Fourier Tables
--------------


Table of pairs
==============

.. csv-table::
   :header: Signal :math:`x(t)`, Fourier transform :math:`X(\\omega)`
   :align: center
           
   :math:`\delta(t)`, :math:`1`
   :math:`1`, :math:`2\pi\delta(\omega)`
   :math:`\mathrm{rect}(\frac{t}{a})`, :math:`a\mathrm{sinc}(\frac{\omega a}{2})`
   :math:`\frac{a}{\pi}\mathrm{sinc}(at)`, :math:`\mathrm{rect}(\frac{\omega}{2a})`
   :math:`e^{-at}u(t)`, :math:`\frac{1}{j\omega + a}`
   :math:`e^{j\omega_0t}`, :math:`2\pi\delta(\omega-\omega_0)`
   :math:`\mathrm{cos}(\omega_0t)`, :math:`\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]`
   :math:`\mathrm{sin}(\omega_0t)`, :math:`\frac{\pi}{j}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]`


Table of properties
===================

.. csv-table::
   :header: Signal :math:`x(t)`, Fourier transform :math:`X(\\omega)`
   :align: center
           
   :math:`x(t)`, :math:`X(\omega)`
   :math:`x(t-t_0)`, :math:`e^{-jwt_0}X(\omega)`
   :math:`e^{jw_0t}x(t)`, :math:`X(\omega-\omega_0)`
   :math:`x(at)`, :math:`\frac{1}{|a|}X(\frac{\omega}{a})`
   :math:`\frac{d}{dt}x(t)`, :math:`j\omega X(\omega)`
   :math:`tx(t)`, :math:`j\frac{d}{d\omega}X(\omega)`
   :math:`x_1(t) \ast x_2(t)`, :math:`X_1(\omega) + X_2(\omega)`
   :math:`x_1(t)x_2(t)`, :math:`\frac{1}{2\pi}X_1(\omega) \ast X_2(\omega)`
   :math:`x(t)^*`, :math:`X^*(-\omega)`
   :math:`X(t)`, :math:`2\pi x(-\omega)`
   :math:`\int_{-\infty}^tx(\tau)d\tau`, :math:`\frac{X(\omega)}{j\omega}+\pi X(0) \delta(\omega)`