This letter studies the massive multiple-input multiple-output (MIMO) analog uncoded over-the-air computation (AirComp), which is a promising solution in Internet-of-Things (IoT) networks for fusion centers (FCs) to compute functions of data distributed in a large number of wireless devices (WDs). To avoid the heavy overhead of the perfect real-time channel state information (CSI) estimation, we develop a two-timescale hybrid beamforming (THB) scheme in AirComp. Under this setup, we aim to minimize the transmit power over time, while ensuring a minimum required computation distortion in terms of the mean-squared error (MSE). We propose a constrained stochastic successive convex approximation (CSSCA) based algorithm to solve the formulated stochastic non-convex optimization problem for the THB design. Specifically, the short-timescale receive digital beamforming matrix at the FC is optimized at each time slot based on the real-time effective low-dimensional CSI matrices by a sum-minimization-mean-squared-error (sum-MMSE) receiver, which can better deal with the signal misalignment. By contrast, the long-timescale transmit beamforming matrix at WDs and the receive analog beamforming matrix at the FC are designed based on the available channel statistics and updated in a frame-based manner, where a frame contains a fixed number of time slots. Numerical results show that the proposed THB design algorithm achieves similar transmit power performance as that achieved by the design when the perfect real-time CSI is available.

中文翻译：

---

具有两时标混合波束成形的大规模MIMO无线计算的功率最小化

这封信研究了大规模多输入多输出（MIMO）模拟未经编码的空中计算（AirComp），这是物联网（IoT）网络中用于融合中心（FC）计算功能的有前途的解决方案分布在大量无线设备（WD）中的数据。为了避免完美的实时信道状态信息（CSI）估计的沉重开销，我们在AirComp中开发了一种两时标混合波束成形（THB）方案。在这种设置下，我们的目标是使随时间变化的发射功率最小化，同时确保在均方误差（MSE）方面确保所需的最小计算失真。我们提出了一种基于约束随机连续凸逼近（CSSCA）的算法来解决THB设计中所制定的随机非凸优化问题。具体来说，总和最小均方误差（sum-MMSE）接收器基于实时有效的低维CSI矩阵在每个时隙上对FC处的短时接收数字波束成形矩阵进行优化处理信号失准。相比之下，WD处的长时间尺度发送波束成形矩阵和FC处的接收模拟波束成形矩阵是基于可用信道统计信息设计的，并以基于帧的方式进行更新，其中帧包含固定数量的时隙。数值结果表明，所提出的THB设计算法在可获得理想的实时CSI时，具有与设计相似的发射功率性能。