Adaptive Filters

CHAPTER 16 Performance of LMS

We now move on to illustrate the use of the variance relation (15.40) in evaluating the steady-state performance of adaptive filters. We start with the simplest of algorithms, namely, LMS.

16.1 VARIANCE RELATION

Thus, assume that the data {d(i), ui} satisfy model (15.16) and consider the LMS recursion

(16.1)

for which

(16.2)

Relation (15.40) then becomes

(16.3)

Several terms in this equality get cancelled. We shall carry out the calculations rather slowly in this section for illustration purposes only. Later, when similar calculations are called upon, we shall be less detailed.

To begin with, the expression on the left-hand side of (16.3) expands to

(16.4)

where we used the fact that v(i) is independent of both tt, and e_a(i) (recall Lemma 15.1), so that the cross-terms involving {v(i), e_a(i), ui} cancel out. We also used the fact that

Similarly, the expression on the right-hand side of (16.3) simplifies to 2E |e_a(i)|², which is simply 2ζ^LMS as i → ...