## 7 Secrecy Capacity Analysis
Thus, the average secrecy capacity (in bits per second) can be obtained as equations (38)-(41) in [39]:  
\[C_{s} =\int_{0}^{\infty}\int_{0}^{\infty}C_{s}(x,y)f_{\gamma_{m}}(x)f_ {\gamma_{e}}(y)dxdy\] \[=W\log_{2}(e)[\mathcal{I}_{1}+\mathcal{I}_{2}-\mathcal{I}_{3}], \tag{16}\]where \(f_{\gamma_{m}}(\cdot)\) and \(f_{\gamma_{e}}(\cdot)\) denote the PDF of the instantaneous SNR in the main and eavesdrop-dropper links, respectively, and  
\[\mathcal{I}_{1} =\int_{0}^{\infty}\ln(1+x)F_{\gamma_{e}}(x)f_{\gamma_{m}}(x)dx \tag{17}\] \[\mathcal{I}_{2} =\int_{0}^{\infty}\ln(1+x)F_{\gamma_{m}}(x)f_{\gamma_{e}}(x)dx\] (18) \[\mathcal{I}_{3} =\int_{0}^{\infty}\ln(1+x)f_{\gamma_{e}}(x)dx \tag{19}\]  
with \(F_{\gamma_{m}}(\cdot)\) and \(F_{\gamma_{e}}(\cdot)\) denoting the CDF of the instantaneous SNR in the main and eavesdrop-dropper links, respectively. Based on the expressions shown above and capitalising on the effective SNR models proposed in Section 5, this section analyses the secrecy capacity of CA when used as a diversity technique. Similar to Section 6, analytical results will be provided for the average model of the effective SNR. The equivalent expressions for the ideal effective SNR model can be obtained by simply replacing \(\overline{\gamma}_{m}/N\) and \(\overline{\gamma}_{e}/N\) with \(\overline{\gamma}_{m}\) and \(\overline{\gamma}_{e}\), respectively (for the homogeneous SNR scenario), and by replacing \(\overline{\gamma}_{m,n}/N\) and \(\overline{\gamma}_{e,n}/N\) with \(\overline{\gamma}_{m,n}\) and \(\overline{\gamma}_{e,n}\), respectively (for the heterogeneous SNR scenario). **Theorem 4**. : _Secrecy capacity of single carrier scenario The secrecy capacity of the single carrier scenario is given by_  
\[C_{s} =B(1-\alpha)\log_{2}(e)\biggl{\{}\exp\biggl{(}\frac{1}{\overline{ \gamma}_{m}}\biggr{)}E_{1}\biggl{(}\frac{1}{\overline{\gamma}_{m}}\biggr{)}\] \[\quad-\,\exp\biggl{(}\frac{1}{\overline{\gamma}_{m}}+\frac{1}{ \overline{\gamma}_{e}}\biggr{)}E_{1}\biggl{(}\frac{1}{\overline{\gamma}_{m}}+ \frac{1}{\overline{\gamma}_{e}}\biggr{)}\biggr{\}} \tag{20}\]  
_where \(E_{1}(\cdot)\) is the exponential integral function._  
Proof. : Since the control signalling overhead introduced by a single carrier requires a capacity equivalent to a bandwidth \(\alpha B\), the bandwidth effectively available for data transmission is \(W=B(1-\alpha)\). Introducing (1)-(2) in (16)-(19) and noting that \(\log_{2}(x)=\log_{2}(e)\ln(x)\), the resulting integrals can be solved with the help of equation (4.337.2) in [34] and equation (5.1.7) in [37]. **Theorem 5**. : _Secrecy capacity of CA with homogeneous SNR The secrecy capacity of CA with homogeneous SNR_  
\[C_{s} =B(1-\alpha N)\log_{2}(e)\Biggl{\{}\exp\biggl{(}\frac{N}{ \overline{\gamma}_{m}}\biggr{)}\sum_{n=1}^{N}\frac{\Gamma\biggl{(}n-N,\frac{N} {\overline{\gamma}_{m}}\biggr{)}}{\biggl{(}\frac{N}{\overline{\gamma}_{m}} \biggr{)}^{n-N}}\] \[\quad-\exp\biggl{(}N\biggl{[}\frac{1}{\overline{\gamma}_{m}}+ \frac{1}{\overline{\gamma}_{e}}\biggr{]}\biggr{)}\sum_{n=1}^{N}\frac{(n+N-2)!}{( n-1)!(N-1)!}\] \[\quad\times\biggl{(}\frac{N}{\overline{\gamma}_{m}\overline{ \gamma}_{e}}\biggr{)}^{n+N-1}\biggl{(}\overline{\gamma}_{m}^{n-1}\overline{ \gamma}_{e}^{N}+\overline{\gamma}_{e}^{n-1}\overline{\gamma}_{m}^{N}\biggr{)}\] \[=W\log_{2}(e)[\mathcal{I}_{1}+\mathcal{I}_{2}-\mathcal{I}_{3}], \tag{16}\]0  
_where \(\Gamma(\cdot,\cdot)\) represents the upper incomplete gamma function._  
Proof. : The overhead introduced by \(N\) CCs requires a capacity equivalent to a bandwidth \(\alpha NB\), thus \(W=B(1-\alpha N)\). Introducing (8)-(9) in (16)-(19), substituting the lower incomplete gamma function with its equivalent form in equation (8.352.1) in [34] and noting that \(\log_{2}(x)=\log_{2}(e)\ln(x)\), integrals of the form of equation (15.24) in [29] are obtained, which can be resolved with the help of equation (15B.7) in [29]. After reorganising and grouping terms, the expression shown in (21) is obtained.