A New Perspective on Understanding Resolution Limit Via an Asymptotic Study of Christoffel-Darboux Kernel Based Spectrum Estimator

Abstract

This paper aims to provide a new perspective on understanding the resolution limit by examining a recently proposed spectrum estimator based on the Christoffel-Darboux Kernel. Different from other estimators, this estimator possesses a definite criterion of resolvability building on the well known Szegö extremum on the unit circle and the source estimates are allowed to merge. For a uniform linear array of length M and two uncorrelated sources, we show that there is a sharp transition of the limiting extremum at a separation of order O(M<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–1.5</sup>). Our result sheds light on understanding the fundamental difference between on-grid and gridless models, the meaning of super-resolution region, and the benefit of sparse arrays that have been widely exploited in recent literature.