# MIMO Over-the-Air Computation for Distributed Estimation
## 5 Performance Evaluation  
This section investigates how the proposed MIMO-AirComp framework performed for distributed estimation. For numerical experiments, we considered the conventional localization scenarios of mobile vehicles, where the target vehicle determined its position by measuring the direction and/or range to nearby beacon nodes whose positions were known [27]. The target vehicle estimated the position using the received signal strength of the wireless signals transmitted by \(K\) beacon nodes [28]. We placed the target vehicle at \(\mathbf{x}=\left[20\ 10\right]^{\top}\), whereas the \(K\) beacon nodes were randomly deployed in the range of \(100\ \mathrm{m}\times 100\ \mathrm{m}\) around the target position. The linear observation model of the position estimation of beacon \(k\) is  
\[\mathbf{\hat{y}}_{k}= \begin{bmatrix}-1&0\\ 0&-1\end{bmatrix} \mathbf{x}+\mathbf{b}_{k}+\mathbf{v}_{k},\quad\forall k=1, \ldots,K, \tag{15}\]  
where \(\mathbf{b}_{k}\in\mathds{R}^{2}\) is the position vector of beacon \(k\), and \(\mathbf{v}_{k}\in\mathds{R}^{2}\) is the measurement noise. By defining the debiased measurement state \(\mathbf{y}_{k}=\mathbf{\hat{y}}_{k}-\mathbf{b}_{k}\), Equation (15) has the form comparable to Equation (10). Each beacon only multiplied the \((2\times 2)\) matrix with the \((2\times 1)\) vector as the local preprocessing and ran the element-wise division of the \((2\times 1)\) vector as the source encoder. In addition, a sliding window average was used to reduce the impact of the measurement noise on the position estimation. To minimize the computation error, the differential geometry was used to derive the close-to-optimal transmit and receive beamforming gains of the MIMO-AirComp [15]. We used the solution of the weighted centroid of points on a Grassmann manifold as the joint beamforming matrix of the MIMO-AirComp framework. We compared the proposed approach to a conventional separated communication and computation approach, where the global estimation of Equation (11) uses all individually received measurements through the Time Division Multiple Access (TDMA) scheme. The link reliability of the TDMA scheme was set to the block error rate of the 5G physical uplink control channel [29]. We note that this scheme provided extremely reliable communication performance for our considered range of network parameters. Furthermore, the measurement collection delay using the TDMA scheme was proportional to the number of nodes. For a fair comparison, the TDMA scheme allowed updating the position states when it received more than three beacon signals by considering the previously received information. For the numerical analysis, we defined the performance metric called the service delay based on the settling time of a dynamical system [30]. The service delay \(T_{s}\) is the elapsed time for the system curve to reach and remain within a specified error norm bound \(\delta\) to the ideal value \(\mathbf{r}\):  
\[T_{s}=\min\{t>0:\left\|\mathbf{x}(t^{\prime})-\mathbf{r}\right\|\leq\delta \|\mathbf{r}\|\,,\ \forall t^{\prime}\geq t\}. \tag{16}\]  
The inverse of the service delay \(1/T_{s}\) is analogue to the computation throughput. The network parameters such as the reliability and node density heavily affect the service delay. Furthermore, we showed the performance of the state estimation in terms of the root mean square error (RMSE) of the estimated state to the target position after 5 s.  
We analyzed the service delay and the RMSE of the proposed framework for the various number of nodes, antenna configurations, channel noise, and measurement noise. We considered various channel noises with variance \(\sigma_{\mathrm{on}}^{2}=1,0.5\), which corresponded to the average SNR between 0 dB and 3 dB with the maximum transmit power. We also considered various measurement noises with covariance matrix \(\mathbf{R}_{v,k}=\sigma_{\mathrm{sn}}^{2}\mathbf{I}\).