On the Extended $\eta$-$\mu$ Model: New Results and Applications to IRS-Aided Systems

Recently, an extension to the $\eta$ - $\mu$ fading model was proposed that generalizes the classic $\eta$ - $\mu$ one with the introduction of cluster imbalance. This feature renders the Extended $\eta$ - $\mu$ fading model to accommodate more diverse fading scenarios, ranging from very severe to more favorable ones. On the other hand, some statistics, such as level crossing rate, outage probability, and bit error probability, are found in terms of more elaborate formulations, e.g., double hypergeometric functions. Even though some of these functions are available in mathematical packages, such as Wolfram Mathematica, their evaluation still hinders usability. In this paper, capitalizing on the use of discrete values of the number of multipath clusters, we derive relevant envelope statistics. These statistics are obtained in closed-form expressions, written in terms of either elementary or simple functions. Furthermore, we derive formulations for the product of an arbitrary number and the ratio of two Extended $\eta$ - $\mu$ variates. These expressions are used to evaluate the following: (i) the exact coverage probability, (ii) the outage probability, and (ii) the bit error probability for coherent binary modulation schemes, all applied to systems aided by intelligent reflective surfaces (IRS). Simulations results show adherence to the proposed analytical framework.