  We consider an Internet-of-Things (IoT) system in which an energy harvesting
powered sensor node monitors the phenomenon of interest and transmits its
observations to a remote monitor over a Gaussian channel. We measure the
timeliness of the signals recovered by the monitor using age of information
(AoI), which could be reduced by transmitting more observations to the monitor.
We evaluate the corresponding distortion with the mean-squared error (MSE)
metric, which would be reduced if a larger transmit power and a larger source
coding rate were used. Since the energy harvested by the sensor node is random
and limited, however, the timeliness and the distortion of the received signals
cannot be optimized at the same time. Thus, we shall investigate the
timeliness-distortion trade-off of the system by minimizing the average
weighted-sum AoI and distortion over all possible transmit powers and
transmission intervals. First, we explicitly present the optimal transmit
powers for the performance limit achieving save-and-transmit policy and the
easy-implementing fixed power transmission policy. Second, we propose a
backward water-filling based offline power allocation algorithm and a genetic
based offline algorithm to jointly optimize the transmission interval and
transmit power. Third, we formulate the online power control as an Markov
Decision Process (MDP) and solve the problem with an iterative algorithm, which
closely approach the trade-off limit of the system. Also, we show that the
optimal transmit power is a monotonic and bi-valued function of current AoI and
distortion. Finally, we present our results via numerical simulations and
extend results on the save-and-transmit policy to fading sensing systems.
