An Asymptotic Maximum Likelihood for Joint Estimation of Nominal Angles and Angular Spreads of Multiple Spatially Distributed Sources

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper proposes a large-sample approximation of the maximum likelihood (ML) criterion for the joint estimation of nominal directions and angular spreads in the presence of multiple spatially spread sources. The key idea is the concentration on the exact likelihood function by replacing the parametric nuisance estimate, which depends on all unknown parameters at the critical point, by another estimate relying on only the angles of interest, such as nominal angles and angular spreads. Rather than the <formula formulatype="inline"><tex Notation="TeX">$(3N_{S} + 1)$</tex> </formula>-dimensional optimization required by the exact ML estimator, the proposed large-sample approximation allows <formula formulatype="inline"><tex Notation="TeX">$2N_{S}$</tex></formula>-dimensional search, where <formula formulatype="inline"><tex Notation="TeX">$N_{S}$</tex></formula> is the number of sources. To demonstrate the proposed estimator, numerical results are conducted for the illustration of estimation error variance. In the non-asymptotic region, the proposed estimator outperforms previous approaches adopting the <formula formulatype="inline"><tex Notation="TeX"> $2N_{S}$</tex></formula>-dimensional search. </para>