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{{merge|Parseval’s identity|date=December 2023}} {{Short description|Theorem in mathematics}}In mathematics, Parseval’s theoremParseval 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). 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.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. 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.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.

Statement of Parseval’s theorem

Suppose that A(x) and B(x) are two complex-valued functions on mathbb{R} of period 2 pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series and respectively. Then{{Equation box 1|indent =|title=

:

{{EquationRef|Eq.1}}}}|cellpadding= 6|border|border colour = #0073CF|background colour=#F5FFFA}}where i is the imaginary unit and horizontal bars indicate complex conjugation. Substituting A(x) and overline{B(x)}: begin{align}sum_{n=-infty}^infty a_noverline{b_n}&= frac{1}{2pi} int_{-pi}^pi left( sum_{n=-infty}^infty a_ne^{inx} right) left( sum_{n=-infty}^infty overline{b_n}e^{-inx} right) , mathrm{d}x [6pt]&= frac{1}{2pi} int_{-pi}^pi left(a_1e^{i1x} + a_2e^{i2x} + cdotsright) left(overline{b_1}e^{-i1x} + overline{b_2}e^{-i2x} + cdotsright) mathrm{d}x [6pt]&= frac{1}{2pi} 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}{2pi} 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} + cdotsright) mathrm{d}xend{align}As is the case with the middle terms in this example, many terms will integrate to 0 over a full period of length 2pi (see harmonics): begin{align}sum_{n=-infty}^infty a_noverline{b_n} &= frac{1}{2pi} 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 + cdotsright] _{-pi} ^{+pi} [6pt]&= frac{1}{2pi} left(2pi a_1 overline{b_1} + 0 + 0 + 2pi a_2 overline{b_2} + cdotsright) [6pt]&= a_1 overline{b_1} + a_2 overline{b_2} + cdots [6pt]end{align}More generally, if A(x) and B(x) are instead two complex-valued functions on mathbb{R} of period P that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series A(x)=sum_{n=-infty}^infty a_ne^{2pi nileft(frac{x}{P}right)} and B(x)=sum_{n=-infty}^infty b_ne^{2pi nileft(frac{x}{P}right)} respectively. Then{{Equation box 1|indent =|title=

:

{{EquationRef|Eq.2}}}}|cellpadding= 6|border|border colour = #0073CF|background colour=#F5FFFA}}Even 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(G) and L2(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 mathbb{R}, G^ is also mathbb{R} 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 f(x) is a square-integrable function over [-pi, pi] (i.e., f(x) and f^2(x) are integrable on that interval), with the Fourier series f(x) simeq frac{a_0}{2} + sum_{n=1}^{infty} (a_n cos(nx) + b_n sin(nx)). ThenBOOK, 439, Arthur E. Danese, Advanced Calculus, 1, Allyn and Bacon, Inc., Boston, MA, 1965, BOOK, 519, Wilfred Kaplan, Advanced Calculus,archive.org/details/advancedcalculus00kapl_841, limited, Addison Wesley, Reading, MA, 1991, 4th, 0-201-57888-3, Wilfred Kaplan, BOOK, 119, Georgi P. Tolstov, Fourier Series,archive.org/details/fourierseries00tols, registration, Silverman, Richard, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962, frac{1}{pi} int_{-pi}^{pi} f^2(x) ,mathrm{d}x = frac{a_0^2}{2} + sum_{n=1}^{infty} left(a_n^2 + b_n^2 right). Notation used in engineering In electrical engineering, Parseval’s theorem is often written as: int_{-infty}^infty | x(t) |^2 , mathrm{d}t = frac{1}{2pi} int_{-infty}^infty | X(omega) |^2 , mathrm{d}omega = int_{-infty}^infty | X(2pi f) |^2 , mathrm{d}f where X(omega) = mathcal{F}_omega{ x(t) } represents the continuous Fourier transform (in normalized, unitary form) of x(t), and omega = 2pi f 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: sum_{n=-infty}^infty | x[n] |^2 = frac{1}{2pi} int_{-pi}^pi | X_{2pi}({phi}) |^2 mathrm{d}phi where X_{2pi} is the discrete-time Fourier transform (DTFT) of x and phi represents the angular frequency (in radians per sample) of x.Alternatively, for the discrete Fourier transform (DFT), the relation becomes: sum_{n=0}^{N-1} | x[n] |^2 = frac{1}{N} sum_{k=0}^{N-1} | X[k] |^2 where X[k] is the DFT of x[n], both of length N.We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of X[k], we can derive begin{align} frac{1}{N} sum_{k=0}^{N-1} | X[k] |^2&= frac{1}{N} sum_{k=0}^{N-1} X[k]cdot X^*[k] = frac{1}{N} sum_{k=0}^{N-1} left[sum_{n=0}^{N-1} x[n],expleft(-jfrac{2pi}{N}k,nright)right] , X^*[k] [5mu]&= frac{1}{N} sum_{n=0}^{N-1} x[n] left[sum_{k=0}^{N-1} X^*[k],expleft(-jfrac{2pi}{N}k,nright)right] = frac{1}{N} sum_{n=0}^{N-1} x[n] (N cdot x^*[n]) [5mu]&= sum_{n=0}^{N-1} | x[n] |^2,end{align}where * 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 Parseval’s Theorem on Mathworld