Sparsity based off-grid blind sensor calibration

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Sparsity

based

off-grid

blind

sensor

calibration

Sedat Camlica

a

,

∗

,

Imam Samil Yetik

b

,

Orhan Arikan

c a_{AselsanA.S.,Ankara,06370Turkey}

b_{DepartmentofElectricalandElectronicsEngineering,TOBBUniversityofEconomicsandTechnology,Ankara06560,Turkey} c_{DepartmentofElectricalandElectronicsEngineering,BilkentUniversity,Ankara06800,Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline25October2018 Keywords: Blindcalibration Sparse Compressivesensing Off-grid Directionfinding Frequencyestimation

Compressive Sensing (CS)basedtechniques generallydiscretizethe signalspaceand assume that the signalhasasparsesupportrestrictedonthediscretizedgridpoints.Thisrestrictionofrepresentingthe signalonadiscretizedgridresultsintheoff-gridproblemwhichcausesperformancedegradationinthe reconstruction ofsignals. Sensorcalibrationisanotherissuewhichcancauseperformance degradation if notproperly addressed.Calibrationaimstoreducethe disruptiveeffects ofthephase andthe gain biases.Inthispaper,aCSbasedblindcalibrationtechniqueisproposedforthereconstructionofmultiple off-gridsignals.Theproposedtechniqueiscapableofestimatingtheoff-gridsignalsandcorrectingthe gainandthephasebiasesduetoinsufficientcalibrationsimultaneously.Itisappliedtooff-gridfrequency estimation anddirectionfindingapplications usingblindcalibration.Extensive simulationanalysesare performedforbothapplications.Resultsshowthattheproposedtechniquehassuperiorreconstruction performance.

©2018ElsevierInc.Allrightsreserved.

1. Introduction

Sparsity basedCompressiveSensing(CS) methodshavegained highimportancein recentyears,sincethey play acentral rolein manypractical applicationareas.CS methods makeit possibleto operate under Nyquist rate if the signal is sparse in a perfectly knowntransform domainsatisfyingcertain constraints [1–3]. Be-cause of very promising properties and possible advantages, CS based techniques have received considerable attention in radar, sensorandcivilianapplications[4,5].

In radar and sensor systems, calibration is an important is-sue,which directlyaffects thesystemperformance thusmust be addressed carefully. Aging and/or environmental effects such as temperature changes introduce system errors making calibration capability an unavoidable need for a given system. Furthermore, imperfect knowledge of the system parameters may cause cali-bration errors[6–8]. Guidedcalibration solutions can be used to achieve desiredperformance inwhich a knowntraining signal is used.However,suchatechniqueincreasesthecostandcomplexity oftheunderlyingsystem. Insystems suchassensorand/or radar systems,phased array antennas,MIMO systems,blind calibration

*

Correspondingauthor.

E-mailaddresses:scamlica@aselsan.com.tr(S. Camlica),syetik@etu.edu.tr (I.S. Yetik),oarikan@ee.bilkent.edu.tr(O. Arikan).

isan alternativetechniquethat estimatesthesignal firstand per-formsthecalibrationusingtheestimatedsignal.Itisbeneficialin termsofreducingmaintenancecostsand/orcomplexityofthe sys-tem;aswellasavoidingtheneedforatrainingsignal.

Intheliterature,therearevariousmethodsforblindsensor cal-ibration. In [9], sparsity based convex optimization methods for blindsensorcalibrationarestudied.Signalreconstructionand cal-ibration are performedsimultaneously. Phase only, gain onlyand joint phase andgain blind calibration methods are proposed [9]. In[6],blind calibrationofsensornetworksisperformedusing or-dinarysensormeasurements.In[7],sensornetworksarecalibrated usingsignalsourceswithknownlocations.Sensorpositions,phase and gain errors and mutual coupling between sensors are esti-mated. Asparsity basedconjugategradient algorithm isused for blind sensor calibration in[10]. In [11], sparsity basedblind cal-ibration of sensor networks is studied using total least squares approach. Sparsity based blind sensor calibration is also studied in [12] for phased array sensors.In [13], amoments basedblind calibration method is studied for mobile sensor networks. It is assumedthat thesensorsare locatedinthesameregionand ob-served signals havealmost identical statistics which is exploited for calibration.In [14], a non-convexblind calibration method is presented for linear random sensing models with positive non-complex gain errors.A similar compressive sensing basedgreedy blind calibration method for positive non-complex gain errors is alsopresentedin[15].

https://doi.org/10.1016/j.dsp.2018.10.005

Asmentioned,therearesparsitybasedmethodswhichare ca-pableofjointlyestimatingthesignalandcorrectingthephaseand thegain errors.However, anothersource oferrorthat invalidates theexact sparsity assumptions andaffectthereconstruction per-formanceofsparsitybasedtechniquesisthebasismismatch.These techniquesdiscretizethesignal spaceandassume thatthe signal issparseonthediscretizedgrid.Duetocontinuous natureofthe signals,representingthesignalonadiscretizedgridresultsinthe off-gridproblemwhichcausesperformancedegradation.IntheCS literaturetheoff-gridproblemalsoexistsinfrequencyestimation [16],angleofarrival(AOA)estimation[16],delay-Doppler imaging [17],andSARimaging [18].Therearevarioustypesofsolutionsto off-gridtargetprobleminCS[18,17,19–22].

The current sparsity based blind sensor calibration methods, as to best of our knowledge, do not deal with the off-grid sig-nals.Therefore, development ofsuch a method,whichis capable of jointly estimating the signal andhandling the phase and the gaincalibrationandoff-gridphaseerrors,ismandatory.

In[18], our group developeda sparsity basedSpotlight mode SyntheticAperture Radar imagingand autofocus method for off-grid sparse scenes. From a mathematical point of view, the pro-posedmethod[18] estimatesatwodimensionaloff-gridsignaland performsphaseerrorcalibrationsimultaneously.

In this work, our method in [18] is expanded to blind sen-sorcalibrationwithmultiplesnapshotsandanovelsparsitybased blind sensor calibrationmethod foroff-grid signals isdeveloped.

We also apply the proposed method to two different problems

and show the effectiveness for these example application areas. The proposed method handles the off-grid problem usinga gra-dientsearch basedoptimizationalgorithmwhilebeingcapableof estimatingand calibrating phase and gain errors aswell as off-grid perturbations simultaneously for multiple measurements as a novelcontribution. The currentblind calibration techniquesdo not take care of off-grid signal errors, and the proposed tech-nique,astobestofourknowledge,isthefirsttechniquethatdoes blind sensor calibration under both phase, gain and off-grid sig-nal errors jointly. The proposed off-grid blind sensor calibration algorithm(PerturbedCalibrationOrthogonalMatchingPursuit, PC-OMP) is first derived for the frequency estimation problem and correspondingsimulationanalysisisgivenindetail.Then,PC-OMP algorithm is adopted to off-grid direction finding problem with certainrequiredvariationsandsimulationanalysisisalsogivenfor thiscase. Althoughthe proposed algorithm isapplied tothe fre-quencyestimationanddirectionfindingproblems,itcanbeeasily adopted to any other application concerning array processing as longasaconvenientmeasurementmodelcanbe constructed.For example,theproposed technique can be applied to phasedarray radarsystems,ElectronicWarfarereconnaissance systemssuchas frequencyspectrum sensing, direction finding,target localization. Simulationresultsshowthattheproposedmethodshavesuperior performanceintermsofthecalculatedmetrics.

Theorganizationofthispaperisasfollows:theoff-gridsignal modelwith calibration errors is givenin Section 2.The detailed derivationoftheproposedmethodforthegeneralcaseisgivenin Section3andsimulationsaregiveninSection4.Off-griddirection findingandblindcalibrationmethodisexplainedinSection5with simulationsforthiscaseandconclusionsaredrawninSection6.

Thenotationthroughoutthetext isasfollows:lowercasebold letters (i.e. x) indicates column vectors, uppercase bold letters (i.e. G) indicates matrices,

.

indicates l2 normand

|.|

indicates absolutevalue.

2. Signalmodel

Asensorarray isconsidered inwhichindividual sensors sam-plesthereceivedsignalatthesametime,thenthemeasurements

are processed together. It is assumed that each sensor has cali-brationerrors(i.e.phaseandgain errors)whichareconstantover time andindependentfromother sensors.Forsuch an array sys-tem, let yi

(

n

)

be the signal measured by nth sensor attime

in-stancei.ForK sources, yi

(

n

)

canbeexpressedas;

yi

(

n

)

=

dnejφn K



k=1

α

i,kgn

(

f_io_,k

)

+

wi

,

(1)

wheredn and

φ

n arethegainandphasecalibrationerrors

respec-tively,

α

i,k are the complex signal gains, gn

(.)

is the equivalent

basisvector,and f_io_,kisthesignalparameter.

ForNynumberofsensors,writingthemeasurementsinvector

formyields yi

=



yi

(1)

yi

(2) ...

yi

(

Ny

)



T

. Theon-grid based tech-niquesassumesthatthesignalparametersreliesexactlyonagrid, and using the grid parameters they populate the relevant basis vectorswhich(byassumption)spans thesignalspacecompletely. Thus ameasurementmodelmatrixcan beconstructed.Usingthe vector form notation and measurement model matrix, the blind sensor calibration signal model for the on-grid case can be ex-pressedas;



y1 y2

. . .

yD



= 

G



x1 x2

. . .

xD



+

w

,

(2)

whereD isthenumberofmeasurements;xiandyiarethedesired

signal to be estimated and corresponding measurement vectors, whosedimensionsare Nx

×

1 andNy

×

1,respectively. Ny isthe

number ofsensors and w is theadditive zero-mean whitenoise Ny

×

D matrix. Let



be the measurement model matrix, then

G

= 

A, where A is the basis matrix on which the signal xi is

sparse, and



is Ny

×

Ny diagonalmatrixwhose entriescontain

calibrationerrors.Then,



isdefinedas:



=

⎡

⎢

⎢

⎢

⎣

d1ejφ1 d2ejφ2

. .

_.

dNye jφN y

⎤

⎥

⎥

⎥

⎦

.

(3)



being adiagonal matrixmakes the calibrationerrors indepen-dent among different sensors. In blind calibration, the error pa-rameters dn and

φ

n are to be jointly estimatedand corrected in

theproposedmethodPC-OMP.

Forthe off-grid signal case, thesignal parameters fo i,k are

as-sumed to be relyon anywhere in the signal spacerather thana knowngridlocations.Thus,apredefinedmeasurementmodel ma-trixcannotbeconstructed,insteaditmustalsobeestimatedalong withother desiredsignalparameters.Intheoff-gridcaseyiis

ex-pressedas: yi

= 

K



k=1

α

i,kg

(

fio,k

)

+

wi

,

(4)

where

α

i,k arethecomplexsignal gains, g(.)the equivalentbasis

vectorusedtorepresentthesignalanddefinedasg(fk

)

= 

a(fk

).

Here, f_io_,k causes the off-gridproblem andis the continuous pa-rameterforith measurementandkth signalcomponent.

Weproposetosolvethefollowingoptimizationproblemto es-timate the unknown parameters. The cost function is defined as thesumofthesquarederrors:

min αi,k,fi,k,δfi,k,φn,dn



i







yi

− 

K



k=1

α

i,ka

(

fi,k

+ δ

fi,k

)







2 2

,

such that:



n dn

=

c

,

|δ

fk

| < /

2 , 1

≤

i

≤

D , 1

≤

n

≤

Ny

,

(5)

Table 1

Proposedmethod.

Initialize: y0_=y

,T0= {} ,k=1 WHILE k<=K

OMP Projection and atom selection: FOR i=1:D,

j∗=arg maxj|g(fi,j)Hri,k−1|

Ti,k=Ti,k−1∪ {fj∗}

Perform blind calibration using (16). END FOR

Perform perturbation estimation using (6); FOR i=1:D

(αi,[δfi,1, ..., δfi,k]) = S(yi,Ti,k)

END FOR

Repeat blind calibration with updated signal estimates using (16). k=k+1

Output:α,δfi,1, ..., δfi,k 

,Ti,k, 

where



is the grid cell size, fi,k isthe closest grid point,

δ

fi,k

isthe corresponding perturbation whichcauses off-gridproblem, andc is a positive number. There can be found infinite number ofsolutionsfory

=

dy in

ˆ

amplitudecalibration.Toavoidthe am-biguity, without loss ofgenerality the constraintof



_ndn

=

c is

introduced. Simultaneous estimation of the complex signal gains

(

α

i,k

),

thesignalparametersof fi,kand

δ

fi,k,andthegainandthe

phase calibration errors

(φ

n

,

dn

)

is the main contribution of this

work.

Aswillbe detailedinthenextsection, theoptimization prob-lemin(5) issolvediteratively.Thesignalparameters

α

i,k, fi,kand

δ

fi,k are estimatedfirst.Then, using theseestimatedparameters,

thephase

φ

n andthe gain errorsdn are estimatedandcorrected

blindly.Theiterationisperformeduntilastoppingcriterionismet. Forinstance,a maximumnumberofiterations ora thresholdfor the residualerror or sparsity level can be used toterminate the iterations.

3. Theproposedmethod

The optimization problem in (5) is non-convex and poses a significantchallenge tosolvedirectly. Therefore,asub-optimal it-erativemethodwhichalternativelysolvesforthecalibrationerrors andthe signal perturbations within a greedytechnique is devel-oped.Theoutershelloftheproposedmethodisbasedon Orthog-onalMatchingPursuit(OMP)[23] andgiveninTable1.

Atthekth iterationoftheproposed algorithm,foreachyi,the

selectedk grid locations Ti,k and themeasurements are input to

aninneralgorithmabstractlyshownas:



α

i

,



δ

fi,1

, ..., δ

fi,k



= S



yi

,

Ti,k

,

(6)

withasolver

S(·)

whichproducesasolutionto(5) fortheselected grid parameters andoutputs corresponding complexreflectivities andperturbations. It is importantto note that the solver

S(·)

is notspecificallydependentonOMPandcanbeintegratedintoany algorithmthatprovidesasuitableestimationoftheinitialgrid pa-rameters.

Evenafteranestimateforthek-gridpositionsareprovidedby theOMP,theoptimizationproblemin(5) isstillnon-convexanda carefulconsiderationisneededtoobtainestimatesforthecomplex reflectivities

α

i,perturbationsfromthegivengridcenters

δ

fi,kand

calibration errors

φ

j and dj. For this purpose, an iterative

opti-mizationapproachisdevelopedwhereeachoftheseparametersis optimizedseparately while keepingothersfixed andthisis done iterativelyuntilaconvergencecriterionismet.

First,thecomplexamplitudevector

α

l isfoundusing;

α

l

=

arg min_α





y

−

k



p=1

α

pg

(

fp,l

)





2 2

,

(7)

where l is the iteration index of

S(·)

and fp,l is the fp vector

in lth iteration. Then, the perturbations are performed, fp,l+1

=

fp,l

+ δ

fp,l,and



δ

f1,l

. . . δ

fk,l



iscalculatedbythefollowing mini-mization[17,18]; min δfp:|δfp|≤1/2



ˆ

y

−

k



p=1

α

p,lg

(

fp,l

+ δ

fp

)





2 2

.

(8)

The solution to Eq. (7) is found using the well known least squaresmethod.Ontheother hand,Eq.(8) isa constrained non-linear optimization problem. A possible approach to solving this problemisto usea gradientdescent technique.Forthispurpose, the cost function in Eq. (8) can also be linearized around fp,l.

Further, the expression g(fp,l

+ δ

fp

)

can be approximated using

thefirst orderTaylorseries[17,18].Usingfirst orderTaylorseries makesitpossibletouselinearexpressionsandsolvetheunknown parameterswithagradientdescentapproach.

g

(

fp,l

+ δ

fp

)

≈

g

(

fp,l

)

+

∂

g

∂

fp,l

δ

fp

.

(9)

Using Eq. (9) and dropping constraints, expression (8) can be rewrittenas:



δ

f1,l

. . . δ

fk,l



=

arg min u JR

(

u

) ,

(10)

=

arg min u





rl

−

Blu





2 2

,

where rl

= ˆ

y

−

k



p=1

α

p,lg(fp,l

)

is theorthogonalresidual andBl

∈

C

N×k_{containsthepartialderivativesatthelinearizationpoint;}

Bl

=



α

1,l

∂

g

∂

f1,l

, ...,α

k,l

∂

g

∂

fk,l



.

(11)

Hereu

=

[δf1

, ..., δ

fk]T

∈ 

k×1containstheupdatesatthelth

iter-ation.When fp,l iscalculated,Bl isalsoupdated ateachiteration,

andthesearchisperformedinthedirectionofnegative gradient. Aftercalculationoftheperturbations,theparametersareupdated andtheupdatesarelimitedtosingleresolutioncellofthegrid.

Forthecostfunction inEq.(10),thenegative gradientof J at u

=

0 isgivenas

−∇

uJ

(u)

|

u=0

=

Re



2B_lHrl



,whichcanbeusedin thefollowingiterativesolutionoftheoptimizationprobleminthe Eq.(8):

α

l

=



g



f1,l

g



f2,l

...

g



fk,l



† y

,

fp,l+1

=

fp,l

+

μ

p,lRe



BH_l rl



,

(12)

where

μ

p,l istheproperly chosen stepsizeparameter. The value

ofthe stepsize canbe chosen asasufficiently smallratioofthe initial grid sizeofthe OMP structuregivenin Table1.Note that, settingatoosmallvaluewillincreasetheconvergencetime,anda toolargevaluewillincreasetheestimationerror.

In theproposed method, thereis no assumptionon the mea-surement sets. Amplitudes and the perturbation parameters are calculatedindependentlyforeachmeasurementyi.Thenusingthe

calculated parameters, the blind calibration procedure is applied jointlytoallmeasurements.

Table 2

ProposedSolverS(·).

Input:fi,1, ...,fi,k,yi,μ Initialize: l=0, fp,0=fp, p=1:k

UNTIL Stopping Condition is met,

Gl=  gfi,1,l gfi,2,l ...gfi,k,l  , αl=G†lyi, r=yi−Glαl Construct Blusing (11) FOR p=1:k fi,p,l+1=fi,p,l+μp,lRe  BH lrl fi,p,l+1within a grid cell?

δfi,p=fi,p,l+1−fi,p,0 END FOR

Output:α,δfi,1, ..., δfi,k

Intheprocessflowofthealgorithm giveninTable1,foreach measurement, a grid point is selected using OMP [23] and it is addedtothecluster Ti,k.Using Ti,k andyi,signalestimation and

blindcalibrationisperformed.Thenusingthecalibratedsignaland OMPselectedgridpoints,perturbationprocedure,giveninTable2, is applied. Finally, using perturbed parameters, blind calibration procedureis re-appliedfor furtherimprovement inestimationof thephaseandthegainerrors.

Theperformance of theiterative methodgivenin Table1 de-pendsonthechoiceofstoppingcriterion fortheiterations.When thestoppingcriterion isnotproperlychosen,itdegradesthe per-formanceof the algorithm.To avoidthisissue andachieve more clarified performance analysis, the source number, the sparsity levelofthesignals,isassumedtobeknown.Theiterationsare ter-minatedwhentheestimatedsignalsattainacertainlevelof spar-sity.Intheliterature,therearemethodsforestimationofnumber ofsources [24–26].One ofthesetechniquescan be applied prior totheiterationsoftheproposedtechnique.

Estimationandcorrectionofthecalibrationerrors

Inthissection,estimationofthegainandthephasecalibration errorsisexplained.Thephaseandthegainerrorestimationis per-formedminimizing the cost function JC

(

dn

,

φ

n

)

which is defined

asl2 norm of the error between the measurements y and their estimatey:

ˆ

JC

(

dn

, φ

n

)

=

arg min d,φ



y

(

n

)

−

dejφy

ˆ

(

n

)



2₂

.

(13) UsingthecalculatedparametersinEq.(4),estimatedmeasurement vectory

ˆ

iisconstructedasyi

=



K

k=1

α

i,kg(fio,k

).

Then usingthese

estimates,y(n

)

thesamplesofthenth sensorandthenth rowof they is

ˆ

constructedandisgivenasy(

ˆ

n

)

=



y

ˆ

1

(

n

),

y

ˆ

2

(

n

), . . . ,

y

ˆ

D

(

n

)



wherey

ˆ

i

(

n

)

isthenth elementofthey

ˆ

i.

Toestimate thephaseandthegain calibrationerrors,the par-tialderivativesofthecostfunctionin(13) withrespectto

θ

andd areformedandequatedtozeroindependentlyforeachparameter.

Thenthephaseerror

φ

n isestimatedas:

ˆφ

n

=

y

ˆ

(

n

)

Hy

(

n

).

(14)

UsingEq.(14),thegainerrorisestimatedas;

ˆ

dn

=

0

.

5



ejˆφy

(

n

)

Hy

ˆ

(

n

)

+

e−jˆφy

ˆ

(

n

)

Hy

(

n

)

ˆ

y

(

n

)

H_y

_ˆ

₍

_n

₎



.

(15)

Usingtheestimates in(14) and (15) thecalibrationis performed as:

yc

(

n

)

=

e−jˆφny

(

n

)/ˆ

dn

,

(16)

where yc is the calibrated measurement. That is, a part of our

optimization problemhas a closed form solution and is used as

such.

2

In following sections the proposed PC-OMP (Perturbed Cali-brationOrthogonalMatching Pursuit)techniqueis appliedtotwo differentproblems.In Section 4,theproposed PC-OMP technique without any modification is applied to the frequency estimation andjointblindcalibration,andsimulationsstudiesforthiscaseare alsogiven inthesame section. InSection 5,a variantofthe PC-OMPalgorithmPerturbedCalibrationOrthogonalMatchingPursuit Direction Finding (PC-OMP-DF) is introduced for the off-grid di-rectionfindingandjointblindcalibrationproblem,thesimulation studiesforthiscasearealsogiveninSection5.

4. Applicationtofrequencyestimation

Inthissection,simulationstudiesareperformedinorderto

an-alyze the performance of the proposed method PC-OMP for the

frequency estimation problem. In this case, xi contains the

fre-quencydomaincomponentsofthecorrespondingmeasurementyi.

A becomestheinverseDFTmatrixwithapropersize.Withoutloss ofgenerality



ischosen to be a complexrandom Gaussian ma-trixwhose elements are drawnfrom N

(0,

1),although anyother

proper



matrix can be used with the proposed PC-OMP

tech-nique.

In addition to the proposed method PC-OMP, a state of the arttechnique, ScalableComplete Calibration technique (CCAL) [9] is used in simulations forcomparison. Signal reconstruction and blindcalibrationperformanceisdependentonvariousparameters suchassensornumber,measurementnumber,SignaltoNoise Ra-tio(SNR),sparsitylevelandintensityofthecalibrationerrors.For the varying values of these parameters, different scenarios and measurement sets are generated synthetically, and Monte-Carlo runsareperformed.Resultsareanalyzedandcommentsaredrawn. In thesimulations, the proposed off-gridblind sensor calibra-tion algorithm PC-OMP givenin Table 1,the proposed algorithm without perturbation capability (Calibration Orthogonal Matching Pursuit, C-OMP) andthe Scalable Complete Calibration algorithm (CCAL) [9] are used forcomparison. In the simulations, Matlab™ codes forthe CCAL that are available atthe websitegiven in[9] areused.

Inthesimulations,20Monte-Carlorunsareperformedforeach case ofthe scenarios. Calibrated signal reconstruction aswell as thephaseandthegainestimationperformancesareanalyzed.Itis importanttonotethatthetruesparsesignalscontainsvaluesthat can be anywhereon a continuous parameterspacerather thana discretized grid. While compared techniquesreconstruct the sig-nal on a discrete grid, the output of the proposed algorithm is calibratedcomplexamplitudesandfrequencyestimatesofoff-grid signals. Hence classical mean-square-error (MSE) metric is not a properwayofevaluatingtheperformanceofanyalgorithmunder off-gridtargetscenarios.Instead,Earthmoverdistance(EMD)[27,

28],whichisalsousedtomeasuresparseoff-gridreconstructions invarious applications [29,18], is usedin quantitative analysisof signalreconstructions.Inaddition,MSEforthephaseandthegain estimationsare alsocalculated.The EMDmetricisdefinedas fol-lows:

•

Earth Movers Distance (EMD): P

= (

pi

,

ui

)

mi=1 and Q

=

(

qj

,

vj

)

nj=1 are the reconstructed andthe original signals re-spectively. pi and qj denote amplitudes, ui and vj denote

values of the ith and jth frequencies respectively. The EMD value is the minimum work that is requiredfor mass trans-portation needed for the reconstructed signal to match the originalsignal.TheEMDisdefinedas[27,28];

E M D

(

P

,

Q

)

=

min F=ˆfi,j 



i,j

ˆ

fi,jdi,j



i,j

ˆ

fi,j (17) with constraints



_j

ˆ

fi,j

≤

pi,



i

ˆ

fi,j

≤

qi,



i,j

ˆ

fi,j

=

min



_ipi

,



jqj



and

ˆ

fi,j

≥

0.di,j isthedifferencebetween

frequenciesi and j. F

=



ˆ

fi,j



denotessetofflows.Theflow

ˆ

fi,j denotes the masstransported from the ith frequencyto

the jth.EMDiscalculatedasdefinedin[28].

Since the y

=

dy can

ˆ

haveinfinitely many solutions, the gain errorestimates can be ambiguous especially forlow SNR values. When this issue is not addressed, the calculated metric will be misleading.Toovercomethisproblem,therealandtheestimated signal values are normalizedby its l2 norm before EMD calcula-tion.Forthesamereason,thesumofthegainerrorestimatesare equalizedtotherealgainerrorsumbeforeMSEcalculation.

Forthefrequency estimation,syntheticdata time domain sig-nals with off-grid frequency components are generated, then, phaseandgainerrorsareinjectedtothedataasexplainedlaterin detail.Simulationsareperformedusingthisdata.

4.1. ComparativestudyofvariousSNRlevels

Forcomparative analysis, simulations are performed with on-gridandoff-gridsignalsatdifferentSNRlevels.ForeachSNRlevel, syntheticdata forthe on-gridand theoff-grid signalsare gener-ated and average results of the Monte-Carlo runs are plotted. It mustbenotedthatCCALisimplementedusingthealternating di-rection method of multipliers (ADMM) [9], and its performance is dependent to the proper choice of the ADMM parameter

μ

. Althoughthe defaultvalue isset to

μ

=

50 in thecodes supple-mented in [9], in thereported simulations the best performance ofCCALisobtainedfor

μ

=

0.01.Monte-Carlorunsareperformed andresultsareplottedforPC-OMP,C-OMPandCCALfor

μ

=

0.01 and

μ

=

50 forcomparisons. Thescenario parameters areset as:

the number of sensors Ny

=

50, the number of measurements

M

=

50,thegridsize(i.e.thelengthofxi) Nx

=

50 and the

spar-sity level K

=

3. Each simulated signal has K frequency compo-nents. The gain errorsare drawn from N

(1,

0.05),and thephase errorsare drawn froma uniformdistribution between

[

0,0.02

π

]

for each sensor. As a benchmark EMD metric is also calculated fortheleastsquaresestimator(LSE) forthesignal reconstruction withoutperforminganyphaseorgainerrorcalibration.

Forthe on-gridsignal case, the resultsof thecompared tech-niques are giveninFig. 1.It can beseen that C-OMPhas higher performance. CCAL gives better resultsfor higher SNR levels, al-thoughCCALwith

μ

=

50 isveryclosetotheLSEvalues.PC-OMP

hasa performancebetweenCCALandC-OMP.TheC-OMPhas

ad-vantageofusingtheexactknowledge ofthe gridpoints resulting better performance than PC-OMP which wouldnot be applicable tooff-gridsignals.

Fortheoff-grid signal case, theresultsof thecompared tech-niquesaregiveninFig.2.CCALwith

μ

=

50 beingslightlybetter

than LSE, the performances of CCAL and C-OMP degrades

com-pared to the on grid case especially for the EMD metric. Mean-while,PC-OMP hassimilar performance compared tothe ongrid case.ThisresultsshowstheeffectivenessoftheproposedPC-OMP algorithmfortheoff-gridsignals.SinceC-OMPconsiders only on-gridparametersolution,itsperformancedecreasesforoff-grid sig-nals.Ontheotherhand,PC-OMPusesparameterperturbationsto estimatetheoff-gridsignalparameterswhichresultsinbetter per-formancethanC-OMP.Notethat,theperformanceoftheproposed PC-OMPdecreases belowSNR

= −

10 dBwhichcan beconsidered asaperformancebound.

In terms of the estimation of the phase and the gain errors, PC-OMPhasbetterresultsthanC-OMP,whilebothtechniqueshave higherperformancecomparedtoCCAL.

Runtime of the compared techniques are also calculated for computational complexity analysis. Simulation studies are per-formed on a PCwithIntel® Core™ i5-6200u processorand12GB of RAM with Matlab® 2016b. Although the runtime of an algo-rithm dependsontheimplementation,theyare calculatedonthe

average asfor C-OMP 0.028, PC-OMP 8.462 and CCAL 15.739 in

seconds.NotethatimplementationsofC-OMPandPC-OMPfollows very closelyto Table 1 and Table 2. Main algorithmic difference

between C-OMP andPC-OMP isthat PC-OMP uses perturbations

tosolveoff-gridsignalparameters.RuntimeofC-OMPisnegligible compared toPC-OMP, thus itcan be saidthat almost all compu-tational complexity of PC-OMP originates fromperturbation pro-cess.Notethat,computationalcomplexityoftheproposedPC-OMP method mainly dependson the sparsity of the signal. When the sparsitylevelincreases,theruntimeofPC-OMPdecreases.To clar-ify theperformance ofthe methods,algorithms are implemented inastraightforwardmanner,thusotherfactors(suchascalibration errors)donoteffectthecomputationalcomplexity.

4.2. Phasecalibrationerroranalysis

Anothersimulationstudyisperformedforthephasecalibration errorperformanceanalysis.Thesamescenarioparametersusedas theSNR analysiswithSNR

=

30 dB,arelativelyhighSNR valueis selected to focus on the phase calibration.The phase calibration error isdrawn fromauniformdistribution between

[

0,

φ

]

where

φ

isthecorrespondingscenarioparameter.

ResultsareprovidedinFig.3.PC-OMPandC-OMPperformance degrades as phase calibration errorincreases while CCAL results have small fluctuationsaround a mean. CCAL

μ

=

0.01 has bet-ter resultsthan CCAL

μ

=

50. Proposed PC-OMP hasthe highest performanceamongstcomparedtechniques.

Note that CCAL method includes a parameter

μ

considerably affecting theperformance withno particularmethod ofselecting it.

Since its performance decreasesafter 0.3

π

,thephase calibra-tion error below this level can be considered as save operating regimefortheproposedPC-OMPtechnique.

4.3. Gaincalibrationerroranalysis

Gain errorperformances ofthe compared techniques are also analyzedinthesimulationstudies.Thesamescenarioparameters usedastheSNR analysiswithSNR

=

30 dB.Monte-Carloruns de-pending on gain errorstandard deviationare performedfor two different cases with normal and log-normal distributed parame-ters. The standard deviationof the gain erroris varied, then the scenariosaregeneratedandresultsarecalculated.

The results are given in Fig. 4(a) and (b). For Fig. 4(a). The gain errors are drawn from N

(10,

dS T D

)

where dS T D is the gain

error standard deviation scenario parameter. Although, PC-OMP gives better results amongst the compared techniques,results of the compared techniques do not change considerably depending on thestandard deviationwithnormaldistribution.Thus, to em-phasizetherobustnessoftheproposedPC-OMPtechnique,another case of simulations are performed with log-normally distributed gain calibrationerror withmean 1and standard deviationbeing scenarioparameter.TheresultsaregiveninFig.4(b),although PC-OMPtechniqueperformsbetterthancomparedtechniquesits per-formance degradesforgain errorstandarddeviationvalueslarger than0.5.

Fig. 1. Monte-Carlo results for the on grid case with varying SNR=[−20,10] (dB). (a) EMD, (b) Gain Error Estimation MSE, (c) Phase Error Estimation MSE. 4.4.Effectsofnumberofmeasurements

Simulationscenariosareperformedtoanalyzetheeffectsofthe

number of the measurements M (i.e. the number of the

snap-shots) and the number of the sensors Ny. The same scenario

parameters used as the SNR analysis with SNR

=

30 dB.

Scenar-ios are generated with varying measurement and sensor

num-bers.Theresultsarecalculatedforcomparedtechniquesandgiven in Fig.5.

The results for different measurement numbers are given in Fig.5(a). Performances of the compared techniques increasesfor

highermeasurement numberlevels andbecomes convergentand

nearly constant after M

=

20. The results for the varying sensor numberare giveninFig.5(b).It mustbenotedthatfor Ny

=

10, Ny

/

Nx

=

0.2 andforNy

=

50,Ny

/

Nx

=

1.Resultsofthecompared

techniquesalso becomes betterfor thehigher valuesof the sen-sor number. For both cases, PC-OMP technique yields improved results.

4.5.Resolutionperformanceanalysis

Resolution performance of the compared techniques are also analyzed. The same scenario parameters used as the SNR

anal-ysis with SNR

=

30 dB. Two signal sources having similar ab-solute magnitudes with varying frequency difference levels are generated. The results are given in Fig. 6. PC-OMP performs

better amongst compared techniques. Although PC-OMP

perfor-mance increases slightly after the frequency difference level of 0.2, it increasesnotably after level of1 which is the initial grid cell size. It is important to emphasize that this results indi-cates the resolution performance of the proposed PC-OMP tech-nique, it means that initial grid cell size effects the PC-OMP re-sults.

4.6. Sparsitylevelanalysis

Simulationscenariosareperformedundervaryingsparsity lev-els(K/Nx)between[2%,10%].Thesamescenarioparametersused

astheSNRanalysiswithSNR

=

30 dB andNx

=

50.

TheresultsaregiveninFig.7.Itisseenthatperformanceofthe proposed PC-OMP technique decreases slowly andconvergestoa limitforincreasingsparsitylevels,andPC-OMPgivesbetterresults comparedtoothertechniques.

Notethat,underidealconditions,whenthesparsity(K )is con-stant,increasing Nx willresultalower K

/

Nx ratio,i.e.lower

Fig. 2. Monte-Carlo results for the off-grid case with varying SNR=[−20,10] (dB). (a) EMD, (b) Gain Error Estimation MSE, (c) Phase Error Estimation MSE.

Fig. 3. EMD results for the Phase Calibration Error between [0,0.5π].

4.7. Effectofthesparsityparameter

Inthissection,theeffectontheperformance oftheerroneous choiceofthesparsityparameterisinvestigated.Thesamescenario parameters used as theSNR analysiswithSNR

=

30 dB. PC-OMP andC-OMPare drivenon Monte-Carlorunswitherroneous spar-sityparameterandresultsaregathered.EMDforsignal reconstruc-tion andMSE forcalibrationerrorestimations are calculatedand plotted. The resultsare giveninFig.8.Note thatthe phaseerror estimationMSEisinradian2.

FortheactualsparsitylevelofK

=

3,theEMDresultsaregiven inFig.8(a).TheEMDvaluesdecreasesrapidlyforK

≤

3.ForK

=

1 and K

=

2,althoughPC-OMPhasslightlybetterperformance, the EMDresultsofC-OMPandPC-OMPareveryclose.Thedifferenceis atthehighestvalueforthetruesparsity levelinput atK

=

3.For the inputs higher than K

=

3 signal reconstruction performance decreasesasEMDincreases.WhilePC-OMPandC-OMPresultsare closefor K

≥

4,PC-OMPhasbetterperformance.

The calibration error performances are given inFig. 8(b). The phaseandgainerrorestimationMSEdecreasesrapidlywhenK

≤

3 for both techniques. Note that unlike the EMD, calibration error estimation MSE decreases for K

≥

4. For K

≥

4, the OMP selects noise-generated atoms which decreases EMD values while

cali-Fig. 4. EMD results for the Gain Calibration Error Standard Deviation for (a) Normal Distributed [0,1], (b) Log-Normal Distributed [0.01,3].

Fig. 5. EMD results for (a) Measurement number [5,50], (b) Sensor number [10,50].

Fig. 6. EMD results for the Frequency Difference levels between [0.2,2π].

bration performance is not affected. This shows that the signal reconstruction andtherefore the blind calibration are performed robustly even with theerroneous choice of the sparsity level. In addition, PC-OMP has better resultsat all the erroneoussparsity levels.

2

Inthissection,theperformanceoftheproposedtechnique PC-OMP is demonstrated through comprehensive simulationstudies, andits effectivenessis representedusing differentscenario cases amongst compared techniques. Inthe next section, the proposed techniqueisappliedtothedirectionfindingproblem,andits per-formanceisinvestigatedforthiscase.

5. Applicationtodirectionofarrivalestimation

There are differentstudies onblind sensor calibrationapplied to direction finding in the literature. In [30], a high resolution direction finding and blind calibration method is proposed. Iter-atively,thedirectionsareestimatedfirstusingEigenvalue decom-position,thentheblind calibrationisperformed.Mutualcoupling betweensensors is alsoestimatedaswell as thegain andphase errors[30].In[31],anon-iterativealgebraicsolutionisdeveloped

fortheblindsensorcalibrationandbeamforming.Inthismethod, thesensor locationsareassumedto beknown, andthegain and phasecalibrationerrorsofthesensorsareestimated.Similarly,the signal estimation andthe blind sensor calibration are performed separately in[32].In[33], ablind calibrationmethodusing inde-pendentcomponent analysisis studied which considers the gain andthephaseerrorsforalineararray.In[34],compressivesensing basedarray self-calibrationalgorithms are proposed for direction findinginwhichthesignalparametersandunknowncomplex sen-sorgainsareestimatedinanalternatingmanner.In[35],asparse DOAestimationapproach isproposed usingarrayswithunknown perturbationsonsensorlocations.DOAandperturbationvaluesare solvediteratively.In[36],asparsity basediterativemethodis de-velopedforthe off-gridDOA estimationwithco-prime arrays. In [37],a methoddirectionofarrivalestimationandgainandphase errorcorrectionisproposedbasedontheexploitingtheknowledge onthesignaleigenvectors.

Highresolutionandsubspacemethodsforblindsensor calibra-tionanddirectionfindingdonot exploitthecompressive sensing instruments, while CS based array calibration methods generally donotconsiderthe off-gridproblem.Toaddresstheseissues,we

Fig. 7. EMD results for different sparsity levels between [2%,10%].

proposeacompressivesensingbasedoff-griddirectionfindingand blindcalibrationmethod.

In thissection, a variant of the PC-OMP algorithm Perturbed

Calibration Orthogonal Matching Pursuit Direction Finding

(PC-OMP-DF) is introduced for the off-grid direction finding and jointblindcalibrationproblem.Themethodiscapableof estimat-ing the off-grid directionof the signals while jointlyperforming blind calibration to estimate andcorrect the gain and the phase errorsofthesensors.Inthenextsectionthealgorithmofthe pro-posedPC-OMP-DFisexplainedandsimulationanalysisisgiven.In thenextsubsection,thePC-OMP-DFtechniqueisexplainedandits variationsfromPC-OMPisgivenindetail.

5.1. Sparsitybasedoff-griddirectionfindingandblindcalibration DirectionfindingproblemissolvedusinganUniformLinear Ar-ray(ULA).Assumingthesourcesaresparseindirectiondimension

and using ULA, the equivalent measurement model matrix G in

Eq. (2) becomesthearraymanifoldmatrixandexpressedas[38]:

G

=



a

(θ

1

)

a

(θ

2

) ...

a

(θ

K

)



,

(18)

where a(θk

)

=

[1 exp(

−

ja

)

exp(

−

j2a

) ...

exp(

−

j

(

N

−

1)a

)]

T is

array response at angle

θ

k, and a

=

2

πd

/λ

sin(θk

).

a and d the

phasedifferenceanddistancebetweensuccessiveelementsinULA respectively;

λ

thewavelength.

The proposed PC-OMP algorithm can be applied to direction finding problemusingtheequivalent measurement modelmatrix in Eq.(18) without anyother variation. Onthe other hand,with a sufficientlysmalltotalmeasurement time,itis possibleto con-siderthatthedirectionsareconstantoversnapshots,sincetheydo notchangeconsiderably.Usingtheconstantdirectionsassumption, in thissection, theproposed PC-OMP algorithmisadopted to di-rectionfindingproblemwithsomevariations.Ifthedirectionsare expressedoveragridhavingacertainresolution(i.e.,1◦),the con-stant assumeddirectionsofdifferentsnapshotswillformagroup overthegrid.

In the proposed PC-OMP-DF technique, the atom selection in OMPisperformedjointlyforallexploitingthegrouppropertyand using the group [39] and the block [40] sparsity ideas, the fol-lowingprojection techniqueisproposed fortheatom selectionin OMP;

ˆ

p

=

max D



i=1





GHyi



 .

(19)

Fig. 9. Uncalibrated and calibrated MUSIC spectrums.

The same atom p which

ˆ

is the maximum of the projection in Eq. (19) is used for every snapshot. The perturbation process is performed similar to PC-OMP with small variations. After each perturbation,angularaverageoftheperturbeddirectionsare calcu-latedyieldingsingledirectionresultsforeachatomselectedusing Eq. (19). The resulting directions are registered to corresponding snapshots.Thisprocedureisrepeateduntilastoppingcriterion is met.TheblindcalibrationisappliedsimilartoPC-OMP.

5.2. Simulations

Simulations are performed to investigate the performance of theproposedPC-OMP-DFtechnique.AvariantofPC-OMP-DF with-out perturbation capability, Calibration OMP-DFmethod (C-OMP-DF)isusedinsimulations todemonstratetheeffectivenessofthe perturbationmoreclearly.

Syntheticdataisgeneratedusingauniformlineararray(ULA). Array beampattern, resolutionandaccuracy isdependenton the array size and the directionof the signals [38]. Toreduce these effects,aconstantnumberofsignalsK

=

3 isusedwithstationary directionsof

−

25◦,9.57◦and42.48◦whicharerandomlyselected withnosignificanteffectonperformance.

In simulations, several metrics are used to evaluate the per-formance oftheproposed techniques.The EarthMoversDistance

Fig. 11. Results for varying SNR (a) EMD, (b) Calibration Error Est. MSE (dB), (c) MUSIC Spectrum Max/Min Ratio, (d) MUSIC Spectrum Mean.

(EMD) betweentrueandestimateddirections are calculated.The MSEofthegainandthephaseerrorsarealsocalculated.

Inaddition to mentionedmetrics, the MUSICspectrum ofthe measuredandcalibratedsignalscanalsobeexploitedasa perfor-manceindicator.AsampleMUSICspectrumoftrueandcalibrated signalsisgiveninFig.9withSNR

=

20 dB andanULAwith40 ele-ments.Itisseenthatcalibratedspectrumsaremuchmoresharper thanthe uncalibrated spectrum.The calibrationincreasesthe co-herent signal processing gain which results insharper spectrum. Using this result, the following indicators are also calculated for theuncalibratedandthecalibratedspectrums;

•

Themeanvalueofthespectrum,

•

Theratioofthe maximumandminimum valuesofthe spec-trum.

5.2.1. SNRanalysis

Forthe DF performance analysis, first caseof simulations are performed for varying SNR levels. For each scenario parameter, 50 Monte-Carlo runs are evaluated andresults are plotted.

Sen-sor and snapshot numbers are chosen as 40, number of the

sources is K

=

3. The gain calibration errorof the sensors is

in-dependently drawn from N

(1,

0.1), and thephase calibration er-ror isdrawn independently froma uniformdistribution between

[

0,0.2

π

]

.

EMD direction finding (DF) results for varying SNR are given in Fig. 10(a). The performances of the compared techniques im-provesforhigherSNRvalues.ItisseenthatPC-OMP-DFtechnique hasbetterperformancecomparedtoC-OMP-DF.Thisresult demon-strates the effectiveness of the perturbation procedure. The gain andthe phase calibrationerror estimationperformance increases forhigherSNRlevels.Forthecalibrationerrorestimation, PC-OMP-DFalsoyieldsimprovedresults.

Theresultscalculatedusingtheuncalibratedandthecalibrated signals’ MUSIC spectrums are given in Fig. 10(c) and (d) respec-tively.ThemaximumandminimumvalueratiooftheMUSIC spec-trums are givenin Fig.10(c). Higher value of thisratioindicates better calibration. It is seen that the ratioswith calibrated

spec-trums using PC-OMP-DFandC-OMP-DF are higherthan those of

the uncalibratedspectrum.Thisresultshowstheimprovementof the calibration by proposed techniques. Similarly, the mean val-uesoftheuncalibratedandthecalibratedspectrums aregivenin Fig. 10(d). Smaller value indicates better calibration for this pa-rameter. It is seen that calibration results inbetter performance.

Fig. 12. EMD results for gain error standard deviation [0.1,0.3].

Bothparametershavebetterresultscomparedtotheuncalibrated casefor increasing SNR forboth in Fig. 10(c) and(d). For all of thecases,PC-OMP-DFhashigherperformance thanC-OMP-DF al-gorithm,andtheperformance differenceincreasesforhigherSNR. Thisisalsoanotherprooffortheeffectivenessoftheproposed

PC-OMP-DFmethod.

Runtime of the compared techniques are also calculated for computational complexity analysis. Simulation studies are per-formedonaPCwithIntel® _Core™_{i5-6200u processorand12 GB}

of RAM with Matlab® 2016b. In seconds, runtime of C-OMP-DF

is calculated as 0.028, and for PC-OMP-DF 2.375. Since runtime of C-OMP-DF is negligible, it can be said that almost all com-putationalcomplexityofPC-OMP-DForiginatesfromperturbation process.SimilartoPC-OMP, computationalcomplexity ofthe pro-posedPC-OMP-DFmethodmainlydependsonthesparsity ofthe signal.

5.2.2. Phasecalibrationerroranalysis

Inthissection,simulationresultsforthevaryinglevelofphase calibrationerroraregiven.Scenarioparametersarethesameasin Section5.2.1withSNR

=

20

(dB).

Phasecalibrationerrorisdrawn independentlyfroma uniform distributionbetween 0 and corre-spondingscenarioparameter.

The results for varying phase calibration error are given in Fig. 11. It is seen that for the phase calibration errors larger than 0.4

π

, the performances of the compared techniques de-grades considerably. Although results are very close, PC-OMP-DF givesbetterperformanceunderthephasecalibrationerrorlevelof 0.4

π

.

5.2.3. Gaincalibrationerroranalysis

Inthissection,simulation resultsforthevarying levelof gain calibrationerrorstandarddeviationaregiven.Scenarioparameters are the same as in Section 5.2.1 with SNR

=

20

(dB).

The gain errors are drawn from N

(1,

dS T D

)

where dS T D is the gain error

standarddeviationscenarioparameter.

Results are given in Fig. 12. Although the performances of the compared techniques does not change considerably for dif-ferent SNR levels, it can be seen that PC-OMP-DF gives better results.

6. Conclusions

Compressed Sensing (CS) based techniques representthe sig-nalon adiscretized grid whichresults in,dueto continuous

na-ture of the signals, the off-grid problem, which significantly af-fects the performance. Impropercalibration is also another issue which can cause performance degradation. In this work, a spar-sitybasedoff-gridblindcalibrationmethodisproposed(Perturbed CalibrationOrthogonalMatchingPursuit,PC-OMP)andwithsome variationsitisappliedtofrequencyestimationanddirection find-ing problems. The proposed techniques are capable of estimat-ing the off-grid signal parameters (such as frequency or direc-tion) and complex gains while performing the blind calibration to estimate and calibrate the gain and the phase errors jointly. The signal estimation and blind calibration procedures are per-formed alternately in an iterative manner. The methods use a gradient descent approach to solve the off-grid signal parame-ters. The simulation results show that the proposed techniques PC-OMPandPC-OMP-DFprovidesignificantperformance improve-ments.

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SedatCamlica wasbornin1982Izmir,Turkey.In 2006, he receivedthe B.Sc. degree in Telecommuni-cations Engineering from Istanbul Technical Univer-sity,Istanbul,Turkey.HereceivedtheM.Sc.degreein 2009 in Electrical and Electronics Engineering from theMiddleEast TechnicalUniversity,Ankara,Turkey. HereceivedthePh.D.degreein2017inElectricaland Electronics Engineering at TOBB University of Eco-nomics and Technology, Ankara, Turkey. Since 2006, heisserving atASELSANA.S.,Radar& ElectronicWarfareSystems Busi-ness Sector, Ankara, Turkey. His research interests includecompressive sensingapplications,arrayandradarsignalprocessing,targettracking, re-motesensingandimaging.

ImamSamilYetik received the B.S. degree from BogaziciUniversity,Istanbul,Turkey,in1998,theM.S. degreefromBilkentUniversityin2000,andthePh.D. degreeinelectricalengineeringfromtheUniversityof IllinoisatChicago,Chicago,IL,USA,in2004.Between 2005and2006,hewasapost-doctoralatthe Univer-sityofIllinois-ChicagoandUniversityofCaliforniaat Davis.HeservedasafacultymemberatIllinois Insti-tuteofTechnologybetween2006and2011.Currently, heisan AssociateProfessorwiththeDepartmentofElectricaland Elec-tronics EngineeringatTOBBETU.Hisresearchinterestsare intheareas ofsignal and imageprocessing withstatistical approaches andmachine learningtechniquesappliedtobiomedicineanddefensesystems.

OrhanArikan wasbornin1964inManisa,Turkey. In1986,hereceivedhisB.Sc.degreeinElectricaland ElectronicsEngineeringfromtheMiddleEast Techni-cal University, Ankara, Turkey. Hereceived bothhis

M.S. and Ph.D. degrees in Electrical and Computer

Engineering from the Universityof Illinois, Urbana-Champaign,in1988and1990,respectively.Following hisgraduatestudies,hewasemployedasaResearch ScientistatSchlumberger-DollResearchCenter, Ridge-field,CT.In1993hejoinedtheElectricalandElectronicsEngineering De-partment ofBilkent University,Ankara, Turkey.Since 2006, heis a full professor at Bilkent University.Since 2011, he hasbeen serving as the chairmanofdepartment.Hiscurrentresearchinterestsincludestatistical signalprocessing,time–frequencyanalysisandremotesensing.Dr.Arikan hasservedaschairmanofTurkeychapterofIEEE.