Mathematics and Statistics Vol. 9(6), pp. 1019 - 1033
DOI: 10.13189/ms.2021.090618
Reprint (PDF) (654Kb)

Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications

Habti Abeida ^*
Department of Electrical Engineering, College of Engineering, Taif University, Al-Haweiah, 21974, Saudi Arabia

ABSTRACT

Absolutely Continuous non-singular complex elliptically symmetric distributions (referred to as the nonsingular CES distributions) have been extensively studied in various applications under the assumption of nonsingularity of the scatter matrix for which the probability density functions (p.d.f's) exist. These p.d.f's, however, can not be used to characterize the CES distributions with a singular scatter matrix (referred to as the singular CES distributions). This paper presents a generalization of the singular real elliptically symmetric (RES) distributions studied by Díaz-García et al. to singular CES distributions. An explicit expression of the p.d.f of a multivariate non-circular complex random vector with singular CES distribution is derived. The stochastic representation of the singular non-circular CES (NC-CES) distributions and the quadratic forms in NC-CES random vector are proved. As special cases, explicit expressions for the p.d.f's of multivariate complex random vectors with singular non-circular complex normal (NC-CN) and singular non-circular complex Compound-Gaussian (NC-CCG) distributions are also derived. Some useful properties of singular NC-CES distributions and their conditional distributions are derived. Based on these results, the p.d.f's of non-circular complex t-distribution, K-distribution, and generalized Gaussian distribution under singularity are presented. These general results degenerate to those of singular circular CES (C-CES) distributions when the pseudo-scatter matrix is equal to the zero matrix. Finally, these results are applied to the problem of estimating the parameters of a complex-valued non-circular multivariate linear model in the presence either of singular NC-CES or C-CES distributed noise terms by proposing widely linear estimators

KEYWORDS
Non-singular CES Distributions, Singular CES Distributions, Circular Complex Random Vector, Non-circular Complex Random Vector, Circular/Non-circular Quadratic Forms, Circular/Non-circular Complex-valued Linear Model

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Habti Abeida , "Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications," Mathematics and Statistics, Vol. 9, No. 6, pp. 1019 - 1033, 2021. DOI: 10.13189/ms.2021.090618.

(b). APA Format:
Habti Abeida (2021). Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications. Mathematics and Statistics, 9(6), 1019 - 1033. DOI: 10.13189/ms.2021.090618.