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Parseval's theorem
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Parseval's theorem

Creator
Creator
Seonglae ChoSeonglae Cho
Created
Created
2023 Sep 13 3:28
Editor
Editor
Seonglae ChoSeonglae Cho
Edited
Edited
2023 Oct 13 2:45
Refs
Refs
Conservation of Energy

The average power in the time domain is the sum of average powers in all harmonics

So
Trivial
total energy is
 

Parseval’s Relation

For each coefficient
notion image
the average power of the th harmonic component case
proven because of
DTFS Time shift
,
CTFS Time operation
 
 
 
 
 
Parseval's theorem
In mathematics, Parseval's theorem[1] usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.[2]
Parseval's theorem
https://en.wikipedia.org/wiki/Parseval's_theorem
 
 

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Parseval's theorem
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