$H$ -Transforms for Wireless Communication

The $H$ -transforms are integral transforms that involve Fox’s $H$ -functions as kernels. A large variety of integral transforms can be put into particular forms of the $H$ -transform since $H$ -functions subsume most of the known special functions including Meijer’s $G$ -functions. In this paper, we embody the $H$ -transform theory into a unifying framework for modeling and analysis in wireless communication. First, we systematize the use of elementary identities and properties of the $H$ -transform by introducing operations on parameter sequences of $H$ -functions. We then put forth $H$ -fading and degree-2 irregular $H$ -fading to model radio propagation under composite, specular, and/or inhomogeneous conditions. The $H$ -fading describes composite effects of multipath fading and shadowing as a single $H$ -variate, including most of typical models such as Rayleigh, Nakagami- $m$ , Weibull, $ \alpha - \mu $ , $N\ast $ Nakagami- $m$ , (generalized) $K$ -fading, and Weibull/gamma fading as its special cases. As a new class of $H$ -variates (called the degree- $ \zeta $ irregular $H$ -variate), the degree-2 irregular $H$ -fading characterizes specular and/or inhomogeneous radio propagation in which the multipath component consists of a strong specularly reflected or line-of-sight (LOS) wave as well as unequal-power or correlated in-phase and quadrature scattered waves. This fading includes a variety of typical models such as Rician, Nakagami- $q$ , $ \kappa - \mu $ , $ \eta - \mu $ , Rician/LOS gamma, and $ \kappa - \mu $ /LOS gamma fading as its special cases. Finally, we develop a unifying $H$ -transform analysis for the amount of fading, error probability, channel capacity, and error exponent in wireless communication using the new systematic language of transcendental $H$ -functions. By virtue of two essential operations—called Mellin and convolution operations—involved in the Mellin transform and Mellin convolution of two $H$ -functions, the $H$ -transforms for these performance measures culminate in $H$ -functions. Using the algebraic asymptotic expansions of the $H$ -transform, we further analyze the error probability and capacity at high and low signal-to-noise ratios in a unified fashion.