Parseval's Theorem

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## Statement of Parseval's theorem

## Notation used in physics

## See also

## Notes

## References

## External links

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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, the most general form of this property is more properly called the Plancherel theorem.^{[3]}

Suppose that and are two square integrable (with respect to the Lebesgue measure), complex-valued functions on of period 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 *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 , *G^* is also 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.

Parseval's theorem can also be expressed as follows: Suppose is a square-integrable function over (i.e., and are integrable on that interval), with the Fourier series

Then^{[4]}^{[5]}^{[6]}

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

where represents the continuous Fourier transform (in normalized, unitary form) 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 (DTFT) of and represents the angular frequency (in radians per sample) of .

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

where is the DFT of , both of length .

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

**^**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 savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants 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 à l'etude de la representation d'une fonction arbitraire par les integrales définies,"*Rendiconti del Circolo Matematico di Palermo*, vol. 30, pages 298-335.**^**Arthur E. Danese (1965).*Advanced Calculus*.**1**. Boston, MA: Allyn and Bacon, Inc. p. 439.**^**Wilfred Kaplan (1991).*Advanced Calculus*(4th ed.). Reading, MA: Addison Wesley. p. 519. ISBN 0-201-57888-3.**^**Georgi P. Tolstov (1962).*Fourier Series*. Translated by Silverman, Richard. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 119.

- 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.

- Parseval's Theorem on Mathworld

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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