## 4 Performance Analysis
### Secrecy Outage Probability (SOP)  
#### 4.2.1 Exact Analysis  
These are the most vital performance indices which describe the probability that the instantaneous secrecy capacity falls below a predefined threshold rate of confidential information \(R_{s}\) and can be defined mathematically as follows [6]:  
\[SOP=\int_{0}^{\infty}F_{\alpha q}(\Theta\gamma)f_{e}(\gamma)d\gamma, \tag{23}\]  
where \(\Theta=e^{R_{s}}\). By invoking (12) and (22) into (23), the system SOP can be expressed as:  
\[SOP=I_{1}+I_{2}. \tag{24}\]  
Thus,  
\[I_{1}=\frac{\xi^{2}\tau^{\alpha+\beta-2}\eta exp(-K)}{2\pi^{r-1}(\Gamma( \alpha)\Gamma(\alpha))}\sum_{d=0}^{\infty}\frac{K^{l}}{(l! )^{2}}G_{r+1,3r+1}^ {3r,2}\bigg{(}\frac{(\alpha\beta)^{r}}{\eta r^{2}\mu_{r}}\bigg{|} \begin{array} []{c}-l,1,\,\psi_{1}\\ \psi_{2},0\end{array} \bigg{)}d\gamma. \tag{25}\]  
By utilizing the integral identity detailed in Reference [38] (Equation (7.813(1))), the \(I_{1}\) term of (25) can be expressed as:  
\[I_{1}=\frac{\xi^{2}\tau^{\alpha+\beta-2}exp(-K)}{2\pi^{r-1}(\Gamma(\alpha) \Gamma(\alpha))}\sum_{d=0}^{\infty}\frac{K^{l}}{(l! )^{2}}G_{r+2,3r+1}^{3r,2} \bigg{(}\frac{(\alpha\beta)^{r}\Theta}{\eta r^{2}\mu_{r}}\bigg{|} \begin{array} []{c}-l,1,\,\psi_{1}\\ \psi_{2},0\end{array} \bigg{)}. \tag{26}\]  
Similarly,  
\[ \begin{array}{c}I_{2}=\omega\eta exp(-K)\sum_{l=0}^{\infty}\sum_{q_{1}=0}^{m _{h}-1}\cdots\sum_{q_{N_{h}}=0}^{m_{h}-1}\frac{(K\eta)^{l}\Xi(N_{u})\Delta_{H} ^{-\lambda}}{(l!)^{2}}\int_{0}^{\infty}\gamma^{l}\exp(-\eta\gamma)\times\\ H_{1,03,1;3,2}^{0,11,20,3}\left(\begin{array}{c}(1,-1,\frac{1}{r})\\ -\end{array}\bigg{|}\begin{array}{c}(1-\lambda,1)(0,1)(1,1)\\ (0,1)\end{array}\bigg{|}\begin{array}{c}(1-\xi^{2},1)(1-\alpha,\,1)(1-\beta, 1)\\ (-\xi^{2},1)\Big{(}0,\,\,\frac{1}{r}\Big{)}\end{array}\bigg{|}\begin{array}{c} \frac{1}{\Delta_{H}\Gamma},\frac{\mu^{1/r}}{(\gamma\Theta)^{1/r}M}\end{array} \right)d\gamma.\end{array}  \tag{27}\]  
By converting the H-Fox function to integral form using the identity detailed in Reference [45], (27) can be expressed as:  
\[ \begin{array}{c}I_{2}=\omega\eta exp(-K)\sum_{l=0}^{\infty}\sum_{q_{1}=0}^{ m_{h}-1}\cdots\sum_{q_{N_{h}}=0}^{m_{h}-1}\frac{(K\eta)^{l}\Xi(N_{u})\Delta_{H} ^{-\lambda}}{(l!)^{2}(2\pi l)^{2}}\int_{L_{1}}\int_{L_{2}}\Gamma\big{(}\frac{t }{r}-s\big{)}\frac{\Gamma(\lambda+s)\Gamma(-s)\Gamma(s+1)}{\Gamma(1-s)}\times \\ \frac{\Gamma\big{(}\xi^{2}+l\Gamma(s+t)\Gamma(\beta+t)\big{)}}{\Gamma(\xi^{2} +1+t)\Gamma\big{(}\frac{t}{r}+1\big{)}}(\Delta_{H}C)^{-s}\bigg{(}\frac{\Theta^ {1/r}M}{\mu^{1/r}}\bigg{)}^{-t}\underbrace{\int_{0}^{\infty}\gamma^{l-t/r} \exp(-\eta\gamma)d\gamma}_{l_{3}}dsdt.\end{array}  \tag{28}\]  
By applying the integral identity defined in Reference [38] (Equation (3.326(2))), the \(I_{3}\) term of (28) can be solved as:  
\[I_{3}=\Gamma\bigg{(}l-\frac{t}{r}+1\bigg{)}\eta^{-(l-\frac{t}{r}+1)}. \tag{29}\]  
Putting (29) into (28) and apply the identity given in Reference [45], the \(I_{2}\) term of (28) can be obtained as:\[ \begin{array}{c}I_{2}=\omega exp(-K)\sum_{l=0}^{\infty}\sum_{q_{1}=0}^{m_{l}-1} \cdots\sum_{q_{N_{k}}=0}^{m_{m}-1}\frac{K^{2}(\Sigma(N_{k})\Delta_{l}^{-1})}{(l )^{2}}\times\\ H_{1,0,3:1,3}^{0,1:1:2,1,3}\left(\begin{array}{c}(1,-1,\frac{1}{r})\\ -\end{array}\right)\left|\begin{array}{c}(1-\lambda,1)(0,1)(0,1)\\ (0,1)\end{array}\right|\begin{array}{c}(1-\xi^{2},1)(1-\alpha,\,1)(1-\beta,1) \\ (-\xi^{2},1)\Big{(}0,\,\,\frac{1}{r})\end{array}\right|\frac{1}{\Delta_{l}\Gamma^ {\ast}},\frac{(\eta\mu)^{1/r}}{\Theta^{1/r}M}\right).\end{array}  \tag{30}\]  
By substituting (26) and (30) into (24), the system SOP can be obtained as:  
\[\begin{array}{c}SOP=\frac{\xi^{2}r^{\phi+\beta-2}exp(-K)}{2r^{\sigma-1}( \Gamma(a)\Gamma(a))}\sum_{l=0}^{\infty}\frac{K^{l}}{(l)^{2}}G_{r+2\beta+1}^{ 3\sigma+1}\left(\frac{(a\beta)^{\  
secrecy performances, it is depicted by the results that heterodyne detection offers the system better performance than IMDD detection.