Optimal Estimation Over the Collision Channel

Consider a distributed sensing system that comprises two sensors, each observing a random variable, and a remote estimator. The goal of the remote estimator is to produce estimates of the random variables based on information transmitted to it by the sensors. The random variables are independent and information is transferred from the sensors to the estimator via a collision channel, which can only convey a single packet. Each sensor has the authority to decide what and when to transmit, and simultaneous transmissions result in a collision event to be detected at the estimator. In our formulation, there is no communication between the sensors, which precludes the use of coordinated strategies. Our results characterize the structure of policies at the sensors and the remote estimator that are optimal with respect to a mean squared error criterion. More specifically, we show that there exist optimal policies at the sensors that use deterministic threshold strategies to decide when to transmit. This structural result is independent of the distributions of the observed random variables. In our analysis, we prove that the computation of a person-by-person optimal threshold policy can be recast as a one-bit optimal quantization problem for which the cost is nonuniform across representation symbols. Based on that observation, we provide an iterative procedure akin to the Lloyd-Max algorithm that can be used to compute locally optimal solutions. In the Gaussian case, our iterative method converged to an optimal solution in all numerical examples we have tried. We provide several examples that show the optimality of asymmetric threshold policies even when the overall framework is symmetric, such as when both random variables are Gaussian with zero mean and the same variance.