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.

One important property of the DFT is its **periodicity**. The DFT has a period of :math:`N`.
Hence, all :math:`X_k` such that :math:`k > \frac{N}{2}` corresponds to negative frequencies:

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

   &= \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi (n - \frac{n}{N})}

   &= \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi n} \cdot e^{j2\pi\frac{n}{N}}

   &= \sum_{n=0}^{N-1} x_n \cdot e^{j2\pi\frac{n}{N}} \mathrm{~~as~~} \forall n: e^{-j2\pi n} = 1

   &=X_{-1}

This explanation is extracted from `here <https://dsp.stackexchange.com/questions/50505/why-is-the-second-half-of-the-fft-negative-frequencies>`__.
Note that, the frequency corresponding to :math:`\frac{N}{2}` is called `Nyquist_rate <https://en.wikipedia.org/wiki/Nyquist_rate>`__.
As frequencies above :math:`\frac{N}{2}` cannot be captured by the signal due to the `Nyquist-Shannon <https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem>`__, a reasoning phenomenon starts to arise.
This leads to a capturing effect of the negative frequencies.
See `here <https://www.youtube.com/watch?v=nOLuVNhBIg0>`__ for a visual explanation.
   
Inverse DFT
===========
   
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
=================

Let's take a discrete signal :math:`\vec{x}=\{1,2,3,4\}`, sampled at a frequency :math:`f_s=10 Hz`.

.. literalinclude:: code/dft.R