Adaptive Filters

CHAPTER 18 Performance of Sign-Error LMS

We now illustrate the use of the variance relation (15.40) in evaluating the steady-state performance of the sign-error LMS algorithm,

(18.1)

for which

(18.2)

where the data {d(i),u_i,v(i)} are assumed to satisfy model (15.16). To proceed, we need to distinguish between real-valued data and complex-valued data. This is because the definition of the sign function is different in both cases. The final expressions for the EMSE, however, will turn out to be identical except for a scaling factor.

18.1 REAL-VALUED DATA

In this case the sign function is defined as

Using the fact that g²(x) = 1 almost everywhere on the real line, the variance relation (15.40) becomes

(18.3)

In order to arrive at the value of the EMSE we need to evaluate the expectation on the right-hand side. For this purpose, we shall rely on the following assumption:

(18.4)

This condition is violated in general. For example, even if we assume that v(i) and u_i in model (15.16)