Optimal and Suboptimal Broad-Band Source Location Estimation

Maximum likelihood (ML) parameter estimation algorithms are popular because they tend to be asymptotically efficient. For the problem of estimating direction of arrival (DOA) of N superimposed, far-field signals of unknown spectral levels in Gaussian noise using an array of M sensors, two different ML estimators have been considered. One is based on the assumption that the sources radiate stochastic-Gaussian signals and therefore is called the stochastic-Gaussian ML (SGML) estimator; the other, using estimates of the actual signals (not their assumed distribution), is called the conditional ML (CML) estimator. In this paper we discuss the two ML estimators for broad-Band sources with unknown spectral parameters. Neither is efficient if the source spectral parameters are completely arbitrary and unknown. This problem is fundamental for the CML estimator, but can be avoided for a version of the SGML estimator if the signal and noise spectra are known to satisfy certain smoothness conditions. While this version of the SGML is formally superior to the CML, we demonstrate that the performance difference is small under conditions not infrequently encountered in practice (e.g., high signal-to-noise ratio). When these are satisfied, the computationally simpler CML can be used without significant loss. The required conditions become more stringent as the source separation decreases or correlation between sources increases. In addition, the paper provides a closed-form analytic expression for the small-error variance of the CML estimator of the DOA of the nth source in the presence of N - 1 other sources. This result, derived under the assumption that all sources radiate uncorrelated, zero-mean Gaussian signals, is expressed in terms of physically meaningful parameters. Under the conditions mentioned above it can serve as an approximation of the stochastic-Gaussian Cramer-Rao lower bound, and therefore provides insight into the inherent limitations of the broad-Band DOA estimation problem.