工程表格/傅立葉變換表格 2

	訊號	傅立葉變換 酉，角頻率	傅立葉變換 酉，普通頻率	備註
	$g(t)\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,$	$G(\omega )\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,$	$G(f)\!\equiv$ $\int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,$
10	$\mathrm {rect} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{|a|}}\cdot \mathrm {sinc} \left({\frac {f}{a}}\right)$	矩形脈衝和歸一化 sinc 函式
11	$\mathrm {sinc} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {rect} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{|a|}}\cdot \mathrm {rect} \left({\frac {f}{a}}\right)\,$	規則 10 的對偶。矩形函式是理想化的低通濾波器，sinc 函式是這種濾波器的非因果脈衝響應。
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 是三角函式
13	$\mathrm {tri} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{|a|}}\cdot \mathrm {sinc} ^{2}\left({\frac {f}{a}}\right)\,$	規則 12 的對偶。
14	$e^{-\alpha t^{2}}\,$	${\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}$	${\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi f)^{2}}{\alpha }}}$	表明高斯函式 $\exp(-\alpha t^{2})$ 是其自身的傅立葉變換。為了使之可積，我們必須有 $\mathrm {Re} (\alpha )>0$.
	$e^{iat^{2}}=\left.e^{-\alpha t^{2}}\right|_{\alpha =-ia}\,$	${\frac {1}{\sqrt {2a}}}\cdot e^{-i\left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$	${\sqrt {\frac {\pi }{a}}}\cdot e^{-i\left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$	光學中常見
	$\cos(at^{2})\,$	${\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	${\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)$
	$\sin(at^{2})\,$	${\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	$-{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)$
	$e^{-a|t|}\,$	${\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}$	${\frac {2a}{a^{2}+4\pi ^{2}f^{2}}}$	a>0
	${\frac {1}{\sqrt {|t|}}}\,$	${\frac {1}{\sqrt {|\omega |}}}$	${\frac {1}{\sqrt {|f|}}}$	變換後的函式本身
	$J_{0}(t)\,$	${\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$	J0(t) 是第一類零階貝塞爾函式，rect 是矩形函式
	$J_{n}(t)\,$	${\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2(-i)^{n}T_{n}(2\pi f)\mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$	這是之前變換的推廣形式；Tn (t) 是第一類切比雪夫多項式。
	${\frac {J_{n}(t)}{t}}\,$	${\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,$   $\cdot \ {\sqrt {1-\omega ^{2}}}\mathrm {rect} \left({\frac {\omega }{2}}\right)$	${\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,$   $\cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f)$	Un (t) 是第二類切比雪夫多項式