Performance of distributed estimation over unknown parallel fading channels

We consider distributed estimation of a source in additive Gaussian noise, observed by sensors that 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 and transmitted sensor observations, where the total power is fixed. In the second phase we consider both an equal power scheduling among sensors and an optimized choice of powers. We also optimize the percentage of total power that should be allotted for training. We prove that 50% training is optimal for equal power scheduling and at least 50% is needed for optimized power scheduling. For both equal and optimized cases, a power penalty of at least 6 dB is incurred compared to the perfect channel case to get the same mean squared error performance for the source estimator. However, the diversity order is shown to be unchanged in the presence of channel estimation error. In addition, we show that, unlike the perfect channel case, increasing the number of sensors will lead to an eventual degradation in performance. We approximate the optimum number of sensors as a function of the total power and noise statistics. Simulations corroborate our analytical findings.