# Distributed MIMO Measurements for Integrated Communication and Sensing in an Industrial Environment
## 6 Analysis and Discussion
### Doppler-Delay Positioning and Tracking
In the following, we assume that the agent is moving on a plane parallel to the \(xy\)-plane at a known height; hence, we aim for two-dimensional (2D) positioning and velocity estimation in this work, well-knowing that Equation (22) is also suitable for three-dimensional (3D) positioning. We analyze the Bartlett spectrum around an observation step \(n=2544\) and hence employ a window of received signals \(\bar{n}\in\{2395,\ldots,2544\}\). To evaluate the impact of only the temporal array response on the Bartlett spectrum, we choose \(\mathbf{c}:=\mathbf{1}_{(N_{\
u}\times 1)}\) denoting a \(N_{\
u}\)-dimensional vector of all ones and evaluate Equation (22), which results in the spectrum depicted in Figure 17. Due to the limited bandwidth of \(BW=35\,\mathrm{MHz}\) and the respective delay resolution of approximately \(8.6\,\mathrm{m}\), the resulting Bartlett spectrum is rather flat around the _true_ agent position (\(\mathbf{\cdot}\)). Furthermore, imperfections in the delay calibration lead to a bias of the maximum \(\arg\max_{\mathbf{p}}\{\hat{P}(\mathbf{p})\}\) with respect to the true position \(\mathbf{p}_{n}^{(13)}\). To evaluate the impact of only the Doppler array response on the Bartlett spectrum, we choose \(\mathbf{b}:=\mathbf{1}_{(N_{\
u}\times 1)}\) and evaluate Equation (22), resulting in the spectrum shown in Figure 18. At the current time step \(n\), the agent velocity is \(\|\mathbf{v}_{n}\|\approx 0.77\,\mathrm{m}\,\mathrm{s}^{-1}\). At this speed, the Doppler array response is already much more informative (i.e., it exhibits a higher curvature) around the _true_ agent position than is the temporal array response at the chosen window size. Hence, with a moving agent, the Doppler information quickly dominates over the delay information. Another benefit of the Doppler array response in Equation (21) is that it also depends on the agent velocity \(\mathbf{v}_{n}\), and therefore \(\arg\max_{\mathbf{p},\mathbf{v}}\{\hat{P}(\mathbf{p},\mathbf{v})\}\) is a joint position-velocity estimator. Figure 19 shows the resulting Bartlett spectrum in the velocity domain. At the given speed, the Doppler array response likewise causes a distinct peak around the _true_ agent velocity \(\mathbf{v}_{n}\) (\(\mathbf{\cdot}\)). Figure 17: Bartlett spectrum in the position domain exploiting only delay information. Although reasonable position and velocity estimates can be extracted from a single snapshot of data, state filtering over all snapshots along the trajectory achieves much better results. To showcase initial results, we employ the empirical Bartlett spectrum from Equation (22) in a particle filter together with a nearly constant velocity state-space model (cf. [45] p. 274). Figure 20 shows the resulting estimatesin the position domain compared to the ground truth trajectory.It is observable that the estimates slowly converge to the true trajectory and follow it closely until the agent performs its 180\({}^{\circ}\) turn. In the curve, the velocity decreases, as does the information from the Doppler array response in the position domain because the _sensitivity_ of a Doppler frequency shift with respect to the position; i.e., \(\partial v_{n}^{(h13)}/\partial p_{n}^{(13)}\) depends linearly on the (radial) velocity of the agent relative to the \(l\)th unit. The estimation accuracy decreases for a moment until the agent moves at maximum speed and the position estimates converge again. Using the Bartlett beamformer-based implementation, we achieve a positioning mean-square error (MSE) of 18.4 cm with respect to our ground truth. These initial results highlight the potential of the dataset for positioning and tracking and set the stage for future work on more elaborate estimators. Figure 19: Bartlett spectrum in the velocity domain exploiting only Doppler information. Figure 18: Bartlett spectrum in the position domain exploiting only Doppler information.