Parseval's theorem


In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum 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.
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.

Statement of Parseval's theorem

Suppose that and are two complex-valued functions on of period that are square integrable over intervals of period length, with Fourier series
and

respectively. Then
where is the imaginary unit and horizontal bars indicate complex conjugation.
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 and L2 When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line, G^ is also and the unitary transform is the Fourier transform on the real line. 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 is a square-integrable function over , with the Fourier series
Then

Notation used in physics

In physics and engineering, Parseval's theorem is often written as:
where represents the continuous Fourier transform of, and is frequency in radians per second.
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 of and represents the angular frequency of.
Alternatively, for the discrete Fourier transform, the relation becomes:
where is the DFT of, both of length.