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
Highlights
Non-singular CES distributions have recently been the focus of active research in engineering applications involving non-Gaussian data models [1,2,3,4,5,6,7,8]
This paper presents a generalization of the singular real elliptically symmetric (RES) distributions studied by Dıaz-Garcıa et al to singular CES distributions
This paper presents the derivation of explicit expressions for the p.d.f.’s of multivariate singular circular CES (C-CES), NCCES, circular complex normal (C-CN) and non-circular complex normal (NC-CN) distributed random variables (r.v.’s) following the reasoning proposed in [26]
Summary
Non-singular CES distributions have recently been the focus of active research in engineering applications involving non-Gaussian data models [1,2,3,4,5,6,7,8]. A problem that has not been tackled completely is related to the p.d.f. of singular C-CES, NC-CES, C-CN, and NC-CN distributions, which are not unusual in theoretical and practical engineering problems It was proved by [26] that the singular RES distributions have p.d.f’s on a subspace of smaller dimension and equal to the rank of the scatter matrix. The linear and widely linear minimum mean-squared error (LMMSE and WLMMSE) estimators introduced in [27,28] work under the assumption that the covariance matrix of measurement is non-singular The performance of these estimators are compared in [29, 30] and widely used in many practical applications [31,32,33,34]. These results are applied to the problem of estimating the parameters of a complex-valued linear model in the presence of either singular NC-CES or CCES distributed noise terms and followed by the derivation of widely linear estimators
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