Power-bandwidth-distortion scaling laws for sensor networks
The goal of a class of sensor networks is to monitor an underlying physical reality at the highest possible fidelity. Sensors acquire noisy measurements and have to communicate them over a power- and possibly bandwidth-constrained interference channel to a set of base stations. The goal of this paper is to analyze, as a function of the number of sensors, the trade-offs between the degrees of freedom of the underlying physical reality, the communication resources (power, temporal and spatial bandwidth), and the resulting distortion at which the physical reality can be estimated by the base stations. The distortion can be expressed as the sum of two fundamentally different terms. The first term reflects the fact that the measurements are noisy. It depends on the number of sensors and on their locations, but it cannot be influenced by the communication resources. The second contribution to the distortion can be controlled by the communication resources, and the key question becomes: What resources are necessary to make it decay at least as fast as the first distortion term, as a function of the number of sensors? This question is answered threefold: First, a lower bound to the power-bandwidth trade-o is derived, showing that at least a constant to linearly increasing total power is required for typical cases (as a function of M). But is this also sufficient? In the second answer, communication strategies are considered where each sensor applies the best possible distributed compression algorithm, followed by capacity-achieving channel codes. For such a separation strategy, it is shown for typical cases that the power must increase exponentially as a function of the number of sensors, suggesting that the lower bound derived in this paper is far too optimistic. However, in the third answer, it is shown that this is not the case: For some example scenarios, the power requirements of the lower bound are indeed achievable, but joint source-channel coding is required. Finally, the problem of sensor synchronization is considered, and it is shown that the scaling laws derived in this paper continue to hold under a Rician fading model.