The Periodogram

 

The discrete version of Parseval’s Theorem says that the total power of a time domain series is closely related to the total power of its frequency domain representation:

 

,

 

where represent the original series in the time domain and the are the Fourier coefficients of the series in the frequency domain.

The power of each of the frequencies of the Fourier spectrum is defined as

 

 

which is called the Periodogram and the sum of these powers is equal to the total power in the original series.