beamforming

SpatialSignalProcessing(Beamforming)WhatIsBeamforming?•Beamformingisspatialfiltering,ameansoftransmittingorreceivingsoundpreferentiallyinsomedirectionsoverothers.•Beamformingisexactlyanalogoustofrequencydomainanalysisoftimesignals.•Intime/frequencyfiltering,thefrequencycontentofatimesignalisrevealedbyitsFouriertransform.•Inbeamforming,theangular(directional)spectrumofasignalisrevealedbyFourieranalysisofthewaysoundexcitesdifferentpartsofthesetoftransducers.•Beamformingcanbeaccomplishedphysically(shapingandmovingatransducer),electrically(analogdelaycircuitry),ormathematically(digitalsignalprocessing).BeamformingRequirements•Directivity–Abeamformerisaspatialfilterandcanbeusedtoincreasethesignal-to-noiseratiobyblockingmostofthenoiseoutsidethedirectionsofinterest.•Sidelobecontrol–Nofilterisideal.Mustbalancemainlobedirectivityandsidelobelevels,whicharerelated.•Beamsteering–Abeamformercanbeelectronicallysteered,withsomedegradationinperformance.•Beamformerpatternfunctionisfrequencydependent:–Mainlobenarrowswithincreasingfrequency–Forbeamformersmadeofdiscretehydrophones,spatialaliasing(“gratinglobes”)canoccurwhenthethehydrophonesarespacedawavelengthorgreaterapart.ASimpleBeamformerαplanewavesignalwavefrontsdh1h20planewavehaswavelengthλ=c/f,wherefisthefrequencycisthespeedofsoundh1h1aretwoomnidirectionalhydrophonesAnalysisofSimpleBeamformer•GivenasignalincidentatthecenterCofthearray:•Thenthesignalsatthetwohydrophonesare:where•Thepatternfunctionofthedipoleisthenormalizedresponseofthedipoleasafunctionofangle:)t(ie)t(R)t(sω⋅=αλπφsind)(nn1−=⎟⎠⎞⎜⎝⎛=+=αλπαsindcossss)(b21)t(i)t(iiiee)t(R)t(sφω⋅=BeamPatternofSimpleBeamformer-150-100-50050100150-60-50-40-30-20-100α,degreesPatternLoss,dBPatternLossvs.AngleofIncidenceofPlaneWaveForTwoElementBeamformer,λ/2ElementSpacing0-10-20-30-40-50-180-165-150-135-120-105-90-75-60-45-30-150153045607590105120135150165PolarPlotofPatternLossFor2ElementBeamformerλ/2ElementSpacingBeamPatternofa10ElementArray-150-100-50050100150-60-50-40-30-20-100α,degreesPatternLoss,dBPatternLossvs.AngleofIncidenceofPlaneWaveForTenElementBeamformer,λ/2Spacing0-10-20-30-40-50-165-150-135-120-105-90-75-60-45-30-150153045607590105120135150165180PolarPlotofPatternLossFor10ElementBeamformerλ/2SpacingBeamforming–AmplitudeShading•Amplitudeshadingisappliedasabeamformingfunction.•Eachhydrophonesignalismultipliedbya“shadingweight”•Effectonbeampattern:–Usedtoreducesidelobes–ResultsinmainlobebroadeningBeamPatternofa10ElementDolph-ChebychevShadedArray-80-60-40-20020406080-60-50-40-30-20-100α,degreesPatternLoss,dBComparisonBeamPatternOfA10ElementDolph-ChebychevBeamformerWith-40dBSideLobesAndλ/2ElementSpacingWithAUniformlyWeighted10ElementBeamformerDolph-ChebychevBeamformerUniformBeamformerAnalogyBetweenSpatialFiltering(Beamforming)andTime-FrequencyProcessingGoalsofSpatialFiltering:1.IncreaseSNRforplanewavesignalsinambientoceannoise.2.Resolve(distinguishbetween)planewavesignalsarrivingfromdifferentdirections.3.Measurethedirectionfromwhichplanewavesignalsarearriving.GoalsofTime-FrequencyProcessing:1.IncreaseSNRfornarrowbandsignalsinbroadbandnoise.2.Resolvenarrowbandsignalsatdifferentfrequencies.3.Measurethefrequencyofnarrowbandsignals.ψ1ψ0SpatialangleψAmbientnoiseangulardensityPlanewaveatψ1Planewaveatψ0Narrowspatialfilteratψ0f1f0FrequencyBroadbandnoisespectrumSinewaveatf1Sinewaveatf0Narrowbandfilteratf0Time-FrequencyFilteringandBeamformingSNRCalculation:Time-FrequencyFilteringresponsepowerFilter(W/Hz)densityspectralpowerNoise(W/Hz)densityspectralpowerSignal202≡≡≡−)f(H)f(N)ff(δαDefineSignalPowerIs:(watts)202202)f(Hdf)f(H)ff(Psαδα=−=∫∞∞−SNRCalculation:Time-FrequencyFiltering(Cont’d)⎪⎩⎪⎨⎧≤−≤−=otherwise022102,ff,)f(HββIfweassumeIdealizedrectangularfilterwithbandwidthβThenthenoisepoweris:(watts)02βNdf)f(H)f(NPN=⋅=∫∞∞−N0isthenoiselevelinbandAndSNRis:βα02NPPSNRNs==SNRCalculation:SpatialFilteringresponsepowerangularfilterSpatialan)(W/steradidensityangularpowerNoisean)(W/steradidensityangularpowerSignal202≡≡≡−)(G)(N)(ψψψψδαDefineSignalPowerIs:(watts)2024202)(Gdf)(G)(Psψαψψψδαπ=−=∫SNRCalculation:Time-FrequencyFiltering(Cont’d)⎪⎩⎪⎨⎧≤−≤−=otherwise022102,,)(GββψψψψψIfweassumeIdealized“cookiecutter”beampatternwithwidthψβThenthenoisepoweris:(watts)42Kd)(G)(NPNβπψΩψψ=⋅=∫KisthenoiseintensityinbeamAndSNRis:KPPSNRNsβ2ψα==ArrayGainandDirectivityCalculationsOHArraySNRSNRGainArray=DefineThen∫∫∫=⋅−==πππΩΩαΩΩΩΩΩΩΩδα42424202d)(Nd)(G)(Nd)(G)(PPSNROHOHNsOHAssume•planewavesignal•arbitrarynoisedistributionΩΩallfor12=)(GFortheomnidirectionalhydrophone,ArrayGainandDirectivityCalculations(Cont’d)Forthearray,assumeitissteeredinthedirectionofΩ0andthat∫∫∫⋅=⋅−==πππΩΩΩαΩΩΩΩΩΩΩδα422424202d)(G)(Nd)(G)(Nd)(G)(PPSNRarrayarrayarrayNsarrayPuttingthesetogetheryields1Ω202=)(GThen∫∫⋅=ππΩΩΩΩΩ424d)(G)(Nd)(NAGarrayArrayGainandDirectivityCalculations(Cont’d)Ifthenoiseisisotropic(thesamefromeverydirection)ThentheArrayGain(AG)becomestheDirectivityIndex(DI),aperformanceindexForthearraythatisindependentofthenoisefield.K)(N=Ω∫=π42ΩΩπ4d)(GDIarrayArrayGainandDirectivityIndexareusuallyex