% This is a MATLAB script file which creates and displays
% the sum of two complex-conjugate complex-exponentials
% as an animation.  Very cool...
% That is, the animation shows how 
%
%      exp(j*t)+exp(-j*t)=2*cos(t).
%
% I have found that this is a concept which students often
% struggle with in Linear Systems (soon to be Signals and Systems)
% and which is the basis of all Fourier analysis.
%
% The moving blue circle represents exp(j*t).
% The moving red circle represents exp(-j*t).
% The moving green circle represents the sum exp(j*t)+exp(-j*t)=2*cos(t).
% The moving yellow is simply a time marker on the zero axis.
%
% Be patient, the frames must load, then they play fast.
% You can reduce the memory requirements by increasing the
% SKIP parameter and by increasing the step size in t. 
%
% Author: Dr. Russell C. Hardie 10/22/97
% University of Dayton

%---------------%
% Generate Data %
%---------------%

figure(1)
step=.01;
t=[0:step:4*pi];
x=exp(i*t);
plot3(real(x),t,imag(x))        
hold on
xc=exp(-i*t);                   
%% plot3(real(xc),t,imag(xc),'r-') % plot conjugate path
plot3(real(xc+x),t,imag(xc+x),'g-') 
Z=zeros(size(t));
plot3(Z,t,Z,'y-') 
title('The formation of cos(t) from complex exponentials');
xlabel('Real Axis');
ylabel('Time Axis');
zlabel('Imaginary Axis');
grid

%----------------------------------%
% Create Movie One Frame at a Time %
%----------------------------------%

SKIP=20;  % controls how many points in t to move between each frame
M=moviein(length(1:SKIP:length(t)));
count=1;
for n=1:SKIP:length(t)

  plot3(real(x(n)),t(n),imag(x(n)),'bo');
  plot3(real(xc(n)),t(n),imag(xc(n)),'ro');
  plot3(real(x(n)+xc(n)),t(n),imag(x(n)+xc(n)),'go');
  plot3(0,t(n),0,'yo');
  M(:,count)=getframe;
  count=count+1;
  drawnow

  hold off

  plot3(real(x),t,imag(x))
  hold on
  %% plot3(real(xc),t,imag(xc),'r-')  % plot conjugate path
  plot3(real(xc+x),t,imag(xc+x),'g-')
  plot3(Z,t,Z,'y-') 
  title('The formation of cos(t) from complex exponentials');
  xlabel('Real Axis');
  ylabel('Time Axis');
  zlabel('Imaginary Axis');
  grid

end

%------------%
% Play Movie %
%------------%

% frames per second
fps=10;  % movie speed (10 is max).
movie(M,-5,fps)