# Engineering Tables/Fourier Transform Table 2

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
${\displaystyle g(t)\!\equiv \!}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,}$
${\displaystyle G(\omega )\!\equiv \!}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,}$
${\displaystyle G(f)\!\equiv }$

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
10 ${\displaystyle \mathrm {rect} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} \left({\frac {f}{a}}\right)}$ The rectangular pulse and the normalized sinc function
11 ${\displaystyle \mathrm {sinc} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {rect} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {rect} \left({\frac {f}{a}}\right)\,}$ Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12 ${\displaystyle \mathrm {sinc} ^{2}(at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {tri} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {tri} \left({\frac {f}{a}}\right)}$ tri is the triangular function
13 ${\displaystyle \mathrm {tri} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} ^{2}\left({\frac {f}{a}}\right)\,}$ Dual of rule 12.
14 ${\displaystyle e^{-\alpha t^{2}}\,}$ ${\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}}$ ${\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi f)^{2}}{\alpha }}}}$ Shows that the Gaussian function ${\displaystyle \exp(-\alpha t^{2})}$ is its own Fourier transform. For this to be integrable we must have ${\displaystyle \mathrm {Re} (\alpha )>0}$.
${\displaystyle e^{iat^{2}}=\left.e^{-\alpha t^{2}}\right|_{\alpha =-ia}\,}$ ${\displaystyle {\frac {1}{\sqrt {2a}}}\cdot e^{-i\left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}}$ ${\displaystyle {\sqrt {\frac {\pi }{a}}}\cdot e^{-i\left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}}$ common in optics
${\displaystyle \cos(at^{2})\,}$ ${\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$ ${\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$
${\displaystyle \sin(at^{2})\,}$ ${\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$ ${\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$
${\displaystyle e^{-a|t|}\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}}$ ${\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}f^{2}}}}$ a>0
${\displaystyle {\frac {1}{\sqrt {|t|}}}\,}$ ${\displaystyle {\frac {1}{\sqrt {|\omega |}}}}$ ${\displaystyle {\frac {1}{\sqrt {|f|}}}}$ the transform is the function itself
${\displaystyle J_{0}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}$ ${\displaystyle {\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}}$ J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function
${\displaystyle J_{n}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}$ ${\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi f)\mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}}$ it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
${\displaystyle {\frac {J_{n}(t)}{t}}\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,}$

${\displaystyle \cdot \ {\sqrt {1-\omega ^{2}}}\mathrm {rect} \left({\frac {\omega }{2}}\right)}$

${\displaystyle {\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,}$

${\displaystyle \cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f)}$

Un (t) is the Chebyshev polynomial of the second kind