Neyman-Pearson Detection of a Gaussian Source using Dumb Wireless Sensors

We investigate the performance of the Neyman-Pearson detection of a stationary Gaussian process in noise, using a large wireless sensor network (WSN). In our model, each sensor compresses its observation sequence using a linear precoder. The final decision is taken by a fusion center (FC) based on the compressed information. Two families of precoders are studied: random iid precoders and orthogonal precoders. We analyse their performance in the regime where both the number of sensors k and the number of samples n per sensor tend to infinity at the same rate, that is, k/n tends to c in (0, 1). Contributions are as follows. 1) Using results of random matrix theory and on large Toeplitz matrices, it is proved that the miss probability of the Neyman-Pearson detector converges exponentially to zero, when the above families of precoders are used. Closed form expressions of the corresponding error exponents are provided. 2) In particular, we propose a practical orthogonal precoding strategy, the Principal Frequencies Strategy (PFS), which achieves the best error exponent among all orthogonal strategies, and which requires very few signaling overhead between the central processor and the nodes of the network. 3) Moreover, when the PFS is used, a simplified low-complexity testing procedure can be implemented at the FC. We show that the proposed suboptimal test enjoys the same error exponent as the Neyman-Pearson test, which indicates a similar asymptotic behaviour of the performance. We illustrate our findings by numerical experiments on some examples.