Complex elliptically symmetric distributions and their applications in signal processing

Tutorial T13 presented in the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-14, Firenze, ItalyComplex elliptically symmetric (CES) distributions have been widely used in various engineering applications where non-Gaussian models are called for. In this tutorial, circularly symmetric 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 and real-word examples, simulations and analysis of real radar data. A particular emphasis is put on maximum likelihood (ML) estimation of the scatter (covariance) matrix parameter. General conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are discussed in detail. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex t-distribution, K-distribution, the generalized Gaussian distribution and the recently proposed generalized inverse Gaussian distribution. Also the closely related angular central Gaussian distribution is discussed. A generalization of ML-estimators, the M-estimators of scatter matrix, are also discussed and asymptotic analysis is provided. The tutorial also contains new results on regularized (penalized) M-estimators of scatter, which are needed e.g., in low sample support scenarios. Applications of CES distributions and the adaptive signal processors based on conventional and regularized ML- and M-estimators of the scatter matrix are illustrated in diverse applications such as radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) and beamforming in low sample support cases. Also graphical models and applications in image processing are discussed. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given. The tutorial is partially based on our recent article (IEEE Trans. SP, vol. 60, no. 11, pp. 5597-5625) and to our other recent (also unpublished) works in this field