Discrete Fourier Transforms

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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 discrete series of delta functions x(n) instead of the continuous function x(t).

What is it used for?

The DFT is used in a wide variety of areas. Not only is it useful for one dimensional applications such as signal processing, it is also used for 2D applications such as CAT scans. The DFT allows data to be transformed from one domain into another domain so that minipulation of the data is easier, much the same as a Fourier Transform, only in a discrete setting. Then, when the minipulations are finished, the IDFT (Inverse DFT) can be used to convert back to the original domain if that is required.