# Fourier Transform

## Introduction

The Fourier Transform is a generalization of the Fourier Series. Strictly speaking it only applies to continous and aperiodic functions, but the use of the impulse function allows the use of discrete signals.

The Fourier transform is defined as

The inverse transform is defined as

If we take the Fourier transform of a square pulse,

and apply the Fourier transform to it we get:

Since the pulse is zero everywhere except in the range  we can rewrite the equation as:

Note how the limits have changed from +/- infinity to +/- T/2, and that f(t) has disappeared because between the limits of +/- T/2 it has the value of one. Substituting the limits in then gives us

Using Euler's expressions  we can rewrite this as:

This is the sinc(x) function that is shown below. This function appears very frequently in Fourier transforms. It shows the main frequency content based around zerp frequency (d.c.) with progressively less and less energy in the higher frequencies.

Now that we have one example of the Fourier tansform, what do you think we get when we take a fourier transform of a exponentially decaying signal, say, the electromagnetic power radiated by an atom?