Distributed estimation over parallel fading channels with channel estimation error

We consider distributed estimation of a source observed by sensors in additive Gaussian noise, where the sensors are connected to a fusion center with unknown orthogonal (parallel) flat Rayleigh fading channels. We adopt a two-phase approach of (i) channel estimation with training, and (ii) source estimation given the channel estimates, where the total power is fixed. We prove that allocating half the total power into training is optimal, and show that compared to the perfect channel case, a performance loss of at least 6 dB is incurred. In addition, we show that unlike the perfect channel case, increasing the number of sensors will lead to an eventual degradation in performance. We characterize the optimum number of sensors as a function of the total power and noise statistics. Simulations corroborate our analytical findings.