Parseval's theorem

We now prove another very important theorem using the Convolution Theorem. We first give its statement:

The Parseval's theorem states that the inner product between signals is preserved in going from time to the frequency domain.

i.e. *

where X(f), Y(f), are the Fourier Transforms of x(t), y(t) respectively.

If we take x(t) = y(t), 

This is interpreted physically as “Energy calculated in the time domain is same as the energy calculated in the frequency domain”.  

  |X(.)|² is called the “Energy Spectral Density”.

Proof:

Hence Proved.