On the Angular Resolution Limit Uncertainty Under Compound Gaussian Noise

Maria S Greco (1) , Remy Boyer (2, 3)

1 University of Pisa - Università di Pisa
2 CentraleSupélec
3 L2S - Laboratoire des signaux et systèmes

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.

Domaines: Traitement du signal et de l'image [eess.SP]; Statistiques [stat]; Autres [stat.ML]

Maria S Greco, Remy Boyer. On the Angular Resolution Limit Uncertainty Under Compound Gaussian Noise. Signal Processing, 2019, 164, pp.217-224. ⟨10.1016/j.sigpro.2019.05.029⟩. ⟨hal-02144024⟩