## 6 Ergodic Capacity Analysis  
The concept of channel capacity is a fundamental concept in communication theory, particularly in wireless communication systems. The capacity of a communication channel represents the maximum rate at which information can be transmitted reliably over that channel. The instantaneous capacity of a channel (in bits per second) is given by \(W\log_{2}(1+\gamma)\), where \(W\) is the channel bandwidth and \(\gamma\) is its instantaneous SNR. Since the instantaneous SNR of a wireless communication channel fluctuates randomly, the instantaneous channel capacity will fluctuate accordingly. The concept of ergodic capacity extends the notion of channel capacity to account for the statistical variations in channel conditions over time. Instead of considering the capacity of the channel at a specific time instant, the ergodic capacity evaluates the average capacity of the channel over an extended period, assuming that the channel conditions vary randomly according to certain statistical distributions. Therefore, the ergodic channel capacity is a measure of the achievable data rate or capacity of a communication channel in the long term, considering statistical variations in the channel conditions, such as those caused by the presence of fading. Based on the concept of effective SNR, the ergodic capacity of a CA scenario can be obtained in bits per second as  
\[C=W\int_{0}^{\infty}\log_{2}(1+x)f_{\gamma_{\text{CA}}}(x)dx \tag{12}\]  
where \(W\) is the total aggregated bandwidth effectively available for data transmission and \(f_{\gamma_{\text{CA}}}(x)\) is the effective SNR PDF. In this section, analytical results will be provided only for the average SNR model presented in Section 5.2. The counterparts for the ideal model of Section 5.1 can be obtained by simply replacing \(\overline{\gamma}/N\) with \(\overline{\gamma}\) (homogeneous SNR scenario) and \(\overline{\gamma}_{n}/N\) with \(\overline{\gamma}_{n}\) (heterogeneous SNR scenario). **Theorem 1**. : _Ergodic capacity of single carrier scenario The ergodic capacity of the single carrier scenario is given by_  
\[C=B(1-\alpha)\log_{2}(e)\exp\left(\frac{1}{\overline{\gamma}}\right)\text{E}_ {1}\left(\frac{1}{\overline{\gamma}}\right) \tag{13}\]  
_where \(\text{E}_{1}(x)=\int_{x}^{\infty}e^{-t}t^{-1}dt\) denotes the exponential integral function, equation (5.1.1) in [37]._  
Proof. : Since the signalling overhead introduced by a single carrier requires a capacity equivalent to a bandwidth \(\alpha B\), the bandwidth effectively available for data is \(W=B(1-\alpha)\). Introducing (1) in (12) and noting that \(\log_{2}(x)=\log_{2}(e)\ln(x)\), the resulting integral can be solved with the help of equation (4.337.2) in [34] and equation (5.1.7) in [37]. **Theorem 2**. : _Ergodic capacity of CA with homogeneous SNR The ergodic capacity of CA with homogeneous SNR is_  
\[C=B(1-\alpha N)\log_{2}(e)\exp\left(\frac{N}{\overline{\gamma}}\right)\sum_{n= 1}^{N}\frac{\Gamma\left(n-N,\frac{N}{\overline{\gamma}}\right)}{\left(\frac{N }{\overline{\gamma}}\right)^{n-N}} \tag{14}\]  
_where \(\Gamma(\cdot,\cdot)\) represents the upper incomplete gamma function defined in equation (8.350.2) in [34]._  
Proof. : The control signalling overhead introduced by \(N\) CCs requires a capacity equivalent to a bandwidth \(\alpha NB\), thus \(W=B(1-\alpha N)\). Introducing (8) in (12) and noting that \(\log_{2}(x)=\log_{2}(e)\ln(x)\), an integral of the form of equation (15.24) in [29] is obtained, which can be resolved with the help of equation (15B.7) in [29] to obtain the expression shown in (14). **Theorem 3**. : _Ergodic capacity of CA with heterogeneous SNR The ergodic capacity of CA with heterogeneous SNR is_  
\[C=B(1-\alpha N)\log_{2}(e)\sum_{n=1}^{N}\Omega_{n}\exp\left(\frac{N}{ \overline{\gamma}_{n}}\right)\text{E}_{1}\left(\frac{N}{\overline{\gamma}_{n} }\right). \tag{15}\]  
Proof. : For \(N\) CCs, \(W=B(1-\alpha N)\). Introducing (10) in (12) yields a sum of integrals, each of which can be resolved as discussed in the proof of Theorem 1. **Remark 2**.