## 4 System Model  
Let \(B\) denote the total bandwidth of a contiguous block of spectrum whose size is appropriate to be exploited as a single 4G LTE or 5G NR carrier. The mobile operator divides the bandwidth \(B\) into \(N\) adjacent sub-blocks, each of which is physically exploited as a separate CC and combined with the rest of sub-blocks at the receiver by means of CA. (Note that an operator may divide the total bandwidth into CCs with the same or different bandwidths. This work does not make any specific assumptions on the bandwidth allocated to each individual CC and the analytical results here presented do not take this aspect into account. The study of the impact of equal/unequal CC bandwidth allocations and the optimum configuration that maximises the total aggregated capacity is out of the scope of this work and is left as future work. However, the simulation results in [26] suggest that a homogeneous bandwidth allocation where the total bandwidth is equally divided across the used CCs may maximise the total aggregated capacity.) The transmitter's MAC layer splits the sequence of data packets generated at higher layers into \(N\) data flows, each of which is transmitted over one of the \(N\) available CCs, as shown in Figure 1. Each CC in general carries a different data stream. The CA-enabled receiver reorders the recovered packets at the MAC layer so that the process is transparent to the higher layers. The total data rate observed at higher layers is the sum of the individual rates experienced in each CC.  
When CA is employed, each CC runs its own instance of the MAC and PHY layers and their associated processes, which are dynamically adapted to the instantaneous channel quality conditions experienced in each CC. This requires the use of L1/L2 control signalling to exchange control and feedback information for each CC. The signalling overhead introduced by each CC together with the guard bands required between adjacent CCs can both be jointly characterised by a fraction \(\alpha\) (\(0<\alpha\ll 1\)) of the total bandwidth. As a result, a bandwidth \(\alpha B\) needs to be sacrificed for each employed CC and the total bandwidth \(W\) available for data transmission when \(N\) CCs are used is \(W=B(1-\alpha N)\). Note that the overhead parameter \(\alpha\) refers to the fraction of total available bandwidth \(B\) and not to the bandwidth allocated to each individual CC. In order to obtain a positive data rate, the requirement \(B(1-\alpha N)>0\) must be met, which implies that the number of CCs that can be used is constrained by the upper bound \(N\leq N_{max}=\lfloor 1/\alpha\rfloor\).  
The data transmitted through each CC experience an independent Rayleigh fading process. The probability density function (PDF) of the instantaneous SNR at the receiver is, therefore, given by equation (2.7) in [29]:  
\[f_{\gamma_{n}}(x)=\frac{1}{\overline{\gamma}_{n}}\exp\left(-\frac{x}{ \overline{\gamma}_{n}}\right),\ \ \text{for}\ n=1,\ldots,N \tag{1}\]  
and its cumulative distribution function (CDF) is obtained by integrating (1), which yields  
\[F_{\gamma_{n}}(x)=1-\exp\left(-\frac{x}{\overline{\gamma}_{n}}\right),\ \ \text{for}\ n=1,\ldots,N \tag{2}\]  
where \(\overline{\gamma}_{n}\) is the average SNR in the \(n\)th CC. 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 SNR (\(\overline{\gamma}_{n}\
eq\overline{\gamma}_{k}\) for \(n\
eq k\)), with an overall average SNR \(\overline{\gamma}=(1/N)\sum_{n=1}^{N}\overline{\gamma}_{n}\). The main motivation to consider these two scenarios regarding the SNR observed in different CCs is to provide a more comprehensive evaluation of the performance of CA as a diversity technique in scenarios with varying levels of frequency diversity.