## 2 System Model
From (6), we can formulate the instantaneous signal-to-interference-plus-noise ratio (SINR) obtained at PR, as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]0  
From (8), the TAS/SC algorithm can be written as follows:  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]1  
Equation (9) implies that PT and PR cooperate to choose the best transmit and receive antennas to maximize the SINR obtained in this time slot. For SR\({}_{b}\), since \(d_{\text{SS}_{1}\text{SR}}\leq d_{\text{SS}_{2}\text{SR}}\), and hence \(P_{\text{SS}_{1}}\geq P_{\text{SS}_{2}}\) (see Section 3.2), and SR\({}_{b}\) has to decode \(x_{1}\) first. From (7), the SINR obtained for decoding \(x_{1}\) is calculated as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]2  
Then, if \(x_{1}\) can be correctly decoded, SR\({}_{b}\) can be removed \(\sqrt{P_{\text{SS}_{1}}}h_{\text{SS}_{1}\text{SR}_{b}}x_{1}\) in \(y_{\text{SR}_{b}}\)[20; 21; 22; 23; 24; 25]. Then, the signal used to decode \(x_{2}\) is given as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]3  
From (11), the SINR obtained for decoding \(x_{2}\) is written as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]4  
**Remark 3**. : _If SR\({}_{b}\) only correctly decodes \(x_{1}\), it will only send \(x_{1}\) to SS\({}_{2}\) at the second time slot. It is worth noting that if SR\({}_{b}\) cannot successfully decode \(x_{1}\), it cannot perform ICT to remove \(x_{1}\), and hence \(x_{2}\) is also not decoded. Moreover, in the case where SR\({}_{b}\) cannot correctly obtain \(x_{1}\) and \(x_{2}\), it will do nothing at the second time slot._  
Let us consider the secondary time slot, and assume that SR\({}_{b}\) can access the licensed band to transmit the data \(x^{*}\) (\(x^{*}\in\{x_{\oplus},x_{1}\}\)). In this time slot, assume that PT uses the \(w\)-th antenna to send the \(x_{\text{P}}\) to PR. Then, the received signals at the \(z\)-th antenna of PR, at SS\({}_{1}\) and at SS\({}_{2}\) can be written, respectively, as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]5  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]6  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]7  
where \(P_{\text{SR}_{b}}\) is the transmit power of SR\({}_{b}\) (\(P_{\text{SR}_{b}}\) which will be derived in the next section), and \(n_{\text{B}}\) denotes the Gaussian noises at the receiver B whose zero mean and unit variance with \(\text{B}\in\{\text{PR}_{z},\text{SS}_{1},\text{SS}_{2}\}\). From (13), the SINR received at \(\text{PR}_{2}\) is calculated as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]8  
Similarly to (9), the TAS/SC algorithm in the second time slot can be written as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]9  
**Remark 4**. : _To realize the TAS/SC algorithms in (9) and (17), in the set-up phase, PT, \(\text{SS}_{1}\), \(\text{SS}_{2}\), and \(\text{SR}_{b}\) send pilot signals to PR. Then, PR estimates the channel coefficients \(h_{\text{PT}_{p},\text{PR}_{q}}\), \(h_{\text{SS}_{1},\text{PR}_{q}}\), \(h_{\text{SS}_{2},\text{PR}_{q}}\), and \(h_{\text{SR}_{b}\text{PR}_{q}}\) to calculate the instantaneous SINRs \(\varphi_{\text{PT}_{p},\text{PR}_{q}}\). Using (9) and (17), PR can determine the best transmitting antennas at PT and its best receiving antennas, in the first and secondary slots. Finally, PR feedbacks the index of these antennas to PT.