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

Meta-surfaces intend to improve significantly the performance of future wireless networks by controlling the wireless propagation and shaping the radio waves according to the generalized Snell's law. A recent application of meta-surfaces is reconfigurable intelligent surfaces which are practically limited by the requirement for perfect knowledge of the user's position. For the case where the user's position cannot be obtained, we introduce randomly reconfigurable surfaces (RRSs) aiming to diffuse the incoming wave. A RRS is defined as a reconfigurable meta-surface that each of its elements induces a randomly selected time-variant phase shift on the reflected signal. To facilitate the performance analysis of a RRS-assisted system, first, we present novel closed-form expressions for the probability density function, the cumulative distribution function, the moments, and the characteristic function of the distribution of the sum of double-Nakagami-m random vectors, whose amplitudes follow the double-Nakagami-m distribution, i.e., the distribution of the product of two Nakagami-m random variables, and phases follow 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 these expressions to investigate the performance of the RRS-assisted composite channel, assuming that the two links undergo Nakagami-m fading and the equivalent phase follows the circular uniform distribution. 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. These metrics are also derived for the practical special case where one of the two links undergoes Rayleigh fading.

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