A unified framework for the statistical characterization of the SNR of amplify-and-forward multihop channels

In this paper, we present a unified approach to analyze the exact statistical characteristics of the harmonic mean of N ≥ 2 statistically independent and non-identically distributed random variables (RVs), which we term the N-normalized harmonic distribution (i.e., NHD distribution), for the purpose of modeling the amplify-and-forward multihop relay channels. We present exact statistical metrics for the moments-generating function (MGF), moments (Mellin moments-generating function), probability density function (PDF) and cumulative distribution function (CDF) of the NHD distribution. Aside from unifying past results based on the geometric-mean approximation of the harmonic-mean, our approach relies on the algebraic combination of Mellin and Laplace transforms to obtain exact single integral expressions which can be easily computed using the Gauss-Laguerre quadrature rule or can be readily expressed in terms of the multivariable Meijer's G of Fox's H functions. Numerical and simulation results, performed to verify the correctness of the proposed formulation, are in perfect agreement. The proposed formulation can be used to analyze the performance measures of the amplify-and-forward multihop relay channels such as outage probability, outage capacity, average capacity and average bit error probabilities.