Dictionary:Parseval’s theorem

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FIG. F-22. Fourier transform theorems.

(par’ s∂ v∂lz) For two aperiodic functions h₁(t) and h₂(t) with respective Fourier transforms H₁(f) and H₂(f),

$\phi _{12}(0)=\int h_{1}(t)h_{2}(t)dt=\int H_{1}(f)H_{2}(f)df=\Phi _{12}(0).$

Thus the zero-lag value of the crosscorrelation ₁₂(0), the two values on the left, equals the integral of the cross-product spectrum and the cross-power-spectral amplitude at zero frequency Φ₁₂(0), the two terms on the right. Both equal the cross-energy in the time domain. See Figure F-22. Named for Marc-Antoinne Parseval des Chenes (1755–1836), French mathematician.

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