On the Angular Resolution Limit uncertainty under compound Gaussian noise

Abstract

The Angular Resolution Limit (ARL) is a fundamental statistical metric to quantify our ability to resolve two closely-spaced narrowband far-field complex sources. This statistical quantity, is defined as the minimal angular deviation between the two sources to be separated for a prefixed detection-based performance. In this work, we assume that the sources of interest are corrupted by a compound-Gaussian noise. In the standard literature, denoting with δ the true distance between the two sources, the derivation of the ARL is based on the statistical distribution of the Generalized Likelihood Ratio Test (GLRT) for a binary test where there is only one source under the null hypothesis (i.e., δ=0) and two sources under the alternative hypothesis δ ≠ 0. In literature, the true angular distance (TAD) is generally considered as an unknown deterministic parameter, then a maximum likelihood-based estimation of δ is exploited in the GLRT. In this paper, breaking away from existing contributions, we suppose that the TAD is a random variable, Gaussian distributed, meaning that δ∼N(δ0,σδ2). The TAD uncertainty can have many causes as for instance moving sources or/and platform, antenna calibration error, etc. In this work, a generic and flexible (but common) statistical model of the uncertain knowledge of the TAD is preferred instead of a too much specified error model. The degree of randomness (or uncertainty) is quantified by the ratio ξ=δ02/σδ2. The standard framework of the GLRT is no longer feasible for our problem formulation. To cope with the compound Gaussian noise modeling and the random model of the TAD, a powerful upper bound from information/geometry theory is exploited in this work. More precisely, a new expected Chernoff Upper Bound (CUB) on the minimal error probability is introduced. Based on the analysis of this upper bound, we show that the expected-CUB is highly dependent on the degree of uncertainty, ξ. As a by-product, the optimal s-value in the Chernoff divergence for which the expected-CUB is the tightest upper bound is analytically studied and the role of the mean value δ0 in the ARL context is analyzed.