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)
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  ¡
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.
Exam zone o ( z) H ¤ opt ( z) = ² 2 min = R dd  ¡ P xx ( z) = ¾ H opt ( z) = H H X X X w opt [k ]R dx [k ] w[k ]x [n ¡ k ]: w[k ]x [n ¡ k...