# MIMO Over-the-Air Computation for Distributed Estimation
## 1 Introduction  
Many estimation applications rely on real-time data collection and processing from spatially deployed wireless devices, which are used for time-critical inference and decisions [1; 2; 3]. In particular, some emerging services, such as surveillance, networked control, and distributed artificial intelligence, are interested in computing a target function using distributed measurements rather than collecting all individual measurements from an enormous number of devices [4; 5; 6]. Furthermore, some latency-critical control systems require a very low delay to estimate the system state in the order of 1-10 ms with a large number of nodes \(10^{2}\)-\(10^{3}\)[7]. Thus, these services primarily need high computation throughput, defined as the rate of computing outputs from nodes to a Fusion Center (FC), rather than the communication throughput of conventional wireless networks [8]. The FC (the nodes, respectively) can be an access point, base station, parameter server, or central controller (the user equipment, edge device, and sensor, respectively), depending on the application domain [1; 9]. Even with new techniques, a conventionally separated communication and computation principle encounters fundamental challenges due to the stringent throughput/delay requirements and the heavy computational load due to the limited spectrum bandwidth with resource-constrained devices [2; 5; 10]. For instance, the 5G standardization has recently introduced new wireless communication and computing techniques such as massive Multiple-Input Multiple-Output (MIMO), millimeter-wave, network slicing, and edge computing [4; 11]. However, despite these breakthrough technologies, practical 5G networks only achieve an average data rate of around 150-200 Mb/s with a delay of 10-100 ms [12]. One of the interesting information-theoretical results shows that the superposition property of the multiple access channel enables efficient aggregation of the distributedsignals [5; 13]. For instance, the simultaneous transmission automatically computes a linear combination of the measurements, exploiting the interference among the signals. Thus, the wireless channel operates as a distributed computer for various target functions of emerging applications. The over-the-air computation (AirComp) has been developed to leverage the inherent broadcast nature of the wireless channel to efficiently fuse the distributed data [14]. In particular, the recently proposed MIMO-AirComp enables multifunction computation through spatial multiplexing while improving the link reliability under a harsh channel condition [15; 16]. The MIMO-AirComp is a new communication and computation design approach to improve the computation throughput while reducing the spectrum usage and the delay for the massive Internet of Things (IoT). This article bridges the gap between the MIMO-AirComp and distributed estimation applications. We present how the representative target function of the distributed estimation problem is converted to the nomographic function with the proper processing function. The proposed framework consists of the preprocessing function of each node, the collaborative aggregation of the wireless channel, and the postprocessing function of the FC. The analysis and preliminary results demonstrate that such a design improves the computation throughput of the distributed estimation. Furthermore, we show the operating condition for using the proposed MIMO-AirComp compared to the conventional separated design approach for the overall system performance. Most existing works investigate the optimization problem of the transmit and receive beamforming gain to minimize the computation error of the AirComp [15; 17; 18; 19]. Some recent surveys have only provided a general overview of the AirComp for various applications [12; 20]. On the other hand, several distribution estimation techniques have mainly focused on the design of the local processing function without considering the communication aspects [21; 22].