File:SG RLS LMS chan var.png

clear all; close all; clc 

%% Initialization

% channel parameters
sigmaS = 1; %signal power
sigmaN = 0.01; %noise power

% CSI (channel state information):
% the channel for the transmission of the first NS1 training symbols
channel1 = [0.722 - 0.779i; -0.257 - 0.722i; -0.789 - 1.862i];
% the channel for the transmission of the next NS2 training symbols
channel2 = [-0.831 - 0.661i;-1.071 - 0.961i; -0.551 - 0.311i];

M = 5; % filter order

% step sizes
mu_LMS = [0.01,0.07];
mu_SG = [0.01,0.07];

% symbols / ensembles
NS1 = 500;
NS2 = 500;
NS = NS1+NS2;
NEnsembles = 1000; %number of ensembles

%% Compute Rxx and p

%the maximum index of channel taps (l=0,1...L):
L = length(channel1) - 1;  
H = convmtx(channel1, M-L); %channel matrix (Toeplitz structure)
Rnn = sigmaN*eye(M); %the noise covariance matrix

% Inline functions:
calc_Rxx = @(channel) ...
sigmaS*(convmtx(channel, M-L)*convmtx(channel, M-L)')+sigmaN*eye(M);

calc_p = @(channel) sigmaS*(convmtx(channel,M-L))*[1; zeros(M-L-1, 1)];

Rxx = zeros(M,M,2);
p = zeros(M,2);
A = calc_Rxx(channel1);

Rxx(:,:,1) = calc_Rxx(channel1);
Rxx(:,:,2) = calc_Rxx(channel2);
p(:,1) = calc_p(channel1);
p(:,2) = calc_p(channel2);

% An inline function to calculate MSE(w) for a weight vector w
calc_MSE = @(w, ch) real(w'*Rxx(:,:,ch)*w - w'*p(:, ch) - p(:, ch)'*w + sigmaS);

%% Adaptive Equalization
N_test = 2;
MSE_LMS = zeros(NEnsembles, NS, N_test);
MSE_SG = zeros(NEnsembles, NS, N_test);
MSE_RLS = zeros(NEnsembles, NS, N_test);

for nEnsemble = 1:NEnsembles
	%initial symbols:
	symbols1 = sigmaS*sign(randn(1,NS1));
    symbols2 = sigmaS*sign(randn(1,NS2));
	%received noisy symbols:
	X1 = convmtx(channel1, M-L)*hankel(symbols1(1:M-L),[symbols1(M-L:end),zeros(1,M-L-1)]) + ...
		sqrt(sigmaN)*(randn(M,NS1)+1j*randn(M,NS1))/sqrt(2);

	X2 = convmtx(channel2, M-L)*hankel(symbols2(1:M-L),[symbols2(M-L:end),zeros(1,M-L-1)]) + ...
		sqrt(sigmaN)*(randn(M,NS2)+1j*randn(M,NS2))/sqrt(2); 

	X = [X1, X2];
	symbols = [symbols1, symbols2];
	for n_mu = 1:N_test
		w_LMS = zeros(M,1);
		w_SG = zeros(M,1);
		p_SG = zeros(M,1);
		R_SG = zeros(M);
		for n = 1:NS
			if n <= NS1, curh = 1; else curh = 2; end 
			%% LMS - Least Mean Square
			e = symbols(n) - w_LMS'*X(:,n);
			w_LMS = w_LMS + mu_LMS(n_mu)*X(:,n)*conj(e);
			MSE_LMS(nEnsemble,n,n_mu)= calc_MSE(w_LMS, curh);
			
			%% SG - Stochastic gradient
			R_SG = 1/n*((n-1)*R_SG + X(:,n)*X(:,n)');
			p_SG = 1/n*((n-1)*p_SG + X(:,n)*conj(symbols(n)));
			w_SG = w_SG + mu_SG(n_mu)*(p_SG - R_SG*w_SG);
			MSE_SG(nEnsemble,n,n_mu)= calc_MSE(w_SG, curh);
		end
	end
	
	%RLS - Recursive Least Squares
	lambda_RLS = [0.8; 1]; %forgetting factors
	for n_lambda=1:length(lambda_RLS)
		%Initialize the weight vectors for RLS
		delta = 1; 
		w_RLS = zeros(M,1);
		P = eye(M)/delta; % (n-1)-th iteration, where n = 1,2...
		PI = zeros(M,1); % n-th iteration
		K = zeros(M,1);
		for n=1:NS
			if n <= NS1, curh = 1; else curh = 2; end
			% the recursive process of RLS
			PI = P*X(:,n);
			K = PI/(lambda_RLS(n_lambda)+X(:,n)'*PI);
			ee = symbols(n) - w_RLS'*X(:,n);
			w_RLS = w_RLS + K*conj(ee);
			MSE_RLS(nEnsemble,n,n_lambda)= calc_MSE(w_RLS, curh);
			P = P/lambda_RLS(n_lambda) - K/lambda_RLS(n_lambda)*X(:,n)'*P;
		end
	end
end

%% Wiener Solution
MSE_Wiener(1:NS1) = calc_MSE(Rxx(:,:,1)\p(:,1),1);
MSE_Wiener(NS1+1:NS) = calc_MSE(Rxx(:,:,2)\p(:,2),2);

MSE_LMS_1 = mean(MSE_LMS(:,:,1));
MSE_LMS_2 = mean(MSE_LMS(:,:,2));
MSE_SG_1 = mean(MSE_SG(:,:,1));
MSE_SG_2 = mean(MSE_SG(:,:,2));
MSE_RLS_1 = mean(MSE_RLS(:,:,1));
MSE_RLS_2 = mean(MSE_RLS(:,:,2));

figure(1)
n = 1:NS;
m= [2 4 6 10 30 60 100 300 600 1000];

semilogy(m, MSE_LMS_1(m),'+','linewidth',2, 'color','blue');
hold all;
semilogy(m, MSE_LMS_2(m),'o','linewidth',2, 'color','blue');
semilogy(m, MSE_SG_1(m),'+','linewidth',2, 'color','red');
semilogy(m, MSE_SG_2(m),'o','linewidth',2, 'color','red');
semilogy(m, MSE_RLS_1(m),'+','linewidth',2, 'color','green');
semilogy(m, MSE_RLS_2(m),'o','linewidth',2, 'color','green');

semilogy(n, MSE_Wiener(n), 'color','black','linewidth',2);
semilogy(n, MSE_LMS_1(n),'linewidth',2, 'color','blue');
semilogy(n, MSE_LMS_2(n),'linewidth',2, 'color','blue');
semilogy(n, MSE_SG_1(n),'linewidth',2, 'color','red');
semilogy(n, MSE_SG_2(n),'linewidth',2, 'color','red');
semilogy(n, MSE_RLS_1(n),'linewidth',2, 'color','green');
semilogy(n, MSE_RLS_2(n),'linewidth',2, 'color','green');
grid on
xlabel('Ns');
ylabel('MSE');
title(['LMS, SG, RLS, \sigma_N= ' num2str(sigmaN) ', \sigma_S= '...
    num2str(sigmaS) ', M= ' num2str(M) ', L= ' num2str(L) ]);
legend(['LMS, \mu=' num2str(mu_LMS(1))],['LMS, \mu=' num2str(mu_LMS(2))],...
    ['SG, \mu=' num2str(mu_SG(1))],['SG, \mu=' num2str(mu_SG(2))],...
    ['RLS, \lambda=' num2str(lambda_RLS(1))],['RLS, \lambda=' ...
    num2str(lambda_RLS(2))],'Weiner solution',2);
axis([0 NS 0.002 1])