The LMS Adaptive Filter Algorithm **new** 2003/07/01 14:37:34.995 GMT-5 2003/08/04 17:08:45.570 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Kyle Clarkson kclarks@rice.edu Recall the Weiner filter problem

x k , d k jointly wide sense stationary Find W minimizing ε k 2 ε k d k y k d k i M 1 0 w i x k - i d k X k W k X k x k x k - 1 ⋮ x k - M + 1 W k w 0 k w 1 k ⋮ w M - 1 k The superscript denotes absolute time, and the subscript denotes time or a vector index. the solution can be found by setting the gradient 0 ∇ k W ε k 2 2 ε k X k -2 d k X k W k X k 2 d k X k X k X k W -2 P 2 R W ⇒ W opt R P Alternatively, W opt can be found iteratively using a gradient descent technique W k + 1 W k μ ∇ k In practice, we don't know R and P exactly, and in an adaptive context they may be slowly varying with time. To find the (approximate) Wiener filter, some approximations are necessary. As always, the key is to make the right approximations! Approximate R and P: ⇒ RLS methods, as discussed last time. Approximate the gradient! ∇ k W ε k 2 Note that ε k 2 itself is a very noisy approximation to ε k 2 . We can get a noisy approximation to the gradient by finding the gradient of ε k 2 ! Widrow and Hoff first published the LMS algorithm, based on this clever idea, in 1960. ∇ k W ε k 2 2 ε k W d k W k X k 2 ε k X k 2 ε k X k This yields the LMS adaptive filter algorithm The LMS Adaptive Filter Algorithm y k W k X k i 0 M 1 w i k x k - i ε k d k y k W k + 1 W k μ ∇ k W k μ -2 ε k X k W k 2 μ ε k X k ( w i k + 1 w i k 2 μ ε k x k - i ) The LMS algorithm is often called a stochastic gradient algorithm, since ∇ k is a noisy gradient. This is by far the most commonly used adaptive filtering algorithm, because it was the first it is very simple in practice it works well (except that sometimes it converges slowly) it requires relatively litle computation it updates the tap weights every sample, so it continually adapts the filter it tracks slow changes in the signal statistics well

Computational Cost of LMS To Compute ⇒ y k ε k W k + 1 = Total multiplies M 0 M 1 2 M 1 adds M 1 1 M 2 M So the LMS algorithm is O M per sample. In fact, it is nicely balanced in that the filter computation and the adaptation require the same amount of computation. Note that the parameter μ plays a very important role in the LMS algorithm. It can also be varied with time, but usually a constant μ ("convergence weight facor") is used, chosen after experimentation for a given application.

Tradeoffs large μ: fast convergence, fast adaptivity small μ: accurate W → less misadjustment error, stability