  This paper considers a sequential sensor scheduling and remote estimation
problem with one sensor and one estimator. The sensor makes sequential
observations about the state of an underlying memoryless stochastic process and
makes a decision as to whether or not to send this measurement to the
estimator. The sensor and the estimator have the common objective of minimizing
expected distortion in the estimation of the state of the process, over a
finite time horizon. The sensor is either charged a cost for each transmission
or constrained on transmission times. As opposed to the prior work where
communication between the sensor and the estimator was assumed to be perfect
(noiseless), in this work an additive noise channel with fixed power constraint
is considered; hence, the sensor has to encode its message before transmission.
Under some technical assumptions, we obtain the optimal encoding and estimation
policies in conjunction with the optimal transmission schedule. The impact of
the presence of a noisy channel is analyzed numerically based on dynamic
programming. This analysis yields some rather surprising results such as a
phase transition phenomenon in the number of used transmission opportunities,
which was not encountered in the noiseless communication setting.
