Channel Matrix 

 

 

 

 

 

 

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

 

< Code 1 >

 

n = 0;

w = n * pi/40;

 

a = 0.5;

 

e11 = exp(j * w * 0);

e12 = exp(j * w * 0);

e21 = a * exp(j * w);

e22 = a * exp(-j * w);

 

H = 1/2 * [e11 e12;e21 e22];

 

[u, s, v] = svd(H);

r = rank(H);

c = cond(H);

d = det(H)

 

hFig = figure(1,'Position',[300 300 700 600]);  

 

% ======================= Printing Matrix Information =========================

 

subplot(3,3,[1 3]);

hold on;

plot([0.0],[0.0]);

 

x0 = 0.01;

y0 = 0.0;

tStr = sprintf('%0.02f%+-0.02f j    %0.02f%+-0.02f j\n%0.02f%+-0.02f j    %0.02f%+-0.02f j', ...

               real(H(1,1)),imag(H(1,1)),real(H(1,2)),imag(H(1,2)),...

               real(H(2,1)),imag(H(2,1)),real(H(2,2)),imag(H(2,2)));

text(x0+0.0,y0+1.0,tStr,'FontSize',14,'color','black');

line([x0-0.01 x0-0.01],[y0+0.8 y0+1.2],'LineWidth',1,'Color','black');

line([x0+0.31 x0+0.31],[y0+0.8 y0+1.2],'LineWidth',1,'Color','black');

 

x0 = 0.01;

y0 = -0.2;

text(x0-0.04,y0+0.69,"=",'FontSize',14,'color','black');

tStr = sprintf('%0.02f%+-0.02f j    %0.02f%+-0.02f j\n%0.02f%+-0.02f j    %0.02f%+-0.02f j', ...

               real(u(1,1)),imag(u(1,1)),real(u(1,2)),imag(u(1,2)),...

               real(u(2,1)),imag(u(2,1)),real(u(2,2)),imag(u(2,2)));

text(x0+0.0,y0+0.7,tStr,'FontSize',14,'color','black');

line([x0-0.01 x0-0.01],[y0+0.5 y0+0.9],'LineWidth',1,'Color','black');

line([x0+0.32 x0+0.32],[y0+0.5 y0+0.9],'LineWidth',1,'Color','black');

 

tStr = sprintf('%0.02f%+-0.02f j    %0.02f%+-0.02f j\n%0.02f%+-0.02f j    %0.02f%+-0.02f j', ...

               real(s(1,1)),imag(s(1,1)),real(s(1,2)),imag(s(1,2)),...

               real(s(2,1)),imag(s(2,1)),real(s(2,2)),imag(s(2,2)));

text(x0+0.35,y0+0.7,tStr,'FontSize',14,'color','black');

line([x0+0.34 x0+0.34],[y0+0.5 y0+0.9],'LineWidth',1,'Color','black');

line([x0+0.65 x0+0.65],[y0+0.5 y0+0.9],'LineWidth',1,'Color','black');

 

tStr = sprintf('%0.02f%+-0.02f j    %0.02f%+-0.02f j\n%0.02f%+-0.02f j    %0.02f%+-0.02f j', ...

               real(v(1,1)),imag(v(1,1)),real(v(1,2)),imag(v(1,2)),...

               real(v(2,1)),imag(v(2,1)),real(v(2,2)),imag(v(2,2)));

text(x0+0.68,y0+0.7,tStr,'FontSize',14,'color','black');

line([x0+0.67 x0+0.67],[y0+0.5 y0+0.9],'LineWidth',1,'Color','black');

line([x0+1.00 x0+1.00],[y0+0.5 y0+0.9],'LineWidth',1,'Color','black');

 

tStr = sprintf('Rank = %d',r);

text(x0-0.01,y0+0.32,tStr,'FontSize',14,'color','black');

tStr = sprintf('Condition Number = %d',c);

text(x0-0.01,y0+0.12,tStr,'FontSize',14,'color','black');

tStr = sprintf('Determinant = %0.02f%+-0.02f j',real(d),imag(d));

text(x0-0.01,y0-0.1,tStr,'FontSize',14,'color','black');

 

axis([0.0 1.01 0 1.2]);

set(gca,'Visible','off');

hold off;

 

 

% ======================= Plot Constellation =========================

 

N = 500;

rand ("seed", 100);

 

n1 = 0.05*(randn(1,N) + j*randn(1,N));

n2 = 0.05*(randn(1,N) + j*randn(1,N));

c1 = (2*randi([0 1],1,N)-1) + j*(2*randi([0 1],1,N)-1);

c2 = (2*randi([0 1],1,N)-1) + j*(2*randi([0 1],1,N)-1);

 

c1 = c1 + n1;

c2 = c2 + n2;

 

n = 1;          

Tx = [c1; c2];             

 

Rx = H * Tx;

 

 

subplot(3,3,4);

plot(real(Tx(1,:)),imag(Tx(1,:)),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',4);

axis([-2 2 -2 2]);

title('Tx(1)');

grid on;

box on;

 

subplot(3,3,7);

plot(real(Tx(2,:)),imag(Tx(2,:)),'mo','MarkerFaceColor',[1 0 1],'MarkerSize',4);

axis([-2 2 -2 2]);

title('Tx(2)');

grid on;

box on;

 

subplot(3,3,6);     

hold on;

plot(real(Rx(1,:)),imag(Rx(1,:)),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',4);

axis([-2 2 -2 2]);

title('Rx(1)');

hold off;

grid on;

box on;

 

subplot(3,3,9);     

hold on;

plot(real(Rx(2,:)),imag(Rx(2,:)),'ko','MarkerFaceColor',[0 0 0],'MarkerSize',4);

axis([-2 2 -2 2]);

title('Rx(2)');

hold off;

grid on;

box on;

 

 

subplot(3,3,[5 8]);

hold on;

plot([0.0],[0.0]);

 

x0 = 0.1;

y0 = 0.0;

text(x0+1.0,y0+1.4,"H",'FontSize',14,'color','black');

tStr = sprintf('%0.02f%+-0.02f j    %0.02f%+-0.02f j\n%0.02f%+-0.02f j    %0.02f%+-0.02f j', ...

               real(H(1,1)),imag(H(1,1)),real(H(1,2)),imag(H(1,2)),...

               real(H(2,1)),imag(H(2,1)),real(H(2,2)),imag(H(2,2)));

text(x0+0.0,y0+1.0,tStr,'FontSize',11,'color','black');

line([x0-0.1 x0-0.1],[y0+0.8 y0+1.2],'LineWidth',1,'Color','black');

line([x0+2.28 x0+2.28],[y0+0.8 y0+1.2],'LineWidth',1,'Color','black');

 

axis([0.0 2.4 0 2.4]);

set(gca,'Visible','off');

hold off;