# MIMO Over-the-Air Computation for Distributed Estimation
## 5 Performance Evaluation
For simplicity, we considered the block fading channels, where the wireless channels remained unchanged within each time slot but may change from one slot to another. We assumed that the channel coefficients over different time slots were generated from a stationary and ergodic stochastic process. In each time slot, the channel coefficients were generated following Rayleigh fading with unit variance. We set the maximum beamforming gain \(\mu\leq 1\), theMIMO configuration \(N_{\text{tx}}=N_{\text{rx}}=3\), and the error norm bound \(\delta=0.01\), unless otherwise stated. We calculated the service delay and the RMSE using 1000 Monte Carlo simulation results, where each simulation ran 10 s with 10 ms unit time. Figure 2 plots the estimated position states \(\hat{x}_{1}\) and \(\hat{x}_{2}\) by using MIMO-AirComp and TDMA with different numbers of beacon nodes \(K=4,20\) over time. We set \(\sigma_{\text{cn}}^{2}=1\) and \(\sigma_{\text{sn}}^{2}=0.9\). The solid bold lines present the actual position in two-dimensional space. Both schemes stabilized in all scenarios but at different rates, heavily depending on the node density. For the small number of nodes \(K=4\), the overall estimated states of the TDMA were slightly better than those using the MIMO-AirComp. On the other hand, the MIMO-AirComp provided a faster convergence of the estimated states compared to that of the TDMA scheme when the number of nodes \(K=20\) was large. Moreover, the MIMO-AirComp considerably reduced the stationary error without having significant oscillation. In each time step, the MIMO-AirComp efficiently aggregated the sensing data of many beacon nodes. The TDMA, on the other hand, was almost independent of the node population, since it updated the position states in each received beacon information. The MIMO-AirComp showed a significant improvement in the state estimation performance with \(K=20\). We analyzed further details of the static estimation performance on the service delay and RMSE by considering the impact of the channel and measurement noise, the node population, and the antenna configuration. Figure 3 shows the average service delay and RMSE using MIMO-AirComp and TDMA with different parameters \(\sigma_{\text{cn}}^{2}=0.5,1\) and \(\sigma_{\text{sn}}^{2}=0.3,0.9\) for the various numbers of nodes \(K=4,\ldots,20\). One interesting observation was that the MIMO-AirComp and TDMA provided considerably different behavior as a function of different node populations. While the MIMO-AirComp improved the service delay and the RMSE to meet the error bounds as \(K\) increased, the TDMA scheme showed an approximately constant service delay and RMSE, even after adding more nodes. This was because the MIMO-AirComp with a large \(K\) essentially suppressed the noise variance due to the wireless channel and the measurement in each time slot. On the other hand, the TDMA scheme mainly reduced the channel noise at the cost of the time delay. Figure 2: Estimated position states \(\hat{x}_{1}\) and \(\hat{x}_{2}\) of the MIMO-AirComp and TDMA with different numbers of beacon nodes \(K=4,20\) over time. The higher amount of sensing noise \(\sigma_{\text{sn}}^{2}=0.9\) naturally degraded the service delay and the RMSE of the MIMO-AirComp and the TDMA more than with the lower amount of sensing noise \(\sigma_{\text{sn}}^{2}=0.3\). Under both types of sensing noise \(\sigma_{\text{sn}}^{2}=0.3,0.9\), the TDMA scheme outperformed the MIMO-AirComp for the small number of nodes \(K\leq 10\), while the MIMO-AirComp performed better for a large number of nodes \(K>10\). The TDMA scheme provided a similar service delay and RSME even with different amounts of channel noise \(\sigma_{\text{cn}}^{2}=0.5,1\), since it achieved extremely reliable link performance under the harsh channel condition \(\sigma_{\text{cn}}^{2}=1\). On the other hand, the low amount of channel noise \(\sigma_{\text{cn}}^{2}=0.5\) considerably reduced the service delay and the RMSE of the MIMO-AirComp compared to that with the higher amount of channel noise \(\sigma_{\text{cn}}^{2}=1\).