Probability of Resolution of Partially Relaxed DML an Asymptotic Approach

Abstract

The Partial Relaxation approach has recently been proposed to solve the Direction-of-Arrival estimation problem [1], [2]. In this paper, we investigate the outlier production mechanism of the Partially Relaxed Deterministic Maximum Likelihood (PR-DML) Direction-of-Arrival estimator using tools from Random Matrix Theory. Instead of applying a single source approximation to multi-source estimation criteria, which is the case for the MUSIC algorithm, the conventional beamformer, or the Capon beamformer, the Partial Relaxation approach accounts for the existence of multiple sources using a non-parametric modification of the signal model. In this paper, an accurate description of the probability of resolution for the PR-DML estimator is provided by analyzing the asymptotic behavior of the PR-DML cost function, assuming that both the number of antennas and the number of snapshots increase without bound at the same rate. The finite dimensional distribution of the PR-DML cost function is shown to be Gaussian in this asymptotic regime and this result is used to compute the probability of resolution.

Asymptotic Stochastic Analysis of Partially Relaxed DML

“…In [14] it was shown shown that the PR-DML cost functionη(θ) in (2) becomes asymptotically close to its deterministic counterpart…”

“…Let us consider the more general case of the covariance in (7) for p,q=1,...,L. Instead of numerically computing the complex double contour integrals in (7) as it was done in [14], we express the logarithm log(1−Ω(ω1,ω2)) of the integrand in (7) as a Taylor expansion and only consider the terms of the integrand that involve 1/N or 1/N 2 . This is motivated by the closed-form solution of the asymptotic variance obtained in (8) which only involves expressions that vanish with N at rates 1/N and 1/N 2 .…”

“…This region is typically characterized by a systematic appearance of outliers in the DoA estimates, which are mainly caused by the incapability of resolving closely spaced sources. In [14], the probability of resolution of the PR-DML method was investigated by studying the stochastic behavior of the corresponding cost function for Gaussian distributed observations. The analysis in [14] is asymptotic in both the number of antennas and the number of snapshots.…”

“…In [14], the probability of resolution of the PR-DML method was investigated by studying the stochastic behavior of the corresponding cost function for Gaussian distributed observations. The analysis in [14] is asymptotic in both the number of antennas and the number of snapshots. In this paper, we follow a similar approach as in [14] and utilize tools from Random Matrix Theory (RMT) to compute the asymptotic stochastic behavior of the PR-DML cost function whereby both the number of snapshots and the number of antennas are large quantities but their quotient converges to a fixed finite value.…”

“…The analysis in [14] is asymptotic in both the number of antennas and the number of snapshots. In this paper, we follow a similar approach as in [14] and utilize tools from Random Matrix Theory (RMT) to compute the asymptotic stochastic behavior of the PR-DML cost function whereby both the number of snapshots and the number of antennas are large quantities but their quotient converges to a fixed finite value. In contrast to [14] where the second order asymptotic behavior of the PR-DML cost function is computed numerically, we provide a closed-form expression for the variance and a suitable approximation for the closed-form expression of the covariance.…”