# Effects of Timing Noise on Square-Wave
## 2 Methods
In as much as \(n,n^{\prime}\in\mathbb{Z}^{+}\), we never encounter \(n=-n^{\prime}\), and so, only the terms in the second line of Equation (5) contribute to \(S_{f}(\omega)\). Collecting terms containing common factors and using the stationarity of \(\varphi(t)\), we have  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]0  
with \(\Phi(t,t^{\prime})=\varphi(\tau)-\varphi(0)\). Note that the assumed stationarity implies that averages depend only on \(|\tau|\). We henceforth replace \(T\) in averages by \(0\). It proves convenient to write Equation (6) as  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]1 \[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]2  
\(\mathcal{F}(\tau)\) is the acf of \(f(t)\), _viz._,  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]3  
We can see from Equation (6) that in the limit \(\varphi(t)\to 0\),  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]4  
as might not be surprising. To evaluate the integral in Equation (6) in the presence of timing noise, we need to evaluate \(\langle\exp[\pm in\omega_{0}\Phi(\tau)]\rangle\). The average of \(\Phi(\tau)\) depends only on \(|\tau|\), which means \(\langle\exp[in\omega_{0}\Phi(\tau)]\rangle=\langle\exp[-in\omega_{0}\Phi(\tau )]\rangle\). In the following, we consider cases (A)-(C). The results for \(\langle\exp[in\omega_{0}\Phi(\tau)]\rangle\) are given in Refs. [17; 50]. Explicit results are given below, where we consider cases (A)-(C). _A reference oscillator is absent_. In this case, \(\varphi(t)\) obeys a one-dimensional diffusion equation. One finds [45]  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]5  
where \(\Lambda\) determines the rate at which \(\varphi(t)\) wanders. The integration in Equation (6) is easily carried out,  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]6  
The PSD line shapes centered at the modulation harmonics \(n\omega_{0}\) are Lorentzians [45] with width linear in \(n\). As for \(\langle\exp[in\omega_{0}\varphi(\tau)]\rangle=0\), this implies \(\langle f(t)\rangle=0\), which is a consequence of the fact that the absolute timing over times \(t\gg\Lambda^{-1}\) wanders uniformly over the period \(\omega_{0}^{-1}\). _An ideal reference oscillator resonant with \(\omega_{0}\) is present_. Due to synchronization with the reference oscillator, \(\varphi(t)\) obeys a one-dimensional diffusion equation in a sinusoidal potential, which in the low-noise limit can be approximated by a harmonic potential. From the resulting Uhlenbeck-Ornstein model [51], one finds [17; 50]  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]7where \(\sigma_{\varphi}^{2}=\langle\varphi(t)\varphi(t)\rangle\) is the (time-independent) variance of \(\varphi(t)\) and \(\tilde{\tau}\) is the autocorrelation time of \(\langle\varphi(\tau)\varphi(0)\rangle\). In this case  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]8  
Note that the last exponential factor in the integral causes it to diverge; however, the integral can be regularized by the following procedure. Write  
\[\langle a(t)\rangle=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^ {\hat{T}/2}\!\!dt\ a(t).\]9 \[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]0  
Clearly, the first term gives rise to a Dirac-\(\delta\) function upon performing the \(\tau\) integration while the second term is integrable. The exponentials in the summation over \(m\) can now be easily integrated.