SIAM REVIEWVol. 29, No. 3, September 1987

(C) 1987 Society for Industrial and Applied Mathematics001

INVERSE SCATTERING FOR DISCRETE TRANSMISSION-LINE MODELS*

A. M. BRUCKSTEIN’ AND T. KAILATHf

Abstract. This paper presents several methods for the identification of the impedance and reflectioncoefficient profile of a nonuniform, discrete transmission-line from its response to a given forcing function.This problem is the prototype for a wealth ofone-dimensional inverse scattering problems arising in variousfields. A unified, straightforward approach is presented, providing all known inversion procedures and somenovel ones. The derivations readily follow from causality of signal propagation on the transmission-line. Itis shown that a direct exploitation of the signal propagation model leads to recursive, computationallyefficient and reliable inversion algorithms. In particular, the relationships between layer-peeling (Schurtype, difference equation) and layer-adjoining (Levinson type, integral equation) algorithms are clearlydisplayed.

Key words, fast algorithms, Schur recursions, layer-peeling methods, Gelfand-Levitan equations,lossy media

AMS(MOS) subject classifications. 15-02, 15A06, 15A90, 45A05, 45E10, 81F99, 93E11

A mathematical theory is not to be considered complete until you have made it so clear that youcan explain it to the first man whom you meet on the street.

David Hilbert (quoting an old French Mathematician),in Mathematical Problems, AMS Bulletin 8, 1901-02.

1. Introduction. Motivated by numerous applications, physicists, electrical en-gineers, geophysicists and mathematicians have studied a wide range of inversescattering problems. It turns out that inverse spectral problems for Schr6dingeroperators and vibrating strings, algorithms for transmission-line synthesis, determi-nation ofthe vocal tract area function in speech research, identification ofthe acousticimpedance or conductivity profiles in layered-earth models, the design of digital filtersin cascade form and the derivation of fast lattice-form linear least-squares predictionalgorithms, can all be interpreted as sharing a common mathematical foundation.They all require a procedure that identifies from the so-called scattering data (whichare "excitation-response" pairs of time-functions measured at the boundary) thedepth-dependent parameters of a one-dimensional, highly structured signal prop-agation model, whose prototype is, for electrical engineers, a lossless nonuniformtransmission-line.

A famous paper of Gelfand and Levitan, published in 1955, reduced the solutionof the (Schr6dinger) inverse scattering problem to the solution of a parametrized setoflinear integral equations. Subsequent work ofMarchenko, Krein and others, furtherstrengthened the belief that inverse scattering is equivalent to solving linear integralor, in the discrete case, linear matrix equations (see e.g. [1 ]-[7]).

However, in some quite early work, mostly in geophysics, a different approachemerged. This approach, more directly concerned with a local analysis of acoustic

* Received by the editors September 1, 1984; accepted for publication (in revised form) November25, 1985. This work was supported in part by the U.S. Army Research Office under contract DAAG29-79-C-0215 and by the Air Force Office of Scientific Research under contract AF49-620-79-C-0058.

Information Systems Laboratory, Stanford University, Stanford, California 94305.This author gratefully acknowledges the support provided by an Erna and Jakob Michael Visiting

Chair in Theoretical Mathematics at the Weizmann Institute of Science, Rehovot, Israel, during SpringQuarter, 1984.

Present address, Faculty of Electrical Engineering, TECHNION, Haifa, Israel.

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360 /. M. BRUCKSTEIN AND T. KAILATH

wave propagation, provided recursive methods, called dynamic deconvolution, ordownward continuation, algorithms (see e.g. [8]-[ 13]), for the identification oflayered-earth models which are the geophysical counterpart of nonuniform transmission lines.Though by now fairly well known in some ofthe geophysics and engineering literature,such direct methods were not always regarded as viable, since the effect of noise inthe scattering data was intuitively expected to be disastrous. Most researchers thereforereturned to integral or matrix-equation-based approaches with the hope that suchmethods would provide some averaging of the noise and thus have better numericalbehavior. Though this expectation was never justified, since both classes of methodsproduce the same results assuming infinite precision computations and also havesimilar numerical stability properties (see [14]-[15]), almost all effort in "inversescattering" research has been focused on exploration of linear equation formulations,providing alternative derivations, interpretations, etc. (see e.g. [16]-[21]). Only re-cently was the balance somewhat restored as more work has been devoted to a carefulanalysis of direct, differential methods (see e.g. [22]-[27]). In so doing, it has beennoticed that, though both direct and indirect methods are numerically stable, the verystructure, and the simplicity of derivation, of the direct methods indicates intelligent"error control" techniques that allow more effective medium identification in thepresence of noise in the scattering data (see [28], [29]).

The main goal of this paper is to use a discretized form of the nonuniformtransmission-line equations as the vehicle for simple, physically based derivations andcomparisons of the direct (downward continuation type) and indirect (or linearequations based) methods. We briefly indicate some of the new insights and newresults of this paper here.

As a first example we mention that in our approach (see 4) it becomes clear thatthe linear equations of the inverse scattering problem are direct consequences of(only) the causality and symmetry properties of the transmission-line model. In factwe obtain an apparently new, general set of linear equations for solving the inverseproblem. This set of equations is based on arbitrary scattering data (i.e. the responseto an arbitrary excitation function) and we show that the discrete analogues of theclassical equations ofGelfand-Levitan, Marchenko and Krein are obtained by makingspecial choices of excitation-response pairs.

More centrally, our theme is that formulation via the discrete transmission-linemodel shows (see 3) that a simple layer-peeling inverse scattering procedure arisesmore directly than do the linear equations; this procedure only invokes causality ofsignal propagation and a simple rule for recursively computing the discrete waveformsat increasing depths in the scattering medium. For lossless media, the layer-peelingmethod is similar to the dynamical deconvolution (downward continuation) methoddiscovered by geophysicists (see [9], [10], [29]). Our simple approach highlights thefact that the layer-peeling method also applies not only to lossless scattering mediabut to several more general models, with nonsymmetric and even nonlinear interac-tions arising in applications such as the minimal partial realization problem of systemtheory [30], the development of efficient decoding algorithms for error correctingcodes [31 ], and in solving certain nonlinear convolution-type integral equations arisingin biological modeling problems (see [32]).

Moreover it is fascinating to see (3) that, in the lossless case, one version of thedirect inversion algorithm, when expressed analytically via the so-called z-transform(or generating function) notation for discrete signal analysis, turns out to be the sameas a recursive procedure devised by I. Schur in 1917 (see [33], [34]), for testing if apower series in z- is bounded outside the unit disc in the complex plane. Thisconnection of the Schur algorithm to inverse scattering was perhaps first explicitly

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INVERSE SCATTERING FOR DISCRETE MODELS 361

made by P. Dewilde (ca. 1977), who had encountered the Schur algorithm in studiesof cascade synthesis procedures for stable digital filters (see [35]-[38] and also [39]).As mentioned above, geophysicists had unknowingly rederived the Schur algorithmin their development ofthe downward continuation (dynamic deconvolution) method[9]-[13].

The direct inversion methods have a complexity of O(]V2), i.e., proportional toN computations are needed to identify the medium up to depth N. The solution vialinear equations, when those are solved in a straightforward way by, say, Gaussianelimination, would require O(N3) computations. The computational efficiency of thedirect methods is due to the fact that these fully exploit the assumed structure of thescattering medium. Linear-equations-based methods can be made to yield computa-tionally efficient algorithms if the special structure of their coefficient matrices iscleverly taken into account; this was done, for example, in [17], [18], [40] for theimportant special equations of Gelfand-Levitan, Marchenko and Krein. We shall seethat the understanding of how the medium structure leads to the general linearequations for inverse scattering also provides the appropriate (general) fast algorithmsincluding as particular cases the algorithms derived in the literature for the Krein orMarchenko and Gelfand-Levitan equations.

In brief, we show in this paper that the transmission-line model clearly, bringsout the reasons for various differences between the direct (layer-peeling) and linearequations (which, as we shall see, may be called layer-adjoining) methods and in factreadily suggests generalizations of various existing results. Furthermore, we can easilyrecognize the possibility of implementing layer-peeling methods with parallel com-putation, requiting O(N) time with an array of N processors, or with doublingtechniques that require O(N log N) time on a single processor.

We should emphasize that the transmission-line model on which the argumentsare based is quite general: by appropriate transformations, a variety of other inverseproblems can be recast in this form, e.g., the discrete versions of the vibrating stringproblem [41], the inverse Schr6dinger problem [5]-[7], and the acoustic vocal tractproblem [42]. The results for the corresponding continuous problems can be obtainedby a limiting procedure, or by following similar, though somewhat less elementary,"propagation of singularities" arguments [20], [25], [27].

However we should note that the problems solvable by the approaches discussedin this paper are inherently one-dimensional. Inverse scattering in more than onespatial dimension is as yet an unsolved problem in all but a few particular cases withspecial symmetries that effectively reduce the dimensionality. It is however a problemof much interest in geophysics and materials science, concerned with nondestructivetesting and in medical imaging with ultrasound waves, and therefore a lot of researchis dedicated to it. In stratified media with general geometries, even the direct scatteringproblem, i.e. the analysis ofwave propagation, is in many cases too difficult to tackle.Solving these problems requires in most cases complicated ray-tracing methods andalso accounting for the possibility of having guided waves along internal layers inwhich propagation speed is higher. As far as we know, important problems such aswhat is the scattering data that has to be gathered in order to make a high-dimensionalinversion problem well-posed are not completely solved today. It is, however, possiblethat a layer-peeling idea combined with some ray-tracing method would lead to asuccessful inversion algorithm for certain particular cases of the multi-dimensionalinverse scattering problem. Some work in this direction is described in [43].

This paper is organized as follows. A preliminary discussion of basic facts onsignal propagation on lossless, discrete transmission lines is given in 2. Several directalgorithms for inverse scattering and their relationships to the work of Schur are then

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362 A.M. BRUCKSTEIN AND T. KAILATH

derived in 3. In 4, we use the transmission-line picture to develop linear equationapproaches to the inverse problem. Several remarks on related results in factorizationand estimation theory conclude our discussion of inverse scattering methods.

2. Discrete lossless transmission lines. Let us review some basic facts aboutnonuniform and lossless transmission-line models. The propagation of signals on alossless transmission line can be described by a set of symmetrized, so-called "teleg-rapher’s equations" [25], [44], [45],

0v(x, t)

(2.1) -x -Zxi(x, t),

0i(x, t) -Z-;

0v(x’t)

where v(x, t) is the voltage at point x on the line, at time t, i(x, t) is the current atpoint x at time t, and Zx is the (local) impedance at the point x. By assumption Zx isstrictly positive and finite, and, without loss of generality, we shall set Zo 1.Normalizing the voltage and current variables to

(2.2) V(x, t) v(x, t)(Zx)-/2 and I(x, t)= i(x, t)(Zx)/2

at each point on the line, we get for the {V(x, t), I(x, t)} pair of variables the equation

(2.3) -x I(x, t) J -OlOt kx J I(x, t) Jwhere kx is the so-called local "reflectivity" parameter given by

ld(2.4) k=xIn Zz.

It is useful to derive a modified set of equations relating the so-called left- and fight-propagating wave variables defined via the transformations

(2.5) W(x, t)=V(x, t) + I(x, t)

and Wz(x, t)V(x, t) I(x, t)

2 2

For these quantities the evolution equations, obtained by combining (2.3) and (2.5),are

(2.6) 0-- W.(x, t) -kx O/Ot WL(X, t)

This set of equations is interpreted as describing the propagation of waves to the leftand right, the intensity of their local interaction being parametrized by the reflectivitykx. Indeed, suppose that over a portion of the line, say [Xa, Xb], the local impedanceis constant. Then, by (2.4), the local reflectivity is zero and the above equationsprovide

(2.7)

Xa<X<Xb

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INVERSE SCATTERING FOR DISCRETE MODELS 363

with general solutions of the form: WR(X, t)= W(X-t) and Wz(x, t)= Wz(x + t),describing noninteracting (left and fight) propagating waves. Therefore, any portionof a transmission line with constant local impedance acts on the wave variables as apure time-delay and time-advance operator. When k is nonzero, the propagatingwaves do interact, part of each wave being "backscattered" and added to the onepropagating in the opposite direction.

2.1. Discrete wave propagation equations. With this background, let us considera discrete transmission line with sections of constant impedance over intervals oflength 1/2 (see Fig. 1). We shall define

(2.8) Z= Z,,_, x [(n 1)/2, n/2)

as the impedance of the nth section, and recall our assumption that Z0 1. Becauseof the constancy of the local impedance over a section, we can write (see (2.7))

(2.9) [ W(x, t) t)WL(X, t) ]x=[(n_l)/21+

where A is a "time-delay" operator defined by

(2.10a) Af(t)=f(t--1/2)and correspondingly/x- is a "time-advance" operator

(2. lOb) A-f(t)=f(t+1/2).Note that the time-delay operator is physically meaningful for fight-going waves,while the time-advance operator is so for left-going waves.

72

7oZ5

1,,

0.5 1.5 2 2.5

FIr. 1. Typical impedance profilefor a discrete transmission line.

Wave interactions, reflections and transmissions, will occur at the points wherethere are changes in the local impedance. To compute these interactions, we returnto the original unnormalized voltage and current variables {v, ], which by (2. l) haveto be continuous at all points of the line, i.e., we have

(2.11) v(x,t) t)i(x,t)lx=t,/2)+ [v(x,L i(x, t)

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364 A. M. BRUCKSTEIN AND T. KAILATH

Using (2.2) and (2.5), we see that

(2.12) [W(x,t)] l[Z;1/2

WL(X, t -Zlx/2J L i(x, t) E(Zx)L i(x, t)

say. Therefore we obtain, combining (2.11) and (2.12),

WR *[Z "_,-l[Z 1) "-’On(2.13) W. x=(n/2)+ ’’ n! n- WL x=(n/2)- WL x==(n/2)-

Note that we wrote Wg instead of Wg(x, t), and so on, the independent variablesbeing clearly understood from the context. Now referring back to (2.9), we can writea complete discrete wave-evolution equation,

where

"--1 Zn’Zn-I(2.15) On--..(Zn).. (Zn_l)-m(ZnZn_l)_l/2 _Zn_Zn_l

Define the nth local "reflection coefficient" as

Zn.- Zn_

Zn "t- Zn-I

(2.16) k, (Zn Zn-l)/(Zn t- Zn_l).

Since Z, > 0 for all n, it follows that

(2.17) Ik.l_-< 1.

The introduction of the reflection coefficient sequence {k} allows rewriting of thematrix O in the form

(2.18)

and it is easy to check that such matrices are J-orthogonal, i.e.

where superscript T denotes transpose. This relationship, expressing energy conser-vation, can also be derived in a more physical way, as will be seen below.

For simplicity of notation we shall define

(2.20) W(n, t) W x;(/+

so that the relation (2.14) will be written

WL(n, t) O. WL(n-- 1, t)

A pictorial signal-flow graph "representation" of the wave propagation implied by(2.21) is shown in Fig. 2(a). (See Appendix A for a review of signal-flow graphrepresentations.) We use the quotation marks to emphasize that, due to the presence

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INVERSE SCATTERING FOR DISCRETE MODELS 365

t)n Kn)

WR(n,’)

(a)(n+l (Kn/ I)

WR (n+/,.)

WLln*l,-)

WR(n-I,.).----

WLjn’I")

_ W(n,.)

n(Kn) n/l(Kn/l(b)

WR(n+l,-)

WL(n +1,’)

FIG. 2. The wave propagation model (a) transmission diagram (b) causal scattering picture.

of the advance operator A-, this scheme does not correspond to a causal/physicalsignal flow.

A physically meaningful representation can be obtained by reversing the directionof signal flow for the wave W, with the effect that the "time-advance" operator in(2.2 l) becomes a causal, "time-delay" operator A acting on the left propagating wave.Of course, the wave interaction matrix 0 will have to be modified so as to preservethe algebraic relationships between W and W at each point of the line. The propermodification can be obtained by standard flow-graph manipulation rules or by simplealgebraic rearrangement (see e.g. [46] or Appendix A). The result is that (2.21) can berewritten as a causal wave scattering equation

(2.22) [Wg(n,t)W(n_ 1, t)]=[ 0m]n[ O1][Wg(n-l’t)]W(n, t)where the scattering or interaction-gain matrix ; is given by

(2.23) Zn (1 --/2n)1/2k,, (1 -k2n)1/2

A sequence of such causal wave scattering relationships is graphically represented bythe now physically meaningful discrete transmission-line model shown in Fig. 2(b).

The matrix is called the scattering matrix of the nth section of theline since it describes the physical interaction between the incident wavesWg(n, t), W(n + 1, t)l and the reflected waves Wg(n + 1, t), W(n, t)} at that section.

It is easy to check that n is unitary, i.e.,

(2.24) ,Tn ,n-" I"- ,n,Tn

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366 A.M. BRUCKSTEIN AND T. KAILATH

which corresponds to the physical property ofenergy conservation or losslessness, i.e.,(llf(’)ll denoting 1 norm)

W(n- 1,. )112 + W(n,. )112 Wa(n,. )112 + Wz(n- 1,. )112,or incident energy equals reflected energy. Note also that k, can indeed be interpretedas a local reflection coefficient because from (2.14) we have

(2.26) knW(n 1, t)

WR(n- l,t--1)if Wz(n, r)= 0 for r < t.

An immediate consequence of the energy conservation relation (2.25) is that

(2.27) WR(n,. )11 =- WL(n,. )112= W(n- 1,. )11 =- WL(n-- 1,. )112,which explains why the matrix O obeys the previously noted identity

The matrices I1 are often called chain scattering or transfer matrices. The transferrepresentation of a wave-scattering process, although not always corresponding to aphysical signal flow, is very useful because the natural cascade composition rule fortransfer matrices is simply the usual matrix multiplication. For scattering matrices12:nl, the cascade composition requires a more involved computation known asRedheffer’s star-product rule [46].

Among the more significant consequences of energy conservation relations areresults on fast algorithms for the triangular factorization of a class of matrices thatcan be regarded as natural generalizations of the much-studied Toeplitz matrices, see[47]. We shall not pursue these interesting connections here (see also [48]), but shallturn instead to evolution equations for the voltage and current variables.

2.2. Voltage-current evolution equations. Though the physical evolution of thewave variables {W, Wz.} has attracted most attention so far, it should be noted thatthere are similar evolution equations for the voltage and current variables {v, I, whichare equally useful in inverse scattering (as we shall see in 3.1).

Since the {v, i} variables are continuous along the whole line, it suffices toconsider their propagation within a single section. The propagation equations can beobtained by combining (2.9) and (2.12), and we obtain

.% (Zn-lx=(n=l)/2

(2.28)=n/2 -l (Z,I

where E(Z) is from (2.12)

[Z-1/(2.29a) (z)=-Lz_l/2This matrix has the scattering representation

(2.29b) Ez(Z) /2 Z-1/2Z-L

Z1/2__21/2

and therefore the voltage-current propagation can be described pictorially as inFig. 3, either as a noncausal signal transfer flow-graph (3(a)) or as the corresponding,more physical, scattering diagram (3(b)).

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INVERSE SCATTERING FOR DISCRETE MODELS 367

v(n, .)

i(n,.)

(Zn)(a)

,-(Zn)

v(n+l.)

i(n+l, ")

v(n,’)

i(n,.)

(b)’(Zn)

v(n+l.)

i(n + I,

FIG. 3. The current-voltage propagation model (a) noncausal signal transfer (b) scattering diagram.

For completeness, we also note here that from (2.16) we can express the localimpedance of the discrete line in terms of the local reflection coefficient sequence as

(2.30) Z," l+ki+k.z._

-kn -ki

where we again recall our assumption that Zo 1.Gain-delay characterizations. In summary, note that the various formulations

given above provide a so-called gain-delay characterization of the transmission-line.The line is seen to be a layered, or cascade, system composed of static "gain" sectionsseparated by delays. Each gain section is specified completely by a wave interactionmatrix that depends on a local parameter, which can either be the reflection coefficientor the local impedance. We shall see that this characterization immediately suggestsa direct, time-domain recursive procedure for solving the inverse problem for losslesslines. Moreover, the same inversion method will be applicable to more general wavepropagation models having similar gain-delay structure.

3. Layer-peeling algorithms for inverse scattering. The inverse scattering prob-lem for transmission-lines is to determine the line given the input and responsefunctions, W(0, t) and W(0, t), under the assumption that the line was initiallyquiescent. This assumption means that the response recorded is entirely due to thecausal propagation of the probing signal on the line.

There are many ways to attack this problem. The most widely known are methodsusing special choices of input sequences, based on which the inversion problem isshown to be equivalent to the solution of sets of linear equations, discrete analoguesof certain well-known integral equations of continuous inverse scattering, associated

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368 A.M. BRUCKSTEIN AND T. KAILATH

with the names of Gelfand and Levitan [1 ], [4], Krein [3], Marchenko [2], Gopinathand Sondhi [45] (see also [17], [20], [25], [27]).

The linear matrix equations required to determine the line up to length N are ofincreasing size, up to Nx N, so that their solution would generally require O(N3)elementary operations. However, the special properties of the transmission-line im-pose further structure on the linear equations, e.g., making their coefficient matricesToeplitz (Krein, Gopinath-Sondhi), Toeplitz plus Hankel (Gelfand-Levitan) orHankel (Marchenko, Burridge). These special structures can be exploited to reducethe computational burden by an order of magnitude to O(N2), leading to so-calledfast inverse scattering algorithms (see e.g. [17], [18], [40]). It turns out, however, thatthis is a rather indirect route to fast inverse scattering algorithms. The special propertiesof transmission-lines, analyzed in detail in 2, directly suggest a family of fast inversescattering algorithms, which were apparently first discovered by geophysicists andcalled "dynamic deconvolution" algorithms (see [8]-[ 10]).

In this section, we develop several direct inversion algorithms in a form thatimmediately suggests extensions to more general inverse scattering problems. First weshall show that the gain-delay transmission-line structure, described in 2, can be usedto obtain a general direct inversion algorithm. This algorithm uses the scattering dataas given, and there is no need for a "deconvolution" step to put the data in any specialform. The intimate relationship of the direct inversion procedure to certain results ofSchur [33] is then discussed in some detail. Finally we show how these ideas canreadily be extended to more general inverse problems for lossy and even a class ofnonlinear scattering media.

3.1. "Layer-peeling" algorithms for lossless lines. We start with a discretetransmission-line, i.e., one with a piecewise constant impedance profile. The line isassumed to be quiescent at t 0, i.e., v(x, 0) 0 i(x, 0), when we apply a currentinput at x 0 and measure the voltage response at that point, say v(0, t). The problemis to recover the parameters of the transmission-line, e.g., the local reflection coeffi-cients or the local impedances at all depths x in this scattering medium. Of course,we can easily go from one parametrization to the other by using the relations (2.16)and (2.30); however it turns out to be just as easy, and more advantageous in practice,to determine each of them directly.

Recovery ofthe local reflection coefficients. The first step is to transform the datainto the wave variables at x 0 (cf. (2.5)). We get the equivalent scattering data

(3.1) WR(0, t)= V(0, t) + I(O,t)]/2= Iv(0, t) + i(O,t)]/2,Wz(O, t)= [V(O, t) I(0, t)]/2 [v(O, t) + i(O, t)]/2,

since, by assumption, Zo 1.Now from (2.22) (see Fig. 2(b)) and because the line is assumed to be at rest at

t 0, there will be no left propagating wave response from the line for at least onetime unit, i.e.,

(3.2) W(O, t) O, 0 _-< < 1.

On the other hand, we have

(3.3) W(O,t)=k, W(O,t- 1), -_<t<2,

since, by causality and the delay structure of the line, it will take at least two timeunits for any left-going incident wave to appear as an input to the first section.

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369

anyt[2, 3)

INVERSE SCATTERING FOR DISCRETE MODELS

Therefore we can identify the first reflection coefficient k as

W(O, t(3.4) k W(0, t) anyt[l,2)

By the same argument we observe that, since it will take at least three time units fora left-going wave to be incident on the second section,

Wz(1, t)(3.5) kz-- WR(i t) anyt[2,3)"

(Recall our convention, (2.20), that Wz(1, t)= Wz(x, t)l=/z+.) The identification ofk from the scattering data is thus immediate; however we do not have at handW.(1, t) and W(1, t), which are needed to determine k. But, by using the chain-scattering (or transfer) matrix 0, which is completely specified once we now knowk, we can compute

(3.6) W(1 t) (1 k)-1/ (0, t-- 1- W(O,t+

and now k2 can be computed.In effect, once we have determined the reflection coefficient of the first section,

we use its associated O matrix to "peel off" the effect of the first layer of themedium and put ourselves in the same position as before with a line parametfized by{k2, k3, 1. In other words, using kl and the ofiNnal scattering data we are able toproduce the "aificial scattering data" for the line extending over [2, ).

We can continue in this way, successively determining a reflection coefficientand peeling off the associated layer, to determine as much as we wish of the (infinite)extent of the transmission-line. For reasons to be explained in 3.2, this procedurewill also be called a Schur type algorithm.

Recovery of the impedance function. Once the {ki} have been deteined, the{Zi} can be found by appeal to (2.30). However, direct recoveff of the impedanceprofile by a different layer-peeling algorithm is also possible. Causality argumentsapplied to the signal propagation in the gain-delay structure shown in Fig. 3(b), similarto those used in the previous section, show that we can deteine Z as

v(1,t)(3.7) Zl-

Of course v(0, t) i(0, t) for < 1, since Zo 1, and using (2.28) will yield the signalsat depth (this initialization is in fact a trivial layer-peeling step). After determiningZl we can again use (2.28) to "peel off" the second medium layer and compute thescattering data {v(2, t), i(2, t)}. We have

fro kio ohti

(3.9) Z2 i(2, t)

and so on.

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370 A. M. BRUCKSTEIN AND T. KAILATH

This procedure is the discrete analogue of methods recently presented for thecontinuous case by Santosa and Schwetlick [23] and by Sondhi and Resnick [24] (seealso [], [251, [271).

Remarks on numerical aspects. A count ofthe numerical operations in the abovealgorithms shows that O(N2) computations are needed for the identification processto reach a depth N on the line. The numerical properties of such recursive procedureshave also been recently analyzed and the algorithms were found to be stable in thebackwards sense of numerical analysis ([14], [15]).

In 14], 15] and in all our analysis so far, it was assumed that the given scatteringdata are exact. The effects of noisy data on the recovery of transmission-line para-meters are studied in [28] (see also [29], [49]), and it is shown that the mediumparameters induce a certain inherent conditioning of the inverse problem thatdetermines the rate of error growth in the identification process. The error accumu-lation with depth is determined by a running product of terms of the form(1 + [k[)/(1 ki[), which shows that if the reflection coefficients are never close to

in absolute value, the medium identification error due to noisy data will not growexplosively (see [28]). In many practical examples we may have prior information onthe range or distribution of reflection coefficients. In particular, as may often happen,if the reflection coefficients are small most of the time, the medium can be accuratelyrecovered to a substantial depth before the error due to noisy data reaches a destructivelevel (see [28], [29]).

3.2. Linear fractional maps and an algorithm of I. Schur. In our previous analysisof the discrete transmission line, the impedance profile was assumed to be piecewisecontinuous but the input and output waveforms were functions of a continuous timeparameter. In the inversion procedure we saw that only one pair of values from thescattering data is needed to determine the next reflection coefficient. This being thecase we may choose the probing current waveform, i(0, t), to be such that only onenumber per unit time will be sufficient to describe it completely, and this will then bethe case for the response (0, t) too. One such choice is a piecewise constant signal;another is a sequence of impulses or, in particular, a single impulse. It is easy torealize (see e.g. Fig. 3) that the response to a piecewise constant signal is also piecewiseconstant, and that an impulsive forcing current will elicit a voltage response that is aweighted series of impulses, say

(3.10) v(O,t)=i(t)+ 2 E hib(t- i) when i(O,t)=i(t).i=1

Converting this data to fight- and left-propagating wave variables, we have thefollowing scattering data pair:

(3.1 l) Wn(O, t) =/i(t) + E h,(t- i) and WL(O, t) , hit3(t- i).i=1 i=l

Since the transmission-line sections operate linearly on the propagating signals, theimpulse response data in (3.10), or equivalently in (3.11), will completely characterizethe behavior of the semi-infinite line. (In fact, from (3.11) we can also obtain a causalimpulse response that relates the reflected wave Wd0, t) to the probing wave WR(O, t).This function will play an important role later on.) In the sequel we shall thereforeconsider the nonuniform line as a discrete, time-invariant linear system acting onsequences of numbers, bearing in mind that we in fact operate on sequences ofimpulses or, alternatively, on piecewise constant waveforms.

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INVERSE SCATTERING FOR DISCRETE MODELS 371

We formally define the corresponding z-transforms of the time sequences arisingin (3.10) and (3.11) as

(3.12) i(0, z) 1, v(0, z)= + 2 Y h2z-=l + 2H(z),i=1

say, and accordingly

(3.13) W(0, z) + H(z) and W(0, z) H(z).

We next introduce a function often encountered in the scattering literature, thedepth-n left reflection function, defined as

Wz(n, z)(3.14) Rn(z)

Wg(n, z)

Therefore, R,,(z) is the z-transform ofthe left-propagating response to a probing, fight-propagating impulse (i.e. the WzJWg transfer function), for the transmission lineextending from the beginning of section n to infinity (see Fig. 4). In particular, forn 0, we have the wave-domain impulse response, or reflection function

(3.15) Ro(z)WI.(O, z) H(z) H(z) 2 hz-.Wg(O, z) + H(z) j=l

Let us now try to relate the elements of the sequence of functions {Rn(z)} to theparameters of the transmission line. Using the evolution equations (2.21) in thetransform domain, or directly referring to Fig. 4, we obtain, after some easy algebra,the recursion

zRn-l(Z)- k,,(3.16) R,(z) -knzRn-(z)"This recursion, which is a linear fractional map, or a discrete Riccati equation,provides the evolution of the reflection function as we proceed deeper and deeperalong the transmission line, provided we are given the reflection coefficients k,. Fromthe discussion at the beginning of this section, we know that we can determine therequired reflection coefficients recursively as we propagate (3.16); merely note that k,can be determined from past results as

(3.17) kn=limzR,,_(z).

For example, we have that

w,(0, 1)(3.18) k lira zRo(z)

z- w(0,0)

because the first nonzero response to an impulsive probing function Wg(0, t) will bean impulse with amplitude k. The same is true at any depth, and therefore we havean alternative inversion algorithm, which recursively propagates (3.16) together withthe identification formula (3.17). However, it can be verified that, when translated interms of time-series coefficients, the linear fractional recursion (3.16) is simply afunctional description of the steps in the layer-peeling algorithm described in 3.1.More precisely, we can "linearize" the fractional map (3.16) by determining inter-locked recursions for the numerator and denominator of the function R,(z) (putR,(z) U,(z)/Vn(z), and using (3.16) it is easy to see that Un and Vn obey the recursion(2.21), with Z-1/2 -" A).

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372 A. M. BRUCKSTEIN AND T. KAILATH

Rn(Z)

FIG. 4. The definition and evolution ofthe left-reflectionfunctions.

The continuous counterpart of the inverse scattering algorithm (3.16), (3.17)involves the propagation of a nonlinear Riccati equation together with a boundarycondition for the identification of the local reflectivity function. This method forsolving continuous inverse scattering problems was first derived independently byBruckstein, Levy and Kailath [25] and by Corones, Davison and Krueger [50].Numerical implementations of the nonlinear recursions for continuous case "reflec-tion kernels" R(t) are clearly possible (see e.g. [50]). However, these recursions havea complexity ofO(N3), which is, quite unnecessarily, as high as a nonetflcient approachvia matrix equations. This happens because, to propagate the reflection kernels,implicit deconvolutions (to provide the impulse response of the medium at depth x)have to be performed. The linearized layer-peeling methods, on the other hand, usethe medium model directly to compute the waves propagating into the medium, and,as we have seen, these waves provide information on the local reflectivity in a moredirect and efficient way.

Let us now define a power series (in -) as

(3.19) Sn(z) zR.(z)= s + s’lz-1

so that the recursion (3.16) becomes

(3.20) S(z) z-knSn-l(Z)"

The reason for doing this is that (3.20) then turns out to be identical to a recursionpresented by I. Schur in 1917 for checking whether a power series in z-L, S(z), isbounded by one outside the unit circle of the complex plane, i.e.,

(3.21) IS(z)l < for Izl > 1.

Schur’s test (see [33], [34]) is precisely that the sequence of numbers kn implicitlydefined by the recursion (3.20), and

(3.22) lim S_l(Z) k,,

should satisfy

(3.23) Iknl< 1.

This result also shows that the left reflection function associated with any transmission-line will be bounded by one outside the unit circle of the complex plane, a fact oftenderived differently in the scattering literature.

The algorithm of Schur has been encountered in many recent engineeringanalyses, e.g., in network and digital filter synthesis, and in the derivation of fast

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INVERSE SCATTERING FOR DISCRETE MODELS 373

algorithms for linear estimation and fast Cholesky factorization of Toeplitz andrelated matrices [35]-[37], [39], [51]-[53]. Our discussion in 2 and 3 shows thatthis algorithm may be regarded as describing the propagation ofwaves along a passivetransmission line, and consequently as providing a direct inverse scattering algorithmfor determining the parameters ofthe line given the special input and output sequences(3.13), + y]o hz- and y]o hz_. We may remark that this is an important type ofscattering data, being encountered in the so-called "marine seismograms," where themedium has a perfect reflection at the boundary (for example, the sea-air interface)(see Fig. 5).

WR=[I,h=h 2, hN...]

Scottering medium

FIG. 5. The perfect reflection experiment.

Despite the prevalence of the special scattering data (3.13), we should stress thatthe inversion procedures of 3.1 work with arbitrary input and output sequencesWg(0, z) and Wd0, z); of course, by linearity, there must exist a power series f(z)(in z-) such that

(3.24) Wg(O, z) =f(z)(1 + H(z)), WI.(O, z) =f(z)H(z)

corresponding to exciting the medium not with a current impulse i(0, z) but withthe function f(z). In works on inverse scattering based on the classical linear equationmethods (see 4), it is sometimes advocated that a "deconvolution" step be firstperformed to remove the effect of the excitation f(z) and go back to the situation ofan impulse input. Our layer-peeling formulation shows clearly that such a preliminarydeconvolution is unnecessary. This is also clear from the linear fractional form (3.16)of the Schur recursion, since Ro(z) and So(z) are the same no matter what the non-zero function f(z) is. We remark that, in the application to matrix factorizationresults, the above remarks provide a nice interpretation of certain congruence rela-tionships between Toeplitz matrices and matrices of displacement inertia {1, 1}(see [47], [48]).

Impedance-domain recursions. We note that we may associate with the voltage-current evolution equations (2.28) a different linear fractional transformation. Indeed,defining the depth-n impedancefunction

(3.25) A.(z)

we obtain for its evolution the recursion

(3.26) A,,+(z)

v(n,z)i(n,z)

+ z)A.(z) + z)Z,,(1 z)ZIAn(z) q" (1 + z)

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374 A.M. BRUCKSTEIN AND T. KAILATH

This functional recursion, together with the local impedance identification formula

(3.27) Z,= lim An(z),Z---OO

provides an alternative inversion process, similar to the Schur recursions. Clearly wehave the initialization Ao(z)= + 2H(z), and the above algorithm implicitly testswhether this function has positive real part outside the unit circle, since this is theproperty equivalent to the boundedness ofRo(z) (see e.g. [34]). The function is positivereal provided the Z, sequence implicitly defined via (3.26) and (3.27) is strictly positiveand bounded. We note that this result, a corollary of the boundedness test of Schur,is apparently new.

3.3. Extensions to other types of media. Up to this point we assumed that theunderlying scattering medium is a piecewise constant and lossless transmission-linemodel. However, it is not hard to see that the layer-peeling algorithm can be usedwhenever each section of the line is uniquely specified by the local left reflectioncoefficient, without necessarily being lossless. That is, the scattering matrix of eachsection could have the general form

(3.28) (k) (all(k)kalE(k)]a2a(k)J

with ai; arbitrary and 0"22 nonzero. The invertibility of 0"22 is required for the existenceof the chain-scattering or transfer matrix representation O(k) of the above scatteringmatrix. This matrix is given by the following (exchange) rule which can be obtainedby some simple algebra (see e.g. [46] and Appendix A)

(3.29) O(k)=[rl(k)- k(k)l_(k) J(k)(k)]-ka:(k) a;(k) J"

In this context, we do not generally have lossless propagation; neither can we expecta simple transmission-line model to correspond to the wave propagation equations.We are merely solving an inverse scattering problem for a more complicated twocomponent difference equation, through the solution of an initial value problem thatinvolves propagation of the model equations together with a formula yielding thenext required parameters from the already computed quantifies. In this context, it isnot known in general how to derive Gelfand-Levitan or Krein or Marchenko typematrix equations-based approaches (for reasons that will be evident in 4).

A particular case (3.28) of interest is the simple propagation model described by

This yields an input-output map, specified by the impulse response R0 Yr kz-,and the layer-peeling inverse scattering algorithm then reduces to a simple recursivedeconvolution process. Another propagation model that is quite interesting arises inthe solution of the partial realization problem. The minimal partial realizationproblem is to determine a lowest order linear system whose impulse response matchesa given sequence up to a certain lag, or depth. Clearly we could realize any sequenceby an almost arbitrary propagation model; however, the lowest order requirementforces the choice of a propagation model with a minimal number of delay sections.In [30], inverse scattering principles, associated with a slightly more general layeredmodel, were applied to analyze the partial realization problem. The layer-peeling ideathen directly led to a relatively recent partial realization procedure, the generalized

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INVERSE SCATTERING FOR DISCRETE MODELS 375

Lanczos algorithm; the inverse scattering framework also provided several alternativesolutions. Those subsequently found an application in devising efficient architecturesfor decoding Reed-Solomon and BCH error-correcting codes (see [31]).

Analyzing layer-peeling inverse scattering algorithms, it becomes clear that thecrucial property allowing a recursive identification of the medium is the possibility todetermine the nearest scattering layer from a causal input-output pair recorded at itsleft boundary. In gain-delay type media such as the ones encountered above, we seethat the initial portion of the response can be attributed solely to the action of thenearest layer and this is sufficient for its identification.

To make an even stronger case for the easily derived and understood layer-peeling methods, we may note that it will work even on a nonlinear gain-delay typescattering model with a special structure. Indeed, suppose we are given a model forthe local behavior that has alternating delay sections with nonlinear interactions,described by the scattering operator

W,(n + 1, t) [ Fk,[ WR(n, t-- 2)} + W(n + 1,(3.30) W(n, t) [F,.{ Wg(n, t- 1)] + F,,{ W(n + 1, t- 2)}

where F,{. }, F,{. }, F,{. and Ff,{. are static nonlinear functions, parametrizedby k and assumed to have the following properties.

1) They all pass through the point (0, 0) for all k, i.e., Fk(O) 0;2) F,{. is an invertible transformation for all k;3) Given v and the value of Fk[V}, k is uniquely determined.

With these assumptions we have the following straightforward inversion algorithm:

(a) From F,,{ W(n, n/2)} W(n, n/2 + 1) determine k,.(b) Compute the waves at n + using first

-1

(3.31a) WL(n+ 1,t)=Fk, IWL(n,t+1/2)-Fk,lWR(n,t-1/2)llthen

(3.31b) R(n+ 1, t)= Uk, (n, t--1/2)l + Fkl(n+ l, t)l.

Note that (3.3 a, b) effectively yield the nonlinear transfer operator corre-sponding to the scattering layer described by (3.30).

(c) Return to step (a) with the forward propagated scattering data W(n + l, t)and W(n + l, t).

The above algorithm is guaranteed to proceed successfully provided the scatteringdata was genuine, i.e., in fact produced by the model (3.30). The algorithm abovemay be viewed as an example of nonlinear inverse scattering via downward continu-ation. We also note that in the nonlinear case we do not have an equivalent fractionaltransformation-based method, recursively propagating for the impulse responses ofthe media considered over the intervals [n, ). In fact the nonlinear system describedabove is completely determined by its impulse because, as we saw above, from it themedium parameters can be recovered. However, in order to obtain the response ofthe medium to an arbitrary input, one has no alternative but to propagate this inputthrough the scattering structure.

At a first glance, the ’above generalization might seem rather unmotivated.However some recent papers (see e.g. [32]) addressed the issue of numerically solvingnonlinear Volterra equations of the convolution type, which appear in modelingbiological and neurophysiological processes. The discretized version of such integral

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376 A.M. BRUCKSTEIN AND T. KAILATH

equations can be written in our notation as follows:

(3.32) Wd0, t)= Y F{k_, W()} or Y F{k, WR(t-- )}.o 0

It is not hard to realize that the above equation describes the input/output relation ofa nonlinear scattering medium with the following scattering description

[W(n+l,t)] [ W(n,t-1) ](3.33) Wz(n, t) f{kn, W(n, t- 1)1 + Wz(n + 1, t)

which is a particular case of (3.30). Therefore the solution of (3.32), i.e., the deter-mination of the function k, for all n, can proceed, under some mild conditions onthe nonlinear function involved, via a fast layer-peeling algorithm. In the analysisdone by Hairer et al. [32], the conclusion is that some even faster (O(Nlog2 N))algorithms can be obtained to solve nonlinear convolution equations of the Volterratype. Very fast algorithms already exist for the linear deconvolution problem(O(Nlog N), via the fast Fourier transform technique). More importantly, severalrecently obtained results for fast factorizations of structured matrices [53]-[55] alsoindicate that such O(Nlog2 N) algorithms can be devised for the inverse scatteringproblem too, via a so-called doubling technique. Although we shall not discuss thisissue in detail, we briefly note that the idea is the following. Suppose we only use Nlags ofthe scattering data to recover the medium up to depth N. The identified portionof the medium is a linear system relating the original scattering data to the waves atdepth N in the medium, via a matrix transfer function (corresponding to the cascadeof the N determined sections of the scattering medium (see {}4)). Thus to compute thewaves at depth Nwe have to perform convolutions of the original scattering data withthe impulse response matrix of the already determined layers. The doubling step is touse a portion of the medium of depth N to propagate for only 2N lags of the signalsat depth N, and to do the convolutions involved via the fast Fourier transform. Thefirst N lags of the waves at depth N will, of course, be zero. By causality, however, thenext N lags will be a causal pair of scattering data for the portion of the scatteringmedium starting at depth N. Therefore one can now identify the medium for Nadditional steps in depth, use the 2N portion of the medium to propagate for 4N lagsof the signals at depth 2N and so on. A complexity count for this process quicklyyields that the number of operations required is O(N log2 N).

We also mention that further generalizations of layer peeling procedures to gain-delay type media with n-dimensional vectors as outputs and reflected waves arepossible and, .implicitly, have found applications in designing digital filters wellmatched to VLSI implementations and to the development of fast, structured esti-mation algorithms for nonstationary processes [391, [51]-[53].

4. Linear equation methods for inverse scattering. The classical approaches tothe solution of inverse scattering problems are via sets of matrix equations withincreasing dimensionality. In this section we show that all the classical methods canbe derived by a straightforward analysis ofthe matrix transfer functions correspondingto cascades of gain-delay sections. We shall first derive, using causality and symmetryarguments, an apparently new, general linear equation, specializations of which fordifferent types of scattering data yield the various previously used equations.

Suppose we have the matrix transfer function corresponding to the cascade ofthe first n sections of the transmission-line. Therefore we have, using the z-transform

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INVERSE SCATTERING FOR DISCRETE MODELS 377

notation,

(4.1) [ W(n, z) m21(n, z) m22(n, z) W(0, z)

where recall that, for simplicity, we denote by W/(n, z) the sequence of values of thewave variables at depth x (n/2)+. The {mo} are sometimes known as the "influence"or Green functions associated with the propagation equations (2.14). It readily followsfrom the stcture of the transmission line that

(4.2) mll(n,z) ml2(n,z) =(kn)As(Z) (k)As(Z)= [(ki)As(Z)}m2,(n, z) mz2(n, z)

where we have used the following z-transfo representation for the delay operator:

There is complete symmet in the foard transmission-line picture of Fig. 2(a),in that we can replace by - and W by W without affecting the relationshipsbetween the propagating signals. We therefore have the following useful identities:

(4.4) m22(n,z)=mll(n,z-) and ml2(n,z)=m21(n,z-l).At this point a cosmetic step of redefining the unit delay is helpful in simplifying thesubsequent arguments. Setting z- z-/2 in (4.3) redefines the time scale to t 2t.Therefore we shall regard, conceptually, the time functions as sequences of numberspadded with zeros at alternate time lags. Because of the stcture of the "relativeshift" matrix

(4.5/ s(z)

which contains both time-delay and time-advance operators, the time suppo of thefunctions/sequences mo(n, z)l extends from =-n to time t n (in terms of therescaled time), with exactly n + nonzero lags. Figure 6 shows the suppo of thesignals of interest in the (x, t) lane. Also note that by causality, at deNh x n/2 wehave (see Fig. 2(b))

(4.6) W(n,t)=O fort<n, W(n,t)=O fort<n+

while for t n we have

(4.7) W(n, n) (1 k)1/ W(O, 0).

We can rewrite the basic relation (4.1) in the time domain as

W(n, t) W(O,t). mll(X,t)+ W(0, t) *(4.8/

where stands for (sequence) convolution. Now define the function

(4.9) (n, t) ml l(n, t) + ml(n, t),

which has a time span of [-n, n]. Because of (4.4), it turns out that

(4.10) (n,-t)= m(n, t) + mid(n, t).

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378 A. M. BRUCKSTEIN AND T. KAILATH

0 0

() 0

II 0 0

(1 0 ,O

ql 0 ,I 0

() ,’ 0

0

/

n(x:n/2)

%

FIG. 6. Support ofinfluencefunctions and scattering data in the space-time plane.

Therefore, adding the two equations of (4.8), we have

(4.1 l) WR(O,t)* (n,t)+ W(O,t). (n,-t) Wa(n,t)+ W(n,t).

Writing out the convolutions explicitly for t _-< n, and using the causality relations(4.6), we obtain

(4.12) Y W(O, t- i)(n, i) + Y W(O, t- i)(n, -i) 0 for t< n,_, _, W(n, n) for t= n.

In (4.12) the signals {Wg(0, t), W(0, t)} are causal funtions (i.e., their supportis t => 0). Therefore we can rewrite the above result as the following set of linearequations:

00

(4.13) ILr[W] + Lr[WTl’[lav"=

W(n,n)

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INVERSE SCATTERING FOR DISCRETE MODELS 379

where1) xI, is a column vector of dimension n + stacking all the nonzero elements

of the sequence xI,(n, t) from the span t [-n, n] in increasing order of thetime index;

2) Lr[X] is a lower triangular Toeplitz matrix with first column X;3) WT/ [W/(0, 0) W/.(O, 1)... W/(O, n)];4) I is a matrix with ones on the antidiagonal and acts as a time reversal operator.

The above general linear equation was not derived earlier in the inverse scatteringliterature. Usually, equations corresponding to special choices of scattering datasequences W(0, t), W(0, t)} (or corresponding special choices of {i(0, t), v(0, t)}) areobtained by various, specifically tailored approaches that successfully conceal thegeneral causality structure leading to the result above. Let us now proceed to derivefrom (4.13) the classical linear equations that appear in the inverse scattering literature.

4.1. The Gelfand-Levitan, Marchenko and Krein equations. Suppose that thescattering data corresponds to the current impulse response, i.e. that we are given

(4.14) i(O,t)=6(t), v(O, t) + 2 Yh,6(t- i)

or correspondingly

(4.15) W(O,t)=6(t)+ Yh6(t- i),

Then (4.13) reduces to the form

Wz(O, t)= Y. h,6(t- i).

(4.16) {Lr[E+ H"] + Lr[HnllIe/n=

00

Hn(1 k)1/2

a linear system of equations with a Toeplitz + Hankel coefficient matrix. This is adiscretized version of the Gelfand-Levitan equation. Note that we defined Hn to be[0, hi, h2, h,]r and E [1, 0, ..., 0] r.

Multiplying both sides of (4.16) by /, adding the direct and time-reversedequations and then using some simple algebraic tricks, such as the simple observationthat (I + I-)(I + I-) 2(1 + I-), results in a completely symmetric equation for the vectorn + 1"" which corresponds to (n, t) + xI,(n, -t). This equation is

(4.17) f }I+ (I+ T)Lr(H")(I+ ) [n + ’,1

"II.(l --k/2) I/2

0

0H.(I -k/2) 1/2

(4.18) W(0, t) i(t) and Wz(0, t) Y r6(t- i).i--I

and it should be considered as the "true" discretized Gelfand-Levitan system, sincethe continuous counterpart also has a completely symmetric kernel (see e.g. [25]).

Assume next that the scattering data is the left reflection function Ro(z), orequivalently that we are givenD

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380 A.M. BRUCKSTEIN AND T. KAILATH

Substituting this into the general equation (4.13), leads to

0

(4.19) {I+ Lr[R]’[}=II (

with R" =[0, r, r2 r,] r, which can be recognized as a discrete Marchenko equation(see e.g. 171).

The Gelfand-Levitan and Marchenko equations have coefficient matrices thatare respectively Toeplitz + Hankel and Hankel matrices. There is a somewhat lesserknown formulation in the inverse scattering literature due to Krein (see [3], [5], [25],[38]), which has a purely Toeplitz cofficient matrix. To derive this equation we usethe special scattering data sequences that led to the Gelfand-Levitan equation

(4.20) W(0, t) 6(t) + Y. hir(t- i), W(O, t) Y h6(t- i)

and rewrite (4.8) as

Wn(n, t)= m,(n, t) + (Y h6(t- i)) [m, l(n, t) + m12(n, t)],(4.21) W(n,t)=m2(n,t)+(Yh(t-i)),[m,(n,t)+m,z(n,t)].

If we now define

(4.22) q,(n, t)= m(n, t) + m2(n, t)= mzz(n,-t) + m2(n, -t),

then write the second equation above for reversed time and add these equationsyielding W(n, t) and W(n, t) over <-_ n, we obtain the Krein equation

0

(4.23) 1I+ Lr[H"] + TLr[/P]/}"

This equation has a symmetric Toeplitz matrix of coefficients, defined by the sequence[1, h, h_, ..., hn], that will be denoted by Sr[E + Hn]. Recall that in deriving (4.23)we again used the causality relations (4.6).

4.2. Fast algorithms for matrix equation based approaches. To obtain the inversescattering methods associated with the above derived sets oflinear equations, in formscommonly encountered in the literature, e.g., [6], [7], [13], [16], [17], we have toconvert the left-hand side ofthe above equations to [0 0 1]r, which is independentof the reflection coefficient to be recovered. This is done by multiplying both sides byW(n, n) I-I’ (1 + k)-/2. In this case we have to solve the matrix equations for thevectors qn l’I]’ (1 k2i)-/2 and 9, l-i]’ (1 kEi)-/2. To recover the parameters of themedium from the solutions of these equations we may then invoke two identities,

(4.24) ’I (1- k2i )-l/2 ,n-- -I1. _. . 1-k, II(ln k/Z)-l/2-nn’for which we give a very simple proof in Appendix B. From these results it becomesobvious that the ratio of the sums of entries for consecutive solutions readily providesthe sequence of reflection coefficients.

We also note that a multiplication of both sides of the discrete Krein equationby 1-I 1, (1 k)/2 would yield another particular form to its solution. It would haveunit leading coefficient (t=n) and k, for the value at =-n (see Appendix B). This

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INVERSE SCATTERING FOR DISCRETE MODELS 381

form of the Krein linear system ofequations is well known in linear prediction theoryas the set of "normal equations" for the predictor coefficients. The solution of suchequations via the fast, O(N2), Levinson-Durbin recursive method opened up the fieldof fast lattice algorithms in estimation theory, (see e.g. [47], [51]-[55]). Among otherthings, research in this field provided fast algorithms for classes of near-Toeplitzand near-Hankel matrices, which themselves have interesting connections withtransmission-lines (see e.g. [51], [52]).

To derivefast algorithms that recursively solve the above-derived linear equationsfor increasing values of n, we shall use the fact that the solution vectors are related tothe transfer representations of the medium over the intervals [0, n/2]. In particularwe have to determine

(4.25) q/(n,t)=m(n,t)+m2(n,t) and (n,t)=ml(n,t)+m2(n,t)

recursively for increasing values of n.Recalling (4.2), we have for the entries of the transmission representation (which

determine the vector solutions of the matrix equations) the recursions

(4.26) [mz(n, z) [_mz(n- 1, z)

with initial conditions m(0, z)= and m2(0, 2r)-----0. Suppose now that we havealready computed the (influence or Green) functions m(., t) and m2(., t) to depthx n/2. These functions are not causal but have time support I-n, n] at depthx n/2, and therefore we have computed up to this stage about n(n- 1)/2 numbers.Together with the scattering data, the Green functions at n provide, via a convolutionoperation, the causally propagated waves W(n, t) and W.(n, t). To proceed with therecursions above, we need the value of kn+ which (recalling the results of 3.1) isgiven by

W.(n,n+ 1) [W(0,t), mz(n,-t)+ Wz(O,t), ml(n,-t)]=,+(4.27) k,+,= Wn(n,n)= 1-I,(1-k,2.)/2

In order to recursively compute the vectors involved in the equations of inversescattering, we therefore need to compute at each depth two inner products (convolu-tions). The resulting algorithms, when based on (4.26), require about 4n multiplica-tions to go from depth n to depth n + 1, and therefore the recursive solution of thematrix equations of inverse scattering will take O(N2) time.

The above approach is general enough to work for all sets of scattering data. Inthe Toeplitz (or normal equations) case, corresponding to the data H(z), the resultingalgorithm is the well-known Levinson-Durbin recursion. Indeed, note that whenWn(0, t) W.(0, t) for >- 1, we have by (4.27) that only one inner product has to becomputed to determine the next reflection coefficient and proceed with the algorithm.This is indeed what the Levinson-Durbin algorithm does.

In the inverse scattering literature Berryman and Greene obtained a fast algorithmproviding the solutions of the discrete Marchenko equation [17]. Their algorithm,modeled on the Levinson-Durbin procedure, was derived by exploiting the Hankelstructure ofthe coefficient matrix. Both these algorithms are now seen to be particularcases of our general method compactly described by (4.26) and (4.27).

We note again that the above presented general structure was not recognized inprevious work on inverse scattering algorithms. As a result the fast algorithms thatwere derived for the various matrix-equations-based identification methods were only

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382 A.M. BRUCKSTEIN AND T. KAILATH

considered as clever ways to exploit the structure ofthe matrices involved, rather thana direct consequence of the scattering medium structure.

It is also interesting to point out that, while the Schur-type algorithms can beviewed as layer identification and peeling processes, the linear equations basedmethods should be regarded as methods of recursively building up the mediumrepresentations for increasing depth by adjoining successively identified layers (Fig.7). In this sense the latter methods are duals of the Schur type procedures. We notethat if we are only interested in an inverse scattering method, the Schur recursionsare preferable, since they work directly on the scattering data in a simple way andprovide the reflection coefficients with one division, with no need to compute innerproducts. This latter property is a definite advantage when parallel processing of thedata is possible. In parallel implementations, computing the sums ofthe inner products(as, e.g., in (4.27)) is a processing-time bottleneck and even with Nprocessors available,the inversion via the fast matrix-equations algorithms takes O(Nlog N) time. Thelayer-peeling algorithms do not have inner products and can thus yield an O(N) timespeedup with an array N processors (see e.g. [31 ], [61 ]). As discussed in [28], the layerpeeling algorithms are also better suited for intelligently incorporating prior infor-mation to improve the performance of the medium identification process with noisydata.

identifiedmedium

0

M(n,z)

-Layer removed

R..,(z)

yet unidentified

i medium

L Leyer oddedFIG. 7. Layer-peeling vs. layer-adjoiningforfast inversion algorithms.

We also note that the doubling algorithm for inverse scattering discussed in 3.3computes the Green functions m(n, t) for n 1, 2, 4, 8 and so on, and in factmay be viewed as an accelerated layer adjoining procedure. This algorithm, we recall,has a computational complexity of O(Nlog2 N).

It should be clear from the above discussion of the special Gelfand-Levitan,Marchenko and Krein equations that a host of special equations can be obtained byvarious choices of W(0, t), W(0, t)}. Indeed, several papers in the literature addressthe case of step response or even ramp response data. We shall not pursue this here,but, because they are widely known, we further discuss another type of inversionprocedure, based on a particular method of solving integral equations that is also due

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INVERSE SCATTERING FOR DISCRETE MODELS 383

tO Krein (see [62]). Such methods were first presented for the continuous case byGopinath and Sondhi, and later by Burridge [20], [45]. Recently, Caflisch worked outthe details ofthe Gopinath-Sondhi inversion procedure for the discrete case [63]. Ourderivations, based on an adjoint lemma, seem to be much simpler and better suitedto establishing the connections to the classical inversion methods discussed above.

4.3. Gopinath-Sondhi type inversion procedures. Suppose that, instead of solv-ing the Krein equations, we obtain the vectors fl that are solutions of

(4.28) Sr[E+H]ft,=[1 1] r.Such solutions have a special meaning in the theory of transmission-lines (as forcingfunctions that yield a propagating step of voltage on the line, see [40]) and also in thetheory of nonuniform vibrating strings. This led Krein to consider a special methodof solving Volterra type integral equations, via nested solutions having the unitfunction as the fight-hand side (see [63]). The derivations that follow consider thediscrete version of the Volterra-type integral equations first obtained by Krein andlater used by Gopinath and Sondhi.

A question that arises naturally is whether we can relate, in a simple way, thesolutions of (4.28) to those of the "original" Krein equation (4.23). The answer isobtained via the following simple adjoint lemma.

LEMMA. IfR is a symmetric nonsingular matrix, then we have that

(4.29) RX= Yi for i= 1,2 implies (X, Y2)= (X2, Y)

where (., ) denotes the usual inner product. The proofis via the elementary calcula-tion below

(Xl, Y2) (XI RNX2 (RNX X2 ) (Y,X2) (X2 Y )

Now it is easy to obtain a relation between the solutions of symmetric equationswith fight-hand sides [0 0 ]r and ]r. Indeed, by (4.29)

(9,,[0 11)--l-I(1-k,2.)-’/2(,,[1 11),

which explicitly reads as

(4.30) ft,(n) {I, (1- k/2)-’/2} _n (n, i) I, (1+ k)-.This expression could already be used to successively obtain the parameters [k};however, in his paper presenting the Gopinath-Sondhi inverse scattering algorithmfor the discrete case, Caflisch does not derive or use (4.30) but, paralleling thecontinuous case derivations, proceeds to define and use the so-called "central masssequence" of Krein (see e.g. [42], [45], [63]). This is a sequence of positive numberso,, associated with the solutions ft, and defined as

(4.31) l=fl,(i)=(9,,[1 11).Pn iffiO

It is then proved, through energy considerations, that, in complete analogy with thecontinuous case result we have (see [63] and also [64]) that

(4.32)o, 0

from which one can recover the impedance profile {}.

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384 A.M. BRUCKSTEIN AND T. KAILATH

We stress however that there is absolutely no need to introduce the central masssequence in order to solve the inverse problem, since fl(n) already contains all thenecessary information for inversion. In fact we could derive other inversion methodsbased on solving nested sets of Toeplitz systems of equations with almost arbitrarilychosen left-hand side vectors (see [64], [65]).

Since the adjoint lemma (4.29) holds for any symmetric operator, we can readilyderive a counterpart of the Gopinath-Sondhi method that is based on the discreteMarchenko equation. In the continuous case this was already done in a different wayby Burridge [20], but for the discrete case we have not yet encountered the correspond-ing result. Suppose that we have the solutions Fn for the equations

(4.33) {I+Lr[O,r,r2,...,r,]}I’,=[1 1].Invoking the adjoint lemma (the matrix {I+Lr[O, r, r2, ..., r,]/--] is symmetric), weobtain

(4.34) I’n(n) 1-I (1 k-1/ I,(n, i)= 1-[ (1 + k)-This immediately provides an apparently new discrete inverse scattering algorithm.To go even further, we can find yet another inversion method by starting from thesymmetrized Gelfand-Levitan equation. There we would obtain that the sum of thefirst and last elements of the solution corresponding to the [1 1]r right-handside is a simple function of the reflection coefficients, which can therefore be recur-sively determined from consecutive solutions of increasing order. We shall not givethe details since the process is clearly exemplified by the previous derivations.

A further discussion of the connections between the parameters appearing in theabove calculations and the central mass sequence as well as the relation of inversescattering to matrix factorization can be found in [65]. The reader can see again thatdiverse variations of the above procedures and identities are available. We shall notpursue any more here. However, we mention that a common link between variousapproaches to inverse scattering problems may be found in the fact that an inversionalgorithm implicitly produces triangular factorizations for inverses of the coefficientmatrices involved. The uniqueness of triangular factorization is at the heart of manyof the special identities and relations between various parametrizations used in thetheory of inverse scattering (see, e.g., [57], [64], [65]).

5. Concluding remarks. We have presented a unified and comprehensive viewof the theory dealing with scattering of time sequences propagating in space throughlayered media modeled by transmission-line equations. Causality principles provedto be the crucial factor in solving the inverse scattering problem. Causality of thescattering representation of the medium layers leads to the possibility of derivingrecursive layer-peeling algorithms for inversion, whereas the causal propagationthrough a cascade of symmetric layers is recognized as the reason for the existence ofapproaches based on linear equations.

The novel results that were discussed in this paper were the applicability of layer-peeling inverse scattering algorithms to certain classes of nonlossless media, a unifiedand simple derivation of all classical linear-equation based methods including a newgeneral linear equation for inversion with arbitrary scattering data, and (by exploitingan adjoint lemma) a new discrete version of a Gopinath-Sondhi type inversionmethod. It was further stressed in this paper that the recursive layer-peeling methodsof inverse scattering are efficient inversion algorithms, and that those are obtained bydirectly exploiting the structure of the scattering medium. We also showed that the

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INVERSE SCATTERING FOR DISCRETE MODELS 385

derivation of fast algorithms in connection with classical matrix-equations-basedmethods is not so much a result of the structure of the coefficient matrices involvedas it is the direct consequence of the scattering medium structure. The derivations ofthe classical linear-equations-based inversion methods requires the exploitation ofsymmetry properties that are implicit in lossless media; and it should be clear thatsuch equations can be derived for any media with similar properties. Both alternativesare applicable only when the medium is symmetric and suitably parametdzed by thelocal reflection coefficient and lossless media are such. In this sense the direct, layer-peeling approaches that we presented are more general.

The problem of one-dimensional inverse scattering has been an active researchsubject for more than two decades already and the long list of references that followsis therefore far from being comprehensive. Many ofthe results presented in this paperhave been discovered and often rediscovered in the various fields in which inverseproblems arise: physics and mathematics, geophysics, network theory, linear estima-tion, acoustics and voice synthesis. Our paper offers what we hope is a simple butcomprehensive synthesis and overview of many known results, and some that appearto be new, based on the understanding that rather general causality and symmetryproperties of scattering media underlie all of them.

Alpendix A. The basics of signal-flow graph representations. A signal flowdiagram is a simple pictorial representation of relations between time functions orsequences, via a directed graph. The edges ofthe graph represent either a multiplicationof the function by a constant gain factor (specified on the edge) or the action of anoperator acting on the time-function (specified by a box with a label). The nodes ofthe graph are either summing points, so that the unique outgoing flow is the sum oftwo time-functions entering that node, or branching points in which the uniqueincoming function is replicated on the outgoing edges. These definitions are summa-rized in Diagram A. in Fig. 8.

The direction of flow on any edge of a diagram may be reversed together withthe inversion of the operator corresponding to that edge (assuming that the operatoris indeed invertible). This procedure will clearly leave the relationship between thetime functions at the ends of the edge unchanged. If an inverted edge left a summingor branching node, we will also have to invert the flow on another incoming edge atthat node. At a summing node we will also have to multiply by -1 the flow on thebranch that still enters the node (see Diagram A.2 in Fig. 8). This leads us to the issueof inverting the flow on a directed path. The Mason rule for inverting the flow on adirected path is: invert each edge and, at summing nodes, change the sign of theincoming flow. This rule quickly proves the result on converting scattering diagramsto transfer representations and vice versa (see Diagram A.3 in Fig. 8).

Alpendix B. Some properties of the I, and 6 functions. For the functions definedbefore, xI,(n, t) and q,(n, t), we have the following results. In the transform domain

[q(n, z) m2(n,q(n,z-)]=[ 1][.ma(n,z) m(n,z-(l knl/2[z-’ z] [_m:l(n- 1,z)

setting z leads to

(B.2) (n,z= 1)= Z xI,(n,t)=t=-,, ,, + k

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386 A.M. BRUCKSTEIN AND T. KAILATH

k kf(t)f(t)

(A,)

T(kf(t))

T(k f(t))

T(kf(tJ)+g(t

h(t)-g(t)

T-t(h(t)-g(t))hit)

inverted path

(At) (t)

path to be inverted

-Im2mmll-m 21

-I 111 -|m. m|1m m22m

inverted path

FIG. 8

For the alternate functions (n, t) we have

[ (n,z)1 [m,(n,z) ml(n,) ] [1](B.3) L(n,z_)j=Lml2(n,z_) ml(n,z_)]This provides the recursion

2 1/2(a.4) (n, z) (1 k.)- [z- (n l, z) k.z(n l, z-)]

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INVERSE SCATTERING FOR DISCRETE MODELS 387

which, after some easy algebraic manipulations, yields the following results for(n, t):

(n, n)= II (1 lc23-/

Note also that we have

(B.6)

(n,z= 1)= q(n,t)= (1-ki] ’/2

,---. . l+k]

2 -1/2(n,-n) -k. 1-I (1 ki)n

(n, t) + (n,-t) ,b(n, t) + (n, t)= E mo(n, t).i,j

Acknowledgments. It is a pleasure to thank Dr. Hanoch Lev-Ari of Stanford,and Prof. Bernard C. Levy of the Massachusetts Institute of Technology, Cambridge,Massachusetts, for many interesting and useful discussions.

REFERENCES

1] I. M. GELFAND AND B. M. LEVITAN, On the determination ofa differential equationfrom its spectralfunction, Amer. Math. Soc. Transl., (1955), pp. 253-304.

[2] Z. S. AGRANOVlCH AND V. A. MARCHENKO, The Inverse Problem ofScattering Theory, Gordon andBreach, New York, 1963.

[3] M. G. KREIN, On a methodfor the effective solution of the inverse boundary value problem, Dokl.Akad. Nauk SSSR, 94 (1954), pp. 987-990 (in Russian).

[4] L. D. FADDEEV, The inverse problem of quantum theory of scattering, J. Math. Phys., 4 (1963),pp. 72-104.

[5] K. CHADAN AND P. C. SABATIER, Inverse Problems in Quantum Scattering Theory, Spdnger-Vedag,New York, 1977.

[6] K. M. CASE AND M. KAC, A discrete version of the inverse scattering problem, J. Math. Phys., 14(1973), pp. 594-603.

[7] K. M. CASE, The discrete inverse scattering problem on one dimension, J. Math. Phys., 15 (1974),pp. 143-146.

[8] C. L. PEKERIS, Direct method of interpretation in resistivity prospecting, Geophysics, 5 (1940),pp. 31-42.

[9] E. A. ROBINSON, Dynamic predictive deconvolution, Geophys. Prosp., 23 (1975), pp. 779-797.[10] V. BARDAN, Comments on dynamic predictive deconvolution, Geophys. Prosp., 25 (1977), pp. 569-

572.11 E. A. ROmNSON AND S. TREITEL, Geophysical Signal Processing, Prentice-Hall, Englewood Cliffs, NJ,

1980, pp. 492-505.[12] P. HUBRAL, On getting reflection coefficientsfrom waves, Geophys. Prosp., 26 (1978), pp. 627-630.13] E. A. ROalrqSON, Spectral approach to geophysical inversion by Lorentz, Fourier and radon transforms,

Proc. IEEE, 70 (1982), pp. 1039-1054.[14] G. CYaErqKO, The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of

equations, SIAM J. Sci. Statist. Comput., (1980), pp. 303-319.15] A. BULTHEEL, Error analysis ofincoming and outgoing schemesfor the trigonometric momentproblem,

Proc. Conference on Rational Approximation, Amsterdam, Springer-Vedag, New York, 1981.[16] J. A. WARE AND K. AKI, Continuous and discrete inverse scattering problems in a stratified elastic

medium I: plane waves at normal incidence, J. Acoust. Soc. Amer., 45 (1969), pp. 911-921.[l 7] J. G. BERRYMAN AND R. R. GREENE, Discrete inverse methods for elastic waves in layered media,

Geophysics, 45 (1980), pp. 213-233.18] J. M. MENDEL AND F. HABIBI-ASHRAFI, A survey ofapproaches to solving inverse problemsfor lossless

layered media systems, IEEE Trans. Geosci. Remote Sensing, GE-18 (1980), pp. 320-330.

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388 A.M. BRUCKSTEIN AND T. KAILATH

19] R. G. NEWTON, Inversion ofreflection datafor layered media: A review ofexact methods, Geophys. J.Royal Astr. Soc., 65 (1981), pp. 191-215.

[20] R. BURRIDGE, The Gelfand-Levitan, the Marchenko and the Gopinath-Sondhi integral equations ofinverse scattering theory regarded in the context of inverse impulse response problems, WaveMotion, 2 (1980), pp. 305-323.

[21] P. DEIFT AND E. TRUBOWITZ, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), pp.121-251.

[22] W. SYMES, Stable solution ofthe inverse reflection problemfor a smoothly stratified medium, SIAM J.Math. Anal., 12 (198 l), pp. 421-453; see also Impedance profile inversion via thefirst transportequation, J. Math. Anal. Appl., 94 (1983), pp. 435-453.

[23] F. SANTOSA AND n. SCHWETLICK, The inversion of acoustical impedance profiles by methods ofcharacteristics, Wave Motion, 4 (1982), pp. 99-110.

[24] M. M. SONDHI AND J. R. RESNICK, The inverse problem for the vocal tract: numerical methods,acoustical experiments and speech synthesis, J. Acoust. Soc. Amer., 73 (1983), pp. 985-1002.

[25] A. M. BRUCKSTEIN, B. C. LEVY AND T. KAILATH, Differential methods in inverse scattering, ISLReport, Dept. of Electrical Engineering, Stanford Univ., Stanford, CA, 1983; SIAM J. Appl.Math, 45 (1985), pp. 312-335.

[26] A YAGLE AND B. C. LEVY, The Schur algorithm and its applications, LIDS Report, MassachusettsInst. of Technology, 1983; Acta Appl. Math., 3 (1985), pp. 255-284.

[27] K. P. BUBE AND R. BURRIDGE, The one-dimensional inverse problem of reflection seismology, thisJournal, 25 (1983), pp. 497-559.

[28] A. M. BRUCKSTEIN, I. KOLTRACHT AND T. KAILATH, Inverse scattering with noisy data, prepdnt, ISL,Stanford Univ., 1984; SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1331-1349.

[29] G. ARSENE, Z. CEAUSESCU, F. POTRA AND O. TIMOTIN, A new algorithm for detecting reflectioncoefficients in layered media, Ann. Geophysicae, l, 4-5 (1983), pp. 285-290.

[30] T. K. CITRON, A. M. BRUCKSTEIN AND T. KAILATH, An inverse scattering interpretation ofthe partialrealization problem, Proc. 23rd CDC Conference, Las Vegas, 1984.

[31] T. K. CITRON AND T. KAILATH, Euclid’s algorithm, scattering theory and a VLSI architecture fordecoding Reed-Solomon codes, IEEE Trans. Inform. Theory, submitted for publication, 1985.

[32] E. HAIRER, CH. LUBICH AND M. SCHLICHTE, Fast numerical solution ofnonlinear Volterra convolutionequations, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 532-541.

[33] I. SCHUR, Ober Potentzreihen, die im Innern des Einheitskreises Beschrankt sind, J. Reine Angew.Math., 147 (1917), pp. 205-232.

[34] N. I. AKHIEZER, The Classical Moment Problem, Hafner, New York, 1965.[35] P. DEWILDE, A. C. VIEIRA AND T. KAILATH, On a generalized Szego-Levinson realization algorithm

for optimal linear predictors based on a network synthesis approach, IEEE Trans. Circuits andSystems, CAS-25 (1978), pp. 663-675.

[36] P. DEWILDE AND n. DYM,. Schur recursions, errorformulas and convergence ofrational estimatorsforstationary stochastic processes, IEEE Trans. Inform. Theory, IT-27 (1981), pp. 446-461.

[37] , Lossless chain scattering matrices and optimum linear prediction, Circuit Theory Appl., 9(1981), pp. 135-175.

[38] P. DEWILDE, J. T. FOKKEMA AND I. WIDYA, Inverse scatteringand linearprediction, the time continuouscase, in Stochastic Systems: Mathematics of Filtering and Identification Applications, D. Reidel,Dordrecht, The Netherlands, 1981.

[39] S. K. RAO AND T. KAILATH, Orthogonal digitalfiltersfor VLSI implementation, IEEE Trans. Circuitsand Systems, CAS-31 (1984), pp. 993-945; see also VLSI arrays for digital signal processing,Part I: A model identification approach to digital filter realizations, IEEE Trans. Circuits andSystems, 32 (1985), pp. 1105-1118.

[40] B. D. O. ANDERSON AND T. KAILATH, Fast algorithms for the integral equations of the inversescattering problem, Integral Equations Operator Theory, 1/1 (1978), pp. 132-136.

[41] H. DYM AND H. P. MCKEAN, Gaussian Processes, Function Theory and the Inverse ScatteringProblem, Academic Press, New York, 1976.

[42] H. J. LANDAU, The inverseproblemfor the vocal tract and the moment problem, SIAM J. Appl. Math.,14 (1983), pp. 1019-1035.

[43] A. E. YAGLE AND B. C. LEVY, Layer-stripping solutions of multi-dimensional inverse scatteringproblems, LIDS Report LIDS-P- 1475, Electrical Engineering Dept., Massachusetts Inst. of Tech-nology, 1985; also J. Math. Phys., 27 (1986), pp. 1701-1710.

[44] L. WOHLERS, Lumped and Distributed Passive Networks, Academic Press, New York, 1969.[45] B. GOPINATH AND M. M. SONDHI, Inversion ofthe telegraph equation and the synthesis ofnonuniform

lines, Proc. IEEE, 59 (1971), pp. 383-392.

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INVERSE SCATTERING FOR DISCRETE MODELS 389

[46] R. REDHEFFER, On the relation oftransmission-line theory to scattering and transfer, J. Math. Phys.,41 (1962), pp. 1-41.

[47] T. KAILATH, A. M. BRUCKSTEIN AND D. MORGAN, Fast matrixfactorizations via discrete transmissionlines, Linear Algebra Appl., 75 (1986), pp. 1-25.

[48] T. KAILATH AND H. LEV-ARI, On mappings between covariance matrices andphysical systems, Proc.AMS Summer Conference, Bowdoin College, Brunswick, ME, 1984.

[49] W. W. SYMES AqD G. ZIMMERMArq, Experiments in impedance profile inversion using noisy andbandlimited data, AMOCO Research Report, 1982.

[50] J. P. CORONES, M. E. DAVISON AND R. J. KRUEGER, Direct and inverse scattering in the time domainvia invariant imbedding equations, J. Acoust. Soc. Amer., 74 (1983), pp. 1535-1541.

[51] T. KAILATH, Time-variant and time-invariant latticefiltersfor nonstationary processes, in Mathemat-ical Tools and Models for Control Systems Analysis and Signal Processing, Vol. 2, EditionsCNRS, Pads, 1982, pp. 417-464.

[52] H. LEV-ARI AND T. KILATH, Latticefilter parametrizations and modeling ofnonstationary processes,IEEE Trans. Inform. Theory, IT-30 (1984), pp. 2-16.

[53] J. M. DELOSM AqD M. MORF, A tree classification ofalgorithms for Toeplitz and related equationsincluding generalized Levinson and doubling type algorithms, Proc. 19th C.D.C., Albuquerque,1980.

[54] D. MORGAN, Orthogonal triangularization methodsforfast estimation algorithms, Ph.D. thesis, Dept.Electrical Engineering, Stanford Univ., 1985.

[55] B. R. MUSlCUS, Levinson and fast Cholesky algorithms for Toeplitz and almost Toeplitz matrices,Internal Report, Electrical Engineering Dept., Massachusetts Inst. of Technology, November1981.

[56] T. KAILATH, L. LJUqG AND M. MORF, Generalized Krein-Levinson equationsfor efficient calculationofFredholm resolvents of nondisplacement kernels, in Topics in Functional Analysis, essays inhonor ofM. G. Krein, Advances in Mathematics Supplementary Studies, Vol. 3, Academic Press,New York, 1978, pp. 169-184.

[57] T. KAmATS, B. C. LEVY, L. LJUqG AND M. MORF, Thefactorization and representation ofoperatorsin the algebra generated by Toeplitz operators, SIAM J. Appl. Math., 37 (1979), pp. 467-484.

[58] M. MORF, Fast algorithms for multivariable systems, Ph.D. thesis, Electrical Engineering Dept.,Stanford Univ., 1974.

[59] J. RSSArqE, Algorithms for triangular decomposition of block Hankel and Toeplitz matrices withapplications tofactoring positive matrix polynomials, Math. Comp., 27 (1973), pp. 147-154.

[60] I. GOHBERG, T. KAILATH AND I. KOLTRACHT, Fast matrix factorization, ISL Technical Report,Electrical Engineering Dept., Stanford Univ., 1986; see also Efficient solution oflinear systems ofequations with recursive structure, Linear Algebra Appl., 80 (1986), pp. 81-113.

[61] S Y. KUqG ArqD H. HU, A highly concurrent algorithm and pipelined architecturefor solving Toeplitzsystems, IEEE Trans. Acoust. Speech Signal Process., ASSP-31/1 (1983), pp. 66-76.

[62] I. GOHnERG AtqD M. G. KREN, Theory and applications of Volterra operators in Hilbert space, Trans.Math. Monographs, 24, American Mathematical Society, Providence, 1970.

[63] R. CAFLISCH, An inverseproblemfor Toeplitz matrices and the synthesis ofdiscrete transmission lines,Linear Algebra Appl., 38 (1981), pp. 207-225.

[64] A. M. BRUCgSTEIq AqD T. KALATIa, Some matrixfactorization identities in inverse scattering, LinearAlgebra Appl., 74 (1986), pp. 157-172.

[65],Spatio-temporal scattering and inverseproblems, ISL Technical Report, Electrical EngineeringDept., Stanford Univ., 1983.

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