Adaptive Filtering

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):

$w(n+1)=w(n)+\text{μ}(n)e(n)x(n)$

(9.1)

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

$e_{\text{p}s}(n)=d(n)-w^T(n+1)x(n)$

(9.2)

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

$e_{ps}(n)=\left[1-\text{μ}(n)x^T(n)x(n)\right]e(n)x^T(n)x(n)={\Vert x(n)\Vert }^2={\displaystyle \sum _{i=0}^{M-1}|}x(n-i){|}^2$

(9.3)

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

$\text{μ(n)}=\frac{1}{{\Vert x(n)\Vert }^2}$

(9.4)

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