Abstract
The Cramer-Rao Bound (CRB) for direction of arrival (DOA) estimation has been extensively studied over the past four decades, with a plethora of CRB expressions reported for various parametric models. In the literature, there are different methods to derive a closed-form CRB expression, but many derivations tend to involve intricate matrix manipulations which appear difficult to understand. Starting from the Slepian-Bangs formula and following the simplest derivation approach, this paper reviews a number of closed-form Gaussian CRB expressions for the DOA parameter under a unified framework, based on which all the specific CRB presentations can be derived concisely. The results cover three scenarios: narrowband complex circular signals, narrowband complex noncircular signals, and wideband signals. Three signal models are considered: the deterministic model, the stochastic Gaussian model, and the stochastic Gaussian model with the a priori knowledge that the sources are spatially uncorrelated. Moreover, three Gaussian noise models distinguished by the structure of the noise covariance matrix are concerned: spatially uncorrelated noise with unknown either identical or distinct variances at different sensors, and arbitrary unknown noise. In each scenario, a unified framework for the DOA-related block of the deterministic/stochastic CRB is developed, which encompasses one class of closed-form deterministic CRB expressions and two classes of stochastic ones under the three noise models. Comparisons among different CRBs across classes and scenarios are presented, yielding a series of equalities and inequalities which reflect the benchmark for the estimation efficiency under various situations. Furthermore, validity of all CRB expressions are examined, with some specific results for linear arrays provided, leading to several upper bounds on the number of resolvable Gaussian sources in the underdetermined case.
Highlights
The Cramér-Rao Bound (CRB), which provides a lower bound on the variance of any unbiased estimator, has been extensively studied in the context of direction of arrival (DOA) estimation using sensor arrays during the past four decades, and it still attracts substantial research interest with the development of novel DOA estimation methods and array design techniques
The deterministic CRBs and the stochastic ones without a priori knowledge exist only in the overdetermined case, regardless of the array geometry. Those stochastic CRBs employing the a priori knowledge of uncorrelated sources can exist in the underdetermined case
A number of closed-form Gaussian CRB expressions for DOA estimation under various model assumptions were reviewed under a unified framework, with some new supplementary results reported
Summary
The Cramér-Rao Bound (CRB), which provides a lower bound on the variance of any unbiased estimator, has been extensively studied in the context of direction of arrival (DOA) estimation using sensor arrays during the past four decades, and it still attracts substantial research interest with the development of novel DOA estimation methods and array design techniques. A closed-form CRB expression offers a clear interpretation of the CRB, and allows the comparison with the asymptotic covariance matrix of estimation errors It supports the understanding of the source/array configuration and provides physical insights into the underlying problem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
