Noise covariance matrix I've been trying to understand this paper.  
The paper seems to be about analyzing noise properties of a superconducting coplanar waveguide microwave resonator.  
They use an IQ mixer to perform measurements and it returns in phase and out of phase quadrature amplitudes. $\xi=[I,Q]^T $  and $\delta\xi=[\delta I,\delta Q]^T $, the latter of which is shown here:

$$\langle \delta \xi(\nu)\delta \xi ^\dagger  (\nu') \rangle
= S(\nu)\delta(\nu - \nu'), \, S(\nu) =
\left(
\begin{array}{cc}
S_{II}(\nu) & S_{IQ}(\nu) \\
S_{IQ}^*(\nu) & S_{QQ}(\nu)
\end{array}
\right) \tag{1}
$$
  where $\delta \xi (\nu)$ is the Fourier transform of the time-domain data, the dagger represents Hermitian conjugate, $S_{II}$ and $S_{QQ}$ are the auto-power spectra, and $S_{IQ}$ is the cross-power spectrum.
  The matrix $S$ is Hermitian and may be diagonalized using a unitary transformation; however, we find that the imaginary part of $S_{IQ}$ is negligible and that an ordinary rotation applied to the real part $\text{Re} S$ gives almost identical results. The eigenvectors are calculated at every frequency:
  $$O^T(\nu)\text{Re} S(\nu) O(\nu)=
\left(
\begin{array}{cc}
S_{aa}(\nu) & 0 \\
0 & S_{bb}(\nu)
\end{array}
\right) \tag{2}$$

First off, I noticed $\langle \delta \xi \delta\xi^\dagger \rangle$, which is just the standard covariant matrix.
Aside from that, I wasn't able to get much out of it though.
I would like to understand where they got the first two equations shown below.
I think that would be a decent starting point for me. 
I was hoping someone could point me in the direction of references that could help me understand the paper.
 A: In general, a wide-sense stationary (wss) process, say $\textbf{x}(t)$, does not have a Fourier transform because it is not square integrable (it has finite power but not finite energy). But let us ignore this mathematical detail, instead proceed formally and write fearlessly:
$$\textbf{X}(u)=\int_{-\infty}^{\infty}\textbf{x}(t)e^{-\iota 2 \pi ut}dt \\ \textbf{x}(t)=\int_{-\infty}^{\infty}\textbf{X}(t)e^{\iota 2 \pi ut}du$$ 
Now calculate, again formally, the correlation of the spectral amplitudes at different frequencies by fearlessly interchanging the integration and the statistical expectation operator $\mathbf{E}$, etc.:
$$\mathbf{E}[\textbf{X}(u)\textbf{X}^{*}(u')] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathbf{E}[\textbf{x}(t)\textbf{x}^{*}(t')]e^{-\iota 2 \pi (ut-u't')}dtdt'$$
Here $K_x(t-t')=\mathbf{E}[\textbf{x}(t)\textbf{x}^{*}(t')]$ is the (time) auto-correlation of the process itself, so with this you can write now 
$$\mathbf{E}[\textbf{X}(u)\textbf{X}^{*}(u')] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}K_x(t-t')e^{-\iota 2 \pi (ut-u't')}dtdt'$$
Next, change the integration variables to $\tau=t-t'$ followed by integration with respect to $\tau$ and $t'$ after which you arrive to 
$$\mathbf{E}[\textbf{X}(u)\textbf{X}^{*}(u')] = S_x(u)\delta(u-u')$$
where the spectral density $S_x(u)$ is defined as the Fourier transform of 
the auto-correlation function $S_x(u)=\int_{-\infty}^{\infty}K_x(\tau)e^{-\iota 2 \pi u\tau}d\tau$.
Although here it was defined and not derived the relationship is usually referred to as the Wiener-Khinchin theorem (see comment of @IamAStudent above).
The same can be "derived" for a matrix process as was used in the paper you quoted.
