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Fourierren transformaketa - Wikipedia

Fourierren transformaketa

Wikipedia(e)tik

f(x) "denboraren eremuko" funtzioa izanik, fren Fourier transformatua deritzo (Jean Baptiste Joseph Fourierren omenez) \hat f funtzioari,

\hat f(\xi)=\int_{-\infty}^\infty f(x)e^{-2\pi ix\xi} dx,

bezala definitzen dena. Berau f funtzio integragarriarentzat definitua dagoelarik, non

\int_{-\infty}^\infty |f(x)|dx < \infty.

Transformatu honen bidez funtzioa "maiztasun eremura" aldatzen da denboraren eremuan argi azaltzen ez den informazioa lortzeko.

\hat f transformatua funtzio jarrai eta bornatu bat da. \hat f-k \int_{-\infty}^\infty |\hat f(\xi)|d\xi < \infty, betezten badu, bere alderantzizko transformatua:

f(x)=  \int_{-\infty}^\infty \hat f(\xi)e^{2\pi i\xi x} d\xi izango da.

Bere propietateak direla eta:

\widehat{\frac{df}{dx}}(\xi) = 2\pi i\xi \hat f(\xi) \quad \mbox{ y }\quad \widehat{xf}(\xi) = -\frac{1}{2\pi i} \frac{d}{d\xi}\hat f(\xi),

Fourier transformatua oso garrantzitsua da ekuazio diferentzialen soluzioak lortzeko.

[aldatu] Ikus, gainera

  • Fourier transformatu diskretua
  • FFT
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