## 4 An Investigation for a Remaining Issue
### Outage Probability Analysis  
The system outage probability is defined as the probability of the event that the instantaneous system capacity, \(C=\frac{1}{2}\text{log}_{2}\left(1+\gamma_{SR_{k}D}\right)\), is less than a predefined threshold \(R_{0}\) in \(\text{bits}/\text{s}/\text{Hz}\)  
\[P_{out} =\Pr\left[C<R_{0}\right]\] \[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\] \[=F_{\gamma_{SR_{k}D}}\left(2^{2R_{0}}-1\right), \tag{14}\]  
Figure 3: A multiple half-duplex AnF system in the considered correlated environment. where \(R_{0}\) is the system spectral efficiency in bits/s/Hz. As deriving an exact expression for the system outage probability is cumbersome, for a more tractable analysis, we start with the bounds on the end-to-end received SNR given in Equation (13). The CDF of min \(\left(\gamma_{SR_{k}},\gamma_{R_{k}D}\right)\), \(F_{\min}^{k}\left(\gamma\right)\), is formulated as follows  
\[F_{\min}^{k}\left(\gamma\right) =\Pr\left[\min\left(\gamma_{SR_{k}},\gamma_{R_{k}D}\right)\leq \gamma\right]\] \[=2-\Pr\left[\gamma_{SR_{k}}>\gamma\right]-\Pr\left[\gamma_{R_{k}D }>\gamma\right]-\Pr\left[\gamma_{SR_{k}}\leq\gamma,\gamma_{R_{k}D}\leq\gamma \right], \tag{15}\]  
where the marginal probabilities \(\Pr\left[\gamma_{SR_{k}}>\gamma\right]\) and \(\Pr\left[\gamma_{R_{k}D}>\gamma\right]\) are equal to \(e^{-\frac{\gamma}{\lambda_{k}}}\). Making use of Equation (16) in [13], we obtain  
\[\Pr\left[\gamma_{SR_{k}}\leq\gamma,\gamma_{R_{k}D}\leq\gamma\right] =1+e^{\frac{-2\gamma}{\lambda_{k}\left(1-\rho_{k}^{2}\right)}}I_{ 0}\left(\frac{2\rho_{k}\gamma}{\lambda_{k}\left(1-\rho_{k}^{2}\right)}\right)-\] \[2e^{-\frac{\gamma}{\lambda_{k}}}Q\left(\sqrt{\frac{2\gamma}{ \lambda_{k}\left(1-\rho_{k}^{2}\right)}},\rho_{k}\sqrt{\frac{2\gamma}{\lambda _{k}\left(1-\rho_{k}^{2}\right)}}\right), \tag{16}\]  
where \(\lambda_{k}=P\delta_{k}^{2}/N_{0}\), \(Q\left(a,b\right)\) is the first order Marcum Q-fuction and \(I_{0}\left(\cdot\right)\) is the modified Bessel function of the first kind of order zero [24]. Plugging Equation (16) into Equation (15) gives  
\[F_{\min}^{k}\left(\gamma\right) =1-2e^{-\frac{\gamma}{\lambda_{k}}}-e^{-\frac{2\gamma}{\lambda_{ k}\left(1-\rho_{k}^{2}\right)}}I_{0}\left(\frac{2\rho_{k}\gamma}{\lambda_{k} \left(1-\rho_{k}^{2}\right)}\right)\] \[+2e^{-\frac{\gamma}{\lambda_{k}}}Q\left(\sqrt{\frac{2\gamma}{ \lambda_{k}\left(1-\rho_{k}^{2}\right)}},\rho_{k}\sqrt{\frac{2\gamma}{ \lambda_{k}\left(1-\rho_{k}^{2}\right)}}\right). \tag{17}\]  
Additionally, the CDF of \(\frac{1}{2}\min\left(\gamma_{SR_{k}},\gamma_{R_{k}D}\right)\), \(F_{\frac{1}{2}\min}^{k}\left(\gamma\right)\), can be directly obtained as  
\[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\]0  
Now, let us define \(\gamma_{L}\) and \(\gamma_{U}\) as follows:  
\[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\]1 \[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\]2  
Then, it is readily to show that  
\[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\]3  
By exploiting Equation (21), we can derive lower and upper bounds on the system outage probability as follows:  
\[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\]4  
where \(\gamma_{th}=2^{2R_{0}}-1\). Although the lower and upper bounds on the system outage probability are useful for readily evaluating the system performance without doing complicated computer simulations, they are in general too complex to gain insight. Motivated by this, we now look into the high SNR region and derive asymptotic approximations for the lower and upper bounds, which enables the characterization of the achievable DO of the system. For that purpose, we will consider the case that \(\left\{\lambda_{k}\right\}_{k=1}^{K}=\lambda\rightarrow\infty\) and \(\left\{\rho_{k}\right\}_{k=1}^{K}=\rho\). Then, by using the Taylor approximation of the Marcum Q-function [25] (first order only), we obtain  
\[=\Pr\left[\gamma_{SR_{k}D}<2^{2R_{0}}-1\right]\]5  
which holds if \(\lambda\left(1-\rho^{2}\right)\gg 1\).