parseval's theorem
DESCRIPTION
Parseval's TheoremTRANSCRIPT
Parseval’s theorem
In mathematics, Parseval’s theorem [1] usually refers tothe result that the Fourier transform is unitary; loosely,that the sum (or integral) of the square of a function isequal to the sum (or integral) of the square of its trans-form. It originates from a 1799 theorem about series byMarc-Antoine Parseval, which was later applied to theFourier series. It is also known asRayleigh’s energy the-orem, or Rayleigh’s Identity, after John William Strutt,Lord Rayleigh.[2]
Although the term “Parseval’s theorem” is often used todescribe the unitarity of any Fourier transform, espe-cially in physics and engineering, the most general formof this property is more properly called the Planchereltheorem.[3]
1 Statement of Parseval’s theorem
Suppose that A(x) and B(x) are two square integrable(with respect to the Lebesgue measure), complex-valuedfunctions on R of period 2π with Fourier series
A(x) =∞∑
n=−∞ane
inx
and
B(x) =
∞∑n=−∞
bneinx
respectively. Then
∞∑n=−∞
anbn =1
2π
∫ π
−π
A(x)B(x) dx,
where i is the imaginary unit and horizontal bars indicatecomplex conjugation.More generally, given an abelian topological group Gwith Pontryagin dual G^, Parseval’s theorem says thePontryagin–Fourier transform is a unitary operator be-tween Hilbert spaces L2(G) and L2(G^) (with integrationbeing against the appropriately scaled Haar measures onthe two groups.) WhenG is the unit circle T,G^ is the in-tegers and this is the case discussed above. WhenG is the
real line R, G^ is also R and the unitary transform is theFourier transform on the real line. When G is the cyclicgroup Z , again it is self-dual and the Pontryagin–Fouriertransform is what is called discrete Fourier transform inapplied contexts.
2 Notation used in engineering andphysics
In physics and engineering, Parseval’s theorem is oftenwritten as:
∫ ∞
−∞|x(t)|2 dt = 1
2π
∫ ∞
−∞|X(ω)|2 dω =
∫ ∞
−∞|X(2πf)|2 df
where X(ω) = Fω{x(t)} represents the continuousFourier transform (in normalized, unitary form) of x(t),and ω = 2πf is frequency in radians per second.The interpretation of this form of the theorem is that thetotal energy of a signal can be calculated by summingpower-per-sample across time or spectral power acrossfrequency.For discrete time signals, the theorem becomes:
∞∑n=−∞
|x[n]|2 =1
2π
∫ π
−π
|X(eiϕ)|2dϕ
where X is the discrete-time Fourier transform (DTFT)of x and Φ represents the angular frequency (in radiansper sample) of x.Alternatively, for the discrete Fourier transform (DFT),the relation becomes:
N−1∑n=0
|x[n]|2 =1
N
N−1∑k=0
|X[k]|2
where X[k] is the DFT of x[n], both of length N.
3 See also
Parseval’s theorem is closely related to other mathemati-cal results involving unitarity transformations:
1
2 5 EXTERNAL LINKS
• Parseval’s identity
• Plancherel’s theorem
• Wiener–Khinchin theorem
• Bessel’s inequality
[1] Parseval des Chênes, Marc-Antoine Mémoire sur lesséries et sur l'intégration complète d'une équation aux dif-férences partielles linéaire du second ordre, à coefficientsconstants” presented before the Académie des Sciences(Paris) on 5 April 1799. This article was published inMé-moires présentés à l’Institut des Sciences, ettres et Arts, pardivers savans, et lus dans ses assemblées. Sciences, math-ématiques et physiques. (Savans étrangers.), vol. 1, pages638–648 (1806).
[2] Rayleigh, J.W.S. (1889) “On the character of the com-plete radiation at a given temperature,” PhilosophicalMagazine, vol. 27, pages 460–469. Available on-linehere.
[3] Plancherel, Michel (1910) “Contribution a l'etude de larepresentation d'une fonction arbitraire par les integralesdéfinies,” Rendiconti del Circolo Matematico di Palermo,vol. 30, pages 298–335.
4 References• Parseval,MacTutor History of Mathematics archive.
• George B. Arfken and Hans J. Weber,MathematicalMethods 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 (Pren-tice Hall: Upper Saddle River, NJ, 1999) p 60.
• WilliamMcC. Siebert, Circuits, Signals, and Systems(MIT Press: Cambridge, MA, 1986), pp. 410–411.
• David W. Kammler, A First Course in Fourier Anal-ysis (Prentice–Hall, Inc., Upper Saddle River, NJ,2000) p. 74.
5 External links• Parseval’s Theorem on Mathworld
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6 Text and image sources, contributors, and licenses
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