Parseval's theorem

Creator

Creator

Created

Created

2023 Sep 13 3:28

Editor

Editor

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

total energy is

\sum|x[n]|^2 = E_\infty

Parseval’s Relation

\frac{1}{T}\int_T |x(t)|^2 = \sum_ {k = -\infty} ^\infty |a_k|^2

\frac{1}{N}\sum|x[n]|^2 = \sum|a_k|^2

For each coefficient

notion image

the average power of the

k

th harmonic component case

\frac{1}{T}\int_T|a_ke^{jk\omega_0t}|^2 = |a_k|^2

\frac{1}{N}\sum_{k=<N>}|a_ke^{j\omega_0kn}|^2 = \frac{1}{N}\sum_{k=<N>}|a_k|^2 = |a_k|^2

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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