Frequency and Phase Synchronization in Distributed Antenna Arrays Based on Consensus Averaging and Kalman Filtering

High Accuracy Distributed Kalman Filtering for Frequency and Phase Synchronization in Distributed Phased Arrays

“…representing the design parameters of a quartz crystal oscillator [12], [13]; the phase δθ f n in (1) denotes the phase drift due to the temporal variation of the frequency drift δf n between the update intervals, and it is computed by δθ f n = −πT δf n [13]; finally, the phase δθ n in (1) represents the phase jitter of an oscillator which is modeled as δθ n ∼ N (0, σ θ ). The standard deviation σ θ can be mathematically modeled as σ θ = √ 2 × 10 A/10 in which A is the integrated phase noise power of an oscillator that can be computed from its phase noise profile.…”

“…The standard deviation σ θ can be mathematically modeled as σ θ = √ 2 × 10 A/10 in which A is the integrated phase noise power of an oscillator that can be computed from its phase noise profile. In this work, we set A = −53.46 dB to model a typical high phase noise voltage controlled oscillator [11], [13]. To initialize the process in (1), we assume that f n (0) ∼ N (f c , σ 2 ), in which f c is the nominal carrier frequency and σ = 10 −4 f c denotes a crystal clock accuracy of 100 parts per million (ppm), whereas θ n (0) ∼ U(0, 2π) which represents the initial phase offset due to the hardware and the oscillator.…”

“…In a distributed phased array, the temporal variation in the electrical states of a node can be modeled using a state-space model wherein the process noise at the update time can be described by the oscillator-induced frequency and phase offset errors, and the measurement noise can be defined using the estimation errors of the electrical states. Using this approach, a distributed Kalman filtering based KF-DFPC algorithm was proposed in [13], where it was shown that the use of local KF with consensus averaging significantly reduces the residual phase error of the array, particularly when fast update rates are used for the array synchronization 1 . In the KF-DFPC algorithm, the nodes iteratively share their KF-updated frequency and phase estimates as well as the error covariances with their neighbors, and compute a weighted average of the shared values in each iteration to reach a consensus in a CEEC scheme.…”

“…For open-loop distributed phased array, centralized coordination approaches have been demonstrated previously [10], [11], however such approaches are challenging to scale and suffer from single points of failure. Decentralized consensus-based algorithms are more robust to node failures and are more easily scalable [12], [13]. Consensus-based algorithms rely only on iteratively broadcasting the electrical state information between the neighboring nodes, where each node computes an average of its local state and that of its neighbors to update its state in each iteration.…”

“…Our prior work has investigated the use of a Kalman filtering (KF) based decentralized frequency and phase consensus (KF-DFPC) algorithm to predict the temporal variation of the frequency and phase drift in a distributed phased array and thereby reduce the residual phase error of the array [13]. In this algorithm, the nodes iteratively share their KF-updated frequency and phase estimates as well as the error covariances with their neighbors, and compute a weighted average of the shared values in each iteration to reach a consensus, which is referred to here as a consensus on estimates and error covariances (CEEC) approach.…”