PMD theory; S21 measures; NGMI theory

Functions/Theory/NGMI/calculate_gmi_example.m

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+% Parameters
+symbols = [0, 1, 2, 3];                  % PAM-4 symbols
+P_X = [0.25, 0.25, 0.25, 0.25];         % Uniform probabilities
+sigma2 = 0.1;                           % Noise variance
+received_samples = [0.2, 1.1, 1.9, 2.8];% Received symbols (example)
+gray_bits = [0 0; 0 1; 1 1; 1 0];       % Gray coding (bits per symbol)
+m = size(gray_bits, 2);                 % Bits per symbol
+N = length(received_samples);           % Number of received samples
+
+% Conditional probability function for AWGN
+q_Y_given_X = @(y, x) (1 / sqrt(2 * pi * sigma2)) * exp(-(y - x).^2 / (2 * sigma2));
+
+% Entropy term
+H_X = -sum(P_X .* log2(P_X)); % Entropy of input distribution
+
+% GMI computation
+noise_impact_term = 0;
+for k = 1:N
+    y_k = received_samples(k); % Current received sample
+    [~, closest_symbol_idx] = min(abs(symbols - y_k)); % Closest symbol index
+    closest_symbol = symbols(closest_symbol_idx);     % Closest symbol
+    
+    for i = 1:m
+        % Extract i-th bit for each symbol
+        bit_mask = gray_bits(:, i); % Binary column for i-th bit of all symbols
+        matching_symbols = symbols(bit_mask == gray_bits(closest_symbol_idx, i));
+
+        % Numerator: Sum over x in x_{b_{k, i}}
+        numerator = sum(q_Y_given_X(y_k, matching_symbols) .* P_X(ismember(symbols, matching_symbols)));
+
+        % Denominator: Sum over all x
+        denominator = sum(q_Y_given_X(y_k, symbols) .* P_X);
+
+        % Logarithmic contribution
+        noise_impact_term = noise_impact_term + log2(numerator / denominator);
+    end
+end
+
+% Normalize the noise impact term by N
+noise_impact_term = noise_impact_term / N;
+
+% GMI
+GMI = H_X + noise_impact_term;
+NGMI = GMI / m;
+
+
+% Display the result
+fprintf('GMI: %.4f bits\n', GMI);
+fprintf('NGMI: %.4f bits\n', NGMI);

Functions/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);