Exam zone

Overview of FIR Wiener Filtering For the optimal FIR Wiener ﬁlter, the estimate of the takes the form: LX¡ 1 dˆ[n ]= w[k ]x [n ¡ k ]:

signal of inter est (SOI)

k =0

The corresponding estimation error is given via: LX¡ 1

e[n ]= d[n ] ¡

w[k ]x [n ¡ k ]: k =0

The orthogonality principle associated with this MMSE optimization problem is given by: E f e[n ]x ¤ [n ¡ l ]g =0 ; 0 · l · L ¡ 1: The Wiener–Hopf equationsfortheoptimalFIRWiener ﬁlterderivedfromthe orthogonality principle are given by: LX¡ 1

wopt [k ]R xx [l ¡ k ]= R dx [l ]; 0 · l · L ¡ 1 k =0

Reformulated into a matrix system the normal equations can be written as: R xw

opt

= r dx ;

where R x istheautocorrelationmatrixoftheobservationsand r dx isthecrosscorrelation vector between the SOI and the observations. If a unique solution for the optimal FIR ﬁltering problem is to exist, r dx 2 Range(R x ), i.e., the matrix R x must be invertible and from the properties of covariance and correlation matrices, also has to be positive–deﬁnite: aTR ax > 0; a 2 R L : The corresponding minimum me an-squar e d err (MMSE) or for the optimal FIR ﬁlter is given via: T T ² 2min = R dd [0] ¡ w opt r dx = R dd [0] ¡ w opt R xw

opt

= R dd [0] ¡ r Tdx R ¡x 1 r dx :

In the context of linear transversal ﬁltering seen in digital communication applications, the linear FIR–MMSE ﬁlter is sometime referred to as the ﬁnite tap e qualizer . Note that eventhough the FIR Wiener ﬁlter is suboptimal compared to the non-causal and causal IIR ﬁlters, it is more practical for real-time applications such as equalization, echo-cancellation, and interference mitigation. Furthermore there are no stability problems associated with the FIR–MMSE ﬁltering that one encounters with IIR ﬁltering.

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prob theory & stochastic6
prob theory & stochastic6

Exam zone The corresponding estimation error is given via: w opt [k ]R xx [l ¡ k ]= R dx [l ]; 0 · l · L ¡ 1 X X X e[n ]= d[n ] ¡ w[k ]x [n...