# Distributed MIMO Measurements for Integrated Communication and Sensing in an Industrial Environment
## 6 Analysis and Discussion
### Local Scattering Function  
In the case of a user moving in an industrial environment, the surroundings are usually cluttered with many objects that can have a considerable impact on the behavior of the propagation of wireless signals. Hence, the fading process is nonstationary, which means that the wireless channel can be approximated by a piecewise stationary stochastic process where statistical parameters are valid locally (i.e., in small regions) [14]. To extract parameters from nonstationary channels, we utilize the local scattering function defined in [34; 35]. This time-frequency-bounded function covers a stationarity region where the wireless channel can be well approximated by a W  
Figure 8: Using maximum ratio transmission (MRT), we clearly see a channel hardening effect where the deep fading dips are canceled. For visualization, a subset of the snapshots from the _ref_ scenario is selected, and only one, antenna 8 is plotted for comparison with MRT. To the left is MRT for a section of scenario _ref_, and on the right, scenario _loop_ is shown. First, collect all snapshots between the antenna pair \((hh^{\prime})\) in the matrix \(\mathbf{H}^{(hh^{\prime})}\)  
\[\mathbf{H}^{(hh^{\prime})}=\left[\mathbf{r}_{1}^{(hh^{\prime})},\ldots,\mathbf{r}_{N_{\text{ ss}}}^{(hh^{\prime})}\right],\quad\in\mathbb{C}^{(N_{\text{t}}\times N_{\text{ss}})}\,, \tag{12}\]  
Then, following the methodology outlined in [35], we denote the index of the sliding window in time and frequency with \(k_{t}\) and \(k_{f}\), respectively. The size of the stationarity region is denoted with \(M\) snapshots in the time domain and \(N\) samples in the frequency domain. Following [19], we use \(N=N_{\text{f}}\) and henceforth drop \(k_{f}\). The local scattering function is estimated as in [35]:  
\[\left[\mathbf{C}^{(hh^{\prime})}\right]_{k_{t};m^{\prime},p}=\frac{1}{IJ}\sum_{w=0 }^{IJ-1}\biggl{|}\left[\left(\mathbf{H}^{(hh^{\prime})}\right)^{w}\right]_{k_{t};m ^{\prime},p}\biggr{|}^{2} \tag{13}\]  
where \(n^{\prime}\in\{0,\ldots,N-1\}\) is the delay index, and \(p\in\{-M/2,\ldots,M/2-1\}\) is the Doppler index. The local scattering function at \(k_{t}\) represents the center value of the time-frequency region. Let \(\left(\mathbf{H}^{(hh^{\prime})}\right)^{w}\) be the windowed time-variant channel transfer function between each pair of antennas \(h\) and \(h^{\prime}\) in the stationarity region \(k_{t}\). \[\left[\left(\mathbf{H}^{(hh^{\prime})}\right)^{w}\right]_{k_{t};m^{\prime},p}=\sum _{m^{\prime}=-M/2}^{M/2-1}\sum_{q^{\prime}=-N/2}^{N/2-1}\left[\mathbf{H}^{(hh^{ \prime})}\right]_{m^{\prime}-k_{t},q^{\prime}}\cdot\left[\mathbf{G}^{w}\right]_{m^ {\prime}q^{\prime}}\cdot\mathrm{e}^{-\mathrm{j}2\pi\left(\frac{m^{\prime}}{M }-\frac{m^{\prime}q^{\prime}}{N}\right)}, \tag{14}\]  
where \(m^{\prime}\) and \(q^{\prime}\) are the relative time and frequency indices within each stationarity region. The relationship between the absolute time index \(n\) and the relative time index \(m^{\prime}\) is \(n=(k_{t}-1)\Delta_{t}+m^{\prime}+M\) for \(k_{t}\in\left\{1,\ldots,\lfloor\frac{N_{\text{s}}-M}{\Delta_{t}}\rfloor\right\}\), where \(\Delta_{t}\) corresponds to the time shift between two consecutive regions of stationarity. The taper functions \(\left[\mathbf{G}^{w}\right]_{m^{\prime}q^{\prime}}=u_{i}[m^{\prime}+M/2]\hat{q}_ {i}[q^{\prime}+N/2]\) are the (separable) band-limited discrete prolate spheroidal sequences (DPSSs) [37], which are well localized within the region \([-M/2,M/2-1]\times[-N/2,N/2-1]\). The sequences \(u_{i}\) and \(\hat{u}_{j}\) are indexed by \(i\in\{0,\ldots,I-1\}\) and \(j\in\{0,\ldots,J-1\}\), respectively, with \(w=ij+j\), which is the summation index in Equation (13). For our measurements, we set \(I=1\) and \(J=2\) following the recommendations of [35]. We choose \(M=75\), as the region of the minimum-time-stationarity region that corresponds to a duration of \(375\,\mathrm{ms}\).