  We consider the problem of decentralized frequency and phase synchronization
in distributed phased arrays via local broadcast of the node electrical states.
Frequency and phase synchronization between nodes in a distributed array is
necessary to support beamforming, but due to the operational dynamics of the
local oscillators of the nodes, the frequencies and phases of their output
signals undergo the random drift and jitter in between the update intervals.
Furthermore, frequency and phase estimation errors contribute to the total
phase errors, leading to a residual phase error in the array that degrades
coherent operation. Recently, a classical decentralized frequency and phase
synchronization algorithm based on consensus averaging was proposed with which
the standard deviation of the residual phase errors upon convergence was
reduced to $10^{-4}$ degrees for internode update intervals of $0.1$ ms,
however this was obtained for arrays with at least $400$ nodes and a high
connectivity ratio of $0.9$. In this paper, we propose a message passing based
average consensus (MPAC) algorithm to improve the synchronization of the
electrical states of the nodes in distributed arrays. Simulation results show
that the proposed MPAC algorithm significantly reduces the residual phase
errors to about $10^{-11}$ degrees, requiring only $20$ moderately connected
nodes in an array. Furthermore, MPAC converges faster than the DFPC-based
algorithms, particularly for the larger arrays with a moderate connectivity.
