# Distributed MIMO Measurements for Integrated Communication and Sensing in an Industrial Environment
## 2 Signal Model  
We consider \(H_{\text{a}}\) transceiver units distributed in the environment, and their positions are given as \(\mathbf{p}_{n}^{(h)}=[p_{\times,n}^{(h)},p_{y,n}^{(h)},p_{z,n}^{(h)}]^{\text{T}} \in\mathbb{R}^{3\times 1}\), with \(h\in\mathcal{N}_{\text{a}}\triangleq\{1,\ldots,H_{\text{a}}\}\). In our setup, each transceiver unit supports two independent radio frequency (RF) chains, each connected to a single omnidirectional antenna. It should be noted that a switched--possibly distributed--array can also be connected to the RF chains for even larger setups. In the following signal model, we limit ourselves to the single antenna case for the sake of brevity in notation, but it can easily be extended to the switched array channel sounding system. The \(H_{\text{a}}\)th unit \(\mathbf{p}_{n}^{(H_{\text{a}})}\) represents the mobile agent, and the other units indexed by \(h\in\{1,\ldots,H_{\text{a}}-1\}\) are the single antenna anchors at known positions. At each time, the \(h^{\prime}\)th unit acts as a transmitter and emits a periodic signal \(\bar{s}(t)\), and the other units \(\mathbf{p}_{n}^{(h)}\) with \(h\in\mathcal{N}_{\text{r}}\triangleq\{1,\ldots,H_{\text{a}}\}\setminus h^{\prime}\) act as receivers. Signals are represented by their complex envelopes with respect to a center frequency \(f_{\text{c}}\). The signal received at the \(h\) th antenna at the discrete observation time \(n\) reads  
\[\mathbf{r}_{n}^{(hh^{\prime})}=\exp\Bigl{(}\mathsf{j}\,2\pi\eta_{n}^{ (hh^{\prime})}t_{hh^{\prime}}\Bigr{)}\exp\Bigl{(}-\mathsf{j}\,2\pi f_{\text{c }}\epsilon^{(hh^{\prime})}\Bigr{)}\] \[\quad\times\sum_{l=1}^{L_{n}}\alpha_{l,n}^{(hh^{\prime})}\exp \Bigl{(}\mathsf{j}\,\eta^{(hh^{\prime})}\Bigr{)}\exp\Bigl{(}-\mathsf{j}\,2\pi f _{\text{c}}\tau_{l,n}^{(hh^{\prime})}\Bigr{)}\exp\Bigl{(}\mathsf{j}\,2\pi\upsilon _{l,n}^{(hh^{\prime})}t_{hh^{\prime}}\Bigr{)}\mathbf{s}+\mathbf{w}_{n}^{(hh^{\prime})}, \tag{1}\]  
where the first term comprises \(L_{n}\) multipath components (MPCs), \(l\in\{1,\ldots,L_{n}\}\), with each being characterized by its complex amplitude \(\alpha_{l,n}^{(hh^{\prime})}\in\mathbb{C}\) and propagation delay \(\tau_{l,n}^{(hh^{\prime})}\). Hardware impairments and imperfect synchronization are also characterized in the signal model. More specifically, \(\mu_{n}^{(hh^{\prime})}\) denotes the frequency offset between the \(h\)th unit and the \(h^{\prime}\)th unit, \(\eta^{(h)}\) denotes the unknown phase offset of the \(h\)th unit relative to a reference unit, \(\epsilon^{(h)}\) denotes the time shift due to the clock offset of the \(h\)th unit, and \(\upsilon_{l,n}^{(hh^{\prime})}\) represents the Doppler shift at the time instant \(t_{hh^{\prime}}\) when the channel between the \(h^{\prime}\)th transmit antenna and the \(h\)th receive antenna of the snapshot \(n\) is measured. Note that we omit the frequency dependency of the hardware impairment characteristics, given that a limited signal bandwidth of 40 MHz is used. Assuming we are transmitting on \(N_{\text{f}}\) subcarriers, the vector \(\mathbf{s}\in\mathbb{C}^{N_{\text{f}}\times 1}\) accounts for the system response \(\mathbf{g}\in\mathbb{C}^{N_{\text{f}}\times 1}\) and the baseband signal spectrum \(\mathbf{s}_{\text{f}}\in\mathbb{C}^{N_{\text{f}}\times 1}\); that is, \(\mathbf{s}\triangleq\mathbf{g}\odot\mathbf{s}_{\text{f}}\). The system response is usually measured by a back-to-back calibration procedure. The noise measurement processes \(w_{n}^{(h)}\)in Equation (1) are independent additive white Gaussian noise (AWGN) with double-sided power spectral density \(N_{0}/2\).