lax theorem that there willbe an isometric and causal transfer operatorand an inputsequence such that . A state spacerealization for is easily derived from the realization for,we just have to complete the local isometric matrices so thatthey become unitary:

(33)

shares with , and remarkably, also the defect space.

The isometric will hence possess a Darlington embeddingiff its defect space is zero. This abstract condition translates toa concrete criterion, in the following important special case:

Theorem 7: The locally finite and causal isometric transferoperator will have a Darlington embedding if it has anisometric ue-stable realization.

Proof: An isometric realization has automatically a unit(and hence strictly nonsingular) observability Gramian. If,on top of that, also , then the embedding given by (33)

actually produces

with invertible and its unitarity can be verifiedby direct calculation as before in Lemma 1.

As in the LTI case, the Darlington synthesis is produced viaa right coprime factorization for :

and the algorithm is a version of the square root algorithmpresented before.

VII. D ARLINGTON EMBEDDING OF A CONTRACTIVE

TIME-VARYING TRANSFER FUNCTION

We start out with a strictly contractive time-varying operatorin state space form:

(34)

and we assume, in addition, that the realization is ue-stable,. We try to find a solution to the Darlington

embedding problem, i.e. a such that is inner and embeds:

Before formulating the realization theorem, we search for thesolution by following an inductive approach partly inspiredby [22]. The Darlington idea is to augment input and outputspaces so that the resulting operator becomes unitary. A unitaryoperator will have a unitary realization, so let us try toconstruct a realization for of the form:

(35)

and which has the additional property that there is a statetransformation which makes the transformed realization

(36)

unitary. The diagonal matrix together with theentries and must then satisfy at least the followingequations:

(37)

(The other entries will follow easily once these equationsare solved, see further.) Observe that and can beeliminated further when is invertible,to produce the famous discrete time Riccati equation:

(38)


We shall not handle this equation directly, but the theory thatwe shall give produces a positive definite solution to it andgives conditions for the required invertibility as well. For anextensive study of Riccati equations and means to solve themdirectly, see the book [23]).

Remarkably, the solution to (37), if it exists, has a closedform for the positive definite and the related and ,which is found by a physical analysis of the properties ofthe Darlington embedding. The secret lays in the study of the‘future operator’ connected to the embedded system which willhave to be unitary when the Darlington embedding exists, andwhich we introduce now. Thefuture operatorof a system isfound by decomposing inputs as and outputsas , according to the decomposition of spaces

. When presented with the input , thesystem produces a state given by—a diagonal operatorand the output , as given by the ‘map of the past’:

(39)

in which is the reachability map, and , the map ,a restriction of to signals ‘in the past’. These expressions canbe made concrete by decomposing the signals and operatorsin diagonals, see Appendix A for information and details. Thestate is in this case also a diagonal, which collects thestates obtained at each time point. Next, together with

produce the output according to the ‘relations for thefuture’:

(40)

in which

and

The resulting maps look as in Fig. 5. Now we want to augmentinputs and outputs, and allow for a bounded and boundedlyinvertible state transformation, so that the resulting systembecomes unitary. We call the additional inputs andand the state transformation and obtain the arrangement ofFig. 6 and the equations:

(41)

in which and are new operators (anobservability operator and three restriced transfer operatorsrespect.) to be determined and wherebyis the transformedstate.

Suppose now that a solution does exist indeed, and let uscompute the consequences. Unitarity requires at the least that

(42)

Fig. 5. The decomposition of the operatorS as a past and a future operatorconnected by the state.

Fig. 6. The augmented system has a unitary map linking the state and thefuture input to the future output.

Hence, in particular, with

(43)

Since we assumed thatis strictly contractive, the same holdstrue for , because it is a restriction of, and isan invertible operator. Since

we may conclude that will be an invertible operator aswell (it still has to be determined), and one must have


and

Finally, we obtain the expression for :

(44)

in terms of originally known data!Returning to the original embedding equations (37), we see

(with some calculations, see [21]) that

will be isometric, if given by (44) is invertible.In that case, the transformed will be ue-stabletogether with , and the realization can be augmented point-wise to a locally unitary realization with a ue-stable transitionoperator. will be invertible iff the system is uniformlyobservable, i.e. iff has a bounded inverse. If, on the otherhand, the system is not uniformly observable, then, althougha unitary embedding for the realization can be derived,it willusually not correspond to an inner embedding, because therewill be nontrivial defect spaces. In that case, the transformedtransition matrix will not be ue-stable.This defective case is considerably more difficult to analyzethan the straight case where the given systemis ue-stableand uniformly observable. Nonetheless, we have obtained aDarlington theorem for time-varying systems:

Theorem 8: If is a strictly contractive transfer operatorwith ue-stable locally finite realization which is uniformlyobservable, then possesses an inner embedding. If theue-stable realization for is given by , thenthe embedding has the unitary realization given by (36), inwhich the state transformation operator is obtained from

as , which is also apositive definite solution of the Riccati equation (38).

Proof—Sketch:Suppose that is strictly positive definite,then it can be factored as in which is a boundedand boundedly invertible operator. Then the state realizationas given in Eqs. (36) turns out be orthogonal, which is verifiedby writing Eqs. (43) out in terms of the realizations. Sinceisbounded and boundedly invertible, the orthogonal realizationgiven by (36) is ue-stable and is therefore, as stated in Lemma1, the realization of an inner transfer operator.

Square root algorithms to solve equations of the type (37)and the connected Riccati equation (38) have been knownfor a long time (for a literature survey, see [24]). Theirinterpretation as representative for factorization problems ofthe external or inner-outer type have been well documentedalso in the recent literature, see [22], [25]. In the context ofthe Darlington theory they get a particularly nice interpretation,since, as we have argued, the Darlington embedding problemreduces to a factorization problem. Conversely, to solve theDarlington embedding problem numerically, a square rootalgorithm would provide for the appropriate means. We derivethe algorithm in Appendix B, further information on itsnumerical properties can be found in the book [21].

VIII. D ISCUSSION

The focus of the present paper is the existence of theDarlington embedding, and we have seen that it is depended onthe absence of a nontrivial ‘defect’ space in the input or outputnullspace of the Hankel operator connected to the transferoperator of the system to be embedded. This is an unexpectedconclusion, if one considers that the original problem statementas given by Darlington and the early researchers on thetopic, Belevitch [6], Oono-Yasuura [26] and Youla [27], allconsidered it to be a problem in rational matrix factorization.The import of system theory for Darlington theory was mostclearly realized by Vongpanitlerd and Anderson in their book[5], which gives a systematic account of the solution of clas-sical circuit synthesis problems using state space formalism.However, only in the early 1970’s some insight in the roleplayed by external factorization and/or the Hankel operator(the two are closely related) in the problem started to dawn,and necessary and sufficient conditions for the existence of theDarlington or inner embedding appeared [2], [3], [19]. Later,a fairly complete picture for the time-varying case emergedalong the same lines of reasoning, but the time-varying theoryis considerably more delicate that the time invariant [21].

To conclude this paper, I want first to make the connectionof the abstract embedding theory as presented in this paperwith the classical cascade multiport synthesis problem, whichis based on the factorization of the Chain Scattering Matrix,and has considerable interest in its own right. I concludethe discussion with some remarks concerning the practicaluse of Darlington theory. Darlington’s original goal was toextend the cascade synthesis theory for a connecting networkby factoring out a transmission zero at a point in the-plane which was not located on the imaginary (for which anextraction theory already existed known as ‘extraction of aBrune section’). This gave rise to the so-called ‘Darlingtonsection’ [28], [29]. The cascade extraction theory can fairlyeasily be generalized to the LTI-multiport and even time-varying case. A recursive factorization theory as well as anequivalent cascade state space realization theory are availablefor each. In each instance, the most appealing way is toinvoque thechain scattering matrixrelated to the embedding

, and either to factor it directly into elementary sectionsin the -plane or the -plane, or to consider its state spacedescription in an algebraically reduced form. If the innerscattering operator is given by the input-output description

(45)

then the chain scattering description is defined by

(46)

Theta will be well defined if or, equivalently, isboundedly invertible, and it is then given by

(47)


Fig. 7. A multiport cascade realization is obtained through the factorizationof the Chain Scattering Matrix, whereby each section realizes one transmissionzero.

We see that, in case of an LTI system, the poles ofare theconjugates of the poles of . When is inner, then hasimportant properties, some of which are:

• for the signature matrix:

is para- -unitary, meaning that:

(in the LTV-case, a similar property will hold but with amore complex signature operator);

• inside the unit disc of the complex plane, will be-contractive, i.e.

(remember that denotes the Hermitian transform).This relation follows from the relation (dropping the

-dependence):

obtained from the connection betweenand .

From the para--unitarity of , we see that the zeros of(i.e. the poles of its inverse) are actually the conjugates of itspoles, and vice versa. Hence, shares zeros with . Thispoint may be exploited for algebraic benefit, by writing theembedding formula for as

(48)

A cascade synthesis of will now consist in extractingelementary sections from , since in practical problemsit is usually the transfer scattering matrix that isgiven. It turns out that such a multiport, recursive extractionof elementary zeros using-contractive, lossless sections isalways possible [30], [31] and results in a cascade synthesisas shown in Fig. 7.

Alternatively, one can produce a formula with , inwhich case an attractive embedding formula follows from theexpression for

(49)

We shall exploit this formula in Appendix B to obtain a squareroot algorithm for the embedding in state space formalism.

The elementary sections for to be used here have thefollowing general forms—see in this respect the general workon multiport inner and Blaschke factors by Potapov [32], wegive them for the sake of completeness:

• If the zero of is located in and (multiportDarlington section):

in which is a coefficient of modulus one and is aconstant matrix such that for some integer

(the Smith–McMillan degree of such a sectionis exactly ).

• If the zero of is located at (multiport Schursection):

• if the zero of is located at (multiport Brunesection):

(50)

in which is a strictly positive real number and isa nonsingular, so called neutral matrix: (thecolumns of span a so called isotropic space, i.e. a spaceof elements that are all -orthogonal to each other);

• if the zero of is located at a point outside theunit disc:

(51)

in which and now for someinteger ;

• and, finally, if the zero of is at infinity:

(52)

where and for some integersuch that .

The network-theoretical literature has been very much con-cerned with the properties of these various sections and certaincascades of them. If a strict, minimal degree synthesis isdesired, then at each elementary extraction, the degree shouldgo down by a number equal to the degree of the section thatis being extracted, for the sections given above. However,it is also possible to extract sections without the degreereduction, but respecting the passivity of the remainder circuit.In that case, correct approximating networks are obtained whenappropriate interpolation conditions are satisfied. There is anextensive literature on the properties of the approximationsobtained, it is of interest for circuit reduction in the VLSImodeling context [33].

I finish with a brief discussion of three important practicalproblems which have found elegant solutions thanks to thegeneralized Darlington theory, but whose discussion fallsoutside the context of this paper.


Broadband Matching:Darlington theory provides the keyingredient in broadband matching theory, as it succeeds in rep-resenting both the source impedance and the load impedance asa Darlington connecting network, and the broadband matchingproblem then is converted to the design of a connectingsection. See the literature for various approaches [34], [35],[36].

Minimal Algebraic Computations and Canonic Represen-tations of Systems:Darlington synthesis leads to numericalrealizations of operators withminimal algebraic complexity,see [21] for an extensive discussion of this topic. The keyingredient here is the fact that a unitary representation can berealized with a number of elementary rotations which is alge-braically minimal, i.e. equal to the number of free algebraicparameters. Although other methods of obtaining algebraicallyminimal realizations are possible, the one based on unitaryrealizations and elementary rotations is particularly attractivebecause it combines algebraic minimality with numericaloperations that have minimal error propagation. Algebraicminimality can be very important at least for two reasons.One is that a variation of a parameter in an algebraicallyminimal realization will not change the class of the realization(for example: a lossless circuit will stay lossless). This facthas been extensively exploited by the very elegant realizationtheory for Wave Digital Filters discovered and pioneered byA. Fettweiss [37]. The second reason is that algebraicallyminimal realizations can be build in such a way that they forma continuum for all possible realizations of a given degree andhence can be used for circuit optimalization by continuousvariation of parameters.

Inversion of Systems:Recently, numerical inversion theoryof systems of equations has been extended to handle a prettygeneral type of infinite systems which are described witha finite set of parameters. The class for which the bestsolution have been obtained consists of systems which aretime invariant for but are varying in between. Itappears that inner-outer factorization, the main mechanism forDarlington synthesis, again plays the key role in solving theinversion problem.

APPENDIX APAST-FUTURE DECOMPOSITION OF ACAUSAL OPERATOR

The concrete representation of operators such asand in (39) and (40) hinge on the choice of bases in theinput and output spaces of these operators. In our case, thehandiest choice for the input spaceand the output spaceare decompositions in diagonals as

in which is the -th diagonal of , properly positioned. Letus also represent the operatorby its diagonal decomposition:

This is a basis decomposition in which the constituting ele-ments are diagonal operators. Hence, viewed as series,and

can now be represented by series of diagonals:

The operator maps toand, taking notice of the rule

, it will be represented by

......

......

Likewise, maps to , and will also be represented,in the diagonal bases chosen, by an operatorconsisting ofdiagonals. Using the rule , it will be givenby:

......

.... . .

The connecting operators and generate the statethrough the action of the Hankel operator, which, as expected,factors out, since

, as

......

... ...

...

Hence, ‘concrete’ representations forand in the diagonalalgebra are given by

...

It should be clear, then, that is indeed given by the diagonal:


APPENDIX BSQUARE ROOT ALGORITHM FOR

THE DARLINGTON EMBEDDING

The square root algorithm for the Darlington embedding asderived by [22] can be interpreted as a state space expressionof the Darlington embedding equation (49), which itself is aspecial instance of an inner-outer factorization equation. If theoriginal scattering operator has a realization ,then the outer spectral factor of will sharethe with , and will have and determined bythe condition that there should exist a bounded and boundedlyinvertible state transformation operatorsuch that

is co-isometric (this is the dual case of the factorization caseconsidered in the main text of this paper, for good reason,see the brief discussion at the end of this section). Expressingthe embedding (49) in terms of these diagonal state spaceoperators now takes the form

(53)

in which is actually a -unitary realization matrix for theanticausal transfer function . These equations form thebasis of the square-root algorithm to compute the embedding.They state that, given the original state space realization forand the value of at a given point in the recursion, onefinds and by annihilating the middle rows of

using a -unitary transformation matrix, with

to produce

The existence of the Darlington embedding actually insures theexistence of the transformation matrix (if it had been unitary,it would always exist, but -unitarity is a different case). Forfurther details on implementation and numerical properties,see [21].

The square-root algorithm shown here computes an outercompanion and a causal embedding to . This cor-responds to the traditional Darlington embedding method

whereby a transfer scattering operator is first convertedto an input scattering operator , which is then furtherrealized using cascaded sections with the correct transmissionzeros. The dual factorization method given in Section VIIIis in a sense more interesting: it proceeds directly with thefactorization of the given , and produces both anddoing so.

ACKNOWLEDGMENT

The author wishes to thank D. Pik of the Department ofMathematics and Computer Sciences of the the Free Uni-versity of Amsterdam, Amsterdam, The Netherlands, for hissuggestions, comments and proofreading of the manuscript.

REFERENCES

[1] S. Darlington, “Synthesis of reactance 4-poles which produce a pre-scribed insertion loss characteristics,”J. Math. Phys., vol. 18, pp.257–355, 1939.

[2] P. Dewilde, “Roomy scattering matrix synthesis,” Tech. Rep., Dept. ofMathematics, Univ. of California, Berkeley, 1971.

[3] D. Arov, “On the Darlington method in the theory of dissipativesystems,”Dokl. Akad. Nauk USSR,vol. 201, no. 3, pp. 559–562, 1971.

[4] H. Helson, Lectures on Invariant Subspaces. New York: Academic,1964.

[5] B. Anderson and S. Vongpanitlerd,Network Analysis and Synthesis.Englewood Clifffs, NJ: Prentice Hall, 1973.

[6] V. Belevitch, “Elementary applications of the scattering formalism tonetwork design,”IRE Trans. Circuit Theory, vol. CT-3, pp. 97–104,June 1956.

[7] , Classical Network Theory. San Francisco, CA: Holden Day,1968.

[8] J. Neirynck and P. V. Bastelaer, “La synthese des filtres par la factor-ization de la matrice de transfert,”Rev. MBLE, vol. X, no. 1, pp. 5–32,1967.

[9] D. Youla, “Cascade synthesis of passiven-ports,” Polytechnic Inst. ofBrooklyn, NY, Tech. Rep., June 1965.

[10] V. B. P. Dewilde and R. Newcomb, “On the problem of degree reductionof a scattering matrix by factorization,”J. Franklin Inst., vol. 291, no.5, pp. 387–401, May 1971.

[11] P. Dewilde, “Cascade scattering matrix synthesis,” Ph.D. dissertation,Stanford University, CA, June 1970.

[12] W. Rudin, Real and Complex Analysis. New York: McGraw-Hill,1966.

[13] K. Hoffman, Banach Spaces of Analytic Functions. Englewood Cliffs,NJ: Prentice-Hall, 1962.

[14] L. C. D. C. Youla and H. Carlin, “Bounded real scattering matrices andthe foundation of linear passive network theory,”IRE Trans., vol. CT-4,no. 1, pp. 102–124, Mar. 1959; corrections,Ibid., p. 317, Sept. 1959.

[15] D. Youla and P. Tissi, “n-port synthesis via reactance extraction—PartI,” IEEE Int. Conf. Rec., 1966, vol. 14, no. 7, pp. 183–205.

[16] P. M. P. and N. Wiener, “The prediction theory of multivariablestochastic processes,”Acta Math., vol. 98, pp. 111–150, 1957; andActaMath., vol. 99, pp. 93–137, 1958.

[17] G. Szego, “Beitrage zur Theorie der Toeplitzen Formen (Ersten Mit-teilung),” Math. Zeit, vol. 6, 1920.

[18] R. Saeks, “The factorization problem—A survey,”Proc. IEEE,vol. 64,pp. 90–95, Jan. 1976.

[19] P. Dewilde, “Input-output description of roomy systems,”SIAM J.Contr. Opt., vol. 14, pp. 712–736, July 1976.

[20] R. Kalman, P. Falb, and M. Arbib, “Topics in mathematical systemtheory,” Int. Series in Pure and Applied Math.New York: McGraw-Hill, 1970.

[21] P. Dewilde and A.-J. van der Veen,Time-Varying Systems and Compu-tations. Amsterdam, The Netherlands, Kluwer, 1998.

[22] A. van der Veen, “Time-varying system theory and computational mod-eling: Realization, approximation, and factorization,” Ph.D. dissertation,Delft Univ. of Technology, Delft, The Netherlands, June 1993.

[23] P. Lancaster and L. Rodman,Algebraic Riccati Equations. Oxford,U.K.: Oxford Univ. Press, 1995.

[24] M. Morf and T. Kailath, “Square-root algorithms for least-squaresestimation,”IEEE Trans. Automat. Contr., vol. 20, no. 4, pp. 487–497,1975.


[25] A. Halanay and V. Ionescu, “Time-Varying discrete linear systems,”Operator Theory: Advances and Applications. Germany: Birkhauser,vol. 68, 1994.

[26] Y. Oono and K. Yasuura, “Synthesis of finite passive 2n-terminalnetworks with prescribed scattering matrices,”Mem. Fac. Eng. KyushuUniv., vol. 14, no. 2, pp. 125–177, May 1954.

[27] D. Youla, “A new theory of cascade synthesis,”IRE Trans. CircuitTheory, vol. 8, pp. 244–260, Sept. 1961; also, “Correction,”IRE Trans.Circuit Theory, vol. CT-13, pp. 90–91, Mar. 1966.

[28] A. Belevitch, Classical Network Theory. San Francisco, CA: HoldenDay, 1968.

[29] R. Newcomb,Linear Multiport Synthesis New York: McGraw Hill,1966.

[30] P. Dewilde, “Cascade scattering matrix synthesis,” Ph.D. dissertation,Stanford University, Stanford, CA, 1970.

[31] P. Dewilde and H. Dym, “Lossless inverse scattering, digital filters, andestimation theory,”IEEE Trans. Inform. Theory, vol. 30, pp. 644–662,July 1984.

[32] V. Potapov, “The multiplicative structure ofJ-contractive matrix func-tions,” Amer. Math. Soc. Transl. Ser. 2, vol. 15, pp. 131–243, 1960.

[33] P. Dewilde and Z.-Q. Ning,Models for Large Integrated Circuits,Boston: Kluwer, 1990.

[34] J. W. Helton, “Broadbanding: Gain equalization directly from data,”IEEE Trans. Circuits Syst., vol. CAS-28, Dec. 1981.

[35] H. Carlin and J.-C. C. Wu, “Amplitude selectivity versus constantdelay in minimum-phase lossless filters,”IEEE Trans. Circuits Syst.,vol. CAS-23, pp. 447–455, July 1976.

[36] Y. Chan and E. Kuh, “A general matching theory and its applicationsto tunnel diode amplifiers,”IEEE Trans. Circuit Theory, vol. CT-13, no.1, pp. 6–18, 1966.

[37] A. Fettweiss, “Digital filter structures related to classical filter net-works,” A.E.U., vol. 25, pp. 79–89, 1971.

Patrick Dewilde (S’69–M’73–SM’81–F’82) re-ceived the degree of electrical engineering fromthe University of Leuven in 1966, the License inMathematics from the Belgian Central ExaminationCommission in 1968 and the Ph.D. degree inelectrical engineering from Stanford Universityin 1970.

He has held various research and teachingpositions at the University of California in Berkeley,the University of Lagos in Nigeria and theUniversity of Leuven, Leuven, Belgium. In 1977 he

became full professor of Electrical Engineering at the Technical Universityof Delft (the Netherlands). In 1981 he was named Fellow of the IEEE forhis work on Scattering Theory. His research interests include the design ofintegrated circuits (VLSI) especially in the area of signal processing, largescale computational problems, theoretical topics in system theory and signalprocessing, and information management. He has been a project leader ofmajor European projects in Microelectronics. The NELSIS design system,which pioneered a unique design information management methodology,was developed under his direction. In 1993, he became the ScientificDirector of DIMES, the Delft Institute of Microelectronics and SubmicronTechnology, which employs more than 300 researchers active in advancedMicroelectronics. He is the author of a large number of scientific publications,a book entitledLarge Scale Modeling of Integrated Circuits(Kluwer, 1988)and one entitledTime Varying Systems and Computations(Kluwer, 1998).

Dr. Dewilde was elected as a regular member of the Dutch Royal Academyof Science in 1993.

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