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);

Functions/Theory/PMD_maxwellshape/dgd_considerations.m

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+
+if 0
+pmd             = 0.2 * ( 1e-12 / sqrt(1e3) ) ;   % 0.1 ps/sqrt(km) -> 1e-9 -> s/sqrt(m)
+L               = 1000;     % m
+corr_len        = 100;
+num_wave_plates = 100;
+wp_len          = L / num_wave_plates; %waveplate length
+dgd             = pmd * sqrt(L); 
+fdac            = 120e9;    % GHz
+
+fsim            = fdac * 512 ; %oversampled simulation frequency
+dt              = 1/fsim;   % sample time
+nt              = 4;      % sampled signal length
+omega           = 2*pi*[(0:nt/2-1),(-nt/2:-1)]/(dt*nt) ;  %angular frequency vector for optical signal of length nt
+omega           = 2*pi*fsim ;  %angular frequency vector for optical signal of length nt
+
+for rlz = 1:20000
+
+    db0 = [];
+    db1 = [];
+    delta_beta = [];
+    
+    db0             = (rand(num_wave_plates,1)*2*pi - pi); % normal distr. between -pi <-> +pi
+
+    db1             = sqrt(3*pi/8)*(dgd/fsim)/ num_wave_plates .* omega; % linear increasing delta beta 1
+    
+    %loop over waveplates
+    for n_wp = 1:length(db0)
+        delta_beta(n_wp,:)      = (db1+db0(n_wp))./corr_len;
+    end
+    
+    dphi            = delta_beta*wp_len;       %phase shift due to propagation constant = beta * L
+
+    if rlz == 1
+        figure()
+        plot(cumsum( delta_beta ));
+    end
+    
+    dphi_end(rlz)   = sum( dphi );
+
+end
+
+figure;
+histogram((dphi_end),100);
+
+deltaT = mean(abs(dphi_end));
+
+%H               = exp(-1j*dphi);       % Filter to apply phase shift in freq. domain
+end
+
+
+PMD         = 0.1 * ( 1e-12 / sqrt(1e3) ); %PMD s/sqrt(m)
+L           = 10e3;                   %m
+sigma_dgd   = PMD * sqrt(L);       % in sec.
+disp(['variance of DGD in pico seconds: ',num2str(sigma_dgd*1e12)]);
+disp(['variance of DGD in pico seconds: ',num2str(sigma_dgd*1e12)]);
+
+dgd = [];
+for i = 1:4
+    dgd(i,:)       = DGD(sigma_dgd,10000);
+end
+
+rms =  PMD * sqrt(40e3) * 1e12 ;       % in sec.
+t = [0:0.01:100];
+z2 = sqrt(2/pi).*(t.^2)/(rms.^3).*exp(-t.^2/(2*rms^2)) ;
+figure;
+histogram(dgd*1e12,1000,"Normalization","pdf");
+hold on
+plot(t(1:200),z2(1:200),'r');
+hold off
+
+function z = DGD(sigma_dgd,N)
+
+    %PMD = sqrt( (PMD*1e12)^2/3 );
+
+   % PMD = PMD * 1/( 1e-12 / sqrt(1e3) );
+
+    sigma_dgd = sigma_dgd*1e12;
+
+    x = [0:N];
+
+    almaga = randn(3,N)*sigma_dgd;
+
+    y = almaga.^2;
+
+    z0 = sqrt(sum(y,1));
+
+   
+    disp(mean(z0));
+    disp(std(z0));
+  % disp(var(z0));
+
+    z1 = [];
+
+    for l = 0:0.1:N
+        z1 = [z1 sum(z0(1,:)>l & z0(1,:)<l+0.1) / N ];
+    end
+
+     z = z0 *1e-12;
+
+    rms = sigma_dgd;
+    t = [0:0.01:N];
+    z2 = sqrt(2/pi).*(t.^2)/(rms.^3).*exp(-t.^2/(2*rms^2)) ;
+
+    figure;
+    histogram(z0,1000,"Normalization","pdf");
+    hold on
+    plot(t(1:200),z2(1:200),'r');
+
+    return
+
+end
+
+
+
+

Functions/Theory/PMD_maxwellshape/pmd_maxwell.m

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+
+PMDcoeff    = 0.1 * ( 1e-12 / sqrt(1e3) ); %PMD s/sqrt(m)
+
+
+L_40        = 40e3;                   %m
+mean_dgd    = PMDcoeff * sqrt(L_40);       % in sec. E(tau) == PMD value 
+
+disp(['defined mean of DGD in pico seconds: ',num2str(mean_dgd*1e12)]);
+
+dgd_40km = [];
+for i = 1:4000
+    dgd_40km(i,:) = getDGDrealization(mean_dgd,1);
+end
+
+disp(['simulated mean of DGD in pico seconds: ',num2str(mean(dgd_40km)*1e12)]);
+
+%%% 2 test for segmented link %%%%
+
+L_10        = 10e3;                   %m
+mean_dgd    = PMDcoeff * sqrt(L_10);       % in sec. E(tau) == PMD value 
+
+dgd_10km = [];
+for j = 1:4
+    for i = 1:1000
+        dgd_10km(i,j) = getDGDrealization(mean_dgd,1);
+    end
+end
+
+a = sum(dgd_10km,2) ;
+
+%%% calc maxwell curve
+mean_dgd =  PMDcoeff * sqrt(L_40) * 1e12 ; % in sec.
+q = sqrt(pi/8) * mean_dgd;
+t = [0:0.01:100];
+
+maxwell = sqrt(2/pi).* (t.^2)/(q.^3) .* exp(-t.^2/(2*q^2)) ;
+
+
+%%% plot everything 
+
+figure;
+histogram(dgd_40km*1e12,1000,"Normalization","pdf",'EdgeColor','none','FaceColor','b','FaceAlpha',0.6);
+hold on
+histogram(a*1e12,1000,"Normalization","pdf",'EdgeColor','none','FaceColor','g','FaceAlpha',0.6);
+plot(t(1:200),maxwell(1:200),'r','LineWidth',3);
+xline(mean_dgd,'LineWidth',3,'Color','magenta')
+hold off
+
+function z = getDGDrealization(mean_dgd,N)
+
+    mean_dgd = mean_dgd; %to pico seconds
+    q = sqrt(pi/8) * mean_dgd;
+    threenormaldists = randn(3,N)*q; %3x normal dist around [-1,1] with a deviation of mean_dgd 
+
+    y = threenormaldists.^2;
+
+    z = sqrt(sum(y,1));
+  
+end