Second Order Channel Statistics: An Analysis for Optimal Single-user RIS Systems

A key challenge facing reconfigurable intelligent surfaces (RISs) is channel state information acquisition. Passive RISs cannot generate pilot signals or process data, making rapid temporal changes in the channel problematic. Additionally, the impact of spatial changes in RIS channels has not been thoroughly investigated. Therefore, in this work, we use second order statistics to investigate the spatio-temporal behaviour of a single-user (SU) RIS system. Assuming a line-of-sight (LoS) RIS to base station (BS) link, we derive an exact expression for the level crossing rate (LCR) of the RIS link (user equipment (UE)-RIS-BS path) and propose a numerically stable approximation for the LCR of the global UE-BS channel. Each LCR expression attained is then utilised to find the corresponding average fade duration (AFD). The temporal signal-to-noise ratio (SNR) correlation is also derived assuming an LoS RIS-BS link. Assuming a Ricean RIS-BS link, expressions for the spatial correlation matrix of the global channel and the mean SNR loss due to channel ageing are derived. All of the analyses are verified by simulation, and the impact of key system parameters is investigated. We show that the use of an RIS does not significantly amplify changes in the channel.