  We consider a wireless sensor network (WSN), consisting of several sensors
and a fusion center (FC), which is tasked with solving an M-ary hypothesis
testing problem. Sensors make M-ary decisions and transmit their digitally
modulated decisions over orthogonal channels, which are subject to Rayleigh
fading and noise, to the FC. Adopting Bayesian optimality criterion, we
consider training and non-training based distributed detection systems and
investigate the effect of imperfect channel state information (CSI) on the
optimal maximum a posteriori probability (MAP) fusion rules and optimal power
allocation between sensors, when the sum of training and data symbol transmit
powers is fixed. We consider J-divergence criteria to do power allocation
between sensors. The theoretical results show that J-divergence for coherent
reception will be maximized if total training power be half of total power,
however for non coherent reception, optimal training power which maximize
J-divergence is zero. The simulated results also show that probability of error
will be minimized if training power be half of total power for coherent
reception and zero for non coherent reception.
