On the Distribution of the Sum of Double-Nakagami-$m$ Random Vectors and Application in Randomly Reconfigurable Surfaces

Meta-surfaces intend to improve the performance of future wireless networks significantly by controlling the wireless propagation and shaping the radio waves according to the generalized Snell's law. A recent application of meta-surfaces are reconfigurable intelligent surfaces that are mainly proposed for the reflection and steering of the impinging signal. In this article, we introduce randomly reconfigurable surfaces (RRSs) aiming to diffuse the incoming wave and characterize the performance of an RRS-assisted communication network. To facilitate the performance analysis of an RRS-assisted network, first, we present novel closed-form expressions for the probability density function, the cumulative distribution function, the moments, and the characteristic function of the amplitude of the distribution of the sum of double-Nakagami-
$m$
random vectors, whose amplitudes follow the double-Nakagami-
$m$
distribution, and phases following the circular uniform distribution. We also consider a special case of this distribution, namely the distribution of the sum of Rayleigh-Nakagami-m random vectors. Then, we exploit the obtained expressions to investigate the RRS-assisted composite channel, assuming that the two links undergo Nakagami-
$m$
fading. Specifically, closed-form expressions for the outage probability, the average received signal-to-noise ratio, the ergodic capacity, the bit error probability, the amount of fading, and the channel quality estimation index are provided to evaluate the performance of the considered system. Finally, these metrics are also derived for the practical special case where one of the two links undergoes Rayleigh fading, implying that this link is non-line-of-sight.