BEACON: an adaptive set-membership filtering technique with sparse updates

2928 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 11, NOVEMBER 1999

BEACON: An Adaptive Set-MembershipFiltering Technique with Sparse Updates

Shirish Nagaraj,Student Member, IEEE, Sridhar Gollamudi,Student Member, IEEE,Samir Kapoor,Member, IEEE, and Yih-Fang Huang,Fellow, IEEE

Abstract—This paper deals with adaptive solutions to the so-called set-membership filtering (SMF)problem. The SMF method-ology involves designing filters by imposing a deterministic con-straint on the output error sequence. A set-membership decisionfeedback equalizer (SM-DFE) for equalization of a communica-tions channel is derived, and connections with the minimum meansquare error (MMSE) DFE are established. Further, an adaptivesolution to the general SMF problem via a novel optimal bound-ing ellipsoid (OBE) algorithm called BEACON is presented. Thisalgorithm features sparse updating, wherein it uses about 5–10%of the data to update the parameter estimates without any lossin mean-squared error performance, in comparison with theconventional recursive least-squares (RLS) algorithm. It is shownthat the BEACON algorithm can also be derived as a solution toa certain constrained least-squares problem. Simulation resultsare presented for various adaptive signal processing examples,including estimation of a real communication channel. Further,it is shown that the algorithm can accurately track fast timevariations in a nonstationary environment. This improvement isa result of incorporating an explicit test to check if an updateis needed at every time instant as well as an optimal data-dependent assignment to the updating weights whenever anupdate is required.

I. INTRODUCTION

T HE PROBLEM of designing a linear-in-parameter filter,given knowledge of the input and the corresponding de-

sired output, is studied in this paper. Traditional methodologiesinclude the minimum mean square error (MMSE) filters andtheir deterministic counterparts [the least-squares error (LSE)filters], which seek a filter by minimizing the 2-norm of theerror sequence [1]. Performance of the MMSE filters relies onthe accurate knowledge of the statistics of the input and outputobservations, whereas an off-line LSE procedure requiresdata processing in batches, which is not computationallyattractive. On-line, or recursive, methods to iteratively achievethe same goals include the least-mean squares (LMS) andthe recursive least-squares (RLS) algorithms, which have beenstudied extensively in the literature; see, e.g., [1] and [2].

Manuscript received January 27, 1998; revised April 6, 1999. This work waspresented in part at the 1997 Conference on Information Sciences and Systems,Baltimore, MD, and at the 1997 International Symposium on Nonlinear Theoryand its Applications Hawaii. This work was supported in part by the NationalScience Foundation under Grant MIP-9705173, in part by the Center forApplied Mathematics, University of Notre Dame, and in part by the TellabsResearch Center, Mishawaka, IN. The associate editor coordinating the reviewof this paper and approving it for publication was Prof. Chi Chung Ko.

S. Nagaraj, S. Gollamudi, and Y.-F. Huang are with the Laboratory forImage and Signal Analysis, Department of Electrical Engineering, Universityof Notre Dame, Notre Dame, IN 46556 USA.

S. Kapoor is with NEC USA, Inc., Princeton, NJ 08840 USA.Publisher Item Identifier S 1053-587X(99)08310-5.

In conventional system identification problems, we havesomea priori knowledge about the system to be estimated.Incorporating this in the form of constraints in the estimationprocedure leads to solutions that are consistent with thatknowledge. In a more general framework of filter design,constraints can be imposed in the estimation procedure if weneed assurance ofgoodperformance on a deterministic (point-wise) basis. As before, these requirements then have to beincorporated in the design of the estimator/filter to ensure suchan acceptable performance. Moreover, by taking the structureand constraints of the problem into account, it is likelythat computationally attractive recursive algorithms emerge aspossible solutions. With these motivations in mind, the authorshave recently introduced, in [3], a methodology for the designof filters that bound the worst-case error achieved by the filter.This method is termedset-membership filtering (SMF), and theresulting filter is called an SM filter [3]. SMF owes its nameto the so-called set-membership identification (SMI) technique[4]–[8], which is applicable only for identifying a linear-in-parameter plant with output corrupted by additive boundednoise. It is straightforward to show that SMI is a special caseof the SMF problem.

The objective of an SM filter is to estimate a memberof a so-calledfeasibility set. This set defines the SM filter’sperformance specification. SMF requires that this specificationbe met for every possible input-desired output pair of datathat come from a certaindesign space. Any closed-formsolution of an SMF problem requires accurate characterizationof the design space over which the filter is required to meetthe specification. Such a description of the design spacewould, in general, need knowledge of a functional relationshipbetween the input and desired outputs. In problems where suchknowledge is unknown or is not accurate enough, we requirea tool to estimate a point in the feasibility set in a recursivefashion. It has been shown that one such method is givenby the class of optimal bounding ellipsoid (OBE) algorithms.These algorithms were originally developed for SMI [9] andhave gained much attention in the past decade [7], [8], [10]due to some of their attractive features. Among others, theOBE algorithms employ adiscerning updaterule, i.e., theyuse the data selectively in updating the parameter estimates.

In this paper, we present a novel OBE algorithm calledthe Bounding Ellipsoidal Adaptive CONstrained least-squares(BEACON) algorithm, which shares many of the desirable fea-tures exhibited by the various OBE algorithms [5] developedto date. In addition, however, the proposed algorithm features

1053–587X/99$10.00 1999 IEEE

NAGARAJ et al.: BEACON: AN ADAPTIVE SET-MEMBERSHIP FILTERING TECHNIQUE 2929

Fig. 1. Set-membership filtering.

some favorable characteristics that are unique and differentfrom traditional OBE algorithms. The BEACON algorithmis found to update much less frequently in comparison withthe existing OBE algorithms such as the one developed byDasgupta and Huang called DH-OBE [4], whereas it exhibitsbetter mean-squared error (MSE) performance than both DH-OBE and the weighted RLS. This paper examines someissues regarding the updating mechanism and provides anovel geometrical interpretation. Moreover, it investigates thebehavior of this algorithm when the initial conditions areincorrectly chosen in such a way that the basic premise ofbounding ellipsoids fails. Surprisingly, it turns out that acertain point estimate obtained via the BEACON recursionsis insensitive to thesemodel violations, and some convergenceresults also hold. It is also observed that BEACON exhibitsvery good tracking capabilities for time-varying systems.

The above features exhibited by the point estimate motivatethe development of a least-squares-like estimation scheme thatis constrained on the specification of bounded errors. Thisapproach, although it is decoupled from the OBE method, isshown to lead to the same recursions for the parameter (point)estimate as before. Moreover, it also establishes a clear linkbetween least-squares and OBE estimation paradigms.

The next section first overviews the concept of set-membership filtering and then addresses an application fora special but important problem in digital communications,namely, that of channel equalization. Specifically, conditionsfor the existence of thefeasibility setfor a decision feedbackequalizer (DFE)will be derived. Relations between the so-called SM-DFE and the commonly used MMSE-DFE arestudied. Section III introduces the BEACON algorithm asa recursive solution to SMF, gives an explicit formula tocompute the weights for updating, and presents some analysisof its performance. Simulation results are also provided inthis section. The alternative recursive solution method forSMF, via a constrained least-squares method, is developed inSection IV. Finally, Section V concludes the paper.

II. SET-MEMBERSHIP FILTERING

A. Problem Formulation of SMF

Set-membership filtering is a filtering method distinct fromtraditional least-squares filtering in terms of performance crite-rion [3], as depicted in Fig. 1. The objective is to design a filterwhose output error is not greater than a specified value for allpossible input-desired output pairs of data. In this paper, werestrict our attention to linear-in-parameter filters of the form(1). To formalize the above idea, assume that the input, whichis denoted by , and the corresponding desired outputcomefrom a certain design space C C, whereC is the

-dimensional complex Euclidean space. The problem can beposed as follows:Given and a designer-specified positivereal number , design a filter, C C C that (linearly)maps the pairs to the output error1

(1)

where C The objective of SMF is to choose a parametervector such that

(2)

In other words, find a parameter vector that meets the errorspecification for all input-desired output pairs in Therefore,the SMF problem can be cast in the form of the following twoparts:

• Does there exist a such that

(3)

• If so, find a such that(3) is satisfied with

In contrast, traditional methodologies like LSE and MMSEminimize the squared error either in a deterministic or astochastic setting. The question of the existence of an SMfilter for a given specification is an important issue.

In general, the bounded error specification (2) can be met bya setof filters, which is referred to as thefeasibility set, sinceany one member of this set is a valid SM filter. Therefore,the objective in SMF is to estimate this feasibility set or amember of it. This set is given by

C

C

The feasibility set is an ensemble intersection over all thepossible input-desired output pairs that come from the designspace. A few properties of this set are listed below.

• is a convex set.• for any• , where• If , then , where

C

It may happen that there exists no parameter vector that canmeet the performance criterion for a particular choice ofand , as mentioned earlier. Therefore, once the design spaceis fixed, the chance of finding a nonempty increaseswith and , which is intuitively well understood. In addition,a larger design space leads to a smaller feasibility set. Thatis, as we increase the number of data pairs for which thespecification is to be met, there are fewer filters that can

1For a vectorxxx 2 Cn; kxxxk1 and kxxxk2, denote the standard 1- and 2-norms whilexxxT ; xxx

�; xxxH stand for the transpose, conjugate, and Hermitian,

respectively.


Fig. 2. Discrete-time model of a communication system with a DFE.

meet the criterion. The following observation is utilized fordesigning an SM filter.

In order to determine if the feasibility set isnonempty or not, it is enough to check iffor all , where

argC

(4)

where is as defined in (1). That is, is nonemptyif and only if the above condition is met.In the subsequent sec-tions, the applicability of the concept of SMF is demonstratedfor the problem of channel equalization in communicationsusing a decision feedback equalizer structure. For a linearequalizer, certain sufficient and necessary conditions for theexistence of a nonempty feasibility set have been derived in[11]. The set-membership decision feedback equalizer (SM-DFE) is introduced in this paper using the DFE structure[12] with the resulting equalizer designed according to anSMF criterion. Conditions for the existence of a nonemptyset of SM-DFE coefficients, and importantly, a result showingconnections to the well-known MMSE-DFE are derived. It isalso shown that the SM linear equalizer is a special case ofthe SM-DFE.

B. Set-Membership Decision Feedback Equalization

To set up the model for channel equalization, considerthe equivalent complex baseband discrete-time channel model[12], as depicted in Fig. 2. The channel consists of the trans-mitting filter, modulator, the physical channel, the demodula-tor, and the receiving filter. Let be the transmittedsymbol sequence and be the additive noise atthe channel output. The discrete-time sampled output of thechannel is given by

(5)

where is the (finite length and sampled) channelimpulse response.

Consider a decision feedback equalizer whose outputattime is given by

(6)

where is the vector of coefficientsof the feedforward filter, and is the(strictly causal) feedback filter tap weight vector. Dependingon the sampling rate, this model can be shown to encompassfractionally or symbol-spaced equalizers. The vectoris theregressor input to the feedforward equalizer, i.e., the vector

comprising of the channel outputs ,which, from (5), can be written as

(7)

where

The channel convolution matrix is

......

..... .

. . ... .

. . ....

.. .. . .

...

In addition, is the regressor vector of the previously decodedbits

Let the transmitted symbols come from a constellationIn the case of equalization, it is intuitively appealing to fixthe upper bound on the output error at , whereis the minimum Euclidean distance of two distinct symbolsfrom the constellation In other words, by such a design,the SM-DFE will have no decoding errors whenever the datapairs come from the design space

In order to guarantee that the output error be boundedby some , we need to define the design space forwhich the specification can hold. The input to the DFE canbe represented by a concatenated vector ,whereas the desired output is From (7), it follows thatdepends only on and if is assumed known. The vector

is not only specified by and but also depends on anonlinear function of the parameter vectorsand In order toremove this dependence and to make the analysis tractable, theso-calledDFE assumption[12] is commonly invoked, namely,that the previous decoded bits are correct. This implies

where

Here, refers to a zero matrix of size byand to an identity matrix of size by

The concatenated input to the DFE is now completelyspecified by the pair of vectors and The desired output


Fig. 3. Set-membership equalization criterion for a 16-QAM system. Theequalized symbol is constrained to lie in a circle of radius centered at thetransmitted symbol.

of the equalizer can be expressed as , whereis the unit vector of dimension with

a “1” in the st position. Therefore, the pairis equivalent to

As a result, the design space, which consists of the input-desired output pairs, can be equivalently described in terms ofthe Cartesian product space of the vectorsand for all ,where is taken from and The setconsists of all noise components that are bounded

C (8)

for some noise bound Designing requires fixing ofthe symbol constellation, the value of, and the order of theequalizer. Rewriting (6) for the SM-DFE, we have [13]

The dependence on the time variable has been dropped in thisequation because the equation has to hold for all possible pairsfrom the design space and not just for a particular sequence.

The aim here is to ensure that the maximum Euclideandistance between the transmitted and equalized outputs (beforedecision) is upper bounded by a specified value for allthe data sets that belong to the design space, i.e.,

(9)

Condition (9) requires that the equalized output remain in acircle of radius centered at the transmitted symbol, asshown in Fig. 3 for 16-QAM signaling with

We define the set of all feed-forward and feedback equalizerweight vectors of length and , respectively, that

satisfy (9) as thefeasible SM-DFE set

C C

where Further, (9) can be rewritten as

(10)

It should be noted that the SM-DFE feasibility set is convexonly when the DFE assumption is true, i.e., when all thepast decisions are correct. The following proposition providesconditions for the existence of a nonempty parameter set

For simplicity of discussion, assume that the inputsymbols and noise come from a zero-mean independent andidentically distributed (i.i.d.) sequence (hence white) and thatthe noise and input symbols are uncorrelated.

Proposition 1—Existence of a Feasible Set of SM-DFE FilterWeights: Consider a linear channel model (5) and a DFEequalizer as described above. The following are true:

1) Sufficiency: is nonempty if the channel satisfies

(11)

where is the maximum amplitude in the constellationand

, where consists of the torows of If (11) is satisfied, then

2) Necessity:If is nonempty, then

(12)

where is the variance of the symbols in the constel-lation ,and , where is as before.

Remark 1: Similar conditions have been derived for a linearequalizer in [11]. It can be shown that the conditions derivedhere apply to a linear equalizer as a special case since thelinear equalizer is obtained by setting the feedback weightsof the DFE to zero. The equivalent conditions for a linearequalizer can be obtained by setting Moreover, thenecessary condition (12) is an improvement over a similarresult in [11] when linear equalization is considered.

Remark 2: If the sufficient condition (11) is met, thenis a member of the feasibility set, and hence, it

is a valid SM-DFE filter. The structure of this filter is verysimilar to that of the MMSE-DFE [12] in that the feedbackfilter cancels part of the combinedpost-cursor intersymbol in-terference (ISI)effects [12] of the channel and the feedforwardfilter. The best cancellation is achieved by setting Forthe remaining part of the discussion, we shall assume

It turns out that the MMSE-DFE is a member of the feasibleSM-DFE set of filters under some conditions. If the sufficient


condition (11) holds with , where

with being the variance of the additive (Gaussian) noise,then is the MMSE-DFE [14] and belongs to thefeasibility set.

Remark 3: The above result implies that if the sufficientcondition (11) holds, then the MMSE-DFE assures that theequalized outputs lie within a radius of the true transmittedsymbol whenever the squared magnitude of each componentof the channel noise vector is less than Further, if (11)holds with , then the MMSE-DFE makes no errorsin decoding whenever the data comes from the design space.

C. SM-DFE Design Methodology and an Example

Designing an SM-DFE filter that results in error-free equal-ization for a subset of the data sets would require testingfor condition (11) to hold with The LHS ofthis equation depends only on the parametersand andis denoted by There are a set of values of theseparameters that satisfy the condition Withinthis set, we would like to choose a value for the noise bound

and order of the filter (proportional to ), which resultsin the largestdesign space It is shown that this requirementcan be cast as the optimization of a single objective function.This goal is to maximize the probability that the data comesfrom the design space. We have

Prob

Prob Prob

Prob (13)

The random variable is Rayleigh distributedsince the real and imaginary parts of the noise component,given by and , respectively, are independent Gaussianrandom variables [15]. The constrained optimization problemis

Maximize Prob

with respect to

Subject to:

In general, it is not easy to obtain an analytical solution to thisproblem. In our case, since the maximum order of the filteris limited by the computational constraint on the designer, asimple exhaustive search procedure can be used. Practically,the maximum value of can be assumed to be of the orderof half the channel impulse response length (equal to 9 inthe example used below). Moreover, it can be shown that thefunction is approximately linear with respect tofor values in the range of interest. For each value of,there is only a single that meets the constraint. This can

Fig. 4. Plot of sufficiency condition versus � and Nf = 2 for designof SM-DFE with = dmin=2 for a 20-tap microwave radio channel withQPSK signaling.

be seen easily from Fig. 4, where A practical designmethodology using these ideas is as follows.

1) Given a certain constellation, set2) Check if If not, increase the value of

until the condition is met. Fix this to be the specifiederror bound.

3) For each , find suchthat with associated cost

4) Find the value of for which this cost is maximum.

Intuitively, we might expect to be as large and to be assmall as possible. Smaller order filters are also highly desirablefrom a computational point of view.

To illustrate this method, consider a microwave radio chan-nel obtained from actual field measurements [16]. A symbol-spaced equalizer is used and also the channel impulse responseis truncated to 20 significant coefficients (i.e., ). AQPSK signaling is assumed, resulting in and

The value of was set equal to The signal-to-noise ratio (SNR) at the receiver input was taken to be 20dB. The design procedure described above is followed with

The optimal values were found to be

For this SNR of 20 dB, the probability that the data comesfrom the design space was 0.999 999 982, leading to an upperbound on the probability of error of , which isconsistent with the results shown in Fig. 5. The probability oferror curves for different values of SNR (using the appropriateoptimal values of and ) are plotted for both the SM andMMSE-DFE’s in Fig. 5. It was observed that the performanceof the SM-DFE was very similar, and almost identical, to thatof the MMSE-DFE in the range of 0–20 dB of SNR.

In many practical situations, however, the design spaceis not known a priori or might be time varying. For

example, in most wireless communication systems, the channelcharacteristics cannot be assumed to be knowna priori,especially due to time variations. Adaptive equalization is thenrequired to mitigate the effects of ISI. Two approaches toformulating a recursive (adaptive) algorithm for the general


Fig. 5. Plot of probability of error for SM-DFE versus signal-to-noise ratio(SNR).

SMF problem that estimate a member of the feasibility set arederived Sections III and IV. It is shown that both approachesyield the same estimates on a pointwise basis. The firstapproach is based on the concept of optimal bounding ellipsoid(or OBE) algorithms [9], which gives a set, as well as a pointestimate. The second approach employs a least-squares-likestrategy, which is shown to give the same point estimates asthe former approach.

III. B EACON: A RECURSIVE

ELLIPSOIDAL BOUNDING APPROACH

An OBE algorithm called BEACON is presented first.Explicit rules for updating as well as assignments to theweighting sequence are derived. A number of results on thepoint estimates obtained by BEACON are established thatmotivate and lead to the least-squares formulation discussedin Section IV.

A. Recursions and Updating Mechanism

One of the important considerations in developing real-timeestimation methods is that of keeping computational expenseslow without compromising significantly on performance (mea-sured in terms of, e.g., convergence speed and asymptoticestimation error). The family of OBE algorithms, developedfor SMI, use outer bounding ellipsoids to capture thetrueparameter. In the case of SMF, however, we do not necessarilyneed to assume the existence of a true system model (that is,linear-in-parameters with bounded output noise). However, itis straightforward to show that OBE algorithms can be usedto approximate the feasibility set in the context of SMF.

For developing the recursions, the performance specificationis imposed on the real-time input-desired output pairs. Thus,the filter model takes the form

(14)

where C is the desired filter output, C is the error,C is the known input sequence, and

Fig. 6. OBE procedure.

C is an estimate of a point in SMF requiresthat be such that

for the given error boundThe above equation combined with (14) results in a set of

all the possible parameters at each instantthat are consistentwith the data observed at that time. Obviously, liesin this so-calledobservation-induced set(which is denoted by

and given by

C

The above equation describes a degenerate ellipsoid in theparameter space. Given observations , we definethemembership set as the intersection over time, up to andincluding the present instant, of all the observation-inducedsets. That is, Since for all

, it follows that for allThe basic idea of OBE algorithms is to outer bound the

membership set at each instant by a mathematically tractableellipsoid Specifically, given an ellipsoid that outerbounds , we obtain an ellipsoid such that

This process is depicted in Fig. 6. Given an initial ellipsoidC with

some properly chosen initial parameter estimateandsuch that , the algorithm

establishes a recursive procedure for computing the sequenceof ellipsoids The feasibility set lies in , where

is obtained as an outer bounding ellipsoid of the intersectionof with The recursions for the BEACON algorithmare presented in the result below [17].

Proposition 2—The Update Equations:Given the obser-vation-induced set and the ellipsoid , the feasibility set


lies in , where

(15)

(16)

(17)

for any , where theprediction erroris ,and

The proof is on similar lines as in [4]. For a properly chosenand , we have for all The centroid of

the ellipsoid at each instant can be considered to be a pointestimate at that instant if need be. In the actual implementationof the algorithm, the matrix , rather than its inverse, isupdated. That is

This update can be derived easily from the matrix inversionlemma [4]. Each value of yields a different boundingellipsoid. To compute , we adopt a measure of optimalitysimilar to the one used in DH-OBE [4]. The difference betweenDH-OBE and this approach in terms of formulation is themanner in which the previous ellipsoid is combined with thedata constraint set at the current instant. This difference resultsin a simpler update checking rule and assignment to the time-varying weights in our algorithm. The objective, therefore, isto minimize with respect to under the constraint that

When the minimum of occurs at , it resultsin no update [17]. Thediscerning updatecapability of theproposed algorithm follows as a result.

The optimal value of , denoted by , is obtained bymaximizing the following function:

This yields

if

if (18)

It has been noted in [5] that , as a measure of opti-mality in OBE algorithms, is not a physically interpretablemeasure of the size of the ellipsoid. The minimum volumeand minimum trace criteria proposed by Fogel and Huang [9],which was also employed in other OBE algorithms [7], [10],are easily interpretable optimality measures since they havedirect relation to some notion of the geometrical size of theellipsoid. The volume of the ellipsoid is proportional todet , and the sum of squares of the semi-axes is givenby trace The justification for minimization has beenthat it is a natural bound on the Lyapunov function in theconvergence analysis in [4] and that it bears relations to thevolume and trace criteria asymptotically [18]. Although the

OBE algorithms derived using the Fogel–Huang criterion alsoupdate selectively, this does not result in any computationalsavings. This is so because the test for checking whether anupdate is needed or not is itself an operation, whichis the same as the computation required in carrying out theactual update. The minimization strategy leads to ancheck for updating. With BEACON’s sparse updating prop-erty, this implies significant computational savings. It is furthershown in this paper that by minimizing , BEACON yieldssequences of volume and trace measures of the ellipsoids thatare monotone nonincreasing. This feature is not shared by DH-OBE, which is another minimizing OBE algorithm [4].Therefore, the size of the ellipsoids generated by BEACONdecreases with time, even though the decrease may not beoptimal at every time step. For the next result and much ofthe subsequent analysis, the following definitions are needed.

Definition 1: Thebounding hyperplanesto the observation-induced set are denoted by , where , andgiven by

C (19)

In particular, if , then there are only two boundinghyperplanes given by and Moreover, theinterior of is

C

From the above construction, we haveDefinition 2: The nearest bounding hyperplaneto a vector,

say at time instant denoted by , is

(20)

By the assignment to , the following result for thea

posteriori error can be shown, from whichwe obtain a geometrical interpretation to the update rule.

Proposition 3: The a posteriorierror given by is alwaysless than or equal to the noise boundin magnitude. That is

if is an updating instantif is not an updating instant.

Geometrically, the update rule can be interpreted as—Noupdate is needed if else, update in the directionof the vector such that

B. Effect of Initial Conditions and Convergence Issues

The choice of the initial conditions, specifically that ofand (where ), is treated in this section. The

motivation for the following analysis is two fold.

1) It has been observed that various choices oflead tosignificantly different performance in DH-OBE

2) A problem that may occur in tracking time-varyingparameters is that may become negative for some


By a certain result pertaining to robustness of the estimateswith respect to choice of initial conditions, we can effec-tively explain the good tracking properties of the BEACONalgorithm.

In the absence of anya priori knowledge regarding theparameter, the origin forms the initial estimateof Theoretically, the condition to be satisfied whilechoosing and is

(21)

Since this cannot be implemented in practice, alarge enoughand are picked to start the algorithm. First, the effect of

any arbitrary scaling on , keeping the product constantsuch that (21) is satisfied, is investigated.

Proposition 4: Consider two realizations of the algorithmwith different initial conditions: 1) an initial condition

and 2) for anyThen, for all , the following are true.

• The sequence of updating instants are the same.• , and• Consequently, In addition, and are

identical.

Remark 4: For any volume or trace minimizing the OBE al-gorithm, these results are trivially true. In minimizing OBEalgorithms, however, it is not essential that the performancebe invariant to such scaling. In fact, it has been observed thatnone of the above results are true for DH-OBE.

It is possible to generalize the above result to the potentiallycatastrophic case of a wrong choice of and such that(21) is not satisfied. In such a scenario, the basic assumptionthat is a bounding ellipsoid fails, leading to what we termas amodel violation. Extending the above result shows thatthe BEACON algorithm, given by (15)–(18), is robust withrespect to such model violations [17]. It is important to notethat other types of deviations in the assumed system model, inthe form of, e.g., decision errors in a DFE, are not addressedin this work. We address the case when the initial ellipsoidfails to outer bound the feasibility set, leading to a violationin the assumed model that the OBE algorithms are based on.Specifically, we hve the following.

For a fixed , the sequence of updating instants, weights, and parameter estimates are the same for all

Therefore, the point estimates are not affected, even ifUnder model violations, the algorithm no longer

operates according to bounding ellipsoid principles. The resultsuggests that the point estimate given by (15)–(18) might havea more general interpretation. An alternative solution methodto the SMF problem, resulting from an explicit least-squares-like cost function optimization, is addressed in Section IV.

Remark 5: It is pertinent to note that although the central(point) estimate is independent of the the value of, theellipsoidal set estimate will change if is varied with

held constant.Convergence issues in OBE algorithms for set-membership

identification were addressed by Dasgupta and Huang in [4]and Nayeri et al. in [19], which established convergence

results for most known OBE algorithms along with conditionsto guarantee point convergence to the true parameter in astochastic setting. In this paper, we analyze some convergenceproperties for the BEACON algorithm in the SMF contextand prove asymptotic convergence of the prediction error evenunder model violations. First, it is shown that if the estimateconverges, then in the limit as , the estimate is apoint on theboundaryof the feasibility set that is denotedby That is, if , and

for someProposition 5: Consider the sequence of estimates.

• If there exists some such that , then, i.e.,

• Assume that andThen,

• Let be bounded, that is, for allThen, for any , .

In the special case of system identification with boundedoutput noise (i.e., SMI), it is shown that using an overestimatedvalue for the noise bound in the BEACON recursions leads to abiased estimate, assuming the estimates converge. As a degreeof confidence, convergence of the estimate to a region aroundthe true parameter is also shown. The SMI model assumes thatthe output noise in the system to be identified is bounded bysome That is, there exists and such that

(22)

Proposition 6: Choose some in theOBE recursions. Further, assume the sequence of estimatesconverge to some Then, the following are true.

• , where is the true parameter.• , where

is the noise variance, and is the minimum positivesingular value of , which is the autocorrelation matrixof the (stationary) input.

C. Simulation Studies

Some simulation studies using the BEACON algorithm aredescribed, and a comparison with the traditional least-squaresalgorithm like RLS is made. Superior tracking capabilities ofthe BEACON algorithm for time-varying systems is demon-strated via an example. Simulation results for the followingcases are presented:

• identification of an eigth-order FIR filter under uniformlydistributed noise;

• estimation of an actualmicrowave radio communicationschannelwith truncated Gaussian noise;

• tracking of a fast time-varying second-order FIR filter.

Minute details of the simulation are omitted for the sakeof brevity. The results are shown in Figs. 7–10. For all casesconsidered, the SNR was 15 dB, and the input was taken froma binary alphabet.

The performance of the BEACON algorithm is similar toRLS for Case 1. Importantly, the number of updates wasaround 30 out of the total of 1000 samples. It is pertinent


Fig. 7. Plot of mean squared prediction error for estimation of an eight-tapFIR filter with uniform noise.

Fig. 8. Estimation of a microwave radio channel.

to note that such performance is obtained even though thealgorithm uses very few inputs for updating.

For Case 2, it can be observed that the rate of convergenceof BEACON is slightly slower than RLS, although the finalMSE is comparable in spite of sparse updates. The selectiveupdating feature is one of the main reasons for the somewhatslower convergence. The value ofcan be varied to trade offbetween achievable performance and computational complex-ity. A higher value of would result in fewer updates and,consequently, a higher MSE (since the algorithm essentiallystops taking inputs for updating once the prediction errorgoes below in magnitude). On the other hand, a lower,leading to a more stringent error specification, would imply alower MSE at the expense of an increased number of updates.When multiple filters need to be estimated simultaneously, theselective update feature can be used to share a small numberof updating processors among many filters, leading to a savingin hardware requirements [11]. The BEACON algorithm is anideal candidate for such sharing as its frequency of updatingis one of the least among the known OBE algorithms.

In Case 3, for tracking time-varying parameters, identifi-cation of a second-order FIR filter was simulated similar to

Fig. 9. Tracking performance comparison.

Fig. 10. Two-tap FIR filter—evolution of the ellipsoids and the centralestimate through time,k (the x- and y-axes represent the two componentsof the parameter vector).

Case 1, but the coefficients varied randomly, thus making ita time-varying system. The time variations in the parametersare introduced by having random, but bounded, jumps at every15 samples. It can be seen from Fig. 9 that the BEACONalgorithm outperforms weighted RLS (with an exponentialforgetting factor of 0.9) in terms of tracking while using onlyaround 25 data points out of 100 for updating. This superiortracking capability is an outcome of the data-dependent choiceof weights as well as the selective updating mechanism.In comparison with RLS, the sparse updating by BEACONresults in a slight loss in convergence speed in a stationaryenvironment. On the other hand, in the presence of timevariations, it is this selective updating capability along withthe dependence of the weights on the prediction error thatresult in the excellent tracking characteristics of BEACON.

Finally, Fig. 10 is a plot of the evolution of the ellipsoidsthrough time for the case of identifying a second-order FIR


filter under uniform noise. The “” denotes , whereas“ ” denotes It can be seen that in just 25 samples, thecentral estimate seems to converge to the true parameter andthat the associated ellipsoid shrunk to a much smaller size,albeit not yet a point. This can be explained from the fact thatthe centroid of the BEACON algorithm can also be lookedupon as the solution of a certain least-squares like problemunder the constraint of bounded errors. The robustness tomodel violations of the centroid also indicates the existenceof an alternative estimation approach, which leads to the samerecursions as (15)–(17) that is not hinged upon the choice of

The next section is devoted to the formulation of such amethodology.

IV. L EAST-SQUARES FILTERING UNDER BOUNDED ERRORS

The idea here is to formulate a least-squares cost function,given the knowledge that the errors are bounded but otherwiseunknown. The problem of least-squares estimation with adead zone has been treated in [20]. We derive the least-squares method as a constrained optimization problem, andthe resulting weights are shown to be optimal in the senseof satisfying the constraint at every instant. Similar algorithmshave been derived in [21] for the problem of model validation,where the motivation is to develop least-squares estimatesthat (asymptotically or otherwise) lie in the membership set.Optimality issues in identification for systems with boundedconstraints on the error sequences are addressed in [22].

A. Problem Formulation

Assume that at time , a certain quadratic cost function isconstructed, given information up to and including time instant

, that is, Denote this function by ,which is defined as

(23)

for some appropriately chosen weighting sequenceThe choice of the time-varying and

(possibly) data-dependent weights will be explained shortly.Note that the additive constants appearing in the cost functiondo not affect the optimization procedure. They only facilitateconnections to the OBE derivation. The first term in (23)represents the confidence on the initial guess.

Consider the data set at timegiven by the pairThe information contained by the new data, due to the

requirement of the bounded errors, is characterized byThe basic idea of deriving this least-

squares estimator is the following.Given the cost function,and the observation-induced set, find the new

estimate, say, , that minimizes subject toIn other words, the strategy is to move from to a newestimate to incur the least increase in the cost functionunder the constraint that the new estimate has to lie in theobservation-induced set

Fig. 11. Contour plot ofVi�1(�) along with geometric representation ofupdating process.

Remark 3: When , there is no need to moveto a new estimate as itself is the global minimum of

This results inno updateof the parameter estimate.Moreover, the condition for no update is exactly the sameas for BEACON, given by Proposition 3. Therefore, thediscerning update rule for this least-squares-type problemis established and shown to coincide with the condition inBEACON.

This estimation strategy is cast as a constrained optimizationproblem. At each instant

Minimize

(24)

Subject to (25)

The parameter vector minimizing this problem is the newestimate, which is denoted by

Henceforth, assume that because if it did, wehave shown that there is no update and that is indeed thesolution. Since the cost function is convex and the constraintset is also a convex set, it is easy to establish that the solutionto the problem (24) and (25), which is denoted by, belongsto This, in turn, is given by , where is thephase of the prediction error (see Definition 2). Therefore,the phase of thea posteriori error is the same as the phaseof the prediction error. Recall that the above is identical tothe result obtained in Proposition 3, which was derived usingOBE methods. With this, the problem constraint becomes anequality, rather than an inequality, as in (25).

B. The Algebraic Solution

Construct an auxiliary function

where is the Lagrangian multiplier. The minimizer is theparameter at which the normal vectors to the surfaces of

, and the constraint function are colinear. This isdepicted in Fig. 11.


Proposition 7: Given data set and the cost functionat time , the solution to the optimization problem

posed in (24) and (25) is given by

(26)

where, as before,Remark 7: By the above formulation, the recursive solution

for the parameter estimate, the update mechanism, and theassignment to the time-varying weights are seen to be the sameas for the centroid in the BEACON algorithm. In addition,the sequence of weights in the BEACON recursions given by

are the the same as the Lagrangian multipliers in theconstrained optimization problem It is this weightingstrategy that leads to the satisfaction of the constraint at eachinstant. The direction of movement from given byis essentially a compromise between reaching the observation-induced set at each instant and incurring the least increase onthe error surface

A geometrical solution to the same problem is presentednext without resorting to the standard Lagrange multipliertechnique.

C. The Geometric Solution

Transforming the parameter space by the (invertible) matrixresults in a transformation of the ellipsoid to a-

dimensional spheroid. That is, a transformation of the kind, where is a vector from the transformed domain.

If the singular value decomposition of is given byfor some unitary matrix and a diagonal matrix

of positive elements, then the transformation by can belooked upon as consisting of two operations—transformationby and then scaling by The first results in a rotation(under norm invariance) of the axes, aligning all the semi-axesof the ellipsoid along the (new) coordinates. The next operationamounts to normalizing the lengths of the semi-axes to unity,resulting in a spheroid in the new coordinates.

The input is also transformed to, say, , wheresuch that the product By this, the

observation-induced set remains unchanged. In addition,. Now, the optimization problem is

as follows:Minimize the cost function given by

under the constraint that the solution belongs togiven byThis problem statement is equivalent

to (24) and (25).We know that the optimum solution lies on the nearest

bounding hyperplane to and that the solution occurs atthe point where the normal direction to coincideswith that of Since the contours of are spheroids,this amounts to moving along the direction of to reachthe nearest bounding hyperplane.

Proposition 8: The solution to the above problem is

which results in the same recursion (after transforming backto the original coordinates) as in Proposition 7. This resultcan be shown as follows: Let ,where is a constant such that Thisimplies Some rearrangementand transforming back to thespace yields the above result.

We also derive an explicit recursion for the cost function attime , namely , given and the data pairThe property that should be possessed by this function is thatit is quadratic and of the same form as In addition,

should be the global minimizer of this function.Effectively, the question is with regard to the assignment to

the weight , as in (23). Therefore

It turns out that the only assignment to, which is consistentwith the above formulation, is given by from (26), and thisdefines the updates for the matrix and the scalar Sincewe know that , these recursions are the same as (15)and (17).

Thus, we have formulated a least-squares type cost functionat each instant and obtained a solution constrained on the needfor the estimate to lie in the observation-induced set at thatinstant. The solution was found by the Lagrange multipliertechnique and by a geometrical argument. Additional con-straints imposed on the model, in the form of a bound onthe error process, allowed for a discerning method to pick theweighting sequence in a recursive least-squares-like algorithm.

D. Discussions on the Optimality Measure andTracking Properties of BEACON

In a manner independent of one another, the OBE derivationwas shown to coincide with the constrained least-squaresderivation. Note that the set of all the parameters that resultin describe an -dimensional ellipsoid in theparameter space. Moreover, the non-negative quantityisthe depth of the point of minimum of (which occursat ). It can also be be proved that minimizing the costfunction at each instant (24) reduces to the problem of findingan extremum of To show this, assume now that theestimator has been partially determined by setting the gradientwith respect to to zero. Then, recognizing the dependenceon , we denote the estimate by

From the above equation, by using the equation for andsome tedious algebra, we obtain

Therefore, the process of obtaining an extremum of withrespect to , in order to satisfy the constraint (25) and find


the Lagrange multiplier, is the same as finding the extremumof with respect to Moreover, with denoting thevalue of , which achieves this extremum, it turns out that

for This implies that is aminimizer of Hence, the problem of finding the extremumof the auxiliary cost function with respect to the Lagrangemultiplier is the same as finding the minimum of over all

As pointed out earlier, the geometrical significance ofis that it defines the depth of the minimum point of

below the -space.As far as tracking is concerned, one of the singular problems

encountered in OBE algorithms is the case when the intersec-tion of with becomes void at some time instantAtsuch instants, becomes negative, and the OBE algorithmshave to resort to certain rescue procedures so that tracking isrestored [6]. However, as has been shown in the earlier section,

becoming negative does not affect the updating recursions(15)–(17), although the OBE interpretation breaks down. In theabsence of rescue procedures, other OBE estimators tend tolose track of the parameter for abrupt and significant changesin the parameter, while such is not the case with the BEACONalgorithm, which explains the good tracking characteristicsobserved. From the least-squares perspective, this fact is evenmore apparent. The only objective in that formulation is tooptimally move to the observation-induced set at the presenttime instant.

However, if we wish the estimator to operate according toOBE principles, a scheme using ideas from set-membershipstate estimation can be implemented for tracking, thus giv-ing a concrete method to circumvent the need for rescuemechanisms. For more details on the state estimation viaset-membership principles, see [23] and [24].

V. CONCLUSION

This paper dealt with a filtering problem known as set-membership filtering (SMF). The formulation imposed convexset constraints on the filter by requiring the resulting errorsequences to be bounded in magnitude by a designer-specifiedvalue. Results for a special case of filtering, namely, decisionfeedback equalization of communication channels, was studiedin this framework. Its relation to the existing MMSE-DFE wasexplored wherein it was shown that, in certain circumstances,the MMSE-DFE belongs to the class of SM-DFE filters. Next,a recursive algorithm called BEACON was derived accordingto an optimal bounding ellipsoid criterion as an adaptivesolution to the SMF problem. This algorithm was shown tofeature a highly selective update mechanism, wherein a largepercentage of the data (greater than 90%) was not used toupdate the parameter estimate. Moreover, a least-squares-likecriterion was used to derive the same algorithm. The analysispresented allowed for connections between OBE and least-squares estimation methods. The alternative formulation alsoshed light on the observed insensitivity to model violationsand robust performance under time-varying conditions. Theability of the BEACON algorithm to track fast time variationswas demonstrated via simulations that indicate much promisein emerging wireless communication systems where high

performance and low complexity adaptive signal processingmethods are imperative.

APPENDIX

PROOFS OFRESULTS

Proposition 1—Proof:“ :” The supremum of the LHS in(10) is equal to

and is achieved when , whereis such that An upper boundon the first term in the above expression is

which leads to a condition of the form (11). Since

because there exists a constantsuch that Byminimizing this upper bound over all , we obtain theexpression for and

“ :” The necessary condition is also obtained in a similarmanner by noting that

since Further, since , we obtaina lower bound on (10) as

and are the filters that minimize this function,leading to the necessary condition (12).

Proposition 3—Proof:First, if updating instants, thenElse, from the update equation

for , we have that thea posteriori error is

where the second equality comes from the optimal assignmentof Thus, the phase of is the same as that of , and

, which implies thatProposition 4—Proof:The result is proved by induction.

Assume that is the same in both the realizations. Then, allthe equalities above are true at time Further, let theabove results be true at a time instant Since

is independent of the scale factor, also does notdepend on From (18), this implies thatsince by assumption. This immediately

leads to the fact that from (15) and from(16), and it is easy to conclude that The scalingof can also be shown similarly, which also implies that


Proposition 5—Proof:

• If such that , thensince This implies

there is no update at time , and Fromthis, it follows that for all

• The result does not hold in the trivial case when(since there would be no updates at all, and

Otherwise, assume the contrary: LetThis is possible only if such

that C , and[note that if , then by the previousresult, there would be no more updates]. This implies that

such thatsince the error magnitude is at updatinginstants. Note that lies strictly inside Thatis, it belongs to Therefore, there aresome parameter vectors (which alsobelong to the feasibility set). Since thesedo not belongto , which is not possiblebecause also belongs to and

Hence,• Assume that there is no model violation and that

is not true. Then, there exists ansuch that

(27)

In other words, for infinitely many At suchinstants , by substituting the value of in the expression for

(17), we have

where for all , and This, coupledwith (27), implies that for some large enough, we have

, which is not possible. However, by changing thevalue of by any factor does not change the sequence ofestimates. Therefore, this result is extendible to arbitrary

Proposition 6—Proof:

• When the actual noise bound is overestimated, then thereis a feasible set of plants that result in This setis nonempty since belongs to it. This is so because

for all generated by (22). Since , we knowthat However, even if the estimatesconverge to a point in , we have shown thatthey converge to the boundary of the feasibility set (fromProposition 5). This means that

• Using the fact that and lowerbounding the LHS by the expected value of the squaredprediction error, the result follows after substituting theexpression for from (22).

Proposition 7—Proof:The gradient of the auxiliary func-tion with respect to gives

The satisfaction of the constraint impliesFrom this, we obtain

From the above results,, which gives the desired result. It is straight-

forward to verify that this vector is a point of minimum.

ACKNOWLEDGMENT

The authors wish to thank Prof. J. R. Deller, Jr., Depart-ment of Electrical Engineering, Michigan State University,East Lansing, for valuable discussions. The data for themicrowave radio channel was obtained from the web sitehttp://spib.rice.edu/spib/microwave.html.

REFERENCES

[1] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: Prentice-Hall, 1996.

[2] B. Widrow and S. D. Stearns,Adaptive Signal Processing. EnglewoodCliffs, NJ: Prentice-Hall, 1985.

[3] S. Gollamudi, S. Nagaraj, S. Kapoor, and Y. F. Huang, “Set-membershipfiltering and a set-membership normalized LMS algorithm with anadaptive step size,”IEEE Signal Processing Lett., vol. 5, May 1998.

[4] S. Dasgupta and Y. F. Huang, “Asymptotically convergent modifiedrecursive least-squares with data dependent updating and forgettingfactor for systems with bounded noise,”IEEE Trans. Inform. Theory,vol. IT-33, pp. 383–392, May 1987.

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[6] A. K. Rao and Y. F. Huang, “Tracking characteristics of an OBEparameter estimation algorithm,”IEEE Trans. Signal Processing, vol.41, pp. 1140–1148, Mar. 1993.

[7] J. P. Norton, Ed.,Int. J. Automat. Contr. Signal Process.(Special issueson Bounded-Error Methods in System Identification), vol. 8, Jan.–Feb.1993–1994.

[8] E. Walter and H. Piet-Lahanier, “Estimation of parameter bounds frombounded-error data: A survey,”Math. Comput. Simul., vol. 32, pp.449–468, 1990.

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[18] S. Kapoor, S. Gollamudi, S. Nagaraj, and Y. F. Huang, “Tracking oftime-varying parameters using optimal bounding ellipsoid algorithms,”in Proc. 34th Ann. Allerton Conf. Commun., Contr. Comput., Oct. 2–4,1996.


[19] M. Nayeri, M. S. Liu, and J. R. Deller, “Do optimal bounding ellipsoidalgorithms converge? Parts I and II,” inProc. IFAC/IFOSR Conf. Syst.Ident., Budapest, Hungary, 1994.

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[23] S. Gollamudi, S. Nagaraj, S. Kapoor, and Y. F. Huang, “Set-membershipstate estimation with optimal bounding ellipsoids,” inProc. Int. Symp.Inform. Theory Appl., Victoria, B.C., Canada, Sept. 17–20, 1996, pp.262–265.

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Shirish Nagaraj (S’97) received the B.Tech. degreein electrical engineering from the Indian Instituteof Technology, Bombay, in May 1995 and theM.S. degree from the University of Notre Dame,Notre Dame, IN, in May 1997, also in electricalengineering. He is currently pursuing the Ph.D.degree at the University of Notre Dame.

He worked as a research intern at Lucent Tech-nologies–Bell Laboratories, Whippany, NJ, fromJune to December 1988 in the Wireless Signal Pro-cessing Laboratory. His research interests include

adaptive signal processing, statistical estimation and detection theory, set-membership filtering, and multiuser detection for CDMA communications.

Mr. Nagaraj was a recipient of the Michael J. Birck fellowship from 1996to 1997 and was a graduate fellow of the Center for Applied Mathematics,University of Notre Dame, from 1997 to 1998.

Sridhar Gollamudi (S’96) received the B.Tech.degree in electrical engineering from the IndianInstitute of Technology, Bombay, in 1994 and theM.S.E.E. degree from University of Notre Dame,Notre Dame, IN, in 1996. He is currently pursuingthe Ph.D. degree at the Department of ElectricalEngineering, University of Notre Dame.

From May to December 1997, he was an intern atLucent Technologies–Bell Laboratories, Whippany,NJ, where he worked on interference suppressionfor wireless CDMA systems. His research interests

are in the areas of statistical and adaptive signal processing, communicationtheory, multiuser communications, and antenna array processing.

Mr. Gollamudi was awarded the Center for Applied Mathematics Fellow-ship at the University of Notre Dame from 1996 to 1997. He is a memberof Eta Kappa Nu.

Samir Kapoor (M’98) received the B.Tech., M.S.,and Ph.D. degrees in electrical engineering from theIndian Institute of Technology, Bombay, in 1992,Washington State University, Pullman, in 1994, andUniversity of Notre Dame, Notre Dame, IN, in 1998,respectively.

From 1994 to 1998, he was with Tellabs Re-search Center in Mishawaka, IN. He is currentlywith NEC USA, Inc., Princeton, NJ. His researchinterests include digital communications, adaptivesignal processing, and array processing.

Dr. Kapoor is a member of Tau Beta Pi and Eta Kappa Nu and serves asan Associate Editor for the IEEE TRANSACTIONS ON MULTIMEDIA .

Yih-Fang Huang (F’95) received the B.S. degreein electrical engineering from National TaiwanUniversity, Taipei, Taiwan, R.O.C., in 1976, theM.S.E.E. degree from University of Notre Dame,Notre Dame, IN, in 1980, and the Ph.D. degree inelectrical engineering from Princeton University,Princeton, NJ, in 1982.

Since August 1982, he has been on the Facultyat University of Notre Dame, where he is currentlyProfessor and Chair of Electrical Engineering. InSpring 1993, he received the Toshiba Fellowship

and was Toshiba Visiting Professor at Waseda University, Tokyo, Japan, in theDepartment of Electrical Engineering. His research interests are in the generalareas of statistical communications and signal processing. He also worked onimage source coding and, more recently, joint source-channel coding.

Dr. Huang has served as Associate Editor for the IEEE TRANSACTIONS

ON CIRCUITS AND SYSTEMS from 1989 to 1991 and for EXPRESSLETTERS forthe same journal from 1992 to 1993. From 1995 to 1996, he was Chairmanfor the Digital Signal Processing Technical Committee of the IEEE Circuitsand Systems Society. He was a Co-Chairman for Workshops/Short Coursesfor the 1997International Symposium on Circuits and Systems. He servedas Regional Editor of America for theJournal of Circuits, Systems, andComputers. He also served as Vice President—Publications, of the IEEECircuits and Systems Society from 1997 to 1998.

BEACON: an adaptive set-membership filtering technique with sparse updates

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