diff --git a/source/index.rst b/source/index.rst index ebbcb45..ae46ff6 100644 --- a/source/index.rst +++ b/source/index.rst @@ -26,6 +26,7 @@ Welcome to ScienceNotes's documentation! :caption: Signal Processing signal_processing/fourier_tables.rst + signal_processing/dft.rst Indices and tables ================== diff --git a/source/signal_processing/dft.rst b/source/signal_processing/dft.rst new file mode 100644 index 0000000..7e6107a --- /dev/null +++ b/source/signal_processing/dft.rst @@ -0,0 +1,25 @@ +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 `__ 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 +================= +