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, σ 2 δ ). 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 = δ 2 0 /σ 2 δ . 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.

Refinement and derivation of statistical resolution limits for circular or rectilinear correlated sources in CES data models

“…The second approach rests on hypothesis test using the generalized likelihood ratio test [6], [7], on Rao's test [8] or on Bayesian approach [9]. The third approach is based on the information theory and, more precisely, either on Stein's lemma which relates the false alarm probability resulting from the Neyman-Pearson decision criterion to the relative entropy between two hypothesis [10], on Chernoff upper bound [11], or on the mutual information between DOA, scattering properties and the received signal [12]. The last approach capitalizes on the CRB where two criteria were proposed.…”