Adaptive Filters

CHAPTER 39 Fast Fixed-Order Filters

The arguments so far do not only provide an interpretation for the columns of but they also motivate other efficient RLS implementations. These alternative implementations are not in the form of array algorithms, as was the case in Chapter 37, but in terms of explicit sets of recursions. We start with the Fast Transversal Filter (FTF) form.

39.1 FAST TRANSVERSAL FILTER

Define the normalized gain vector

and observe that the RLS recursion (37.2) can be used to time-update the weight vectors as follows:

(39.1)

(39.2)

where c_M,_i-i is used in the recursion for instead of c_M,_i, since the forward projection problem (38.19) is based on H_M,i–1while the backward projection problem (38.10) is based on H_M,i

Moreover, writing expression (37.10) in terms of {C_M,i,C_M,i–1}, and using the low-rank factorization (38.25) at time (i – 1), we obtain

(39.3)

in terms ...