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Estim, Filter&Detect

by: Ophelia Ritchie

Estim, Filter&Detect EECS 564

Ophelia Ritchie
GPA 3.8


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This 5 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 564 at University of Michigan taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/231538/eecs-564-university-of-michigan in Engineering Computer Science at University of Michigan.

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Date Created: 10/29/15
EECS 564 ESTIMATION IN WGN Winter 1999 GIVEN GOAD Observe Mt sta wt 0 g t g T 8t a is a known function of unknown parameter a wt WGN White Gaussian Noise 0 mean Sww 02 Compute MAPRt given a priom39 pdf paA K L praltRlA Let q i be any complete orthonormal basis 7quot fOT rtq itdt Sim fOT 8t owlHalt Project r siawiil2 7quot7quot17quot2 H 02 e Ri SiA2202 gt HeQRiSiA S A202 neglecting terms independent of A don t affect argmaX Using Parseval s theorem f xtytdt Z Luzy and logparltAR logpraltRA 10gpaltA 10gp7 ltR gt argmax A 2 fOT RtstAdt fOT sta2dtgt 2o 2 logpaA arger naX ZfT Rlttgt8lttAdtfTSlttya2dt l o o l Mt astwt a N N0 03 linear Gaussian prior fOT Rtstdt fOT 32tdt A2 203 an 2 fOT RtAstdt fOT A282tdt Matched lter again this time for estimation an 2 fOT RtAstdt fOT A232tdtgt 2o 2 gt amp fOT Rtstdt 0T 32tdt Gaussian gt MAP ef cient gt MAP LS EM Z EEabove expression E f rtstdt Ea afa82t wtstdt 03 f82tdt at EU fa82t wtsta32u wusudt du aid 32 tdt2 02 f82tdt Plug these in above 11m 39 A i ag ooaMAp aMLEnoaprior1 ag oaMAp i 0 Why EECS 564 LINEAR LEAST SQUARES ESTIM Winter 1999 O MEAN NOTE ALL random variables are assumed to be O rnean J Orthogonality Principle of LeastSquares Estim Let be LSE of x from Let c 10 1 Then Eeyt 0 for each value t of s PROOF Eleylttl Eytl EElylt8lylttl EyElxylttylt8l EElxylttylt8ll 0 using EylElxylyll EylyElxlyll INTERP spanys all linear combs of all possible linear estimators of 1projection of 10 on spanys eprojection errorlspanys EX1 Estirnate vector 10 from vector y 1 Ay linear form J Z AyyTl Katy AKy 0 gt 5 nyKy ly Much easier than previous derivations EX2 Estirnate t frorn ysT S 8 S Tf FORM t htsysds linear compare to 1 Ay Elt fht7uyuduy8l nyltt78f ht7uKyU78dU Solve this integral equation over T2 3 t Tf see below oo In nite smoothing lter Ti gt ooTf gt oo xtyt jointly WSS gt ht s ht s tirne invariant gt Kant s ff ht uKyu sdu 0 7 gt SxyW Hw5yw gt 4101 SxywSyw39 Special case yt xt 11t and Etvs 0 gt Elty8l EMU 188 08l Elt8l and KM 8 Elmt Ut8 08l Km Kw gt Sm w w and Syw 833W Sq Substitute gt 0Hw wSxw Svwo Note noncausal 53 Sg EECS 564 CAUSAL WIENER FOR WHITE RPs Winter 1999 WANT 1tys oo lt s lt t H gt ooTf t causal xtyt O rnean jointly WSS CAN T use 00 smoothing LEMMA THEN NOTE PROOF yi yN 0mean With Eiyz39yjl 52 y y1leT 30y1 Projections on J add NOT TRUE UNLESS 1 O rnean 2 62339 My Kwaly Eixy1ExyNnIy 2 Emily LEMMA 1tys 00 lt s lt t ffoo ht sysds Where i Kant fortgt0 ht 7 0 for t lt 0 promdedyt is white Kmyu S fht uKyu sdu ffoo ht u6u sdu Kant s ht s for s lt t and ht causal QED What is the transfer function for this ht 39 Now use Laplace xt ff xtestdt Xs XltSlsjw Xjw ft 2 sided Laplace X 8 2 Sign partial fraction expansion Xs REELO 33 302 sum over poles in lhp Xs Xs Xsrealizableunrealizable parts Note xt Xs gt tt gt 0 0t lt 0 Xs EX NOTE 3 i 1 1 3 7 1 m i m a gt m i m Use this later 1s2 2 ett gt 0 6 t lt 0 2 sided exponential 1Sia eatt lt 0 0t gt 0 for a gt 0 note signsl yt White gt Hs Km3Jr Where Kans nyt Use this after prewhz tcm ng lter gt causal Wiener lter EECS 564 RATIONAL WHITENING FILTERS Winter 1999 GOAL A lter ht so that wi yt gtllt ht is White t is innovations process associated With w ht must be causal and causally invertible ht has causal inverse lter h1i ht h1t 5t WHY Know 1118 00 lt s lt i Hknow ys 00 lt s lt t So 1tys 00 lt s lt i 1iws oo lt s lt ht causalcausally invertiblee ht minimum phase Hjw zi H jw zi 2 39 HOW Syjw Hjwpi Hjwpi Syw Replace 8 you 7quot gt L Sys Sy 82 Watch sign of 32 i ITS Pi Let Hs 7 W H taken over RE 215m lt 0 Then H spreWhitening lter for yt o ht 5 1 causal since RE lt 0 o h1i 1HS causal since RE pl lt 0 Sys SJsSys spectral factorization Hs EX Syjw M Syw2 Compute the lter w425w2144 ylt82 s4sj 55ss224t1444 32 w2 Mt 5 1 6t seat 2627 for i 2 0 using11s Z yi gt ht gt wi gt Kwtt gt 0 gt am 501 Sww SmywHw gt Sws SmysH s 2 mamas heat gt 0 1 ML Sic Sm EECS 564 EXAMPLES OF WIENER FILTERING Winter 1999 EX1 SPECS Observe yt rtvt Want fctys 00 lt s lt Elmlttvltsl 0 1 w21 NOTE 2 2211 W2 32 From before irreeo Saillt3 8333 1332 s1 1 1 Sxjw has poles at 1 Wiener lter has pole at 2 EX2 Find 00 smoothing lter for EX1 H w WW 3ltw21gt 1 W i 3 gt Mt 2ltl noncausal EX3 HQ 4 W 1W W Etv8l 0 Evtv8l W 8 Causal Wiener lter gt oHs 1 5 if 8333 strictly proper polesgtzeros PROOF SW S 1 i 1 i 1 5y ltSy 14 5 SmySmSy 1 gt 1 Saw SJ 5 5 strictly gt proper since SQ S ro er gt S ro er gt 4 ro er 2 y 3y strictly since poleszeros gtpartial fraction1 proper EX1 s2i 1 1 1s1 i 32 checks Recall yt gt gt wt INNOV ATIONS 1 W Mt gt Mt W W W W here oInnovationsWhitened ytprediction error for Kalrnan lter uses this to form innovations process


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