  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.
