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Discrete-Time, Causal Wiener Filter

Module by: Douglas L. Jones

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

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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=0M−1 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=0M−1 w l x k - l 2=E d k 2−2∑l=0M−1 w l E d k x k - l +∑l=0M−1∑m=0M−1 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 0−2∑l=0M−1 w l r dx l+∑l=0M−1∑m=0M−1 w l w m r xx l−m 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 l−m=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 0−2PWT+WTRW ε 2 r dd 0 2 P W W R W where P= r dx 0 r dx 1⋮ r dx M−1 P r dx 0 r dx 1 ⋮ r dx M 1 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 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 ∇≐∂∂ w 0 ε2∂∂ w 1 ε2⋮∂∂ w M - 1 ε2 ≐ ∇ w 0 ε 2 w 1 ε 2 ⋮ w M - 1 ε 2 ∇=-2P+2RW ∇ 2 P 2 R W (recall ddWATW=AT W A W A , ddWWMW=2MW W W M W 2 M W for symmetric MM) setting the gradient equal to zero ⇒ WoptR=P⇒Wopt=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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