Bounds for maximum likelihood regular and non-regular DoA estimation in $K$-distributed noise

TL;DR

This paper analyzes the bounds of maximum likelihood DoA estimation in K-distributed noise, revealing conditions for unbounded Fisher information and proposing estimators with faster convergence rates under certain conditions.

Contribution

It provides new explicit expressions for the Fisher information matrix in K-distributed noise and characterizes when it becomes unbounded, along with estimator analysis.

Findings

FIM can be unbounded in certain K-distributed noise conditions.

The covariance part of the FIM is always bounded.

Estimators can converge faster than the usual rate when FIM is unbounded.

Abstract

We consider the problem of estimating the direction of arrival of a signal embedded in $K$-distributed noise, when secondary data which contains noise only are assumed to be available. Based upon a recent formula of the Fisher information matrix (FIM) for complex elliptically distributed data, we provide a simple expression of the FIM with the two data sets framework. In the specific case of $K$-distributed noise, we show that, under certain conditions, the FIM for the deterministic part of the model can be unbounded, while the FIM for the covariance part of the model is always bounded. In the general case of elliptical distributions, we provide a sufficient condition for unboundedness of the FIM. Accurate approximations of the FIM for $K$-distributed noise are also derived when it is bounded. Additionally, the maximum likelihood estimator of the signal DoA and an approximated version are derived, assuming known covariance matrix: the latter is then estimated from secondary data using a conventional regularization technique. When the FIM is unbounded, an analysis of the estimators reveals a rate of convergence much faster than the usual $T^{-1}$. Simulations illustrate the different behaviors of the estimators, depending on the FIM being bounded or not.