Joint Sparsity and Beamforming Design for RDARS-Aided Systems

Imagine you are trying to shout a message to a group of friends in a noisy, crowded stadium. You have a megaphone (the Base Station), but the crowd is so thick that your voice gets blocked. To help, you hire a team of assistants (the RDARS system) standing on the sidelines.

This paper introduces a smart new way to organize these assistants to make your message louder and clearer for everyone. Here is the breakdown using simple analogies:

1. The Problem: The "Crowded Stadium"

In the future of wireless internet (6G), we need to send massive amounts of data. But buildings and people block signals.

Old Way (RIS): Imagine a wall of mirrors. They can reflect your voice, but they can't talk back to you. They are passive.
The New Way (RDARS): Imagine a wall of assistants who can do two things:
1. Reflect: They act like mirrors, bouncing your voice around obstacles.
2. Connect: They can grab a microphone, listen to you, and shout the message directly to your friends.

The magic of this new system is that each assistant can switch between being a mirror or a microphone instantly.

2. The Challenge: Too Many Choices

The problem is, if you have 500 assistants, deciding exactly which ones should be mirrors and which should be microphones is a nightmare. It's like trying to solve a puzzle with millions of pieces. If you try to calculate the perfect setup for every single person, your computer will overheat, and the system will be too slow to be useful.

3. The Solution: The "Sparse Array" (The Spacing Trick)

The authors came up with a clever shortcut. Instead of lining up all 500 assistants shoulder-to-shoulder (a "compact array"), they spread them out with gaps in between.

The Analogy: Imagine a row of 100 people holding hands. If you want to see the whole group, you need all 100. But if you only need to hear a specific direction, you might only need 20 people spaced far apart.
The Benefit: By spacing them out (creating a "sparse array"), the team covers a much wider area (a larger "aperture") without needing as many people to be active at once. This simplifies the math needed to decide who does what.

4. The Strategy: Finding the "Sweet Spot"

The paper asks: "How far apart should we space these assistants?"

Scenario A (One Friend): If you are only talking to one person, it doesn't matter how you space the assistants. As long as they are there, the message gets through.
Scenario B (Two Friends): If you are talking to two friends standing in different spots, spacing matters a lot!
- If they are too close together, their voices mix up (interference).
- If you space the assistants just right, you can create a "beam" that hits Friend A without hitting Friend B. It's like using a flashlight to shine on one person in a dark room without blinding the person next to them.
- The paper provides a mathematical formula to find this perfect spacing instantly, rather than guessing.

5. The Algorithm: The "Smart Manager"

For a stadium with 20 or 100 friends (users), the math gets too hard to solve perfectly. So, the authors created a "Smart Manager" algorithm (called WA).

How it works: Instead of trying to solve the whole puzzle at once, the manager takes small steps:
1. "Okay, let's fix the spacing first."
2. "Now, let's adjust the microphones."
3. "Now, let's tweak the mirrors."
4. "Repeat until everyone is happy."
The Result: This method is incredibly fast. The paper shows it runs 60% faster than previous methods while delivering the same (or better) quality of service.

6. The Bottom Line

This research proves that you don't need to use every single antenna in a massive system to get great performance. By strategically spacing out the active antennas and letting the system switch modes between "mirror" and "microphone," we can:

Send data faster.
Reduce interference between users.
Save massive amounts of computing power.

In short: It's about organizing a team of helpers so they aren't crowded together, but spread out just enough to cover the whole stadium efficiently, ensuring everyone hears you clearly without the computer getting a headache.

Here is a detailed technical summary of the paper "Joint Sparsity and Beamforming Design for RDARS-Aided Systems".

1. Problem Statement

The paper addresses the challenge of optimizing Reconfigurable Distributed Antennas and Reflecting Surfaces (RDARS) in multi-user communication systems. While RDARS offers a "selection gain" by dynamically switching elements between connection mode (active, linked via RF-over-Fiber) and reflection mode (passive), existing optimization methods for mode selection and beamforming suffer from high computational complexity (e.g., $O(N^3)$ or $O(N^{3.5})$ ), making them impractical for large-scale deployments.

Furthermore, the paper identifies a gap in characterizing the performance of the Connected Element Array (CEA). Specifically, it is unclear how to optimally determine the sparsity level (the spacing between connected elements) to maximize system performance while simplifying the mode configuration. The core problem is formulated as a Sum Rate Maximization task by jointly optimizing:

Active beamforming at the Base Station (BS).
Passive beamforming (phase shifts) at the RDARS.
The sparsity level ( $\eta$ ) of the uniform sparse array formed by the connected elements.

2. Methodology

The authors propose a hybrid approach combining theoretical analysis for special cases and an iterative algorithm for general scenarios.

A. System Model

Architecture: A BS with a Uniform Linear Array (ULA) communicates with $K$ single-antenna User Equipments (UEs) via an RDARS.
RDARS Configuration: The RDARS has $N$ total elements. $a$ elements are in connection mode (forming a uniform sparse array with spacing $\eta d$ ) and $N-a$ elements are in reflection mode.
Optimization Goal: Maximize the sum rate $R(V, \Phi, \eta)$ subject to power constraints, unit-modulus constraints for passive elements, and discrete sparsity constraints.

B. Theoretical Analysis (Special Cases)

To derive insights, the authors analyze two specific scenarios:

Single UE: The analysis reveals that the Signal-to-Noise Ratio (SNR) is independent of the sparsity level of the CEA. Any placement of connected elements yields the same maximum rate.
Two UEs: The performance depends on minimizing the Squared-Channel Correlation Coefficient (CSCC) between users to reduce Inter-User Interference (IUI). The authors derive closed-form expressions for the optimal sparsity ( $\eta_{opt}$ $η_{o pt}$ ) under three conditions:
- RDARS-UE link dominant: Optimal sparsity is derived based on the spatial frequency difference between users.
- BS-RDARS link dominant: Sparsity has negligible impact; any valid sparsity works.
- Average Phase Reference: Optimal sparsity is found by numerically minimizing the CSCC.

C. Proposed Algorithm: Weighted MMSE Alternating Optimization (WA)

For an arbitrary number of UEs ( $K > 2$ ), where closed-form solutions are intractable, the authors propose a Weighted Minimum Mean-Square Error (WMMSE)-based Alternating Optimization (AO) algorithm:

Transformation: The non-convex sum rate problem is transformed into a WMMSE minimization problem using auxiliary variables.
Iterative Steps:
- Active Beamforming: Optimized in closed-form given fixed passive beamforming and sparsity.
- Passive Beamforming: Optimized using a Power Iteration algorithm to handle unit-modulus constraints.
- Sparsity Optimization: Since the objective function is complex with respect to $\eta$ , the algorithm performs an exhaustive search over the feasible set of sparsity levels ( $\mathcal{F}$ ).
Convergence: The algorithm iterates until the fractional increase in the objective value falls below a threshold ($10^{-4}$).

3. Key Contributions

Novel Architecture: Introduction of a Uniform Sparse Array for the connected elements of RDARS to simplify mode configuration and enhance spatial degrees of freedom (DoF).
Closed-Form Insights: Derivation of optimal sparsity designs for single and two-user scenarios, revealing that sparsity primarily aids in distinguishing users spatially (reducing CSCC) rather than boosting SNR in single-user cases.
Low-Complexity Algorithm: Development of the WA algorithm, which decouples the complex joint optimization problem. Crucially, the sparsity optimization step reduces complexity from cubic ( $O(N^3)$ ) in previous methods to linear with respect to the number of elements ( $O(\lfloor \frac{N-1}{a-1} \rfloor)$ ).
Performance-Complexity Trade-off: Demonstrates that the proposed low-complexity sparsity design achieves performance comparable to high-complexity benchmark algorithms (like MM and PWM) but with significantly reduced runtime.

4. Numerical Results

Simulations were conducted with parameters such as $N=128$ , $K=20$ , and $f_0=28$ GHz.

Sum Rate Gains: For a two-user scenario, the proposed optimal sparsity design achieved 86.98% rate improvement over a compact array ( $\eta=1$ ) and 17% improvement over random sparsity at 30 dBm transmit power.
Sparsity Impact: Results show that increasing sparsity initially improves performance by enlarging the physical aperture and reducing interference. However, excessive sparsity leads to grating lobes, which worsen interference, confirming the need for optimization rather than arbitrary selection.
Complexity Reduction: Compared to benchmark algorithms (MM and PWM), the proposed WA algorithm reduced computational runtime by 62.72% and 29.75%, respectively, while maintaining similar sum rate performance.
Robustness: The algorithm demonstrated robustness under imperfect Channel State Information (CSI).

5. Significance

This paper bridges the gap between theoretical RDARS potential and practical implementation by addressing the computational bottleneck of mode selection. By proving that sparse array configurations for connected elements can significantly enhance spatial resolution and sum rate with low computational cost, the work provides a viable pathway for deploying RDARS in future 6G networks. It shifts the paradigm from complex, quantization-based codebook designs to efficient, optimization-based joint beamforming and sparsity control.