In mathematics, Parseval's theorem 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.
[1] This theorem was first developed at the University of Alberta.
It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series.
[2] Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.
that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series and respectively.
is the imaginary unit and horizontal bars indicate complex conjugation.
that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series and respectively.
Then Even more generally, given an abelian locally compact group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.)
When G is the unit circle T, G^ is the integers and this is the case discussed above.
When G is the cyclic group Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts.
Parseval's theorem can also be expressed as follows: Suppose
are integrable on that interval), with the Fourier series Then[4][5][6] In electrical engineering, Parseval's theorem is often written as: where
represents the continuous Fourier transform (in non-unitary form) of
The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For discrete time signals, the theorem becomes: where
is the discrete-time Fourier transform (DTFT) of
represents the angular frequency (in radians per sample) of
Alternatively, for the discrete Fourier transform (DFT), the relation becomes: where
We show the DFT case below.
For the other cases, the proof is similar.
represents complex conjugate.
Parseval's theorem is closely related to other mathematical results involving unitary transformations: