Fourier transform math

This page describes some of the math you need for the Fourier transform, as well as deriving some of the formulas you might see thrown around. I'm not a math person, so this is not meant to be as accurate or precise as you will probably see elsewhere. I am including many links to tutorials I found useful in this process.

Fourier Series

Any repeating waveform can be written as an infinite sum of sine and cosine waves at frequencies from 0 toinfinity. Each sine and cosine wave at each frequency will have a coefficient, or amplitude, that shows how much that sine and cosine wave contributes to describing the repeating waveform. For example, suppose we have a waveform that has a lot of "sine wave at f = 3" character. The coefficient of the sine wave at f = 3 will thus have a large coefficient.

The whole 0 to infinity number of frequencies looks kind of freaky, but it's basically just saying that the more sine and cosine waves you add to your Fourier series, the better you can describe the repeating waveform. [This] tutorial has a nifty flash program that lets you make a repeating waveform and then reconstruct it with a bunch of sine and cosine waves. The more waves at different frequencies you add, the more the your Fourier series looks like the repeating waveform.

Written as phase shifted cosine

Sometimes the Fourier series is written like this: