New bivariate gamma types with MIMO application

ABSTRACT

In this paper a bivariate gamma type distribution emanating from the diagonal elements of an inverse Wishart type distribution is developed; which in turn originates from the complex matrix variate elliptical class. From this, a bivariate Weibullised gamma type distribution is also presented, of which the bivariate Nakagami-m type is a special case. The derived results may be applied as decision statistics for a MIMO (multiple input multiple output) system with two transmit antennas. It is proposed that under this elliptical umbrella some performance measures such as the outage probability of MIMO systems can be analyzed in broad generality.

KEYWORDS:

• Complex inverse Wishart
• Complex t
• Elliptical
• Nakagami-m
• Outage probability

MATHEMATICS SUBJECT CLASSIFICATION:

• 62E15
• 60E05
• 94A99

Acknowledgments

The authors would like to thank the anonymous referee for the constructive comments which improved the presentation of the paper. The authors acknowledge the support of the StatDisT group. This work is based upon research supported by the National Research Foundation, South Africa (ref. CPRR13090132066 grant nr 91497 & ref. CPRR160403161466 grant nr. 105840). M. Arashi’s research is supported in part by the National Research Foundation of South Africa (grant nr. 109214).

Notes

1 H denotes Hermitian transpose.

2 For a matrix X, X > 0 indicates the matrix is positive definite.

3 ${\mathbb{R}}^{+}$ denotes the positive real line.

4 a* denotes the conjugate transpose of a.

5 ${\Gamma }_2\left(m\right)={\prod }_{i=1}^2\pi \Gamma (m-i+1)=\pi \Gamma \left(m\right)\Gamma (m+1)$

6 $\mathrm{etr}(·)$ defines exp (tr( · )).