Parseval's relation

The name Parseval refers to mathematician Marc-Antoine Parseval (April 27, 1755 – August 16, 1836).

Statement of Parseval's relation

For functions $f(t)$ and $g(t)$ such that the Fourier transform $F(\omega )$ and $G(\omega )$ exist

$\int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{2\pi }}F(\omega ){\overline {G(\omega )}}\;d\omega .$

Here ${\overline {G(\omega )}}$ and ${\overline {g(t)}}$ are the respective complex conjugates of $G(\omega )$ and $g(t),$ respectively.

Proof

We write the respective Fourier transform representations of $f(t)$ and $g(t)$

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )e^{-i\omega t}\;d\omega$

$g(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }g(\Omega )e^{-i\Omega t}\;d\Omega$

and the complex conjugate of the second espression is

${\overline {g(t)}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\overline {g(\Omega )}}e^{i\Omega t}\;d\Omega$

Forming the inner product

$\int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{(2\pi )^{2}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\;d\omega \;\int _{-\infty }^{\infty }\;d\Omega \;F(\omega ){\overline {G(\Omega )}}e^{-i(\omega -\Omega )t}\;dt.$

Rearranging the order of the integrations, we may write

$\int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\;d\omega \;\int _{-\infty }^{\infty }\;d\Omega \;F(\omega ){\overline {G(\Omega )}}\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{-i(\omega -\Omega )t}\;dt.\right]$

Recognizing that the factor in $[...]$ is the Fourier representation of the Dirac delta function $\delta (\Omega -\omega )$ we may perform the $\Omega$ integration via the sifting property of the delta function to yield Parseval's relation

$\int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega ){\overline {G(\omega )}}\;d\omega .$

When $f(t)$ and $g(t)$ are the same function,

$\int _{-\infty }^{\infty }|f(t)|^{2}\;dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|F(\omega )|^{2}\;d\omega ,$

where $|...|$ indicate that the modulus of a complex valued function is being squared and integrated.