On the role of feedback in large sensor networks

Feedback can considerably simplify the task of approaching and achieving the performance limits predicted by information theory. This paper determines some of the potential offered by feedback in a typical sensor network situation that could be termed monitoring: the task is to monitor an underlying physical reality at the highest possible fidelity. Since the sensed signals are often analog, and the communication channels noisy, it will not generally be possible to exactly communicate the sensed signals. Rather, such sensor network scenarios involve both a compression and a communication problem. It is well known that in general, these two tasks must be addressed jointly for optimal performance, but optimal performance is generally unknown. Instead, this paper focuses on the scaling behavior of performance, i.e., its dependence on the number of nodes in the network. For a Gaussian sensor network situation, where the goal of the data collector is to estimate the underlying sources to within mean-squared error distortion, we determine an explicit lower bound to the achievable distortion as a function of the number of sources, sensors, and base stations, as well as the sensor transmit power and the bandwidth of the communication channel. Our lower bound allows for feedback, and for arbitrary collaboration between the sensors. We then argue that a simple amplify-and-forward strategy achieves the optimum scaling law for a class of sensor networks, as long as the (spatio-temporal) source bandwidth is equal to the (spatio-temporal) channel bandwidth, leaving no scaling-law relevant role for feedback. When the channel bandwidth is larger than the source bandwidth, such a simple strategy is no longer sufficient to exploit the additional resources. In this paper, we analyze a simple feedback coding strategy for the sensors that permits to match the source bandwidth to the channel bandwidth whenever the latter is larger. We establish that this strategy exploits the additional channel bandwidth optimally in the scaling sense.