# Distributed MIMO Measurements for Integrated Communication and Sensing in an Industrial Environment
## 6 Analysis and Discussion
### Doppler-Delay Positioning and Tracking  
To show how well our data set is suited for positioning and to hint at what accuracy can be achieved, we present the initial positioning and tracking results. We focus on an uplink positioning task where the agent transmits signals from which the D-MIMO infrastructure infers its position. For this purpose, we focus on scenario _ref_, where our aim is to track the agent (\(\star\)) at a \"true\" unknown position \(\mathbf{p}_{n}^{(13)}\), moving on a (ground truth) trajectory (\(\blackarrow\)) based on its uplink signals \(\mathbf{r}_{n}^{(h13)}\) received by the D-MIMO antennas (\(\bullet\)) at the \"true\" known positions \(\{\mathbf{p}^{(h)}\mid 1\leq h\leq 12\}\). Before solving the positioning task, we first analyze the data. For this purpose, we collect the available received snapshots until the current step \(n\) along the trajectory into overlapping windows with a length of \(N_{\
u}=150\) and assemble them in matrices as follows:  
\[\mathbf{H}_{n}^{(h13)}=\left[\mathbf{r}_{n-N_{\
u}+1}^{(h13)},\ldots,\mathbf{r}_{n}^{(h13) }\right]\quad\in\mathbb{C}^{(N_{\
u}\times N_{\
u})}\,, \tag{19}\]  
where we perform a rough delay calibration to account for the time shifts \(\epsilon^{(h13)}\) introduced by the clock offsets of the receiving units \(h\) w.r.t. the agent. We formulate the \((N_{\text{f}}\times 1)\) temporal array response in the frequency domain through its elements as follows:  
\[\left[\mathbf{b}\left(\mathbf{p}_{n}^{(13)}\right)\right]_{n_{f}}=\exp\left(-\,\text{ j}\,2\pi f_{n_{f}}\mathbf{r}_{n}^{(h13)}\right), \tag{20}\]  
with \(f_{n_{f}}\) denoting \(n_{f}\)th frequency bin of the signal in the complex baseband and \(\mathbf{\gamma}_{n}^{(h13)}\) modeling the hypothetical propagation delay from the agent at \(\mathbf{p}_{n}^{(13)}\) to the \(h\)th receiving antenna at \(\mathbf{p}^{(h)}\). We further formulate the \((N_{\
u}\times 1)\) Doppler array response in the time domain through its elements as follows:  
\[\left[\mathbf{c}\left(\mathbf{p}_{n}^{(13)},\mathbf{v}_{n}\right)\right]_{\hat{n}}=\exp \left(\,\text{j}\,2\pi t_{n}\mathbf{v}_{n}^{(h13)}\right), \tag{21}\]  
where \(t_{\hat{n}}\in\{0,\ldots,(N_{\
u}-1)/f_{\text{rep}}\}\) corresponds to time instances of the current window of snapshots, and \(\mathbf{v}_{n}^{(h13)}\) models the hypothetical Doppler frequency shift depending on the agent position \(\mathbf{p}_{n}^{(13)}\) and velocity \(\mathbf{v}_{n}\) relative to the \(h\)th receiving antenna. Note that we omit the dependence on \(\text{MPCs}\,\,l>1\) in Equations (20) and (21). In our position and velocity estimator, we model LoS propagation only where NLoS paths enter as disturbance. Since the LoS amplitudes are likely to be large compared to the NLoS amplitudes \(\{\alpha_{i,n}^{(h13)}\mid l>1\}\) and some of the receiving units \(h\) will have the LoS conditions (refer to Figures 5 and 7), the D-MIMO units are likely to jointly agree on the true agent position, even in such an  
Figure 16: The normalized Doppler spectral density (DSD) estimated with MUSIC for link (4, 13). The red line is the theoretical LoS Doppler. unfavorable industrial environment. We compute the _nonphase-coherent_ empirical Bartlett spectrum (for brevity, we omit the normalization term in the denominator of the classical Bartlett spectrum from Equation (A4)) (see Appendix A for a derivation). \[\hat{P}(\mathbf{p},\mathbf{v})=\sum_{h=1}^{12}\left|\mathbf{b}^{\mathrm{H}}\mathbf{H}^{(h13)} _{\bar{n}}\mathbf{c}^{*}\right|^{2} \tag{22}\]  
such that the contributions of all receiving antennas \(h\in\{1,\ldots,12\}\) are summed noncoherently as powers instead of complex-valued amplitudes because we do not have an accurate phase calibration available between our single-antenna receiving units.