On Performance Limit for Bandwidth-Constrained Distributed Parameter Estimation Systems

We consider a bandwidth-constrained distributed parameter estimation problem, where each sensor makes a noisy observation of an unknown random source $\theta$. Each sensor is unaware of $\theta$'s prior distribution and the actual dynamic range of its observation, and simply assumes that its observation is limited to a finite interval [$\tau_{k}, \tau_{k}$]. Each sensor quantizes its observation using a multi-bit uniform quantizer, where the quantization step size is chosen according to $\tau_{k}$. Sensors send their quantized observations to a fusion center (FC), that is tasked with estimating $\theta$ based on the received data from the sensors. We derive the Bayesian Fisher information, which is the inverse of the Bayesian Cramer-Rao lower bound, for two types of random $\theta$, namely Gaussian and Laplacian $\theta$. To quantify the amount of information loss on $\theta$ when the FC uses the quantized observation for estimating $\theta$, due to both limited dynamic ranges at the sensors and uniform quantization, we examine the derived Fisher information at the asymptotic regimes, when the quantization rate and $\tau_{k}$ go to infinity. We also provide two accurate approximations of the Fisher information for two cases of (i) binary and (ii) high rate multi-bit quantizers. Through simulations we explore the conditions under which the information loss on $\theta$ is negligible and demonstrate the accuracy of the provided approximations.