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Square Wave from Sine Waves

This example shows how the Fourier series expansion for a square wave is made up of a sum of odd harmonics.

Start by forming a time vector running from 0 to 10 in steps of 0.1, and take the sine of all the points. Plot this fundamental frequency.

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

Figure contains an axes. The axes contains an object of type line.

Next add the third harmonic to the fundamental, and plot it.

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

Figure contains an axes. The axes contains an object of type line.

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);

Figure contains an axes. The axes contains an object of type line.

For a finale, go from the fundamental all the way to the 19th harmonic, creating vectors of successively more harmonics, and saving all intermediate steps as the rows of a matrix.

Plot the vectors on the same figure to show the evolution of the square wave. Note that the Gibbs effect says it will never quite 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')

Figure contains an axes. The axes with title The building of a square wave: Gibbs' effect contains 5 objects of type line.

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