Exam zone
Overview of Causal Wiener Filtering For the optimal causal IIR filter, the estimate of the takes the form: X1 ˆ d[n ]= w[k ]x [n ¡ k ]:
signal of inter est(SOI)
k =0
Notethatthesummationlimitsarefrom0to 1 . Thecorrespondingestimation error is given via: X1 e[n ]= d[n ] ¡ w[k ]x [n ¡ k ]: k =0
Also note that the space of optimal causal IIR filters is a subset of the more general space of optimal IIR filters and consequently is a suboptimal filter as compared to the general optimal IIR filter. Assuming that the observation process x [n ] corresponds to a zero-mean, WSS, finite variance random sequence that has a rational PSD, it has a powerspectral factorization of the form: µ ¶ 1 1 2 ¤ Pxx ( z) = ¾o H ( z) H ; ½< j zj < ; z¤ ½ where H ( z) isthemonic-minimum-phasepartof Pxx ( z). Theoptimalrealizable IIR filter is terms of this factorization is: Ã ! 1 Pdx ( z) ( c) ¡ ¢ H opt ( z) = 2 ; ¾o H ( z) H ¤ z1¤ +
where the + su±x denotes the time–causal part. The corresponding minimum me an-squar e d err(MMSE) or for this causal IIR filter is given via: ² 2min = R dd [0] ¡
X1 wopt [k ]R dx [k ] k =0
Notethatonlypositive-sidedcorrelationsbetweentheSOIandtheobservations is being subtracted o® in the expression for the MMSE. Consequently if there does exist a correlation between the SOI and the observations for negative indices, thesearenottakenintoaccount. Comparingtheexpressionofthecausal IIR filter to the form of the optimal non-causal IIR filter: ( nc )
H opt ( z) =
Pdx ( z) ¡ ¢ ¾o2 H ( z) H ¤ z1¤
we can see the the di®erence is essentially in the time–causal part of the causal IIR filter.
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