Frequency tracking with intermittent wrapped phase measurement using the Rao-Blackwellized particle filter

In this paper, we consider the problem of frequency and phase tracking with intermittent measurements. While this problem is of fundamental interest, we are primarily motivated by the coherence requirements of distributed MIMO (DMIMO) applications in which cooperating nodes emulate a virtual antenna array and frequency synchronization among transmitting nodes is achieved by each node detecting and compensating for its frequency offset from a reference pilot that is broadcast periodically by a master node. Since accurate one-shot frequency estimation requires long measurement epochs, in the interest of minimizing overhead, we consider the problem of frequency/phase tracking using phase-only measurements. While the evolution of frequency and unwrapped phase can be modeled using a standard linear state space model, the measurement model is nonlinear, since the phase measurements are wrapped to a 2π interval. We develop a systematic design framework based on the concept of a Rao-Blackwellized particle filter: a particle filter is utilized to deduce the unwrapped phase from the wrapped phase measurements, while a Kalman filter estimates the frequency offset and provides phase predictions that are used to update particle weights. We show that the proposed filter effectively tracks the frequency and phase of on oscillator as long as the resync interval is smaller than a minimum that depends on the oscillator´s drift statistics.