HW
What is a Fourier Transform?
Unfortunately, the Fourier Transform isn't a Transform-er.
(This handsome fellow is named Roolbar.)
Check any of the other pages on this site to find fifty different ways to explain what a Fourier Transform is. If you already know what it is, or you're too lazy to look at the other pages, here's my super trite description: A Fourier Transform is a bunch of sinusoids of different frequencies and time offsets added together create a just about any function you want. Also, you can say that a Fourier Transform is the way of representing a function in the frequency domain instead of the time domain.
Instead of describing the Fourier Transform itself once again, the following pages describe how Fourier Transforms are analyzed and utilized in a computer.
Fourier Transform Applications
The "Fast" Fourier Transform
What is a Fast Fourier Transform? (FFT)
It's an algorithm that can compute the discrete Fourier transform faster than other algorithms. In digital systems, continuous Fourier Transforms are sampled, turning them into discrete Fourier Transforms which then can be computed and manipulated using Digital Signal Processing.
An intuitive brute force way of computing a Fourier Transform means rearranging the the summation so that you don't compute the transform in sequential order - you group similar elements together and simplify before combining them. This cuts down the adding and multiplying, thus cutting computation time down by about 100 times.
Cooley-Turkey Algorithm
Three popular algorithms are the Cooley Tukey algorithm (radix-r algorithms), the Good-Thomas Algorithm, and the Winograd FFT algorithm. The Cooley Tukey Algorithm Radix-2 algorithm is the most popularone