Minor changes

source/signal_processing/dft.rst

@@ -2,7 +2,7 @@ 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:
+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}
@@ -12,6 +12,25 @@ The DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:
    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>`__.
+   
+Inverse DFT
+===========
+   
 The inverse DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:
 
 .. math::
@@ -23,3 +42,5 @@ The inverse DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1
 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`.
+