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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.

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  • $\begingroup$ Equation 1 looks like a generalization of the Wiener-Khinchine theorem (mathworld.wolfram.com/Wiener-KhinchinTheorem.html) $\endgroup$
    – wcc
    Aug 1 '18 at 21:35
  • $\begingroup$ I'd really like to help with this question, but I'm not sure where to start. Can you perhaps narrow down what you'd like to know? $\endgroup$
    – DanielSank
    Aug 2 '18 at 0:33
  • $\begingroup$ @DanielSank Thank you for the reply. I think it is too big a jump to attempt to fully understand the paper. So I thought I could start with just figuring out where the equation 1 came from. (later in the paper, it's stated that once you take the real part of the matrix and diagonalize it, the diagonal terms tell you the terms tell you the noise tangent and normal to the circle) $\endgroup$
    – Blackwidow
    Aug 2 '18 at 13:46
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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.

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  • $\begingroup$ could you recommend a good reference where this derivation is done more rigorously? $\endgroup$
    – wcc
    Aug 2 '18 at 17:51
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    $\begingroup$ @IamAStudent I have a personal technical note written on this topic. Email me if you want and I can send. Alternatively build this TeX file from my GitHub repo. $\endgroup$
    – DanielSank
    Aug 2 '18 at 19:43
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    $\begingroup$ @IamAStudent I suggest read Chapter 3 in Middleton: An Introduction to Statistical Communications Theory or if you do not want that much "engineering" then read Chapter 4 in Priestley:Spectral Analysis and Time Series; both are excellent in their own way without Doob/Wiener style abstraction. $\endgroup$
    – hyportnex
    Aug 2 '18 at 21:07

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