- Parseval's theorem
In
mathematics , Parseval's theorem [Parseval des Chênes, Marc-Antoine "Mémoire sur les séries et sur l'intégration complète d'une équation aux differences partielle linéaire du second ordre, à coefficiens constans" 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).] usually refers to the result that theFourier 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 a1799 theorem about series byMarc-Antoine Parseval , which was later applied to theFourier series . It is also known as Rayleigh's energy theorem, afterJohn 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.]Although the term "Parseval's theorem" is often used to describe the unitarity of "any" Fourier transform, especially in
physics andengineering , the most general form of this property is more properly called thePlancherel theorem . [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.]Statement of Parseval's theorem
Suppose that "A"("x") and "B"("x") are two
Riemann integrable , complex-valued functions on R of period 2π with (formal)Fourier series : and
respectively. Then
:
where "i" is the
imaginary unit and horizontal bars indicatecomplex conjugation .Parseval, who apparently had confined himself to real-valued functions, actually presented the theorem without proof, considering it to be self-evident. There are various important special cases of the theorem. First, if "A" = "B" one immediately obtains:
:
from which the unitarity of the Fourier series follows.
Second, one often considers only the Fourier series for real-valued functions "A" and "B", which corresponds to the special case: real, , real, and . In this case:
:
where denotes the
real part . (In the notation of theFourier series article, replace and by .)Applications
In
physics andengineering , 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 (notangular 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 theangular frequency (inradians 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".
Equivalence of the norm and inner product forms
We shall refer to :as the
inner product form, and to:as the norm form. It is not difficult to show that they are (pointwise ) equivalent. One can use thepolarization identity :which is true for all complex numbers "a" and "b", and thelinearity of both integration and the Fourier transform.See also
*
Bessel's inequality
*Parseval's identity Notes
References
* [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Parseval.html Parseval] , "MacTutor History of Mathematics archive".
* George B. Arfken and Hans J. Weber, "Mathematical Methods for Physicists" (Harcourt: San Diego, 2001).
* Hubert Kennedy, " [http://home.att.net/~clairnorman/Eight_Mathematical.pdf 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
* [http://mathworld.wolfram.com/ParsevalsTheorem.html Parseval's Theorem] on Mathworld
*In the movie "Good Will Hunting ", the theorem that Professor Lambeau finishes writing on the classroom chalkboard just after we first see him is Parseval's theorem. [http://www.maplesoft.com/applications/app_center_view.aspx?AID=77]
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