Linear Detection of a Weak Signal in Additive Cauchy Noise

Abstract

The detection of a weak signal in additive Cauchy noise is of great importance in many applications. A locally optimum detector (LOD) exists for such a scenario; however, it is non-linear in nature. In general, implementation of non-linear detectors is difficult in practice, and linear detectors with good properties, such as high asymptotic relative efficiency (ARE) with respect to the LOD, are often desirable. In this paper, we propose a linear detector for a weak signal in additive Cauchy noise. The proposed test statistic is a linear combination of order statistics. For the special case of a constant signal in additive Cauchy noise, we prove the asymptotic normality of the trimmed linear detector, and show that the ARE of the trimmed linear detector with respect to the LOD is unity. Extensive simulation results are provided to demonstrate that the loss in the performance of the linear detector is very small compared with the non-linear LOD. We also discuss the hardware complexities of the LOD and the linear detector, and demonstrate the advantages of the linear detector over the LOD, in terms of hardware implementation.