%% Square Wave from Sine Waves
% The Fourier series expansion for a square-wave is made up of a sum of odd
% harmonics.  We show this graphically using MATLAB.
%
% Copyright 1984-2002 The MathWorks, Inc.
% $Revision: 1.14 $  $Date: 2002/04/09 17:18:11 $

%%
% We start by forming a time vector running from 0 to 10 in steps of 0.1, and
% take the sine of all the points.  Let's plot this fundamental frequency.

t = 0:.1:10;
y = sin(t);
plot(t,y);

%%
% Now add the third harmonic to the fundamental, and plot it.

y = sin(t) + sin(3*t)/3;
plot(t,y);

%%
% Now use the first, third, fifth, seventh, and ninth harmonics.

y = sin(t) + sin(3*t)/3 + sin(5*t)/5 + sin(7*t)/7 + sin(9*t)/9;
plot(t,y);

%%
% For a finale, we will go from the fundamental to the 19th harmonic, creating
% vectors of successively more harmonics, and saving all intermediate steps as
% the rows of a matrix.
%
% These vectors are plotted on the same figure to show the evolution of the
% square wave.  Note that Gibbs' effect says that it will never really get
% there.

t = 0:.02:3.14;
y = zeros(10,length(t)); 
x = zeros(size(t));
for k=1:2:19
    x = x + sin(k*t)/k;
    y((k+1)/2,:) = x;
end
plot(y(1:2:9,:)')
title('The building of a square wave: Gibbs'' effect')

%%
% Here is a 3-D surface representing the gradual transformation of a sine wave
% into a square wave.

surf(y);
shading interp
axis off ij