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
Famous quotes containing the word discussion:
“It was heady stuff, recognizing ourselves as an oppressed class, but the level of discussion was poor. We explained systemic discrimination, and men looked prettily confused and said: “But, I like women.””
—Jane O’Reilly, U.S. feminist and humorist. The Girl I Left Behind, ch. 2 (1980)
“We cannot set aside an hour for discussion with our children and hope that it will be a time of deep encounter. The special moments of intimacy are more likely to happen while baking a cake together, or playing hide and seek, or just sitting in the waiting room of the orthodontist.”
—Neil Kurshan (20th century)
“The whole land seems aroused to discussion on the province of woman, and I am glad of it. We are willing to bear the brunt of the storm, if we can only be the means of making a break in that wall of public opinion which lies right in the way of woman’s rights, true dignity, honor and usefulness.”
—Angelina Grimké (1805–1879)