Two-Dimensional Direction-of-Arrival Estimation Using Stacked Intelligent Metasurfaces

TL;DR

This work presents a novel SIM-based framework for 2D DOA estimation by embedding a 2D DFT computation into the wave domain. It formulates and solves a gradient-descent optimization to realize the 2D DFT within a cascaded metasurface, enabling the receiver to read the angular spectrum via simple energy detection and without per-element RF chains. A practical estimation protocol reconfigures the zeroth layer across snapshots to generate orthogonal frequency-bin DFT matrices, yielding on-grid DOA estimates with a theoretical MSE bound that is validated by extensive simulations. The approach promises ultra-fast, hardware-efficient DOA sensing suitable for high-frequency platforms, with potential extensions to multi-source scenarios and angular-domain processing. The results demonstrate competitive accuracy (MSE ~ 10^-4 in favorable conditions) and highlight design tradeoffs in SIM thickness, layer count, and metasurface density, guiding practical deployments.

Abstract

Stacked intelligent metasurfaces (SIM) are capable of emulating reconfigurable physical neural networks by relying on electromagnetic (EM) waves as carriers. They can also perform various complex computational and signal processing tasks. A SIM is fabricated by densely integrating multiple metasurface layers, each consisting of a large number of small meta-atoms that can control the EM waves passing through it. In this paper, we harness a SIM for two-dimensional (2D) direction-of-arrival (DOA) estimation. In contrast to the conventional designs, an advanced SIM in front of the receiver array automatically carries out the 2D discrete Fourier transform (DFT) as the incident waves propagate through it. As a result, the receiver array directly observes the angular spectrum of the incoming signal. In this context, the DOA estimates can be readily obtained by using probes to detect the energy distribution on the receiver array. This avoids the need for power-thirsty radio frequency (RF) chains. To enable SIM to perform the 2D DFT, we formulate the optimization problem of minimizing the fitting error between the SIM's EM response and the 2D DFT matrix. Furthermore, a gradient descent algorithm is customized for iteratively updating the phase shift of each meta-atom in SIM. To further improve the DOA estimation accuracy, we configure the phase shift pattern in the zeroth layer of the SIM to generate a set of 2D DFT matrices associated with orthogonal spatial frequency bins. Additionally, we analytically evaluate the performance of the proposed SIM-based DOA estimator by deriving a tight upper bound for the mean square error (MSE). Our numerical simulations verify the capability of a well-trained SIM to perform DOA estimation and corroborate our theoretical analysis. It is demonstrated that a SIM having an optical computational speed achieves an MSE of $10^{-4}$ for DOA estimation.

Figures (12)

• Figure 1: A SIM-aided array system.
• Figure 2: The top and front views of the SIM-based array systems.
• Figure 3: The proposed SIM-based DOA estimation protocol.
• Figure 4: The convergence behavior of the proposed gradient descent algorithm for optimizing a SIM to fit a 2D DFT matrix with $\left ( 2,2 \right )$ grid points.
• Figure 5: The convergence behavior of the proposed gradient descent algorithm for optimizing a SIM to fit a 2D DFT matrix with $\left ( 4,4 \right )$ grid points.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 1
• Remark 5
• Remark 6
• Remark 7
• Theorem 2
• Lemma 1