Abstract
This paper focuses on optimal beamforming to maximize the mean signal-to-noise ratio (SNR)
for a passive reconfigurable intelligent surface (RIS)-aided multiple-input single-output (MISO) downlink
system. We consider a realistic setting where both the direct and indirect (through RIS) links to the user
equipment (UE) experience correlated Rician fading. Such a general fading model is particularly important to
capture the impact of line-of-sight (LoS) and correlated multipath fading that may occur due to the compact
placement of a large number of RIS elements. The assumption of passive RIS imposes the unit modulus
constraint, which makes the beamforming problem non-convex. To tackle this issue, we apply semidefinite
relaxation (SDR) to obtain the optimal phase-shift matrix and propose an iterative algorithm to obtain
a statistically optimal solution for the transmit beamforming vector and RIS-phase shift matrix. Further,
to measure the performance of the proposed beamforming scheme, we analyze key system performance
metrics such as outage probability (OP) and ergodic capacity (EC). Similar to the existing works, the OP
and EC evaluations rely on the numerical computation of the proposed iterative algorithm. However, this
is not conducive to revealing the functional dependence of system performance on key parameters such as
line-of-sight (LoS) components, correlated fading, number of reflecting elements, number of antennas at the
base station (BS), and fading factor. To overcome this major limitation, we derive closed-form expressions
for the optimal beamforming vector and phase shift matrix for various special cases of the above general
fading model. These fixed-point solutions aid in deriving a closed-form solution for OP that further provides
a direct evaluation of EC. These mathematical expressions are then used to gain useful insights into the
system’s performance. We analytically establish the fact that correlated fading is more beneficial than the
independent and identically distributed (i.i.d.) case when the LoS components are blocked. Further, we also
analytically demonstrate that the maximum mean SNR improves linearly/quadratically with the number of
RIS elements in the absence/presence of LoS component under i.i.d. fading.