Furijeova transformacija

Furijeova transformacija razlaže funkciju vremena (signal) u frekvencije koje ga čine, na sličan način kao što muzički akordi mogu biti izraženi kao frekvencije njegovih sastavnih nota.

Istorija

Žozef Furije je 1822. godine pokazao da neke funkcije mogu biti zapisane kao beskonačna suma harmonika.

Definicija

Furijeova transformacija signala $f(t)$računa se na sledeći način:

$F(j\omega )=\int \limits _{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

$F(j\omega )$ je kompleksna veličina. Njen moduo naziva se spektralna gustina amplituda, a argument spektralna gustina faza.

Inverzija

Inverzna Furijeova transformacija je:

$f(t)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega$

Osobine Furijeove transformacije

Linearnost

Za bilo koje kompleksne brojeve $a$ i $b$, ako je $h(x)=af(x)+bg(x)$, važi da je $H(j\omega )=aF(j\omega )+bG(j\omega )$.

Translacija

Za bilo koji realan broj $x_{0}$, ako je $h(x)=f(x-x_{0})$, važi da je $H(j\omega )=e^{j\omega tx_{0}}F(j\omega )$.

Vidi još

• Diskretna Furijeova transformacija
• Brza Furijeova transformacija
• Furijeov red

Literatura

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Spoljašnje veze

• Encyclopedia of Mathematics
• Weisstein, Eric W. „Fourier Transform”. MathWorld.