Fundamental Tradeoffs between Sparsity, Sensing Diversity and Sensing Capacity

A fundamental problem in sensor networks is to determine the sensing capacity, i.e., the minimum number of sensors required to monitor a given region to a desired degree of fidelity based on noisy sensor data. This question has direct bearing on the corresponding coverage problem, wherein the task is to determine the maximum coverage region with a given set of sensors. In this paper we show that sensing capacity is a function of SNR sparsity-the inherent complexity/dimensionality of the underlying signal/information space and its frequency of occurrence-and sensing diversity, i.e., the number of independent paths from the underlying signal space to the multiple sensors. We derive fundamental tradeoffs between SNR, sparsity, diversity and capacity. We show that the capacity is a monotonic function of SNR and diversity. A surprising result is that as sparsity approaches zero so does the sensing capacity irrespective of diversity. This implies for instance that to reliably monitor a small number of targets in a given region requires an disproportionally large number of sensors.