CLEANUP - changes to folder structure

Theory/NGMI/mutual_infomation_undertsanding_gpt.m

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+% estimate_entropies.m
+clear; rng(0);
+
+%% 1) Parameters
+M        = 1e5;          % number of Monte-Carlo samples
+EbNo_dB  = 0;           % SNR per bit in dB
+EbNo     = 10^(EbNo_dB/10);
+sigma2   = 1/(2*EbNo);   % noise variance per real dimension
+
+% 4-QAM constellation (row vector)
+X  = [1+1j, 1-1j, -1+1j, -1-1j];  
+N  = numel(X);
+PX = ones(1,N)/N;        % uniform PMF
+
+%% 2) Generate transmit symbols and AWGN
+idx = randi(N,1,M);             % 1×M random symbol indices
+s   = X(idx);                   % 1×M transmitted symbols
+n   = sqrt(sigma2)*(randn(1,M) + 1j*randn(1,M));
+y   = s + n;                    % 1×M received samples
+
+%% 3) Compute p_{Y|X}(y|x) for each constellation point
+%    Create an N×M matrix where row n is |y - X(n)|^2
+d2     = abs(bsxfun(@minus, X(:), y)).^2;        % N×M
+pYgX   = (1/(pi*sigma2)) * exp(-d2 / sigma2);    % N×M
+
+%% 4) Estimate H(Y) = -E[ log2 p_Y(Y) ]
+%    Mixture density p_Y(y_m) = sum_n PX(n)*pYgX(n,m)
+pY     = PX * pYgX;                 % 1×M
+HY      = -mean(log2(pY));          % bits
+
+%% 5) Estimate H(Y|X) = -E[ log2 p(Y|X) ]
+%    For each m, pick the row corresponding to the true idx(m)
+linearIdx = sub2ind([N, M], idx, 1:M);
+pYgX_true = pYgX(linearIdx);        % 1×M
+HYgX     = -mean(log2(pYgX_true));  % bits
+
+%% 6) Mutual information
+I = HY - HYgX;
+
+%% 7) Display
+fprintf('Estimated H(Y)      = %.4f bits\n', HY);
+fprintf('Estimated H(Y|X)    = %.4f bits\n', HYgX);
+fprintf('Estimated I(X;Y)    = %.4f bits\n', I);