Applied Mathematics/Parseval's Theorem

Parseval's theorem

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where   represents the continuous Fourier transform of x(t) and f represents the frequency component of x. The function above is called Parseval's theorem.

Derivation

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Let   be the complex conjugation of  .

 
 
 
 

Here, we know that   is equal to the expansion coefficient of   in fourier transforming of  .
Hence, the integral of   is

 
 
 
 
 

Hence