⚡ electrical engineering

Gridless Full-Space DOA Estimation for STAR-RIS-Assisted Wireless Systems

This paper proposes a gridless full-space direction-of-arrival (DOA) estimation framework for STAR-RIS-assisted systems that leverages a structured low-rank recovery approach via proximal gradient descent to achieve high-accuracy angle retrieval without angular discretization.

Original authors: Ziming Liu, Tao Chen, Muran Guo, Francesco Verde

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ziming Liu, Tao Chen, Muran Guo, Francesco Verde

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 figure out where people are shouting from in a giant, circular room. Some people are standing on the same side of the room as you (the "Reflection" side), and others are on the opposite side, behind a wall (the "Transmission" side).

In the past, smart walls (called RIS) could only "listen" to the people on their own side. To hear the people on the other side, you'd need a second wall or a second set of ears, which is expensive and complicated.

This paper introduces a new, super-smart wall called a STAR-RIS. Think of it as a magical window that can both reflect sound (like a mirror) and transmit sound (like a clear window) at the exact same time. This allows a single listener (a base station with just one "ear") to hear everyone in the entire 360-degree room at once.

However, there's a catch: because the wall is doing two things at once, the sounds get mixed up in a complex way. If you try to listen to the "mirror" side and ignore the "window" side, the ignored sounds act like confusing static, making it impossible to pinpoint exactly where the shouters are, no matter how loud they get.

The Problem: The "Pixelated" Map

Most current methods for finding where sounds come from use a "grid." Imagine trying to find a lost cat in a forest by checking every single square foot of a map. If the cat is standing between the squares, your method gets confused and says the cat is in the wrong spot. This is called "grid mismatch." It's like trying to draw a smooth circle using only square Lego bricks; it will always look jagged.

The Solution: A Continuous Flow

The authors propose a new way to listen that doesn't use a grid at all. They realized that the way the STAR-RIS mixes the sounds creates a special mathematical pattern (called a Finite Rate of Innovation or FRI structure).

The Analogy of the "Magic Filter":
Imagine the mixed-up sounds are a complex song. The authors found that if you apply a specific "magic filter" (an annihilating polynomial) to this song, the parts that don't belong cancel each other out perfectly, leaving only the true locations of the shouters.

Instead of checking a grid of squares, their method solves a puzzle to find the exact, smooth location of the shouters. It's like using a high-resolution camera instead of a pixelated one.

Two Ways to Tune the Wall

The paper tackles two different ways this smart wall might be built:

The Uniform Wall (Scenario 1): Imagine every tiny piece of the wall is identical. They all reflect and transmit sound in the exact same ratio. The authors created a streamlined algorithm for this. It's like a well-oiled machine that works perfectly when everything is uniform.
The Custom Wall (Scenario 2): Imagine each tiny piece of the wall is different. Some reflect more, some transmit more, depending on what's needed. This is more realistic but messier. The authors created a more complex "paired" algorithm for this. It's like having a conductor who can manage a choir where every singer has a slightly different voice, ensuring everyone is still heard clearly.

How They Do It (The "Proximal Gradient" Dance)

To solve the math puzzle, they use a method called Proximal Gradient Descent with Alternating Projections.

The Dance: Imagine you are trying to find the lowest point in a foggy valley. You take a step in the direction that feels like "down" (Gradient Descent).
The Correction: But you also have a rule: you must stay on a specific path (the low-rank structure). If your step takes you off the path, you are gently pushed back onto it (Alternating Projections).
The Result: You repeat this dance until you find the perfect spot. This allows them to recover the exact angles even with very few measurements and in noisy conditions.

The Results

The paper tested this against older methods (like the "grid" methods and standard scanning techniques).

Accuracy: Their method found the angles much more precisely, often getting within less than one degree of the true location.
No "High-SNR Floor": Old methods hit a "ceiling" where making the signal louder didn't help because the grid mismatch was the problem. The new method keeps getting better as the signal gets clearer.
Speed: While the math is complex, the computer time required is comparable to, or only slightly higher than, the standard methods, making it practical for real use.

The Bottom Line

This paper proves that by treating the STAR-RIS as a single, unified system that couples reflection and transmission, we can pinpoint the location of users all around a building with incredible accuracy. It removes the need for expensive multiple receivers and avoids the "jagged" errors of grid-based systems, offering a smooth, continuous, and highly accurate way to "see" the world using radio waves.

Problem Statement

The paper addresses the challenge of accurate Direction-of-Arrival (DOA) estimation in Integrated Sensing and Communication (ISAC) systems utilizing Simultaneously Transmitting and Reflecting Reconfigurable Intelligent Surfaces (STAR-RIS). Unlike conventional reflection-only RISs limited to a single hemisphere (0°–180°), STAR-RISs enable full-space (0°–360°) coverage by simultaneously reflecting and transmitting signals.

The core difficulty lies in the coupled observation model. In STAR-RIS systems, the received signal is a superposition of contributions from both the reflection space (RS) and transmission space (TS). Conventional DOA estimators (e.g., MUSIC, ESPRIT) or sparse reconstruction methods (e.g., OMP, SBL) typically process these subspaces independently or treat the opposite subspace's contribution as unmodeled interference. This leads to a structured bias that cannot be eliminated by increasing the Signal-to-Noise Ratio (SNR) or refining the angular search grid. Furthermore, the system operates with a single RF sensing chain, which limits the availability of repeated stationary snapshots required for covariance-based methods, as the STAR-RIS configuration changes over time slots.

Methodology

The authors propose a gridless approach that exploits the Finite Rate of Innovation (FRI) structure inherent in the STAR-RIS signal model.

Signal Modeling:
- The received signal is modeled as a superposition of complex exponentials indexed by the unknown DOAs.
- The authors derive a deterministic mapping between the reflection and transmission coefficients of the STAR-RIS elements, showing that the aggregated output admits a structured low-rank representation.
- Two scenarios are considered:
  - Element-wise Uniform: All elements share identical energy-splitting parameters.
  - Element-wise Nonuniform: Energy-splitting parameters vary across elements (a more general and practical case).
Algebraic Structure (Annihilating Filters):
- The problem is cast as recovering a latent FRI vector from compressed measurements.
- The signal vector satisfies a set of linear equations defined by an annihilating polynomial (or annihilating filter). The roots of this polynomial correspond to the DOAs.
- This structure allows the construction of a block-Hankel matrix where the rank is determined by the number of sources ( $K$ ).
Recovery Algorithms:
- Algorithm 1 (Uniform Case): Uses a Proximal Gradient Descent (PGD) framework with alternating projections. It enforces a rank- $K$ constraint on a stacked block-Hankel matrix derived from a single unified signal model.
- Algorithm 2 (Nonuniform Case): Introduces a paired lifting strategy. Instead of a single unified model, it constructs a horizontally concatenated Hankel matrix that separately models the reflection and transmission subspaces while enforcing a joint low-rank constraint. This preserves the algebraic structure even when energy-splitting coefficients are non-uniform.
- Both algorithms recover the latent signal vector and then extract DOAs via root-finding on the annihilating polynomial, selecting roots closest to the unit circle.
Performance Bounds:
- A Ziv-Zakai Bound (ZZB) is derived for the coupled full-space model. This bound explicitly accounts for the cross-subspace coupling term induced by simultaneous transmission and reflection, providing a theoretical benchmark for the minimum mean square error (MSE) across the full SNR range.

Key Contributions

Unified Full-Space Sensing Model: The paper establishes a unified model that rigorously accounts for the coupling between reflection and transmission responses, covering both uniform and nonuniform energy-splitting configurations.
Gridless FRI-Based Estimation: It demonstrates that the coupled STAR-RIS model admits an FRI structure, enabling gridless angle estimation without angular discretization or covariance estimation.
Structured Recovery Framework: The authors develop PGD-based solvers with alternating projections onto block-Hankel sets. They provide sufficient local stability conditions for the step-size and lifting parameters to ensure convergence.
Extension to Nonuniform Configurations: A novel block-structured lifting strategy is introduced for nonuniform settings, resolving the high-SNR bias floor that affects methods ignoring the coupled full-space structure.
Theoretical Benchmarking: The derivation of a coupled full-space ZZB provides a performance limit that captures the benefits of cross-subspace information coupling.

Numerical Results

Simulations were conducted with $N=16$ elements and $K=4$ users (2 in RS, 2 in TS).

Accuracy: The proposed methods (M1 for uniform, M2 for nonuniform) consistently outperform grid-based baselines (FFT, OMP, SBL). They achieve sub-degree accuracy within $\pm 60^\circ$ of boresight.
High-SNR Performance:
- In the uniform scenario, both proposed methods approach the ZZB and show no error floor.
- In the nonuniform scenario, M2 (paired lifting) significantly outperforms M1 and baselines. While M1 degrades due to model mismatch, M2 maintains accuracy, though a mild high-SNR error floor persists due to the approximate nature of the paired-Hankel structure in nonuniform settings.
- Baselines (FFT/OMP/SBL) exhibit a persistent high-SNR error floor even at 30 dB SNR because they treat the opposite subspace as interference, a bias that grid refinement cannot remove.
Robustness: The methods maintain high success probability (near 1.0) for angles within $[-60^\circ, 60^\circ]$ (semi-space representation). Performance degrades near grazing angles (coplanar case), consistent with classical array limitations.
Computational Cost: The proposed methods have comparable or slightly higher computational costs than grid-based methods but remain efficient (converging within ~45 iterations). M2 is more computationally demanding than M1 due to the larger lifted matrix size.
Aperture Dependence: The proposed FRI/Hankel-based methods show a stronger dependence on the number of elements ( $N$ ) than baselines. Larger apertures significantly improve their performance by strengthening the low-rank constraint, whereas baselines show only marginal improvements once $N > K$ .

Significance and Claims

The paper claims that the proposed framework is particularly significant for low-cost STAR-RIS sensing architectures utilizing a single RF chain, where traditional covariance-based methods fail due to a lack of stationary snapshots.

Overcoming Bias: The primary contribution is the elimination of the structured bias caused by treating the coupled full-space observation as two independent subspaces. By explicitly modeling the reflection-transmission coupling, the proposed methods avoid the high-SNR error floors inherent in conventional approaches.
Practicality: The work addresses both idealized (uniform) and practical (nonuniform) hardware configurations, offering a unified solution that adapts to the specific energy-splitting design of the STAR-RIS.
Limitations: The authors modestly acknowledge two limitations:
1. Performance degradation near grazing angles, which is a fundamental limitation of array-based DOA estimation.
2. A residual error floor in highly nonuniform scenarios, attributed to the approximate nature of the paired-Hankel structure rather than noise.
Future Work: The paper notes that the current framework assumes a single-path model and slot-invariant gains. Extending the approach to multipath environments and time-varying channels is identified as a natural direction for future research.