Merge branch 'main' of cau-git.rz.uni-kiel.de:nt/mitarbeiter/silas/imdd_simulation

Functions/Theory/Dispersion/power_fading_gif.m

@@ -0,0 +1,68 @@
+%% ============================================================
+%  IM/DD Power Fading Evolution GIF (1 km -> 20 km)
+%  Uses the provided GifWriter (serial mode)
+% ============================================================
+
+clear; close all; clc;
+
+%% Fiber and system parameters
+lambda0 = 1310e-9;       % zero-dispersion wavelength [m]
+lambda  = 1275e-9;       % operating wavelength [m]
+S0      = 0.09;          % dispersion slope [ps/(nm^2·km)]
+c       = physconst('lightspeed');
+
+%% Derived quantities (length-independent)
+D_lambda = (S0/4) * (lambda*1e9 - (lambda0*1e9)^4/(lambda*1e9)^3); % ps/(nm·km)
+D_si     = D_lambda * 1e-6;                                        % s/m^2
+b2       = -D_si * lambda^2 / (2*pi*c);                            % s^2/m
+
+%% Frequency grid
+f_max = 150e9;
+f = linspace(0, f_max, 4000);  % [Hz]
+
+%% Figure setup (keep it stable for nicer GIFs)
+fig = figure('Color','w');
+ax = axes(fig); %#ok<LAXES>
+hold(ax,'on'); grid(ax,'on'); box(ax,'on');
+xlabel(ax,'Frequency [GHz]');
+ylabel(ax,'Magnitude [dB]');
+ylim(ax,[-30 0]);
+xlim(ax,[0 f_max/1e9]);
+
+%% GIF writer (serial mode; simplest)
+g = GifWriter('Name','power_fading_evolution', 'DelayTime',0.12, 'Parallel',false);
+
+%% Loop: 1 km to 20 km
+L = [1:20,19:-1:1];
+for L_km = L
+    L_meter = L_km * 1e3; % [m]
+
+    % IM/DD transfer function (power fading)
+    phi = 2*pi^2 * b2 * f.^2 * L_meter;
+    H   = abs(cos(phi));
+    HdB = 10*log10(max(H, 1e-12)); % avoid -Inf for deep notches
+
+    % Clear and redraw (stable axes)
+    cla(ax);
+
+    plot(ax, f/1e9, HdB, 'LineWidth', 1.8, 'Color','black');
+
+    % Analytic first-null frequency marker
+    f_null = sqrt(c*(0.5)/(abs(D_si)*lambda^2*L_meter));
+    xline(ax, f_null/1e9, 'r--', 'LineWidth', 1.2, ...
+        'Label', sprintf('f_{null}=%.1f GHz', f_null/1e9), ...
+        'LabelOrientation','horizontal', ...
+        'LabelVerticalAlignment','bottom');
+
+    title(ax, sprintf('Power Fading for: %.0f km @ 1275 nm', L_km));
+
+    drawnow;
+
+    % Add frame to GIF
+    g.addFrame(fig);
+end
+
+%% Done
+g.compile(fig.Number);
+
+disp(fullfile(g.OutputDir, sprintf('%s_fig_%d.gif', g.Name, fig.Number)));

Functions/Theory/Dispersion/power_fading_gif_lambda_variation.m

@@ -0,0 +1,69 @@
+%% ============================================================
+%  IM/DD Power Fading Evolution vs Wavelength (L = 10 km)
+% ============================================================
+
+clear; close all; clc;
+
+%% Fixed fiber parameters
+lambda0 = 1310e-9;       % zero-dispersion wavelength [m]
+S0      = 0.09;          % dispersion slope [ps/(nm^2·km)]
+L       = 10e3;          % fiber length FIXED [m]
+c       = physconst('lightspeed');
+
+%% Frequency grid
+f_max = 150e9;
+f = linspace(0, f_max, 4000);  % [Hz]
+
+%% Figure setup (stable axes for clean GIF)
+fig = figure('Color','w');
+ax = axes(fig);
+hold(ax,'on'); grid(ax,'on'); box(ax,'on');
+xlabel(ax,'Frequency [GHz]');
+ylabel(ax,'Magnitude [dB]');
+ylim(ax,[-30 0]);
+xlim(ax,[0 f_max/1e9]);
+
+%% GIF writer
+g = GifWriter('Name','power_fading_vs_wavelength', ...
+              'DelayTime',0.12, ...
+              'Parallel',false);
+
+%% Wavelength sweep (around ZDW)
+lambda_vec = linspace(1260e-9, 1360e-9, 25);  % 1260–1360 nm
+
+for k = 1:length(lambda_vec)
+
+    lambda = lambda_vec(k);
+
+    %% Dispersion for current wavelength
+    D_lambda = (S0/4) * (lambda*1e9 - (lambda0*1e9)^4/(lambda*1e9)^3); % ps/(nm·km)
+    D_si     = D_lambda * 1e-6;                                       % s/m^2
+    b2       = -D_si * lambda^2 / (2*pi*c);                           % s^2/m
+
+    %% Power fading transfer function
+    phi = 2*pi^2 * b2 * f.^2 * L;
+    H   = abs(cos(phi));
+    HdB = 10*log10(max(H, 1e-12));
+
+    cla(ax)
+    plot(ax, f/1e9, HdB, 'LineWidth',1.8,'Color','black');
+
+    %% First-null frequency
+    if abs(D_si) > 0
+        f_null = sqrt(c*(0.5)/(abs(D_si)*lambda^2*L));
+        xline(ax, f_null/1e9, 'r--', 'LineWidth',1.2, ...
+            'Label', sprintf('f_{null}=%.1f GHz', f_null/1e9), ...
+            'LabelOrientation','horizontal', ...
+            'LabelVerticalAlignment','bottom');
+    end
+
+    title(ax, sprintf('Power Fading for: 10 km @ %.0f nm', lambda*1e9));
+
+    drawnow;
+    g.addFrame(fig);
+end
+
+%% Compile GIF
+g.compile(fig.Number);
+
+disp(fullfile(g.OutputDir, sprintf('%s_fig_%d.gif', g.Name, fig.Number)));

Functions/Theory/Dissertation/loss_curve_digitized.m

@@ -1,120 +0,0 @@
-function loss_curve_digitized(filename)
-% PLOT_WPD_DATASETS Reads and plots scattering data from a CSV file.
-%   filename: String containing the path to the CSV file (e.g., 'wpd_datasets.csv')
-%
-%   This function expects a CSV with the following structure:
-%   Row 1: Dataset Names (every 2nd column)
-%   Row 2: Variable Names (X, Y, X, Y...)
-%   Row 3+: Numeric Data (potentially with NaNs for unequal lengths)
-
-    % --- 1. Import Data ---
-    if nargin < 1
-        filename = 'wpd_datasets.csv'; % Default filename
-    end
-
-    % Read the numeric data, skipping the first 2 header lines
-    % 'TreatAsMissing' ensures empty cells become NaNs
-    raw_data = readmatrix(filename, 'NumHeaderLines', 2);
-
-    % Read the first line separately to parse Dataset Names
-    fid = fopen(filename, 'r');
-    if fid == -1
-        error('Could not open file: %s', filename);
-    end
-    header_line = fgetl(fid);
-    fclose(fid);
-    
-    % Split header by comma to get names
-    raw_names = split(header_line, ',');
-    
-    % Extract non-empty names (assuming names are in col 1, 3, 5...)
-    dataset_names = raw_names(~cellfun('isempty', raw_names));
-    
-    % --- 2. Setup Plot ---
-    figure('Color', 'w', 'Position', [100, 100, 800, 600]);
-    ax = gca;
-    hold(ax, 'on');
-   
-    
-    line_styles = {'-', '-', '-', '-'};
-    
-    % --- 3. Iterate and Plot Each Dataset ---
-    num_datasets = length(dataset_names);
-    
-    for i = 1:num_datasets
-        % Calculate column indices for X and Y
-        % Dataset 1: Cols 1,2 | Dataset 2: Cols 3,4 | etc.
-        col_x = (i-1)*2 + 1;
-        col_y = (i-1)*2 + 2;
-        
-        % Extract data
-        if col_y > size(raw_data, 2)
-            warning('Data columns missing for dataset %d', i);
-            break;
-        end
-        
-        X = raw_data(:, col_x);
-        Y = raw_data(:, col_y);
-
-        
-        
-        % Remove NaNs (missing data due to unequal lengths)
-        valid_mask = ~isnan(X) & ~isnan(Y);
-        X = X(valid_mask);
-        Y = Y(valid_mask);
-
-        [X, sortIdx] = sort(X);
-        Y = Y(sortIdx);
-        
-        % 3. Smooth the Data
-
-        if numel(X) > 20
-        if 1
-            Y = smoothdata(Y, 'sgolay', 15); 
-        else
-            Y = movmean(Y, 15); 
-        end
-        end
-
-        
-        % Plotting
-        % Using semilogy because scattering data often spans orders of magnitude
-        % (Adjust to 'plot' if linear scale is preferred)
-        p = plot(ax, X, Y, ...
-            'LineStyle', line_styles{mod(i-1, length(line_styles)) + 1}, ...
-            'Marker', 'none', ...
-            'Color', 'black', ...
-            'LineWidth', 1.5, ...
-            'MarkerSize', 6, ...
-            'DisplayName', dataset_names{i});
-            
-        % Optional: Fill marker faces for better visibility
-        % p.MarkerFaceColor = p.Color;
-        % p.MarkerFaceAlpha = 0.3; % Semi-transparent fill
-    end
-    
-    % --- 4. Styling and Formatting ---
-    
-    % Axis Labels (Inferred from typical scattering plots)
-    xlabel(ax, 'Wavelength (\mu m)', 'FontSize', 12, 'FontWeight', 'bold');
-    ylabel(ax, 'Intensity / Cross-Section (a.u.)', 'FontSize', 12, 'FontWeight', 'bold');
-    
-    % Title
-    title(ax, 'Dataset Comparison', 'FontSize', 14);
-    
-    % Legend
-    legend(ax, 'Location', 'best', 'Interpreter', 'none', 'Box', 'on');
-    
-    % Grid
-    grid(ax, 'on');
-    ax.GridAlpha = 0.3;
-    ax.MinorGridAlpha = 0.1;
-    
-    % Set Log Scale for Y (likely required for this data type)
-    set(ax, 'YScale', 'log'); 
-    
-    % Enhance axis appearance
-    set(ax, 'Box', 'on', 'LineWidth', 1.2, 'FontSize', 10);
-    
-    hold(ax, 'off');
-end

Functions/Theory/Dissertation/wpd_datasets.csv

@@ -1,189 +0,0 @@
-Rayleigh,,Experimental,,Infrared Absorption,
-X,Y,X,Y,X,Y
-0.7066005680911753,3.651009696525016,0.7072188355785799,4.848577786727532,1.4943918372804705,0.009324755400827079
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-,,1.626903638312196,0.2467330049135315,,
-,,1.6281904819904773,0.25383038758817866,,
-,,1.6410748057322797,0.28430548752659124,,
-,,1.660414530477476,0.29244621369149987,,
-,,1.6642816810785659,0.29661578287603696,,
-,,1.6707152375133467,0.34423590359793155,,
-,,1.6739290369259265,0.3828666146961718,,
-,,1.6758540067903511,0.4228270071256609,,
-,,1.690028483993558,0.4702407837690221,,
-,,1.6964732936909575,0.48374976893430716,,
-,,1.6996923887565347,0.5083600695634991,,
-,,1.7003318388559272,0.5380344919079171,,

Functions/Theory/calcFWM/FWM_products.m

@@ -0,0 +1,69 @@
+
+
+
+w0 = [1290:2:1290+15*2]';
+w0 = [1290:2:1290+15*2]';
+w0 = [ 1302        1304        1306        1308]';
+%w0 = [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16]';
+m = 0.5*(numel(w0)^3 - numel(w0)^2);
+a = [1,1,1]';
+
+w = w0;
+
+
+
+
+for o = 2:3
+    
+    p = nchoosek(w,3);
+    
+    q = [];
+    parfor i = 1:size(p,1)
+
+        q_ = perms(p(i,:));
+        q = [q;q_];
+        
+    end
+    p = q;
+    %p = unique(q,"rows");
+    w_ = p(:,1) + p(:,2) - p(:,3);
+    a_ = ones(size(w_)).* 1/o;
+    
+    w = [w ; w_];
+    a = [a ; a_];
+%    w = unique(w);
+%     a = unique(a);
+
+    m(end+1) = 0.5*(numel(w)^3 - numel(w)^2);
+
+end
+
+figure(11)
+hold on 
+
+lambda = min(w):max(w);
+
+gen = sum(lambda == w,1);
+gen(gen==0) = NaN;
+stem(lambda,gen,"filled",'LineWidth',1,'Marker','o','MarkerSize',2,'LineStyle',':')
+
+initial = sum(lambda == w0,1);
+initial(initial==0) = NaN;
+stem(lambda,initial,"filled",'LineWidth',1.5,'MarkerSize',5,'Marker','^');
+
+xlabel('Wavelength');
+ylabel('number of FWM products');
+
+grid minor
+legend('Generated Products', 'Initial Channel Position')
+AxesMain = gca;
+fig = gcf;
+fontsize(AxesMain,8,"points")
+
+fig.Units = "centimeters";
+fig.Position = [2 2 8.5 7];
+
+
+
+
+

Functions/Theory/calcFWM/JLT_statistics_laser_and_zdw.m

@@ -0,0 +1,43 @@
+%% Laser Offset Statistics
+
+ figure
+for i = 1
+    res = 1.7e6;
+    n_chann = 16;
+    df_T_exact = (-n_chann/2+0.5:n_chann/2).* 200e9;
+    
+    for key = 1:100
+        laser_frequency_imperfection(key,:) = res .* round(randn(1,n_chann)*i*100);
+        df_T(key,:) = df_T_exact + laser_frequency_imperfection(key,:);
+    end
+    
+    hold on
+    histogram(laser_frequency_imperfection.*1e-6,100,"Normalization","probability","EdgeColor","none","FaceAlpha",0.3,'DisplayName',['Std. Dev.: ',num2str(mean(std(laser_frequency_imperfection))*1e-6),' MHz']);
+    xlabel('Laser Frequency Offset in MHz');
+    ylabel('Probability');
+    title(['Laser deviations from exact grid.'])
+
+end
+
+%% ZDW Statistics
+
+for k = 1:1000
+
+  % Set parameters
+  meanUniformMin = 1309;
+  meanUniformMax = 1315;
+  meanValue = 1310;
+  sigma = 2;
+
+  % Seed the random number generator (assuming Mersenne Twister)
+  rng(k);
+
+  % Generate normally distributed random numbers
+  randomNumbers(k,:) = normrnd(meanValue, sigma, [n_chann, 1]).';
+
+end
+
+figure;
+histogram(randomNumbers,100,"Normalization","probability","EdgeColor","none");
+xlabel('ZDW in nm');
+ylabel('Probability');

Functions/Theory/calcFWM/analytical_calculation_paper.m

@@ -0,0 +1,65 @@
+%%FWM analysis from "Analytical Calculation of the Number of
+%%Four-Wave-Mixing Products in Optical Multichannel Communication Systems"
+
+N_ = [4,8,16];
+
+df_hz = 400e9;
+center_nm = 1310;
+
+figure()
+for i = 1:length(N_)
+
+    vec = (2*N_(i)-1:-1:2-N_(i)) -(N_(i)/2+0.5);
+    channelplan_hz = nm2hz(center_nm) + (vec * df_hz) ;
+    channelplan_nm = hz2nm(channelplan_hz);
+
+    [Mndg,Mdg] = getProducts(N_(i));
+    total(i) = sum(Mndg) + sum(Mdg);
+    subplot(1,length(N_),i)
+    xline(calcWavelengthPlan(N_(i), df_hz, center_nm));
+    hold on
+    stem(channelplan_nm,(Mndg+Mdg),'filled','LineWidth',1,'Marker','o','MarkerSize',2)
+    stem(channelplan_nm,(Mdg),'filled','LineWidth',1,'Marker','none');
+    ylim([0,100])
+    xlim([1260, 1365]);
+    grid off
+    xlabel('O-band wavelength region in nm');
+    ylabel('Number of FWM products');
+    title([num2str(N_(i)),' ch.'])
+
+end
+
+
+function [Mndg,Mdg] = getProducts(N)
+ 
+    s = abs(2-N-1) ;
+    
+    for n = 2-N:2*N-1  
+    
+        if n<-N
+            Mdg(n+s) = 0;
+        elseif (-N <= n)&&(n <= 0)
+            Mdg(n+s) = N - ceil((N-n)/2);
+        elseif (1 <= n)&&(n <= N)
+            Mdg(n+s) = N - 1 - floor(n/2) - ceil((N-n)/2);
+        elseif (N < n)&&(n <= 2*N)
+            Mdg(n+s) = N - floor(n/2);
+        elseif n > 2*N
+            Mdg(n+s) = 0;
+        end
+    
+        if n<-N
+            Mndg(n+s) = 0;
+        elseif (-N <= n)&&(n < 1)
+            Mndg(n+s) = ceil((N^2 + n^2 - 2*N - 2*n + 2*N*n)/4);
+        elseif (1 <= n)&&(n <= N)
+            Mndg(n+s) = ceil(((N^2 - 6*N - 2*n^2 + 2*n + 4)/4) + floor((N*n)/2));
+        elseif (N < n)&&(n <= 2*N)
+            Mndg(n+s) = floor(N^2 + n^2 /4 - N*n);
+        elseif n > 2*N
+            Mndg(n+s) = 0;
+        end
+    
+    end
+end
+

Functions/Theory/calcFWM/calcFwmEfficiency.m

@@ -0,0 +1,23 @@
+function [eta, deltaBeta] = calcFwmEfficiency(f_i, f_j, f_k, f_0, Ds, alphaDbPerKm, Lkm)
+% FWM efficiency including phase mismatch and attenuation.
+% alphaDbPerKm is the power attenuation in dB/km, Lkm is the fiber length in km.
+
+deltaBeta = calcPhaseMatching(f_i, f_j, f_k, f_0, Ds);
+
+alphaNpPerM = alphaDbPerKm .* log(10) ./ 10 ./ 1e3;
+Lm = Lkm .* 1e3;
+
+denominator = alphaNpPerM.^2 + deltaBeta.^2;
+term1 = alphaNpPerM.^2 ./ denominator;
+
+loss_term = 1 - exp(-alphaNpPerM .* Lm);
+
+if abs(alphaNpPerM) < eps
+    term2 = 4 .* sin(deltaBeta .* Lm ./ 2).^2 ./ max((alphaNpPerM .* Lm).^2, eps);
+else
+    term2 = 1 + 4 .* exp(-alphaNpPerM .* Lm) .* sin(deltaBeta .* Lm ./ 2).^2 ./ (loss_term.^2);
+end
+
+eta = term1 .* term2;
+
+end

Functions/Theory/calcFWM/calcFwmPower.m

@@ -0,0 +1,48 @@
+function [P_fwm, eta, deltaBeta, Leff] = calcFwmPower( ...
+    f_i, f_j, f_k, f_0, Ds, alphaDbPerKm, Lkm, ...
+    P_i, P_j, P_k, gammaWInvKmInv, degeneracyFactor)
+% Calculate FWM power for a fiber with attenuation and phase mismatch.
+%
+% Inputs:
+%   f_i, f_j, f_k, f_0 : frequencies in Hz
+%   Ds                 : dispersion slope in ps / (nm^2 km)
+%   alphaDbPerKm       : attenuation in dB/km
+%   Lkm                : fiber length in km
+%   P_i, P_j, P_k      : launch powers in W
+%   gammaWInvKmInv     : nonlinear coefficient in 1/(W km)
+%   degeneracyFactor   : typically 3 for degenerate FWM, 6 for non-degenerate
+
+if nargin < 8 || isempty(P_i)
+    P_i = 1;
+end
+if nargin < 9 || isempty(P_j)
+    P_j = P_i;
+end
+if nargin < 10 || isempty(P_k)
+    P_k = 1;
+end
+if nargin < 11 || isempty(gammaWInvKmInv)
+    gammaWInvKmInv = 1;
+end
+if nargin < 12 || isempty(degeneracyFactor)
+    degeneracyFactor = 1;
+end
+
+[eta, deltaBeta] = calcFwmEfficiency(f_i, f_j, f_k, f_0, Ds, alphaDbPerKm, Lkm);
+
+alphaNpPerM = alphaDbPerKm .* log(10) ./ 10 ./ 1e3;
+Lm = Lkm .* 1e3;
+gammaWInvMInv = gammaWInvKmInv ./ 1e3;
+
+if abs(alphaNpPerM) < eps
+    Leff = Lm;
+else
+    Leff = (1 - exp(-alphaNpPerM .* Lm)) ./ alphaNpPerM;
+end
+
+P_fwm = degeneracyFactor .* eta .* ...
+    (gammaWInvMInv .* Leff).^2 .* ...
+    P_i .* P_j .* P_k .* ...
+    exp(-alphaNpPerM .* Lm);
+
+end

Functions/Theory/calcFWM/calcPhaseMatching.m

@@ -0,0 +1,20 @@
+function deltaBeta = calcPhaseMatching(f_i, f_j, f_k, f_0, Ds)
+% Approximate phase mismatch for degenerate FWM close to the ZDW.
+% Inputs are frequencies in Hz.
+%   f_i, f_j : pump frequencies (equal in the degenerate case)
+%   f_k      : signal frequency
+%   f_0      : zero-dispersion frequency
+%   Ds       : dispersion slope in ps / (nm^2 km)
+
+c = physconst('LightSpeed');
+
+pump_frequency = 0.5 .* (f_i + f_j);
+lambda_zdw_m = c ./ f_0;
+dispersion_slope_si = Ds .* 1e3;
+
+deltaBeta = -(2 .* pi .* lambda_zdw_m.^4 ./ c.^2) .* ...
+    dispersion_slope_si .* ...
+    (pump_frequency - f_0) .* ...
+    (pump_frequency - f_k).^2;
+
+end

Functions/Theory/calcFWM/scriptFWM.m

@@ -0,0 +1,104 @@
+clear;
+clc;
+
+% Sweep the degenerate pump frequency around its nominal wavelength.
+pump_detuning_hz = (-800:0.01:800) .* 1e9;
+
+% Degenerate FWM setup: two pump photons at f_p and one signal at f_s
+% generate an idler at f_i = 2*f_p - f_s.
+pump_wavelength_nm = 1310;
+signal_wavelength_nm = 1308;
+zdw_wavelength_nm = 1310;
+
+f_pump_nominal = wavelength2frequency(pump_wavelength_nm, 'nm');
+f_signal_scalar = wavelength2frequency(signal_wavelength_nm, 'nm');
+f_zdw_scalar = wavelength2frequency(zdw_wavelength_nm, 'nm');
+
+f_pump = f_pump_nominal + pump_detuning_hz;
+f_signal = f_signal_scalar .* ones(size(f_pump));
+f_zdw = f_zdw_scalar .* ones(size(f_pump));
+f_idler = 2 .* f_pump - f_signal;
+
+% Fiber parameters
+dispersion_slope_ps_nm2_km = 0.07;
+attenuation_db_per_km = 0.21;
+fiber_length_km = 10;
+
+% Launch powers and nonlinear coefficient
+pump_power_dbm = 10;
+signal_power_dbm = 10;
+pump_power_w = dbm2watt(pump_power_dbm);
+signal_power_w = dbm2watt(signal_power_dbm);
+gamma_w_inv_km_inv = 1.3;
+degeneracy_factor = 3;
+
+[P_fwm, eta, delta_beta, L_eff_m] = calcFwmPower( ...
+    f_pump, f_pump, f_signal, f_zdw, ...
+    dispersion_slope_ps_nm2_km, attenuation_db_per_km, fiber_length_km, ...
+    pump_power_w, pump_power_w, signal_power_w, ...
+    gamma_w_inv_km_inv, degeneracy_factor);
+
+f_pump_thz = f_pump .* 1e-12;
+f_zdw_thz = f_zdw_scalar .* 1e-12;
+f_signal_thz = f_signal_scalar .* 1e-12;
+idler_power_dbm = 10 .* log10(max(P_fwm, realmin) ./ 1e-3);
+
+figure;
+tiledlayout(2,1);
+
+ax1 = nexttile;
+plot(ax1, f_pump_thz, eta, 'LineWidth', 2);
+hold(ax1, 'on');
+xline(ax1, f_zdw_thz, '--r', 'ZDW', 'LineWidth', 1.2, ...
+    'LabelOrientation', 'horizontal', 'LabelVerticalAlignment', 'bottom');
+xline(ax1, f_signal_thz, '--k', 'Signal', 'LineWidth', 1.2, ...
+    'LabelOrientation', 'horizontal', 'LabelVerticalAlignment', 'middle');
+ylabel(ax1, 'FWM efficiency');
+grid(ax1, 'on');
+title(ax1, 'FWM Efficiency and Idler Power versus Pump Frequency');
+
+ax2 = nexttile;
+plot(ax2, f_pump_thz, idler_power_dbm, 'LineWidth', 2);
+hold(ax2, 'on');
+xline(ax2, f_zdw_thz, '--r', 'ZDW', 'LineWidth', 1.2, ...
+    'LabelOrientation', 'horizontal', 'LabelVerticalAlignment', 'bottom');
+xline(ax2, f_signal_thz, '--k', 'Signal', 'LineWidth', 1.2, ...
+    'LabelOrientation', 'horizontal', 'LabelVerticalAlignment', 'middle');
+xlabel(ax2, 'Pump frequency (THz)');
+ylabel(ax2, 'FWM idler power (dBm)');
+grid(ax2, 'on');
+
+fprintf('Pump wavelength        : %.3f nm -> %.6f THz\n', ...
+    pump_wavelength_nm, f_pump_nominal .* 1e-12);
+fprintf('Signal wavelength      : %.3f nm -> %.6f THz\n', ...
+    signal_wavelength_nm, f_signal_scalar .* 1e-12);
+fprintf('ZDW wavelength         : %.3f nm -> %.6f THz\n', ...
+    zdw_wavelength_nm, f_zdw_scalar .* 1e-12);
+fprintf('Pump launch power      : %.2f dBm -> %.4g W\n', ...
+    pump_power_dbm, pump_power_w);
+fprintf('Signal launch power    : %.2f dBm -> %.4g W\n', ...
+    signal_power_dbm, signal_power_w);
+fprintf('Peak FWM efficiency    : %.4g\n', max(eta));
+fprintf('Peak FWM idler power   : %.4g W (%.2f dBm)\n', ...
+    max(P_fwm), 10 .* log10(max(P_fwm) ./ 1e-3));
+fprintf('Effective fiber length : %.4f km\n', L_eff_m ./ 1e3);
+fprintf('Idler wavelength range : %.3f nm to %.3f nm\n', ...
+    min(frequency2wavelength(f_idler, 'nm')), max(frequency2wavelength(f_idler, 'nm')));
+fprintf('Max |delta beta|       : %.4g 1/m\n', max(abs(delta_beta)));
+
+function wavelength = frequency2wavelength(frequency, outputUnit)
+    c = physconst('LightSpeed');
+    wavelength = c ./ frequency;
+
+    switch lower(outputUnit)
+        case 'm'
+        case 'nm'
+            wavelength = wavelength .* 1e9;
+        otherwise
+            error('Unsupported output unit "%s". Use "m" or "nm".', outputUnit);
+    end
+end
+
+function power_w = dbm2watt(power_dbm)
+    power_w = 1e-3 .* 10.^(power_dbm ./ 10);
+end

Functions/Theory/calcFWM/validate_fwm.m

@@ -0,0 +1,124 @@
+%% Validate the analytical FWM product count against a brute-force reference
+% The paper counts channel combinations, not only unique output frequencies:
+%   non-degenerate: i < j, k ~= i, k ~= j, n = i + j - k
+%   degenerate:     i == j, k ~= i,       n = 2*i - k
+
+clear;
+clc;
+
+N_values = [4, 8, 16];
+plot_N = 8;
+
+fprintf('Validating analytical FWM product count from Goebel and Hanik (2008)\n');
+
+for idxN = 1:numel(N_values)
+    N = N_values(idxN);
+    fprintf('\nN = %d\n', N);
+
+    [Mndg_ana, Mdg_ana] = getProducts(N);
+    [Mndg_brute, Mdg_brute, n_values] = getProductsBruteForce(N);
+
+    diff_ndg = Mndg_ana - Mndg_brute;
+    diff_dg = Mdg_ana - Mdg_brute;
+
+    fprintf('  Analytical total   : %d\n', sum(Mndg_ana) + sum(Mdg_ana));
+    fprintf('  Brute-force total  : %d\n', sum(Mndg_brute) + sum(Mdg_brute));
+
+    if all(diff_ndg == 0) && all(diff_dg == 0)
+        fprintf('  Match              : yes\n');
+    else
+        fprintf('  Match              : no\n');
+        fprintf('  Non-degenerate diff: %s\n', mat2str(diff_ndg));
+        fprintf('  Degenerate diff    : %s\n', mat2str(diff_dg));
+    end
+
+    if N == plot_N
+        plotComparison(n_values, Mndg_ana, Mdg_ana, Mndg_brute, Mdg_brute, N);
+    end
+end
+
+function [Mndg, Mdg, n_values] = getProductsBruteForce(N)
+    n_values = (2 - N):(2*N - 1);
+    Mndg = zeros(size(n_values));
+    Mdg = zeros(size(n_values));
+
+    for idx = 1:numel(n_values)
+        n = n_values(idx);
+
+        % Degenerate products: two identical pumps and one different channel.
+        for i = 1:N
+            k = 2*i - n;
+            if isValidChannel(k, N) && (k ~= i)
+                Mdg(idx) = Mdg(idx) + 1;
+            end
+        end
+
+        % Non-degenerate products: unordered pump pair plus one third channel.
+        for i = 1:N
+            for j = (i + 1):N
+                k = i + j - n;
+                if isValidChannel(k, N) && (k ~= i) && (k ~= j)
+                    Mndg(idx) = Mndg(idx) + 1;
+                end
+            end
+        end
+    end
+end
+
+function tf = isValidChannel(channel_idx, N)
+    tf = (channel_idx >= 1) && (channel_idx <= N) && (channel_idx == round(channel_idx));
+end
+
+function plotComparison(n_values, Mndg_ana, Mdg_ana, Mndg_brute, Mdg_brute, N)
+    figure;
+
+    subplot(1,2,1);
+    stem(n_values, Mndg_ana + Mdg_ana, 'filled', 'LineWidth', 1, 'Marker', 'o', 'MarkerSize', 2);
+    hold on;
+    stem(n_values, Mdg_ana, 'filled', 'LineWidth', 1, 'Marker', 'none');
+    title(['Analytical (N=', num2str(N), ')']);
+    xlabel('Product index n');
+    ylabel('Number of FWM products');
+    legend('Total', 'Degenerate');
+    grid on;
+
+    subplot(1,2,2);
+    stem(n_values, Mndg_brute + Mdg_brute, 'filled', 'LineWidth', 1, 'Marker', 'o', 'MarkerSize', 2);
+    hold on;
+    stem(n_values, Mdg_brute, 'filled', 'LineWidth', 1, 'Marker', 'none');
+    title(['Brute force (N=', num2str(N), ')']);
+    xlabel('Product index n');
+    ylabel('Number of FWM products');
+    legend('Total', 'Degenerate');
+    grid on;
+end
+
+function [Mndg, Mdg] = getProducts(N)
+    s = abs(2 - N - 1);
+
+    for n = 2 - N:2*N - 1
+        if n < -N
+            Mdg(n + s) = 0;
+        elseif (-N <= n) && (n <= 0)
+            Mdg(n + s) = N - ceil((N - n)/2);
+        elseif (1 <= n) && (n <= N)
+            Mdg(n + s) = N - 1 - floor(n/2) - ceil((N - n)/2);
+        elseif (N < n) && (n <= 2*N)
+            Mdg(n + s) = N - floor(n/2);
+        elseif n > 2*N
+            Mdg(n + s) = 0;
+        end
+
+        if n < -N
+            Mndg(n + s) = 0;
+        elseif (-N <= n) && (n < 1)
+            Mndg(n + s) = ceil((N^2 + n^2 - 2*N - 2*n + 2*N*n)/4);
+        elseif (1 <= n) && (n <= N)
+            Mndg(n + s) = ceil(((N^2 - 6*N - 2*n^2 + 2*n + 4)/4) + floor((N*n)/2));
+        elseif (N < n) && (n <= 2*N)
+            Mndg(n + s) = floor(N^2 + n^2/4 - N*n);
+        elseif n > 2*N
+            Mndg(n + s) = 0;
+        end
+    end
+end

Functions/Theory/calcFWM/wavelength2frequency.m

@@ -0,0 +1,18 @@
+function frequency = wavelength2frequency(wavelength, inputUnit)
+% Convert wavelength to optical frequency.
+% Supported units: m, nm.
+
+c = physconst('LightSpeed');
+
+switch lower(inputUnit)
+    case 'm'
+        wavelength_m = wavelength;
+    case 'nm'
+        wavelength_m = wavelength .* 1e-9;
+    otherwise
+        error('Unsupported input unit "%s". Use "m" or "nm".', inputUnit);
+end
+
+frequency = c ./ wavelength_m;
+
+end

Functions/Theory/mutual information rate TUM/LICENSE.txt

@@ -0,0 +1,25 @@
+Copyright (c) 2019 Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+Institute for Communications Engineering (LNT)
+Technical University of Munich, Germany
+www.lnt.ei.tum.de
+
+All rights reserved.
+
+Permission is hereby granted, free of charge, to any person obtaining a
+copy of this software and associated documentation files (the
+"Software"), to deal in the Software without restriction, including
+without limitation the rights to use, copy, modify, merge, publish,
+distribute, sublicense, and/or sell copies of the Software, and to permit
+persons to whom the Software is furnished to do so, subject to the
+following conditions:
+
+The above copyright notice and this permission notice shall be included
+in all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
+OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN
+NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
+DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
+OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE
+USE OR OTHER DEALINGS IN THE SOFTWARE.

Functions/Theory/mutual information rate TUM/README.md

@@ -0,0 +1,24 @@
+## MI-CG: Numerically Computing Achievable Rates of Memoryless Channels
+
+This repository provides a MATLAB function mi_cg.m to numerically compute achievable rates for memoryless channels. The function uses a conditionally-Gaussian (CG) channel model to obtain a lower bound on the achievable rate of the true channel. The method is well-known, and it is explained in [this short document](https://mediatum.ub.tum.de/node?id=1533663). Two example scripts that compute several achievable rate curves are also provided.
+
+### Citation
+
+This software and the accompanying document are meant as a tutorial to get started with mutual information as a numerical figure of merit for a communications channel. The software is provided under the open-source [MIT license](https://opensource.org/licenses/MIT). If you use the software in your academic work, please cite the accompanying [document](https://mediatum.ub.tum.de/node?id=1533663) as follows:
+
+> F. J. Garcia-Gomez, “Numerically computing achievable rates of memoryless channels,” TUM University Library, 2019. [Online]. Available: https://mediatum.ub.tum.de/node?id=1533663 
+
+The corresponding BibTeX entry is
+```
+@article{garcia2019numerically,
+	author = "Francisco Javier Garcia-Gomez",
+	title = "Numerically Computing Achievable Rates of Memoryless Channels",
+	year = "2019",
+	journal="TUM University Library",
+	url={https://mediatum.ub.tum.de/node?id=1533663}
+}
+```
+
+### Acknowledgment
+
+This work was supported by the German Research Foundation (DFG) under Grant KR 3517/8-2.

Functions/Theory/mutual information rate TUM/air.m

@@ -0,0 +1,156 @@
+function air = air(x,r,idx_tx,Px,M_training)
+
+MAX_MEMORY = 200e6; % maximum allowed size for a matrix
+
+if nargin == 3
+    Px = [];
+    M_training = [];
+end
+
+if nargin == 4
+    M_training = [];
+end
+
+
+% if input is complex, separate into real and imaginary parts
+if any(imag(x(:))~=0) || any(imag(r(:))~=0)
+    x = [real(x); imag(x)];
+    r = [real(r); imag(r)];
+end
+
+D = size(x, 1); % D = 2 if complex x
+N = size(x, 2); % number of constellation points
+M = size(r, 2); % number of samples
+
+% set default training set size
+if isempty(M_training)
+    M_training = ceil(0.3*M);
+end
+
+M_testing = M - M_training;
+
+% Training: estimate parameters of the conditionally Gaussian model
+% sort according to transmit index
+[idx_tx_training, idx_sort] = sort(idx_tx(1:M_training));
+r_training = r(:, idx_sort);
+i_bounds = zeros(1, N+1);
+
+% compute conditional means and covariance matrices
+C_n = zeros(D, D, N);
+det_n = zeros(1, N);
+
+for n=1:N
+    % find how many times x(:, n) was transmitted and update i_bounds
+    N_current_x = find(idx_tx_training((i_bounds(n)+1):end)==n, 1, 'last');
+    if isempty(N_current_x), N_current_x=0; end
+    i_bounds(n+1) = i_bounds(n) + N_current_x;
+
+    if N_current_x > 0
+        % Compute mu_n=E[Y|X=x_n] according to Eq. (14) and store it in
+        % x(:, n) to save space
+        x(:, n) = sum(r_training(:, (i_bounds(n)+1):i_bounds(n+1)), 2)/(i_bounds(n+1)-i_bounds(n));
+
+        % compute C_n=cov[Y|X=x_n] according to Eq. (15)
+        r_meanfree = r_training(:, (i_bounds(n)+1):i_bounds(n+1)) - x(:, n);
+        C_n(:, :, n) = (r_meanfree*r_meanfree')/(i_bounds(n+1)-i_bounds(n));
+        % store also the determinant of C(:, :, n)
+        det_n(n) = det(C_n(:, :, n));
+
+        % if the determinant is 0, or if the matrix is badly conditioned,
+        % regularize by adding a small identity matrix. Note that we do
+        % need the check for 0 determinant, in case a cloud has exactly 0
+        % variance according to the training set
+        if det_n(n)==0 || cond(C_n(:, :, n))>1e16
+            C_n(:, :, n) = C_n(:, :, n) + 5 * eps * eye(D);
+            det_n(n) = (5*eps)^D;
+        end
+
+    end
+end
+
+% uniform input pmf Px if not provided
+if isempty(Px)
+    Px = repmat(1/N, [1, N]);
+end
+
+
+% extract testing set and sort it according to transmit index
+[idx_tx_testing, idx_sort] = sort(idx_tx((M_training+1):M));
+r_testing = r(:, M_training+idx_sort);
+
+% computation of h(Y|X)
+h_Y_X = 0;
+i_bounds_testing = zeros(1, N+1);
+% loop over constellation points to compute h(Y|X)
+for n = 1:N
+    % find how many times x(:, n) was transmitted and update
+    % i_bounds_testing
+    N_current_x = find(idx_tx_testing((i_bounds_testing(n)+1):end)==n, 1, 'last');
+    if isempty(N_current_x), N_current_x=0; end
+    i_bounds_testing(n+1) = i_bounds_testing(n) + N_current_x;
+    % add the corresponding contribution to the mutual information (two
+    % first lines of Eq. (17)). This, together with
+    % D/2*log2(2*pi) after the end of the loop, gives h(Y|X)
+    h_Y_X = h_Y_X + N_current_x * log2(det_n(n))/2+...
+        sum(sum(conj(r_testing(:, (i_bounds_testing(n)+1):i_bounds_testing(n+1))-x(:, n)).*(C_n(:, :, n)\(r_testing(:, (i_bounds_testing(n)+1):i_bounds_testing(n+1))-x(:, n)))))/2/log(2);
+
+    
+end
+h_Y_X = D/2*log2(2*pi) + h_Y_X/M_testing;
+
+% When computing log(py), we might run out of memory. If necessary, we
+% doe the computation in blocks
+logpy = zeros(1, M_testing);
+BLOCK_SIZE = floor(MAX_MEMORY/N);
+N_blocks = ceil(M_testing/BLOCK_SIZE);
+
+% loop over blocks of symbols. This loop can be replaced by parfor to allow
+% parallel computation
+for i_block = 1:N_blocks
+
+    logpy_cur = zeros(1, M_testing);
+
+    % beginning of block
+    i_start = (i_block-1) * BLOCK_SIZE + 1;
+    % end of block
+    i_end = min(M_testing, i_block*BLOCK_SIZE);
+    % block size
+    current_block_size = i_end-i_start+1;
+
+    % compute exponents of third line of (17)
+    exponents = zeros(N, current_block_size);
+    for n = 1:N
+        exponents(n, :) = -log(det_n(n))/2-real(sum(conj(r_testing(:, i_start:i_end)-x(:, n)).*(C_n(:, :, n)\(r_testing(:, i_start:i_end)-x(:, n))), 1))/2;
+        %sum über 2 einträge von r
+    end
+
+    % compute third line of Eq. (17). Use a custom function
+    % that computes log(sum(exp(x))) avoiding overflow errors
+    logpy_cur(i_start:i_end) = math_logsumexp(log(Px(:))+exponents, 1);
+    logpy = logpy + logpy_cur;
+end
+
+% output entropy h(Y)
+h_Y = D/2*log2(2*pi) - mean(logpy)/log(2);%log basis change
+
+% compute mutual information
+air = h_Y - h_Y_X;
+
+
+end
+
+
+function [y] = math_logsumexp(x, dim)
+%[y] = math_logsumexp(x, dim)
+%  Computes log(sum(exp(x), dim)), avoiding overflow errors when one of the
+%  x is large.
+
+if nargin<2 || isempty(dim)
+    m = max(x);
+    y = m + log(sum(exp(x-m)));
+else
+    m = max(x, [], dim);
+    y = m + log(sum(exp(x-m), dim));
+end
+end
+

Functions/Theory/mutual information rate TUM/example_mi_cg_complex_16qam.m

@@ -0,0 +1,124 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Copyright (c) 2019 Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+% Institute for Communications Engineering (LNT)
+% Technical University of Munich, Germany
+% www.lnt.ei.tum.de
+%
+% All rights reserved.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% 
+% Permission is hereby granted, free of charge, to any person obtaining a
+% copy of this software and associated documentation files (the
+% "Software"), to deal in the Software without restriction, including
+% without limitation the rights to use, copy, modify, merge, publish,
+% distribute, sublicense, and/or sell copies of the Software, and to permit
+% persons to whom the Software is furnished to do so, subject to the
+% following conditions:
+% 
+% The above copyright notice and this permission notice shall be included
+% in all copies or substantial portions of the Software.
+% 
+% THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
+% OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+% MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN
+% NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
+% DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
+% OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE
+% USE OR OTHER DEALINGS IN THE SOFTWARE.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%% example_mi_cg_complex_16qam.m %%%%%%%%%%%%%%%%%%%%%%
+%
+% Example usage of the function mi_cg to numerically compute mutual
+% information between two complex sequences. This file simulates a
+% one-dimensional complex AWGN channel with 16-QAM constellation for
+% different SNRs, and then plots the achievable rate, which is equal to two
+% times the 4-PAM curve of Fig. 1 of [1]. 
+%
+% Note that using mi_cg with complex sequences assumes that the channel
+% model q(Y|X) is circularly symmetric for a given X=x (i.e., that the
+% received clouds are circular). This is not the case in a channel with
+% phase noise: see example_it_mi_cg.m for an example of how to deal with
+% non-circularly-symmetric channels.
+%
+% [1] G. David Forney and Gottfried Ungerboeck, "Modulation and Coding for
+% Linear Gaussian Channels", IEEE Trans. Inf. Theory vol. 44, no. 6, pp.
+% 2384-2415, October 1998 
+%
+% Technische Universitaet Muenchen - Lehrstuhl fuer Nachrichtentechnik
+% Date: 12.12.2019
+% Author: Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+clear;
+% close all;
+
+%%% Simulation parameters
+M_QAM=16; % QAM size
+var_w=1; % noise variance
+SNR_dB_values=(-5):2:25; % SNR values in dB
+N=8000; % number of Monte-Carlo points
+
+%%% Derived parameters
+n_SNR=length(SNR_dB_values);
+SNR_values=10.^(SNR_dB_values/10);
+
+%%% Generation of QAM constellation with power 1
+M_PAM=sqrt(M_QAM); % number of points per real dimension
+X_1d=-(M_PAM-1)+2*(0:(M_PAM-1)); % generate PAM
+X=X_1d+1i*X_1d.'; % transform to QAM
+X=X(:).'*sqrt(3/2/(M_QAM-1)); % set to unit power
+X=[real(X); imag(X)]; % separate real and imaginary parts
+
+%%% Uniformly choose transmit indices
+idx_tx=randi(M_QAM, [1, N]);
+
+%%% AWGN noise
+w=sqrt(var_w)*(randn([2, N]));%+1i*randn([1, N]));
+
+%%% Loop over the SNR values
+
+MI_awgn=zeros(1, n_SNR);
+fig = figure(1);
+for i_SNR=1:n_SNR
+
+    % transmitted points
+    s=sqrt(SNR_values(i_SNR)*var_w)*X(:, idx_tx);
+    
+    % AWGN channel
+    r_awgn = s+w;
+
+    %%%%%%%%%%%%
+    clf
+    hold on
+    xlim([-20, 20]);
+    ylim([-20, 20]);
+    %received signal
+    scatter(r_awgn(1,:),r_awgn(2,:),1,"red",'.');
+    %tx signal constellation
+    scatter(s(1,:),s(2,:),4,"black",'o','filled');
+    %uni power transmit constallation
+    scatter(X(1,:),X(2,:),5,'blue','o','filled');
+    drawnow
+    pause(0.1)
+    %%%%%%%%%%%%
+
+    % Compute MI
+    MI_awgn(i_SNR)=air(X, r_awgn, idx_tx);
+    % The following also works but is slower
+%     MI_awgn(i_SNR)=mi_cg(s, r_awgn);
+end
+
+
+I_shannon=log2(1+SNR_values);
+
+figure(2);
+hold on
+plot(SNR_dB_values, I_shannon, '-', 'DisplayName', 'log_2 (1+SNR)');
+hold on;
+plot(SNR_dB_values, MI_awgn, '--', 'DisplayName', [num2str(M_QAM) '-QAM, AWGN']);
+hold off;
+xlabel('SNR (dB)'); ylabel('Achievable rate (bits/complex dimension)');
+title(['Achievable rate of ' num2str(M_QAM) '-QAM in AWGN']);
+legend('Location', 'NorthWest');

Functions/Theory/mutual information rate TUM/example_mi_cg_real_2D_awgn_vs_phasenoise.m

@@ -0,0 +1,140 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Copyright (c) 2019 Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+% Institute for Communications Engineering (LNT)
+% Technical University of Munich, Germany
+% www.lnt.ei.tum.de
+%
+% All rights reserved.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% 
+% Permission is hereby granted, free of charge, to any person obtaining a
+% copy of this software and associated documentation files (the
+% "Software"), to deal in the Software without restriction, including
+% without limitation the rights to use, copy, modify, merge, publish,
+% distribute, sublicense, and/or sell copies of the Software, and to permit
+% persons to whom the Software is furnished to do so, subject to the
+% following conditions:
+% 
+% The above copyright notice and this permission notice shall be included
+% in all copies or substantial portions of the Software.
+% 
+% THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
+% OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+% MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN
+% NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
+% DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
+% OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE
+% USE OR OTHER DEALINGS IN THE SOFTWARE.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%% example_mi_cg_real_2D_awgn_vs_phasenoise %%%%%%%%%%%%%%%%%
+%
+% Example usage of the function mi_cg to numerically compute mutual
+% information between two sequences. This file simulates a two-dimensional
+% AWGN channel with 4-PAM constellation (or, equivalently, a complex AWGN
+% channel with 16-QAM constellation) for different SNRs, and then plots 
+% the achievable rate, reproducing the 4-PAM curve of Fig. 1 of [1]. Note
+% that this curve can also be reproduced by simulating a 1-dimensional
+% channel with 4-PAM: this file is an example of how to deal with multiple
+% dimensions.
+%
+% This file also simulates a complex phase-noise channel:
+%                  r=s.*exp(1i*theta)+w
+% where w is complex AWGN and theta is i.i.d. real Gaussian. As q(Y|X) for
+% this channel is not circularly-symmetric, this complex channel needs to
+% be separated into two real dimensions to compute the mutual information.
+% The resulting achievable rate is, as expected, below the rate for the
+% AWGN channel.
+%
+% [1] G. David Forney and Gottfried Ungerboeck, "Modulation and Coding for
+% Linear Gaussian Channels", IEEE Trans. Inf. Theory vol. 44, no. 6, pp.
+% 2384-2415, October 1998 
+%
+% [2] 
+%
+% Technische Universitaet Muenchen - Lehrstuhl fuer Nachrichtentechnik
+% Date: 12.12.2019
+% Author: Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+clear;
+close all;
+
+%%% Simulation parameters
+D=1; % number of complex dimensions
+M_QAM=16; % QAM size
+var_w=1; % noise variance
+E_theta=0.2; % phase offset for the phase noise channel
+var_theta=0.05; % phase noise variance
+SNR_dB_values=(-5):2:25; % SNR values in dB
+N=8000; % number of Monte-Carlo points
+
+%%% Derived parameters
+n_SNR=length(SNR_dB_values);
+SNR_values=10.^(SNR_dB_values/10);
+
+%%% Generation QAM constellation with power 1
+M_PAM=sqrt(M_QAM); % number of points per real dimension
+X_1d=-(M_PAM-1)+2*(0:(M_PAM-1)); % generate PAM
+X=X_1d+1i*X_1d.'; % transform to QAM
+X=X(:).'*sqrt(3/2/(M_QAM-1)); % set to unit power
+X=[real(X); imag(X)]; % separate real and imaginary parts
+
+%%% Uniformly choose transmit indices
+idx_tx=randi(M_QAM^D, [1, N]);
+
+%%% AWGN noise (real and imaginary parts)
+w=sqrt(var_w/2)*randn([2*D, N]);
+%%% phase noise
+theta=E_theta+sqrt(var_theta)*randn([D, N]);
+
+%%% Loop over the SNR values
+MI_awgn=zeros(1, n_SNR);
+MI_phasenoise=zeros(1, n_SNR);
+fig = figure(1);
+for i_SNR=1:n_SNR
+    % transmitted points
+    s=sqrt(SNR_values(i_SNR)*var_w)*X(:, idx_tx);
+    
+    % AWGN channel
+    r_awgn=s+w;
+    
+    % Compute MI
+    MI_awgn(i_SNR)=air(X, r_awgn, idx_tx)/D;
+    % The following also works but is slower
+%     MI_awgn(i_SNR)=air(s, r_awgn)/D;
+    
+    % Phase-noise channel
+    s_complex=s(1:2:end, :)+1i*s(2:2:end, :); % transform to complex
+    r_phasenoise_complex=s_complex.*exp(1i*theta); % phase noise and AWGN
+    r_phasenoise=zeros(2*D, N); % transform to real
+    r_phasenoise(1:2:end, :)=real(r_phasenoise_complex);
+    r_phasenoise(2:2:end, :)=imag(r_phasenoise_complex);
+    r_phasenoise=r_phasenoise+w; % add AWGN
+
+    clf
+    fig = scatter(r_phasenoise(1,:),r_phasenoise(2,:),1,'.');
+    xlim([-20, 20]);
+    ylim([-20, 20]);
+    drawnow
+    pause(0.1)
+    
+    % Compute MI
+    MI_phasenoise(i_SNR)=air(X, r_phasenoise, idx_tx)/D;
+    % The following also works but is slower
+%     MI_phasenoise(i_SNR)=air(s, r_phasenoise)/D;
+end
+
+% Shannon capacity of the AWGN channel
+I_shannon=log2(1+SNR_values);
+
+figure;
+plot(SNR_dB_values, I_shannon, '-', 'DisplayName', 'log_2 (1+SNR)');
+hold on;
+plot(SNR_dB_values, MI_awgn, '--', 'DisplayName', '16-QAM, AWGN');
+plot(SNR_dB_values, MI_phasenoise, '-.', 'DisplayName', ['16-QAM, Phase noise, \sigma_\Theta^2=' num2str(var_theta)]);
+hold off;
+xlabel('SNR (dB)'); ylabel('Achievable rate (bits/complex dimension)');
+title(['Achievable rate of 16-QAM in AWGN and in a phase noise channel with \theta~N(' num2str(E_theta) ', ' num2str(var_theta) ')']);
+legend('Location', 'NorthWest');

Functions/Theory/mutual information rate TUM/example_silas.m

@@ -0,0 +1,164 @@
+colored_noise = true;
+I_shannon=[];
+for cmplx = [0,1]
+    cnt = 1;
+    for M_PAM = [4,8,16]
+
+
+        complex_constellation = cmplx;
+
+        %%% Simulation parameters
+        % M_PAM=2; % QAM size
+        var_w=1; % noise variance
+        SNR_dB_values=(-5):1:35; % SNR values in dB
+        N=8000; % number of Monte-Carlo points
+
+        %%% Derived parameters
+        n_SNR=length(SNR_dB_values);
+        SNR_values=10.^(SNR_dB_values/10);
+
+        %%% Generation of QAM constellation with power 1
+        M=sqrt(M_PAM); % number of points per real dimension
+
+        if complex_constellation
+            X_ = qammod(0:M_PAM-1,M_PAM,"gray");
+        else
+            X_ = pammod(0:M_PAM-1,M_PAM,0,'gray');
+        end
+
+        X_ = X_ ./ rms(unique(X_));
+        X_=[real(X_); imag(X_)];
+
+        %%%%%%%%%%%%
+        figure(10);
+        clf
+        hold on
+        scatter(X_(1,:),X_(2,:),15,'red','x','DisplayName','Matlab');
+        %%%%%%%%%%%%
+
+        %%% Uniformly choose transmit indices
+        idx_tx=randi(M_PAM, [1, N]);
+
+        %%% AWGN noise
+        w=sqrt(var_w)*(randn([2, N]));
+
+        %%% Loop over the SNR values
+
+        MI_awgn=zeros(1, n_SNR);
+        fig = figure(2);
+
+        if colored_noise
+            for i_SNR = 1:n_SNR
+                % Transmitted signal for the given indices
+                signal = X_(:, idx_tx);
+    
+                % Generate white Gaussian noise of the same size as the signal
+                noise_white = randn(size(signal));
+    
+                % Define the filter coefficient for colored noise (adjust as needed)
+                a = 0;  % A higher value gives more correlation
+    
+                % Filter the white noise to create colored noise.
+                % Here we filter each row (dimension) independently.
+                noise_colored = zeros(size(signal));
+                noise_colored(1,:) = filter(1, [1, -a], noise_white(1,:));
+                noise_colored(2,:) = filter(1, [1, -a], noise_white(2,:));
+    
+                % Compute signal and unscaled noise power for proper scaling.
+                signal_power = mean(abs(signal(:)).^2);
+                noise_power = mean(abs(noise_colored(:)).^2);
+    
+                % Convert desired SNR from dB to linear scale.
+                SNR_linear = 10^(SNR_dB_values(i_SNR)/10);
+    
+                % Scale the colored noise so that signal_power / noise_power equals SNR_linear.
+                scaling_factor = sqrt(signal_power / (SNR_linear * noise_power));
+                noise_colored_scaled = scaling_factor * noise_colored;
+    
+                % Received signal is the sum of the signal and the scaled colored noise.
+                r_colored = signal + noise_colored_scaled;
+    
+                % For SNR measurement, compute the noise actually added.
+                measured_noise = r_colored - signal;
+                snr_meas_(i_SNR) = snr(signal(1,:) + 1i*signal(2,:), ...
+                    measured_noise(1,:) + 1i*measured_noise(2,:));
+    
+                % Plotting the transmitted and received signal
+                clf;
+                hold on;
+                xlim([-4, 4]);
+                ylim([-4, 4]);
+                scatter(signal(1,:), signal(2,:), 3, "black", 'o', 'DisplayName','Tx Constellation');
+                scatter(r_colored(1,:), r_colored(2,:), 1, "red", 'x', 'DisplayName', 'Colored Noise Mapping');
+                drawnow;
+    
+                % Compute Mutual Information (or any other metric) with the new channel
+                MI_awgn_(i_SNR) = air(X_, r_colored, idx_tx);
+            end
+        else
+
+
+            for i_SNR=1:n_SNR
+    
+    
+                s_ = sqrt(SNR_values(i_SNR)*var_w) * X_(:, idx_tx);
+                % r_awgn_ = s_ + w;
+    
+                r_awgn_ = awgn(X_(:, idx_tx),SNR_dB_values(i_SNR),"measured",10);
+    
+                w_awgn_matlab = r_awgn_ - X_(:,idx_tx);
+                snr_meas_(i_SNR) = snr(X_(1, idx_tx)+1i*X_(2, idx_tx) , w_awgn_matlab(1,:)+1i*w_awgn_matlab(2,:));
+    
+                %%%%%%%%%%%%
+                clf
+                hold on
+                xlim([-4, 4]);
+                ylim([-4, 4]);
+                %received signal
+                scatter(X_(1, idx_tx),X_(2, idx_tx),3,"black",'o','DisplayName','Tx Constellation');
+    
+    
+                scatter(r_awgn_(1,:),r_awgn_(2,:),1,"red",'x','DisplayName','Matlab Mapping');
+    
+                drawnow
+                %%%%%%%%%%%%
+    
+                MI_awgn_(i_SNR)=air(X_, r_awgn_, idx_tx);
+                % The following also works but is slower
+                %     MI_awgn(i_SNR)=mi_cg(s, r_awgn);
+            end
+        end
+
+        figure(3);
+        hold on
+
+        if isempty(I_shannon)
+            I_shannon=log2(1+db2pow(snr_meas_));
+            plot(snr_meas_, I_shannon, '-', 'DisplayName', 'log_2 (1+SNR)','Color','black');
+        end
+
+
+        cols = [    0.9047    0.1918    0.1988
+            0.2941    0.5447    0.7494
+            0.3718    0.7176    0.3612
+            1.0000    0.5482    0.1000
+            0.8650    0.8110    0.4330
+            0.6859    0.4035    0.2412];
+        if complex_constellation
+            plot(snr_meas_, MI_awgn_, ':', 'DisplayName', ['',num2str(M_PAM) '-QAM, AWGN'],'LineWidth',1,'Color',cols(cnt,:));
+        else
+            plot(snr_meas_, MI_awgn_, '-', 'DisplayName', ['',num2str(M_PAM) '-PAM, AWGN'],'LineWidth',1,'Color',cols(cnt,:));
+        end
+
+        hold off;
+        ylim([0,8]);
+        xlim([min(SNR_dB_values) max(SNR_dB_values)])
+        xlabel('SNR (dB)'); ylabel('Achievable rate (bits/complex dimension)');
+        title(['Achievable rate of ' num2str(M_PAM) '-QAM in AWGN']);
+        legend('Location', 'NorthWest');
+        yline(log2(M_PAM),'LineStyle',':','HandleVisibility','off','Color','black');
+
+        cnt = cnt+1;
+
+    end
+end

Functions/Theory/mutual information rate TUM/mi_cg.m

@@ -0,0 +1,104 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Copyright (c) 2019 Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+% Institute for Communications Engineering (LNT)
+% Technical University of Munich, Germany
+% www.lnt.ei.tum.de
+%
+% All rights reserved.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% 
+% Permission is hereby granted, free of charge, to any person obtaining a
+% copy of this software and associated documentation files (the
+% "Software"), to deal in the Software without restriction, including
+% without limitation the rights to use, copy, modify, merge, publish,
+% distribute, sublicense, and/or sell copies of the Software, and to permit
+% persons to whom the Software is furnished to do so, subject to the
+% following conditions:
+% 
+% The above copyright notice and this permission notice shall be included
+% in all copies or substantial portions of the Software.
+% 
+% THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
+% OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+% MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN
+% NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
+% DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
+% OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE
+% USE OR OTHER DEALINGS IN THE SOFTWARE.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%% example_mi_cg_complex_16qam.m %%%%%%%%%%%%%%%%%%%%%%
+%
+% Example usage of the function mi_cg to numerically compute mutual
+% information between two complex sequences. This file simulates a
+% one-dimensional complex AWGN channel with 16-QAM constellation for
+% different SNRs, and then plots the achievable rate, which is equal to two
+% times the 4-PAM curve of Fig. 1 of [1]. 
+%
+% Note that using mi_cg with complex sequences assumes that the channel
+% model q(Y|X) is circularly symmetric for a given X=x (i.e., that the
+% received clouds are circular). This is not the case in a channel with
+% phase noise: see example_it_mi_cg.m for an example of how to deal with
+% non-circularly-symmetric channels.
+%
+% [1] G. David Forney and Gottfried Ungerboeck, "Modulation and Coding for
+% Linear Gaussian Channels", IEEE Trans. Inf. Theory vol. 44, no. 6, pp.
+% 2384-2415, October 1998 
+%
+% Technische Universitaet Muenchen - Lehrstuhl fuer Nachrichtentechnik
+% Date: 12.12.2019
+% Author: Francisco Javier Garcia-Gomez <javier.garcia@tum.de>
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+clear;
+close all;
+
+%%% Simulation parameters
+M_QAM=16; % QAM size
+var_w=1; % noise variance
+SNR_dB_values=(-5):2:25; % SNR values in dB
+N=8000; % number of Monte-Carlo points
+
+%%% Derived parameters
+n_SNR=length(SNR_dB_values);
+SNR_values=10.^(SNR_dB_values/10);
+
+%%% Generation of QAM constellation with power 1
+M_PAM=sqrt(M_QAM); % number of points per real dimension
+X_1d=-(M_PAM-1)+2*(0:(M_PAM-1)); % generate PAM
+X=X_1d+1i*X_1d.'; % transform to QAM
+X=X(:).'*sqrt(3/2/(M_QAM-1)); % set to unit power
+
+%%% Uniformly choose transmit indices
+idx_tx=randi(M_QAM, [1, N]);
+
+%%% AWGN noise
+w=sqrt(var_w/2)*(randn([1, N])+1i*randn([1, N]));
+
+%%% Loop over the SNR values
+MI_awgn=zeros(1, n_SNR);
+for i_SNR=1:n_SNR
+    % transmitted points
+    
+    s=sqrt(SNR_values(i_SNR)*var_w)*X(:, idx_tx);
+    
+    % AWGN channel
+    r_awgn = s+w;
+    
+    % Compute MI
+    MI_awgn(i_SNR)=air(X, r_awgn, idx_tx);
+    % The following also works but is slower: MI_awgn(i_SNR)=air(s, r_awgn);
+end
+
+
+I_shannon=log2(1+SNR_values);
+
+figure;
+plot(SNR_dB_values, I_shannon, '-', 'DisplayName', 'log_2 (1+SNR)');
+hold on;
+plot(SNR_dB_values, MI_awgn, '--', 'DisplayName', [num2str(M_QAM) '-QAM, AWGN']);
+hold off;
+xlabel('SNR (dB)'); ylabel('Achievable rate (bits/complex dimension)');
+title(['Achievable rate of ' num2str(M_QAM) '-QAM in AWGN']);
+legend('Location', 'NorthWest');

Functions/Theory/mutual information rate TUM/silas_example_mi_cg_pam.m

@@ -0,0 +1,135 @@
+
+clear;
+I_shannon=[];
+for cmplx = [1,0]
+    cnt = 1;
+    for M_PAM = [2,4,8,16]
+
+        % close all;
+        usegarcia = 0;
+
+        complex_constellation = cmplx;
+
+        %%% Simulation parameters
+        % M_PAM=2; % QAM size
+        var_w=1; % noise variance
+        SNR_dB_values=(-5):1:35; % SNR values in dB
+        N=8000; % number of Monte-Carlo points
+
+        %%% Derived parameters
+        n_SNR=length(SNR_dB_values);
+        SNR_values=10.^(SNR_dB_values/10);
+
+        %%% Generation of QAM constellation with power 1
+        M=sqrt(M_PAM); % number of points per real dimension
+
+        if complex_constellation
+            X_ = qammod(0:M_PAM-1,M_PAM,"gray");
+        else
+            X_ = pammod(0:M_PAM-1,M_PAM,0,'gray');
+        end
+
+        X_ = X_ ./ rms(unique(X_));
+        X_=[real(X_); imag(X_)];
+
+        %%%%%%%%%%%%
+        figure(10);
+        clf
+        hold on
+        scatter(X_(1,:),X_(2,:),15,'red','x','DisplayName','Matlab');
+        %%%%%%%%%%%%
+
+        %%% Uniformly choose transmit indices
+        idx_tx=randi(M_PAM, [1, N]);
+
+        %%% AWGN noise
+        w=sqrt(var_w)*(randn([2, N]));
+
+        %%% Loop over the SNR values
+
+        MI_awgn=zeros(1, n_SNR);
+        fig = figure(2);
+        for i_SNR=1:n_SNR
+
+            if usegarcia
+
+
+                %scale transmitted points acc. to SNR condition (this is not correct I think, the papaer also show diff results)
+                % s=sqrt(SNR_values(i_SNR)*var_w)*X(:, idx_tx);
+
+                %scale nosie acc. to snr condition
+                noise_power = mean(abs(X_(1, idx_tx)+1i*X_(2, idx_tx)).^2) / SNR_values(i_SNR); % Noise power
+                w_ = sqrt(noise_power) .* w / sqrt(2);
+
+                % AWGN channel
+                r_awgn = X(:, idx_tx)+w_;
+                % snr_meas(i_SNR) = snr(s(1,:)+1i*s(2,:),w(1,:)+1i*w(2,:));
+                snr_meas(i_SNR) = snr(X_(1, idx_tx)+1i*X_(2, idx_tx),w_(1,:)+1i*w_(2,:));
+            end
+
+            s_ = sqrt(SNR_values(i_SNR)*var_w) * X_(:, idx_tx);
+            % r_awgn_ = s_ + w;
+
+            if cmplx
+                r_awgn_ = awgn(X_(:, idx_tx),SNR_dB_values(i_SNR),"measured",10);
+                w_awgn_matlab = r_awgn_ - X_(:,idx_tx);
+                snr_meas_(i_SNR) = snr(X_(1, idx_tx)+1i*X_(2, idx_tx) , w_awgn_matlab(1,:)+1i*w_awgn_matlab(2,:));
+            else
+                r_awgn_ = awgn(X_(1, idx_tx),SNR_dB_values(i_SNR),"measured",10);
+                w_awgn_matlab = r_awgn_ - X_(1,idx_tx);
+                snr_meas_(i_SNR) = snr(X_(1, idx_tx) , w_awgn_matlab(1,:));
+            end
+
+            %%%%%%%%%%%%
+            clf
+            hold on
+            xlim([-4, 4]);
+            ylim([-4, 4]);
+            %received signal
+            scatter(X_(1, idx_tx),X_(2, idx_tx),3,"black",'o','DisplayName','Tx Constellation');
+
+
+            if cmplx
+                scatter(r_awgn_(1,:),r_awgn_(2,:),1,"red",'x','DisplayName','Matlab Mapping');
+            else
+                scatter(r_awgn_,zeros(size(r_awgn_)),1,"red",'x','DisplayName','Matlab Mapping');
+            end
+            drawnow
+
+            MI_awgn_(i_SNR)=air(X_, r_awgn_, idx_tx);
+            % The following also works but is slower
+            %     MI_awgn(i_SNR)=mi_cg(s, r_awgn);
+        end
+
+        figure(4);
+        hold on
+
+        if isempty(I_shannon)
+            I_shannon=log2(1+db2pow(snr_meas_));
+            plot(snr_meas_, I_shannon, '-', 'DisplayName', 'log_2 (1+SNR)','Color','black');
+        end
+
+        if usegarcia
+            plot(SNR_dB_values, MI_awgn, '--', 'DisplayName', ['GARCIA ',num2str(M_PAM) '-QAM, AWGN']);
+        end
+
+        cols = linspecer(6);
+        if complex_constellation
+            plot(snr_meas_, MI_awgn_, ':', 'DisplayName', ['',num2str(M_PAM) '-QAM, AWGN'],'LineWidth',1,'Color',cols(cnt,:));
+        else
+            plot(snr_meas_, MI_awgn_, '-', 'DisplayName', ['',num2str(M_PAM) '-PAM, AWGN'],'LineWidth',1,'Color',cols(cnt,:));
+        end
+
+        hold off;
+        ylim([0,8]);
+        xlim([min(SNR_dB_values) max(SNR_dB_values)])
+        xlabel('SNR (dB)'); ylabel('Achievable rate (bits/complex dimension)');
+        title(['Achievable rate of ' num2str(M_PAM) '-QAM in AWGN']);
+        legend('Location', 'NorthWest');
+        yline(log2(M_PAM),'LineStyle',':','HandleVisibility','off','Color','black');
+
+        autoArrangeFigures;
+        cnt = cnt+1;
+
+    end
+end