Detection of unknown and arbitrary sparse signals against noise

The detection of sparse signals against background noise is difficult since the information in the signal is only carried by a small portion of it. Prior information is usually assumed to ease detection. This study considers the general unknown and arbitrary sparse signal detection problem when no prior information is available. Under a Neyman-Pearson hypothesis-testing problem model, a new detection scheme referred to as the likelihood ratio test with sparse estimation (LRT-SE) is proposed. The SE technique from the compressive sensing theory is incorporated into the LRT-SE to achieve the detection of sparse signals with unknown support sets and arbitrary non-zero entries. An analysis of the effectiveness of LRT-SE is first given in terms of the characterisation of the conditions for the Chernoff-consistent detection. A large deviation analysis is then given to characterise the error exponent of LRT-SE with respect to the signal-to-noise ratio and the angle between the sparse signal and its estimate. Numerical results demonstrate superior detection performance of the proposed scheme over existing asymptotically optimal sparse detectors for finite signal dimensions. In addition, the simulation shows that the error probability of the proposed scheme decays exponentially with the number of observations.