Abstract
This paper establishes a rigorous mathematical framework for analyzing and optimizing secure communications in wireless systems aided by Reconfigurable Intelligent Surfaces (RIS) using Regularized Zero-Forcing (RZF) precoding. Wireless communication is inherently vulnerable to eavesdropping; Physical Layer Security (PLS) exploits channel physics to mitigate this. While RIS can enhance signal strength at intended users, traditional Maximum Ratio Transmission (MRT) precoding often leads to significant information leakage in multi-user environments. By leveraging Random Matrix Theory (RMT), we derive closed-form expressions for secrecy rates and propose a two-timescale optimization algorithm. Our results demonstrate that RZF significantly outperforms MRT, particularly in high-SNR regimes, by effectively balancing signal gain and interference suppression.
Introduction
The broadcast nature of wireless signals makes them vulnerable to unauthorized interception. Reconfigurable Intelligent Surfaces (RIS) have emerged as a transformative technology for PLS, allowing for the proactive shaping of the electromagnetic environment. By adjusting the phase shifts of reflecting elements, an RIS can constructively combine signals at a legitimate receiver while destructively interfering at an eavesdropper.
However, the choice of precoding at the Base Station (BS) is critical. Maximum Ratio Transmission (MRT), though computationally simple, fails to manage inter-user interference, which acts as "information leakage" to internal eavesdroppers. Regularized Zero-Forcing (RZF) precoding offers a superior alternative by suppressing interference. Despite its benefits, the matrix inversions required by RZF complicate the analysis of the concatenated BS-RIS-User channel. This work provides a comprehensive performance analysis and optimization framework for such systems.
Mathematical Framework and Methodology
To analyze the system performance without exhaustive Monte Carlo simulations, we employ Random Matrix Theory (RMT). We derive "deterministic equivalents" of the Ergodic Secrecy Sum Rate (ESSR), which are closed-form expressions valid in the large-system limit. These expressions allow for efficient system design and optimization without the need for millions of realizations.
The notations and expressions used in our RMT derivations are summarized in the tables below, providing the basis for the deterministic equivalent of the signal-to-interference-plus-noise ratio (SINR).
Two-Timescale Design
A practical challenge in RIS systems is the overhead associated with estimating instantaneous Channel State Information (CSI) for a large number of reflecting elements. We propose a two-timescale approach:
- Long-term (Statistical CSI): RIS phase shifts are optimized based on statistical channel features, which change slowly.
- Short-term (Instantaneous CSI): The BS precoder is updated based on the effective instantaneous channel, which accounts for fast fading.
Alternating Optimization (AO)
We developed an AO algorithm to jointly optimize the RZF regularization factor and the RIS phase shifts. This ensures the system operates at the optimal point for secrecy maximization, balancing the trade-off between signal power and leakage suppression.
Key Insights
Our analysis reveals several critical properties of RIS-aided RZF systems:
- Single-Hop Convergence: When the number of RIS elements ($L$) is much larger than the number of antennas ($M$) and users ($N$), the complex two-hop channel begins to behave mathematically like a simpler single-hop Rayleigh channel.
- Secrecy Scaling: Unlike MRT, which hits a "secrecy floor" at high SNR because leakage increases proportionally with power, RZF allows the secrecy rate to scale logarithmically with SNR.
- Optimal Regularization: In high-SNR regimes, the optimal regularization factor follows a $1/\text{SNR}$ scaling law, providing a simple rule of thumb for engineering design.
Results and Discussion
Numerical simulations validate our theoretical derivations. The closed-form ESSR expressions show an almost perfect match with Monte Carlo results, proving the robustness of the RMT approach. In scenarios with 24 BS antennas and 32 RIS elements, RZF and ZF precoding maintained high secrecy rates where MRT performance plateaued. Furthermore, the proposed two-timescale AO algorithm achieved performance levels near those of schemes requiring full instantaneous CSI, but with significantly reduced signaling overhead.
Conclusion
This paper establishes a rigorous framework for optimizing secure communications in RIS-aided systems using RZF precoding. By combining RMT-based analysis with a practical two-timescale optimization approach, we demonstrate that RZF is a robust solution for mitigating information leakage and enhancing physical layer security in multi-user wireless networks. Future work will address hardware constraints such as discrete phase shifts and the presence of active eavesdroppers.