## Fourier Transforms, An Introduction

### Introduction

It would be difficult to overstate the importance of the fourier transform.
It is used in almost every area of science and engineering,
and it even pops up in pure mathematics when you least expect it,
e.g. deriving the first billion digits of π.
The computer you are using to browse this website
probably has 4 different implementations of the fourier transform,
in various applications,
plus a couple more in hardware and/or microcode.
When Joseph Fourier
(biography)
developed the transform in 1805
to study the flow of heat through a conductor,
he couldn't have imagined the value of his new technique.
Audio filters, image processing,
seismology, AC power distribution,
and rf communications
were all in the distant future.
In its simplest terms,
the fourier transform takes a periodic wave form,
such as the sound of a ringing bell,
and converts it into a sum of sines and cosines.
After all, sines and cosines repeat with certain frequencies;
perhaps they can be combined to describe the ringing of a bell,
or the tone of an oboe,
or the vibrations of an earthquake.
In two dimensions, replicate an image across the xy plane,
like a checkerboard,
then express the brightness levels of the image as sums of sines and cosines.
When A sound, or image, is represented as frequencies,
one can adjust some of the frequencies and rebuild the original wave.
Turn down the high frequencies and take the high notes out of a symphony.
Turn up the high frequencies and increase the contrast on an image.
And so much more.