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

Reconfigurable intelligent surfaces (RISs) intend to improve significantly the performance of future wireless networks, by controlling the wireless propagation medium through elements that can shift the phase of the reflected signals. Although ideally the signals reflected from a RIS are added coherently at the receiver, this is very challenging in practice due to the requirement for perfect channel state information (CSI) at the RIS and phase control. To facilitate the performance analysis of more practical RIS-assisted systems, 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 random variables following the Nakagami-m distribution, and phases are circular uniformly distributed. 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 RIS-assisted composite channel, assuming that the two links undergo Nakagami-m fading and the equivalent phase follows the uniform distribution, which corresponds to the case where CSI is not available at the RIS and leads to a lower bound of the performance of a system with CSI. 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.