Performance bounds for the rate-constrained universal decentralized estimators in sensor networks

We consider the decentralized estimation of a noise corrupted deterministic parameter using a bandwidth constrained sensor network with a fusion center (FC). The sensor noises are assumed to be additive, zero mean, and spatially uncorrelated. Assuming that each sensor sends to the FC a one-bit message per sample, we derive a Cramer-Rao lower bound (CRLB) for the rate-constrained decentralized estimators using the noise probability distribution functions (pdfs) and local quantization rules. We then optimize this CRLB with respect to the noise pdfs and local quantization rules to obtain a lower bound for the mean squared error (MSE) performance of a class of universal decentralized estimators [Z-Q. Luo (2004), A. Ribeiro et al. (2004)]. Our results show that if the noises and the parameter to be estimated both have finite range in [-U, U], then the minimum MSE performance of any rate-constrained universal decentralized estimator is at least in the order of U²/(4K), where K is the total number sensors. This bound implies that the recently proposed universal decentralized estimators [Z.-Q. Luo (2004), A. Ribeiro et al. (2004)] are optimal up to a constant factor (of 16).