51 lines
2.1 KiB
ReStructuredText
51 lines
2.1 KiB
ReStructuredText
Discrete Fourier Transform
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Discrete Fourier Transform (DFT) is used to perform the Fourier transform of a discrete signal.
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The DFT of a discrete complex/real signal :math:`\vec{x}=\{x_0,...,x_{N-1}\}` is:
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.. math::
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X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi\frac{k}{N}n}
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.. note::
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In this case, :math:`\frac{k}{N}` can be seen as the frequency of the wave used to determined
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the :math:`k^{th}` frequency :math:`X_k`.
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See `here <https://www.youtube.com/watch?v=mkGsMWi_j4Q&t=282s>`__ for more infos.
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One important property of the DFT is its **periodicity**. The DFT has a period of :math:`N`.
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Hence, all :math:`X_k` such that :math:`k > \frac{N}{2}` corresponds to negative frequencies:
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.. math::
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X_{N-1} &= \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi\frac{N-1}{N}n}
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&= \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi (n - \frac{n}{N})}
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&= \sum_{n=0}^{N-1} x_n \cdot e^{-j2\pi n} \cdot e^{j2\pi\frac{n}{N}}
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&= \sum_{n=0}^{N-1} x_n \cdot e^{j2\pi\frac{n}{N}} \mathrm{~~as~~} \forall n: e^{-j2\pi n} = 1
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&=X_{-1}
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This explanation is extracted from `here <https://dsp.stackexchange.com/questions/50505/why-is-the-second-half-of-the-fft-negative-frequencies>`__.
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Note that, the frequency corresponding to :math:`\frac{N}{2}` is called `Nyquist_rate <https://en.wikipedia.org/wiki/Nyquist_rate>`__.
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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.
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This leads to a capturing effect of the negative frequencies.
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See `here <https://www.youtube.com/watch?v=nOLuVNhBIg0>`__ for a visual explanation.
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Inverse DFT
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===========
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The inverse DFT of a discrete complex/real signal :math:`\vec{x}={x_0,...,x_{N-1}}` is:
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.. math::
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x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{j2\pi\frac{k}{N}n}
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.. note::
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If the signal is *real*, all the complex terms of the inverse DFT are suppose to cancel out.
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In depth example
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=================
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Let's take a discrete signal :math:`\vec{x}=\{1,2,3,4\}`, sampled at a frequency :math:`f_s=10 Hz`.
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.. literalinclude:: code/dft.R
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