  A decentralized approach for joint frequency and phase synchronization in
distributed antenna arrays is presented. The nodes in the array share their
frequencies and phases with their neighboring nodes to align these parameters
across the array. Our signal model includes the frequency drifts and phase
jitters of the local oscillators as well as the frequency and phase estimation
errors at the nodes and models them using practical statistics. A decentralized
frequency and phase consensus (DFPC) algorithm is proposed which uses an
average consensus method in which each node in the array iteratively updates
its frequency and phase by computing an average of the frequencies and phases
of their neighboring nodes. Simulation results show that upon convergence the
DFPC algorithm can align the frequencies and phases of all the nodes up to a
residual phase error that is governed by the oscillators and the estimation
errors. To reduce this residual phase error and thus improve the
synchronization between the nodes, a Kalman filter based decentralized
frequency and phase consensus (KF-DFPC) algorithm is presented. The total
residual phase error at the convergence of the KF-DFPC and DFPC algorithms is
derived theoretically. The synchronization performances of these algorithms are
compared to each other in light of this theoretical residual phase error by
varying the duration of the signals, connectivity of the nodes, the number of
nodes in the array, and signal to noise ratio of the transmitted signals.
Simulation results demonstrate that the proposed KF-DFPC algorithm converges in
fewer iterations than other algorithms including DFPC. Furthermore, for shorter
intervals between local information broadcasts, the KF-DFPC algorithm
significantly outperforms the DFPC algorithm in reducing the residual total
phase error, irrespective of the signal to noise ratio of the received signals.
