## 8 Results  
The performance of CA as a diversity technique is evaluated numerically in this section based on the analytical results obtained in Sections 6 and 7, considering as a benchmark the case of single-carrier transmission over the same amount of bandwidth used by the considered CA-based transmission scheme. The main aim is to determine whether the mathematical model and closed-form expressions derived in this work can correctly predict the trends observed by simulations in [22] and to determine the impact of various relevant parameters on the system performance. Note that a direct quantitative comparison of the numerical results obtained in this section with those obtained from simulations in [22] would not be meaningful, not only because of the approximated nature of the effective SNR models considered in this work but also the practical limitations of the simulator employed in [22], which is based on 4G LTE, and therefore, supports up to five CCs only. For this reason, the comparison with [22] is carried out in a qualitative manner by verifying that the main conclusions derived from the study presented in [22], which are summarised in Section 3, are also reached when evaluating the mathematical modelling framework developed in this work. This approach is sufficient for the main objective of this section, which is to demonstrate that the developed mathematical model can characterise the operation of CA as a diversity technique and provide a theoretic basis that supports and explains the findings of the simulation study presented in [22].  
As discussed in Section 4, two frequency diversity scenarios are considered, namely, a homogeneous SNR scenario, where all CCs experience the same average SNR (\(\overline{\gamma}_{n}=\overline{\gamma}\) for \(n=1,\ldots,N\)), and a heterogeneous SNR scenario, where each CC experiences a different average SNR (\(\overline{\gamma}_{n}\
eq\overline{\gamma}_{k}\) for \(n\
eq k\)), with an overall average SNR \(\overline{\gamma}=(1/N)\sum_{n=1}^{N}\overline{\gamma}_{n}\). In the heterogeneous SNR scenario, the average SNR of each individual CC is selected from the interval \(10\log_{10}\overline{\gamma}_{n}\in[10\log_{10}\overline{\gamma}-\varepsilon, 10\log_{10}\overline{\gamma}+\varepsilon]\), where \(\varepsilon\) (in dB) is a spread parameter that determines the level of potential frequency diversity (the larger the value of \(\varepsilon\), the higher the level of diversity in the propagation scenario). For simplicity, the performance is mainly investigated for the ideal model of effective SNR. The numerical results obtained for such a model show more pronounced trends that illustrate more clearly the impact of different relevant parameters. Some examples illustrating the impact of considering the average model of effective SNR are also shown, corroborating that the same trends are observed (just with different numerical values) and the same conclusions are therefore obtained. For convenience, all the capacity performance results presented in this section are given in terms of the ratio \(C/B\), which represents the capacity per unit bandwidth (in bit/s/Hz), and therefore, does not require the consideration of a specific channel bandwidth for numerical evaluation.