My question relates to trying to understand the ways to characterize cyclostationary processes. Reference to the literature would be helpful!
My question has the following parts:
Part I: Phase Noise
Given the following definitions:
$$ \mathcal{L}(f) = {1 \over A^2} \int_{-\infty}^{\infty} d \tau R_{vv} (\tau) e^{-j (2 \pi f- \omega_0) \tau} $$ where $A$ is the normalization for dBc per Hz. $$ \mathcal{S}_\phi (f) = \int_{-\infty}^{\infty} d \tau R_{\phi \phi} (\tau) e^{-j 2 \pi f \tau} $$ $$v(t) = \cos( \omega_0 t + \phi(t))$$ $$R_{vv}(\tau) = \frac{1}{T}\int_{0}^T dt E[ v(t+ \tau) v(t)]$$ where $T = {2 \pi \over \omega_0}$ $$R_{\phi \phi} (\tau) = E[ \phi(t+ \tau) \phi(t)]$$
What is the relationship between $\mathcal{L}(f)$ and $S_\phi(f)$? Under what conditions are they similar, and under what conditions do they differ?
For this part at least, we can assume $\phi$ is Wide-Sense Stationary, although I am curious how this generalizes for cyclostationary $\phi$.
Part II: Jitter
Let us define the time series $ t_0, t_1,...$ as the deviation of the 0 crossing points times of the waveform v(t) from an ideal one with $\phi = 0$. Remember that v(t) is modulated by the stationary (or maybe even cyclostationary with period ${2 \pi \over \omega_0}$) process $\phi(t)$. We are interested in calculating the value $\sigma_t$, which is the standard deviation of the time series.
What is the expression that relates $\mathcal{L}(f)$ to $\sigma_t$? What is the expression that relates $\mathcal{S_\phi}(f)$ to $\sigma_t$? Under what conditions do these expressions break down?
Wiener Khintchine seems to suggest that $\sigma_t \propto \frac{1}{ \omega_0} \sqrt{\int_0^\infty \mathcal{S_\phi}(f) df}$. However I am not sure the integration limits if I replace $S_\phi$ with $\mathcal{L}(f)$
Also, what is the analog to Wiener Khintchine for cyclostationary $\phi(t)$?