setmem_rls.m

System identification with adaptive filter using full and partial-update Recursive-Least-Squares

function [e,w,u] = setmem_rls(x,d,N,M,w0,lambda,epsilon,gamma)
%setmem_rls(x,d,N,M,w0,lambda,epsilon,gamma) implements set-membership
%partial-update RLS algorithm
%   ----------------
%   input parameters
%   ----------------
%   x : Lx1 input signal
%   d : Lx1 desired response 
%   N : filter length
%   M : number of coefficients to be updated
%   w0 : Nx1 initialization
%   lambda : exponential forgetting factor
%   epsilon: regularization parameter 
%   gamma : error bound
%   ----------------
%   function outputs
%   ----------------
%   e : Lx1 output error vector
%   w : LxN coefficients vectors
%   u : Lx1 update indicator

x = x(:);
d = d(:);
w0 = w0(:);

L = length(x);
w = zeros(L,N);
e = zeros(L,1);
u = zeros(L,1);

w(1,:) = w0';
xvec = zeros(N,1);
P = (1/epsilon) * eye(N);
invlambda = 1/lambda;

for i = 1:L-1
    xvec = [x(i);xvec(1:N-1)];
    e(i) = d(i)-w(i,:)*xvec;    %error
    
    [xs,ix] = sort(-abs(xvec));     %form IM
    de = zeros(1,N);
    de(ix(1:M)) = 1;
    IM = diag(de);
    
    %PM(k) computation
    aa = 1/(1+invlambda*xvec'*P*IM*xvec);
    P = invlambda*P - (invlambda^2*P*IM*xvec*xvec'*P)*aa;
    
    if abs(e(i))>gamma,
        upd = (xvec/norm(xvec)^2*(aa - gamma/abs(e(i))) + P*IM*xvec)*e(i);
        w(i+1,:) = w(i,:) + upd';
        u(i) = 1;
    else
        w(i+1,:) = w(i,:);
    end
end
xvec = [x(L);xvec(1:N-1)];
e(L) = d(L)-w(L,:)*xvec;