Fourier Transform

The Fourier Transform takes a curve and finds the the frequencise it is made of.

(graphic)

How does it do this magic?

Well, we can test a frequency by mulitplying it by the function. As you can see below this works nicely

(app)

So we can see how it works, but how do we write that down with mathematics?

(formula)

Whoa, steady on there! What does all that mean?

Let us try something a little simpler. Instead of e^iw... let us just use sin()

(formula with sin())

This can work in certain circumstances but is troubled by PHASE

Phase is the left or right shifting on the graph. When two wave are in phase they myultiply to soemting bigger, byt wehn out of ophase things go terribly wrong

Try this:

(app with phase setting on)

So, we need somnething more powerful, and that is here the e^iw... comes in.

To understand e^-w... please read Euler's Identity

So this gives us a "full circle" which solves the pahse problem

It is like we take our function and wrap it in a circle and then find out how far the center of the area is away form the center.

(graphic)

And this is the reult.

Neat huh?