Exam zone

Overview of Optimal Wiener Filtering For the optimal Wiener ﬁlter, the estimate of the signal of inter est (SOI) takes the form: X1

dˆ[n ]=

w[k ]x [n ¡ k ]: k=

The corresponding estimation error is given via: X1 e[n ]= d[n ] ¡

w[k ]x [n ¡ k ]: k=

The Wiener–Hopf equations for the optimal Wiener ﬁlter are given by: X1 wopt [k ]R xx [l ¡ k ]= R dx [l ]; l 2 Z : k=

The optimal Wiener ﬁlter for the estimation of the SOI is given by: H opt ( z) =

Pdx ( z) : Pxx ( z)

This optimal ﬁlter can be reformulated in terms of the power spectral factorization of Pxx ( z) as: ! µ ¶Ã 1 Pdx ( z) ¡ ¢ : H opt ( z) = ¾o2 H ( z) H ¤ z1¤ The corresponding minimum me an-squar e d err (MMSE) or for the optimal IIR ﬁlter is given via: ² 2min = R dd [0] ¡

X1 wopt [k ]R dx [k ] k=

In the frequency–domain this MMSE expression can be reformulated as: µ ¶ Z¼ 1 j Pdx ( ej! ) j 2 2 j! ² min = Pdd ( e ) 1 ¡ d!: 2¼ ¡ ¼ Pdd ( ej! ) Pxx ( ej! ) In terms of the spectral–coherence parameter½xy ( ej! ) the MMSE is given by: Z¼ ¡ ¢ 1 ² 2min = Pdd ( ej! ) 1 ¡j ½xy ( ej! ) j 2 d!: 2¼ ¡ ¼ Whenthespectral-coherence iszero, i.e., thereisnostatistical correlation betweenthe spectral components of the SOI and x [n ] we have: j ½( ej! ) j =0 ()

dˆ[n ]=0 ()

² 2min = R dd [0]:

When the spectral-coherence is unity, i.e., then the spectral components of the SOI and the observations x [n ] are perfectly correlated then: j ½( ej! ) j =1 ()

Pdx ( ej! ) = Pxx ( ej! ) Pdd ( ej! ) ()

1

² 2min =0 :

prob theory & stochastic2

Exam zone The optimal Wiener ﬁlter for the estimation of the SOI is given by: For the optimal Wiener ﬁlter, the estimate of the signal of in...