Chapter utorial: The Kalman Filter
Chapter11 lter[1]haslongbeenregardedastheoptimalso lutiontomanytrackinganddatapredictiontas ks,[2]. lterisconstructedasameansquarederrormini miser,butanalternativederivationofthe lterisalsoprovidedshowinghowthe lteringistoextracttherequiredinformation fromasignal, nethegoalofthe ;yk=akxk+nk( )where;ykisthetimedependentobservedsigna l,akisagainterm, erencebetweentheestimateof^xkandxkitself istermedtheerror;f(ek)=f(xk ^xk)( )Theparticularshapeoff(ek)isdependentupo ntheapplication,howeveritisclearthatthef unctionshouldbebothpositiveandincreasemo notonically[3].Anerrorfunctionwhichexhib itsthesecharac-teristicsisthesquarederro rfunction;f(ek)=(xk ^xk)2( )133Sinceitisnecessarytoconsidertheabili tyofthe ltertopredictmanydataoveraperiodoftimeam oremeaningfulmetricistheexpectedvalueoft heerrorfunction;lossfunction=E(f(ek))( )Thisresultsinthemeansquarederror(MSE)fu nction; (t)=E e2k ( ) , ningthegoalofthe lterto ndingthe^ ;max[P(yj^x)]( )AssumingthattheadditiverandomnoiseisGau ssiandistributedwithastandarddeviationof kgives;P(ykj^xk)=Kkexp (yk ak^xk)22 2k ( ) ;P(yj^x)=YkKkexp (yk ak^xk)22 2k ( )Whichleadsto;logP(yj^x)= 12Xk (yk ak^xk)2 2k +co
y that his solution uses b oth the auto correlation and the cross correlation of the receiv ed signal with the original data, in order to deriv e an impulse resp onse for the lter. Kalman also presen ted a prescription of the optimal MSE lter. Ho w ev er Kalman's has some adv an tages o v er W einer's; it sidesteps the need to determine impulse ...
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