Online Expectation-Maximization Based Frequency and Phase Consensus in Distributed Phased Arrays

A distributed phased array is comprised of separate, smaller antenna systems (nodes) that coordinate with each other to support coherent beamforming. However, due to the frequency drift and phase jitter of the oscillators on the nodes, as well as the frequency and phase estimation errors induced in the synchronization process, there exists a decoherence between the nodes that degrades their beamforming gain. A decentralized frequency and phase consensus (DFPC) algorithm was proposed in a prior work for undirected array networks in which the nodes locally share their frequencies and phases with their neighboring nodes to reach a synchronized state. Kalman filtering (KF) was also integrated with DFPC, and the resulting KF-DFPC showed significant reduction in the total residual phase error upon convergence. One limitation of the DFPC-based algorithms is that, due to relying on the average consensus protocol which assumes undirected networks, they do not generally converge for directed array networks with column stochastic weighting matrix. For directed networks, a push-sum protocol based frequency alignment algorithm was recently proposed, but it does not take into account both frequency and phase variations in-between the update intervals. In this paper, we model the variations using a first-order Markov process and propose a push-sum based frequency and phase consensus (PsFPC) algorithm which is accordingly a modified version of the push-sum protocol. The residual phase error of PsFPC upon convergence is theoretically derived as well. Kalman filtering is also integrated with PsFPC and the resulting KF-PsFPC algorithm shows significant reduction in the residual error of the array. A limitation of KF-PsFPC is that the model parameters, i.e., the measurement noise and the process noise covariance matrices, are assumed to be known a priori. Since they may not be known in practice, we develop an online expectation maximization (EM) based algorithm that iteratively computes the maximum likelihood (ML) estimates of these matrices in an online manner. EM is integrated with KF-PsFPC and the resulting algorithm is referred to as the EM-KF-PsFPC algorithm. Simulation results are included where the performances of the PsFPC-based algorithms are analyzed for different distributed array networks, and are compared to the DFPC-based algorithms as well as to the earlier proposed hybrid consensus on measurement and consensus on information (HCMCI) algorithm.