Over-the-air computation (AirComp) is a known technique in which wireless devices transmit values by analog amplitude modulation so that a function of these values is computed over the communication channel at a common receiver. The physical reason is the superposition properties of the electromagnetic waves, which naturally return sums of analog values. Consequently, the applications of AirComp are almost entirely restricted to analog communication systems. However, the use of digital communications for over-the-air computations would have several benefits, such as error correction, synchronization, acquisition of channel state information, and easier adoption by current digital communication systems. Nevertheless, a common belief is that digital modulations are generally unfeasible for computation tasks because the overlapping of digitally modulated signals returns signals that seem to be meaningless for these tasks. This paper breaks through such a belief and proposes a fundamentally new computing method, named ChannelComp, for performing over-the-air computations by any digital modulation. In particular, we propose digital modulation formats that allow us to compute a wider class of functions than AirComp can compute, and we propose a feasibility optimization problem that ascertains the optimal digital modulation for computing functions over-the-air. The simulation results verify the superior performance of ChannelComp in comparison to AirComp, particularly for the product functions, with around 10 dB improvement of the computation error.

翻译：空中计算（AirComp）是一种已知技术，无线设备通过模拟幅度调制传输数值，从而在公共接收器处通过通信信道计算这些数值的函数。其物理原理在于电磁波的叠加特性，能够自然生成模拟数值的和。因此，AirComp的应用几乎完全局限于模拟通信系统。然而，利用数字通信进行空中计算将带来诸多优势，例如纠错、同步、信道状态信息获取，以及更容易被当前数字通信系统采用。尽管如此，普遍观点认为数字调制一般不适合计算任务，因为数字调制信号的重叠会返回看似对计算任务无意义的信号。本文突破这一观点，提出一种全新的计算方法——ChannelComp，可通过任意数字调制实现空中计算。具体而言，我们提出了能够计算比AirComp更广泛函数类别的数字调制格式，并构建了一个可确定用于空中函数计算最优数字调制的可行性优化问题。仿真结果验证了ChannelComp相对于AirComp的优越性能，特别是在乘积函数方面，计算误差改善了约10 dB。