DFTJEW

Discrete Fourier Transform

The Fourier Transform is a powerful tool to convert a continuous function from the time domain into the frequency domain. The Fourier transform, however, is an integral transform; it is done by integration. This cannot be done with a discrete function. The Discrete Fourier Transform (DFT) allows us to transform a discrete function from the time domain into the frequency domain.

If $x(n)$ is a discrete function in time, then the DFT of $x(n)$ would be:

$\mbox{DFT}[x(n)] \equiv X(m) \equiv \sum_{n=0}^{N-1} x(n) e^{-j \frac{2 \pi n m}{N} }$

When dealing with a continuous function, an inverse Fourier Transform can be used to go back from the frequency domain into the time domain. The discete analog of the inverse Fourier Transform is the IDFT (Inverse Discrete Fourier Transform) which is defined below:

$\mbox{IDFT}[X(m)] \equiv x(k) \equiv \frac{1}{N} \sum_{m=0}^{N-1} X(m) e^{j \frac{2 \pi k m}{N} }$

One thing to note about the DFT is that it assumes periodicity. Let us, for example, consider a DFT on the interval of $-m$ to $+m$. The DFT will expect the data at +$m$ to be equal to the data at -$m$.

Principle author: Jeffrey Wonoprabowo