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 lead to different maximum-likelihood (ML) methods (termed CML and UML, respectively) and different Cramer-Rao bounds (CRB) on DOA estimation accuracy (B_c and B_u, respectively). An explicit expression is derived for the covariance matrix of the UML and for B_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: (a) CML is statistically less efficient than UML; (b) MODE is asymptotically equivalent to UML; (c) UML and MODE achieve the unconditional CRB, B_u; and (d) B_u is a lower bound on the asymptotic statistical accuracy of any (consistent) DOA estimate based on the data sample covariance matrix; B_c cannot be attained. It is also proven that B_u and B_c decrease monotonically as the number of sensors or snapshots increases and increase monotonically as the number of sources increases.