Adaptive Filters

CHAPTER 19 Performance of RLS and Other Filters

We now examine the performance of the RLS algorithm and comment on the performance of several other adaptive filters.

19.1 PERFORMANCE OF RLS

We consider the RLS algorithm of Sec. 14.1, namely,

(19.1)

(19.2)

with initial condition P _–1 = ∈⁻¹I and 0 << λ < 1. The scalar ∈ is a small positive number and λ is usually close to one. We are using a boldface letter P_i to indicate that P_i is a random variable due to its dependence on the regressors {u_j} . Also, recall from (14.3) and (14.5) that

(19.3)

which shows that P _i > 0 for all finite i.

Compared with the update recursion (15.23), which was seen to be useful in studying the performance of several adaptive filters in the previous sections, we now find that the RLS update differs in a special way; it includes the matrix factor P _i, which appears multiplying from the left in (19.2). Still, the energy-conservation approach of Sec. 15.3 can be extended to treat this case, and even some other more general cases (see, e.g., Prob. IV.9 as well as Part V (Transient Performance) and the problems ...