In this paper, we consider a sensor scheduling and remote estimation problem over an additive noise channel. At each time, the sensor observes the state of a one-dimensional stochastic process, and then decides whether to transmit its observation to the remote estimator or not. The sensor is restricted to transmit its observations for no more than a fixed number of times. If the sensor decides to transmit its observation, it sends the observation to the encoder, who then sends an encoded message over the communication channel. The remote estimator generates real time estimates on the source based on the information received from the encoder. Different from problems considered in prior work, which assumed that the encoder has a constraint on the stagewise encoding power, we consider in this paper a more general setting where the encoder has a constraint on the total power over the time horizon. Under some technical assumptions, we obtain team optimal decision strategies. Moreover, we uncover a rather surprising result that under optimal strategies the encoding power allocated to each stage is the same.