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

Stacked intelligent metasurfaces (SIMs) are capable of emulating reconfigurable physical neural networks by utilizing electromagnetic (EM) waves as carriers. They can also perform various complex computational and signal processing tasks. An SIM is constructed 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 an SIM for two-dimensional (2D) direction-of-arrival (DOA) estimation. In contrast to conventional designs, an advanced SIM in front of a receiver array can be designed to automatically compute the 2D discrete Fourier transform (DFT) as the incident waves propagate through it. As a result, a receiver array can directly observe the angular spectrum of the incoming signal, and it can estimate the DOA by simply using probes to detect the energy distribution on the receiver array. This avoids the need for power inefficient radio frequency chains. To enable an SIM to perform the 2D DFT in the wave domain, we formulate an optimization problem that minimizes the mean square error (MSE) between the SIM’s EM response and the 2D DFT matrix. Then, a gradient descent algorithm is customized for iteratively updating the phase shift applied by each meta-atom of the SIM. To further improve the DOA estimation accuracy, we configure the phase shifts of the input 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 MSE. Extensive numerical simulations verify the capability of an optimized SIM to perform DOA estimation and corroborate the theoretical analysis. Specifically, we show that an SIM is capable of performing DOA estimation with an MSE of the order of $10^{-4}$ .