Slepian-Bangs formula and Cramér Rao bound for circular and non-circular complex elliptical symmetric distributions

International audienceThis letter is mainly dedicated to an extension of the Slepian-Bangs formula to non-circular complex elliptical symmetric (NC-CES) distributions, which is derived from a new stochastic representation theorem. This formula includes the non-circular complex Gaussian and the circular CES (C-CES) distributions. Some general relations between the Cramér Rao bound (CRB) under CES and Gaussian distributions are deduced. It is proved in particular that the Gaussian distribution does not always lead to the largest stochastic CRB (SCRB) as many authors tend to believe it. Finally a particular attention is paid to the noisy mixture where closed-form expressions for the SCRBs of the parameters of interest are derived

Direct derivation of the stochastic CRB of DOA estimation for rectilinear sources

International audienceSeveral direction of arrival (DOA) estimation algorithms have been proposed to exploit the structure of rectilinear or strictly second-order noncircular signals. But until now, only the compact closed-form expressions of the corresponding deterministic Cram´er Rao bound (DCRB) have been derived because it is much easier to derive than the stochastic CRB (SCRB). As this latter bound is asymptotically achievable by the maximum likelihood (ML) estimator, while the DCRB is unattainable, it is important to have a compact closed-form expression for this SCRB to assess the performance of DOA estimation algorithms for rectilinear signals. The aim of this paper is to derive this expression directly from the Slepian-Bangs formula including in particular the case of prior knowledge of uncorrelated or coherent sources. Some properties of these SCRBs are proved and numerical illustrations are give