This is machine translation

Translated by Microsoft
Mouse over text to see original. Click the button below to return to the English verison of the page.

라이선스가 부여된 사용자만 번역 문서를 볼 수 있습니다. 번역 문서를 보려면 로그인하십시오.

Square Wave from Sine Waves

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

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);

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

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

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;

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;
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.

shading interp
axis off ij

Was this topic helpful?