Thefrequency
generated by high frequency oscillators contains a small butsignificant
noise component known as phase noise, also known asoscillator noise
or phase jitter. The phase noise belongs tothe family of stochastic
processes with spectra $1/f^\alpha$, which exhibitsscaleinvariance
(or self-similarity) and a long-term correlation structure that decayspolynomially
in time. Both the phase and thermal noises causeerrors in receivers
that contain the oscillators. In particular, theycause losses of
lock in phase tracking systems such asthe phase locked loop in
coherent systems, which include cellularphones, global positioning
systems (GPS), and radar (e.g., synthetic apertureradar (SAR)), and
in the delay locked loop (DLL), whichis an important component of
code division multiple access receiversand interface to modern
memory modules, such as double datarate synchronous dynamic random
access memory. The mean time tolose lock (MTLL) is well known to be
an importantdesign objective for various tracking loops. The
evaluation of theMTLL is known in the mathematical literature as
the exitproblem for a dynamical system driven by noise, which isthe problem of calculating the mean time for the noisytrajectories
to reach the boundary of the domain of attractionof a stable point
of the noiseless dynamics. In thispaper we develop an analytic
approach to the evaluation ofthe leading order term for MTLL of a
second orderDLL, due to both the non-Markovian $1/f^\alpha$ noise
and tothermal white noise. The method is applicable to more generalsystems driven by a wide class of phase noises. Thekeys to
the solution of this exit problem are theconstruction of a series
of higher order Markovian processes thatconverge to the
non-Markovian $1/f^\alpha$ noise and the asymptotic solutionto a
multidimensional elliptic boundary value problem that the meanfirst
passage time (MFPT) satisfies.
UCLA - Biblioteca de Ciencias y Tecnologia Felix Morales Bueno
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