Outlie etectio heory Chapter7. etermiistic Sigals with Ukow Parameters afiseh S. Mazloum ov. 3th Backgroud Importace of sigal iformatio Ukow amplitude Ukow arrival time Siusoidal detectio Classical liear model Backgroud Statistical ecisio heory I PF for assumed hypothesis is completely kow. pproaches: eyma Pearso theorem Bayesia approach based o miimizatio of Bayes risk Sigal model dl etermiistic sigal Radom sigal Backgroud (Cot. Statistical ecisio heory II PF has ukow parameters (i this chapter determiistic sigals pproaches: UMPdoes otusually exist Geeralized Likelihood Ratio est (GLR ecide if p( x;, LG ( x p( x;, Bayesia approach ecide if (, ( p( x; p x p d p( x; p(, p( d x
Importace of Sigal Iformatio ssume there is o kowledge of sigal : x [ ] ω[ ],,..., : x [ ] s [ ] + ω [ ],,...,,, ω[ ] ~ WG with variace s [ ]~ determiistic ad completely ukow GLR decides if p( x; s[],..., s[ ], p ( x ; where s [ ]is MLE of s [ ]for,,...,,, Importace of Sigal Iformatio (Cot. Uder the PF of p ( x; exp [ ] x ( π Uder the PF of hus, x x s [ ] x [ ] p ( x ; s [],..., s [ ], exp ( [ ] [ ] x s ( π ( π exp x [ ] ( π eergy detector ( X x [ ] xs [ ] [ ] Importace of Sigal Iformatio (Cot. etectio performace ( ( P Q Q P d Loss i performace d E d MF dmf log log db d E Matched filter coheretly combies data Eergy detector icoheretly combies data Ukow mplitude etectig a determiistic sigal with ukow amplitude i WG UMP test GLR Bayesia approach : [ ] [ ],,...,,, : x [ ] s [ ] + ω[ ],,..., :ukow amplitude ω[ ] ~ WG with variace s [ ]: determiistic ad completely kow
Ukow mplitude (Cot. UMP test: LR decides if exp ( [ ] [ ] xs p( x; ( π p( x; exp [ ] x ( π x [ ] s [ ] > UMP does ot exist xs [ ] [ ] > > xs [ ] [ ]< < Ukow mplitude (Cot. GLR decides if ( ; px, ( π px ( ; is MLE of uder ( X x[ ] s[ ] exp ( x[ ] s[ ] exp x [ ] ( π xs [ ] [ ] s [ ] or xs [ ] [ ] > Ukow mplitude (Cot. Ukow mplitude (Cot. GLR detector performace P Pr u( x ; { } { } P Pr u( x ; xs [ ] [ ] > GLR detector performace ( ( / ( / ( P Q Q P d + Q Q P + d where d, s [ ] uder xs ux ( [ ] [ ] s [ ], s [ ] uder
Ukow mplitude (Cot. Ukow mplitude (Cot. Bayesia approach, uder x s + w x + w ( [ s[], s[],..., s[ ] ] radom variable with PF μ, s w WG with variace (, I P test statistic μ ( x ( x s + ( s s x s + s s + correlator Squared correlator P test decides if (results i Sectio 5.6 ( ( x x C + Cω μ + xc C ( C + C x> ω ω μ μ ω C C I if (kow amplitude case ( x x μ s ( x ( x s ss If (ukow amplitude μ Usig Bayesia detector we require kowledge of ad. Ukow rrival ime Ukow rrival ime (Cot. etectig a determiistic sigal with ukow arrival time i WG : [ ],,..., : x [ ] s [ ] + ω[ ],,..., GLR decides if p( x;, p( ( x ; is MLE of uder + M max xs [ ] [ ] ukow delay ω[ ] WG with variace s [ ] detemiistic ad kow [, M-] + M Observatio iterval Uder + M p ( x ;, exp [ ] x π M. exp [ ] [ ] π ( x s. exp x [ ] + M π
Ukow rrival ime (Cot. est statistic + M + M ( xs [ ] [ ] s[ ] + M ( x x[ ] s[ ] + M ( x max x[ ] s[ ] [, M ] + Choose ( > max over < M s [ ] x [ ] Ch ( x s [ ] Ukow rrival ime (Cot. etectio performace of GLR is difficult! PF of correlated Gaussia radom variable has to be determied. + M P Pr max [ ] [ ] ; [, M] x s + M P Pr max xs [ ] [ ] γ ; [, M] > elay is less tha samplig iterval ( x max X( f S( fexp( j π f df [, M] * Siusoidal etectio etectio of a siusoid i WG : x [ ] ω[ ],,..., ω [ ],,...,,,, + M,...,, : x [ ] cos( π f + ϕ + ω[ ], +,..., + M GLR approach ukow, ϕ ukow, ϕ, f ukow, ϕ, f, ukow Siusoidal etectio (Cot. etectio of a siusoid i WG with ukow amplitude ad phase, GLR decides if p ( x ;, ϕ, p( x; ad ϕ are MLE of ad ϕ α + α α ϕ arcta α α x [ ]cos π f α x[ ]siπ f
Siusoidal etectio (Cot. We decide if exp ( [ ] cos( ϕ exp [ ] x ( π x π f + ( π x [ ]exp( j π f I( f periodogram detector icoheret or quadrature MF Siusoidal etectio (Cot. GLR detectio performace { } { } P Pr I( f ; P Pr I( f ; I( f ξ + ξ joitly Gaussia ξ ξξ ξ x [ ]cosπ f, I uder ξ x [ ]siπ f ξ cosϕ, I uder si ϕ Siusoidal etectio (Cot. Siusoidal etectio (Cot. GLR detectio performace P Q exp χ γ P Q χ λ ( etectio of a siusoid i WG with ukow GLR decides if p ( x ;, ϕ, f,, p( x;, ϕ, f, P Q l χ ( λ P λ, ϕ, f ad are MLE of, ϕ, f ad α + α α + M ϕ arcta α α M + M x π f M α [ ]cos ( x π f [ ]si (
Siusoidal etectio (Cot. fter simplificatio, p( x;, ϕ, f (,, I f l p( x; where I ( f x[ ]exp( j f We decide if I ( f max >, f + M + M π M + Choose max over, f < x [ ] exp( j π f Classical Liear Model Liear Bayesia model etectio problem with ukow sigal parameters coverts to geeral Gaussia problem P detector Classical liear model Parameters assumed determiistic GLR GLR Classical Liear Model ssume that x + w is kow pobservatio matrix of rak GLR for hypothesis testig problem : b : b is p vector of parameters w is oise vector with PF (, I is r pmatrix of rak r b is r vector bis cosistet set of liear equatios p GLR Classical Liear Model (Cot. We decide if ( b ( ( ( l L ( b x G x ( x is the MLE of uder
GLR Classical Liear Model (Cot. etectio performace P Q χ ( r GLR Classical Liear Model (Cot. Ukow amplitude sigal i WG : [ ] [ ] x ω,,..., : x [ ] s [ ] + ω [ ],,..., P Q χ λ ( ( r ( b ( ( λ b : x [ ] s [ ] + ω[ ],,,..., : x [ ] s [ ] + ω [ ],,,..., x + w : [ s[], s[],..., s[ ]] : GLR Classical Liear Model (Cot. We decide if ( b ( ( ( l LG ( b x x, b ( x ( x x[ s ] [ ] s [ ] x[ s ] [ ] ( x s [ ] GLR Classical Liear Model (Cot. etectio performace P Q χ ( P Q ( χ ( λ r r s [ ] whereλ ( ( / ( / ( P Q Q P d + Q Q P + d where d