Distribution of the amplitude of a sum of singly and doubly scattered fading radio signal

The concept of multiple scattering radio propagation channels - in contrast to the conventional single (Rayleigh) scattering - has been recently proposed. Except for some special cases, expressions for the amplitude distribution of such channels are unknown. In this paper we focus on the double scattering radio channel. Let H₁, H₂, H₃ be identically distributed, independent zero-mean circularly symmetric complex Gaussian random variables with unit complex variance. We derive the distribution functions (pdf and cdf) of the random variable R = |w₁H₁+w₂H₂H₃|, where w₁ and w₂ are real-valued positive weights. The distribution functions are given in integral and series forms. We also derive a computationally simple moment-based estimator for the "power" mixture weights w₁² and w₂². In addition to the study of multiple scattering propagation channels, our results may also be applied in assessing the performance of communication schemes over such media.