Accurate Variable-Order Approximations to the Sum of α – μ Variates With Application to MIMO Systems

Abstract

Sum of fading signals arises in several communication applications such as maximal-ratio combining (MRC) used in multiple antenna systems and eliminating the inter-symbol interference in high speed wireless networks. The exact formulation of the statistics of the sum is known to be very complicated. This article presents two simple closed-form approximations to the probability density function (PDF) and cumulative distribution function (CDF) of the sum of <inline-formula> <tex-math notation= LaTeX >$\\alpha -\\mu $ </tex-math></inline-formula> variates, resembling an urban mobile communication environment. By direct integration, it is demonstrated that the PDF of the sum of <inline-formula> <tex-math notation= LaTeX >$\\alpha -\\mu $ </tex-math></inline-formula> random variables can be approximated by a weighted sum of few identical <inline-formula> <tex-math notation= LaTeX >$\\alpha -\\mu $ </tex-math></inline-formula> distributions including order one coefficients. The proposed approximations are of variable-order to achieve the desired accuracy and also have specific parameters using which, multiple-input multiple-output (MIMO) diversity systems can be explicitly analyzed as simply as a single-input single-output (SISO) system. In addition, a MIMO wireless system utilizing orthogonal space-time block coding technique (OSTBC-MIMO) in an <inline-formula> <tex-math notation= LaTeX >$\\alpha -\\mu $ </tex-math></inline-formula> frequency nonselective fading environment is analyzed, and expressions for outage probability (OP) and bit error rate (BER) of digital binary modulations as well as the capacity are obtained. Moreover, the asymptotic OP and BER expressions are derived from the declared approximate expressions. The asymptotic OP and BER performances clearly indicate a diversity order equal to the product of the <inline-formula> <tex-math notation= LaTeX >$\\frac {\\alpha }{2}$ </tex-math></inline-formula> parameter, the <inline-formula> <tex-math notation= LaTeX >$\\mu $ </tex-math></inline-formula> parameter, the number of transmit antennas and the number of receive antennas. Analytical results are verified by simulation.