Mode, maximum likelihood and Cramer-Rao bound: conditional and unconditional results

Two different types of data model used in estimating the direction-of-arrival (DOA) of narrowband signals using sensor arrays are considered: the conditional model (CM), which assumes the signals to be nonrandom, and the unconditional model (UM), which assumes the signals to be random. These models leased to different maximum-likelihood (ML) methods (termed CML and UML, respectively) and different Cramer-Rao bounds (CRB) on DOA estimation accuracy (B/sub c/ and B/sub u/, respectively). An explicit expression is derived for the covariance matrix of the UML and for B/sub u/. It is shown that CML, UML, and a recently introduced method of direction estimation (MODE), as well as many other DOA estimation methods, have the same asymptotic statistical properties under CM as under UM. It is proven that: CML is statistically less efficient then UNL; MODE is asymptotically equivalent to UML; UML and MODE achieve the unconditional CRB, B/sub u/; and B/sub u/ is a lower bound on the asymptotic statistical accuracy of any (consistent) DOA estimate based on the data sample covariance matrix; B/sub c/ cannot be attained. It is also proven that B/sub u/ and B/sub c/ decrease monotonically as the number of sensors or snapshots increases and increase monotonically as the number of sources increases.<<ETX>>