Product and Ratio of Product of Fisher-Snedecor ℱ Variates and Their Applications to Performance Evaluations of Wireless Communication Systems

This article considers the product, and the ratio of the product of Fisher-Snedecor F random variables (RVs), which can be used in modeling fading conditions that are encountered in realistic wireless transmission. To this end, exact analytical expressions are derived for the probability density function (PDF) and cumulative distribution function (CDF) of the product of N statistically independent, but not necessarily identically distributed, Fisher-Snedecor F RVs. Capitalizing on these, exact analytical expressions are then derived for the outage probability, average channel capacity and average bit error probability over cascaded fading channels. Moreover, some important statistical metrics such as amount of fading, channel quality estimation index, kurtosis, and skewness are also provided, since they provide useful insights on the characteristics of the encountered fading conditions. In addition, with the aid of the central limit theorem, an approximation for the PDF of N*Fisher-Snedecor F RVs is proposed using a lognormal density, and its accuracy is quantified in terms of the resistor-average distance. Finally, novel expressions for the PDF and CDF of the N-fold product ratio of Fisher-Snedecor F RVs are also derived. As a potential application of our new results, a spectrum sharing network is considered, for which exact analytical expressions for the outage probability, delay-limited capacity, and ergodic capacity are derived. For the cascaded fading scenario and the spectrum sharing network, numerical examples are provided to show the impact of different channel-related parameters, such as fading severity, shadowing, peak and average interference power on the system performance, which is rather useful in the design of conventional and emerging wireless communication systems. Monte-Carlo simulation results are provided to corroborate the presented mathematical analysis.