The exit problem in a nonlinear system driven by 1/f noise: The delay locked loop

摘要

The frequency generated by high frequency oscillators contains a small but significant noise component known as phase noise, also known as oscillator noise or phase jitter. The phase noise belongs to the family of stochastic processes with spectra 1/f(alpha), which exhibits scale invariance (or self-similarity) and a long-term correlation structure that decays polynomially in time. Both the phase and thermal noises cause errors in receivers that contain the oscillators. In particular, they cause losses of lock in phase tracking systems such as the phase locked loop in coherent systems, which include cellular phones, global positioning systems (GPS), and radar (e. g., synthetic aperture radar (SAR)), and in the delay locked loop (DLL), which is an important component of code division multiple access receivers and interface to modern memory modules, such as double data rate synchronous dynamic random access memory. The mean time to lose lock (MTLL) is well known to be an important design objective for various tracking loops. The evaluation of the MTLL is known in the mathematical literature as the exit problem for a dynamical system driven by noise, which is the problem of calculating the mean time for the noisy trajectories to reach the boundary of the domain of attraction of a stable point of the noiseless dynamics. In this paper we develop an analytic approach to the evaluation of the leading order term for MTLL of a second order DLL, due to both the non-Markovian 1/f(alpha) noise and to thermal white noise. The method is applicable to more general systems driven by a wide class of phase noises. The keys to the solution of this exit problem are the construction of a series of higher order Markovian processes that converge to the non-Markovian 1/f(alpha) noise and the asymptotic solution to a multidimensional elliptic boundary value problem that the mean first passage time (MFPT) satisfies.