Secrecy Rate Maximization in RIS-Assisted MIMO Systems Using a Practical Hardware Model

1. Problem Statement

The paper addresses the challenge of maximizing the secrecy rate in a Reconfigurable Intelligent Surface (RIS)-assisted Multiple-Input Multiple-Output (MIMO) wireless communication system. The system consists of:

• Alice: A multi-antenna transmitter.
• Bob: A multi-antenna intended receiver (IU).
• Eve: A multi-antenna eavesdropper (NU).
• RIS: A surface with $M$ passive Reflecting Elements (REs) assisting the transmission.

The Core Challenge:
Existing literature typically assumes an ideal (lossless) RIS model, where the magnitude of the reflection coefficient is constant (unit modulus) regardless of the phase shift. However, in practical hardware, REs possess inherent electrical resistance. This resistance causes the reflection magnitude to vary nonlinearly with the phase shift, introducing signal loss and coupling the design variables (transmit precoding and phase shifts).

The objective is to jointly optimize the transmit precoding matrix ($\mathbf{T}$) and the RIS phase shifts ($\boldsymbol{\theta}$) to maximize the secrecy rate ($C_{sec} = R_b - R_e$) under a practical hardware model that accounts for these resistive losses, subject to power constraints and phase shift boundaries.

2. Methodology

The authors propose a comprehensive framework involving two main optimization approaches:

A. Practical System Modeling

• Non-Ideal Reflection Model: The paper adopts a circuit-based model where the reflection coefficient $\phi_m = \beta_m e^{j\theta_m}$. The magnitude $\beta_m$ is a function of the phase shift $\theta_m$ and the element's resistance $R$.
• Coupling: Unlike ideal models, the magnitude and phase are coupled, making the optimization problem non-convex and highly complex.

B. Primary Solution: Adaptive Projected Gradient Method (PGM)

To solve the exact secrecy rate maximization problem (which is non-convex), the authors propose a low-complexity Joint Optimization algorithm using PGM:

1. Simultaneous Updates: In each iteration, both the precoding matrix $\mathbf{T}$ and the phase shift vector $\boldsymbol{\theta}$ are updated simultaneously based on their respective gradients.
2. Projection:
   • $\mathbf{T}$ is projected to satisfy the total transmit power constraint ($Tr(\mathbf{T}\mathbf{T}^H) \leq P$).
   • $\boldsymbol{\theta}$ is projected to satisfy the feasible phase shift range ($\theta_{min} \leq \theta_m \leq \theta_{max}$) and the nonlinear magnitude-phase relationship.
3. Adaptive Step Size Control: To ensure convergence stability and prevent performance degradation, the algorithm employs an adaptive mechanism. If the secrecy rate decreases in an iteration, the step sizes ($\nu_t$ and $\nu_\theta$) are reduced. Furthermore, adaptive initialization strategies are derived for the step sizes to accelerate convergence.

C. Sub-Optimal Solution: Channel Power Difference Maximization (CPDM)

Recognizing that solving the exact secrecy rate is computationally intensive, a simplified surrogate problem is introduced:

• Objective: Maximize the difference in channel power between the Bob link and the Eve link ($P_{diff} = Tr(\mathbf{T}^H (\frac{\hat{H}_b^H \hat{H}_b}{\sigma_b^2} - \frac{\hat{H}_e^H \hat{H}_e}{\sigma_e^2}) \mathbf{T})$).
• Advantage: This avoids the complex matrix determinant operations required for the exact secrecy rate, significantly reducing computational complexity while serving as a tight upper bound when eigenvalues are small.
• Usage: The CPDM solution is used to generate high-quality initial points for the main PGM algorithm, accelerating convergence.

3. Key Contributions

1. Practical Hardware Model: The study is among the first to explicitly model the electrical resistance of RIS elements in a secrecy rate maximization context. It demonstrates that ignoring this resistance leads to suboptimal designs in real-world scenarios.
2. Low-Complexity Joint Optimization: A novel Adaptive PGM algorithm is developed that jointly optimizes $\mathbf{T}$ and $\boldsymbol{\theta}$. Unlike Block Coordinate Descent (BCD) methods which alternate between variables, this method updates them simultaneously, offering better convergence properties.
3. Adaptive Step Size Mechanism: The paper introduces a robust step-size initialization and control mechanism that significantly reduces computational time and improves convergence stability compared to fixed-step-size PGM.
4. Sub-Optimal CPDM Formulation: The introduction of the CPDM problem provides a computationally efficient alternative for large-scale systems and serves as an effective initializer for the main algorithm.
5. Generalizability: The proposed algorithms are designed to work for both ideal ($R=0$) and practical ($R>0$) RIS models, making them versatile for various system designs.

4. Simulation Results

The authors conducted extensive simulations comparing their proposed Practical-PGM against benchmarks: Ideal-PGM (using ideal model on practical hardware), BCD (Block Coordinate Descent), Random-RIS, and No-RIS.

• Impact of Resistance: When the RIS has resistance ($R=2\Omega$), the Practical-PGM significantly outperforms Ideal-PGM and BCD. The ideal models fail to account for reflection losses, leading to a substantial drop in secrecy rate when applied to practical hardware.
• Scalability: The proposed PGM algorithm exhibits linear complexity with respect to the number of RIS elements ($M$), whereas BCD exhibits quadratic complexity ($O(M^2)$). Consequently, PGM is approximately one order of magnitude faster in execution time.
• Step-Size Efficiency: The adaptive step-size PGM achieves higher secrecy rates and converges much faster than basic PGM with fixed step sizes (e.g., 0.1, 0.01).
• CPDM Effectiveness: The CPDM-based initialization allows the main PGM algorithm to converge in fewer iterations (15–20) compared to random initialization or other methods like SDR (Semidefinite Relaxation), which required 45–60 iterations.
• Robustness: As resistance increases, the secrecy rate of all schemes decreases, but the Practical-PGM degrades the slowest, proving its robustness against hardware non-idealities.

5. Significance

This paper is significant for the advancement of Physical Layer Security (PLS) in 5G/6G networks for several reasons:

• Realism: It bridges the gap between theoretical RIS models and practical hardware limitations. It proves that optimizing based on ideal assumptions is insufficient for secure deployment.
• Efficiency: By providing a low-complexity, adaptive algorithm, it makes real-time optimization of large-scale RIS arrays feasible, addressing the computational bottleneck of existing methods.
• Security Assurance: It demonstrates that accounting for hardware imperfections is crucial for maintaining secure communication links, especially in high-power transmission scenarios where the performance gap between ideal and practical models widens.

In conclusion, the study establishes a scalable and robust optimization framework for secure RIS-assisted MIMO communications, emphasizing that practical hardware modeling is essential for achieving optimal security performance in next-generation wireless networks.