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The $\alpha$-$\eta$-$\kappa$-$\mu$ is one of the most generalized andflexible channel models having an excellent fit to experimental data fromdiverse propagation environments. The existing statistical results on theenvelope of $\alpha$-$\eta$-$\kappa$-$\mu$ model contain an infinite seriesinvolving regularized hypergeometric function and generalized Laguerrepolynomial, prohibiting its widespread application in the performance analysisof wireless systems. In this paper, we employ a novel approach to derivedensity and distribution functions of the envelope of the$\alpha$-$\eta$-$\kappa$-$\mu$ fading channel without an infinite seriesapproximation. The derived statistical results are presented using a singleFox's H-function for tractable performance analysis and efficient numericalcomputations, especially for high-frequency mmWave and terahertz wirelesstransmissions. To gain insight into the distribution of channel envelope, wedevelop an asymptotic analysis using a more straightforward Gamma functionconverging to the exact within a reasonable range of channel parameters. Tofurther substantiate the proposed analysis, we present the exact outageprobability and average bit-error-rate (BER) performance of a wireless linksubjected to the $\alpha$-$\eta$-$\kappa$-$\mu$ fading model using a singletri-variate Fox's H-function. We obtain the diversity order of the system byanalyzing the outage probability at a high signal-to-noise (SNR) ratio. We usenumerical and simulation analysis to demonstrate the significance of thedeveloped statistical results compared with the existing infinite seriesrepresentation for the envelope of the $\alpha$-$\eta$-$\kappa$-$\mu$ model.

The $\alpha$-$\eta$-$\kappa$-$\mu$ Fading Model: An Exact Statistical Representation