Applied Mathematics/Parseval's Theorem

Parseval's theoremEdit

 

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.

DerivationEdit

Let   be the complex conjugation of  .

 
 
 
 

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

 
 
 
 
 

Hence