## 8 Results
### Ergodic Capacity Results  
Figure 2 shows the ergodic capacity of CA (in terms of the spectral efficiency, i.e., bit/s/Hz) as a function of the number of CCs for various values of the bandwidth overhead parameter \(\alpha\). The capacity obtained for single-carrier transmission (i.e., \(N=1\)) is plotted as a horizontal solid black line along the width of the figure to provide a reference threshold that determines the range of CCs for which the considered CA-based transmission scheme outperforms the benchmark (i.e., the case of single-carrier transmission). Note that this threshold is different for each case plotted in this figure and, as is the case in this figure and other figures that will be shown later on, the reference lines may be very close to each other. The first relevant observation is that these results corroborate that CA can be effectively employed as a diversity technique to increase the data rate of the system without increasing the available bandwidth, which can be confirmed by noting that the curves for CA can lead to a higher spectral efficiency than the single carrier scenario (in certain regions of the figure if the number of CCs is correctly configured, which will be discussed later on). It can also be noted that the performance of CA tends to be slightly higher in the heterogeneous SNR scenario than in the homogeneous counterpart. This can be explained by the fact that the level of frequency diversity in the heterogeneous scenario is higher and in such a case the proposed method, based on the use of CA as a diversity technique, can benefit from it to a greater extent. The bandwidth overhead parameter \(\alpha\) is also observed to have a significant impact on the resulting performance. As expected, if the use of CA incurs a higher bandwidth penalty (i.e., higher value of \(\alpha\)), a lower amount of net bandwidth is available for data transmission, which leads to lower data rates. Note that in the hypothetical case of \(\alpha=0\) (i.e., if no bandwidth penalty is incurred by using CA), then the capacity would monotonically increase indefinitely with the number of CCs, as this would increase the frequency diversity gain without any bandwidth penalties. However, in a realistic case with \(\alpha>0\), the curves in Figure 2 become convex. This indicates the existence of an optimum number of CCs that maximises the data rate for each value of \(\alpha\). As explained in Section 3, increasing the number of CCs will initially increase the data rate as a result of increasing the frequency diversity in the system. However, this also increases the total amount of signalling traffic and bandwidth reserved for guard bands (i.e., bandwidth penalty), which reduces the total amount of bandwidth available for data transmission and contributes to reducing the data rate. If the number of CCs is sufficiently high, the bandwidth penalty incurred by a high number of CCs will exceed the gain obtained from the frequency diversity, thus effectively reducing the data rate. In fact, if the number of CCs is made arbitrarily large, it can be seen in Figure 2 that the data rate of CA can indeed fall below that of the single carrier scenario, and this will occur sooner (i.e., for a lower number of CCs) when the bandwidth penalty \(\alpha\) is higher. In the extreme case where the number of CCs is set equal to \(N=N_{max}=\left\lfloor 1/\alpha\right\rfloor\) (e.g., see the case \(\alpha=0.05\) and \(N=20\)), then all the available bandwidth is employed to accommodate signalling traffic and guard bands, which effectively reduces the total capacity available for data transmission to zero. The optimum number of CCs, thus, depends on the bandwidth penalty parameter. From Figure 2 it can be determined that the optimum number of CCs that maximises the data rate for \(\alpha=\left\{0.01,0.02,0.03,0.04,0.05\right\}\) is 16, 9, 7, 5 and 4, respectively, which is in line with the observations above.