I'm trying to follow a paper. There is an early equation in it and I'm not quite sure where it comes from.
There is a noise time series $n(t)$. The paper then says:
We assume that the noise is stationary and Gaussian. Under these assumptions, the noise is fully characterised via the one-sided noise power spectral density $S_n(f)$,
$\langle \tilde{n}(f)\tilde{n}^{\ast}(f')\rangle = \frac{1}{2}\delta \left( f-f' \right) S_{n}(f)$.
The $\tilde{n}(f)$ is the Fourier transform of $n(t)$ and the angled brackets denote an ensemble average over many noise realisations.
I don't understand why $\tilde{n}(f)$ is multiplied by its complex conjugate, where the $\frac{1}{2}$ comes from, or even what $S_{n}$ is.
Any pointers in the right direction would be great!