Discrete-Time, Causal Wiener Filter **new** 2003/06/26 14:03:11.105 GMT-5 2003/08/08 11:42:30.522 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Kyle Clarkson kclarks@rice.edu Stochastic L2 optimal filter design; Wiener developed the core if this theory. Stochastic L 2 optimal (least squares) FIR filter design problem: Given a wide-sense stationary (WSS) input signal x k and desired signal d k (WSS ⇔ y k y k d , r yz l y k z k l , k l r yy 0 )

The Wiener filter is the linear, time-invariant filter minimizing ε 2 , the variance of the error. As posed, this problem seems slightly silly, since d k is already available! However, this idea is useful in a wide cariety of applications. active suspension system design

optimal system may change with different road conditions or mass in car, so an adaptive system might be desirable. System identification (radar, non-destructive testing, adaptive control systems)

Usually one desires that the input signal 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 for convenience, we will analyze only the causal, real-data case; extensions are straightforward. y k l 0 M 1 w l x k - l 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 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 d k 2 r dx l d k X k - l r xx l m x k x k + l - m This can be written in matrix form as ε 2 r dd 0 2 P W W R W where 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 To solve for the optimum filter, compute the gradient with respect to the top weights vector W ≐ ∇ w 0 ε 2 w 1 ε 2 ⋮ w M - 1 ε 2 ∇ 2 P 2 R W (recall W A W A , W W M W 2 M W for symmetric M) setting the gradient equal to zero ⇒ ⇒ W opt R P W opt R P Since R is a correlation matrix, it must be non-negative definite, so this is a minimizer. For R positive definite, the minimizer is unique.