Last time: Completed solution to the optimum linear filter in real-time operation

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1 6.3 tochatic Etimatio ad Cotrol, Fall 4 ecture at time: Completed olutio to the oimum liear filter i real-time operatio emi-free cofiguratio: t D( p) F( p) i( p) dte dp e π F( ) F( ) ( ) F( p) ( p) pecial cae: ( p) ( p) i ratioal: I thi olutio formula we ca carry out the idicated itegratio i literal form i the cae i which ( p) i ratioal. I our work, we deal i a practical way oly with ratioal F, i, ad, o thi fuctio will e ratioal if D( p ) i ratioal. Thi will e true of every deired operatio exce a predictor. Thu exce i the cae of predictio, the aove fuctio which will e ymolized a [ ] ca e expaded ito ( p) ( p) + ( p) where [ ] ha pole oly i HP ad [ ] ha pole oly i HP. The zeroe may e aywhere. For ratioal [ ], thi expaio i made y expadig ito partial fractio, the addig together the term defg HP pole to form [ ] ad addig together the term defg HP pole to form [ ]. ctually, oly [ ] will e required. t t t dte dp{ ( p) + ( p) } e f() t e dt + f() t e dt where f() t ( p) e dp, t < f() t ( p) e dp, t > Note that f() t i the ivere traform of a fuctio which i aalytic i HP; thu f() t for t > ad Page of 7

2 6.3 tochatic Etimatio ad Cotrol, Fall 4 t f () te dt lo f() t i the ivere traform of a fuctio which i aalytic i HP; thu f () t for t <. Thu ( ) t t f () te dt f() te dt Thu fially, DF () ( ) i() F () () F () F( ) () I the uual cae, Fi () a tale, mmum phae fuctio. I that cae, F () F (), F () ; that i, all the pole ad zeroe of F () are i the HP. imilarly, F( ). The D () i() F () () Thu i thi cae the oimum trafer fuctio from iput to output i D () i() FH () () ad the oimum fuctio to e cacaded with the fixed part i otaied from thi y divo y F, () o that the fixed part i compeated out y cacellatio. Free cofiguratio prolem: D () i() Oimum free cofiguratio filter: Page of 7

3 6.3 tochatic Etimatio ad Cotrol, Fall 4 We tarted with a cloed-loop cofiguratio: C () H() + CFB ( ) ( ) ( ) H() C () H( ) F( ) B( ) The loop will e tale, ut C () may e utale. pecial commet aout the applicatio of thee formulae: a) Utale F () caot e treated ecaue the Fourier traform of wf () t doe ot coverge i that cae. To treat thi ytem, firt cloe a feedack loop aroud F () to create a tale fixed part ad work with thi tale feedack ytem a F. () Whe the oimum compeatio i foud, it ca e collected with the origial compeatio if deired. ) F () which ha pole o the ω axi i the limitig cae of fuctio for which the Fourier traform coverge. You ca move the pole ut ito the HP y addig a real part + ε to the pole locatio. olve the prolem with thi ε ad at the ed et it to zero. Zeroe of F () o ω axi ca e icluded i either factor ad the reult will e the ame. Thi will permit cacellatio compeatio of pole of F () o the ω axi, icludig pole at the origi. c) I factorig ito, ay cotat factor i ca e divided etwee ad i ay coveiet way. The ame i true of F () ad F( ). d) Prolem hould e well-poed i the firt place. void comiatio of D () ad ( ω ) which imply ifte dt () ecaue that may aume ifte e for ay realizale filter. uch a a differetiator o a igal which fall off a ω. e) The poit at t wa left hagig i everal tep of the derivatio of the olutio formula. Do t other checkig the idividual tep; ut check the fial olutio to ee if it atifie the eceary coditio. Page 3 of 7

4 6.3 tochatic Etimatio ad Cotrol, Fall 4 The Wieer-Hopf equatio require l( τ ) for τ. Thu () hould e aalytic i HP ad go to zero at leat a fat a for large. () F( ) H () F() () F( ) D() () i We have olved the prolem of the oimum liear filter uder the leat mea quared error criterio. Further aalyi how that if the iput, igal ad oie, are Gauia, the reult we have i the oimum filter. Thi i, there i o filter, liear or oliear which will yield maller mea quared error. If the iput are ot oth Gauia, it i almot ure that ome oliear filter ca do etter tha the Wieer filter. But theory for thi i oly egg to e developed o a approximate ai. Note that if we oly kow the ecod order tatitic of the iput, the oimum liear filter i the et we ca do. To take advatage of oliear filterig we mut kow the ditriutio of the iput. Example: Free cofiguratio predictor (real time) () a () The, are ucorrelated. D () e T Page 4 of 7

5 6.3 tochatic Etimatio ad Cotrol, Fall 4 Ue the olutio form t D( p) i( p) dte dp e π ( ) ( p) t i p p( t+ T ) ( ) dte dp e π ( ) ( p) () () + () + a + + a a a + a + a where a +. i( p) ( p) ( a+ )( a ) i( p) ( a ) ( p) ( a+ )( a )( ) ( a+ )( ) + a + a+ a+ Uig the itegral form, a( t T) ( T) e + a+, t T e + > dp a + p+ a, otherwie Page 5 of 7

6 6.3 tochatic Etimatio ad Cotrol, Fall 4 t ( T) ( T) e + a+, t T e + < dp a + p, otherwie t a( t+ T ) at ( + a) t e e dt e e dt a+ a+ at e a + + a at at ( + a) e e a+ ( + a) ( + ) ( a+ ) + where a +. Note the adwidth of thi filter ad the gai ~ etwee t () ad t ( + T) e at which i the correlatio Example: emi-free prolem with o-mmum phae F. Oimum compeator ( c ) ( + ) F () K d () a () The, are ucorrelated. ervo example where we d like the output to track the iput, o the deired operator, D (). Page 6 of 7

7 6.3 tochatic Etimatio ad Cotrol, Fall 4 a + a + a K F () ( + ε )( d + ) F () c c+ F( ) K ε d F( ) c+ ( )( ) ( ) DF () ( ) i() F () () ( ) ( ) ( )( )( )( ) ( ) ( )( )( ) c+ a c+ a+ a c a+ c Page 7 of 7