Abstract
This paper presents the analytical derivation of joint probability density functions (pdfs) of the maximum likelihood (ML) estimates of a real and complex persymmetric correlation matrices (PCM) of multivariate Gaussian processes. It is oriented at the modifications of the classical Wishart’s–Goodman’s pdfs adapted to the ML estimates of the data CMs in a wide class of signal processing (SP) problems in systems with centrally symmetric (CS) receive channels. The importance of the derived modified pdfs for such CS systems could be as great as that of the classical Wishart’s–Goodman’s pdfs for systems with arbitrary receive channels. Some properties of the new obtained joint pdfs are featured.
Highlights
The multivariate statistical analysis of random processes widely uses the Wishart distribution, which describes statistical properties of a maximum likelihood (ML) estimate of the real-valued positively definite correlation matrix (CM) of multivariate Gaussian processes/fields [1,2,3,4,5]
6 Conclusions The main result of this work is the derivation of the pdfs (37) and (49) of random M × M real-valued and complex-valued ML estimates (21) and (36) of persymmetric CMs (2) and (11) of multivariate Gaussian processes and fields
Such CMs arise in numerous practical applications, in the tasks of space-time adaptive signal processing in systems with central symmetry of receive channels [20,21,22,23,24,25,26,27,28,29,30,31,32, 27,28,29,30,31,32,33,34,35,36,37, 38, 16, 17]
Summary
The multivariate statistical analysis of random processes widely uses the Wishart distribution, which describes statistical properties of a maximum likelihood (ML) estimate of the real-valued positively definite correlation matrix (CM) of multivariate Gaussian processes/fields [1,2,3,4,5]. The goal of this paper is twofold: (i) to derive closed form analytical expressions for the pdfs of the ML estimates of persymmetric real and complex CMs of Gaussian processes/fields of various natures and (ii) to feature their usefulness in statistical data characterization and operational performance analysis in applications to SP systems that possess a space-time receive channel СS property. Replacing in (30) matrix BV by its representation (24a) and taking into account (24b), (9), and (8), we obtain The latter formula describes the desired pdf of the real symmetric and persymmetric random matrix Arp of the ML estimate R^ p (21) of the real and persymmetric CM R of an even order M = 2 ⋅ L defined above in (2).
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