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In this technical note we consider the problem of distributed discrete-time state estimation over sensor networks. Given a graph that represents the sensor communications, we derive the optimal estimation algorithm at each sensor. We further provide a closed-form expression for the steady-state error covariance matrices when the communication graph reduces to a directed tree. We then apply the developed theoretical tools to compare the performance of two sensor trees and convert a randomdoi:10.1109/tac.2009.2026851 fatcat:7vfukd455vb7ncpl3af3mvrfum