The fading scenario of many realistic wireless communication transmission systems, such as, multi‐hop communications and spectrum sharing in cognitive radio networks (CRNs), can be modelled by the product and the ratio of the product of the random variables (RVs) of the channel distribution. However, there is no work has been investigated in the literature to provide unified statistics of the product and the ratio of the products that can be used for a wide range of non‐composite and composite fading conditions. Accordingly, in this paper, the statistical properties, namely, probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) of the product and the ratio of the product of independent and non‐identically distributed (i.n.d.) mixture Gamma (MG) RVs are derived. A MG distribution has been widely employed to approximate with high accuracy most of the conventional fading models, for example, Rayleigh, Nakagami‐m, Nakagami‐q (Hoyt), and Nakagami‐n (Rician) as well as the generalised composite fading channels, such as, generalised‐K(KG),α−μ/gamma, κ−μ/gamma, and η−μ/gamma. Hence, the derived PDF, CDF, and MGF are utilized for the Beaulieu–Xie and α−λ−η−μ shadowed fading channels that have not been yet presented by the previous works due to mathematical intractability of their statistics. Thus, the equivalent parameters of a MG distribution for these channels are given. To this end, simple closed‐form mathematically tractable expressions of the performance metrics are obtained. The derived statistics are applied to analyse the outage probability (OP), the average error probability for different modulation schemes, the effective rate (ER) of wireless communication systems and the average area under the receiver operating characteristics (AUC) curve of energy detection over cascaded fading channels. Moreover, the OP of the multi‐hop communications systems with co‐channel interference (CCI), both the lower bound of secure OP (SOPL) and probability of non‐zero secrecy capacity (PNSC) of the physical layer security (PLS), and the outage and delay‐limited capacities of CRNs are studied via using the statistics of the ratio of the product of MG variates. A comparison between the numerical results and the Monte Carlo simulations is presented to verify the validation of our analysis.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.