2, H ) ˆ, σ p (x ; A ˆ1 1 >γ 2 p (x ; σ ˆ 0 , H0 )
Li, Wei
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
2 σ ˆ0 2 ˆ, σ A ˆ1 2 =σ 2. σ ˆ0 ˆ1
2 = Pr {X1 > γ ; H0 }
Example DC level in WGN A unknown, σ 2 unknown. H0 : A = 0, σ 2 > 0 H1 : A = 0, σ 2 > 0 θr = A, θs = σ 2 , r = 1, s = 1 a 2 ⇒ 2 ln LG (x ) ∼ X1 Same as when σ 2 is known-nuisance parameters do not affect threshold but do decrease λ and hence PD . From previous work 2 ln LG (x ) = N ln 1 1−
Unknown
>0
θ=
A σ2
⇒
θr = A θs = σ2
r =1 s=1
Li, Wei
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
In general we test H0 : θ r H1 : θ r (θr 0 = 0 in example) GLRT is to deci 1 , ˆ θ s 1 , H1 ) >γ p (x ; θ r 0 , ˆ θ s 0 , H0 ) = θr 0 , θs = θr 0 , θs
ˆ θs 1 = MLE of θr , θs under H1 θr 1 , ˆ ˆ θs 0 = MLE of θs under H0 (θr = θr 0 is constraint.)
Li, Wei
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
n=0
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x 2 (n )
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
2 p (x ; σ ˆ0 , H0 ) =
1 e −N /2 2 )N /2 (2π σ ˆ0
d 2 = NA2 /σ 2 For given (PFA , PD ) need given d 2 , as N → ∞ can attain given (PFA , PD ) with A → 0 (weak signal).
Li, Wei
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
General Results Given p (x ; θ) θ= θr θs = r ×1 s ×1 θr is to be tested. θs is a nuisance parameter. Example DC level in WGN H0 : A = 0, σ 2 > 0 H1 : A = 0, σ2
2 σ ˆ0 2−x σ ˆ0 ¯2
2 ln LG (x ) = N ln = N ln
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1 2 1−x ¯2 /σ ˆ0
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
Li, Wei
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
T (x ) = = =
1 (N 1 N 1 (N 1 N 1 (N 1 N
n n
w (n))2 w 2 (n )
2 ∼ X1
Li, Wei Decision Theory
x ¯2 2 σ ˆ0
a
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
Now the PDF is only approximate. ⇒ Threshold only approximate but less error as N → ∞ To find threshold: PFA = Pr {2 ln LG (x ) > γ ; H0 }
Under H1 show that ˆ = x A ¯ 1 N −1 2 (x (n ) − x ¯ )2 σ ˆ1 = N
n=0 2 ˆ, σ ⇒ p (x ; A ˆ1 , H1 ) =
1 e −N /2 2 )N /2 (2π σ ˆ1
2 σ ˆ0 2 σ ˆ1 N /2
LG (x ) =
ˆ =x GLRT fits data with signal A ¯ under H1 and finds residual 2 2. power σ ˆ1 and compares to case of no fit or σ ˆ0 2 2 When signal is present σ ˆ1 σ ˆ0 ⇒ LG (x ) 1
2 )2 ∼ X1
Nx ¯2 2σ 2
Asymptotic PDF holds exactly - for any N.
Li, Wei Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
Li, Wei
Decision Theory
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
Example DC level in WGN, A, σ 2 both unknown No UMP test. can’t implement NP since A unknown x ¯ ≶?γ and 2 σ unknown ⇒ can’t set γ . H0 : A = 0, σ 2 > 0 H1 : A = 0, σ 2 > 0 σ 2 unknown- called a nuisance parameter. GLRT decides H1 if LG (x ) =
Example DC level in WGN, A unknown, σ 2 known. H0 : A = 0 H1 : A = 0
2 under H θr = A, no θs , r = 1 ⇒ 2 ln LG (x ) ∼ X1 0 a
But from previous work ln LG (x ) = Under H0 x ¯ ∼ N (0, σ 2 /N ) ⇒ 2 ln LG (x ) = ( x ¯ σ 2 /N
Under H0 u (n) ∼ N (0, 1)
σ u (n))2 σ 2 u 2 (n ) u (n))2 u 2 (n )
PDF of T does not depend on σ 2 under H0 ⇒ GLRT can be implemented.
Li, Wei
Decision Theory
But 0 ≤ N ln
x ¯2 ≤1 2 σ ˆ0
1 monotonically increasing with x 1−x Equivalent test is x ¯2 T (x ) = 2 > γ σ ˆ0 We see that x ¯ has been normalized ⇒ invariant to σ 2 under H0 ⇒ can find γ
= MLE of σ 2 = MLE of A, σ 2
Under H0 Under H1
Note that 2 we maximize To find σ ˆ0 1 1 N −1 2 p (x ; σ , H0 ) = exp (− 2 x (n)) 2σ (2πσ 2 )N /2 n=0
2
N N 1 ln 2π − ln σ 2 − 2 2 2 2σ N 1 ∂ ln p = − 2+ 4 x 2 (n) = 0 2 ∂σ 2σ 2σ 1 N −1 2 2 x (n) ⇒ σ ˆ0 = N ln p = −
Decision Theory Detection Theory II Generalized Likelihood Ratio Test (GLRT)
Large data record performance of GLRT Results valid if: N is large ⇒ signal weak MLE attains asymptotic PDF ˆ θ ∼ N (θ, I −1 (θ)) I (θ) =Fisher Information Matrix NOTE: Recall DC level in WGN PD = Q (Q −1 (PFA ) − √ d 2)
Then as N → ∞ 2 ln LG (x ) ∼ Xr2 Xr