# Effects of Timing Noise on Square-Wave
## 2 Methods
Equation (11) can be evaluated as  
\[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]1 \[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]2 \[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]3  
Thus, \(S_{f}(\omega)\) is composed of two parts: a sequence of Dirac-\(\delta\) functions at modulation frequencies \(n\omega_{0}\) and pedestals centered at those same frequencies. The pedestals are dominated by the \(m=1\) contribution for small \(n\), but additional terms render the line shape non-Lorentzian. For increasing \(n\), the larger-\(m\) terms play a larger role, causing the pedestals to merge to form a continuum. _A reference oscillator resonant with \(\omega_{0}\) is present_. Here we assume the locking oscillator has a finite but small bandwidth that wanders over sufficiently long timescales, so that on short timescales, \(\varphi(t)\) roughly tracks the temporally local value of the clock due to the reference oscillator. In this case, \(\Lambda\) is associated with the reference oscillator. One finds in this case [17; 50]  
\[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]4  
This last expression is also known from the Maxwell-Voigt model of diffusion in viscoelastic fluids [52; 53]. Note that (A) and (B) are special cases of (C), _viz._, (A) is for \(\sigma_{\varphi}^{2}=0\) and (B) for \(\Lambda=0\). Based on the approaches employed above,  
\[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]5 \[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]6 \[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]7Insofar as the reference oscillator is of narrow bandwidth, we can approximate Equation (14) as  
\[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]8 \[S_{f}(\omega)=\lim_{\hat{T}\to\infty}\frac{1}{\hat{T}}\int_{-\hat{T}/2}^{\hat {T}/2}\!\!dt\int_{-\hat{T}/2}^{\hat{T}/2}\!\!\!dt^{\prime}e^{i\omega(t-t^{ \prime})}\mathcal{G}(t,t^{\prime}) \tag{2}\]9 \[f(t)=\frac{1}{2}\sum_{n=1}^{\infty}\Big{\{}c_{n}e^{in\omega_{0}[t+\varphi(t)] }+c_{n}^{\ast}e^{-in\omega_{0}[t+i\varphi(t)]}\Big{\}} \tag{3}\]0 \[f(t)=\frac{1}{2}\sum_{n=1}^{\infty}\Big{\{}c_{n}e^{in\omega_{0}[t+\varphi(t)] }+c_{n}^{\ast}e^{-in\omega_{0}[t+i\varphi(t)]}\Big{\}} \tag{3}\]1  
that is, a sequence of Lorentzians whose width is determined by the narrow-band reference oscillator and a pedestal determined by the NS OEO itself. This result is similar to Equation (13), but with Lorentzians replacing the Dirac-\(\delta\) functions. Notice that the timing error leads to increasing the broadening with \(n\) in all cases. This is because, for a given timing error, the corresponding _phase_ error for the frequency component at \(n\omega_{0}\) increases linearly with \(n\). The remainder of this paper is devoted to a study of an example of the modulation PSD for cases (A)-(C). We treated \(f(t)=\text{III}[t+\varphi(t)]\) in which \(c_{n}=1\) elsewhere [17]. In the next section, we consider a square wave.