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