Recursive linearly constrained minimum variance estimator in linear models with non-stationary constraints

an author's

https://oatao.univ-toulouse.fr/22750

https://doi.org/10.1016/j.sigpro.2018.03.016

Vincent, François and Chaumette, Eric Recursive linearly constrained minimum variance estimator in linear models

with non-stationary constraints. (2018) Signal Processing, 149. 229-235. ISSN 0165-1684

Short

communication

Recursive

linearly

constrained

minimum

variance

estimator

in

linear

models

with

non-stationary

constraints

François

Vincent,

Eric

Chaumette

∗

Department of Electronics, Optronics and Signal, ISAE-SUPAERO, University of Toulouse, 10 Avenue Edouard Belin, Toulouse 31055, France

Article history: Received 11 August 2017 Revised 18 March 2018 Accepted 21 March 2018 Available online 22 March 2018

Keywords:

Parameters estimation

Linearly constrained minimum variance estimator

Robust adaptive beamforming

a

b

s

t

r

a

c

t

Inparameterestimation,itiscommonplacetodesignalinearlyconstrainedminimumvariance estima-tor (LCMVE)totackle the problemofestimatingan unknownparameter vectorinalinearregression model.Sofar,theLCMVEhasbeenmainlystudiedinthecontextofstationaryconstraintsinstationary ornon-stationaryenvironments,givingrisetowell-establishedrecursiveadaptiveimplementationswhen multipleobservationsareavailable.Inthiscommunication,providedthattheadditivenoisesequenceis temporallyuncorrelated,wedeterminethefamilyofnon-stationaryconstraintsleadingtoLCMVEswhich canbecomputedaccordingtoapredictor/correctorrecursionsimilartotheKalmanFilter.Aparticularly noteworthyfeatureoftherecursiveformulationintroducedistobefullyadaptiveinthecontextof se-quentialestimationasitallowsateachnewobservationtoincorporateornotnewconstraints.

© 2018ElsevierB.V.Allrightsreserved.

1. Introduction

In the signal processingliterature dealingwith parameter es-timation, one of the most studied estimation problem is that of identifyingthecomponentsofaN-dimensionalobservationvector (y ) formed froma linearsuperpositionof P individualsignals (x ) tonoisydata(v ): y =Hx +v 1_{,a.k.a.thelinearregressionproblem,} where H isaN-by-Pmatrix and v isa N-dimensionalvector. The importanceofthisproblemstemsfromthefactthat awiderange ofproblemsincommunications,array processing,andmanyother areascanbecastinthisform [1,2].Asin[3,Section5.1],weadopt a jointpropercomplexsignalsassumption for x and v ,which al-lows to resortto standard estimation inthe mean squared error (MSE)sensedefinedontheHilbertspaceofcomplexrandom vari-ableswithfinitesecond-ordermoment.Apropercomplexrandom variableisuncorrelatedwithitscomplexconjugate.Anyresult de-rivedwithjointpropercomplexrandomvectorsarevalidforreal random vectors provided that one substitutes the matrix/vector transpose conjugateforthe matrix/vector transpose. Additionally, it isassumedthat: (a) v iszeromean,(b) x isuncorrelatedwith

∗ _{Corresponding author.}

E-mail addresses: francois.vincent@isae.fr (F. Vincent), eric.chaumette@isae.fr (E. Chaumette).

1 Throughout the present communication, scalars, vectors and matrices are rep- resented, respectively, by italic, bold lowercase and bold uppercase characters. The scalar/matrix/vector transpose conjugate is indicated by the superscript H . [ A B ] and A

B



denotes respectively the matrix resulting from the horizontal and the vertical

v ,(c)themodelmatrix H andthenoisecovariancematrix C v are

eitherknownorspecifiedaccordingtoknownparametric models. Inthissetting,theweightedleastsquaresestimatorof x [4]:2 xb =argmin x



(

y− Hx

)

H _C−1 v

(

y− Hx

)



(1a) =



HH C−1v H



−1 HH C−1v y, (1b)

coincides withthe maximum-likelihood estimator [5], if x is de-terministicand v isGaussian,andisknowntominimizethe MSE matrixamongalllinearunbiasedestimatorsof x ,thatisx b = W bH y

where [6]: Wb =argmin W

E



WH y− x



WH y− x



H





s.t. WH H=I (2a) =C−1v H



HH C−1v H



−1 , (2b)

whatever x isdeterministicorrandom.Furthermore,sincethe ma-trix W b isaswellthesolutionof [2,6]:

Wb =argmin W



WH CvW



s.t. WH H=I, (2c)

x b is also knownasthe minimum variance distortionless re-sponseestimator(MVDRE) [1,2,6].However,itis wellknownthat the performance achievable by the MVDRE strongly depends on the accurate knowledge on the parametric model of the obser-vations, that is on H and C v, and are not particularly robust in 2 The superscript b is used to remind the reader that the value under considera-

tion is the “best” one according to a given criterion. concatenation of A and B. E[ · ] denotes the expectation operator.

the presence of various types of differences between the model and the actual environment [1, Section 6.6], [7, Section 1], [8]. Thuslinearlyconstrainedminimumvariance estimators(LCMVEs)

[6,9,10]havebeendevelopedinwhichadditionallinearconstraints are imposed to make the MVDRE more robust [1, Section 6.7], [7,Section1], [8]: Wb =argmin W



WH CvW



s.t. WH



=



,



=[H



],



=[I

ϒ

], (3a) =C−1_v







H C−1_v





−1



H , (3b)

where



and Y are known matrices of the appropriate dimen-sions,attheexpenseofanincreaseoftheminimumMSEachieved, since additional degrees of freedom are used by the LCMVE in order to satisfy these constraints. However, firstly, the closed-form solution of the LCMVE (3b) requires the inversion of C v,

which can be too computationally complex for numerous real-world applications. Secondly, C v may be unknown and must be

learnedby an adaptive technique. Interestingly enough, if x and

v are uncorrelated, C v can be replaced by C y in (1b), (2b) and (3b), which means that either C v can be learned from auxiliary

datacontainingnoise only,ifavailable,or C ycan beused instead

and learned from the observations. Therefore, when several ob-servations

{

y 1,...,y k

}

are available, adaptive implementations of

theLCMVEhavebeendevelopedresortingtoconstrained stochas-ticgradient [6,11],constrainedrecursive leastsquares [12,13]and constrained Kalman-type [14,15] algorithms. The known equiva-lencebetweentheLCMVEandthegeneralizedsidelobecanceller processor [9,10,16] allows to resort as well to standard (uncon-strained) stochastic gradient or recursive least squares [2]. These recursive algorithms belongs to the set of sequential estimation algorithms compatible with applications where the observations become available sequentially and, immediately upon receipt of newobservations,itisdesirabletodeterminenewestimatesbased upon all previous observations (including the current ones).It is anattractiveformulationforembeddedsystemsinwhich compu-tationaltimeandmemoryareatapremium,sinceitdoesnot re-quirethatallobservationsareavailableforsimultaneous(“batch”) processing.Last,thiscanbecomputationallybeneficialincasesin whichthenumberofobservationsismuchlargerthanthenumber ofsignals [17].

However,theaforementionedrecursivealgorithmscanonly up-datesequentiallytheLCMVE (3b)innon-stationaryenvironments, i.e.whentheobservationmodelchangesovertime (y _l =H _l x +v _l ,

1_{≤ l≤ k}), for a given set of linear constraints W H

_

₌

_

_[2,6,11– 15],whichdefinesthesetofrecursiveLCMVEsforstationary con-straints.AnexampleofarecursiveLCMVEfornon-stationary con-straintsinnon-stationaryenvironmentsisgivenbytheMVDREx b k

of x , based on observations up to and including time k. Indeed, providedthattheadditivenoisesequence

{

v 1,...,v k

}

istemporally

uncorrelated,x b _k follows apredictor/corrector recursionsimilar to theKalmanFilter[17,Section1] [18]:

xb _k =xb _{k −1}+WbH _k



y_k − Hk xb k −1



, xb ₁=



HH ₁C−1_v1H1



−1

HH ₁C−1_v1y1, (4)

where W b _k isanalogous toa Kalman gain attime k.In thiscase, theset ofconstraints (2c) is non-stationarysince itis definedas

W H H _k =I ,where H _k isthematrixresultingfromthevertical con-catenationofkmatrices H 1,... ,H k, and W isan unknown matrix

oftheappropriatedimensions.Off course,fromatheoreticalpoint ofview,ifall theobservations

{

y 1,...,y k

}

arestacked intoa

sin-glevector y T _k=



y T ₁,...,y T _k



,the “batchform”(3b) obtainedfrom

y _k allows toimplementLCMVEswithnon-stationnaryconstraints, which are, unfortunately, hardly likely to be computable as the sizeof y _k increases.Thereforethenovelcontributionofthepresent

communicationistointroduce,providedthattheadditivenoise se-quence

{

v 1,...,v k

}

istemporallyuncorrelated,thefamilyoflinear

constraintsyielding a LCMVE whichcan be computedrecursively in the form of (4) in place of the “batchform” (3b). It appears that thisfamily only containsnon-stationary constraints, includ-ingtheaforementionedMVDRE.Aparticularlynoteworthyfeature ofthe recursive formulationintroduced is tobe fully adaptivein thecontext ofsequentialestimationasitallows ateachnew ob-servation toincorporate ornot newconstraints.The relevance of theproposedrecursiveformulationoftheLCMVEisexemplifiedin

Section3inthecontextofarrayprocessing.

2. Recursive linearly constrained minimum variance estimators

Inthefollowing:a)thevector spaceofcomplexmatriceswith

NrowsandP columnsisdenotedMC

(

N,P

)

,b)thematrix

result-ingfromtheverticalconcatenationofkmatricesA 1,..., A k ofsame

column numberis denoted A _k . We consider the linear measure-ment/observationmodel:

y_k =H_k x+v_k , k≥ 1, (5a)

where x is a P-dimensional complex unknown vector, y _k is a

N_k -dimensional complex measurement/observation vector, H _k ∈

MC

(

Nk ,P

)

andthecomplexnoise sequence{v k }k ≥ 1 iszero-mean

andtemporallyuncorrelated.Then (5a)canbeextendedona hori-zonofkpointsfromthefirstobservationas:

y_k =

⎛

⎝

y1 .. . y_k

⎞

⎠

=

⎡

⎣

H1 .. . H_k

⎤

⎦

x+

⎛

⎝

v1 .. . v_k

⎞

⎠

=H_k x+v_k ,







y_k ,v_k ∈MC

(

Nk ,1

)

H_k ∈MC

(

Nk ,P

)

Nk =kl=1Nl . (5b) Let W _k =



D_k−1 W_k



where D _{k −1}∈MC



Nk −1,P



and W k ∈ MC

(

Nk ,P

)

.Theaimistolookforthefamilyoflinearconstraints: WH_k



_k =



k ,



k =



Hk



k



,



k =[I

ϒ

k ], (6)

where



_k and Y _k are knownmatrices ofthe appropriate dimen-sions,yieldingaLCMVEx b _{k =}W bH _k y _k where (3a)and (3b):

Wb _k =argmin Wk

WH_k CvkWk



s.t. WHk



k =



k (7a) =C−1_v k



k





H kC−1vk



k



−1



H k , (7b)

which can be computedaccordingto a predictor/corrector recur-sionoftheform,

∀

k≥ 2:

xb _k =xb _{k −1}+WbH k



yk − Hk xbk −1



=



I− WbH k Hk



xb k −1+WbH k yk . (8)

Akey point tosolve theproblemat handisto noticethat, since

C v_l, v_k= C v_k

δ

_kl ,thenforany W ksatisfying (6): P_k



W_k



=WH _k CvkWk =D H k −1Cvk−1Dk −1+W H k CvkWk =Pk



D_{k −1},W_k



, (9) whichsuggeststhatsomeadhoclinearconstraints (6)couldyield separablesolutionsfor D _{k −1}andW _k ,whichisinvestigatedinafirst step. •First step If we recast



_{k =}



H _k



_k



as



_{k =}



k−1 k



where



_{k −1=}



H _{k −1}



_{k −1}



and



_k =[H _k



_k ],thenanequivalentformof (6)is:

4. Conclusions

In this communication, for multiple noisy linear observations which noises are uncorrelated, we have introduced the family of linear constraints yielding a LCMVE computable via a predic-tor/corrector recursion similar to the Kalman Filter in place of the “batchform”. It appears that this family only contains non-stationary constraints. Anoteworthy feature of the recursive for-mulation introduced is to be fully adaptive inthe context of se-quential estimationasitallows optionalconstraintsaddition that can be triggered by a preprocessing of each new observation or externalinformationontheenvironment.

Acknowledgment

This work has been partially supported by the DGA/MRIS (2015.60.0090.00.470.75.01).

Supplementary material

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.sigpro.2018.03.016.

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