Detection and Estimation Theory
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1 ESE 54 Detection nd Estimtion heory Joseph A. O Sullivn Smuel C. Schs Professor Electronic Systems nd Signls Reserch Lbortory Electricl nd Systems Engineering Wshington University Urbuer Hll (Lynd nswers) jo@wustl.edu J. A. O'S. ESE 54, Lecture 5, 4/4/9
2 Announcements Finl Exm My Mke-up clss on Fridy April 4, here Other nnouncements? Questions? J. A. O'S. ESE 54, Lecture 5, 4/4/9
3 Outline: Estimtion i in Rndom Processes Problem definition Scope of coverge this semester Log-likelihood for pproximte rndom process Log-likelihood rtio (motivted by detection theory) Limit s number of terms in pproximtion gets lrge Estimtion theory in rndom processes J. A. O'S. ESE 54, Lecture 5, 4/4/9 3
4 Estimtion heory for Rndom Processes rt () = st (; ) wt (), t, E wtw () ( τ ) = δ ( t τ ), Signl with sclr [ ] prmeter in AWG [ ] onrndom prmeter Rndom prmeter Signl with sclr prmeter in G Signl with multiple prmeters Rndom wveform in presence of noise Vector wveform Stte spce model Continuous-time Klmn filter rt () = st (; ) nt (), t, Entn () ( τ) = Kc(, tτ) δ( t τ), rt () = st (; θ) nt (), t, Entn [ () ( τ) ] = Kc (, t τ) δ( t τ), K () t = lim... kφk() t K k = x () t = F () txt () G () tut (), E utu () ( τ ) = qδ ( t τ ), yt () = Ctxt () () wt () [ ] J. A. O'S. ESE 54, Lecture 5, 4/4/9 4
5 Eti Estimtion heory Concentrte on sclr cse Rndom (MMSE, MAP) nd onrndom (ML) prmeters Crmer-Ro bound determined by Fisher informtion Vector cse Grdients insted of derivtives Fisher informtion mtrix Estimtion of rndom processes not covered this semester Exmple of dely-doppler estimtion J. A. O'S. ESE 54, Lecture 5, 4/4/9 5
6 Estimtion heory: Sclr Prmeter Methodology Use finite pproximtion to the rndom process Find CO set of eigenfunctions for the covrince function for the rndom process Expnd the mesurement using the first eigenfunctions Derive the loglikelihood function for the finite pproximtion ormlize the loglikelihood function using fictionl null hypothesis Let get lrge to get the loglikelihood functionl For the rndom cse, dd the log of the prior Find the mximum (set derivtive or grdient to zero) Evlute performnce (compute Fisher informtion, find Crmer-Ro bound) J. A. O'S. ESE 54, Lecture 5, 4/4/9 6
7 Estimtion heory: Finite i Expnsion for Mesurements If dditive WG, the CO set is rbitrry. Use the signl (given the prmeter) to generte the first (or first few) bsis function(s) to find sufficient sttistic. If not white, use eigenfunctions. Derive loglikelihood function for finite pproximtion. For AWG, the finite pproximtion equls the loglikelihood functionl. rt () = st (; ) wt (), t, E[ wtw () ( τ) ] = δ( t τ), rt ( ) = st ( ; ) nt ( ), t, Entn [ ( ) ( τ) ] = Kc ( t, τ) δ( t τ), λ ckφ k () t = K c (, t τ ) φ k ( τ ) dτ, λ k = λ ck r = r, φ = r( t) φ ( t) dt = s ( ) n k k k k k r = = k k = λk WG Cse: φ ( t) = s( t; ), st ( ; ), s ( t ; ) [ r r r ], ln p( r ) lnλ ( r s ( ) ) ( ) ln pr ( ) = ln π r st ( ; ), st ( ; ), J. A. O'S. ESE 54, Lecture 5, 4/4/9 k k 7
8 Estimtion heory: Finite i Expnsion for Mesurements o derive the loglikelihood functionl, first normlize reltive to fictionl null, noise-only only hypothesis. rt () = st (; ) wt (), t, E[ wtw () ( τ) ] = δ( t τ), H : r() t = w(), t t rt () = st (; ) nt (), t, Entn [ () ( τ) ] = Kc (, tτ) δ( t τ), H : r() t = n(), t t p( r ) ln = ln λk ( rk sk( ) ) ln λk ( rk) p( r H) k = λk k = λk p( r ) ln = r k s k ( ) s k ( ) p ( r H ) k = λk pr ( ) WG Cse: lr ( ) = ln = r st (; ), st (; ) st (; ), st (; ), pr ( H) r = ()(; ), ( ) ()(; ) (; rtstdt lr = rtstdt st ) dt st (; ) dt J. A. O'S. ESE 54, Lecture 5, 4/4/9 White noise cse Colored noise cse White noise cse 8
9 Estimtion heory: Loglikelihood lih Functionl Let the dimension of the pproximtion get lrge. Invoke Mercer s theorem nd the inverse of the covrince function. First order necessry condition for the mximum: set derivtive to. rt ( ) = st ( ; ) nt ( ), t, Entn [ ( ) ( τ) ] = Kc( t, τ) δ( t τ), H : r( t) = n( t), t p( r ) ln = rk sk( ) sk( ) p ( r H ) k = λ k l( r ) = lim rk sk( ) sk( ) = r( t) s( t; ) Q( t, τ ) s k = λk (; τ dtd ) τ Qt (, τ) = [ δ( t τ) ho (, tτ) ] l ( r ) s ( τ; ) = ( rt () st (; )) Qt (, τ) dtdτ J. A. O'S. ESE 54, Lecture 5, 4/4/9 9
10 Estimtion heory: Loglikelihood lih Functionl Let the dimension of the pproximtion get lrge. Invoke Mercer s theorem nd the inverse of the covrince function. First order necessry condition for the mximum: set derivtive to. rt ( ) = st ( ; ) nt ( ), t, Entn [ ( ) ( τ) ] = Kc( t, τ) δ( t τ), H : r( t) = n( t), t p( r ) ln = rk sk( ) sk( ) p ( r H ) k = λ k l( r ) = lim rk sk( ) sk( ) = r( t) s( t; ) Q( t, τ ) s k = λk (; τ dtd ) τ Rndom Prmeter Cse Qt (, τ) = [ δ( t τ) ho (, tτ) ] p( A) lr ( A) = rt ( ) sta ( ; ) Qt (, τ) s( τ; Adtd ) τ ln pa ( ) l ( r ) s ( τ; ) = ( rt () st (; )) Qt (, τ) dtdτ lr ( A) s( τ; A) ln pa ( ) J. A. O'S. ESE 54, Lecture = ( 5, rt () 4/4/9 sta (; )) Qt (, τ) dtdτ A A A
11 Aside: Estimtion of Colored Gussin oise Component he form of the inverse covrince function Q is sum of two terms, one of which inverts the white noise covrince, nd one of which is interpreted s n estimtor-subtrctor [ τ ] n(t) nt () = n() t wt (), t, E n() tn( ) = K(, tτ), c c c c E[ w() t w( τ) ] = δ( t τ) MMSE estimte of the colored noise component nˆ () t E n () t n( u), n( u) c [ ] h o (t,τ) = c Q(t,τ) nˆ () t = h (, t τ) n( τ) dτ c o Qt (, τ ) n( τ ) d τ = nt ( ) ho ( t, ) n ( ) d τ τ τ - ^n c (t) J. A. O'S. ESE 54, Lecture 5, 4/4/9
12 Aside: Estimtion of Colored Gussin oise Component Derivtion of optiml i= estimtor in integrl form. View the derivtion s the limit of derivtion for finite pproximtion. he inverse covrince function is then representble s long s the colored noise covrince function stisfies Mercer s theorem (then h o converges). he result is non-cusl implementtion. Cusl versions (fctoriztions of Q) exist. [ τ ] nt () = nc() t wt (), t, E nc() tnc( ) = Kc(, tτ), E[ w() t w( τ) ] = δ( t τ) Krhunen-Loeve expnsion using colored noise covrince function Kc(, t τ) = λciφi() t φi( τ) ni = n, φi = nci wi, ni, λci λci E[ nci n( u), n( u) ] = E[ nci ni ] = ni λci λci MMSE estimte nˆ () ˆ c t = nciφi() t = φi() t ni i= i= λci nˆ () t = h (, t τ) n( τ) dτ c o o λci = i= λci i t i h (, t τ) φ () φ ( τ) Q(, t τ) n( τ) dτ = n() t ho (, t τ) n( τ) dτ λ ci = λ ci λci J. A. O'S. ESE 54, Lecture 5, 4/4/9
13 Exmple: Pulse Amplitude Modultion Consider both the rt () = Est () wt (), t nonrndom nd rndom cses st () dt=, (, σ ) Answers differ only in scle fctor; SR ( ) = ()() definition vries ML estimte is unbised, vrince /E A l A A E r t s t dt A E σ l ( A ) A = E rtstdt ()() AE = A σ ˆ ML = r()() t s t) dt E Eσ MAP E J. A. O'S. ESE 54, Lecture 5, 4/4/9 ˆ = r()() t s t dt σ E 3
14 Exmple: Pulse Amplitude Modultion Performnce CRLB for nonrndom rt () = Est () wt (), t cse: ML estimte is efficient (unbised nd st () dt=, (, σ ) meets the CRLB) MAP estimte is A efficient l( A) = A E r()() t s t dt A E σ Bound for the rndom cse cn be viewed s l( A) A = E r () t s () t dt AE either reducing the A σ prior vrince or reducing the MSE of l( A) E = the ML estimtor. A σ CRLB =, E J. A. O'S. ESE 54, Lecture 5, 4/4/9 = = σ = E Eσ J E σ Eσ Eσ 4
15 Generl lgussin Cse Performnce CRLB for nonrndom cse is bsed on Fisher informtion. Fisher informtion for the rndom cse includes term from the prior nd n expecttion ti over the prmeter. lr ( A) = rt ( ) sta ( ; ) Qt (, τ) s( τ; Adtd ) τ ln pa ( ) lr ( A) s( τ; A) ln pa ( ) = ( rt () sta (; )) Qt (, τ) dtdτ A A A lr ( A ) sta ( ; ) s ( τ ; A ) E A Q(, t τ) dtdτ = A A A CRLB = s(; t A) s(; τ A) Q(, t τ ) dtd τ AA A st A s A pa J = E Q(, t τ) dtdτ A A A (; ) ( τ; ) ln ( ) ln pa ( ) A J. A. O'S. ESE 54, Lecture 5, 4/4/9 5
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