FREETIME;: Fourier

Transformada de Fourier

Informal Desarrollo de la Transformada de Fourier.

$f(x)=\sum_{n=-\infty }^{\infty }\alpha _{n}e^{in\pi x/l}$

donde :
$\alpha _{n}=\frac{1}{2l}\int_{-l}^{l}f(t)e^{in\pi x/l}dt$
si definimos f en toda la recta real
$f(x)=\lim_{l\mapsto \infty }\left [\sum_{n=-\infty }^{\infty }\left (\frac{1}{2l}\int_{-l}^{l}f(t)e^{-in\pi t/l}dt\right )e^{in\pi x/l} \right ]$

$f(x)=\lim_{l\mapsto \infty }\left [\sum_{n=-\infty }^{\infty }\left (\frac{1}{2l}\int_{-l}^{l}f(t)e^{in\pi (x-t)/l}dt\right )\right ]$

$f(x)=\lim_{l\mapsto \infty }\sum_{n=-\infty }^{\infty }\left [\left (\frac{1}{2\pi }\int_{-l}^{l}f(t)e^{\lambda _{n}i(x-t)}dt\right )\right ]\Delta \lambda$

$F_{l}(\lambda )=\frac{1}{2\pi }\int_{-l}^{l}f(t)e^{\lambda i(x-t)}dt\right )\right$

$f(x)=\lim_{l \mapsto \infty }\int_{-\infty }^{\infty }F_{l}(\lambda )d\lambda$

Cuando l tiende al infinito

$f(x)=\frac{1}{2\pi }\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f(t)e^{i\lambda (x-t)}dtd\lambda$

$f(x)=\frac{1}{\sqrt{2\pi} }\int_{-\infty }^{\infty }\left (\frac{1}{\sqrt{2\pi} }\int_{-\infty }^{\infty }f(t)e^{-i\lambda t}dt \right )e^{i\lambda x}d\lambda$

La funcion:
$\widehat{f}(\lambda )=\frac{1}{\sqrt{2\pi} }\int_{-\infty }^{\infty }f(t)e^{-i\lambda t}dt$

Es llamada transformada de Fourier de f.
$f(x)=\frac{1}{\sqrt{2\pi} }\int_{-\infty }^{\infty }\widehat{f}(\lambda )e^{i\lambda x}d\lambda$

FREETIME;

miércoles, 17 de noviembre de 2010

Fourier

Transformada de Fourier

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