Imperfect CSI Analysis in RIS-Aided Wireless Systems

In this paper, we derive an exact mean SNR expression for the optimal single user Reconfigurable Intelligent Surface (RIS) design in the presence of imperfect channel state information (CSI). It is assumed that the user (UE)-RIS and UE- base station (BS) channels experience correlated Rayleigh fading and the RIS-BS link is line of sight (LOS). We present simplified versions of the derived mean SNR expression for the correlation extremes (no correlation and full correlation) in the UE-RIS and UE-BS channels. The derived mean SNR expression is then leveraged to analytically characterise the impact of error variances from the channel estimation process on the mean SNR. We show that as the error variances increase, the mean SNR expression reduces to that of a system with randomly selected RIS phases. Consequently, the growth in mean SNR reduces from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(N^{2}) $ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ \mathcal {O}(N)$ </tex-math></inline-formula>. Numerical results validate the analysis.