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.

翻译：堆叠智能超表面（SIM）能够通过依赖电磁波作为载体来模拟可重构的物理神经网络，并可执行各种复杂的计算与信号处理任务。SIM通过密集集成多个超表面层制成，每层由大量可控制穿透电磁波的小型超原子构成。本文利用SIM实现二维波达方向（DOA）估计。与传统设计不同，接收阵列前端的先进SIM会在入射波传播过程中自动执行二维离散傅里叶变换（DFT），从而使接收阵列直接观测到来波信号的角频谱。在此背景下，可通过使用探针检测接收阵列上的能量分布轻松获得DOA估计值，从而避免对高功耗射频链路的需求。为使SIM能够执行二维DFT，我们构建了最小化SIM电磁响应与二维DFT矩阵之间拟合误差的优化问题，并定制了梯度下降算法以迭代更新SIM中每个超原子的相位偏移。为进一步提升DOA估计精度，我们在SIM第零层配置相位偏移模式，生成一组与正交空间频率仓相关联的二维DFT矩阵。此外，我们通过推导均方误差的严格上界，对基于SIM的DOA估计器性能进行了理论分析。数值仿真验证了经过充分训练的SIM执行DOA估计的能力，并佐证了我们的理论分析。结果表明，具有光学计算速度的SIM可在DOA估计中达到$10^{-4}$的均方误差。