Discrete Fourier Transform
----------------------------

Discrete Fourier Transform (DFT) is used to perform the Fourier transform of a discrete signal.
The DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:

.. math::
   X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi\frac{k}{N}n}

.. note::
   In this case, :math:`\frac{k}{N}` can be seen as the frequency of the wave used to determined
   the :math:`k^{th}` frequency :math:`X_k`.
   See `here <https://www.youtube.com/watch?v=mkGsMWi_j4Q&t=282s>`__ for more infos.

The inverse DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:

.. math::
   x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{j2\pi\frac{k}{N}n}

.. note::
   If the signal is *real*, all the complex terms of the inverse DFT are suppose to cancel out.

In depth example
=================