  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.
