Discrete Fourier Transforms

Paul's DFT Page

One of the major tools used in signal processing is the DFT, which stands for Discrete Fourier Transform. The reason we need to to a DFT instead of a Fourier Transform is that our computers are limited in their abilites. They use sampling, and they have limited memory, so we have to adapt to the computers.

What is a DFT?

A DFT is like doing a Fourier Transform, but instead of doing it with an integral, we do it with discrete values and a sum. A Fourier Transform looks like this:
$X(f)=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}\,dt$
Which uses an integral, while the DFT which looks like this:
$X(f)=\sum _{n=0}^{N-1}x(n)e^{\frac {-j2\pi nm}{N}}$
Which is using a sum and a noncontinous series of delta functions x(n) instead of the continuous function x(t).