M-ary distributed detection in the presence of channel estimation error

We consider a wireless sensor network, consisting of N sensors and a fusion center (FC), tasked with distributed classification of M = 2^L Gaussian sources. Each sensor makes an M-ary decision, maps it to an L-bit sequence, and modulates these bits employing binary PSK (BPSK) or binary FSK (BFSK) modulators. The binary modulated symbols are transmitted over orthogonal channels, subject to noise and (Rayleigh or Rician) flat fading to the FC. To facilitate channel estimation at the FC with a coherent receiver, each sensor also sends one binary modulated training symbol. We derive the optimal fusion rules, given the channel estimates. To investigate the effect of channel estimation error on the system performance, we assume that the sum of transmit powers corresponding to training and data symbols for each sensor is fixed. We show for BPSK modulation error probability is minimized when each sensor allocates its power equally between training and data. For BFSK error is minimized when power is allotted to data symbols only.