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The authors consider a sensor scheduling problem for a scalar system observed by two sensors. The measurements derived from the two local sensors are sent over a bandwidth-limited network to a remote estimator. The bandwidth constraint allows at most one packet can be sent at each time step. Upon receiving the data, the estimator computes the optimal estimate of the state in the minimum mean-squared error sense. In consideration of simplicity as well as practical and efficient implementation, the authors focus on periodic scheduling scheme. The authors assume that the sensor energy is so limited that in each period there exist open-loop predictions at some steps, which makes the problem more challenging. Using some novel tools, it was shown that the `as uniformly as possible' scheme, originally proposed in an early work for minimising the average estimation error covariance when sensor energy is just enough, still holds true in this limited energy case but the optimal steps of open-loop prediction depend on the system parameters. Such a relationship between scheduling rule and system parameters is novel and does not exist in early works where sensor energy is just enough. Numerical examples are provided to demonstrate the key ideas of the proposed work.