39 lines
1.5 KiB
ReStructuredText
39 lines
1.5 KiB
ReStructuredText
Fourier Tables
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Table of pairs
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==============
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.. csv-table::
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:header: Signal :math:`x(t)`, Fourier transform :math:`X(\\omega)`
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:align: center
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:math:`\delta(t)`, :math:`1`
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:math:`1`, :math:`2\pi\delta(\omega)`
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:math:`\mathrm{rect}(\frac{t}{a})`, :math:`a\mathrm{sinc}(\frac{\omega a}{2})`
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:math:`\frac{a}{\pi}\mathrm{sinc}(at)`, :math:`\mathrm{rect}(\frac{\omega}{2a})`
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:math:`e^{-at}u(t)`, :math:`\frac{1}{j\omega + a}`
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:math:`e^{j\omega_0t}`, :math:`2\pi\delta(\omega-\omega_0)`
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:math:`\mathrm{cos}(\omega_0t)`, :math:`\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]`
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:math:`\mathrm{sin}(\omega_0t)`, :math:`\frac{\pi}{j}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]`
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Table of properties
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===================
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.. csv-table::
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:header: Signal :math:`x(t)`, Fourier transform :math:`X(\\omega)`
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:align: center
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:math:`x(t)`, :math:`X(\omega)`
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:math:`x(t-t_0)`, :math:`e^{-jwt_0}X(\omega)`
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:math:`e^{jw_0t}x(t)`, :math:`X(\omega-\omega_0)`
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:math:`x(at)`, :math:`\frac{1}{|a|}X(\frac{\omega}{a})`
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:math:`\frac{d}{dt}x(t)`, :math:`j\omega X(\omega)`
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:math:`tx(t)`, :math:`j\frac{d}{d\omega}X(\omega)`
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:math:`x_1(t) \ast x_2(t)`, :math:`X_1(\omega)X_2(\omega)`
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:math:`x_1(t)x_2(t)`, :math:`\frac{1}{2\pi}X_1(\omega) \ast X_2(\omega)`
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:math:`x(t)^*`, :math:`X^*(-\omega)`
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:math:`X(t)`, :math:`2\pi x(-\omega)`
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:math:`\int_{-\infty}^tx(\tau)d\tau`, :math:`\frac{X(\omega)}{j\omega}+\pi X(0) \delta(\omega)`
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