## 4 Simulation and Discussion
### Simulation Results  
Generally, the sensing threshold of the \(i\)th SU\(\lambda_{i}\), is evaluated under the scenarios as follows. Note that \(\lambda_{i}\) denotes the sensing threshold of the \(i\)th local energy detector, which is different from the sensing threshold \(k_{0}\) at FC.  
1. Scenario I: The SUs make the decisions at the same sensing threshold and the sensing threshold, \(\lambda\), is determined by minimizing the total error rate at the FC.
2. Scenario II: The sensing threshold of the \(i\)th SU\(\lambda_{i}\), is calculated by minimizing the total error rate.
3. Scenario III: \(\lambda_{i}\) is determined by minimizing the local false alarm probability while detection probability is above the minimum requirements of the CR system.  
(4) Scenario IV: All the SUs make the decision by the fixed threshold.  
Figure 2 exhibits sensing threshold comparisons with different sensing times at the FC under various scenarios, where the four scenarios correspond with the mentioned above and \(N_{1}=7\). The time-bandwidth product [24] is the product of the duration of a signal and its spectral width. The time-bandwidth product measures how well we use the available bandwidth for a given channel, which is positively correlated with the sensing time in the simulation due to the fixed bandwidth. As a result, the time-bandwidth product is approximately equivalent to the sampling points of the local SS detectors in this paper. The classical scheme in Figure 2 denotes the scheme in Equations (8) and (9).  
The total error probability is shown in Figure 3 under four scenarios. The red line denotes the proposed scheme, where both the sensing threshold and the minimum total error probability are with the proposed scheme. The black line denotes the minimum total error probability with the classical scheme. The blue line denotes the real minimum total error probability with the proposed sensing threshold. The main difference between the red line and the green line is that the minimum total error probability of the red line is evaluated by the proposed scheme in this paper and the minimum total error probability of the green line is calculated by the evaluated sensing threshold in this paper.  
Figure 2: The sensing threshold comparisons under various scenarios with the different time-bandwidth products. Scenario I, Scenario II, Scenario III and Scenario IV respectively corresponds with the scenarios mentioned above.  
Figure 4 provides the influence of \(SNR2\) and \(N_{1}\) on the threshold error for the same \(SNR1\), where \(x\) denotes the number of the detector with \(SNR1\) and \(y\) denotes the threshold difference between the proposed scheme and the classical one in Figure 3.  
In Figure 5, the error is proved with the variation of \(N_{1}\) and \(SNR2\), where \(z\) denotes the error probability difference between the proposed scheme with the classical scheme. Obviously, the error, z, increases with the error increase of the decision threshold, which corresponds to the theoretical analysis in Section 3.  
Figure 4: The influence from \(SNR2\) and \(N_{1}\) on the threshold error for the same \(SNR1\).  
Figure 3: The total error probability comparisons under various scenarios. Scenario I, Scenario II, Scenario III and Scenario IV respectively corresponds with the scenarios mentioned above.