In this technical note, we consider the problem of periodic sensor scheduling with limited resources. Two sensors are used to measure the state of a discrete-time linear process. We assume that each sensor has a maximum duty cycle and at most one sensor can communicate with a remote estimator at each time step due to the limited communication bandwidth. When a sensor is scheduled to send data, it sends the most recent measurement data to the estimator. Upon receiving the measurement data from the sensors, the estimator computes the optimal estimate of the state of the process. We first present a necessary condition for a periodic sensor schedule to be optimal. Based on this necessary condition, we construct an optimal periodic schedule that minimizes the estimation error at the estimator and at the same time satisfies the energy and communication bandwidth constraints. Examples are provided throughout the technical note to demonstrate the results developed.