Recursive Least Squares Filter - Recursive Algorithm

Recursive Algorithm

The discussion resulted in a single equation to determine a coefficient vector which minimizes the cost function. In this section we want to derive a recursive solution of the form

where is a correction factor at time . We start the derivation of the recursive algorithm by expressing the cross correlation in terms of

where is the dimensional data vector

Similarly we express in terms of by

In order to generate the coefficient vector we are interested in the inverse of the deterministic autocorrelation matrix. For that task the Woodbury matrix identity comes in handy. With

	is -by-
	is -by-1
	is 1-by-
	is the 1-by-1 identity matrix

The Woodbury matrix identity follows

To come in line with the standard literature, we define

where the gain vector is

Before we move on, it is necessary to bring into another form

Subtracting the second term on the left side yields

With the recursive definition of the desired form follows

Now we are ready to complete the recursion. As discussed

The second step follows from the recursive definition of . Next we incorporate the recursive definition of together with the alternate form of and get

With we arrive at the update equation

where is the a priori error. Compare this with the a posteriori error; the error calculated after the filter is updated:

That means we found the correction factor

This intuitively satisfying result indicates that the correction factor is directly proportional to both the error and the gain vector, which controls how much sensitivity is desired, through the weighting factor, .