The coherent operation of a distributed phased array relies on the accurate alignment of the electrical states (i.e., frequencies and phases) of the antenna nodes, which challenging to achieve particularly at microwave frequencies. To address this challenge, a high accuracy distributed Kalman filter (HA-DKF) is proposed wherein the neighboring nodes share their observations, the process noise covariance matrices, and the KF's predicted as well as updated state estimates and error covariance matrices, and perform consensus averaging on the shared information to precisely synchronize the nodes across the array. Simulation results are included to illustrate that our HA-DKF algorithm takes fewer iterations to yield better synchronization of the states compared to the other filtering algorithms.

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.

A decentralized approach for joint frequency and phase synchronization in distributed antenna arrays is presented. The nodes in the array locally broadcast their frequencies and phases to their neighboring nodes, and use consensus averaging to align these parameters across the array. The architecture is fully distributed, requiring no centralization. Each node has a local oscillator and we consider a signal model where intrinsic frequency and phase errors of the local oscillators on each node caused by the frequency drift and phase jitter as well as the frequency and phase estimation errors at the nodes are included and modeled 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 and to the diffusion least mean square (DLMS), the diffusion KF (DKF) algorithm, and the Kalman consensus information filter (KCIF) algorithms, 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 received signals. Simulation results demonstrate that under certain conditions the proposed KF-DFPC algorithm outperforms and converges in fewer iterations than all the algorithms. 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.

We consider downlink channel estimation in massive multiple input multiple output (MIMO) systems using a Bayesian compressive sensing (BCS) approach. BCS exploits the sparse structure of the channel in the angular domain in order to reduce the pilot overhead. Due to limited local scattering, the massive MIMO channel has a block-sparse representation in the angular domain. Thus, we use a conditionally independent and identically distributed spike-and-slab prior to model the sparse vector coefficients representing the channel and a Markov prior to model its support. An expectation propagation (EP) algorithm is developed to approximate the intractable joint posterior distribution on the sparse vector and its support with a distribution from an exponential family. The unknown model parameters which are required by EP, are estimated using the expectation maximization (EM) algorithm. The proposed combination of EM and EP algorithms is reminiscent of variational EM and is referred to as EM-EP. The approximated distribution is then used for estimating the massive MIMO channel. Simulation results show that our proposed EM-EP algorithm outperforms several recently-proposed algorithms in channel estimation.

A semi-blind algorithm based on expectation propagation (EP) is proposed for multi-cell massive MIMO systems with spatially and temporally correlated channels. The performance of the algorithm in channel estimation and data detection is obtained from simulations and compared to other approaches. The results show that with only K pilots, where K is the number of users in the cell, the channel estimation performance of the proposed method approaches a lower bound as the number of BS antennas or the number of data symbols per frame increase. As expected, for time-varying channels, the proposed algorithm clearly outperforms those designed for block-fading channel models.