OPTIMAL AND SUBOPTIMAL BROAD-BAND SOURCE LOCATION ESTIMATION

Maximum likelihood (ML) parameter estimation algorithms are popular be cause they tend to be asymptotically efficient. For the problem of est imating direction of arrival (DOA) of N superimposed, far-field signal s of unknown spectral levels in Gaussian noise using an array of M sen sors, two different ML estimators have been considered. One is based o n the assumption that the sources radiate stochastic-Gaussian signals and therefore is called the stochastic-Gaussian ML (SGML) estimator; t he other, using estimates of the actual signals (not their assumed dis tribution), is called the conditional ML (CML) estimator. In this pape r we discuss the two ML estimators for broad-band sources with unknown spectral parameters. Neither is efficient if the source spectral para meters are completely arbitrary and unknown. This problem is fundament al 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 certai n smoothness conditions. While this version of the SGML is formally su perior to the CML, we demonstrate that the performance difference is s mall under conditions not infrequently encountered in practice (e.g., high signal-to-noise ratio). When these are satisfied, the computation ally simpler CML can be used without significant loss. The required co nditions become more stringent as the source separation decreases or c orrelation 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 ot her sources. This result, derived under the assumption that all source s radiate uncorrelated, zero-mean Gaussian signals, is expressed in te rms of physically meaningful parameters. Under the conditions mentione d above it can serve as an approximation of the stochastic-Gaussian Cr amer-Rao lower bound, and therefore provides insight into the inherent limitations of the broad-band DOA estimation problem.