9

Variants of Least Mean-Square Algorithm

9.1    THE NORMALIZED LEAST MEAN-SQUARE ALGORITHM

Consider the conventional least mean-square (LMS) algorithm with the fixed step-size parameter µ replaced with a time-varying variable µ(n) as follows (we substitute 2µ with µ for simplicity):

w(n+1)=w(n)+μ(n)e(n)x(n)

(9.1)

Next, define a posteriori error, eps(n), as

eps(n)=d(n)wT(n+1)x(n)

(9.2)

Substituting (9.1) in (9.2), and taking into consideration the error equation e(n) = d(n) − wT (n)x (n), we obtain

eps(n)=[ 1μ(n)xT(n)x(n) ]e(n)xT(n)x(n)= x(n) 2=i=0M1|x(ni)|2

(9.3)

Minimizing eps (n) with respect to µ(n) results in (see Problem 9.1.1)

μ(n)=1 x(n) 2

(9.4)

Substituting (9.4) in (9.1), we find ...

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