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Meta-surfaces intend to improve significantly the performance of futurewireless networks by controlling the wireless propagation and shaping the radiowaves according to the generalized Snell's law. A recent application ofmeta-surfaces is reconfigurable intelligent surfaces which are practicallylimited by the requirement for perfect knowledge of the user's position. Forthe case where the user's position cannot be obtained, we introduce randomlyreconfigurable surfaces (RRSs) aiming to diffuse the incoming wave. A RRS isdefined as a reconfigurable meta-surface that each of its elements induces arandomly selected time-variant phase shift on the reflected signal. Tofacilitate the performance analysis of a RRS-assisted system, first, we presentnovel closed-form expressions for the probability density function, thecumulative distribution function, the moments, and the characteristic functionof the distribution of the sum of double-Nakagami-m random vectors, whoseamplitudes follow the double-Nakagami-m distribution, i.e., the distribution ofthe product of two Nakagami-m random variables, and phases follow the circularuniform 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-assistedcomposite channel, assuming that the two links undergo Nakagami-m fading andthe equivalent phase follows the circular uniform distribution. Closed-formexpressions for the outage probability, the average received signal-to-noiseratio, the ergodic capacity, the bit error probability, the amount of fading,and the channel quality estimation index are provided to evaluate theperformance of the considered system. These metrics are also derived for thepractical special case where one of the two links undergoes Rayleigh fading.

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