## 4 Simulation Results
### OP of the Primary Network and Transmit Power of the Secondary Transmitters  
Figures 2 and 3, respectively, present the OP of the primary network and transmit power of the secondary transmitters as a function of \(P_{\text{PT}}\) in dB. In these figures, the secondary nodes are located at (0.3,0), i.e., \(x_{\text{R}}=0.3\). At first, we can see from these figures that at low \(P_{\text{PT}}\) values, the QoS of the primary network is not satisfied (i.e., \(\text{OP}_{\text{PD}}>\varepsilon_{\text{OP}}\)), and hence the secondary transmitters are not allowed to access the spectrum, (i.e., \(P_{\text{SS}_{1}}=P_{\text{SS}_{2}}=P_{\text{SR}_{b}}=0\)). It is worth noting that without the secondary operation, the OP values at PR equals to \(\text{OP}_{\text{PD}}\). Then, let us consider the case where the transmit power \(P_{\text{PT}}\) is high enough and the QoS of the primary network is satisfied. In this case, \(\text{SR}_{b}\), \(\text{SS}_{1}\) and \(\text{SS}_{2}\) can access the licensed band to transmit the data, using the maximum transmit power as given in (30) and (34). Therefore, the OP at PR is equal to \(\varepsilon_{\text{OP}}\), i.e., \(\text{OP}_{\text{P1}}=\text{OP}_{\text{P2}}=\varepsilon_{\text{OP}}\). As seen, when \(N_{\text{T}}=1\) and \(N_{\text{R}}=3\), the secondary network can access the licensed band when \(P_{\text{PT}}\geq 1\)(dB), and when \(N_{\text{T}}=N_{\text{R}}=2\), the secondary network can access the licensed band when \(P_{\text{PT}}\geq-1\)(dB). We also see that \(N_{\text{T}}=N_{\text{R}}=2\) provides better OP performance for the primary network as well as increases the spectrum access possibility for the secondary network, as compared with \(N_{\text{T}}=1\) and \(N_{\text{R}}=3\). Moreover, with \(N_{\text{T}}=N_{\text{R}}=2\), the transmit power of the secondary transmitters is also higher than those with \(N_{\text{T}}=1\) and \(N_{\text{R}}=3\). Looking at Figure 3, we also see that the transmit power of all secondary transmitters increases as \(P_{\text{PT}}\) increases. Moreover, as proven in (35), the transmit power of all secondary transmitters linearly increases at high \(P_{\text{PT}}\) region. From Figure 2, we can observe that the simulation results validate the theoretical ones of \(\text{OP}_{\text{P1}}\), \(\text{OP}_{\text{P2}}\), and \(\text{OP}_{\text{PD}}\) derived in Section 3.  
Figure 3: Transmit power of the secondary transmitters as a function of \(P_{\rm PT}\) (dB) when \(x_{\rm R}=0.3\).  
Figure 2: OP of the primary network as a function of \(P_{\rm PT}\) (dB) when \(x_{\rm R}=0.3\).  
Now, we consider an optimization problem for the primary OP performance, where the total number of transmit and receive antennas is fixed by \(N_{\text{T}}+N_{\text{R}}=N_{\text{tot}}\), and \(N_{\text{tot}}\) is a constant. From (24), (25), and (27), \(\text{OP}_{\text{P1}}\), \(\text{OP}_{\text{P2}}\), and \(\text{OP}_{\text{P0}}\) obtain the lowest value when  
\[N_{\text{T}}=\left\{\begin{array}{ll}\frac{N_{\text{tot}}}{2},\text{ if }N_{\text{tot}}\text{ is even}\\ \frac{N_{\text{tot}}-1}{2}\text{ or }\frac{N_{\text{tot}}+1}{2},\text{ if }N_{\text{tot}} \text{ is odd}\end{array}\right. \tag{56}\]  
For example, in Figures 2 and 3, since \(N_{\text{tot}}=4\), the optimal values of \(N_{\text{T}}\) and \(N_{\text{R}}\) are \(N_{\text{T}}=N_{\text{R}}=2\).