Selection combining, and opportunistic relaying, with many diversity branches: Switching rates for most common fading types

A closed form expression for the switching rate of selection combining, and opportunistic relaying, with any number of diversity branches, is derived for most common types of fading, including, but not limited to - gamma, lognormal, normal, exponential, Rayleigh, Weibull, generalized gamma (or alpha-mu) and Nakagami-m fading, and Rician fading as a close approximation. These types of fading can all be described by the Amoroso distribution, and here we assume that the fading at each branch is independent and identically distributed (i.i.d.). The switching rate is shown to be only dependent upon the number of branches, Doppler spread, and a shape parameter of an underlying four-parameter Amoroso distribution. Hence the switching rate can simply be evaluated from a one-parameter standard gamma distribution. In the limit of the shape parameter, a further simplified closed form approximation to switching rate is provided for lognormal fading, which is only dependent upon the number of diversity branches, and shown to not be dependent on log-mean or log-standard deviation. Further all theoretical evaluations are shown to closely match those of simulated time-selective fading for a range of shape parameters, including shape parameters tending to infinity, relevant to normal and lognormal fading.