訊號與系統/傅立葉變換表

傅立葉變換

$F(j\omega )={\mathcal {F}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

傅立葉逆變換

${\mathcal {F}}^{-1}\left\{F(j\omega )\right\}=f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega$

傅立葉變換表

此表包含一些最常見的傅立葉變換。

	時域	頻域
	$x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}$	$X(\omega )={\mathcal {F}}\left\{x(t)\right\}$
1	$X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt$	$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$
2	$1\,$	$2\pi \delta (\omega )\,$
3	$-0.5+u(t)\,$	${\frac {1}{j\omega }}\,$
4	$\delta (t)\,$	$1\,$
5	$\delta (t-c)\,$	$e^{-j\omega c}\,$
6	$u(t)\,$	$\pi \delta (\omega )+{\frac {1}{j\omega }}\,$
7	$e^{-bt}u(t)\,(b>0)$	${\frac {1}{j\omega +b}}\,$
8	$\cos \omega _{0}t\,$	$\pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,$
9	$\cos(\omega _{0}t+\theta )\,$	$\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,$
10	$\sin \omega _{0}t\,$	$j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,$
11	$\sin(\omega _{0}t+\theta )\,$	$j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,$
12	${\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$	τsinc(τω/(2π))
13	τsinc(τt/(2π))	2πrect(ω/τ)
14	(1 - 2|t|/τ)rect(t/τ)	(τ/2)sinc2(τω/(4π))
15	(τ/2)sinc2(τt/(4π))	2π(1 - 2|ω|/τ)rect(ω/τ)
16	e-a|t|, Re{a} > 0	2a/(a2 + ω2)
註釋	sinc(x) = sin(πx)/(πx) rect(t/τ) 是寬度為 τ 的矩形脈衝函式。 u(t) 是赫維賽德階躍函式。 δ(t) 是狄拉克δ函式。