# Confused about autocorrelation and PSD

1. Apr 18, 2012

### xaratustra

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

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

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

$S_{XX}(\omega)=\int_{-\infty}^{\infty}r_{XX}(\tau)e^{-i\omega\tau}\,d\tau$

Any one knows where this comes from?

thanks.

2. Apr 18, 2012