Abstract
This article considers the product, and the ratio of the product of Fisher-Snedecor $\mathcal {F}$ 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 $\mathcal {F}$ 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 $\mathcal {F}$ 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 $\mathcal {F}$ 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.
Highlights
M ULTIPATH fading and shadowing are two of the most significant factors, which must be taken into account when characterizing wireless communication channels [2].Throughout the literature, a number of models can be found which are used to characterize multipath fading such as the Nakagami-m and Rice fading models, and more recently the κ-μ, η-μ, and α-μ fading models [3]–[5]
Some well-known composite fading models in the open technical literature are the K-distribution, κ-μ shadowed, κ-μ/gamma, η-μ/gamma, and α-μ/gamma, κ-μ/inverse gamma, η-μ/inverse gamma, and extended generalized-K (EGK) distributions, which can be found in [6]–[20]. While these composite models may characterize the underlying fading phenomena in a reasonably accurate manner, quite often their mathematical representation leads to cumbersome, if not intractable, analytic results for most important performance measures of interest. This impacts the formulation of critical statistical metrics such as the probability density function (PDF), the cumulative distribution function (CDF), and the moment generating function (MGF), which are required to evaluate important performance metrics such as the bit/symbol error probability, channel capacity, and outage probability (OP)
Several other metrics of interest were derived which assisted in quantifying the effect of different fading and shadowing conditions
Summary
M ULTIPATH fading and shadowing are two of the most significant factors, which must be taken into account when characterizing wireless communication channels [2]. Significant effort has been devoted to evaluating the performance of both multipath fading and composite fading models in different communication scenarios [1], [21]–[42] and the references therein In this context, the authors in [21] introduced a rather general product distribution known as N ∗Nakagami-m. In the present paper, some important statistics of the product and ratio of product of Fisher-Snedecor F RVs such as the PDF, and CDF are derived. Considering the ratio of the product of Fisher-Snedecor F RVs, Section VII derives some useful statistics, while Section VIII makes use of the derived expressions to analyze the performance of a spectrum sharing network.
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