  In this paper, we consider a status update system, where an access point
collects measurements from multiple sensors that monitor a common physical
process, fuses them, and transmits the aggregated sample to the destination
over an erasure channel. Under a typical information fusion scheme, the
distortion of the fused sample is inversely proportional to the number of
measurements received. Our goal is to minimize the long-term average age while
satisfying the average energy and general age-based distortion requirements.
Specifically, we focus on the setting in which the distortion requirement is
stricter when the age of the update is older. We show that the optimal policy
is a mixture of two stationary, deterministic, threshold-based policies, each
of which is optimal for a parameterized problem that aims to minimize the
weighted sum of the age and energy under the distortion constraint. We then
derive analytically the associated optimal average age-cost function and
characterize its performance in the large threshold regime, the results of
which shed critical insights on the tradeoff among age, energy, and the
distortion of the samples. We have also developed a closed-form solution for
the special case when the distortion requirement is independent of the age,
arguably the most important setting for practical applications.
