# 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 |
| | $J_{0}(t)\,$ | ${\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$ | ${\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$ | J$_0$(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; T$_n$ (t) is the Chebyshev polynomial of the first kind. |
| | ${\frac {J_{n}(t)}{t}}\,$ | ${\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,$  $\cdot \ {\sqrt {1-\omega ^{2}}}\mathrm {rect} \left({\frac {\omega }{2}}\right)$ | ${\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,$  $\cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f)$ | U$_n$ (t) is the Chebyshev polynomial of the second kind |