## 8 Results
### Ergodic Capacity Results
The impact of the bandwidth penalty \(\alpha\) on the optimum number of CCs and the resulting capacity is more clearly illustrated in Figure 3, where three scenarios are considered, namely, a 4G LTE scenario with a maximum of 5 CCs, a 5G NR scenario with a maximum of 16 CCs, and an ideal scenario where the number of CCs that can be aggregated is unconstrained (i.e., unlimited). Table 1 shows the capacity achieved by CA for various levels of bandwidth penalty (\(\alpha\)) and the improvement with respect to the single carrier scenario, where no CA is used. Notice that, according to Figure 3, 4G LTE and 5G NR can achieve the same performance as the unconstrained CA scenario as long as the bandwidth penalty does not exceed the limits \(\alpha<0.04\) and \(\alpha<0.01\), respectively, in which cases the capacity improvement with respect to the single carrier scenario is equal to 64% for 4G LTE and 120% for 5G NR. Nevertheless, even if the bandwidth penalty exceeds these limits and the maximum number of CCs permitted by the 3 GPP standard is taken into account, the use of CA as a diversity technique can still provide substantial capacity improvements with respect to the single carrier scenario, as evidenced by the results shown in Table 1.  
Figure 3: Optimum number of component carriers as a function of the bandwidth overhead parameter \(\alpha\) (**left**) and the resulting ergodic capacity (**right**). Figure 2: Ergodic capacity as a function of number of CCs for various values of bandwidth overhead parameter \(\alpha\) (average SNR = 10 dB, \(\varepsilon\) = 3 dB). Figure 4 shows the ergodic capacity of CA as a function of the number of CCs for various values of the average SNR experienced in the different considered scenarios. As expected, the capacity performance improves when the SNR increases and degrades when the SNR decreases, thus indicating that the dependency on the SNR is adequately captured. This figure also corroborates the main observations made when discussing Figure 2, namely, that (i) CA can be effectively employed as a diversity technique in order to improve the spectral efficiency with respect to the traditional single carrier scenario based on the same communication bandwidth (provided that the number of CCs is correctly configured); (ii) the performance of CA as a diversity technique tends to increase in the heterogeneous SNR scenario compared to the homogeneous SNR counterpart as a result of a richer diversity; and (iii) there exists an optimum number of CCs that maximises the total capacity, in this case, for each considered average SNR, for the same reasons discussed for Figure 2. Regarding the last observation, it can be seen in Figure 4 that the optimum number of CCs decreases as the average SNR increases. More concretely, for the numerical example shown in Figure 4, the data rate is maximised with 6, 4, and 3 CCs when the average SNR is 0, 10, and 20 dB, respectively. This trend can be explained intuitively based on the fact that a higher average SNR can be associated with a shorter communication distance, where the level of diversity can be expected to be lower than in a longer distance link (where the signal can find a larger number of diverse paths between the transmitter and receiver). Thus, a lower diversity gain means that a lower number of CCs can be used while guaranteeing that the bandwidth penalty associated with the number of CCs does not exceed the diversity gain. The optimum number of CCs as a function of the average SNR for a broader range of average SNR values is illustrated in Figure 5 along with the capacity obtained when the optimum number of CCs is used for every average SNR. This figure shows clearly how the use of CA as a diversity technique can significantly improve the spectral efficiency in order to increase the system capacity without increasing the amount of spectrum employed.