# CUDA-Optimized GPU Acceleration of 3 GPP 3D Channel Model Simulations for 5G Network Planning
## 3 The 3 GPP Channel Model for 5G NR
}_{\text{SpendVect}}\end{split} \tag{1}\]  
where \(F_{\text{rx},u,\theta}\), and \(F_{\text{rx},u,\phi}\) are the field patterns of the \(u\)th element of receiving antenna in the direction of the spherical basis vector, \(F_{\text{tx},s,\theta}\), and \(F_{\text{tx},s,\phi}\) are the field patterns of the \(s\)th element of transmitting antenna, \((\theta,\phi)_{\alpha,\beta,\gamma}\) are the elevation angles \(\gamma\) for ray \(\beta\) in cluster \(\alpha\), and \(\hat{r}\) and \(\hat{d}\) are the spherical unit vector and location vector of antenna elements rx, tx, respectively. Similarly, \(\hat{r}_{\text{tx},n,m}\) is the spherical unit vector with azimuth departure angle \(\phi_{n,m,\text{AOD}}\) and elevation departure angle \(\theta_{n,m,\text{ZOD}}\), and XPR is the cross-polarization power ratio matrix. Rays from the same cluster have identical power levels, denoted by \(P_{n}\). The LOS component of the CIR is calculated as in Equation (2). \[ \begin{split} H_{u,s,1}(t)=\underbrace{\begin{bmatrix}F_{\text{ rx},u,\theta}(\theta_{\text{LOS},\text{ZOA}},\phi_{\text{LOS},\text{AOA}})\\ \underbrace{F_{\text{rx},u,\theta}(\theta_{\text{LOS},\text{ZOA}},\phi_{\text{ LOS},\text{AOA}})}_{\text{SpeedVect}^{los}}\end{bmatrix}^{T}\times \begin{bmatrix}1&0\\ 0&-1\end{bmatrix}\times\begin{bmatrix}F_{\text{rx},s,\theta}(\theta_{\text{ LOS},\text{ZOD}},\phi_{\text{LOS},\text{AOD}})\\ F_{\text{rx},s,\theta}(\theta_{\text{LOS},\text{ZOD}},\phi_{\text{ LOS},\text{AOD}})\end{bmatrix}\\ \cdot\underbrace{\exp\left(-j2\pi\frac{d_{\text{SD}}}{\lambda_{0}}\right)\exp \left(j2\pi\frac{p_{\text{rx},\text{LOS}}^{T}d_{\text{rx},s}}{\lambda_{0}} \right)\exp\left(j2\pi\frac{p_{\text{rx},\text{LOS}}^{T}d_{\text{rx},s}}{ \lambda_{0}}\right)\exp\left(j2\pi\frac{p_{\text{rx},\text{LOS}}^{T}\cdot\tilde{d} _{\text{rx},s}}{\lambda_{0}}t\right)}_{ClusterVectlos}\end{split} }_{\text{ ClusterVectlos}}\end{split} \tag{2}\]  
The Doppler frequency component depends on the UE speed \(v\) with velocity vector \(\tilde{v}\), arrival angles (AOA, ZOA), travel elevation angle \(\theta_{v}\), and azimuth angle \(\phi_{v}\)  
\[v_{n,m}=\frac{\hat{r}_{\text{rx},n,m}^{\text{T}}\cdot\tilde{v}}{\lambda_{0}} \tag{3}\]  
Figure 1: Multi-path scattering in 3D channel model. The CIR can be expressed as the sum of the LOS channel coefficient and the NLOS channel impulse response as  
\[H_{u,s}(t)=\underbrace{H_{u,s,1}(t)}_{LOS}+\underbrace{\sum_{n=1}^{N}\sum_{m=1}^{ M}H_{u,s,n,m}(t)}_{NLOS} \tag{4}\]  
In communication over the wireless channel, a transmitted signal \(x(t)\) arrives at the receiver with a time delay of \(x(t-\tau)\). The received signal comprises multiple reflections or refraction, resulting in identical duplicates of the originally transmitted signal. \[y(t)=\sum x(t-\tau)H_{u,s}(t,\tau) \tag{5}\]  
This represents the convolution of the CIR and the transmitted signal, i.e., an finite impulse response (FIR) filter where the CIR represents the set of coefficients, and can be rewritten as  
\[y(t)=(x\mathbin{\textcircled{\char 37}}H)(t) \tag{6}\]  
where \(\mathbin{\textcircled{\char 37}}\) denotes convolution. User equipment (UE) spatial coordinates in a 3D space affect the CIR. Mobile devices require continuous location calculations, so this variability is critical. This model uses delays and ray mapping from [12] (Table 7.5-5). To test the channel model, we used the values in Table 1.  
The total execution time to compute these coefficients and the respective speedup over the CPU baseline are reported in Section 6. Cluster-delay line (CDL) serves as a modeling tool in scenarios where the received signal comprises several delayed clusters. Each cluster is composed of multipath components that share a common delay, albeit exhibiting slight variations in angles of departure and arrival. Various CDL profiles have been defined by 3 GPP for link-level simulations. For NLOS, three CDL profiles, namely, CDL-A, CDL-B, and CDL-C, are defined and CDL-D and CDL-E are constructed for LOS clusters.