An efficient implementation of iterative adaptive approach for source localization

Li Gang, Zhang Hao, Wang Xiqin, Xia, Xia Xiang-Gen

Description

The iterative adaptive approach (IAA) can achieve accurate source localization with single snapshot, and therefore it has attracted significant interest in various applications. In the original IAA, the optimal filter is performed for every scanning angle grid in each iteration, which may cause the slow convergence and disturb the spatial estimates on the impinging angles of sources. In this article, we propose an efficient implementation of IAA (EIAA) by modifying the use of the optimal filtering, i.e., in each iteration of EIAA, the optimal filter is only utilized to estimate the spatial components likely corresponding to the impinging angles of sources, and other spatial components corresponding to the noise are updated by the simple correlation of the basis matrix with the residue. Simulation results show that, in comparison with IAA, EIAA has significant higher computational efficiency and comparable accuracy of source angle and power estimation.

Sujets

• Compressed sensing

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English

Extrait

Lietal.EURASIPJournalonAdvancesinSignalProcessing2012,2012:7
http://asp.eurasipjournals.com/content/2012/1/7
RESEARCH OpenAccess
Anefficientimplementationofiterative
adaptiveapproachforsourcelocalization
1* 1 1 2GangLi ,HaoZhang ,XiqinWang andXiang-GenXia
Abstract
Theiterativeadaptiveapproach(IAA)canachieveaccuratesourcelocalizationwithsinglesnapshot,andthereforeit
hasattractedsignificantinterestinvariousapplications.IntheoriginalIAA,theoptimalfilterisperformedforevery
scanninganglegridineachiteration,whichmaycausetheslowconvergenceanddisturbthespatialestimateson
theimpinginganglesofsources.Inthisarticle,weproposeanefficientimplementationofIAA(EIAA)bymodifying
theuseoftheoptimalfiltering,i.e.,ineachiterationofEIAA,theoptimalfilterisonlyutilizedtoestimatethespatial
componentslikelycorrespondingtotheimpinginganglesofsources,andotherspatialcomponentscorresponding
tothenoiseareupdatedbythesimplecorrelationofthebasismatrixwiththeresidue.Simulationresultsshowthat,
incomparisonwithIAA,EIAAhassignificanthighercomputationalefficiencyandcomparableaccuracyofsource
angleandpowerestimation.
Keywords: sparserecovery;iterativeadaptiveapproach;sourcelocalization.
1 Introduction [],theSparseLearningviaIterativeMinimization(SLIM)
Source localization is a fundamental problem in a wide [], the iterative adaptive approach (IAA) [], etc. Here,
we are interested in IAA, which is able to provide accu-range of applications including communications, radar,
rate source localization with single snapshot and has at-and acoustics, and many algorithms have been presented
tracted significant interest in various applications [–].intheliteratureduringrecentdecades.TheFourier-based
IAA is non-parametric and it achieves accurate estimatesalgorithms suffer from the low resolution and the high
of angles and powers of the sources by iterative opera-sidelobes. Some methods based on subspace processing,
tions []. The spatial component on every potential anglee.g.,Caponbeamforming[],MUSIC[],ESPRIT[],and
is estimated by optimal filtering, which passes the signalother subspace-based algorithms [, ], provide
superfrom the current angle without distortion and fully sup-resolution for uncorrelated sources with sufficient
numpresses the interferences from other angles. The iterationber of snapshots. However, in the case of few snapshots,
is terminated when the norm of the difference betweenthe performances of these subspace-based methods will
two successive spatial estimates is smaller than a
certaindegradesharply.
threshold.However,itistimeconsumingtoperformopti-Recently,thesourcelocalizationproblemhasbeenconmalfilteringonallpotentialangles,sinceingeneralweareverted into a sparse recovery framework, because the
only interested in several angles where the actual sourcesnumber of actual sources of interest is generally much
arelocated.Moreover,theexcessiveestimationofthespa-smaller than the number of potential source locations in
tial components on the angles that are outside the actualthe region to be observed. A kind of algorithms of sparse
source position set may result in a slow convergence. Inrecovery is based on iterative weighted least squares, e.g.,
thisarticle,weproposeanefficientimplementationofIAAthe FOCal Underdetermined System Solver (FOCUSS)
(EIAA)bymodifyingtheuseoftheoptimalfiltering,i.e.,in
eachiteration,theoptimalfilterisonlyutilizedtoestimate
*Correspondence:[e-mail] the spatial components likely corresponding to the actual
1TsinghuaNationalLaboratoryforInformationScienceandTechnology
signalsources,andotherspatialcomponentscorrespond(TNList),DepartmentofElectronicEngineering,TsinghuaUniversity,Beijing
ing to the noise are updated by the simple correlation of100084,China
Fulllistofauthorinformationisavailableattheendofthearticle the basis matrix with the residue. It will be shown that
© 2012 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium,providedtheoriginalworkisproperlycited.Lietal.EURASIPJournalonAdvancesinSignalProcessing2012,2012:7 Page2of6
http://asp.eurasipjournals.com/content/2012/1/7
EIAA has significant faster convergence speed and
comparableaccuracyofsourceangleandpowerestimation.In
[, ], two fast implementations of IAA have been
proposedbyusingthematrixcomputationtechniquesuchas
Gohberg-Semenculdecomposition,etc.Itisnotedthatthe
way of the computational burden reduction in this article
isdifferentfrom[,]:herein,wefocusonreducingthe
numberofrunningoptimalfilteringprocedures,while[,
]focusonimprovingthecomputationalefficiencyofthe
optimal filtering procedure. In addition, similar to the al- Figure1 Thegeometryofsensorsandsources.
gorithms mentioned above, we are only interested in the
unambiguous angle solution, which depends on the ratio
of interelement spacing of the array to the wavelength. In
source number K is rather considered to be the numberthecasethattheangleambiguityoccurs,wereferto[–
ofdiscretizedanglegrids.Assumethatsissparse,i.e.,the]forresolvingtheambiguity.
numberofactualsourcesismuchsmallerthanK.ConsiderTheremainderofthisarticleisorganizedasfollows.The
the line-spectrum model and let P = diag{p ,p ,...,p },  Ksignal model and the original IAA are introduced in
Secwhose kth diagonal element p contains the power at kthktion.The EIAA algorithmisproposedin Section.The
scanninganglegrid.Theproblemofinterestisrecoveringproposed EIAA is evaluated by some simulations in
Secthe spatial components {p ,p ,...,p }, and the positions  Ktion.ConcludingremarksarepresentedinSection.
and the amplitudes of the peaks of {p ,p ,...,p } directly  K
provide the locations and the powers of the sources. IAA2 SignalmodelandIAA
[]achievesthisgoalassummarizedinTable,wherethe
Suppose that K potential far-field narrowband signals
superscript(i)denotestheithiteration.
are impinging on an M-element array from directions
{θ ,θ ,...,θ }.Insinglesnapshotcase,theoutputmea-  K
3 EfficientimplementationofIAA
surementvectorofthearraycanbeexpressedas
Itisnotedthatstep(b)inTablegivesanoptimalfilterin
terms ofθ , which reserves the signal from angleθ with-k k
y=As+e, out distortion and fully suppresses the interferences
(sig-
nalsfromotherangles).Ineachiterationtheoptimalfilterwhere A is the M ×K basis matrix and is defined by A= ingisperformedK timesforallangles {θ ,θ ,...,θ }.This  K
[a(θ ),a(θ ),...,a(θ )], s is the K ×vectordenotingthe  K is computationally extravagant, because in general we are
complexamplitudesofthesources,eistheadditivenoise. only interested in the angle set where the actual sources
Considering an M-element linear array as shown in Fig- arelocated.Moreover,fortheindexk correspondingtoθk
ure,the kth columnof A correspondingtothepotential outside the angle set of actual sources, p most likely de-k
sourcedirectionθ canberepresentedbyk pends on the noise power, and the iterative estimation of
p may increase the actual number of iterations requiredk
–jπx cos(θ )/λ –jπx cos(θ )/λ k k fortheconvergence.Basedontheaboveobservations,wea(θ )= e ,e ,k
modifyIAAasdescribedinTable.T–jπx cos(θ )/λM k...,e , The main difference between the proposed EIAA and
the original IAA lies in the estimation of spatial
comwhere {x ,x ,...,x } are the positions of the M elements ponents that are outside the actual source location set.  M
T (i)ofthearray,respectively,λisthewavelength,(·) denotes As seen from step (b) in Table , {θ withindexk ∈ }k
transpose. In sparse recovery framework, the potential are considered to be likely angle candidates where actual
Table1 IAAalgorithm.
H 2|a (θ )y|(0) kInitialization:pˆ = fork=1,2,...,K.k H 2[a (θ )a(θ )]k k
Repeat:
(i) (i) Hˆ ˆ(a)CalculatethecorrelationmatrixbyR =AP A .
H (i) –1ˆa (θ(i) )·(R ) ·yk 2(b)Estimatethespatialcomponentsbypˆ = | | ,fork=1,2,...,K.
k H ˆ(i) –1a (θ )·(R ) ·a(θ )k k
K (i–1) (i)(i–1) (i) (i) 2ˆ ˆ ˆ ˆ(c)IfthenormofthedifferencebetweenP andP issmallerthanathreshold,i.e.,δ = [p –p ] <ε,theiterationisstopped;otherwisek=1 k k
leti=i+1andgotoa).Lietal.EURASIPJournalonAdvancesinSignalProcessing2012,2012:7 Page3of6
http://asp.eurasipjournals.com/content/2012/1/7
Table2 EIAAalgorithm.
H 2
(0) |a (θ )y| (0)kInitialization:letpˆ = fork=1,2,...,K;lettheresiduer =y;
k H 2[a (θ )a(θ )]k k
Repeat:
(i) (i) Hˆ ˆ(a)CalculatethecorrelationmatrixbyR =AP A ;
(i) (i)Lettheindexsupp