Distributed Kalman filtering for spatially-invariant diffusion processes: the effect of noise on communication requirements

This work analyzes the communication requirements of Kalman filters for spatially-invariant diffusion processes with spatially-distributed sensing. In this setting Kalman filters exhibit an inherent degree of spatial localization or decentralization. We address the fundamental question of whether the statistical properties of process and measurement disturbances, namely variance and spatial-autocorrelations, can further enhance its inherent spatial localization. We show that when disturbances are spatially and temporally uncorrelated, the spatial localization of the filter depends on the ratio of model to measurement error. Building upon this result, we study exponentially-decaying spatially-autocorrelated process and measurement disturbances. We show that certain level of spatial-autocorrelation in the measurement noise reduces the communication burden of the Kalman filter: indeed, the filter is completely decentralized when a matching condition is satisfied. We also show that spatial autocorrelation of the process disturbance has no benefits in terms of communications, as the level of centralization of the filter increases with the autocorrelation length.