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, α-μ, N∗ Nakagami-m, (generalized) K-fading, and Weibull/gamma fading as its special cases. As a new class of H-variates (called the degree-ζ 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, κ-μ, η-μ, Rician/LOS gamma, and κ-μ/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.