Editing Parseval's theorem

{{Short description|Theorem in mathematics}}
In mathematics, '''Parseval's theorem''' 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.<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 savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants étrangers.)'', vol. 1, pages 638–648 (1806).</ref>  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 [https://books.google.com/books?id=izM9AAAAIAAJ&pg=PA268 here].</ref> 

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]].<ref>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.</ref>

== Statement of Parseval's theorem ==

Suppose that <math>A(x)</math> and <math>B(x)</math> are two complex-valued functions on <math>\mathbb{R}</math> of [[periodic function|period]] <math>2 \pi</math> that are [[square integrable]] (with respect to the [[Lebesgue measure]]) over intervals of period length, with [[Fourier series]]

:<math>A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}</math>
and<br />
:<math>B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}</math>

respectively. Then

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk|:|<math>\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, \mathrm{d}x,</math>|{{EquationRef|Eq.1}}}}
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|border colour = #0073CF
|background colour=#F5FFFA}}
where <math>i</math> is the [[imaginary unit]] and horizontal bars indicate [[complex conjugation]]. Substituting <math>A(x)</math> and <math>\overline{B(x)}</math>:

:<math>
\begin{align}
\sum_{n=-\infty}^\infty a_n\overline{b_n}
&= \frac{1}{2\pi} \int_{-\pi}^\pi \biggl( \sum_{n=-\infty}^\infty a_ne^{inx} \biggr) \biggl( \sum_{n=-\infty}^\infty \overline{b_n}e^{-inx} \biggr) \, \mathrm{d}x \\[6pt]
&= \frac{1}{2\pi} \int_{-\pi}^\pi \Bigl(a_1e^{i1x} + a_2e^{i2x} + \cdots\Bigr) \Bigl(\overline{b_1}e^{-i1x} + \overline{b_2}e^{-i2x} + \cdots\Bigr) \, \mathrm{d}x \\[6pt]
&= \frac{1}{2\pi} \int_{-\pi}^\pi \left(a_1e^{i1x} \overline{b_1}e^{-i1x} + a_1e^{i1x} \overline{b_2}e^{-i2x} + a_2e^{i2x} \overline{b_1}e^{-i1x} + a_2e^{i2x} \overline{b_2}e^{-i2x} + \cdots \right) \mathrm{d}x \\[6pt]
&= \frac{1}{2\pi} \int_{-\pi}^\pi \left(a_1 \overline{b_1} + a_1 \overline{b_2}e^{-ix} + a_2 \overline{b_1}e^{ix} + a_2 \overline{b_2} + \cdots\right) \mathrm{d}x
\end{align}
</math>
As is the case with the middle terms in this example, many terms will integrate to <math>0</math> over a full [[Periodic_function|period]] of length <math>2\pi</math> (see [[harmonic|harmonics]]):
:<math>
\begin{align}
\sum_{n=-\infty}^\infty a_n\overline{b_n} &= \frac{1}{2\pi} \left[a_1 \overline{b_1} x + i a_1 \overline{b_2}e^{-ix} - i a_2 \overline{b_1}e^{ix} + a_2 \overline{b_2} x + \cdots\right] _{-\pi} ^{+\pi} \\[6pt]
&= \frac{1}{2\pi} \left(2\pi a_1 \overline{b_1} + 0 + 0 + 2\pi a_2 \overline{b_2} + \cdots\right) \\[6pt]
&= a_1 \overline{b_1} + a_2 \overline{b_2} + \cdots \\[6pt]
\end{align}</math>

More generally, if <math>A(x)</math> and <math>B(x)</math> are instead two complex-valued functions on <math>\mathbb{R}</math> of period <math>P</math> that are [[square integrable]] (with respect to the [[Lebesgue measure]]) over intervals of period length, with [[Fourier series]]

:<math>A(x)=\sum_{n=-\infty}^\infty a_ne^{2\pi ni\left(\frac{x}{P}\right)}</math>
and<br />
:<math>B(x)=\sum_{n=-\infty}^\infty b_ne^{2\pi ni\left(\frac{x}{P}\right)}</math>

respectively. Then

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk|:|<math>\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{P} \int_{-P/2}^{P/2} A(x)\overline{B(x)} \, \mathrm{d}x,</math>|{{EquationRef|Eq.2}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

Even more generally, given an [[Locally compact abelian group|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''<sup>2</sup>(''G'') and ''L''<sup>2</sup>(''G^'') (with integration being against the appropriately scaled [[Haar measure|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 <math>\mathbb{R}</math>, ''G^'' is also <math>\mathbb{R}</math> 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 Fourier transform]] in applied contexts.

Parseval's theorem can also be expressed as follows:

Suppose <math>f(x)</math> is a square-integrable function over <math>[-\pi, \pi]</math> (i.e., <math>f(x)</math> and <math>f^2(x)</math> are integrable on that interval), with the Fourier series
:<math>f(x) \simeq \tfrac12 a_0 + \sum_{n=1}^{\infty} \bigl(a_n \cos(nx) + b_n \sin(nx)\bigr).</math>
Then<ref>{{cite book | page=439 | author=Arthur E. Danese | title=Advanced Calculus | volume=1 | publisher=Allyn and Bacon, Inc. | location=Boston, MA | year=1965 }}</ref><ref>{{cite book |page=[https://archive.org/details/advancedcalculus00kapl_841/page/n533 519] | author=Wilfred Kaplan | title=Advanced Calculus |url=https://archive.org/details/advancedcalculus00kapl_841 |url-access=limited | publisher=Addison Wesley | location=Reading, MA | year=1991 | edition=4th | isbn=0-201-57888-3 |author-link=Wilfred Kaplan }}</ref><ref>{{cite book | page=[https://archive.org/details/fourierseries00tols/page/119 119] | author=Georgi P. Tolstov | title=Fourier Series | url=https://archive.org/details/fourierseries00tols | url-access=registration | translator-last=Silverman | translator-first=Richard | publisher=Prentice-Hall, Inc. | location=Englewood Cliffs, NJ | year=1962 }}</ref>

:<math>\frac{1}{\pi} \int_{-\pi}^{\pi} f^2(x) \,\mathrm{d}x = \tfrac12 a_0^2 + \sum_{n=1}^{\infty} \left(a_n^2 + b_n^2 \right).</math>

== Notation used in engineering ==

In [[electrical engineering]], Parseval's theorem is often written as:

:<math>\int_{-\infty}^\infty \bigl| x(t) \bigr|^2 \, \mathrm{d}t  =  \frac{1}{2\pi} \int_{-\infty}^\infty \bigl| X(\omega) \bigr|^2 \, \mathrm{d}\omega  =  \int_{-\infty}^\infty \bigl| X(2\pi f) \bigr|^2 \, \mathrm{d}f</math>

where <math>X(\omega) = \mathcal{F}_\omega\{ x(t) \}</math> represents the [[continuous Fourier transform]] (in non-unitary form) of <math>x(t)</math>, and <math>\omega = 2\pi f</math> is frequency in radians per second.

The interpretation of this form of the theorem is that the total [[Energy (signal processing)|energy]] of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.  

For [[discrete time]] [[signal (information theory)|signals]], the theorem becomes:

:<math>\sum_{n=-\infty}^\infty \bigl| x[n] \bigr|^2 = \frac{1}{2\pi} \int_{-\pi}^\pi \bigl| X_{2\pi}({\phi}) \bigr|^2 \mathrm{d}\phi</math>

where <math>X_{2\pi}</math> is the [[discrete-time Fourier transform]] (DTFT) of <math>x</math> and <math>\phi</math> represents the [[angular frequency]] (in [[radian]]s per sample) of <math>x</math>.

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

:<math> \sum_{n=0}^{N-1} \bigl| x[n] \bigr|^2  = \frac{1}{N} \sum_{k=0}^{N-1} \bigl| X[k] \bigr|^2</math>

where <math>X[k]</math> is the DFT of <math>x[n]</math>, both of length <math>N</math>.

We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of <math>X[k]</math>, we can derive

:<math>\begin{align}
\frac{1}{N} \sum_{k=0}^{N-1} \bigl| X[k] \bigr|^2
&= \frac{1}{N} \sum_{k=0}^{N-1} X[k]\cdot X^*[k]
 = \frac{1}{N} \sum_{k=0}^{N-1} \Biggl(\sum_{n=0}^{N-1} x[n]\,\exp\left(-j\frac{2\pi}{N}k\,n\right)\Biggr) \, X^*[k]
\\[5mu]
&= \frac{1}{N} \sum_{n=0}^{N-1} x[n] \Biggl(\sum_{k=0}^{N-1} X^*[k]\,\exp\left(-j\frac{2\pi}{N}k\,n\right)\Biggr) 
 = \frac{1}{N} \sum_{n=0}^{N-1} x[n] \bigl(N \cdot x^*[n]\bigr)
\\[5mu]
&= \sum_{n=0}^{N-1} \bigl| x[n] \bigr|^2,
\end{align}</math>

where <math>*</math> represents complex conjugate.

== See also ==
Parseval's theorem is closely related to other mathematical results involving unitary transformations:
*[[Parseval's identity]]
*[[Plancherel's theorem]]
*[[Wiener–Khinchin theorem]]
*[[Bessel's inequality]]

==Notes==
{{reflist}}

==External links==
* [http://mathworld.wolfram.com/ParsevalsTheorem.html Parseval's Theorem] on Mathworld

[[Category:Theorems in Fourier analysis]]