Biến đổi Fourier liên tục
Biến đổi của các hàm thông dụng > Distributions

|  | Signal | Fourier transform   unitary, angular frequency | Fourier transform   unitary, ordinary frequency | Remarks |
| --- | --- | --- | --- | --- |
|  |  g(t)\, |  G(\omega )\!\ {\stackrel {\mathrm {def} }{=}}\ \!   {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\,dt |  G(f)\!\ {\stackrel {\mathrm {def} }{=}}\    \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\,dt |  |
| 301 |  1\, |  {\sqrt {2\pi }}\cdot \delta (\omega )\, |  \delta (f)\, |  \displaystyle \delta (\omega ) denotes the Dirac delta distribution. |
| 302 |  \delta (t)\, |  {\frac {1}{\sqrt {2\pi }}}\, |  1\, | Dual of rule 301. |
| 303 |  e^{iat}\, |  {\sqrt {2\pi }}\cdot \delta (\omega -a)\, |  \delta (f-{\frac {a}{2\pi }})\, | This follows from and 103 and 302. |
| 304 |  \cos(at)\, |  {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!+\!\delta (\omega \!+\!a)}{2}}\, |  {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!+\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\, | Follows from rules 101 and 303 using Euler's formula:  \displaystyle \cos(at)=(e^{iat}+e^{-iat})/2. |
| 305 |  \sin(at)\, |  i{\sqrt {2\pi }}{\frac {\delta (\omega \!+\!a)\!-\!\delta (\omega \!-\!a)}{2}}\, |  i{\frac {\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!-\!\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\, | Also from 101 and 303 using  \displaystyle \sin(at)=(e^{iat}-e^{-iat})/(2i). |
| 306 |  t^{n}\, |  i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\, |  \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(f)\, | Here,  \displaystyle n is a natural number.  \displaystyle \delta ^{n}(\omega ) is the  \displaystyle n-th distribution derivative of the Dirac delta. This rule follows from rules 107 and 302. Combining this rule with 1, we can transform all polynomials. |
| 307 |  {\frac {1}{t}}\, |  -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )\, |  -i\pi \cdot \operatorname {sgn}(f)\, | Here  \displaystyle \operatorname {sgn}(\omega ) is the sign function; note that this is consistent with rules 107 and 302. |
| 308 |  {\frac {1}{t^{n}}}\, |  -i{\begin{matrix}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(\omega )\, |  -i\pi {\begin{matrix}{\frac {(-i2\pi f)^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(f)\, | Generalization of rule 307. |
| 309 |  \operatorname {sgn}(t)\, |  {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\ \omega }}\, |  {\frac {1}{i\pi f}}\, | The dual of rule 307. |
| 310 |  u(t)\, |  {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)\, |  {\frac {1}{2}}\left({\frac {1}{i\pi f}}+\delta (f)\right)\, | Here  u(t) is the Heaviside unit step function; this follows from rules 101 and 309. |
| 311 |  e^{-at}u(t)\, |  {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}} |  {\frac {1}{a+i2\pi f}} |  u(t) is the Heaviside unit step function and  a>0. |
| 312 |  \sum _{n=-\infty }^{\infty }\delta (t-nT)\, |  {\begin{matrix}{\frac {\sqrt {2\pi }}{T}}\end{matrix}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\begin{matrix}{\frac {2\pi }{T}}\end{matrix}}\right)\, |  {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)\, | The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time. |