# Capacity Bounds for Dense Massive MIMO in a Line-of-Sight Propagation Environment
## 6 Simulation Results and Discussion
The figure also shows that the sum-capacity grows linearly and logarithmically with the average SNR for the low- and high-SNR regimes, respectively, confirming what was discussed in Sections 4.4 and 4.5. Figure 5 shows how the spectral efficiency behaves as \(\rho=P/M^{\alpha}\), where \(\alpha=1/2,1\) and \(3/2\), respectively. The following simulation setup parameters were used: \(K=10\) and \(P=80\) [dB]. As expected and stated in Remark 1, when \(\alpha=1\) and \(M\) increases, the capacity becomes constant no matter the number of antennas. However, when \(\alpha=1/2\) the capacity grows logarithmically fast with  
\(M\) when \(M\to\infty\) and tends to zero when \(\alpha=3/2\) and \(M\to\infty\). These results attest that the transmit power of each device can be reduced proportionally to \(M\). Figure 4: Sum-capacity for low- and high-SNR regimes versus average signal-to-interference ratio, \(\rho\). Figure 3: Simulated and analytical sum-capacity considering \(M=300\). In Figure 6, we show the required transmit power per device that is needed to achieve fixed capacities of 1 and 2 bits/s/Hz, respectively. As expected and predicted by Remark 1, the transmit power can be reduced by approximately 3 [dB] by doubling the number of antennas, \(M\), for both cases, i.e., 1 and 2 bits/s/Hz. Figure 5: Demonstration of the power scaling law for different \(\alpha\) values. Figure 6: Required transmit power to achieve 1 and 2 bits/s/Hz as a function of the number of antennas. Figure 7 shows the comparison between the normalized histogram of the random variable \(Z\), where \(Z\) is defined in Lemma A1, and the Gamma PDF for number of devices equal to \(K=10\), \(20\), and \(50\), respectively. These results were obtained with \(M=100\). The histogram, shown in blue, is the histogram of the random variable \(Z=\sum_{k=1}^{K}M\beta_{k}\). As can be seen, the histogram approaches the Gamma PDF as the number of devices increases. Figure 8 shows the comparison of the instantaneous sum-capacity in the high-SNR regime and favorable propagation condition and its approximation with the Gamma PDF. This comparison was obtained with the following parameters \(K=10\) and \(\rho=90\) [dB]. As can be seen, for the high SNR, the Gamma PDF fits the PDF of the instantaneous sum-capacity random variable as the number of antennas grows. Figure 9 shows the comparison between the approximated and simulated outage probabilities in the high-SNR regime and favorable propagation condition. The simulation parameters are the same used for generating the results in Figure 8. As can also be seen, for the high SNR, the Gamma CDF fits the simulated outage probability of the instantaneous sum-capacity random variable as the number of antennas grows. Additionally, we can also notice that the probability of achieving higher capacities increases with the number of antennas. The increased probability of higher capacity is due to the asymptotic favorable propagation offered by the channel as the number of antennas grows, i.e., the channels become asymptotically orthogonal as the number of antennas increases and consequently, the cross-talk interference is asymptotically mitigated. Figure 10 shows the comparison of the instantaneous sum-capacity in the low-SNR regime and favorable propagation condition and its approximation with the Gamma PDF. This comparison was obtained with the following parameters \(M=100\) and \(\rho=30\) [dB]. As can be seen, for the low SNR, the Gamma PDF fits the PDF of the instantaneous sum-capacity random variable as the number of devices simultaneously served through the same time-frequency resources grows. Figure 11 shows the comparison between the approximated and simulated outage probabilities in the low-SNR regime and favorable propagation condition. This comparison was obtained with the following parameters \(M=300\) and \(\rho=30\) [dB]. As can also be seen, for the low SNR, the Gamma  
Figure 7: Gamma probability density function (PDF) for number of devices equal to \(K=10\), \(20\), and \(50\) respectively.