工程表格/傅里叶变换表格 2

	信号	傅里叶变换 酉，角频率	傅里叶变换 酉，普通频率	备注
	$g(t)\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,$	$G(\omega )\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,$	$G(f)\!\equiv$ $\int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,$
10	$\mathrm {rect} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{|a|}}\cdot \mathrm {sinc} \left({\frac {f}{a}}\right)$	矩形脉冲和归一化 sinc 函数
11	$\mathrm {sinc} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {rect} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{|a|}}\cdot \mathrm {rect} \left({\frac {f}{a}}\right)\,$	规则 10 的对偶。矩形函数是理想化的低通滤波器，sinc 函数是这种滤波器的非因果脉冲响应。
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	$\mathrm {tri} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{|a|}}\cdot \mathrm {sinc} ^{2}\left({\frac {f}{a}}\right)\,$	规则 12 的对偶。
14	$e^{-\alpha t^{2}}\,$	${\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}$	${\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi f)^{2}}{\alpha }}}$	表明高斯函数 $\exp(-\alpha t^{2})$ 是其自身的傅里叶变换。为了使之可积，我们必须有 $\mathrm {Re} (\alpha )>0$.
	$e^{iat^{2}}=\left.e^{-\alpha t^{2}}\right|_{\alpha =-ia}\,$	${\frac {1}{\sqrt {2a}}}\cdot e^{-i\left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$	${\sqrt {\frac {\pi }{a}}}\cdot e^{-i\left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$	光学中常见
	$\cos(at^{2})\,$	${\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	${\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)$
	$\sin(at^{2})\,$	${\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	$-{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)$
	$e^{-a|t|}\,$	${\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}$	${\frac {2a}{a^{2}+4\pi ^{2}f^{2}}}$	a>0
	${\frac {1}{\sqrt {|t|}}}\,$	${\frac {1}{\sqrt {|\omega |}}}$	${\frac {1}{\sqrt {|f|}}}$	变换后的函数本身
	$J_{0}(t)\,$	${\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$	J0(t) 是第一类零阶贝塞尔函数，rect 是矩形函数
	$J_{n}(t)\,$	${\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2(-i)^{n}T_{n}(2\pi f)\mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$	这是之前变换的推广形式；Tn (t) 是第一类切比雪夫多项式。
	${\frac {J_{n}(t)}{t}}\,$	${\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,$   $\cdot \ {\sqrt {1-\omega ^{2}}}\mathrm {rect} \left({\frac {\omega }{2}}\right)$	${\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,$   $\cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f)$	Un (t) 是第二类切比雪夫多项式