On the Performance of Polar Codes Over RIS-Aided Wireless Communication Channels

In this paper, we propose a semi-analytical method to design polar codes for the reconfigurable intelligent surface (RIS)-assisted channel with Rayleigh fading on each wireless link, while also providing an approximate closed-form expression for its word error probability (WEP) as a function of the number of RIS elements. Conventional polar code design methods involve numerically evaluated values of the bit-channel transition probabilities used to obtain the WEP upper bound in which the impact of the number of RIS elements, signal-to-noise ratio (SNR), and code rate is challenging to analyze. In contrast, our approach uses exponential approximations to obtain approximate closed-form expressions for the bit-channel transition probabilities. These exponential terms can be succinctly characterized in terms of coefficient pairs (CPs) which can be tracked numerically during code design and are approximately independent of the SNR and the number of RIS elements. This allows for rapid design of polar codes for any combination of the RIS elements and SNR. Numerical and simulation results show that the proposed method efficiently predicts the required number of RIS elements for a target WEP within the range of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{\text {-}2}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{\text {-}6}$ </tex-math></inline-formula>, and can also be used to optimize resource allocation in a multiuser context.