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Loic Guegan 2023-11-01 18:58:15 +01:00
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source/signal_processing/dft.rst
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@ -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`.