PERFORMANCE STUDY OF CONDITIONAL AND UNCONDITIONAL DIRECTION-OF-ARRIVAL ESTIMATION

Two different types of data model are currently used in the applications concerned with estimating the direction of arrival (DOA) of narrow-band signals using sensor arrays: 1) the conditional model (CM) which assumes the signals to be nonrandom, and 2) the unconditional model (UM) which assumes the signals to be random. These two models lead to different maximum likelihood (ML) methods (termed CML and UML, respectively) and different Cramér-Rao bounds (CRB's) on DOA estimation accuracy (Bc and Bu, respectively). It might also be expected that a DOA estimation method will have different statistical properties under the two models. To summarize the results of this paper, let CCML, etc., denote the covariance matrix of the CML, etc. First we derive an explicit expression for CUML and Bu, which was not available in the literature. Moreover, we show that CML, UML, and a newly introduced method of direction estimation (MODE) as well as, in fact, many other DOA estimation methods have the same asymptotic statistical properties under CM as under UM. Next we prove that: a) CML is statistically less efficient than UML; b) MODE is asymptotically equivalent to UML; c) UML and MODE achieve the unconditional CRB, Bu; and d) Bu is a lower bound on the asymptotic statistical accuracy of any (consistent) DOA estimate based on the data sample covariance matrix; Bc cannot be attained. We also prove that Bu and Bc decrease monotonically as the number of sensors or snapshots increases, and they increase monotonically as the number of sources increases. Furthermore, we show that when the signal-to-noise ratio or the number of sensors increases, then all of the matrices CCML, Bu = CUML = CMODE, and Bc tend to the same limit matrix. Finally, we include some more quantitative, numerical comparisons of CCML, Bu, and Bc. © 1990 IEEE