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% 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.
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% The above copyright notice and this permission notice shall be included
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%%%%%%%%%%%%%%%%%%%%%% 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>
%
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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');