MSE in adaptative filter
This scilab script implements a simple LMS adaptive filter.

M = 50;
LMS = zeros(M,N);
LMS(:,1) = zeros(M,1);
ERR = zeros(1,N);
y = zeros(1,N);
mu = 0.0005;

for t=M:N,
    Uwindowed = u(t - [0:M-1]');
    y(t) = LMS(:,t)'*Uwindowed.';
    ERR(t) = d(t) - y(t);
    LMS(:,t+1) = LMS(:,t) + mu*Uwindowed.'*ERR(t);
end;


The inputs to it are u(t), the noisy signal and d(t) the desired signal.  For the standard LMS algorithm to work, you need to have an idea of what the noiseless signal looks like.

From your equations:


$\alpha$ appears to be similar to mu (or mu/2).
u is your s+n.
I am not sure what your d is.
Your equation for the error is, er, erroneous. :-)