## 3 Link Level Parameters and Channel Coefficient Generation
### Coefficient Generation  
Before channel coefficient generation, we need to randomly draw the initial phases \(\left\{\Phi_{n,m}^{\theta\theta}\Phi_{n,m}^{\theta\theta}\Phi_{n,m}^{\theta \theta}\Phi_{n,m}^{\theta\phi}\right\}\). The initial phases are uniformly distributed in the range \((-\pi,\pi)\). The final time-invariant channel model expression will be a matrix-valued function of the variable \(\tau\). In case of single-polarized antenna elements at both the transmitter and receiver, we define for each cluster \(n\) and ray \(m\) the scalar term \(g_{n,m}\) as:  
\[g_{n,m}=F_{\tau,\theta}(\theta_{n,m,ZOA},\phi_{n,m,AOA})\ \exp(j\Phi_{n,m}^{ \theta\theta})\ F_{\tau,\theta}(\theta_{n,m,ZOD},\phi_{n,m,AOD}) \tag{54}\]  
Figure 4: Representation of the main clusters angles for a UT-BS link w.r.t. their powers and directions, in case of _LOS_. we obtain the following \(N_{r}\times N_{t}\) channel matrix \(\mathbf{H}_{n,m,s}^{NLOS}\) relative to the cluster \(n\), ray \(m\) and sector \(s\):  
\[\mathbf{H}_{n,m,s}^{NLOS}=\sqrt{\frac{P_{n}}{M_{ray}}}\ g_{n,m}\ \mathbf{a}_{rx}( \theta_{n,m}^{\prime},\phi_{n,m}^{\prime})\mathbf{a}_{tx,s}^{H}(\theta_{n,m}^{ \prime},\phi_{n,m}^{\prime}) \tag{55}\]  
In case of dual cross-polarized pairs of antenna, for each array element at both the transmitter and receiver, we define for each cluster \(n\) and ray \(m\) the \(2\times 2\) matrix \(\mathbf{G}_{n,m}\) as:  
\[\mathbf{G}_{n,m}= \begin{bmatrix}F_{rx,\theta}^{+\xi}(\theta_{n,m,ZOA}\phi_{n,m, AOA})&F_{rx,\phi}^{+\xi}(\theta_{n,m,ZOA},\phi_{n,m,AOA})\\ F_{rx,\theta}^{-\xi}(\theta_{n,m,ZOA},\phi_{n,m,AOA})&F_{rx,\phi}^{-\xi}(\theta _{n,m,ZOA},\phi_{n,m,AOA})\end{bmatrix}\begin{bmatrix}\exp(j\Phi_{n,m}^{\phi \theta})&\sqrt{\mathcal{K}_{m,m}^{-1}}\exp(j\Phi_{n,m}^{\phi\phi})\\ \sqrt{\mathcal{K}_{n,m}^{-1}}\exp(j\[PERSON]_{n,m}^{\phi\theta})&\exp(j\Phi_{n,m}^ {\phi\phi})\end{bmatrix}  \tag{56}\] \[ \begin{bmatrix}F_{tx,\theta}^{+\xi}(\theta_{n,m,ZOD},\phi_{n,m, AOD})&F_{tx,\theta}^{-\xi}(\theta_{n,m,ZOD},\phi_{n,m, AOD})\\ F_{tx,\theta}^{-\xi}(\theta_{n,m,ZOD},\phi_{n,m, AOD})&F_{tx,\theta}^{-\xi}(\theta_{n,m,ZOD}, \phi_{n,m, AOD})\end{bmatrix} \]  
where the \(F^{+\xi}\) (\(F^{-\xi}\)) indicates the polarized field component with polarization slant angle \(+45^{\circ}\) (\(-45^{\circ}\)). The channel matrix \(\mathbf{H}_{n,m,s}^{NLOS}\) relative to cluster \(n\), ray \(m\), and sector \(s\) becomes of size \(2N_{r}\times 2N_{t}\) and can be computed as:  
\[\mathbf{H}_{n,m,s}^{NLOS}=\sqrt{\frac{P_{n}}{M_{ray}}}\ \Big{[}\mathbf{a}_{rx}( \theta_{n,m,ZOA}^{\prime},\phi_{n,m,AOA}^{\prime})\,\mathbf{a}_{tx,s}^{H}( \theta_{n,m,ZOD}^{\prime},\phi_{n,m,AOD}^{\prime})\Big{]}\otimes\mathbf{G}_{n,m} \tag{57}\]  
For the \(N_{cl}-2\) weakest clusters, so for \(n=3,4,\ldots,N_{cl}\), all rays belonging to the same cluster \(n\) have equal delay \(\tau_{n}\). For the two strongest cluster, \(n=1\) and \(n=2\), rays are spread in delay to three sub-clusters \(R_{1}\), \(R_{2}\) and \(R_{3}\) in the same cluster \(n\), and are mapped as \(R_{1}=\{1,2,3,4,5,6,7,8,19,20\}\), \(R_{2}=\{9,10,11,12,17,18\}\) and \(R_{3}=\{13,14,15,16\}\). For each subcluster a fixed delay offset is applied and a fraction of the cluster power \(P_{n}\), proportional to the number of rays mapped onto it, is assigned. The delays of the sub-clusters are:  
\[\tau_{n,1}=\tau_{n} \tag{58}\] \[\tau_{n,2}=\tau_{n}+1.28\,c_{DS}\] \[\tau_{n,3}=\tau_{n}+2.56\,c_{DS}\]  
where \(c_{DS}\) is the cluster delay spread that can be found in Table 7.5-6 of [21] (e.g., for UMi _LOS_ scenario, \(c_{DS}=5\) ns).