Reconstructing estimates from noisy transmissions with serially-connected Kalman filters

This paper considers the problem of tracking a time-varying variable with serially-connected Kalman filters. Two nodes are assumed to be serially connected to the target such that only node 1 can directly observe a noisy signal from the target. Node 2 can only observe noisy signals from node 1 corresponding to a linear combination of the current observation and current state estimate at node 1. The objective is to find the linear combination at node 1 that minimizes the mean squared error of the state estimates at node 2 under a transmit power constraint for the signals from node 1 to node 2. An augmented state model is developed to facilitate tracking at node 2. Transmission scaling factors are also derived to satisfy the power constraint. Numerical results are presented for two-node serial tracking in two scenarios: scalar parameter tracking and two-state oscillator phase and frequency tracking. In the scalar parameter tracking example, the results demonstrate that a non-trivial combination of the observation and state estimate at node 1 can improve performance at node 2 with respect to a baseline scenario of simply forwarding scaled observations. In the two-state clock tracking example, an optimal transmission strategy is developed which allows node 2 to achieve the same tracking performance as at node 1.