Stacked Intelligent Metasurfaces (SIM) have emerged as a revolutionary architecture for next-generation wireless communications, offering wave-domain signal processing capabilities with significantly reduced hardware complexity compared to conventional systems. However, most existing SIM research assumes continuous phase shifts and perfect instantaneous channel state information (CSI), which are impractical due to hardware discrete phase shift constraints and prohibitive pilot overhead. This paper presents a joint power allocation and discrete phase shift optimization framework for SIM-aided multiuser multiple-input single-output(MISO) downlink systems under statistical CSI. We formulate the achievable sum rate maximization problem considering practical discrete phase constraints and derive a closed-form expression for the average achievable rate under statistical CSI. To tackle the resulting non-convex optimization problem, we decouple the problem by using the weighted minimum mean square error (WMMSE) algorithm and alternating optimization (AO). Subsequently, we utilize the Lagrangian multiplier method and alternating direction method of multipliers (ADMM) to obtain closed-form iterative solutions. Our simulations demonstrate that the proposed algorithm reduces computational complexity by a factor of 50 compared to semi-definite relaxation (SDR) methods, , while maintaining over 85% of the continuous phase shift performance with only 1-bit quantization, highlighting its feasibility for low-cost hardware systems.

翻译：堆叠智能超表面（SIM）作为一种革命性架构，已崛起为下一代无线通信的关键技术，其波域信号处理能力相较于传统系统可显著降低硬件复杂度。然而，现有SIM研究大多假设连续相位偏移和完美的瞬时信道状态信息（CSI），这种假设因硬件离散相位偏移约束和过高的导频开销而难以实现。本文针对统计CSI下的SIM辅助多用户多输入单输出（MISO）下行链路系统，提出了一种联合功率分配与离散相位偏移优化框架。我们在考虑实际离散相位约束的条件下，构建了可达和速率最大化问题，并推导了统计CSI下平均可达速率的闭式表达式。为求解由此产生的非凸优化问题，我们采用加权最小均方误差（WMMSE）算法和交替优化（AO）方法对问题进行解耦。随后，利用拉格朗日乘子法与交替方向乘子法（ADMM）获得闭式迭代解。仿真结果表明，所提算法与半定松弛（SDR）方法相比，计算复杂度降低了50倍；在仅采用1比特量化时，仍能保持连续相位偏移性能的85%以上，凸显了其在低成本硬件系统中的可行性。