Performance of XOR Rule for Decentralized Detection of Deterministic Signals in Bivariate Gaussian Noise

In this paper, we consider the performance of exclusive-OR (XOR) rule in detecting the presence or absence of a deterministic signal in bivariate Gaussian noise. Signals, when present at the two sensors, are assumed unequal, whereas the noise components have identical marginal distribution but are correlated. The sensors send their one-bit quantized data to a fusion center, which then employs the XOR rule to arrive at the final decision. Here we prove that, in the limit as the correlation coefficient $r$ approaches 1, the optimum fusion rule for both parallel and tandem topologies is XOR with identical, alternating partitions (XORAP) of the observations at the sensors. We further quantify the asymptotic decrease of the Bayes error of XORAP towards zero as a constant multiplied by $\sqrt {1-r}$ , as $r$ approaches 1. When compared to the asymptotic Bayes error of CLRT, which decreases to zero exponentially fast, as a function of $1/(1-r)$ , the Bayes error of XORAP decreases to zero much slower.