The integration of intelligent reflecting surface (IRS) into over-the-air computation (AirComp) is an effective solution for reducing the computational mean squared error (MSE) via its high passive beamforming gain. Prior works on IRS aided AirComp generally rely on the full instantaneous channel state information (I-CSI), which is not applicable to large-scale systems due to its heavy signalling overhead. To address this issue, we propose a novel multi-timescale transmission protocol. In particular, the receive beamforming at the access point (AP) is pre-determined based on the static angle information and the IRS phase-shifts are optimized relying on the long-term statistical CSI. With the obtained AP receive beamforming and IRS phase-shifts, the effective low-dimensional I-CSI is exploited to determine devices' transmit power in each coherence block, thus substantially reducing the signalling overhead. Theoretical analysis unveils that the achievable MSE scales on the order of ${\cal O}\left( {K/\left( {{N^2}M} \right)} \right)$, where $M$, $N$, and $K$ are the number of AP antennas, IRS elements, and devices, respectively. We also prove that the channel-inversion power control is asymptotically optimal for large $N$, which reveals that the full power transmission policy is not needed for lowering the power consumption of energy-limited devices.

翻译：将智能反射面（IRS）集成到空中计算（AirComp）中，是一种通过其高无源波束成形增益降低计算均方误差（MSE）的有效方案。现有关于IRS辅助AirComp的研究通常依赖完整的瞬时信道状态信息（I-CSI），但由于其巨大的信令开销，该方法不适用于大规模系统。为解决这一问题，我们提出了一种新颖的多时间尺度传输协议。具体而言，接入点（AP）处的接收波束成形基于静态角度信息预先确定，而IRS的相位偏移则依赖于长期统计CSI进行优化。利用得到的AP接收波束成形和IRS相位偏移，有效低维I-CSI被用于确定每个相干块中设备的发射功率，从而显著降低信令开销。理论分析揭示，可实现的MSE以${\cal O}\left( {K/\left( {{N^2}M} \right)} \right)$的阶数缩放，其中$M$、$N$和$K$分别表示AP天线数、IRS单元数和设备数。我们还证明了信道反转功率控制在$N$较大时渐近最优，这表明对于降低能量受限设备的功耗，无需采用全功率传输策略。