⚡ electrical engineering

RIS-Assisted Sensing: A Nested Tensor Decomposition-Based Approach

This paper proposes a low-complexity, two-stage algorithm based on nested tensor decomposition and rotational invariance techniques to jointly estimate target delay, Doppler, and angular information in RIS-assisted monostatic MIMO sensing scenarios.

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

Published 2026-05-29

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CC BY 4.0

Original authors: Kenneth Benício, Fazal-E-Asim, Bruno Sokal, André L. F. de Almeida, 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 friend in a massive, foggy city where the direct path to them is blocked by a giant wall. You can't see them, and they can't see you. However, you have a special, high-tech mirror (called a Reconfigurable Intelligent Surface, or RIS) that you can control. By tilting this mirror just right, you can bounce a signal off it, hit your friend, and catch the echo bouncing back.

This paper presents a new, clever way to figure out exactly where your friend is, how fast they are moving, and how far away they are, using that echo. Here is how they do it, broken down into simple concepts:

1. The Problem: A Messy Echo

Usually, when you send a signal and get an echo back, the information is all jumbled together. It's like trying to figure out what a song sounds like by listening to a single, noisy recording where the drums, the guitar, and the vocals are all mixed up. You know something is there, but separating the details (like the exact pitch or the timing) is hard.

2. The Solution: The "Multi-Dimensional" Puzzle

The authors realized that the echo signal isn't just a flat line of data; it has a hidden, multi-layered structure. Think of the signal not as a single sheet of paper, but as a 3D block of Jell-O (or a Rubik's cube) with different flavors mixed inside:

Flavor 1: How long the signal took to travel (Distance/Time).
Flavor 2: How the signal changed because the target is moving (Speed/Doppler).
Flavor 3: The direction the signal came from (Angle).

Instead of trying to untangle this mess one piece at a time, the authors propose treating the whole signal as a giant, multi-dimensional puzzle called a Tensor.

3. The Method: Two-Stage Detective Work

To solve this puzzle, they use a two-step process, like a detective solving a case in two phases:

Phase 1: The "Alternating" Sort (ALS)
Imagine you have a pile of mixed-up socks, and you want to sort them by color and size. You can't do it all at once. So, you sort them by color first, then by size, then look at the colors again, then the sizes again. You keep switching back and forth until everything is perfectly organized.

In the paper, this is called the Alternating Least Squares (ALS) algorithm.
The computer looks at the "Jell-O block" of data and keeps switching its focus between the different layers (time, speed, direction) to separate the signal from the noise. It does this over and over until the picture becomes clear.

Phase 2: The "Rotating" Reveal (ESPRIT)
Once the signal is sorted out, the computer needs to read the specific numbers. The authors use a technique called ESPRIT.

Think of this like shining a flashlight on a spinning object. By watching how the light reflects as the object rotates, you can figure out its exact shape and speed without touching it.
This step takes the sorted data and calculates the precise delay (distance), Doppler (speed), and angle (direction) of the target.

4. The Results: Better with More Data

The authors ran simulations to see how well this "Tensor Detective" works. They found a few interesting things:

More Subcarriers = Sharper Vision: If you send more "flavors" of the signal (more subcarriers), the system gets much better at guessing the distance. It's like having more pixels in a camera; the picture gets clearer.
More Blocks = Better Accuracy: Sending the signal in more chunks (blocks) also helps the system get a better read on the target.
Too Many Mirrors Can Be Tricky: Interestingly, adding too many reflecting elements (mirrors) to the RIS didn't always help. If you add too many, the computer gets overwhelmed trying to sort them all, and the accuracy actually drops slightly.
Speed: The method is fast. Even though it does complex math, it doesn't take a supercomputer to run it. It scales up nicely as you add more data.

Summary

In short, this paper introduces a new way to "listen" to radar echoes in a world where the direct path is blocked. Instead of looking at the echo as a messy noise, they treat it as a structured 3D puzzle. By using a two-step process of sorting (alternating) and revealing (rotating), they can pinpoint a target's location, speed, and direction with high accuracy and low computing cost.

Technical Summary: RIS-Assisted Sensing via Nested Tensor Decomposition

Problem Statement
This paper addresses the challenge of target parameter estimation in a monostatic multiple-input multiple-output (MIMO) sensing scenario assisted by a Reconfigurable Intelligent Surface (RIS). In this setup, the direct line-of-sight (LOS) between the Base Station (BS) and the target is blocked; sensing is achieved solely through a virtual LOS path created by the RIS. The system utilizes Orthogonal Frequency Division Multiplexing (OFDM) waveforms. The core difficulty lies in jointly estimating the target's delay ( $\tau$ ), Doppler shift ( $\nu$ ), and angular information (Azimuth and Elevation) from the received echo signal, which possesses a complex, high-dimensional structure due to the interaction of the BS, RIS, and target.

Methodology
The authors propose a Nested Tensor-Based Sensing (NTBS) algorithm that exploits the intrinsic multidimensional structure of the received echo signal. The approach models the signal as a fourth-order nested tensor and employs a two-stage estimation strategy:

System Modeling: The received signal is formulated in the frequency domain, accounting for the BS-RIS channel, the RIS-Target channel, the target-RIS-BS path, and the RIS phase-shift vectors. The signal is vectorized and stacked across subcarriers, OFDM symbols, and transmission blocks to form a data matrix $\mathbf{Y}$ .
Tensor Decomposition (Stage 1): The data matrix is recognized as a 3-mode unfolding of a Tucker-3 tensor. The authors formulate a tensor fitting problem to estimate the factor matrices corresponding to the BS channel, the target signature, and the combined delay-Doppler steering vectors. This is solved using an Alternating Least Squares (ALS) algorithm, which iteratively updates estimates for the channel matrix $\mathbf{H}$ , the delay-Doppler matrix $\mathbf{F}(\tau, \nu)$ , and the target signature matrix $\mathbf{P}$ .
Parameter Extraction (Stage 2): The estimated delay-Doppler matrix $\hat{\mathbf{F}}(\tau, \nu)$ is reshaped into a third-order Tucker tensor. A second ALS-based decomposition is performed to separate the delay steering vector $\mathbf{c}(\tau)$ and the Doppler steering vector $\mathbf{d}(\nu)$ . Finally, high-resolution parameter estimation techniques, specifically ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques), are applied to the extracted steering vectors to determine the precise values of delay, Doppler, and angles.

Key Contributions

Nested Tensor Formulation: The paper generalizes previous SISO-based tensor approaches to the MIMO case, recasting the RIS-assisted monostatic sensing problem as a nested tensor decomposition problem.
Two-Stage Algorithm: The development of the NTBS algorithm, which combines ALS for tensor factorization with ESPRIT for high-resolution parameter extraction, allowing for the joint estimation of delay, Doppler, and angular parameters.
Identifiability Analysis: The authors derive the conditions required for the uniqueness of the solution, establishing that the number of transmission blocks $K$ must satisfy $K \ge N^2$ and the product of subcarriers and symbols $MQ$ must satisfy $MQ \ge L$ (where $N$ is the number of RIS elements and $L$ is the number of BS antennas).
Computational Complexity Analysis: The paper provides a detailed analysis of the algorithm's complexity, showing that it scales linearly with the number of RIS elements but is less sensitive to variations in the number of subcarriers and OFDM symbols.

Simulation Results
The performance of the NTBS algorithm was evaluated using Root Mean Square Error (RMSE) across 5,000 independent channel realizations. Key findings include:

Resolution Improvement: Increasing the number of subcarriers ( $Q$ ) and the number of transmission blocks ( $K$ ) significantly improves estimation accuracy, particularly for delay estimation, due to the increased resolution of the tensor model dimensions.
RIS Element Trade-off: Contrary to intuition, increasing the number of RIS elements ( $N$ ) initially leads to a deterioration in RMSE. The authors attribute this to the increased number of parameters to be estimated in the first stage without a corresponding increase in resolution, leading to propagated errors in the second stage.
Complexity: The computational complexity increases linearly with the number of RIS elements but remains relatively stable with respect to increases in $Q$ and $M$ , suggesting the algorithm is efficient for systems with higher bandwidth or longer observation times.

Significance and Claims
The paper claims that the proposed NTBS method yields accurate estimates of sensing parameters with low computational complexity by fully exploiting the multidimensional structure of the received signal. It positions the work as a generalization of previous SISO tensor-based sensing to the more challenging MIMO scenario. The authors explicitly state that the algorithm effectively recasts the target sensing problem into a structured decomposition problem, avoiding the need for high-dimensional grid searches (such as the four-dimensional search mentioned in prior work [7]). The paper concludes by noting that future work will focus on extending the NTBS algorithm to multi-target scenarios.