  We consider a wireless sensor network, consisting of K heterogeneous sensors
and a fusion center (FC), that is tasked with solving a binary distributed
detection problem. Each sensor is capable of harvesting and storing energy for
communication with the FC. For energy efficiency, a sensor transmits only if
the sensor test statistic exceeds a local threshold {\theta}k, its channel gain
exceeds a minimum threshold, and its battery state can afford the transmission.
Our proposed transmission model at each sensor is motivated by the channel
inversion power control strategy in the wireless communication community.
Considering a constraint on the average energy of transmit symbols, we study
the optimal {\theta}_k's that optimize two detection performance metrics: (i)
the detection probability P_D at the FC, assuming that the FC utilizes the
optimal fusion rule based on Neyman-Pearson optimality criterion, and (ii)
Kullback-Leibler distance (KL) between the two distributions of the received
signals at the FC conditioned by each hypothesis. Our numerical results
indicate that {\theta}k's obtained from maximizing the KL distance are
near-optimal. Finding these thresholds is computationally efficient, as it
requires only K one-dimensional searches, as opposed to a K-dimensional search
required to find the thresholds that maximize P_D.
