This paper studies the distributed state estimation in sensor network, where $m$ sensors are deployed to infer the $n$-dimensional state of a linear time-invariant (LTI) Gaussian system. By a lossless decomposition of optimal steady-state Kalman filter, we show that the problem of distributed estimation can be reformulated as synchronization of homogeneous linear systems. Based on such decomposition, a distributed estimator is proposed, where each sensor node runs a local filter using only its own measurement and fuses the local estimate of each node with a consensus algorithm. We show that the average of the estimate from all sensors coincides with the optimal Kalman estimate. Numerical examples are provided in the end to illustrate the performance of the proposed scheme.
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