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Discrete-Time, Causal Wiener Filter

Module by: Douglas L. Jones. E-mail the author

Summary: Stochastic L2 optimal filter design; Wiener developed the core if this theory.

Stochastic L 2 L 2 optimal (least squares) FIR filter design problem: Given a wide-sense stationary (WSS) input signal x k x k and desired signal d k d k (WSS ⇔ Eyk=Eyk+d y k y k d , r yz l=Eykzk+l r yz l y k z k l , k , l : r yy 0< k l r yy 0 )

Figure 1
Figure 1 (Discrete-Timefig1.png)
The Wiener filter is the linear, time-invariant filter minimizing Eε2 ε 2 , the variance of the error.

As posed, this problem seems slightly silly, since d k d k is already available! However, this idea is useful in a wide cariety of applications.

Example 1

active suspension system design

Figure 2
Figure 2 (fig2Discrete-Time.png)

Note:

optimal system may change with different road conditions or mass in car, so an adaptive system might be desirable.

Example 2

System identification (radar, non-destructive testing, adaptive control systems)

Figure 3
Figure 3 (fig3Discrete-Time.png)

Exercise 1

Usually one desires that the input signal x k x k be "persistently exciting," which, among other things, implies non-zero energy in all frequency bands. Why is this desirable?

Determining the optimal length-N causal FIR Weiner filter

Note:

for convenience, we will analyze only the causal, real-data case; extensions are straightforward.

y k = l =0M1 w l x k - l y k l 0 M 1 w l x k - l argmin w l Eε2=E d k y k 2=E d k l =0M1 w l x k - l 2=E d k 22 l =0M1 w l E d k x k - l + l =0M1 m =0M1( w l w m E x k - l x k - m ) w l ε 2 d k y k 2 d k l M 1 0 w l x k - l 2 d k 2 2 l M 1 0 w l d k x k - l l 0 M 1 m 0 M 1 w l w m x k - l x k - m Eε2= r dd 02 l =0M1 w l r dx l+ l =0M1 m =0M1 w l w m r xx lm ε 2 r dd 0 2 l M 1 0 w l r dx l l M 1 0 m M 1 0 w l w m r xx l m where r dd 0=E d k 2 r dd 0 d k 2 r dx l=E d k X k - l r dx l d k X k - l r xx lm=E x k x k + l - m r xx l m x k x k + l - m This can be written in matrix form as Eε2= r dd 02PWT+WTRW ε 2 r dd 0 2 P W W R W where P=( r dx 0 r dx 1 r dx M1 ) P r dx 0 r dx 1 r dx M 1 R=( r xx 0 r xx 1 r xx M1 r xx 1 r xx 0 r xx 0 r xx 1 r xx M1 r xx 1 r xx 0 ) R r xx 0 r xx 1 r xx M 1 r xx 1 r xx 0 r xx 0 r xx 1 r xx M 1 r xx 1 r xx 0 To solve for the optimum filter, compute the gradient with respect to the top weights vector WW ( ε2 w 0 ε2 w 1 ε2 w M - 1 ) w 0 ε 2 w 1 ε 2 w M - 1 ε 2 =(2P)+2RW 2 P 2 R W (recall dd W ATW=AT W A W A , dd W WMW=2MW W W M W 2 M W for symmetric MM) setting the gradient equal to zero ⇒ W opt R=P W opt =R-1P W opt R P W opt R P Since RR is a correlation matrix, it must be non-negative definite, so this is a minimizer. For RR positive definite, the minimizer is unique.

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