# Performance of Cooperative Relay NOMA with Large Antenna Transmitters
## 2 System Model
#### Proposed 3D GBSM Models
\(d_{r}\) is the distance between the receiving ports of the RX antennas. The location vector of \(s\)th transmit element of the URA is expressed as \(v_{t}\cdot x_{s}=\cos(\phi-\varphi_{s})\sin\theta\) where \(\varphi_{s}=2\pi(n-1)/N\). The array response of the \(s\)th of TX antenna port of URA considering (3) is given by  
\[[a_{TX}(\phi_{n}^{AoD},\theta_{n}^{AoD})]_{s}=\exp\left(ikZ\cos(\phi_{n}^{AoD} -\varphi_{s})\sin\theta_{n}^{AoD}\right) \tag{10}\]  
Similarly, the response of \(u\)th UE antenna port with regard to the 3D channel model is  
\[G_{TX,H}(\varphi^{AoD})=-12\bigg{(}\frac{\varphi^{AoD}}{\varphi_{3 dB}}\bigg{)} ^{2} \tag{5}\]0  
Thus, the 3D GBSM channel model of the URA transmitter configuration of the proposed mMIMO CR-NOMA model in Figure 3 is  
\[H_{s,\mu}^{\text{URA}}=\sqrt{\frac{P_{n}\sigma_{SF}}{M}}\sum_{n=1}^{N}\kappa_{ n}\left[ \begin{array}{c}\sqrt{G_{TX}^{\text{3D}}(\phi_{n}^{AoD},\theta_{n}^{AoD },\theta_{tilt})}\exp\left(ikZ\cos(\phi_{n}^{AoD}-\varphi_{s})\sin\theta_{n}^ {AoD}\right)\\ \times\sqrt{G_{RS}^{\text{3D}}(\varphi_{n}^{AoA},\theta_{n}^{AoA})}\exp\left( ik(u-1)d_{r}\sin\varphi_{n}^{AoA}\sin\theta_{n}^{AoA}\right)\end{array} \right] \tag{12}\]  
where \(P_{n}\) denotes power of the \(n\)th path, \(\sigma_{SF}\) is lognormal shadow fading of the \(n\)th path. \(N\) is the number of propagation path, \(M\) is the number of subpath per path, \(d_{r}\) is the separation between the receiving antenna ports, \(k\) is the wave number and \(G_{RS}^{\text{3D}}\) is the gain of the RS antenna. In the case of CA, Figure 5b, considering the array dimension of CA as \(l=4\lambda\) with radius given by \(\rho=4\lambda/l\), if \(d_{z}=4\lambda/M\) is the wavelength distance in meters between antenna elements on the first and second position in the \(z\) direction. Then \(4\lambda(m-1)/M\) wavelengths may be used to estimate the placement of the third and following antenna components, where \(m=1,...,M\) is the number of antenna elements on a ring in the \(z\)-axis, and \(\lambda\) is the wavelength in meters. The angular position of the \(n\)th element of the \(m\)th of CA on the \(x\)-\(y\) plane is expressed as \(\varphi_{s}=2\pi(n-1)/N\). Consequently, the location vector of  
Figure 5: Geometric structure of (**a**) rectangular array (URA) and (**b**) cylindrical array (CA). the elements is \(v_{t}\cdot x_{s}=\cos(\phi-\varphi_{s})\sin\theta\). Furthermore, the array response of \(s\)th transmit antenna port of the CA is given by  
\[\left[a_{TX}(\phi_{n}^{AoD},\theta_{n}^{AoD})\right]_{s}=\exp\left(ik\rho\frac{4 \lambda(m-1)}{M}\cos(\phi_{n}^{AoD}-\varphi_{s})\sin\theta_{n}^{AoD}\right) \tag{13}\]  
Similarly, the response of the \(u\)th UE antenna port using the 3D channel model is  
\[\left[a_{RX}(\varphi_{n}^{AoA},\theta_{n}^{AoA}\right]_{u}=\exp\left(ik(u-1)d_ {r}\sin\varphi_{n}^{AoA}\sin\theta_{n}^{AoA}\right) \tag{14}\]  
The final 3D GBSM channel model of the proposed model in Figure 3 between a single antenna receiving port \(u\)th and a CA transmitter's transmit antenna port sth is given by [21]  
\[H_{s,u}^{\text{CA}}=\sqrt{\frac{P_{n}\sigma_{SF}}{M}}\sum_{n=1}^{N}\kappa_{n} \left[ \begin{array}{l}\sqrt{G_{TX}^{\text{3D}}(\phi_{n}^{AoD},\theta_{n}^{ AoD},\theta_{tilt})}\\ \times\exp\left(ik\rho(4\lambda(m-1)\right)/M)\cos(\phi_{n}^{AoD}-\varphi_{s} )\sin\theta_{n}^{AoD})\\ \times\sqrt{G_{RS}^{\text{3D}}(\varphi_{n}^{AoA},\theta_{n}^{AoA})}\exp\left( ik(u-1)d_{r}\sin\varphi_{n}^{AoA}\sin\theta_{n}^{AoA}\right)\end{array} \right] \tag{15}\]