Convergence and learning curve of 2nd order RLS predictor

** lab16_1.m * mw * 05/20/2007

Contents

2nd order system - MSE optimum 2nd order prediction

a = [1 -.71 .25];        % denominator coefficients
b = .24192*[1 2 1];      % numerator coefficients
Nh = 51;                 % simulation lenght of impulse response
h = impz(b,a,Nh);        % impulse response
Rhh = conv(h,flipud(h)); % acf
R = [Rhh(Nh) Rhh(Nh+1);
     Rhh(Nh+1) Rhh(Nh)]/Rhh(Nh);     % correlation matrix
r = [Rhh(Nh+1); Rhh(Nh+2)]/Rhh(Nh);  % cross-correlation vector
fprintf('lab16_2 2nd order correlation matrix\n')
fprintf('Rxx[1,1] = %+6.4f ; Rxx[1,2] = %+6.4f \n',R(1,1),R(1,2))
fprintf('Rxx[2,1] = %+6.4f ; Rxx[2,2] = %+6.4f \n',R(2,1),R(2,2))
fprintf('crosscorrelation vector\n')
fprintf('rxd[1] = %+6.4f ; rxd[2] = %+6.4f \n',r(1),r(2))
b_opt = R\r;    % optimum 2nd order prediction coefficients
MSEopt = R(1,1) + b_opt'*R*b_opt - 2*b_opt'*r; % minimum MSE
fprintf('prediction coefficients and error signal power\n')
fprintf('2nd order : b0 = %+6.4f ; b1 = %+6.4f ; MSEopt = %+6.4f \n',b_opt(1),b_opt(2),MSEopt)
lambda = eig(R); % eigenvalues of the correlation matrix
fprintf('Eigenvalues : lambda1 = %+6.4f ; lambda2 = %+6.4f \n',lambda(1),lambda(2))
R_1 = inv(R);  % inverse correlation matrix
fprintf('inverse correlation matrix\n')
fprintf('Rxx^-1[1,1] = %+6.4f ; Rxx^-1[1,2] = %+6.4f \n',R_1(1,1),R_1(1,2))
fprintf('Rxx^-1[2,1] = %+6.4f ; Rxx^-1[2,2] = %+6.4f \n',R_1(2,1),R_1(2,2))

lab16_2 2nd order correlation matrix
Rxx[1,1] = +1.0000 ; Rxx[1,2] = +0.7885 
Rxx[2,1] = +0.7885 ; Rxx[2,2] = +1.0000 
crosscorrelation vector
rxd[1] = +0.7885 ; rxd[2] = +0.3684 
prediction coefficients and error signal power
2nd order : b0 = +1.3168 ; b1 = -0.6700 ; MSEopt = +0.2085 
Eigenvalues : lambda1 = +0.2115 ; lambda2 = +1.7885 
inverse correlation matrix
Rxx^-1[1,1] = +2.6439 ; Rxx^-1[1,2] = -2.0848 
Rxx^-1[2,1] = -2.0848 ; Rxx^-1[2,2] = +2.6439 

sample function of model process

M = 1e5;           % number of iterations
randn('state',0);
w = randn(1,M);    % innovation
d = filter(b,a,w); % IIR filter for process modeling - desired signal

simulation of adaptiv system

initalization

x = [0 d(1:M-1)];   % input signal for adaptive FIR system
p = 2;              % prediction order (FIR filter order = p-1)

RLS algorithm

rls = struct('alpha',1/64,'beta',1/64,'b',b_opt,'x',zeros(p-1,1),'R_1',R_1);
[y,e,rls,SAVEb] = rls_algorithm(x,d,rls);
fprintf('final prediction coefficients\n')
fprintf('(%i) b0 = %6.4f ; b1 = %6.4f\n',M,rls.b(1),rls.b(2))
EccessMSE = sum(e.^2)/M - MSEopt;
fprintf('Misadjustment = %6.4f\n\n',EccessMSE/MSEopt)

final prediction coefficients
(100000) b0 = 1.4700 ; b1 = -0.7348
Misadjustment = 0.0343

Graphics

b = SAVEb(1:p,:);
LC = zeros(1,M);
for n = 1:M
    LC(n) = R(1,1) + b(1:2,n)'*R*b(1:2,n) - 2*b(1:2,n)'*r;   % MSE, learning curve
end
FIG1 = figure('Name','lab16_2 : Desired signal, prediction error and variance estimate ',...
    'NumberTitle','off');
subplot(2,1,1),plot(0:M-1,d,'LineWidth',2), grid, axis([0 M-1 -4 4])
xlabel('n \rightarrow'), ylabel('d[n] \rightarrow')
title('Desired signal')
subplot(2,1,2),plot(0:M-1,e,'LineWidth',2), grid, axis([0 M-1 -4 4])
xlabel('n \rightarrow'), ylabel('e[n] \rightarrow')
title('Error')
FIG2 = figure('Name','lab16_2 : Learning curve','NumberTitle','off');
plot(1:M-1,LC(1:M-1)/MSEopt,'LineWidth',2), grid
axis([1 M 1 5]);
xlabel('n \rightarrow'), ylabel('MSE[n] / MSEopt \rightarrow')
title('Learning curve (MSE)')
% 2nd order prediction error - MSE
c0 = -2:.05:2; c1 = -2:.05:2;
MSE = zeros(length(c0),length(c1));
for k=1:length(c0)
    for l=1:length(c1)
        cc = [c0(k); c1(l)] + b_opt;
        MSE(l,k) = R(1,1) + cc'*R*cc - 2*cc'*r;
    end
end
FIG3 = figure('Name','lab16_2 : Realtiv MSE and convergence of coefficients c[n]',...
    'NumberTitle','off');
c = b(1:2,:) - repmat(b_opt,1,M);
V = [1 1.2 1.5 2 3 4 6];
[cc,h] = contour(c0,c1,MSE/MSEopt,V,'LineWidth',1,'Color','b'); grid
clabel(cc);
xlabel('c_0 \rightarrow'), ylabel('c_1 \rightarrow')
title('Convergence of coefficients and MSE')
hold on
plot(c(1,:),c(2,:),'o','MarkerSize',4,'MarkerFaceColor','b')
hold off

Published with MATLAB® 7.3