ApproachesforAngleofArrivalEstimation

WenguangMao

AngleofArrival(AoA)

• Definition:theelevationandazimuthangleofincomingsignals• Alsocalleddirectionofarrival(DoA)

AoA Estimation• Applications:localization,tracking,gesturerecognition,……• Requirements:antennaarray• Approaches:• Generateapowerprofileovervariousincomingangles• DetermineallAoA 𝜃"

𝜽𝟏 𝜽𝟐

RelatedConcepts

• Syntheticapertureradar(SAR)• Usingamovingantennatoemulateanarray• Alternativewayofusingphysicalantennaarray• NOT anestimationapproachinthecontextofAoA• MostAoA estimationmethodscanbeappliedtobothphysicalantennaarrayandSAR

• Inthispresentation,weonlyfocusonantennaarray• MayrequiresomemodificationwhenappliedtoSAR

RelatedConcepts

• Beamforming• AclassofAoA estimationapproaches

• MUSIC• Aspecificalgorithminsubspace-basedapproaches

AoA Estimationapproaches

BeamformingSubspacemethods

MUSIC

ApproachesforAoA Estimation

• Naïveapproach• Beamformingapproaches• Bartlettmethod• MVDR• Linearprediction

• Subspacebasedapproaches• MUSIC anditsvariants• ESPIRIT

• Maximumlikelihoodestimator• ……

KeyInsights

• Phasechangesoverantennasaredeterminedbytheincomingangle• Far-fieldassumption• Phaseoftheantenna1:𝜙'• Phaseoftheantenna2:𝜙'• Thenthedifferenceisgivenby

𝝓𝟐 − 𝝓𝟏 = 𝟐𝝅𝒅𝒄𝒐𝒔𝜽𝟏

𝝀 + 𝟐𝐤𝝅

Naïveapproach

• DetermineAoA basedonthephasedifferenceoftwoantenna

• Problems:• Worksforonlyoneincomingsignals• Phasemeasurementcouldbenoisy• Ambiguity

• AdoptedandimprovedbyRF-IDraw

𝒄𝒐𝒔𝜽𝟏 = (𝚫𝝓𝟐𝝅 − 𝒌)

𝝀𝒅

UsingAntennaArray

• Receivedsignalsat𝑚-th antenna:

𝒙𝒎 𝒕 = < 𝒔𝒏(𝒕)𝑵

𝒏?𝟏

𝒆𝒋⋅𝟐𝝅⋅𝝉𝒏⋅(𝒎D𝟏) + 𝒏𝒎(𝒕)

𝑠F(𝑡) :n-th sourcesignals

𝜏F =IJKLMN

O:phaseshiftperantenna

𝑁 :thenumberofsources

𝑀 :thenumberofantennas 𝑛S(𝑡) :noiseterms

UsingAntennaArray

• Matrixform:

𝒙𝟏 𝒕 𝑻

𝒙𝟐 𝒕 𝑻

⋮𝒙𝑴 𝒕 𝑻

= 𝒂(𝜽𝟏) 𝒂(𝜽𝟐) ⋯ 𝒂(𝜽𝑵)

𝒔𝟏 𝒕 𝑻

𝒔𝟐 𝒕 𝑻

⋮𝒔𝑵 𝒕 𝑻

+

𝒏𝟏 𝒕 𝑻

𝒏𝟐 𝒕 𝑻

⋮𝒏𝑴 𝒕 𝑻

𝑿 = 𝑨𝑺 + 𝑵

Steeringvector:𝒂 𝜽 = 𝟏𝒆𝒋𝟐𝝅𝝉 𝜽 𝒆𝒋𝟐𝝅𝝉 𝜽 ⋅𝟐 …𝒆𝒋𝟐𝝅𝝉 𝜽 𝑴D𝟏 𝑻

BeamformingattheReceiver

• Definition:amethodtocreatecertainradiationpattern bycombiningsignalsfromdifferentantennaswithdifferentweights.• Willmagnifythesignalsfromcertaindirectionwhilesuppressingthosefromotherdirections

BeamformingattheReceiver

• Signalsafterbeamformingusingaweightvector𝒘

• Byselectingdifferent𝑤,thereceivedsignal𝑌 willcontainthesignalsourcesarrivedfromdifferentdirection.

• Beamformingtechniquesarewidelyusedinwirelesscommunications

𝒀 = 𝒘𝑯𝑿

BeamformingattheReceiver

• Adjusttheweightvectortorotatetheradiationpatterntoangle𝜃• Measurethereceivedsignalstrength𝑃(𝜃)• Repeatthisprocessforany𝜃 in[0,pi]• Plot(𝜃, 𝑃(𝜃))• Peaksintheplotindicatestheangleofarrival

𝜽𝟏 𝜽𝟐

BartlettBeamforming

• Alsocalled:correlationbeamforming,conventionalbeamforming,delay-and-sumbeamforming,orFourierbeamforming• Keyidea:magnifythesignalsfromcertaindirectionbycompensatingthephaseshift• Consideronesourcesignal𝑠 𝑡 arrivedatangle𝜃d• Signalat𝑚-th antenna:xf t = 𝑠(𝑡) ⋅ 𝑒i⋅jk⋅l(Mm)(SD')

• Weightat𝑚-th antenna:wf =𝑒i⋅jk⋅l(M)(SD')

• Onlywhen𝜃 = 𝜃d,thereceivedsignalY = wpX = ∑𝑤S∗ 𝑥S 𝑡 u ismaximized

Phaseshift

BartlettBeamforming

• Weightvectorforbeamformingangle𝜃:

• Signalpoweratangle𝜃:

• UsedbyUbicarse withSAR

𝒘 = 𝒂(𝜽)

𝑷 𝜽 = 𝒀𝒀𝑯 = 𝒘𝑯𝑿 𝒘𝑯𝑿 𝑯 = 𝒘𝑯𝑿𝑿𝑯𝒘 = 𝒘𝑯𝑹𝑿𝑿𝒘 = 𝒂𝑯 𝜽 𝑹𝑿𝑿𝒂(𝜽)

Thisiswhyitiscalledsteeringvector

Covariancematrix

BartlettBeamforming

• Workswellwhenthereisonlyonesourcesignal

• Sufferswhentherearemultiplesources:verylowresolution

MinimumVarianceDistortionless Response(MVDR)

• AlsocalledCapon’sbeamforming• Keyidea:maintainthesignalfromthedesireddirectionwhileminimizingthesignalsfromotherdirection• Mathematically,wewanttofindsuchweightvector𝒘 forthebeamingangle𝜃

𝐦𝐢𝐧 𝒀𝒀𝑯 = 𝐦𝐢𝐧 𝒘𝑯𝑹𝑿𝑿𝒘

s.t.𝒘𝑯(𝒂 𝜽 𝒔 𝒕 𝑻) = 𝒔 𝒕 𝑻

Maintainthesignalsfromangle𝜽

𝒘𝑯𝒂 𝜽 = 𝟏

MVDR

• Weightvectorforbeamformingangle𝜃:

• Signalpoweratangle𝜃:

𝑷 𝜽 = 𝒀𝒀𝑯 = 𝒘𝑯𝑹𝑿𝑿𝒘 =𝟏

𝒂 𝜽 𝑹𝑿𝑿D𝟏𝒂𝑯(𝜽)

𝒘 =𝑹𝑿𝑿D𝟏𝒂𝑯(𝜽)

𝒂 𝜽 𝑹𝑿𝑿D𝟏𝒂𝑯(𝜽)

MVDR

• ResolutionissignificantlyenhancedcomparedtoBartlettmethod• Butstillnotgoodenough• Betterbeamformingapproachesaredeveloped,e.g.,LinearPrediction• Orresorttosubspacebasedapproaches

SubspaceBasedApproaches

• Beamformingisawayofshapingreceivedsignals• CanbeusedforestimatingAoA• Canalsobeusedfordirectionalcommunications

• Subspacebasedapproachesarespeciallydesignedforparameter(i.e.,AoA)estimationusingreceivedsignals• Cannotbeusedforextractingsignalsarrivedfromcertaindirection

• Subspacebasedapproachesdecompose thereceivedsignalsinto“signalsubspace”and“noisesubspace”• LeveragespecialpropertiesofthesesubspacesforestimatingAoA

MultipleSignalClassification(MUSIC)

• Keyideas:wewanttofindavector𝑞 andavectorfunction𝑓(𝜃)• Suchthat𝒒𝑯𝒇 𝜽 = 𝟎 ifandonlyif𝜃 = 𝜃" (i.e.,oneofAoA)• Thenwecanplot𝒑 𝜽 = '

��� M � ='

�� M ����(M)

• ThepeaksintheplotindicatesAoA• Wecanexpectverysharppeak since𝑞�𝑓 𝜃 = 0,sotheinverseofitsmagnitudeisinfinity

Howtofind𝒒 and𝒇(𝜽)

MultipleSignalClassification(MUSIC)

• MUSICgivesawaytofindapairof𝑞 and𝑓(𝜃)• Thesignalsfromantennaarray

• Covariancematrixofthesignals𝑿 = 𝑨𝑺 + 𝑵

𝑹𝑿𝑿 = 𝑬[𝑿𝑿𝑯] = 𝑬[𝑨𝑺𝑺𝑯𝑨𝑯] + 𝑬[𝑵𝑵𝑯]

𝑹𝑿𝑿 = 𝑨𝑬[𝑺𝑺𝑯]𝑨𝑯 + 𝝈𝟐𝑰

𝑹𝑿𝑿 = 𝑨𝑹𝑺𝑺𝑨𝑯 + 𝝈𝟐𝑰

SignaltermsNoiseterms

MUSIC• Considerthesignalterm• 𝑅�� is𝑁×𝑁 matrix,where𝑁 isthenumberofsourcesignals• 𝑅LL hastherankequalto𝑁 ifsourcesignalsareindependent

• 𝐴 is𝑀×𝑁 matrix,where𝑀 isthenumberofantenna• 𝐴 hasfullcolumnrank• Thesignaltermis𝑀×𝑀 matrix,anditsrankis𝑁• Thesignaltermhas𝑁 positiveeigenvaluesand𝑀 −𝑁 zeroeigenvalues,ifM>N• Thereare𝑀 −𝑁 eigenvectors𝑞" suchthat𝐴𝑅��𝐴�𝑞" = 0• Then𝐴�𝑞" = 0,where𝐴 = [𝑎(𝜃') 𝑎(𝜃j) ⋯ 𝑎(𝜃�)]• Then𝑞"�𝑎 𝜃 = 0 if𝜃 = 𝜃" Whatwewant!!!

MUSIC

• 𝑎(𝜃) isthesteeringfunction,soitisknown• Needstodetermine𝑞",whichneedstheeigenvaluedecompositionofthesignalterm.• Wedon’tknowthesignalterm;weonlyknowthesumofthesignaltermandthenoiseterm,i.e.,𝑅��• Allofeigenvectorsofthesignaltermarealsoonesfor𝑅��,andcorrespondingeigenvaluesareaddedby𝜎j

• Onlyneedtofindtheeigenvectorsof𝑅�� witheigenvaluesequalto𝜎j

MUSIC

• Derive𝑅��• Performeigenvaluedecompositionon𝑅��• Sorteigenvectorsaccordingtotheireigenvaluesindescentorder• Selectlast𝑀 −𝑁 eigenvectors𝑞"• Noisespacematrix 𝑄� = [𝑞��'𝑞��j …𝑞�]• 𝑄��𝑎 𝜃 = 0 foranyAoA 𝜃"• Plot𝑝 𝜃 = '

�� M ������(M)

andfindthepeaks

PerformanceComparison

(a)10antennas (a)50antennas

PerformanceComparison

(a)SNR1dB (b)SNR20dBBeamformingapproaches

Musicvariants