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This paper is mainly dedicated to an extension of the Slepian-Bangs formulato non-circular complex elliptical symmetric (NC-CES) distributions, which isderived from a new stochastic representation theorem. This formula includes thenon-circular complex Gaussian and the circular CES (CCES) distributions. Somegeneral relations between the Cramer Rao bound (CRB) under CES and Gaussiandistributions are deduced. It is proved in particular that the Gaussiandistribution does not always lead to the largest stochastic CRB (SCRB) as manyauthors tend to believe it. Finally a particular attention is paid to the noisymixture where closedform expressions for the SCRBs of the parameters ofinterest are derived.

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