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Fourier transform
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=== Functional relationships, one-dimensional === The Fourier transforms in this table may be found in {{harvtxt|Erdélyi|1954}} or {{harvtxt|Kammler|2000|loc=appendix}}. {| class="wikitable" ! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !!Remarks |- | |<math> f(x)\,</math> |<math>\begin{align} &\widehat{f}(\xi) \triangleq \widehat {f_1}(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math> |<math>\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}</math> |<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_3}(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math> |Definitions |- | 101 |<math> a\, f(x) + b\, g(x)\,</math> |<math> a\, \widehat{f}(\xi) + b\, \widehat{g}(\xi)\,</math> |<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math> |<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math> |Linearity |- | 102 |<math> f(x - a)\,</math> |<math> e^{-i 2\pi \xi a} \widehat{f}(\xi)\,</math> |<math> e^{- i a \omega} \widehat{f}(\omega)\,</math> |<math> e^{- i a \omega} \widehat{f}(\omega)\,</math> |Shift in time domain |- | 103 |<math> f(x)e^{iax}\,</math> |<math> \widehat{f} \left(\xi - \frac{a}{2\pi}\right)\,</math> |<math> \widehat{f}(\omega - a)\,</math> |<math> \widehat{f}(\omega - a)\,</math> |Shift in frequency domain, dual of 102 |- | 104 |<math> f(a x)\,</math> |<math> \frac{1}{|a|} \widehat{f}\left( \frac{\xi}{a} \right)\,</math> |<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math> |<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math> |Scaling in the time domain. If {{math|{{abs|''a''}}}} is large, then {{math|''f''(''ax'')}} is concentrated around 0 and{{br}}<math> \frac{1}{|a|}\hat{f} \left( \frac{\omega}{a} \right)\,</math>{{br}}spreads out and flattens. |- | 105 |<math> \widehat {f_n}(x)\,</math> |<math> \widehat {f_1}(x) \ \stackrel{\mathcal{F}_1}{\longleftrightarrow}\ f(-\xi)\,</math> |<math> \widehat {f_2}(x) \ \stackrel{\mathcal{F}_2}{\longleftrightarrow}\ f(-\omega)\,</math> |<math> \widehat {f_3}(x) \ \stackrel{\mathcal{F}_3}{\longleftrightarrow}\ 2\pi f(-\omega)\,</math> |The same transform is applied twice, but ''x'' replaces the frequency variable (''ξ'' or ''ω'') after the first transform. |- | 106 |<math> \frac{d^n f(x)}{dx^n}\,</math> |<math> (i 2\pi \xi)^n \widehat{f}(\xi)\,</math> |<math> (i\omega)^n \widehat{f}(\omega)\,</math> |<math> (i\omega)^n \widehat{f}(\omega)\,</math> |n{{superscript|th}}-order derivative. As {{math|''f''}} is a [[Schwartz space|Schwartz function]] |- |106.5 |<math>\int_{-\infty}^{x} f(\tau) d \tau</math> |<math>\frac{\widehat{f}(\xi)}{i 2 \pi \xi} + C \, \delta(\xi)</math> |<math>\frac{\widehat{f} (\omega)}{i\omega} + \sqrt{2 \pi} C \delta(\omega)</math> |<math>\frac{\widehat{f} (\omega)}{i\omega} + 2 \pi C \delta(\omega)</math> |Integration.<ref>{{Cite web |date=2015 |orig-date=2010 |title=The Integration Property of the Fourier Transform |url=https://www.thefouriertransform.com/transform/integration.php |url-status=live |archive-url=https://web.archive.org/web/20220126171340/https://www.thefouriertransform.com/transform/integration.php |archive-date=2022-01-26 |access-date=2023-08-20 |website=The Fourier Transform .com}}</ref> Note: <math>\delta</math> is the [[Dirac delta function]] and <math>C</math> is the average ([[DC component|DC]]) value of <math>f(x)</math> such that <math>\int_{-\infty}^\infty (f(x) - C) \, dx = 0</math> |- | 107 |<math> x^n f(x)\,</math> |<math> \left (\frac{i}{2\pi}\right)^n \frac{d^n \widehat{f}(\xi)}{d\xi^n}\,</math> |<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math> |<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math> |This is the dual of 106 |- | 108 |<math> (f * g)(x)\,</math> |<math> \widehat{f}(\xi) \widehat{g}(\xi)\,</math> |<math> \sqrt{2\pi}\ \widehat{f}(\omega) \widehat{g}(\omega)\,</math> |<math> \widehat{f}(\omega) \widehat{g}(\omega)\,</math> |The notation {{math|''f'' ∗ ''g''}} denotes the [[convolution]] of {{mvar|f}} and {{mvar|g}} — this rule is the [[convolution theorem]] |- | 109 |<math> f(x) g(x)\,</math> |<math> \left(\widehat{f} * \widehat{g}\right)(\xi)\,</math> |<math> \frac{1}\sqrt{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math> |<math> \frac{1}{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math> |This is the dual of 108 |- | 110 |For {{math|''f''(''x'')}} purely real |<math> \widehat{f}(-\xi) = \overline{\widehat{f}(\xi)}\,</math> |<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math> |<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math> |Hermitian symmetry. {{math|{{overline|''z''}}}} indicates the [[complex conjugate]]. |- <!-- A Symmetry section has been added instead of this. | 111 |For {{math|''f''(''x'')}} purely real and [[even function|even]] | colspan=3 align=center |<math>\widehat f </math> is a purely real and [[even function]]. | |- | 112 |For {{math|''f''(''x'')}} purely real and [[odd function|odd]] | colspan=3 align=center |<math>\widehat f </math> is a purely [[imaginary number|imaginary]] and [[odd function]]. | |--> | 113 |For {{math|''f''(''x'')}} purely imaginary |<math> \widehat{f}(-\xi) = -\overline{\widehat{f}(\xi)}\,</math> |<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math> |<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math> |{{math|{{overline|''z''}}}} indicates the [[complex conjugate]]. |- | 114 | <math> \overline{f(x)}</math>|| <math> \overline{\widehat{f}(-\xi)}</math> || <math> \overline{\widehat{f}(-\omega)}</math> || <math> \overline{\widehat{f}(-\omega)}</math> || [[Complex conjugate|Complex conjugation]], generalization of 110 and 113 |- |115 |<math> f(x) \cos (a x)</math> |<math> \frac{ \widehat{f}\left(\xi - \frac{a}{2\pi}\right)+\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2}</math> |<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}\,</math> |<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}</math> |This follows from rules 101 and 103 using [[Euler's formula]]:{{br}}<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math> |- |116 |<math> f(x)\sin( ax)</math> |<math> \frac{\widehat{f}\left(\xi-\frac{a}{2\pi}\right)-\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2i}</math> |<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math> |<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math> |This follows from 101 and 103 using [[Euler's formula]]:{{br}}<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math> |}
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