# Performance of Cooperative Relay NOMA with Large Antenna Transmitters
## 2 System Model
#### Proposed 3D GBSM Models  
We consider the 3 GPP's standard, based on the GBSM method [90; 93; 99], to provide a new 3D channel model in which the TX antenna is modeled as URA or CA. The main parameters of importance in the 3D channel model are; delay spread (DS), angle of arrival (AoA), azimuth of departure (AoD), elevation angle of arrival (EoA) and departure (EoD) [17; 100; 101]. In addition, the 3 GPP standard under consideration presents the elevation angle of the antenna boresight into the channel and allows for dynamic adaption of the antenna's downtilt angles \(\theta_{tilt}\). This provides various advantages for 3D beamforming leading to significant improvement in system performance [95]. The 3D channel model is illustrated in Figure 4.  
From Figure 4 and the 3 GPP's model, the effective channel between the sth TX antenna port with \(M\) subpath and the \(u\)th receiver antenna port, as stated in [21; 92], is  
\[\left[\mathbf{H}_{\mathrm{s},u}^{\mathrm{3D}}\right]=\sqrt{\frac{P_{\mathrm{ n}}\sigma_{SF}}{M}}\sum_{n=1}^{N}\kappa_{n}\left[ \begin{array}{l}\sqrt{G_{TX}^{ \mathrm{3D}}(\phi_{\pi}^{AoD},\theta_{\pi}^{AoD},\theta_{tilt})}\sqrt{G_{RS}^ {\mathrm{3D}}(\varphi_{\pi}^{AoA},\theta_{\pi}^{AoA})}\\ \times\left[a_{RX}\left(\varphi_{\pi}^{AoA},\theta_{\pi}^{AoA}\right)\right]_{ u}\left[a_{TX}\left(\phi_{\pi}^{AoD},\theta_{\pi}^{AoD}\right)\right]_{s} \end{array} \right] \tag{3}\]  
where \(\kappa_{n}\) is the complex random amplitude of the \(n\)th path and \(s=1,...,N_{TX}\), \(u=1,...,N_{RS}\), \((\phi_{n}^{AoD},\theta_{n}^{AoD})\) are the azimuth and elevation angles-of-departure (AoD), respectively. \((\varphi_{n}^{AoA},\theta_{n}^{AoA})\) are the azimuth and elevation angles-of-arrival (AoA) of the \(n\)th path, respectively. The gain of antenna array at the TX is  
\[G_{TX}^{\mathrm{3D}}\left(\varphi_{n}^{AoD},\theta_{n}^{AoD},\theta_{tilt} \right)\approx G_{TX,H}\left(\varphi_{n}^{AoD}G_{TX,V}(\theta_{n}^{AoD},\theta_ {tilt})\right) \tag{4}\]The responses of the antenna array are; \((a_{RX}\big{(}\varphi_{n}^{AoD},\theta_{n}^{AoD}\big{)})\) and \((a_{TX}\big{(}\varphi_{n}^{AoD},\theta_{n}^{AoD}\big{)})\). Following the approach in [90], the precise 3D antenna radiation pattern in both the horizontal (\(G_{TX,H}\)) and vertical (\(G_{TX,V}\)) planes is provided by [21],  
\[G_{TX,H}(\varphi^{AoD})=-12\bigg{(}\frac{\varphi^{AoD}}{\varphi_{3 dB}}\bigg{)} ^{2} \tag{5}\]  
and  
\[G_{TX,V}(\theta^{AoD},\theta_{tilt})=-12\bigg{(}\frac{\theta^{AoD}-\theta_{tilt }}{\theta_{3 dB}}\bigg{)}^{2} \tag{6}\]  
where \(\varphi^{AoD}\) is the horizontal azimuth angle between the user and the array's boresight, \(\theta^{AoD}\) is the vertical elevation angle between the user and the array's boresight, and \(\varphi_{3 dB}\) and \(\theta_{3 dB}\) are the horizontal and vertical half-power beamwidths, respectively. In this study, we consider URA or CA at the TX serving a single antenna RS with half-duplex connection in the model. The path between TX and RS has a single bounce cluster giving us \(N\) paths. For the URA configuration indicated in Figure 5a, the location vector of the antenna element is as follows. As given in [102], the steering vector of the \(n\)th propagation path using the URA antenna topology shown in Figure 5a is  
\[a_{n}(\theta_{n}^{AoD},\phi_{n}^{AoD})=Z\sin\left(\theta_{n}^{AoD}\sin(\bar{ \beta}+\phi_{n}^{AoD})\right) \tag{7}\]  
where \(\bar{\beta}=\tan^{-1}\left(Z_{z}/Z_{y}\right)\), \(Z\), \(Z_{z}\) and \(Z_{y}\) can be determined from Figure 5a. The figure shows the scalar \(z\) and \(y\) axes, where \(Z\) is defined [21] as,  
\[Z=\sqrt{Z_{z}^{2}+Z_{y}^{2}} \tag{8}\]  
where,  
\[Z_{z} =2\pi dz(n-m)/\lambda, \tag{9}\] \[Z_{y} =2\pi dy(p-q)/\lambda\qquad\text{and}\] \[\varphi_{s} =cos^{-1}(Z_{z}/Z_{y})\]  
Figure 4: 3D channel model. are obtained. \((m,p)\) and \((n,q)\) are the coordinates of the elements shown in Figure 5a. The position of the URA antenna element at the TX can be determined by \(Z\angle\tilde{\beta}\).