푸리에 변환 표 (Fourier transform Table)

Fourier Transform Table

Function	Fourier Transform	Remarks
$$f(x)$$	$$\widehat{f}(\omega) = \int_{-\infty}^\infty f(x)e^{-i\omega x}dx$$	Definition
$$af(x) + bg(x)$$	$$a\widehat{f}(\omega) + b\widehat{g}(\omega)$$	Linearity
$$f(x-a)$$	$$e^{-ia\omega}\widehat{f}(\omega)$$	Shift in Time Domain
$$f(x)e^{iax}$$	$$\widehat{f}(x-a)$$	Shift in Frequency Domain
$$f(ax)$$	$$\frac{1}{|a|}\widehat{f}\left(\frac{\omega}{a}\right)$$
$$\widehat{f}(x)$$	$$\frac{1}{2\pi}f(-\omega)$$	Duality
$$f'(x)$$	$$i\omega\widehat{f}(\omega)$$
$$f^{(n)}(x)$$	$$(i\omega)^n\widehat{f}(\omega)$$
$$xf(x)$$	$$i\frac{d\widehat{f}}{d\omega}(\omega)$$
$$x^nf(x)$$	$$i^n\frac{d^n\widehat{f}}{d\omega^n}(\omega)$$
$$(f*g)(x)$$	$$\widehat{f}(\omega)\widehat{g}(\omega)$$	Convolution
$$f(x)g(x)$$	$$\left(\widehat{f}*\widehat{g}\right)(\omega)$$
$$\overline{f(x)}$$	$$\overline{\widehat{f}(-\omega)}$$
$$f(x)\cos(ax)$$	$$\frac{\widehat{f}(\omega-a) + \widehat{f}(\omega+a)}{2}$$
$$f(x)\sin(ax)$$	$$\frac{\widehat{f}(\omega-a) - \widehat{f}(\omega+a)}{2i}$$
$$\int_{-\infty}^xf(t)dt, \ \textrm{where} \ \int_{-\infty}^\infty f(x)dx = 0$$	$$\frac{\widehat{f}(\omega)}{i\omega}$$
$$\int_{-\infty}^xf(t)dt$$	$$\frac{\widehat{f}(\omega)}{i\omega} + \pi\widehat{f}(0)\delta(\omega)$$

※ 단, $\delta$는 Dirac delta function.

Function	Fourier Transform	Remarks
$$f(x)$$	$$\widehat{f}(\omega) = \int_{-\infty}^\infty f(x)e^{-i\omega x}dx$$	Definition
$$\mathrm{rect}(x)$$	$$\frac{\sin(\omega/2)}{\omega/2}$$	$$\mathrm{rect}(x) = \left\{\begin{array}{ll} 1 & \textrm{if $|x|\leq 1/2$} \\ 0 & \textrm{otherwise} \end{array}\right.$$
$e^{-ax}u(x)$, $a>0$	$$\frac{1}{a+i\omega}$$	$$u(x) = \left\{\begin{array}{ll} 1 & \textrm{if $x>0$} \\ 0 & \textrm{otherwise} \end{array}\right.$$
$$e^{-x^2}$$	$$\sqrt{\pi}e^{-\omega^2/4}$$
$$e^{-|x|}$$	$$\frac{2}{1+\omega^2}$$
$$1$$	$$2\pi\delta(\omega)$$
$$\delta(x)$$	$$1$$
$$\cos(x)$$	$$\pi\left[\delta(\omega-1) + \delta(\omega + 1)\right]$$
$$\sin(x)$$	$$-i\pi\left[\delta(\omega-1) - \delta(\omega+1)\right]$$
$$u(x)$$	$$\frac{1}{i\omega} + \pi\delta(\omega)$$	$$u(x) = \left\{\begin{array}{ll} 1 & \textrm{if $x>0$} \\ 0 & \textrm{otherwise} \end{array}\right.$$
$$\mathrm{sgn}(x)$$	$$\frac{2}{i\omega}$$	$$\mathrm{sgn}(x) = \left\{\begin{array}{ll} 1 & \textrm{if $x>0$} \\ -1 & \textrm{otherwise} \end{array}\right.$$

※ 단, $\delta$는 Dirac delta function.

Wikipedia의 Fourier Transform 항목을 참고하였습니다.

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