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The LMS Adaptive Filter Algorithm

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

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Recall the Weiner filter problem

Figure 1
Figure 1 (Discrete-Timefig1.png)
x k x k , d k d k jointly wide sense stationary

Find WW minimizing E ε k 2 ε k 2 ε k = d k − y k = d k −∑i=0M−1 w i x k - i = d k − X k T W k 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 X k x k x k - 1 ⋮ x k - M + 1 W k = w 0 k w 1 k ⋮ w M - 1 k 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 0

∇ k =∂∂WE ε k 2=E2 ε k - X k =E-2 d k − X k T W k X k =-2E d k X k +E X k TW=-2P+2RW ∇ 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 (1)
⇒ W opt =R-1P ⇒ W opt R P Alternatively, W opt W opt can be found iteratively using a gradient descent technique W k + 1 = W k −μ ∇ k W k + 1 W k μ ∇ k In practice, we don't know RR and PP 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!

Good idea:

Approximate RR and PP: ⇒ RLS methods, as discussed last time.

Better idea:

Approximate the gradient! ∇ k =∂∂WE ε k 2 ∇ k W ε k 2
Note that ε k 2 ε k 2 itself is a very noisy approximation to E ε k 2 ε k 2 . We can get a noisy approximation to the gradient by finding the gradient of ε k 2 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 T X k =2 ε k - X k =-2 ε k X k ∇ 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

Example 1: The LMS Adaptive Filter Algorithm

  1. y k = W k T X k =∑i=0M−1 w i k x k - i y k W k X k i 0 M 1 w i k x k - i
  2. ε k = d k − y k ε k d k y k
  3. W k + 1 = W k −μ ∇ k ̂= W k −μ-2 ε k X k = W k +2μ ε k X 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 +2μ ε k x k - i 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 ̂ ∇ k is a noisy gradient. This is by far the most commonly used adaptive filtering algorithm, because

  1. it was the first
  2. it is very simple
  3. in practice it works well (except that sometimes it converges slowly)
  4. it requires relatively litle computation
  5. it updates the tap weights every sample, so it continually adapts the filter
  6. it tracks slow changes in the signal statistics well

Computational Cost of LMS

Table 1
To Compute ⇒ y k y k ε k ε k W k + 1 W k + 1 = Total
multiplies MM 00 M+1 M 1 2M+1 2 M 1
adds M−1 M 1 11 MM 2M 2 M

So the LMS algorithm is OM 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 μμ plays a very important role in the LMS algorithm. It can also be varied with time, but usually a constant μμ ("convergence weight facor") is used, chosen after experimentation for a given application.

Tradeoffs

large μμ: fast convergence, fast adaptivity

small μμ: accurate WW → less misadjustment error, stability

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