Performance of unknown and arbitrary sparse signal detection using convex programming method with compressive measurements

We consider the detection of arbitrary and unknown sparse signals against background noise. Under a Neyman-Pearson framework, a new detection scheme referred to as the likelihood ratio test with sparse estimation (LRT-SE) is proposed and analyzed. The error probability of LRT-SE is characterized with respect to the signal-to-noise ratio (SNR) and the estimation error under the high SNR regime. For the low SNR regime, it is shown that there exists a detection boundary on the SNR, above which Chernoff-consistent detection is achievable for LRT-SE. The detection boundary can be calculated using fidelity results on the sparse estimation, and it allows the signal to be consistently detected under vanishing SNR. The error exponent of LRT-SE is also characterized and compared with the oracle exponent assuming signal knowledge. Numerical experiments are used to shown that the proposed method performs in the vicinity of the LRT method and the error probability decays exponentially with the number of observations. Results in this paper also have important implications in showing how well the performance of sparse estimation technique transforms into a hypothesis testing setup.