# 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]}

Although the term "Parseval's theorem" is often used to describe the unitarity of *any* Fourier transform, especially in physics and engineering, the most general form of this property is more properly called the Plancherel theorem.^{[3]}

## Contents

## Statement of Parseval's theorem

Suppose that *A*(*x*) and *B*(*x*) are two square integrable (with respect to the Lebesgue measure), complex-valued functions on **R** of period 2π with Fourier series

and

respectively. Then

where *i* is the imaginary unit and horizontal bars indicate complex conjugation.

More generally, given an abelian topological group *G* with Pontryagin dual *G^*, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces *L*^{2}(*G*) and *L*^{2}(*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 real line **R**, **G^** is also **R** and the unitary transform is the Fourier transform on the real line. When *G* is the cyclic group **Z**_{n}, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts.

## Notation used in engineering and physics

In physics and engineering, Parseval's theorem is often written as:

where represents the continuous Fourier transform (in normalized, unitary form) of *x*(*t*) and *f* represents the frequency component (not angular frequency) of *x*.

The interpretation of this form of the theorem is that the total energy contained in a waveform *x*(*t*) summed across all of time *t* is equal to the total energy of the waveform's Fourier Transform *X*(*f*) summed across all of its frequency components *f*.

For discrete time signals, the theorem becomes:

where *X* is the discrete-time Fourier transform (DTFT) of *x* and Φ represents the angular frequency (in radians per sample) of *x*.

Alternatively, for the discrete Fourier transform (DFT), the relation becomes:

where *X*[*k*] is the DFT of *x*[*n*], both of length *N*.

## See also

Parseval's theorem is closely related to other mathematical results involving unitarity transformations:

## Notes

- ↑ Parseval des Chênes, Marc-Antoine "Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in
*Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savans étrangers.)*, vol. 1, pages 638–648 (1806). - ↑ Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature,"
*Philosophical Magazine*, vol. 27, pages 460–469. Available on-line here. - ↑ Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies,"
*Rendiconti del Circolo Matematico di Palermo*, vol. 30, pages 298–335.

## References

- Parseval,
*MacTutor History of Mathematics archive*. - George B. Arfken and Hans J. Weber,
*Mathematical Methods for Physicists*(Harcourt: San Diego, 2001). - Hubert Kennedy,
*Eight Mathematical Biographies*(Peremptory Publications: San Francisco, 2002). - Alan V. Oppenheim and Ronald W. Schafer,
*Discrete-Time Signal Processing*2nd Edition (Prentice Hall: Upper Saddle River, NJ, 1999) p 60. - William McC. Siebert,
*Circuits, Signals, and Systems*(MIT Press: Cambridge, MA, 1986), pp. 410–411. - David W. Kammler,
*A First Course in Fourier Analysis*(Prentice–Hall, Inc., Upper Saddle River, NJ, 2000) p. 74.

## External links

- Parseval's Theorem on Mathworld