## 3 ABER Performance of Transmit Aperture Selection under Jamming  
In this section, a closed-form expression of average BER (ABER) is determined. Let us define an RV as  
\[w\triangleq\sqrt{\frac{E_{j}}{\rho}}h_{j}=\sqrt{N_{j}}h_{j}, \tag{13}\]  
that encapsulates the jamming component at the receiver while the jammer is ON; \(h_{j}\) follows the PDF defined in (8). We can calculate the PDF of RV \(w\) by employing the transformation of RV [35]; thus, we can write  
\[f_{W}(w)=\frac{\zeta_{j}^{2}}{\sqrt{N_{j}}A_{0}^{\zeta_{j}^{2}}}\Bigg{(}\frac{ w}{\sqrt{N_{j}}}\Bigg{)}^{\zeta_{j}^{2}-1}\Gamma\Bigg{(}1-\zeta_{j}^{2},\frac{w}{A_{ 0}\sqrt{N_{j}}}\Bigg{)}U(w). \tag{14}\]  
Here, \(\zeta_{j}\) indicates the PE faced by the jammer at the receiver, and \(U(\cdot)\) is the unit step function. Now, as we know that Gaussian noise is negligible compared to the jamming noise for \((J=1)\), we can re-write the jamming noise component at the receiver in (1) as \(w=y-\sqrt{E_{x}}h_{m}\)\(x\). Substituting \(w\) in (14), we obtain the PDF of the received signal \(y\) for \((J=1)\) as  
\[f_{Y}(y|\sqrt{E_{x}},h_{m},x) = \frac{\zeta_{j}^{2}}{\sqrt{N_{j}}A_{0}^{\zeta_{j}^{2}}}\Bigg{(} \frac{y-\sqrt{E_{x}}h_{m}x}{\sqrt{N_{j}}}\Bigg{)}^{\zeta_{j}^{2}-1}\Gamma \Bigg{(}1-\zeta_{j}^{2},\frac{(y-\sqrt{E_{x}}h_{m}x)}{A_{0}\sqrt{N_{j}}}\Bigg{)} \tag{15}\] \[\times U(y-h_{m}x).\]  
Considering the case of an idle jammer, the additive noise follows the normal distribution with \(\sqrt{E_{x}}h_{m}x\) mean and \(N_{0}\) variance. Hence, for all possible states of the jammer, we can write the comprehensive PDF of the received signal \(y\) as  
\[f_{Y}(y|\sqrt{E_{x}},h_{m},x)=\rho\frac{\zeta_{j}^{2}}{\sqrt{N_{ j}}A_{0}^{\zeta_{j}^{2}}}\bigg{(}\frac{y-\sqrt{E_{x}}h_{m}x}{\sqrt{N_{j}}} \bigg{)}^{\zeta_{j}^{2}-1}\Gamma\bigg{(}1-\zeta_{j}^{2},\frac{(y-\sqrt{E_{x}}h _{m}x)}{A_{0}\sqrt{N_{j}}}\bigg{)}U(y-h_{m}x)\] \[+(1-\rho)\frac{1}{\sqrt{2\pi N_{0}}}exp\Big{(}\frac{-(y-\sqrt{E_ {x}}h_{m}x)^{2}}{2N_{0}}\Big{)}. \tag{16}\]  
This equation can be simplified for the possible values of \(x\) as  
\[f_{Y}(y)= \begin{cases}\rho\frac{\zeta_{j}^{2}}{\sqrt{N_{j}}A_{0}^{\zeta_{j}^{ 2}}}\bigg{(}\frac{y}{\sqrt{N_{j}}}\bigg{)}^{\zeta_{j}^{2}-1}\Gamma\bigg{(}1- \zeta_{j}^{2},\frac{y}{A_{0}\sqrt{N_{j}}}\bigg{)}U(y)\\ +(1-\rho)\frac{1}{\sqrt{2\pi N_{0}}}exp\Big{(}\frac{-y^{2}}{2N_{0}}\Big{)}, \hskip 56.905512 pt\text{for}\hskip 5.690551 ptx=0,\\ \rho\frac{\zeta_{j}^{2}}{\sqrt{N_{j}}A_{0}^{\zeta_{j}^{2}}}\bigg{(}\frac{y-h_{m }}{\sqrt{N_{j}}}\bigg{)}^{\zeta_{j}^{2}-1}\Gamma\bigg{(}1-\zeta_{j}^{2},\frac {y-h_{m}}{A_{0}\sqrt{N_{j}}}\bigg{)}U(y-h_{m})\\ +(1-\rho)\frac{1}{\sqrt{2\pi N_{0}}}exp\Big{(}\frac{-(y-h_{m})^{2}}{2N_{0}} \Big{)},\hskip 56.905512 pt\text{for}\hskip 5.690551 ptx=1.\end{cases}  \tag{17}\]Furthermore, to determine the authentic information at the receiver, we utilize a threshold-based detection technique defined as  
\[y\underset{x=0}{\overset{x=1}{\geq}}y_{th}. \tag{18}\]  
Here, \(y_{th}\) is the threshold which can be obtained by performing the _Likelihood Ratio Test_[36] on (17):  
\[y_{th}=\sqrt{E_{x}}\bigg{(}\frac{h_{m}}{2}\bigg{)}-\frac{\rho}{1-\rho}\frac{N_{0 }}{N_{j}}\sqrt{2\pi N_{0}}. \tag{19}\]  
As discussed earlier, when the jammer is active, the jamming noise is far greater than the Gaussian noise, i.e., \(N_{j}\gg N_{0}\); hence, the second term in (19) becomes negligible and can be neglected in further calculations.