Staked intelligent metasurface (SIM) based techniques are developed to perform two-dimensional (2D) direction-of-arrival (DOA) estimation. In contrast to the conventional designs, an advanced SIM in front of the receiving array automatically performs the 2D discrete Fourier transform (DFT) as the incident waves propagate through it. To arrange for the SIM to carry out this task, we design a gradient descent algorithm for iteratively updating the phase shift of each meta-atom in the SIM to minimize the fitting error between the SIM's response and the 2D DFT matrix. To further improve the DOA estimation accuracy, we configure the phase shifts in the input layer of SIM to generate a set of 2D DFT matrices having orthogonal spatial frequency bins. Extensive numerical simulations verify the capability of a well-trained SIM to perform 2D DFT. Specifically, it is demonstrated that the SIM having an optical computational speed achieves an MSE of $10^{-4}$ in 2D DOA estimation.

翻译：基于堆叠式智能超表面(SIM)的技术被开发用于实现二维(2D)波达方向(DOA)估计。与传统方案不同，接收阵列前端的先进SIM可在入射波传播过程中自动执行二维离散傅里叶变换(DFT)。为使SIM完成该任务，我们设计了梯度下降算法，通过迭代优化SIM中每个超原子的相移，以最小化SIM响应与二维DFT矩阵之间的拟合误差。为进一步提升DOA估计精度，我们配置了SIM输入层的相移，从而生成一组具有正交空间频率单元的二维DFT矩阵。大量数值仿真验证了训练良好的SIM具备执行二维DFT的能力。具体而言，具有光学计算速度的SIM在二维DOA估计中达到了$10^{-4}$的均方误差(MSE)。