⚡ electrical engineering

Multi-Target Estimation via Tensor Decomposition for Beyond Diagonal RIS-Aided Bistatic Sensing

This paper proposes a two-stage tensor decomposition framework called TenDAE for efficient multi-target estimation in bistatic MIMO systems aided by beyond-diagonal reconfigurable intelligent surfaces (BD-RIS), which decouples angle, delay, and Doppler parameter extraction via parallel tensor factorizations while demonstrating superior performance over classical diagonal RIS approaches.

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

Published 2026-04-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kenneth Benício, André L. F. de Almeida, Fazal-E Asim, Bruno Sokal, Gabor Fodor, Behrooz Makki, 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 group of friends hiding in a massive, foggy warehouse. You can't see them directly because the walls are blocking your view. However, you have a special, high-tech wall made of thousands of tiny, adjustable mirrors (this is the RIS or Reconfigurable Intelligent Surface).

Your goal is to figure out exactly where your friends are, how fast they are moving, and what direction they are facing, just by listening to the echoes of a shout you send out.

This paper presents a new, super-smart way to do this using a special kind of "smart wall" and a mathematical trick called Tensor Decomposition. Here is the breakdown in simple terms:

1. The Problem: The "Old" Wall vs. The "New" Wall

The Old Wall (Diagonal RIS): Imagine a wall where every mirror can only tilt up/down or left/right on its own, independently. It's like a choir where everyone sings their own note without listening to the others. It works okay, but it's limited.
The New Wall (BD-RIS): This is a "Beyond Diagonal" wall. The mirrors are connected. If one mirror tilts, it can influence its neighbors. It's like a choir where the singers can harmonize and create complex, unified sound waves. This gives the system much more "freedom" to shape the signal and catch details the old wall would miss.

2. The Challenge: The "Spaghetti" Mess

When you shout, the sound bounces off your friends, hits the smart wall, and comes back to your receiver. Because there are multiple friends (targets) moving at different speeds and angles, the returning signal is a giant, tangled mess of data—like a bowl of spaghetti where every noodle represents a different piece of information (angle, speed, distance).

If you try to untangle this mess with standard tools, it takes forever, or you get the wrong answer.

3. The Solution: The "Two-Stage Detective" (TenDAE)

The authors created a two-step detective method called TenDAE to untangle the spaghetti efficiently.

Stage 1: The "Big Picture" Sort (KSA)

First, the system looks at the whole bowl of spaghetti and asks, "How many distinct strands (friends) are in here?"

It uses a mathematical shortcut (called Kronecker Sum Approximation) to quickly separate the general noise from the actual signals.
Think of this as using a coarse sieve to separate the big noodles from the sauce. It doesn't tell you exactly where the noodles are yet, but it tells you, "Okay, there are 3 distinct signals here."

Stage 2: The "Microscope" (NTFE)

Now that the system knows there are 3 signals, it uses a powerful microscope to look at each one individually. This is where the Tensor Decomposition magic happens.

The Analogy: Imagine the data is a 3D block of Jell-O. The first layer tells you the Angle (where they are left/right/up/down). The second layer tells you the Time Delay (how far away they are). The third layer tells you the Doppler Shift (how fast they are moving).
The Trick: The system uses a "Nested" approach. It peels the Jell-O apart in parallel.
- One part of the brain figures out the Angles (using a method called PARAFAC).
- The other part figures out the Speed and Distance (using a "Nested" PARAFAC).
Because these two tasks are done simultaneously but separately, they don't get in each other's way. It's like having two people solve a puzzle at the same time: one focuses on the blue sky pieces, the other on the green grass pieces. They finish much faster and more accurately than if one person tried to do it all alone.

4. Why is this better?

Blind to the Hardware: The system is so smart that it doesn't even need to know the exact details of how the mirrors are connected. It just figures out the answers based on the echoes.
Saves Resources: Old methods needed a massive amount of data (thousands of shouts) to get a clear picture. This new method gets a crystal-clear picture with very few shouts (small data).
Accuracy: The paper shows that this method is much closer to the "perfect theoretical limit" (Cramér-Rao Lower Bound) than current state-of-the-art methods. It's like hitting a bullseye when other methods are hitting the edge of the target.

5. The Real-World Impact

This technology is a big deal for the future of 6G networks and autonomous driving.

Self-driving cars could use this to see through fog or around corners to detect pedestrians and other cars with extreme precision.
Smart cities could use these "smart walls" to monitor traffic flow or detect emergencies without needing expensive, high-power radar systems everywhere.

In a nutshell: The paper teaches us how to use a super-connected "smart wall" and a clever two-step math trick to untangle a chaotic mess of echoes, allowing us to find, track, and identify multiple moving objects with incredible speed and accuracy, even with very little data.

1. Problem Statement

The paper addresses the challenge of multi-target parameter estimation (specifically delay, Doppler shift, and angle of arrival/departure) in a bistatic MIMO sensing scenario assisted by Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RIS).

Context: Traditional Reconfigurable Intelligent Surfaces (RIS) use diagonal phase-shift matrices (single-connected elements). BD-RIS introduces inter-element coupling, resulting in a non-diagonal phase-shift matrix (fully-connected architecture), offering more degrees of freedom (DoFs).
Challenge: Estimating parameters for multiple targets ( $K$ ) simultaneously in a noisy environment with limited sensing resources (subcarriers, symbols, time slots). Existing methods often rely on sequential optimization or fail to fully exploit the multidimensional structure of the received signal, leading to high computational complexity or poor accuracy, especially when compared to the theoretical limits.
Goal: Develop a framework that jointly estimates delay, Doppler, and angles for multiple targets with high accuracy, low computational cost, and minimal sensing resources, while leveraging the unique capabilities of BD-RIS.

2. Methodology: The TenDAE Framework

The authors propose TenDAE (Tensor Doppler-delay-angle Estimation), a two-stage framework that decouples the estimation of angles from delay/Doppler parameters using tensor decomposition.

Stage 1: Kronecker Sum Approximation (KSA)

Objective: Estimate the effective channel subspace without explicitly knowing the target parameters.
Process: The received signal model for a BD-RIS is formulated as a sum of Kronecker products. The authors apply a right-filtering operation using the pseudo-inverse of the RIS phase-shift matrix.
Solution: The problem is recast as a Rank- $K$ Kronecker Sum Approximation (KSA). This is solved efficiently via a single Singular Value Decomposition (SVD) to extract the composite channel factors ( $\hat{G}$ $\hat{G}$ and $\hat{J}$ $\hat{J}$ ).
- $\hat{G}$ contains information related to the RIS-to-Receiver and RIS-to-Target channels (Angle information).
- $\hat{J}$ contains information related to the Transmitter-to-RIS channel and target delay/Doppler.

Stage 2: Nested Tensor Factorization Estimation (NTFE)

The extracted factors are processed in parallel using two distinct tensor models:

Angle Estimation (PARAFAC Model):
- The factor $\hat{G}$ is modeled as a third-order PARAFAC tensor.
- Algorithm: Alternating Least Squares (ALS) is used to fit the tensor.
- Extraction: 2D-ESPRIT is applied to the factor matrices to estimate the Angle of Arrival (AoA) at the receiver and Angle of Departure (AoD) from the BD-RIS.
Delay-Doppler Estimation (Nested PARAFAC Model):
- The factor $\hat{J}$ is modeled as a Nested PARAFAC tensor. This structure is crucial as it decouples the delay-Doppler domain from the angle domain.
- Algorithm: A bilinear ALS (BALS) approach is used to estimate the channel and the nested tensor factors. A subsequent standard ALS fits the inner PARAFAC tensor to separate delay and Doppler components.
- Extraction: 1D-ESPRIT is applied to extract the delay ( $\tau$ ) and Doppler ( $\nu$ ) parameters for each target.
Complex Gain Estimation:
- Once angles, delays, and Dopplers are known, the complex channel gains ( $\alpha_k$ ) are recovered using a closed-form Least Squares (LS) solution.

3. Key Contributions

Novel BD-RIS Sensing Model: The paper demonstrates that target estimation with a fully-connected BD-RIS can be recast as a Rank- $K$ KSA problem, solvable via a single SVD, which is more efficient than traditional iterative methods.
TenDAE Framework: Introduction of a modular two-stage approach (KSA + NTFE) that combines subspace estimation with tensor fitting. This allows for simultaneous multi-target estimation with minimal resources.
Nested-PARAFAC Decoupling: The proposal of a nested tensor model that successfully decouples the Doppler-Angle and Delay-Angle domains. This prevents error propagation between domains and allows for high-speed target tracking without compromising angle accuracy.
Comparative Analysis (BD-RIS vs. Diagonal RIS):
- The authors show that for a diagonal RIS, the KSA model reduces to a Khatri-Rao Sum Approximation (KRSA).
- Crucially, they prove that BD-RIS enables unique estimation of both arrival and departure angles at the RIS, whereas diagonal RIS architectures suffer from identifiability issues regarding these specific angles.
Theoretical Bounds: Derivation of the Cramér-Rao Lower Bound (CRLB) for the bistatic MIMO multi-target scenario, providing a benchmark for the proposed estimators.
Identifiability Conditions: Detailed analysis of the conditions required for unique parameter recovery (e.g., $MQ \geq \max(L_{ST}, K)$ and $L_{SR}NMQN \geq K$ ).

4. Numerical Results

The authors evaluated TenDAE against state-of-the-art benchmarks (Maximum Likelihood (ML), Orthogonal Matching Pursuit (OMP), and HOSVD-based methods) and the derived CRLB.

Accuracy: TenDAE consistently outperforms ML and OMP benchmarks.
- It achieves an RMSE gap of < 1 dB from the CRLB for delay, Doppler, and channel coefficients in single-target scenarios.
- It outperforms the 4D-ML benchmark from [24] by ~20 dB in parameter estimation accuracy.
Resource Efficiency: TenDAE achieves robust results with very low sensing resources (e.g., $Q=8$ subcarriers, $M=8$ symbols), whereas the benchmark ML method requires significantly more resources ( $Q=256, M=1120$ ) to initialize.
BD-RIS Advantage: BD-RIS significantly outperforms diagonal RIS in delay and Doppler estimation (approx. 5 dB gain) and uniquely resolves RIS angles, which diagonal RIS cannot do.
Complexity: While the iterative ALS approach has higher complexity for large $K$ , the framework offers a flexible trade-off. The HOSVD-based variant provides a lower-complexity alternative with near-optimal performance in multi-target scenarios.

5. Significance

This work represents a significant advancement in Integrated Sensing and Communications (ISAC) and RIS technology:

Hardware Efficiency: It validates the superiority of BD-RIS over traditional diagonal RIS for sensing applications, justifying the hardware complexity of inter-element coupling.
Algorithmic Innovation: The use of Nested Tensor Decomposition provides a mathematically elegant solution to the "curse of dimensionality" in multi-target sensing, effectively separating coupled parameters (delay/Doppler/Angle) without sequential error accumulation.
Practicality: The ability to estimate multiple targets with minimal pilot overhead makes the TenDAE framework highly suitable for future 6G networks where spectrum and time resources are scarce.
Theoretical Rigor: By providing the CRLB and identifiability conditions, the paper offers a complete theoretical foundation for designing BD-RIS aided sensing systems.