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| In [[mathematics]], '''Parseval's theorem''' <ref>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).</ref> usually refers to the result that the [[Fourier transform]] is [[Unitary operator|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 (mathematics)|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.<ref>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 [http://books.google.com/books?id=izM9AAAAIAAJ&pg=PA268&lpg=PA268&source=bl&ots=5stf6mGwJG&sig=UeoeV2c4dEp9JmWUIanqMEhDMmU&hl=en&ei=QTv9SZKTJIvOMrrxjL0E&sa=X&oi=book_result&ct=result&resnum=3 here].</ref>
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| 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]].<ref>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.</ref>
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| == Statement of Parseval's theorem ==
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| 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]]
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| :<math>A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}</math>
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| and<br />
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| :<math>B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}</math>
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| respectively. Then
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| :<math>\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, dx,</math>
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| where ''i'' is the [[imaginary unit]] and horizontal bars indicate [[complex conjugation]].
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| 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''<sup>2</sup>(''G'') and ''L''<sup>2</sup>(''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'''<sub>n</sub>, again it is self-dual and the Pontryagin–Fourier transform is what is called [[discrete-time Fourier transform]] in applied contexts.
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| == Notation used in engineering and physics ==
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| In [[physics]] and [[engineering]], Parseval's theorem is often written as:
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| :<math>\int_{-\infty}^\infty | x(t) |^2 \, dt = \int_{-\infty}^\infty | X(f) |^2 \, df </math>
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| where <math>X(f) = \mathcal{F} \{ x(t) \}</math> represents the [[continuous Fourier transform]] (in normalized, unitary form) of ''x''(''t'') and ''f'' represents the frequency component (not [[angular frequency]]) of ''x''.
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| The interpretation of this form of the theorem is that the total [[Energy (signal processing)|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''.
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| For [[discrete time]] [[signal (information theory)|signals]], the theorem becomes:
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| :<math> \sum_{n=-\infty}^\infty | x[n] |^2 = \frac{1}{2\pi} \int_{-\pi}^\pi | X(e^{i\phi}) |^2 d\phi </math> | |
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| where ''X'' is the [[discrete-time Fourier transform]] (DTFT) of ''x'' and Φ represents the [[angular frequency]] (in [[radian]]s per sample) of ''x''.
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| Alternatively, for the [[discrete Fourier transform]] (DFT), the relation becomes:
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| :<math> \sum_{n=0}^{N-1} | x[n] |^2 = \frac{1}{N} \sum_{k=0}^{N-1} | X[k] |^2 </math>
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| where ''X''[''k''] is the DFT of ''x''[''n''], both of length ''N''.
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| == See also ==
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| Parseval's theorem is closely related to other mathematical results involving unitarity transformations:
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| *[[Parseval's identity]]
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| *[[Plancherel's theorem]]
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| *[[Wiener–Khinchin theorem]]
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| *[[Bessel's inequality]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Parseval.html Parseval], ''MacTutor History of Mathematics archive''.
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| * George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'' (Harcourt: San Diego, 2001).
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| * Hubert Kennedy, ''[http://hubertkennedy.angelfire.com/Eight_Mathematical.pdf Eight Mathematical Biographies]'' (Peremptory Publications: San Francisco, 2002).
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| * Alan V. Oppenheim and Ronald W. Schafer, ''Discrete-Time Signal Processing'' 2nd Edition (Prentice Hall: Upper Saddle River, NJ, 1999) p 60.
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| * William McC. Siebert, ''Circuits, Signals, and Systems'' (MIT Press: Cambridge, MA, 1986), pp. 410–411.
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| * David W. Kammler, ''A First Course in Fourier Analysis'' (Prentice–Hall, Inc., Upper Saddle River, NJ, 2000) p. 74.
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| ==External links== | |
| * [http://mathworld.wolfram.com/ParsevalsTheorem.html Parseval's Theorem] on Mathworld
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| [[Category:Theorems in Fourier analysis]]
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| [[Category:Theorems in harmonic analysis]]
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