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 |
$$\mathrm{rect}(x)$$ |
$$\frac{\sin(\omega/2)}{\omega/2}$$ |
$$\mathrm{rect}(x) = \left\{\begin{array}{ll} |
$e^{-ax}u(x)$, $a>0$ |
$$\frac{1}{a+i\omega}$$ |
$$u(x) = \left\{\begin{array}{ll} |
$$e^{-x^2}$$ |
$$\sqrt{\pi}e^{-\omega^2/4}$$ |
|
$$e^{-|x|}$$ |
$$\frac{2}{1+\omega^2}$$ |
|
$$1$$ |
$$2\pi\delta(\omega)$$ |
|
$$\delta(x)$$ |
$$1$$ |
|
$$\cos(x)$$ |
$$\pi\left[\delta(\omega-1) + \delta(\omega + 1)\right]$$ |
|
$$\sin(x)$$ |
$$-i\pi\left[\delta(\omega-1) - \delta(\omega+1)\right]$$ |
|
$$u(x)$$ | $$\frac{1}{i\omega} + \pi\delta(\omega)$$ | $$u(x) = \left\{\begin{array}{ll} |
$$\mathrm{sgn}(x)$$ | $$\frac{2}{i\omega}$$ | $$\mathrm{sgn}(x) = \left\{\begin{array}{ll} |
※ 단, $\delta$는 Dirac delta function.
Wikipedia의 Fourier Transform 항목을 참고하였습니다.
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