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% Example: System Identification %
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% %
% In this example we have a typical system identification scenario. We want %
% to estimate the filter coefficients of an unknown system given by Wo. In %
% order to accomplish this task we use an adaptive filter with the same %
% number of coefficients, N, as the unkown system. The procedure is: %
% 1) Excitate both filters (the unknown and the adaptive) with the signal %
% x. In this case, x is chosen according to the 4-QAM constellation. %
% The variance of x is normalized to 1. %
% 2) Generate the desired signal, d = Wo' x + n, which is the output of the %
% unknown system considering some disturbance (noise) in the model. The %
% noise power is given by sigma_n2. %
% 3) Choose an adaptive filtering algorithm to govern the rules of coefficient %
% updating. %
% %
% Adaptive Algorithm used here: RLS_IIR %
% %
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% Definitions:
ensemble = 100; % number of realizations within the ensemble
K = 500; % number of iterations
H = [0.32+0.21*j,-0.3+0.7*j,0.5-0.8*j,0.2+0.5*j].';
Wo = real(H); % unknown system
sigma_n2 = 0.04; % noise power
M = 3; % number of coefficients of the adaptive filter
N = 2;
lambda = 0.97; % forgetting factor
delta = 1e-2;
alpha = 1-lambda;
mu = 0.1;
% Initializing & Allocating memory:
theta = zeros(M+1+N,K+1,ensemble); % coefficient vector for each iteration and realization
MSE = zeros(K,ensemble); % MSE for each realization
MSEmin = zeros(K,ensemble); % MSE_min for each realization
% Computing:
for l=1:ensemble,
X = zeros(M+1,1); % input at a certain iteration (tapped delay line)
d = zeros(1,K); % desired signal
x = sign(randn(K,1)); % Creating the input signal (normalized)
sigma_x2 = var(x); % signal power = 1
n = sqrt(sigma_n2)*randn(K,1); % complex noise
for k=1:K,
X = [x(k,1)
X(1:(M),1)]; % input signal (tapped delay line)
d(k) = (Wo'*X(:,1))+n(k); % desired signal
end
S = struct('lambda',lambda,'M',M,'N',N,'delta',delta);
[y,e,theta] = RLS_IIR(d,transpose(x),S);
MSE(:,l) = MSE(:,l)+(abs(e(:,1))).^2;
MSEmin(:,l) = MSEmin(:,l)+(abs(n(:))).^2;
end
% Averaging:
theta_av = sum(theta,3)/ensemble;
MSE_av = sum(MSE,2)/ensemble;
MSEmin_av = sum(MSEmin,2)/ensemble;
% Plotting:
figure,
plot(1:K,10*log10(MSE_av),'-k');
title('Learning Curve for MSE');
xlabel('Number of iterations, k'); ylabel('MSE [dB]');
figure,
plot(1:K,10*log10(MSEmin_av),'-k');
title('Learning Curve for MSEmin');
xlabel('Number of iterations, k'); ylabel('MSEmin [dB]');
fprintf('Adaptive Filter Coefficients (last iteration computed over %d runs): \n',ensemble);
fprintf('Numerator coefficients (direct part): \n ');
theta_av(N+1:end,end).'
fprintf('Denominator coefficients (recursive part): \n ');
theta_av(1:N,end).'
fprintf('Unkown system (FIR): \n ');
Wo.'
