Discussion
The idea behind RLS filters is to minimize a cost function by appropriately selecting the filter coefficients, updating the filter as new data arrives. The error signal and desired signal are defined in the negative feedback diagram below:
The error implicitly depends on the filter coefficients through the estimate :
The weighted least squares error function —the cost function we desire to minimize—being a function of e(n) is therefore also dependent on the filter coefficients:
where is the "forgetting factor" which gives exponentially less weight to older error samples.
The cost function is minimized by taking the partial derivatives for all entries of the coefficient vector and setting the results to zero
Next, replace with the definition of the error signal
Rearranging the equation yields
This form can be expressed in terms of matrices
where is the weighted sample correlation matrix for, and is the equivalent estimate for the cross-correlation between and . Based on this expression we find the coefficients which minimize the cost function as
This is the main result of the discussion.
Read more about this topic: Recursive Least Squares Filter
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