## 3 Dependence of Geometric Placement of Antenna Array Elements in Regards to Radio Link Improvements
### Effects of Circular Antenna Arrays over Doppler Power Spectrum and Coherence Time Metrics  
In order to determinate the effects of circular antenna arrays over the Doppler power spectrum, defined by [PERSON], and the coherence time model proposed by [PERSON], we used the simulation parameters described in Table 2. In addition to the simulation parameters, with the purpose of calculating the maximum Doppler frequency, a mobile user speed used in urban environments was also considered. Figure 5: Circular antenna array models (**a**) CA-21 and (**b**) CA-47 in CST MicroWave Studio Suite. The simulation results, in regard to the Doppler power spectrum, can be observed in Figure 7, where the frequency of the carrier signal increases and the spectrum becomes broader and more extensive due to the effect of the Doppler shift. Power levels are also dependent on the Doppler frequency; in this sense, a higher frequency tends to have lower power levels. This effect can be counteracted by increasing the antenna gain; thus, improving the possibility of receiving the carrier signal with minimal effect from the Doppler effect. The use of an antenna array is beneficial; there were improvements of up to 10 dB the power level of the carrier signal, with respect to the DPS of the Clarke model. The values of DPS for each 5G frequency band analyzed, at \(f/fd=0\), are summarized in Table 3.  
We can observe that, in Table 3, all carrier frequencies analyzed showed an increase in the power levels as the Doppler frequency increases; thus, demonstrating the mitigation of the Doppler effect and obtaining high power levels at the central frequency. With these \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable or Parameter** & **Value** \\ \hline \(f_{c}\): Carrier frequencies & 700 MHz, 3.5 GHz, 26 GHz, 48 GHz \\ \hline \(D_{r,\lambda}\): Normalized radius & 700 \(\lambda\), 560 \(\lambda\), 520 \(\lambda\), 480 \(\lambda\) \\ \hline \(v\): Users velocity & 30 km/h \\ \hline \(R\): Correlation coefficient & 0.5 \\ \hline \(\mu_{r}\): Pointing angle (small) & 1\({}^{\circ}\) \\ \hline \(\mu_{r}\): Pointing angle (not small) & 90\({}^{\circ}\) \\ \hline
**Circular array** & 21 elements & 47 elements \\ \hline \(G_{0}\): Array gain & 16.5 dB & 24.1 dB \\ \hline \(\theta\): Beamwidth & 24.7\({}^{\circ}\) & 11.11\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters considered. Figure 6: 3D radiation pattern yielded by **(a)** CA-21 at 3.5 GHz and **(b)** CA-47 at 26 GHz in CST MicroWave Studio Suite. results, we can verify the improvement of the behavior of the Doppler power spectrum, using a receiver or transmitter with high antenna gains, regardless of the analyzed carrier frequency. For the simulation of the coherence time, the following suitable parameters were considered: a circular geometry and the variation in the number of elements of the phased array of antennas. For the antenna array of 21 elements (CA-21), a beamwidth of \(\theta=24.7^{\circ}\) was attained; for the antenna array of 47 elements (CA-47), a beamwidth of \(\theta=11.11^{\circ}\) was obtained. Four carrier frequencies were considered--two less than, and two higher than 6 GHz. With the idea of establishing a comparison of the coherence time values obtained in our simulation of the modified Va Model (VaMod), we show the behavior of the IEEE model for 5G, reported in [19; 23], which cannot generate directive beams. Figure 8 shows the behavior of the coherence time, taking into account the previous parameters. From Figure 8, it is possible to distinguish, in both cases, LOS and NLOS conditions, very high coherence time values of the order of several seconds in the window of 0 to 30\({}^{\circ}\) beamwidth. However, it is not recommended to generate this beamwidth since it would imply greater complexity in the structure of the circular phased antenna array.