Confused about autocorrelation and PSD

xaratustra

I am confused a bit . I read in a paper that the following property holds, but can't find where it comes from.

[itex]\mathsf{E}\left[X(\omega)X^*(\omega')\right]=2\pi\delta(\omega-\omega')S_{XX}(\omega)[/itex]

it says that the expected value of the Fourier transformed signal is proportional to the spectral density function (PSD) [itex]S_{XX}(\omega)[/itex] which is as usual defined as:

[itex]S_{XX}(\omega)=\int_{-\infty}^{\infty}r_{XX}(\tau)e^{-i\omega\tau}\,d\tau[/itex]

Any one knows where this comes from?

thanks.

Andy Resnick

That looks vaguely like the Wiener-Khintchine theorem:

http://en.wikipedia.org/wiki/Wiener%...inchin_theorem