Estimation Theory. Chapter 4

Transcription

1 Estmato ory aptr 4

2 LIEAR MOELS W - I matrx form Estmat slop B ad trcpt A,,.. - WG W B A l fttg Rcall W W W B A

3 W ~ calld vctor I gral, ormal or Gaussa ata obsrvato paramtr Ma, ovarac KOW p matrx to b stmatd, trmd t Lar Modl.It s mportat bcaus w ca asly fd t MVU stmator ts cas I W p ~ I p

4 for A symmtrc sow ts b But l,, ~ ow, π A A b p I p I

5 I MVU stmator ad ffct s By RLB orm! lp p colum ra full as b ff s vrtblwll ow assum lp I g

6 a ow vrfy l fttg rsults gv prvously attmpt ts at om Asd: For to b full ra, ts p colums must b larly dpdt. W ow tat ts rows ca vr b larly dpdt for >p ot: If s squar ad vrtbl p ad

7 as W s ot Full Ra B A B A B A w caot dtfy paramtrs Ev wt o os, ot xst dos colums larly dpdt Exampl:

8 If I practc, f ll - codtod or s ra dfct aot stmat rlably s B All produc A sam obsrvatos

9 Exampls: Fourr Aalyss Suspct data cossts of M π a cos frqucs ar f strog cyclcal compots os M b s π W,...- /, armocally rlatd Ampltuds a, b to b stmatd a a b b M M As a lar modl w av

10 p p M M π s M π s M π cos π cos M π s π s M π cos π cos < OLUMS lt fd o M M M

11 frqucs du to coc of j, for all f s f cos f s f s f cos f cos Sc j colums ar ortogoal But j j j j M M M j j M M M π π δ π π δ π π : / : j j

12 dscrt Fourr trasform coffcts Usual s cos I M M b a or π π

13 o Fd PF of MVU Estmator UBIASE I WW E E E E E E

14 o Fd PF of MVU Estmator Idpdt, ~ b, ~ â or, ~ For Fourr aalyss, ~ ~ Gaussa sc t s lar fucto of b a I

15 Systm Idtfcato Exampl Smpl modl for lar systm s appd lay L or FIR fltr z z Accoutg for os at output p z U z W W provd U ad masur. Ws to stmat tap wgts

16 Systm Idtfcato Exampl If u s provdd for,.,- Ad u <, w obsrv W p p u u u u u u W u p OR,,... -

17 If Ws WG wt varac, W ad w av lar modl Systm Idtfcato Exampl ot tat ow wll dpds o quvaltly u w ca stmat t tap wgts or quvaltly or ;.. s ~, I MVU stmator ad ffct, j j r uu s auto-corrlato matrx of put squc cos trag squc

18 Problm :oos u to mmz Var,,...p subjct to costrat Soluto : proof comg soo oos uso tat r uu δ - u s fxdwy? u u OR u sould b wtos Actually w us psudo radom os mpuls-l auto-corrlato asy to grat usg sft rgstrs

19 Proof tat optmum ^ Proof sould b dagoal Lt Lt ^ var ξ ^ wr s a postv dft matrx - - sc & - ξ ^ I rfor, ξ ξ But ξ ξ ξ by usg aucy - Scwarz ξ ξ ξ - qualty

20 Proof of Optmum ovarac Matrx cot. c c wc mmzsvarac ff qualty olds var -, - ^ - - ^ ^ ^ ^ ξ ξ

21 Proof of Optmum ovarac Matrx cot. must b qual. all Sc trag squc s assumd statoary p,,, for p,,, for dagoal p c c c c c c

22 Autocorrlato rosscorrlato u r u r uu uu,,..p - : r r ĥ r If Or ĥ w av Udr ts codtos uu ux ux I

23 Also, Varĥ r ε uu : ε trag squc rgy

24 Equalzato Exampl apply Ŝ to a slcr w M - PAM, costllato. g. ft sgal to a S blogs ot : If S S S Z x dtcto problm! symbols t s actually a w stmat t formato stmatg t cal, Aftr symbols guard as zros L assumg - L S os x x x S S L L L L l l l suboptmal

25 Extsos o Lar Modl A mor gral modl assums W, I o fd MVU stmator l p Rpat drvato Wtg approac By wtg approac, Factor - as s - vrtbl matrx Always possbl for postv dft usg olsy dcomposto for xampl

26 , ', ' ' ' ' ' ' t lar modl ts s : W wtg matrx sc f s a I I W W W Lt I WW E E W W W raw a bloc dagram!

27 colord os vromt paramtrs sgal a ow adl stmato of Also, us, ' ' ' ' ' '

28 Exampl  or sc... wr  A s ad ffct of MVU stmator matrx. covarac s colord Gaussa os wt,,..., : W A W W A

29 Exampl ot d d Lt  wtr Rcall du to os corrlato w logr av  ow, Var stmator covarac matrx s -

30 Exampl ot d A d A If os as wll as W frst wt ad t "avrag"wtg cags sgal ' ' umrcal Exampl :

31 umrcal Exampl ot d  ubasd mas ' '  d

32 Summary If W ~ W, wr s p ow,, ad s to b stmatd, t s t MVU stmator ad also ffct ad as covarac s s t Gral Lar Modl - -

33 Exrcs 4.3:SK P. E E E - W - E W -. E W ow, & W ar dpdt of ad E W φ ac otr E

34 ^ E - - E E ;W WW - - E W E W WW - - E W - - ^ E E - - W If ad W wr ot dpdt of b basd ac otr, may

35 Exrcs 4.4 : wt probablty -ε M -M wt probablty ε - x ; o fad - A M- x ; fad M Assumg tat fad locatos ar ow xactly

36 var  E - -ε M M ε ε > Sam as cas of o fadg oly f M or ε. Otrws, varac s crasd. Morovr, t stmator varac dos' t go to zro as bcoms ft wy?