Engineering Tables/Fourier Transform Table 2

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