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In this paper, we derive closed-form exact and asymptotic expressions for the symbol error rate (SER) and channel capacity when communicating over Fox's $H$-function fading channel. The SER expressions are obtained for numerous practically employed modulation schemes in case of single-branch and three multiple-branch diversity receivers: maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC). The derived exact expressions are given in terms of the univariate and multivariate Fox's $H$-functions for which we provide a portable and efficient Python code. Since Fox's $H$-function fading channel represents the most generalized fading model ever presented in the literature, the derived expressions subsume most of those previously presented for all the known simple and composite fading models. Moreover, easy-to-compute asymptotic expansions are provided to easily study the behavior of the SER and channel capacity at high values of the average signal-to-noise ratio (SNR). The asymptotic expansions are also useful in comparing different modulation schemes and receiver diversity combiners. Numerical and simulation results are also provided to support the mathematical analysis and prove the validity of the obtained expressions. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel framework for deriving closed-form exact expressions for the symbol error rate (SER) of $\alpha$– $\mu$ fading channels, assuming single-branch and equal-gain combining and maximal-ratio combining receivers and considering most of the commonly used modulation schemes. The proposed framework is based on Mellin transform, and the SER expressions are given in terms of the univariate and multivariate Fox $H$-functions, which have recently been extensively used in the literature. The proposed framework has the following advantages over previous frameworks: First, it is straightforward and general; therefore, it allows the derivation of the exact SER expressions for cases untreated before in the literature. Second, it enables direct derivation for the asymptotic expressions of the SER for high average signal-to-noise ratios (SNRs). To validate the obtained expressions, we compare the results of the special case of the Nakagami- $m$ fading channel with those reported in the literature. Furthermore, Monte Carlo simulations are conducted, and their results are shown to perfectly match the analytic expressions. Finally, the obtained asymptotic expressions for all the studied modulation schemes and diversity receivers are shown to match the behavior of their corresponding exact values for a wide range of SNR values that are of practical interest. [ABSTRACT FROM AUTHOR]