# Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems +
## 3 A Practical Non-Stationary of 3-D Massive MIMO Channel Model
### Generation of the Cluster/Channel in System Level
#### 3.1.2 Toward the Receive Antenna Elements  
\(qt_{th}\) cluster will be observed to the \(qr_{th}\) cluster at the RX. \(qr=\{1,...,Qr\}\), where \(Qr\) is equal to Cluster\({}_{t}=Qr(t+\Delta t)\) as total clusters, in which arrive to the \(l^{\prime}k^{\prime}{}_{th}\) antenna element with time \((t+\Delta t)\). Similarly, the first cluster with time \(t\) is arrived to the \((0,0)_{th}\) antenna element via \(i_{2\,th}\) path. Additionally, the rest of the clusters are arriving to adjacent element with succeeding time instantaneously \((\Delta t)\) calculating with distance \(d^{UE}\). As an example, cluster \(qt1\) is observed to the cluster \(qr1\), where \(qr1\) is observed at the RX via \(i_{2\,th}\) path to the \((0,0)_{th}\) antenna element and so on. To perform the evolution of outdoor channel coefficient, the clusters in the set of \(\left(Qt(t+\Delta t)\bigcap Qr(t+\Delta t)\right)\) are generated based on the outdoor cluster evolution for outdoor communication. Based on the outdoor model, \((Cl_{n}^{out})\) is the total number of clusters, where \(n=1,...,N\) cluster for outdoor communication can be expressed as  
\[Cl_{n}^{out}=card\left(\bigcup\limits_{lk=1}^{LK}\bigcup\limits_{l^{\prime}k^ {\prime}=1}^{L^{\prime}K^{\prime}}\left(Qt(t+\Delta t)\bigcap(Qr(t+\Delta t) \right)\right) \tag{21}\]  
where the operator card (.) denotes the cardinality of a set [8; 14]. The operator card \(Cl_{n}^{out}\), each cluster, say Cluster\({}_{n}(n=1,...,Cl_{n}^{out})\) is for outdoor models. Based on the above analysis and the summary of key parameter definitions in Table 2, the massive MIMO channel matrix can be expressed as an \(LK\times L^{\prime}K^{\prime}\) complex matrix \(\mathbf{H}(t,\Delta t,\tau)=\left[h_{lk,L^{\prime}k^{\prime}}(t+\Delta t,\tau) \right]_{LK\times L^{\prime}K^{\prime}}\). The amplitude complex gains between the \(lk_{th}\) antenna element at the \(LK\) transmit antenna, \(l^{\prime}k^{\prime}_{th}\) antenna element at the \(L^{\prime}K^{\prime}\) receive antenna and the delay \(\tau\), can be represented as  
\[\mathbf{H}_{lk,l^{\prime}k^{\prime}}(t+\Delta t,\tau)=\sum\limits_{n=1}^{Cl_{n} ^{out}}h_{lk,l^{\prime}k^{\prime},n}(t+\Delta t)\delta(\tau-\tau_{n}) \tag{22}\]  
Figure 2: A more detailed of field experiments of the beam pattern using spherical coordinate system for non-stationary massive MIMO channel. LOS: The \(lk_{th}\) transmit antenna element vector \(\vec{a}(\phi,\theta)_{lk}\) and the vector between \(LOS_{th}\) path \(\mathbf{D}^{BS}_{LOS,i_{1},lk}(t+\Delta t)\) can be expressed as  
\[\mathbf{D}^{BS}_{LOS,i_{1},lk}(t+\Delta t)= \begin{bmatrix}\sin\phi^{AAoD}_{LOS,i_{1}, lk}(t+\Delta t),\cos\theta^{EAoD}_{LOS,i_{1},lk}(t+\Delta t)\\ \sin\phi^{AAoD}_{LOS,i_{1},lk}(t+\Delta t),\sin\theta^{EAoD}_{LOS,i_{1},lk}(t+ \Delta t)\\ \cos\theta^{EAoD}_{LOS,i_{1},lk}(t+\Delta t)\end{bmatrix} ^{T} \tag{28}\]  
Similarly, the \(l^{\prime}k^{\prime}_{th}\) receive antenna element vector \(\vec{a}(\varphi,\theta)_{l^{\prime}k^{\prime}}\) and the vector between \(LOS_{th}\) path \(\mathbf{D}^{UE}_{LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t)\) can be presented as  
\[\mathbf{D}^{UE}_{LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t)= \begin{bmatrix}\sin \varphi^{AAoA}_{LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t),\cos\theta^{EAoA}_ {LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t)\\ \sin\varphi^{AAoA}_{LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t),\sin\theta^{ EAoA}_{LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t)\\ \cos\theta^{EAoA}_{LOS,i_{1},l^{\prime}k^{\prime}}(t+\Delta t)\end{bmatrix} ^{T} +\mathbf{D} \tag{29}\]  
Then, the two random initial phases for outdoor \(LOS_{th}\) cluster is derived as  
\[ \begin{bmatrix}\exp(j\Phi^{p,p}_{LOS})&0\\ 0&\exp(j\Phi^{h,h}_{LOS})\end{bmatrix}  \tag{30}\]  
where \(\Phi^{(p,p)}_{LOS}\), \(\Phi^{(h,h)}_{LOS}\) are two random initial phases of different polarization combination.