Complex Elliptically Symmetric Distributions: Survey, New Results and Applications

Complex elliptically symmetric (CES) distributions have been widely used in various engineering applications for which non-Gaussian models are needed. In this overview, circular CES distributions are surveyed, some new results are derived and their applications e.g., in radar and array signal processing are discussed and illustrated with theoretical examples, simulations and analysis of real radar data. The maximum likelihood (ML) estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are established. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$t$</tex> </formula> -distribution, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex> </formula> -distribution, the generalized Gaussian distribution and the closely related angular central Gaussian distribution. A generalization of ML-estimators, the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$M$</tex></formula> -estimators of the scatter matrix, are also discussed and asymptotic analysis is provided. Applications of CES distributions and the adaptive signal processors based on ML- and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$M$</tex></formula> -estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) estimation and beamforming. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given.