VecComp: Vector Computing via MIMO Digital Over-the-Air Computation

Recently, the ChannelComp framework has proposed digital over-the-air computation by designing digital modulations that enable the computation of arbitrary functions. Unlike traditional analog over-the-air computation, which is restricted to nomographic functions, ChannelComp enables a broader range of computational tasks while maintaining compatibility with digital communication systems. This framework is intended for applications that favor local information processing over the mere acquisition of data. However, ChannelComp is currently designed for scalar function computation, while numerous data-centric applications necessitate vector-based computations, and it is susceptible to channel fading. In this work, we introduce a generalization of the ChannelComp framework, called VecComp, by integrating ChannelComp with multiple-antenna technology. This generalization not only enables vector function computation but also ensures scalability in the computational complexity, which increases only linearly with the vector dimension. As such, VecComp remains computationally efficient and robust against channel impairments, making it suitable for high-dimensional, data-centric applications. We establish a non-asymptotic upper bound on the mean squared error of VecComp, affirming its computation efficiency under fading channel conditions. Numerical experiments show the effectiveness of VecComp in improving the computation of vector functions and fading compensation over noisy and fading multiple-access channels.