Unknown and arbitrary sparse signal detection against background noise

The problem of detecting unknown and arbitrary sparse signals against background noise is considered. Under a fixed hypothesis-testing problem model, a scheme referred to as Likelihood Ratio Test with Sparse Estimate (LRT-SE) is proposed. The relation between the quality of the estimate and the detection performance is quantized through the Kullback-Leibler distance, which shows the performance of LRT-SE is only a function of the angle between the sparse signal and its estimate, thus accurate estimation of signal energy is not necessary. An algorithm of LRT-SE is further proposed. Sufficient conditions on the sparsity level and the angle between the sparse signal and its estimate are given such that Chernoff-consistent detection is achievable. Simulation results show LRT-SE gives close performance to that of likelihood ratio test without knowing the underlying sparse signal.