Introduction

Localization of sources of signals impinging on an array of sensors is a problem of interest in a variety of applications, including Sonar, Radar, biomedical engineering, communications and audio acoustics. A source parameter estimation approach to this problem has been well studied. Starting with received data model assumptions, numerous algorithms for location parameter estimation have been developed. For maximum likelihood (ML) and maximum a posterior (MAP) source location parameter estimation, two basic models have received considerable attention. For the conditional (or deterministic) maximum-likelihood (CML) model [1,2,3,4], source signal amplitudes as well as other secondary or nuisance parameters are considered deterministic and unknown. For the stochastic maximum-likelihood (SML) model [1,3], the covariance of the source signals as well as other nuisance parameters are assumed unknown. The CML model is more generally applicable, while if valid the SML model leads to better location estimator performance. The expectation-maximization (EM), introduced to the signal processing community by Dempster et al. [5], is a general algorithmic approach which can be used to solve ML problems. Its potential advantages include the possibility of development of algorithms which are computationally efficient for ML problems which are intractable to solve directly. For location parameter estimation, examples of the use of the EM approach include Feder and Weinstein [6] for a no secondary-parameter and a SML problem, Miller and Fuhrmann [7] for both CML and SML problems, and Ziskind and Hertz [8] for an AR source signal model problem. In none of the references cited above is prior knowledge of nuisance parameter distributions accounted for.

Compared to joint estimation on both primary and nuisance parameters, marginalization over nuisance parameters can result in improved estimator performance for the primary parameters. This is particularly true in limited-data situations and when the number of nuisance parameters grows with the amount of data. To marginalize over the nuisance parameters, a prior distribution on them must be assumed. Although accuracy of the prior distribution is important, in particular when the prior is restrictive, it has been shown that estimator performance can be improved even when noninformative (or diffuse) priors are assigned to the nuisance parameters. For source localization based on the CML model, see [4] for an illustration of this. In general, a problem with marginalization is the difficulty in solving the resulting optimization problem.

In this paper we consider, within the CML model formulation, incorporation of knowledge of prior distributions on both primary location and secondary source signal parameters. We focus on using priors on the nuisance source signal parameters to marginalize over them, and we introduce a simple EM algorithmic approach to solving the resulting optimization problem. This extends to the array based source location parameter estimation problem our recent work [9,10] on digital communication symbol estimation in the presence of secondary unknown channel parameters.