# Distributed MIMO Measurements for Integrated Communication and Sensing in an Industrial Environment
## 6 Analysis and Discussion
### Collinearity  
The collinearity metric between the local scattering function in two different time instances allows us investigate the extent of the stationarity region in time, \(T_{\mathrm{s}}[n]\); that is, how long the WSSUS assumptions will hold [35]. It should be noted that the stationarity time itself will be time dependent due to the changing environment. Stack the \(N\times M\) elements of \(\left[\mathbf{C}^{(hh^{\prime})}\right]_{k_{l};n^{\prime},p}\) in a column vector \(\mathbf{c}_{k_{l}}\) (without the superscript for readability) and define the collinearity \(R[k_{t_{1}},k_{t_{2}}]\) as follows:  
\[R[k_{t_{1}},k_{t_{2}}]=\frac{\mathbf{c}_{k_{t_{1}}}^{\mathrm{T}}\mathbf{c}_{k_{t_{2}}} }{\|\mathbf{c}_{k_{t_{1}}}\|\|\mathbf{c}_{k_{t_{2}}}\|}. \tag{15}\]  
As in [35], we define the indicator function \(\gamma\left[k_{t}^{\prime},\tilde{\mathbf{\kappa}}_{t}\right]\) as  
\[\gamma\left[k_{t}^{\prime},\tilde{\mathbf{\kappa}}_{t}\right]=\begin{cases}1&:R \left[k_{t}^{\prime},\tilde{\mathbf{\kappa}}_{t}\right]>c_{\mathrm{th}}\,,\\ 0&:\text{otherwise}\,.\end{cases} \tag{16}\]  
where a threshold \(c_{\mathrm{th}}\) is defined. From \(\gamma\), the (time-varying) stationarity time, \(T_{\mathrm{s}}[n]\), can be estimated as the width of the region around the diagonal. In [35], the threshold \(c_{\mathrm{th}}\) was somewhat randomly chosen as 0.9. As seen in Figures 10 and 11, we select two (at random) links from the two scenarios _ref_ and _loop_. We have also plotted how the regions would grow if \(c_{\mathrm{th}}=0.7\) instead.  
In scenario _ref,_ the user was moving down in the middle of the workshop, then returning approximately the same path. In Figure 9(a),b, we can see that on the way back, we move through a region where the time stationarity region is longer. Here, the channel statistics are valid for a longer distance. In Figure 9(c),d, it also looks like the off-diagonal regions indicate that we are actually moving through the same stationarity region on the way back since the collinearity between times 15s and 45s is above the threshold.  
Performing a similar analysis on the collinearity plots of the _loop_ scenario, where the users performed two complete laps around some machinery, we can also indicate that we are in a stationarity region with similar statistics on the second lap. This becomes more apparent if we lower the threshold, \(c_{\mathrm{th}}\), to 0.7; see Figure 11. In general, the stationarity regions seem to become somewhat smaller because of the NLoS conditions.  
Figure 11: The collinearity for two different links of the _loop_ scenario, with different threshold values. (**a**) Link (3, 13) with threshold 0.9, (**b**) link (3, 13) with threshold 0.7, (**c**) link (9, 13) with threshold 0.7, and (**d**) link (9, 13) with threshold 0.7.  
Figure 10: The collinearity for two different links of the _ref_ scenario, with different threshold values. (**a**) Link (3, 13) with threshold 0.9, (**b**) link (3, 13) with threshold 0.7, (**c**) link (9, 13) with threshold 0.7, and (**d**) link (9, 13) with threshold 0.7.  
In Figure 12, we show the corresponding estimated time-varying stationarity regions in meters, \(T_{s}[n]\), for the two scenarios when \(c_{\rm th}=0.7\). The median stationarity region is around 2 m, see Figure 13, which means that the radio channel statistics vary while the UE is moving in the environment. Looking at the recorded statistics and the details of the environment, the rms delay spread, the Doppler power spectrum, and the LoS/NLoS states are changing for just a few meters of movement of the UE, hence the relatively short wide-sense stationarity regions.