傅里叶变换
常用傅里叶变换表 > 平方可积函数

|  | 时域信号 | 角频率表示的 傅里叶变换 | 弧频率表示的 傅里叶变换 | 注释 |
| --- | --- | --- | --- | --- |
|  |  g(t)\!\equiv \!   {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}\mathrm {d} \omega \, |  G(\omega )\!\equiv \!   {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\mathrm {d} t\, |  G(f)\!\equiv    \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\mathrm {d} t\, |  |
| 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) | 矩形脉冲和归一化的sinc函数 |
| 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)\, | 变换10的频域对应。矩形函数是理想的低通滤波器，sinc函数是这类滤波器对反因果冲击的响应。 |
| 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是三角形函数 |
| 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)\, | 变换12的频域对应 |
| 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 }}} | 高斯函数 \exp(-\alpha t^{2})的傅里叶变换是其本身；只有当 \mathrm {Re} (\alpha )>0时，该函数可积的 |
| 15 |  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)} | 光学领域应用较多 |
| 16 |  \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) |  |
| 17 |  \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) |  |
| 18 |  \mathrm {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 |
| 19 |  {\frac {1}{\sqrt {|t|}}}\, |  {\frac {1}{\sqrt {|\omega |}}} |  {\frac {1}{\sqrt {|f|}}} | 变换本身就是一个公式 |
| 20 |  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)是0阶第一类贝塞尔函数。 |
| 21 |  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}}}} | 上一个变换的推广形式; Tn (t)是第一类切比雪夫多项式。 |
| 22 |  {\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 {2\mathrm {i} }{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,     \cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f) | Un (t)是第二类切比雪夫多项式。 |