## Abstract

In this paper, we present a unified framework to analyze the performance of the average bit error probability (BEP) and the outage probability over generalized fading channels. Specifically, we assume that the probability density function (PDF) of the instantaneous signal-to-noise ratio \(\zeta \) is given by the product of: power function, exponential function, and the modified Bessel function of the first kind, i.e., \(f_{\zeta }(\zeta )=\zeta ^{\lambda -1}exp\left( -a\zeta ^{\beta }\right) I_{v}\left( b\zeta ^{\beta }\right) \). Based on this PDF, we obtain a novel closed-form expression for the average BEP over such channels perturbed by an additive white generalized Gaussian noise (AWGGN). Note that other well-known noise types can be deduced from the AWGGN as special cases such as Gaussian noise, Laplacian noise, and impulsive noise. Furthermore, we obtain a novel closed-form expression for the outage probability. As an example of such channels, and without loss of generality, we analyze the performance of the average BEP and the outage probability over the \(\eta \)–\(\mu \) fading channels. Analytical results accompanied with Monte-Carlo simulations are provided to validate our analysis.

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## Notes

- 1.
Note that the BFHF in (15) converges if the following conditions are satisfied [23]: \(\rho _{1}=\sum _{j=1}^{p_{1}}\alpha _{j}+\sum _{j=1}^{p_{2}}\gamma _{j}-\sum _{j=1}^{q_{1}}\beta _{j}-\sum _{j=1}^{q_{2}}\delta _{j}\le 0, \rho _{2}=\sum _{j=1}^{p_{1}}A_{j}+\sum _{j=1}^{p_{2}}E_{j}-\sum _{j=1}^{q_{1}}B_{j}-\sum _{j=1}^{q_{2}}F_{j}\le 0, \Omega _{1}=-\sum _{j=n_{1}+1}^{p_{1}}\alpha _{j}-\sum _{j=1}^{q_{1}}\beta _{j}+\sum _{j=1}^{m_{2}}\delta _{j}-\sum _{j=m_{2}+1}^{p_{2}}\delta _{j}+\sum _{j=1}^{n_{2}}\gamma _{j}-\sum _{n_{2}+1}^{p_{2}}\gamma _{j}>0, \Omega _{2}=-\sum _{j=n_{1}+1}^{p_{1}}A_{j}-\sum _{j=1}^{q_{1}}B_{j}+\) \(\sum _{j=1}^{m_{3}}F_{j}-\sum _{j=m_{3}+1}^{p_{3}}F_{j}+\sum _{j=1}^{n_{3}}E_{j}-\sum _{n_{3}+1}^{p_{3}}E_{j}>0, |\mathrm{arg}~x|<\pi \Omega _{1}/2; \mathrm{and}~|\mathrm{arg}~y|<\pi \Omega _{2}/2\). It is straightforward to show that the parameters of the BFHF in (15) satisfies these sufficient conditions, and therefore the BFHF converges.

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Badarneh, O.S., Aldalgamouni, T. & Almehmadi, F.S. A unified framework for performance evaluation over generalized fading channels.
*Telecommun Syst* **64, **669–678 (2017). https://doi.org/10.1007/s11235-016-0199-6

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### Keywords

- Average bit error probability
- Generalized fading channels
- Generalized Gaussian noise
- Outage probability