Reduced-Dimension Design of MIMO Over-the-Air Computing for Data Aggregation in Clustered IoT Networks

One basic operation of Internet-of-Things (IoT) networks is aggregating distributed sensing-data over wireless-channels for function-computation, called wireless-data-aggregation (WDA). Targeting dense sensors, a recently developed technology called over-the-air computing (AirComp) can dramatically reduce the WDA latency by aggregating distributed data "over-the-air" using the waveform-superposition property of a multi-access channel. In this work, we design multiple-input-multiple-output (MIMO) AirComp for computing a vector-valued function in a clustered IoT network with multi-antenna sensors forming clusters and a multi-antenna access-point (AP) performing WDA. The resultant high-dimensional but low-rank MIMO channels makes it important to reduce channel or signal dimensionality in AirComp to avoid exposure to noise from channel null-spaces. Motivated by this, we develop a framework of reduced-dimension MIMO AirComp, featuring decomposed-aggregation-beamforming (DAB). Consider the case of separable channel-clusters with non-overlapping angle-of-arrival ranges. The optimal DAB has the structure where inner-components extract the dominant eigen-spaces of corresponding channel-clusters and outer-components jointly equalize the resultant low-dimensional channels. Consider the more complex case of inseparable clusters. We propose a suboptimal DAB design where the inner-component performs both dimension-reduction and joint-equalization over clustered-channel covariance matrices and the outer-component jointly equalizes the small-scale fading-channels. Furthermore, efficient algorithms for rank-optimization of individual DAB components and channel-feedback leveraging the AirComp principle are developed.