Information theoretic bounds to sensing capacity of sensor networks under fixed SNR

In this paper we address the problem of finding the sensing capacity of sensor networks for the class of linear observation and fixed SNR models. We define sensing capacity as the maximum number of signal dimensions that can be reliably identified per sensor/observation. In this context the sparsity of the signal plays an important role. Sensing capacity has direct implications for the so called compressed sensing problem, which involves reconstruction of sparse phenomena from random Gaussian projections. In this paper we adopt an information theoretic framework and develop upper and lower bounds for sensing capacity. We first extend Fano´s inequality to incorporate distortion effects as well as continuous signal spaces. We use this modified version of the inequality to derive upper bounds to sensing capacity. Upper bounds to probability of error subject to a distortion criteria are derived under a max-likelihood detection set-up over the set of vector rate-distortion quantization points. Using the upper bound a sufficient condition is identified for reliable reconstruction that provides lower bounds to sensing capacity. For the case of linear observation model with Gaussian ensemble and under fixed SNR, our results show the following interesting behaviors : (A) Sensing Capacity is a function of SNR, distortion and sparsity; (B) Sensing capacity goes down to zero as sparsity goes down to zero, irrespective of SNR. We quantify the effect of sensing diversity (effective coverage per sensor) on sensing capacity and show that sufficiently large diversity can be traded off for SNR and signal sparsity. Also low sensing diversity implies low sensing capacity.