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Practical Issues in Wiener Filter Implementation

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

The weiner-filter, W opt =R-1⁢P W opt R P , is ideal for many applications. But several issues must be addressed to use it in practice.

Exercise 1

In practice one usually won't know exactly the statistics of x k x k and d k d k (i.e. RR and PP) needed to compute the Weiner filter.

How do we surmount this problem?

Solution

Estimate the statistics r xx ⁢l ^≃1N⁢∑ k =0N−1 x k ⁢ x k + l r xx l 1 N k 0 N 1 x k x k + l r xd ⁢l ^≃1N⁢∑ k =0N−1 d k ⁢ x k - l r xd l 1 N k 0 N 1 d k x k - l then solve W^opt= R-1 ^= P ^ W opt R P

Exercise 2

In many applications, the statistics of x k x k , d k d k vary slowly with time.

How does one develop an adaptive system which tracks these changes over time to keep the system near optimal at all times?

Solution

Use short-time windowed estiamtes of the correlation functions.

Equation in Question:

r xx ⁢l ^k=1N⁢∑ m =0N−1 x k - m ⁢ x k - m - l r xx l k 1 N m N 1 0 x k - m x k - m - l
r dx ⁢l ^k=1N⁢∑ m =0N−1 x k - m - l ⁢ d k - m r dx l k 1 N m N 1 0 x k - m - l d k - m and W opt k≃R^k-1⁢P^k W opt k R k P k

Exercise 3

How can r xx k ⁢l ^ r xx k l be computed efficiently?

Solution

Recursively! r xx k ⁢l= r xx k - 1 ⁢l+ x k ⁢ x k - l − x k - N ⁢ x k - N - l r xx k l r xx k - 1 l x k x k - l x k - N x k - N - l This is critically stable, so people usually do (1−α)⁢( r xx kl=α⁢ r xx k - 1 ⁢l+ x k ⁢ x k - l ) 1 α r xx l k α r xx k - 1 l x k x k - l

Exercise 4

how does one choose N?

Tradeoffs

Larger NN → more accurate estimates of the correlation values → better W^opt W opt . However, larger NN leads to slower adaptation.

Note:

The success of adaptive systems depends on xx, dd being roughly stationary over at least NN samples, N>M N M . That is, all adaptive filtering algorithms require that the underlying system varies slowly with respect to the sampling rate and the filter length (although they can tolerate occasional step discontinuities in the underlying system).

Computational Considerations

As presented here, an adaptive filter requires computing a matrix inverse at each sample. Actually, since the matrix RR is Toeplitz, the linear system of equations can be sovled with O⁢M2 O M 2 computations using Levinson's algorithm, where MM is the filter length. However, in many applications this may be too expensive, especially since computing the filter output itself requires O⁢M O M computations. There are two main approaches to resolving the computation problem

  1. Take advantage of the fact that Rk+1 R k 1 is only slightly changed from Rk R k to reduce the computation to O⁢M O M ; these algorithms are called Fast Recursive Least Squareds algorithms; all methods proposed so far have stability problems and are dangerous to use.
  2. Find a different approach to solving the optimization problem that doesn't require explicit inversion of the correlation matrix.

Note:

Adaptive algorithms involving the correlation matrix are called Recursive least Squares (RLS) algorithms. Historically, they were developed after the LMS algorithm, which is the slimplest and most widely used approach O⁢M O M . O⁢M2 O M 2 RLS algorithms are used in applications requiring very fast adaptation.

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