approaches for angle of arrival estimationswadhin/reading_group/slides/aoa.pdf · angle of arrival...
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ApproachesforAngleofArrivalEstimation
WenguangMao
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AngleofArrival(AoA)
• Definition:theelevationandazimuthangleofincomingsignals• Alsocalleddirectionofarrival(DoA)
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AoA Estimation• Applications:localization,tracking,gesturerecognition,……• Requirements:antennaarray• Approaches:• Generateapowerprofileovervariousincomingangles• DetermineallAoA 𝜃"
𝜽𝟏 𝜽𝟐
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RelatedConcepts
• Syntheticapertureradar(SAR)• Usingamovingantennatoemulateanarray• Alternativewayofusingphysicalantennaarray• NOT anestimationapproachinthecontextofAoA• MostAoA estimationmethodscanbeappliedtobothphysicalantennaarrayandSAR
• Inthispresentation,weonlyfocusonantennaarray• MayrequiresomemodificationwhenappliedtoSAR
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RelatedConcepts
• Beamforming• AclassofAoA estimationapproaches
• MUSIC• Aspecificalgorithminsubspace-basedapproaches
AoA Estimationapproaches
BeamformingSubspacemethods
MUSIC
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ApproachesforAoA Estimation
• Naïveapproach• Beamformingapproaches• Bartlettmethod• MVDR• Linearprediction
• Subspacebasedapproaches• MUSIC anditsvariants• ESPIRIT
• Maximumlikelihoodestimator• ……
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KeyInsights
• Phasechangesoverantennasaredeterminedbytheincomingangle• Far-fieldassumption• Phaseoftheantenna1:𝜙'• Phaseoftheantenna2:𝜙'• Thenthedifferenceisgivenby
𝝓𝟐 − 𝝓𝟏 = 𝟐𝝅𝒅𝒄𝒐𝒔𝜽𝟏
𝝀 + 𝟐𝐤𝝅
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Naïveapproach
• DetermineAoA basedonthephasedifferenceoftwoantenna
• Problems:• Worksforonlyoneincomingsignals• Phasemeasurementcouldbenoisy• Ambiguity
• AdoptedandimprovedbyRF-IDraw
𝒄𝒐𝒔𝜽𝟏 = (𝚫𝝓𝟐𝝅 − 𝒌)
𝝀𝒅
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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
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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
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BeamformingattheReceiver
• Definition:amethodtocreatecertainradiationpattern bycombiningsignalsfromdifferentantennaswithdifferentweights.• Willmagnifythesignalsfromcertaindirectionwhilesuppressingthosefromotherdirections
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BeamformingattheReceiver
• Signalsafterbeamformingusingaweightvector𝒘
• Byselectingdifferent𝑤,thereceivedsignal𝑌 willcontainthesignalsourcesarrivedfromdifferentdirection.
• Beamformingtechniquesarewidelyusedinwirelesscommunications
𝒀 = 𝒘𝑯𝑿
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BeamformingattheReceiver
• Adjusttheweightvectortorotatetheradiationpatterntoangle𝜃• Measurethereceivedsignalstrength𝑃(𝜃)• Repeatthisprocessforany𝜃 in[0,pi]• Plot(𝜃, 𝑃(𝜃))• Peaksintheplotindicatestheangleofarrival
𝜽𝟏 𝜽𝟐
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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
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BartlettBeamforming
• Weightvectorforbeamformingangle𝜃:
• Signalpoweratangle𝜃:
• UsedbyUbicarse withSAR
𝒘 = 𝒂(𝜽)
𝑷 𝜽 = 𝒀𝒀𝑯 = 𝒘𝑯𝑿 𝒘𝑯𝑿 𝑯 = 𝒘𝑯𝑿𝑿𝑯𝒘 = 𝒘𝑯𝑹𝑿𝑿𝒘 = 𝒂𝑯 𝜽 𝑹𝑿𝑿𝒂(𝜽)
Thisiswhyitiscalledsteeringvector
Covariancematrix
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BartlettBeamforming
• Workswellwhenthereisonlyonesourcesignal
• Sufferswhentherearemultiplesources:verylowresolution
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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
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MVDR
• Weightvectorforbeamformingangle𝜃:
• Signalpoweratangle𝜃:
𝑷 𝜽 = 𝒀𝒀𝑯 = 𝒘𝑯𝑹𝑿𝑿𝒘 =𝟏
𝒂 𝜽 𝑹𝑿𝑿D𝟏𝒂𝑯(𝜽)
𝒘 =𝑹𝑿𝑿D𝟏𝒂𝑯(𝜽)
𝒂 𝜽 𝑹𝑿𝑿D𝟏𝒂𝑯(𝜽)
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MVDR
• ResolutionissignificantlyenhancedcomparedtoBartlettmethod• Butstillnotgoodenough• Betterbeamformingapproachesaredeveloped,e.g.,LinearPrediction• Orresorttosubspacebasedapproaches
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SubspaceBasedApproaches
• Beamformingisawayofshapingreceivedsignals• CanbeusedforestimatingAoA• Canalsobeusedfordirectionalcommunications
• Subspacebasedapproachesarespeciallydesignedforparameter(i.e.,AoA)estimationusingreceivedsignals• Cannotbeusedforextractingsignalsarrivedfromcertaindirection
• Subspacebasedapproachesdecompose thereceivedsignalsinto“signalsubspace”and“noisesubspace”• LeveragespecialpropertiesofthesesubspacesforestimatingAoA
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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)
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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
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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!!!
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MUSIC
• 𝑎(𝜃) isthesteeringfunction,soitisknown• Needstodetermine𝑞",whichneedstheeigenvaluedecompositionofthesignalterm.• Wedon’tknowthesignalterm;weonlyknowthesumofthesignaltermandthenoiseterm,i.e.,𝑅��• Allofeigenvectorsofthesignaltermarealsoonesfor𝑅��,andcorrespondingeigenvaluesareaddedby𝜎j
• Onlyneedtofindtheeigenvectorsof𝑅�� witheigenvaluesequalto𝜎j
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MUSIC
• Derive𝑅��• Performeigenvaluedecompositionon𝑅��• Sorteigenvectorsaccordingtotheireigenvaluesindescentorder• Selectlast𝑀 −𝑁 eigenvectors𝑞"• Noisespacematrix 𝑄� = [𝑞��'𝑞��j …𝑞�]• 𝑄��𝑎 𝜃 = 0 foranyAoA 𝜃"• Plot𝑝 𝜃 = '
�� M ������(M)
andfindthepeaks
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PerformanceComparison
(a)10antennas (a)50antennas
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PerformanceComparison
(a)SNR1dB (b)SNR20dBBeamformingapproaches
Musicvariants