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푸리에 급수 테이블 , 푸리에 변환 테이블, Fourier Series Table, Fourier Transform Table
출처 : https://ena.etsmtl.ca/pluginfile.php/137982/mod_resource/content/8/Fourier-table.pdf
출처 : https://en.wikipedia.org/wiki/Fourier_transform
Functional relationships, one-dimensional
The Fourier transforms in this table may be found in [Erdélyi 1954] or [Kammler 2000, appendix].
Function | Fourier transform unitary, ordinary frequency |
Fourier transform unitary, angular frequency |
Fourier transform non-unitary, angular frequency |
Remarks | |
---|---|---|---|---|---|
\[f(x)\] | \[\begin{align} &\widehat{f}(\xi) \triangleq \widehat {f_1}(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}\] | \[\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_2}(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}\] | \[\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_3}(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}\] | Definitions | |
101 | \[a\, f(x) + b\, g(x)\] | \[a\, \widehat{f}(\xi) + b\, \widehat{g}(\xi)\] | \[a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\] | \[a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\] | Linearity |
102 | \[f(x - a)\] | \[e^{-i 2\pi \xi a} \widehat{f}(\xi)\] | \[e^{- i a \omega} \widehat{f}(\omega)\] | \[e^{- i a \omega} \widehat{f}(\omega)\] | Shift in time domain |
103 | \[f(x)e^{iax}\] | \[\widehat{f} \left(\xi - \frac{a}{2\pi}\right)\] | \[\widehat{f}(\omega - a)\] | \[\widehat{f}(\omega - a)\] | Shift in frequency domain, dual of 102 |
104 | \[f(a x)\] | \[\frac{1}{|a|} \widehat{f}\left( \frac{\xi}{a} \right)\] | \[\frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\] | \[\frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\] | Scaling in the time domain. If \[{{abs|a}}\] is large, then \[f(ax)\] is concentrated around 0 and \[\frac{1}{|a|}\hat{f} \left( \frac{\omega}{a} \right)\] spreads out and flattens. |
105 | \[\widehat {f_n}(x)\] | \[\widehat {f_1}(x) \ \stackrel{\mathcal{F}_1}{\longleftrightarrow}\ f(-\xi)\] | \[\widehat {f_2}(x) \ \stackrel{\mathcal{F}_2}{\longleftrightarrow}\ f(-\omega)\] | \[\widehat {f_3}(x) \ \stackrel{\mathcal{F}_3}{\longleftrightarrow}\ 2\pi f(-\omega)\] | The same transform is applied twice, but x replaces the frequency variable (ξ or ω) after the first transform. |
106 | \[\frac{d^n f(x)}{dx^n}\] | \[(i 2\pi \xi)^n \widehat{f}(\xi)\] | \[(i\omega)^n \widehat{f}(\omega)\] | \[(i\omega)^n \widehat{f}(\omega)\] | nth-order derivative. As \[f\] is a [[Schwartz space|Schwartz function]] |
106.5 | \[\int_{-\infty}^{x} f(\tau) d \tau\] | \[\frac{\widehat{f}(\xi)}{i 2 \pi \xi} + C \, \delta(\xi)\] | \[\frac{\widehat{f} (\omega)}{i\omega} + \sqrt{2 \pi} C \delta(\omega)\] | \[\frac{\widehat{f} (\omega)}{i\omega} + 2 \pi C \delta(\omega)\] | Integration.[1] Note: \[\delta\] is the [[Dirac delta function]] and \[C\] is the average ([[DC component|DC]]) value of \[f(x)\] such that \[\int_{-\infty}^\infty (f(x) - C) \, dx = 0\] |
107 | \[x^n f(x)\] | \[\left (\frac{i}{2\pi}\right)^n \frac{d^n \widehat{f}(\xi)}{d\xi^n}\] | \[i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}\] | \[i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}\] | This is the dual of 106 |
108 | \[(f * g)(x)\] | \[\widehat{f}(\xi) \widehat{g}(\xi)\] | \[\sqrt{2\pi}\ \widehat{f}(\omega) \widehat{g}(\omega)\] | \[\widehat{f}(\omega) \widehat{g}(\omega)\] | The notation \[f * g\] denotes the [[convolution]] of \[f\] and \[g\] — this rule is the [[convolution theorem]] |
109 | \[f(x) g(x)\] | \[\left(\widehat{f} * \widehat{g}\right)(\xi)\] | \[ \frac{1}{\sqrt{2\pi}} \left(\widehat{f} * \widehat{g}\right)(\omega) \] | \[\frac{1}{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\] | This is the dual of 108 |
110 | For \[f(x)\] purely real | \[\widehat{f}(-\xi) = \overline{\widehat{f}(\xi)}\] | \[\widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\] | \[\widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\] | Hermitian symmetry. \[\overline{z}\] indicates the [[complex conjugate]]. |
113 | For \[f(x)\] purely imaginary | \[\widehat{f}(-\xi) = -\overline{\widehat{f}(\xi)}\] | \[\widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\] | \[\widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\] | \[\overline{z}\] indicates the [[complex conjugate]]. |
114 | \[\overline{f(x)}\] | \[\overline{\widehat{f}(-\xi)}\] | \[\overline{\widehat{f}(-\omega)}\] | \[\overline{\widehat{f}(-\omega)}\] | [[Complex conjugation]], generalization of 110 and 113 |
115 | \[f(x) \cos (a x)\] | \[\frac{\widehat{f}\left(\xi - \frac{a}{2\pi}\right)+\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2}\] | \[\frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}\] | \[\frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}\] | This follows from rules 101 and 103 using [[Euler's formula]]: \[\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.\] |
116 | \[f(x)\sin( ax)\] | \[\frac{\widehat{f}\left(\xi-\frac{a}{2\pi}\right)-\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2i}\] | \[\frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}\] | \[\frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}\] | This follows from 101 and 103 using [[Euler's formula]]: \[\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.\] |
[1] The Integration Property of the Fourier Transform
Function | Fourier transform unitary, ordinary frequency |
Fourier transform unitary, angular frequency |
Fourier transform non-unitary, angular frequency |
Remarks | |
---|---|---|---|---|---|
\[ f(x) \] | \[ \begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align} \] | \[ \begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align} \] | \[ \begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align} \] | Definitions | |
301 | \[ 1 \] | \[ \delta(\xi) \] | \[ \sqrt{2\pi}\, \delta(\omega) \] | \[ 2\pi\delta(\omega) \] | The distribution \[ ''δ''(''ξ'') \] denotes the [[Dirac delta function]]. |
302 | \[ \delta(x) \] | \[ 1 \] | \[ \frac{1}{\sqrt{2\pi}} \] | \[ 1 \] | Dual of rule 301. |
303 | \[ e^{i a x} \] | \[ \delta\left(\xi - \frac{a}{2\pi}\right) \] | \[ \sqrt{2 \pi}\, \delta(\omega - a) \] | \[ 2 \pi\delta(\omega - a) \] | This follows from 103 and 301. |
304 | \[ \cos (a x) \] | \[ \frac{\delta\left(\xi - \frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2} \] | \[ \sqrt{2 \pi}\,\frac{\delta(\omega-a)+\delta(\omega+a)}{2} \] | \[ \pi\left(\delta(\omega-a)+\delta(\omega+a)\right) \] | This follows from rules 101 and 303 using [[Euler's formula]]: \[ \cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}. \] |
305 | \[ \sin( ax) \] | \[ \frac{\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)}{2i} \] | \[ \sqrt{2 \pi}\,\frac{\delta(\omega-a)-\delta(\omega+a)}{2i} \] | \[ -i\pi\bigl(\delta(\omega-a)-\delta(\omega+a)\bigr) \] | This follows from 101 and 303 using \[ \sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}. \] |
306 | \[ \cos \left( a x^2 \right) \] | \[ \sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) \] | \[ \frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) \] | \[ \sqrt{\frac{\pi}{a}} \cos \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right) \] | This follows from 101 and 207 using \[ \cos(a x^2) = \frac{e^{i a x^2} + e^{-i a x^2}}{2}. \] |
307 | \[ \sin \left( a x^2 \right) \] | \[ - \sqrt{\frac{\pi}{a}} \sin \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) \] | \[ \frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) \] | \[ -\sqrt{\frac{\pi}{a}}\sin \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right) \] | This follows from 101 and 207 using \[ \sin(a x^2) = \frac{e^{i a x^2} - e^{-i a x^2}}{2i}. \] |
308 | \[ e^{-\pi i\alpha x^2} \] | \[ \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\pi \xi^2}{\alpha}} \] | \[ \frac{1}{\sqrt{2\pi \alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}} \] | \[ \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}} \] | Here it is assumed \[ \alpha \] is real. For the case that alpha is complex see table entry 206 above. |
309 | \[ x^n \] | \[ \left(\frac{i}{2\pi}\right)^n \delta^{(n)} (\xi) \] | \[ i^n \sqrt{2\pi} \delta^{(n)} (\omega) \] | \[ 2\pi i^n\delta^{(n)} (\omega) \] | Here, \[ n \] is a [[natural number]] and \[ ''δ''{{isup|(''n'')}}\left(\xi\right) \] is the \[ n \]th distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all [[polynomial]]s. |
310 | \[ \delta^{(n)}(x) \] | \[ (i 2\pi \xi)^n \] | \[ \frac{(i\omega)^n}{\sqrt{2\pi}} \] | \[ (i\omega)^n \] | Dual of rule 309. \[ ''δ''{{isup|(''n'')}}\left(\xi\right) \] is the \[ n \]th distribution derivative of the Dirac delta function. This rule follows from 106 and 302. |
311 | \[ \frac{1}{x} \] | \[ -i\pi\sgn(\xi) \] | \[ -i\sqrt{\frac{\pi}{2}}\sgn(\omega) \] | \[ -i\pi\sgn(\omega) \] | Here \[ sgn(\xi) \] is the [[sign function]]. Note that \[ \frac{1}{x} \] is not a distribution. It is necessary to use the [[Cauchy principal value]] when testing against [[Schwartz functions]]. This rule is useful in studying the [[Hilbert transform]]. |
312 | \[ \begin{align} &\frac{1}{x^n} \\ &:= \frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log |x| \end{align} \] | \[ -i\pi \frac{(-i 2\pi \xi)^{n-1}}{(n-1)!} \sgn(\xi) \] | \[ -i\sqrt{\frac{\pi}{2}}\, \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega) \] | \[ -i\pi \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega) \] | \[ \frac{1}{x^n} \] is the [[homogeneous distribution]] defined by the distributional derivative \[ \frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log|x| \] |
313 | \[ |x|^\alpha \] | \[ -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|2\pi\xi|^{\alpha+1}} \] | \[ \frac{-2}{\sqrt{2\pi}}\, \frac{\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} \] | \[ -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} \] | This formula is valid for \[ 0 > \alpha > -1 \]. For \[ \alpha > 0 \] some singular terms arise at the origin that can be found by differentiating 320. If \[ Re \alpha > -1 \], then \[ |x|^\alpha \] is a locally integrable function, and so a tempered distribution. The function \[ \alpha \mapsto |x|^\alpha \] is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted \[ |x|^\alpha \] for \[ \alpha \neq -1, -3, ... \] (See [[homogeneous distribution]].) |
\[ \frac{1}{\sqrt{|x|}} \] | \[ \frac{1}{\sqrt{|\xi|}} \] | \[ \frac{1}{\sqrt{|\omega|}} \] | \[ \frac{\sqrt{2\pi}}{\sqrt{|\omega|}} \] | Special case of 313. | |
314 | \[ \sgn(x) \] | \[ \frac{1}{i\pi \xi} \] | \[ \sqrt{\frac{2}{\pi}} \frac{1}{i\omega} \] | \[ \frac{2}{i\omega} \] | The dual of rule 311. This time the Fourier transforms need to be considered as a [[Cauchy principal value]]. |
315 | \[ u(x) \] | \[ \frac{1}{2}\left(\frac{1}{i \pi \xi} + \delta(\xi)\right) \] | \[ \sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right) \] | \[ \pi\left( \frac{1}{i \pi \omega} + \delta(\omega)\right) \] | The function \[ u(x) \] is the Heaviside [[Heaviside step function|unit step function]]; this follows from rules 101, 301, and 314. |
316 | \[ \sum_{n=-\infty}^{\infty} \delta (x - n T) \] | \[ \frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( \xi -\frac{k }{T}\right) \] | \[ \frac{\sqrt{2\pi }}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right) \] | \[ \frac{2\pi}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right) \] | This function is known as the [[Dirac comb]] function. This result can be derived from 302 and 102, together with the fact that \[ \begin{align} & \sum_{n=-\infty}^{\infty} e^{inx} \\ = {}& 2\pi\sum_{k=-\infty}^{\infty} \delta(x+2\pi k) \end{align} \] as distributions. |
317 | \[ J_0 (x) \] | \[ \frac{2\, \operatorname{rect}(\pi\xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} \] | \[ \sqrt{\frac{2}{\pi}} \, \frac{\operatorname{rect}\left(\frac{\omega}{2}\right)}{\sqrt{1 - \omega^2}} \] | \[ \frac{2\,\operatorname{rect}\left(\frac{\omega}{2}\right)}{\sqrt{1 - \omega^2}} \] | The function \[ J_0(x) \] is the zeroth order [[Bessel function]] of first kind. |