⚡ electrical engineering

Lorentzian-Constrained Holographic Beamforming Optimization in Multi-user Networks with Dynamic Metasurface Antennas

This paper proposes an Adaptive Radius Lorentzian Constrained Holography (ARLCH) algorithm for Dynamic Metasurface Antennas in multi-user MISO networks, which optimizes DMA weights to minimize total transmit power while significantly outperforming existing benchmarks, particularly as the number of users increases.

Original authors: Askin Altinoklu, Leila Musavian

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Askin Altinoklu, Leila Musavian

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to shout a message to a group of friends scattered in a park, but you can't use a megaphone. Instead, you have a giant, high-tech wall made of thousands of tiny, squishy rubber tiles. Each tile can wiggle to change how it bounces your voice, but there's a catch: these tiles are "Lorentzian." That's a fancy way of saying they are stubborn. If you tell a tile to wiggle faster (change the phase), it must also get softer or harder (change the amplitude). You can't just pick a wiggle speed and a volume independently; they are tied together like a pair of dancing partners who refuse to let go.

This is the challenge facing the engineers in this paper. They are working with Dynamic Metasurface Antennas (DMAs), which are like these smart, squishy walls. These walls are amazing because they are cheap and use very little power compared to the massive, expensive computer systems (called "Fully Digital" systems) usually used to beam signals to many people at once. But because the tiles are so stubborn, figuring out exactly how to wiggle them to hit everyone in the park perfectly is a nightmare.

The Problem: The Stubborn Tiles

In the past, researchers tried to solve this by pretending the tiles were perfect (ignoring their stubbornness) and then just "forcing" them to fit their rules later. It's like drawing a perfect circle on a piece of paper and then trying to squash it into a square frame. You get a shape that's close, but it's not quite right, and you waste energy trying to make it work.

The paper argues that the old ways of doing this—specifically a method called LCUSH (which is the most common method used in other studies)—are like trying to force that circle into the square. They work, but they leave a lot of power on the table. The authors show that simply picking a different "center point" for how you squash the shape matters a lot. In fact, they found that a method called LCEH (which uses a different center point) works significantly better than the old standard, LCUSH.

The Big Discovery: The Stretchy Trampoline

But the authors didn't stop there. They asked a bold question: What if we don't just squash the shape into a fixed square frame? What if we let the frame itself stretch?

They introduced a new trick called ARLCH (Adaptive Radius Lorentzian-Constrained Holography). Imagine the "Lorentzian circle" (the rule the tiles must follow) isn't a rigid ring, but a stretchy trampoline. Instead of forcing the perfect signal to land on a fixed ring, ARLCH stretches the trampoline to meet the signal, finds the best landing spot, and then snaps it back to the correct size.

This "stretching" gives the system an extra degree of freedom. It's like having a trampoline that can change its tension to catch a falling acrobat perfectly, rather than just hoping the acrobat lands on a rigid hoop.

What the Numbers Say

The authors ran thousands of computer simulations (they didn't build a physical wall yet, but they modeled it very carefully) to see if this new trampoline trick actually worked. Here is what they found:

Power Savings: When they tested this with a single user, the new ARLCH method used significantly less power than the old methods. As the angle of the user moved further away from the center, the savings got even bigger.
Crowded Parks: When they added more users (up to 8 people in the simulation), the gap between the old methods and the new one grew. For a crowd of 8 users, ARLCH used about 29.1% less power than the next-best method (LCEH).
The "Old Guard": The simulations showed that the most common method used in other papers (LCUSH) was actually quite inefficient. In fact, LCEH (the middle-ground method) was already beating LCUSH by about 17% for a crowd of 8 users. But ARLCH beat even LCEH.
Denser Tiles: They also tested what happens if you pack the tiles closer together (down to $\lambda/6$ spacing). Even with these tricky, crowded setups, ARLCH kept winning, saving about 20% more power than the other methods.

What They Didn't Do (And What They Ruled Out)

It's important to know what this paper didn't do.

No Real Hardware Yet: The authors explicitly state they haven't built this on physical metal and silicon yet. They haven't tested it with real-world "glitches" like broken wires or messy radio waves. They simulated it on a computer.
No "Magic" Fixes: They didn't claim that the stubborn nature of the tiles (the Lorentzian constraint) could be ignored. They didn't say, "Just use a fully digital system and forget the tiles." They accepted the tiles' rules and found a smarter way to dance with them.
No Uplink or Bad Data: They didn't test what happens if the system doesn't know exactly where the users are (imperfect data) or if the users are sending messages to the wall (uplink). They only looked at the wall sending messages to the users (downlink) with perfect knowledge of where everyone is.

The Bottom Line

The paper suggests that by treating the "rules" of these smart antennas as flexible rather than rigid, we can save a massive amount of energy. The ARLCH method is like a master conductor who doesn't just tell the orchestra to play a note, but adjusts the entire room's acoustics to make sure the note sounds perfect with the least amount of effort.

In these simulations, this approach consistently reduced power consumption by over 20% compared to the best existing methods, and the more people you try to talk to, the bigger the savings get. It's a promising step toward making future 6G networks that are not only super fast but also incredibly energy-efficient.

Technical Summary: Lorentzian-Constrained Holographic Beamforming Optimization in Multi-user Networks with Dynamic Metasurface Antennas

Problem Statement
Dynamic Metasurface Antennas (DMAs) offer a promising alternative to fully digital (FD) architectures for 6G and beyond, utilizing low-cost, reconfigurable metasurfaces to achieve hybrid beamforming. However, DMA architectures face a fundamental trade-off: while they significantly reduce the number of required Radio-Frequency (RF) chains and hardware complexity, they impose strict physical constraints on beamforming weights. Specifically, DMA elements behave as Lorentzian resonators, where amplitude and phase are intrinsically coupled rather than independently tunable. This "Lorentzian constraint" restricts available weights to a specific curve (the Lorentzian circle) in the complex plane, unlike the constant-modulus complex circle assumed in conventional arrays.

In multi-user Multiple-Input Single-Output (MISO) networks, this constraint complicates resource allocation. The core problem addressed is minimizing total transmit power while satisfying Signal-to-Interference-plus-Noise Ratio (SINR) constraints for multiple users. Existing literature often employs specific projection methods to map ideal, unconstrained beamforming weights onto the Lorentzian circle, but these methods are typically studied in isolation. There is a lack of systematic comparison between different mapping strategies (e.g., varying the mapping center) and a need for algorithms that can better navigate the trade-off between reduced RF chains and beamforming performance, particularly as the number of users increases.

Methodology
The authors propose a unified framework for optimizing DMA weights and digital precoders using an Alternating Optimization (AO) approach combined with Semi-Definite Relaxation (SDR). The solution process involves two main sub-stages:

Digital Precoder Optimization: With DMA weights fixed, the problem is reformulated as a convex SDR problem to optimize digital precoding vectors.
DMA Weight Optimization: With digital precoders fixed, the problem is again relaxed via SDR to find optimal unconstrained weights. These ideal weights are then projected onto the physically feasible Lorentzian circle.

The paper introduces and formalizes the Generalized Method of Lorentzian-Constrained Holography (GMLCH). This framework standardizes the projection process by parameterizing the mapping center $(x_c, y_c)$ on the Lorentzian circle. It allows for the systematic evaluation of three existing mapping strategies:

LCPH (Lorentzian-Constrained Phase Hologram): Mapping center at $(0, 0)$ .
LCEH (Lorentzian-Constrained Euclidean Hologram): Mapping center at $(0, 0.5)$ .
LCUSH (Lorentzian-Constrained Unitary Shift Hologram): Mapping center at $(0, 1.0)$ .

Building upon GMLCH, the authors propose a novel algorithm: Adaptive Radius Lorentzian Constrained Holography (ARLCH). Unlike previous methods that strictly adhere to a unitary Lorentzian circle diameter, ARLCH introduces an additional degree of freedom by dynamically adjusting the diameter (radius) of the Lorentzian circle during the optimization process. This allows the algorithm to find an optimal mapping center and diameter that minimize the distance between the ideal unconstrained weights and the physically constrained weights before normalizing back to the unitary condition.

Key Contributions

Unified Optimization Framework: The paper develops a novel SDP and AO-based algorithm that integrates GMLCH, enabling the joint optimization of digital precoders and DMA weights within a single framework. This allows for a direct, systematic comparison of different Lorentzian mapping strategies (LCPH, LCEH, LCUSH) under Quality of Service (QoS) guarantees, a gap not previously addressed in literature.
Performance Analysis of Mapping Centers: The study demonstrates that the choice of mapping center significantly impacts performance. Results indicate that LCEH outperforms LCUSH and LCPH, particularly in scenarios with higher user counts, challenging the common reliance on LCUSH in prior works.
Novel ARLCH Algorithm: The authors propose ARLCH, which leverages the relaxation of the Lorentzian circle diameter to achieve enhanced beamforming performance. This method provides an additional degree of freedom, allowing for dynamic adjustment of the mapping center to better align ideal weights with physical constraints.
Scalability and Efficiency: The paper provides extensive numerical results showing that ARLCH significantly reduces transmit power compared to benchmarks, with the performance gain increasing as the number of users grows.

Numerical Results
Simulations were conducted in a near-field multi-user MISO downlink scenario with a 16x16 Uniform Planar Array (UPA) and 28 GHz frequency.

Single-User Scenario: ARLCH reduced transmit power significantly compared to LCUSH and a baseline ADMM-SCA method. For a user at a wide angle ( $80^\circ$ ), ARLCH required approximately half the transmit power of LCUSH to achieve the same SINR.
Multi-User Scenario: As the number of users ( $K$ ) increased, the performance gap between DMA-based and FD-based architectures widened due to degrees of freedom limitations. However, ARLCH consistently narrowed this gap. Compared to the best-performing GMLCH variant (LCEH), ARLCH achieved a transmit power reduction of up to 29.1% for $K=8$ .
Mapping Strategy Comparison: LCEH was found to outperform LCUSH (the most common method in literature) by approximately 17% for $K=8$ . LCPH performed the worst due to its tendency to clip elements in the lower half-plane.
Dense Spacing: The proposed methods remained robust under denser antenna element spacings ( $\lambda/6$ ), with ARLCH maintaining a ~20% power advantage over other GMLCH variants.

Significance and Claims
The paper claims that the primary significance of this work lies in addressing the structural mismatch between ideal beamforming requirements and the physical tunability of DMA elements. By formalizing the projection process through GMLCH and introducing the adaptive radius approach (ARLCH), the authors demonstrate that the performance gap between low-complexity DMA architectures and high-complexity fully digital systems can be significantly reduced.

The authors modestly note that while their optimization approach is computationally more intensive than some baselines due to the SDP formulation, it offers superior flexibility and performance. They emphasize that the results highlight the critical role of selecting the appropriate mapping center and the potential of adaptive radius optimization. The paper concludes that ARLCH provides a scalable solution for future multi-user networks, though it acknowledges that future work must address practical hardware impairments, mutual coupling, and imperfect Channel State Information (CSI) before physical implementation.

The Problem: The Stubborn Tiles

The Big Discovery: The Stretchy Trampoline

What the Numbers Say

What They Didn't Do (And What They Ruled Out)

The Bottom Line

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