  The $\alpha$-$\eta$-$\kappa$-$\mu$ is one of the most generalized and
flexible channel models having an excellent fit to experimental data from
diverse propagation environments. The existing statistical results on the
envelope of $\alpha$-$\eta$-$\kappa$-$\mu$ model contain an infinite series
involving regularized hypergeometric function and generalized Laguerre
polynomial, prohibiting its widespread application in the performance analysis
of wireless systems. In this paper, we employ a novel approach to derive
density and distribution functions of the envelope of the
$\alpha$-$\eta$-$\kappa$-$\mu$ fading channel without an infinite series
approximation. The derived statistical results are presented using a single
Fox's H-function for tractable performance analysis and efficient numerical
computations, especially for high-frequency mmWave and terahertz wireless
transmissions. To gain insight into the distribution of channel envelope, we
develop an asymptotic analysis using a more straightforward Gamma function
converging to the exact within a reasonable range of channel parameters. To
further substantiate the proposed analysis, we present the exact outage
probability and average bit-error-rate (BER) performance of a wireless link
subjected to the $\alpha$-$\eta$-$\kappa$-$\mu$ fading model using a single
tri-variate Fox's H-function. We obtain the diversity order of the system by
analyzing the outage probability at a high signal-to-noise (SNR) ratio. We use
numerical and simulation analysis to demonstrate the significance of the
developed statistical results compared with the existing infinite series
representation for the envelope of the $\alpha$-$\eta$-$\kappa$-$\mu$ model.
