For a second example, we're going to look at how you can build a triangle wave.

So to build a triangle wave, from sine waves, it's similar to a square wave.

Again, the recipe calls for all of the odd harmonics.

So here's the first harmonic over a fundamental f0, the third harmonic, the

fifth harmonic and so on, 7, 9, 11. And the difference here though, is that

the coefficients in front of each of the harmonics has a 1 over n squared.

And so instead of 1 3rd, it's 1 over 3 to the second power, so 1 9th.

And then this is 1 25th and then for the 7th harmonic it would be 1 49th.

So the size of the harmonics gets fairly small much more rapidly than it does in

the square wave. So now we'll look at a MATLAB simulation

of building a triangle wave from a fundamental and its odd harmonics.

In this demonstration it's just like the last one for the square wave.

Except now we're going to build a composite wave form that is a triangle

wave. Another Fourier series recipe for a

triangle wave is also all of the odd harmonics.

But what we're going to do in this case is we're going to add them.

Each harmonic is going to have an amplitude that is 1 over n squared.

So the third harmonic will be 1 3rd squared or 1 9th the amplitude of the

first. And the fifth will be 1 25th and so on,

so here is the same simulation of the starting with the fundamental and then

adding the harmonics.

[NOISE].

You can see that when you get to the end here the amplitudes of those harmonics is

just minuscule. So the 23rd harmonic is 1 over 23

squared, so it's about 1 over 500 or so of the amplitude of the fundamental.

But, there it is. And these higher harmonics, all they're

doing after while is they're really just helping, helping to sharpen up the tip of

the of the triangle wave.