The Exit Problem in a Nonlinear System Driven by 1/f Noise: The Delay Locked Loop

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 scaleinvariance (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 vario...