On sensing capacity of sensor networks for a class of linear observation models

In this paper we derive fundamental information theoretic upper and lower bounds to sensing capacity of sensor networks for several classes of linear observation models under fixed SNR. We define sensing capacity as the number of signal dimensions that can be reliably identified per sensor measurement. The signal sparsity plays an important role in this context. First we derive lower bounds to probability of error by extending the Fano´s inequality to handle arbitrary distortion in reconstruction and continuous signal spaces. It turns out that a necessary condition for signal reconstruction to within an average distortion level is that the rate distortion at the given level of sparsity should be less than the mutual information between the signal and the observations. Through a suitable expansion of the mutual information term we isolate the effect of structure of the sensing matrix on sensing capacity. Subsequently we analyze this effect for several interesting classes of sensing matrices that arise naturally in the context of sensor networks under different scenarios. First we show the effect of sensing diversity - which is related to the field coverage per sensor- on sensing capacity for random ensembles of sensing matrices. We show that low diversity implies low sensing capacity. However sufficiently large diversity can be traded off for SNR and signal sparsity. Then we consider deterministic sensing matrices and evaluate a general upper bound to sensing capacity. As a special case we show that a random LTI filter type structure suffers from low diversity.