⚡ electrical engineering

Decoupled Delay-Doppler and Angle Estimation in BD-RIS Sensing via Nested Tucker Decomposition

This paper proposes the Nested Tensor Factorization and Estimation (NTFE) algorithm, which leverages nested Tucker decomposition to decouple delay-Doppler and angle estimation for single-target localization in group-connected BD-RIS-assisted monostatic networks, demonstrating superior performance over state-of-the-art benchmarks.

Original authors: Kenneth Benício, André L. F. de Almeida, Fazal-E-Asim, Bruno Sokal, Behrooz Makki, Gabor Fodor, A. Lee Swindlehurst

Published 2026-05-28

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CC BY 4.0

Original authors: Kenneth Benício, André L. F. de Almeida, Fazal-E-Asim, Bruno Sokal, Behrooz Makki, Gabor Fodor, A. Lee Swindlehurst

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 find a lost hiker in a dense, foggy forest. You have a powerful flashlight (the transmitter) and a team of helpers (the RIS) standing on a hill. Your goal is to figure out exactly where the hiker is, how fast they are moving, and which direction they are facing, all by listening to the echo of your flashlight beam bouncing off them.

This paper presents a new, smarter way to process those echoes using a technology called a BD-RIS (Beyond-Diagonal Reconfigurable Intelligent Surface). Think of a BD-RIS not as a single giant mirror, but as a wall made of many small, smart tiles. These tiles are grouped together; tiles in the same group can talk to each other and coordinate their reflections, but they don't need to talk to tiles in other groups. This saves energy and hardware complexity while still being very flexible.

Here is how the authors' new method, called NTFE, works, explained through simple analogies:

1. The Problem: A Messy Pile of Data

When the flashlight beam hits the hiker and bounces back, it carries a lot of mixed-up information:

When it came back (Delay).
How fast the hiker is moving (Doppler).
Where the hiker is located (Angle).

Traditional methods try to untangle this mess by looking at the data as one giant, confusing spreadsheet. It's like trying to find a specific needle in a haystack by looking at the whole pile at once. It's slow, and if the data is noisy (like wind blowing leaves), you might get the wrong answer.

2. The Solution: The "Nested Box" Strategy

The authors propose a method called Nested Tucker Decomposition. Imagine the data isn't a messy pile, but a set of Russian nesting dolls (or boxes inside boxes).

The Big Box (The Signal): The raw echo data is a 3D block of information.
The Middle Layer (The Groups): The authors realize the "smart tiles" are grouped. They use this to split the big box into smaller, manageable chunks.
The Inner Layer (The Separation): This is the magic trick. Instead of trying to solve for time, speed, and direction all at once, they separate them into different "compartments."
- One compartment holds the Time/Speed info.
- Another compartment holds the Direction info.

By separating these "compartments," the algorithm can solve for the direction without getting confused by the speed, and vice versa. It's like sorting a mixed bag of red and blue marbles into two separate jars before counting them, rather than trying to count them while they are still mixed.

3. The Two-Step Dance

The algorithm performs a "two-stage dance" to find the hiker:

Step 1: The Rough Sketch (Factorization): First, it uses a mathematical technique (Alternating Least Squares) to get a rough idea of the "shape" of the signal. It doesn't need to know the exact speed or angle yet; it just needs to separate the "Time/Speed" box from the "Direction" box.
Step 2: The Fine-Tuning (Subspace & Closed-Form): Once the boxes are separated, the algorithm uses precise mathematical tools (called ESPRIT) to look inside the "Time/Speed" box to get the exact speed and distance. Then, it looks inside the "Direction" box to get the exact angle. Finally, it calculates how strong the signal is (the complex gain) using a simple division step.

4. Why It's Better

The paper tested this new method against other top-tier methods (like a "Maximum Likelihood" search, which is like trying every possible combination of speed and angle until one fits).

The Result: The new method (NTFE) was significantly more accurate. In their simulations, it reduced errors by about 10 decibels compared to the standard methods.
The Analogy: If the old methods were like trying to find a specific song on a radio by turning the dial randomly, the new method is like having a digital tuner that instantly locks onto the right frequency because it knows exactly how the radio station is structured.

Summary

In short, this paper introduces a clever way to organize messy radar data from a smart surface. By treating the data as a set of nested, separable boxes rather than a jumbled mess, the system can pinpoint a target's location, speed, and direction much faster and more accurately than previous techniques. It proves that understanding the "group structure" of the hardware allows for a much smarter way to listen to the echoes.

Technical Summary: Decoupled Delay-Doppler and Angle Estimation in BD-RIS Sensing via Nested Tucker Decomposition

Problem Statement
The paper addresses the challenge of single-target localization in a monostatic sensing network assisted by a group-connected Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS). Unlike traditional passive RISs that utilize diagonal phase-shift matrices, BD-RIS architectures employ inter-element coupling to realize non-diagonal phase-shift matrices, offering additional degrees of freedom. However, this introduces complexity in channel estimation and parameter extraction. The specific problem is to estimate the target's physical parameters—delay ( $\tau$ ), Doppler shift ( $\nu$ ), and angles of arrival/departure ( $\phi, \theta$ )—in a scenario where the direct line-of-sight path is blocked, and sensing relies on the echo link through the BD-RIS. The system involves a multi-antenna sensing transmitter (ST) and a BD-RIS with $N$ elements partitioned into $K$ groups, where intra-group coupling is preserved but full interconnection is avoided to balance hardware complexity and identifiability.

Methodology
The authors propose the Nested Tucker Factorization and Estimation (NTFE) algorithm, which models the received signal as a 3rd-order nested Tucker tensor. This approach explicitly exploits the block-wise structure induced by the group-connected BD-RIS architecture.

Signal Modeling: The received echo signal is formulated as a 3rd-order nested Tucker tensor decomposition. The model separates the delay-Doppler domain from the angle domain. The received data tensor $\mathbf{Y}$ is expressed as:
$\mathbf{Y} = \mathbf{I} \times_1 \mathbf{G} \times_2 [\mathbf{F}]_{(1)} \times_3 \mathbf{S} \times_4 \mathbf{p}'$
where $\mathbf{G}$ represents the geometric channel, $\mathbf{S}$ the phase-shift matrices, $\mathbf{p}'$ the angle information vector, and $[\mathbf{F}]_{(1)}$ the mode-1 unfolding containing delay-Doppler information. A secondary nested model further isolates the delay and Doppler steering vectors.
Two-Stage Estimation Procedure:
- Stage 1 (Factor Estimation): The algorithm employs an Alternating Least Squares (ALS) approach, specifically a Bilinear Alternating Least Squares (BALS) procedure, to estimate the tensor factors $[\mathbf{F}]_{(1)}$ and $\mathbf{p}'$ . This step decouples the spatial (angle) and temporal (delay-Doppler) components.
- Stage 2 (Parameter Extraction):
  - Delay and Doppler: Using the estimated factor $[\mathbf{F}]_{(1)}$ , the algorithm applies 1D ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) to extract the delay $\tau$ and Doppler $\nu$ .
  - Angles: The delay and Doppler estimates are used to parametrically reconstruct their steering vectors. These refined factors are substituted back into the model to extract the angle information matrix $\mathbf{P}$ . A 2D ESPRIT algorithm is then applied to $\mathbf{P}$ to estimate the azimuth and elevation angles.
  - Complex Gain: Finally, the complex channel gain coefficient $\alpha$ is estimated via a closed-form least squares step using element-wise division, leveraging the full multidimensional dataset to mitigate noise.
Identifiability Analysis: The paper analyzes the uniqueness conditions of the proposed model. Due to the use of a deterministic, known core tensor $\mathbf{I}$ , arbitrary rotations of the estimates are prevented. The remaining ambiguities are limited to trivial scaling factors, which cancel out in the reconstructed tensor or are resolved through the specific structure of the estimation steps.

Key Contributions

Novel Algorithm: The development of the NTFE algorithm, which utilizes a nested Tucker decomposition to model BD-RIS-assisted sensing signals, explicitly accounting for the group-connected architecture.
Decoupling Strategy: A two-stage procedure that successfully decouples the estimation of delay-Doppler parameters from angle parameters, avoiding the high computational cost and error propagation associated with sequential 4D Maximum Likelihood (ML) searches.
Theoretical Guarantees: Derivation of identifiability and uniqueness conditions, demonstrating that the proposed model allows for unique parameter estimation up to trivial scaling ambiguities under specific system dimension constraints (e.g., $LT \geq N$ and $LMQT \geq N^2$ ).
Closed-Form Gain Estimation: A low-complexity method to estimate the complex channel gain coefficient after the structural parameters are resolved.

Simulation Results
The performance of NTFE was evaluated via Monte Carlo simulations against three benchmarks: a sequential 4D ML method, a direct Least Squares (LS) estimate, and a Kronecker Factorization (KF) based method.

Effective Channel Estimation: NTFE demonstrated a significant improvement in Normalized Mean Squared Error (NMSE) for estimating the effective channel. It outperformed the KF method by approximately 25 dB and the direct LS method by roughly 35 dB.
Parameter Accuracy: In terms of Root Mean Squared Error (RMSE) for delay, Doppler, and angle estimation, NTFE achieved a performance gain of approximately 10 dB compared to the standard 4D ML benchmark.
Robustness: The results indicate that NTFE effectively exploits the group-connected BD-RIS structure, providing accurate estimation even in scenarios where sequential ML methods suffer from initialization dependence and error propagation.

Significance and Claims
The paper claims that the NTFE algorithm represents a significant advancement in BD-RIS-assisted sensing by bridging the gap between tensor-based processing and the specific hardware constraints of group-connected BD-RIS. The primary significance lies in the ability to decouple spatial and temporal domains, which leads to superior estimation accuracy and reduced computational complexity compared to state-of-the-art benchmarks. The authors assert that by explicitly modeling the block-wise structure of the BD-RIS, the method achieves identifiability and outperforms existing techniques in both effective-channel reconstruction and physical parameter estimation, making it a viable solution for integrated sensing and communications (ISAC) systems utilizing advanced RIS architectures.