# Effects of Timing Noise on Square-Wave
## 1 Introduction
Various timing-noise studies of stabilization of electronic oscillators [43] and OEOs [31] based on the framework of Refs. [47; 48; 49] can be found in the literature. Our model for timing noise is discussed in Refs. [17; 50]. It is motivated by considering an oscillator physically producing the output signal, in our case, the NS OEO, stabilized by synchronization to a reference oscillator that is much more accurate than the NS OEO. We do not specify physically how this stabilization is achieved; it is characterized by the statistical parameters discussed below. This approach has the virtue that our theoretical treatment remains generic. The NS OEO is a considered to be a _self-sustaining oscillator_ in the sense of Lax [45]. That is, in the absence of stabilization, its _frequency_ fluctuates randomly about its nominal value, leading to the timing of this oscillator to wander being described by a one-dimensional random walk. For the reference oscillator we shall consider three cases specializing our results to the SW OEO: (A) it is absent; (B) it is an ideal error-free reference; and (C) it is itself a self-sustaining oscillator, but with far slower diffusion than the unstabilized NS OEO. For case (B), the NS OEO timing noise generically obeys an Adler equation [47], which is conveniently treated as an Uhlenbeck-Ornstein process of one-dimensional diffusion in a harmonic potential [51]. We shall in fact see that (A) and (B) are special cases of (C). Note that case (C) is essentially the Maxwell-Voigt model developed to describe diffusion in viscoelastic fluids [52; 53]. Certainly, other types of timing noise exist [54; 55; 56; 57]. We do not consider the effects of time-delayed feedback, such as is present in many OEOs [31], nor do we consider filtering of the noise by either linear or nonlinear elements between the primary noise source and the timing noise acting directly on the output signal. In addition, we do not account for colored noise or multiple noise sources. Nonetheless, the model provides a physically reasonable account for an important source of timing noise, namely fluctuations in propagation time within individual elements of the OEO and the approach leads to closed-form expressions for important spectral and time-domain properties. Certainly, additional effects can be incorporated into the model. NS OEO performance is also expected to suffer from phase noise. Because a given timing noise \(\varphi(t)\) results in difference phase noise \(n\omega_{0}\varphi(t)\) for the various Fourier components \(n\) of an NS OEO with period \(\omega_{0}^{-1}\), we use the term timing noise here instead of phase noise. In Section 2, we outline the methods to treat the problem of timing noise in NS OEOs. Taking the nominal modulation signal produced by the NS OEO to be \(f(t)\), we consider the effects of \(\varphi(t)\) on the autocorrelation function (acf) \(\mathcal{F}(\tau)\) of \(f(t)\) and on the PSD \(S_{f}(\omega_{0})\). We consider the three cases, (A)-(C) mentioned above. Section 3 presents results for a case study based on a square-wave modulated signal. In Section 4 we conclude.