⚡ electrical engineering

Deep-Unfolded Wideband ISAC Beamforming for DMA Under Frequency-Selective Lorentzian Model

This paper proposes a deep-unfolded alternating optimization framework for wideband ISAC systems using dynamic metasurface antennas (DMAs) under a frequency-selective Lorentzian model, which significantly improves communication and radar performance while accelerating convergence compared to traditional frequency-flat approximations and standard optimization methods.

Original authors: Abdolrasoul Sakhaei Gharagezlou, Pouya Mobaraki, Mehdi Monemi, Nhan T. Nguyen, Mehdi Rasti, Samad Ali, Matti Latva-aho

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Abdolrasoul Sakhaei Gharagezlou, Pouya Mobaraki, Mehdi Monemi, Nhan T. Nguyen, Mehdi Rasti, Samad Ali, Matti Latva-aho

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 direct a massive, high-tech orchestra (the antenna) to play two different songs at the exact same time: one song for your friends (communication data) and one song for a sonar system to find a hidden object (radar sensing). This is the challenge of ISAC (Integrated Sensing and Communications).

In this paper, the researchers are working with a special type of antenna called a DMA (Dynamic Metasurface Antenna). Think of a DMA not as a single giant speaker, but as a wall covered in thousands of tiny, tunable "flaps" or "valves." Each flap can be adjusted to change how sound (or radio waves) bounces off it.

Here is the breakdown of their discovery, using simple analogies:

1. The Problem: The "One-Size-Fits-All" Mistake

For a long time, engineers designed these antennas assuming that every "flap" on the wall reacts the same way to every note in the song. They treated the antenna like a flat, rigid mirror that reflects all frequencies equally.

The authors say this is a big mistake when dealing with wideband signals (which carry a lot of data, like 5G or future 6G).

The Analogy: Imagine trying to tune a piano by assuming every key is the same. If you press a low note, the string vibrates slowly; if you press a high note, it vibrates fast. If you treat them all as the same, your music will sound out of tune.
The Reality: In wideband systems, the "flaps" on the antenna actually behave differently depending on the frequency (the "note"). They have a specific "resonance" (like a swing that goes higher when pushed at the right time). The old models ignored this, leading to a lot of "noise" and errors.

2. The Solution: The "Smart Tuning" Model

The researchers proposed a new way to describe these flaps using a Frequency-Selective Lorentzian Model.

The Analogy: Instead of treating the flaps as rigid mirrors, they modeled them like tuning forks. A tuning fork has a specific shape and weight that makes it vibrate best at a certain pitch. If you hit it with a different pitch, it doesn't respond the same way.
The Result: By using this "tuning fork" model, they can adjust the antenna's "resonance frequency" and "damping" (how quickly it stops vibrating) to perfectly match the wide range of frequencies being used. This allows them to control both the volume (magnitude) and the timing (phase) of the signal much more accurately.

3. The Challenge: The "Tightrope Walk"

The goal is to balance two competing needs:

Talking to users: Making sure the phone signal is strong and clear.
Finding the target: Making sure the radar "echo" is loud enough to detect an object.

This is like trying to walk a tightrope while juggling. If you focus too much on talking, the radar gets weak. If you focus too much on radar, the phone signal drops. The variables they are juggling are:

Digital Beamforming: The software instructions telling the antenna where to point.
Resonance Frequencies: The "pitch" of the antenna flaps.
Damping Factors: How "bouncy" the flaps are.

These three things are all tangled up together. Changing one affects the others, making the math incredibly difficult to solve.

4. The Method: "Deep Unfolding" (The Smart Coach)

To solve this math puzzle, they first created a standard step-by-step method called Projected Gradient Ascent (PGA).

The Analogy: Imagine you are blindfolded on a hill and want to find the highest peak. You take a step, feel the slope, and take another step. This is the standard method. It works, but it's slow, and you have to guess how big of a step to take. If you take a step that's too big, you might fall off the cliff; too small, and you'll never reach the top.

To fix the slowness, they used Deep Unfolding.

The Analogy: Instead of just walking blindly, they built a smart coach (an AI) that watches the blindfolded walker. The coach learns from thousands of practice runs exactly how big of a step to take at every moment to reach the top as fast as possible.
How it works: They took the standard math steps and turned them into layers of a neural network. They "trained" this network to learn the perfect step sizes.
The Benefit: The "smart coach" method reaches the best solution 20 times faster than the standard method and gets a 7% better result (a stronger signal) without needing to guess the step sizes anymore.

5. The Results: Why It Matters

The paper ran simulations to test their ideas:

Better Accuracy: Using the new "tuning fork" model (Frequency-Selective) instead of the old "flat mirror" model improved performance by about 20%. The old model was so inaccurate in wideband scenarios that it caused huge errors in signal timing and strength.
Faster Speed: The "smart coach" (Deep-Unfolded PGA) found the best solution much faster than the traditional "blind walking" methods.
Better Balance: They successfully balanced the need for high-speed internet and accurate radar sensing simultaneously, outperforming older methods like "Maximum Ratio Transmission" (which just shouts as loud as possible) or "Zero-Forcing" (which tries to cancel out interference but often fails in complex environments).

Summary

The paper says: "We found that the old way of designing these special antennas was too simple for modern, wideband networks. By treating the antenna parts like complex tuning forks instead of flat mirrors, and by using a smart AI coach to speed up the math, we can make these antennas significantly better at doing two jobs at once: talking to your phone and finding objects with radar."

Technical Summary: Deep-Unfolded Wideband ISAC Beamforming for DMA Under Frequency-Selective Lorentzian Model

1. Problem Statement

The paper addresses the design of beamforming for Integrated Sensing and Communications (ISAC) systems utilizing Dynamic Metasurface Antennas (DMAs) in wideband and ultra-wideband scenarios. A critical limitation identified in existing literature is the reliance on the frequency-flat Lorentzian model for DMA elements. While accurate for narrowband systems, this approximation fails to capture the frequency-dependent electromagnetic behavior of DMA elements in wideband contexts. This discrepancy leads to significant magnitude errors (several dB) and phase deviations (tens of degrees), resulting in suboptimal performance.

The core challenge is to jointly optimize the digital beamforming matrices, resonance frequencies, and damping factors of the DMA elements to balance the aggregate Signal-to-Interference-plus-Nois Ratio (SINR) of communication users and the Signal-to-Noise Ratio (SNR) of a radar target. This optimization is complicated by:

The strong coupling between decision variables.
The non-convex nature of the objective function under the frequency-selective Lorentzian model.
The computational intensity and sensitivity to hyperparameters (specifically step sizes) of conventional iterative optimization methods, which often fail to converge within the coherence interval of wideband channels.

2. Methodology

The authors propose a two-stage solution approach:

A. Alternating Projected Gradient Ascent (PGA) Framework
First, the authors develop a model-based optimization framework using alternating optimization. The algorithm iteratively updates three blocks of variables while keeping the others fixed:

Resonance Frequencies ( $f^R$ ): Updated using closed-form gradients derived from the objective function.
Damping Factors ( $\kappa$ ): Similarly updated using derived closed-form gradients.
Digital Beamforming Matrices ( $W_m$ ): Updated via a projected gradient ascent step, followed by a normalization projection to satisfy the total transmit power budget.

The authors derive explicit closed-form gradients for the objective function (a weighted sum of user SINR and radar SNR, plus a penalty term for Quality of Service constraints) with respect to all optimization variables under the frequency-selective Lorentzian response model.

B. Deep-Unfolded PGA Architecture
To overcome the slow convergence and sensitivity to manually tuned step sizes inherent in the conventional PGA, the authors employ deep unfolding.

Architecture: The iterative PGA algorithm is unrolled into a fixed number of trainable layers. Each layer corresponds to one PGA iteration, performing gradient evaluation, variable updates, and projection operations.
Learnable Parameters: The step sizes ( $\mu_{f^R}, \mu_{\kappa}, \mu_{W}$ ) for each variable update are treated as trainable hyperparameters learned from data.
Training: The network is trained in an unsupervised manner to maximize the objective function directly, without requiring labeled ground-truth beamforming solutions. The loss function is defined as the negative of the objective function evaluated at the final layer.
Interpretability: Unlike black-box deep learning, this architecture preserves the physical interpretability of the model-based optimization rules (gradients and projections) while leveraging data to adapt the update trajectory.

3. Key Contributions

Frequency-Selective Modeling: The paper demonstrates that using the frequency-flat approximation in wideband DMA systems introduces significant performance degradation. It proposes the use of the frequency-selective Lorentzian model to accurately characterize the magnitude and phase variations of DMA elements across the bandwidth.
Closed-Form Gradient Derivation: The authors derive closed-form gradients for the non-convex objective function with respect to resonance frequencies, damping factors, and digital beamforming vectors, enabling efficient PGA updates.
Deep-Unfolded Optimization: A novel deep-unfolded PGA framework is introduced. It learns optimal step sizes from data to accelerate convergence and improve the final objective value, avoiding the computational overhead of online line-search methods.
Joint Optimization: The work provides a comprehensive solution for jointly optimizing hybrid beamforming parameters (digital and analog/dynamic metasurface parameters) in a near-field, wideband ISAC setting.

4. Simulation Results

Extensive simulations were conducted using a DMA-based BS with $N_d=5$ microstrips and $N_e=10$ elements per microstrip, operating at 10 GHz with a 1 GHz bandwidth.

Model Accuracy: The frequency-selective Lorentzian model outperformed the frequency-flat approximation by approximately 20% in the objective value, confirming the necessity of accurate wideband modeling.
Convergence Speed: The deep-unfolded PGA method achieved convergence approximately 20 times faster than the classical PGA and 5 times faster than PGA with line-search. It also converged significantly faster than Maximum Ratio Transmission (MRT) and Zero-Forcing (ZF) precoding benchmarks.
Performance Gain: Compared to the classical PGA benchmark, the deep-unfolded method improved the objective value by up to 7%. It also provided roughly 3-fold and 3.75-fold gains over MRT and ZF precoding, respectively.
Scalability: The performance gap between the proposed method and benchmarks widened as the number of DMA elements increased, highlighting the method's effectiveness in larger array configurations.
Complexity: The proposed method avoids the per-iteration overhead of line-search procedures required by conventional PGA, offering a more computationally efficient inference process.

5. Significance and Claims

The paper claims that the proposed approach offers a robust solution for next-generation wideband ISAC systems by bridging the gap between physical accuracy and computational efficiency.

Physical Fidelity: By adopting the frequency-selective Lorentzian model, the system avoids the substantial phase and magnitude mismatches that degrade wideband performance, enabling more reliable communications and resilient sensing.
Algorithmic Efficiency: The deep-unfolded framework addresses the "slow convergence" and "hyperparameter tuning" bottlenecks of traditional iterative solvers. It achieves fast convergence within a fixed number of iterations without sacrificing the interpretability of model-based optimization.
Practical Deployment: The results suggest that this approach facilitates energy- and cost-efficient deployment of large-scale DMA arrays in wideband ISAC networks, providing a viable alternative to fully digital architectures while maintaining high performance.

The authors conclude that combining accurate frequency-selective modeling with data-driven deep unfolding is essential for realizing high-performance, scalable DMA-based wideband ISAC systems.