On the Angular Resolution Limit Uncertainty

Maria S. Greco, Rémy Boyer, Frank Nielsen

Abstract: The Angular Resolution Limit (ARL), denoted by 6, is a key statistical quantity to measure our ability to resolve two closely-spaced narrowband far-field complex sources. In the literature, the ARL, denoted by δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , is systematically assumed to be perfectly known for mathematical convenience. In this work, our knowledge on the ARL is supposed to be only partial, meaning that δ ~N (δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). The degree of uncertainty is quantified by the ratio ξ = δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Based on the Chernoff Upper Bound (CUB) on the minimal error probability, we show that the CUB is highly dependent on the degree of uncertainty, ξ. As by-product, the optimal s-value for which the CUB is the tightest upper bound is analytically studied.