Biến đổi Fourier liên tục
Biến đổi của các hàm thông dụng > Các hàm bình phương khả tích

|  | Signal | Fourier transform   unitary, angular frequency | Fourier transform   unitary, ordinary frequency | Remarks |
| --- | --- | --- | --- | --- |
|  |  g(t)\, |  G(\omega )\!\ {\stackrel {\operatorname {def} }{=}}\ \!   {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\operatorname {d} t\, |  G(f)\!\ {\stackrel {\operatorname {def} }{=}}\    \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\operatorname {d} t\, |  |
| 201 |  \operatorname {rect} (at)\, |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {f}{a}}\right) | The rectangular pulse and the normalized sinc function |
| 202 |  \operatorname {sinc} (at)\, |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {f}{a}}\right)\, | Dual of rule 201. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter. |
| 203 |  \operatorname {sinc} ^{2}(at)\, |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {tri} \left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {f}{a}}\right) | tri is the triangular function |
| 204 |  \operatorname {tri} (at)\, |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {f}{a}}\right)\, | Dual of rule 203. |
| 205 |  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  \operatorname {Re} (\alpha )>0. |
| 206 |  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 |
| 207 |  \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) |  |
| 208 |  \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) |  |
| 209 |  \operatorname {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 |
| 210 |  {\frac {1}{\sqrt {|t|}}}\, |  {\frac {1}{\sqrt {|\omega |}}} |  {\frac {1}{\sqrt {|f|}}} | the transform is the function itself |
| 211 |  J_{0}(t)\, |  {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}} |  {\frac {2\cdot \operatorname {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}} | J0(t) is the Bessel function of first kind of order 0 |
| 212 |  J_{n}(t)\, |  {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}} |  {\frac {2(-i)^{n}T_{n}(2\pi f)\operatorname {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. |
| 213 |  {\frac {J_{n}(t)}{t}}\, |  {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,   \cdot \ {\sqrt {1-\omega ^{2}}}\operatorname {rect} \left({\frac {\omega }{2}}\right) |  {\frac {2\operatorname {i} }{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,   \cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\operatorname {rect} (\pi f) | Un (t) is the Chebyshev polynomial of the second kind |
| 214 |  \operatorname {sech} (at)\, |  {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right) |  {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}f\right) | Hyperbolic secant is its own Fourier transform |