## 4 Simulation Results
### Performance Analysis of Signal Detection  
Figure 1 shows the statistical results of the false detection probability, which is expressed as \(P_{FD}\), and the missed detection probability, which is expressed as \(P_{MD}\), of the proposed method under different \(E_{b}/N_{0}\)s and SIRs. For the purpose of clarity, we plot the results corresponding to \(\text{SIR}=-10\) dB, \(-6\) dB, and \(-2\) dB in Figure 1a-c, respectively. It should be noted that the \(P_{FD}\) and \(P_{MD}\) corresponding to each \(E_{b}/N_{0}\) and SIR are statistical results of \(1\times 10^{5}\) repeated trials. Moreover, due to the use of logarithmic coordinate in the vertical axis, when the calculated \(P_{FD}\) or \(P_{MD}\) is 0, its logarithmic value becomes infinitesimal and cannot be seen in the figure. Thus, some curves in Figure 1a-c are interrupted before reaching the high \(E_{b}/N_{0}\) end. It can be seen from the solid curves marked with \"+\" in Figure 1a-c that the \(P_{FD}\) is almost unchanged when the \(E_{b}/N_{0}\) increases in the range of [\(-5\), 11] dB in each figure, which corresponds to a fixed SIR. Especially, under the circumstance of \(\text{SIR}=-2\) dB and \(E_{b}/N_{0}=-5\) dB, the proposed method can achieve a \(P_{FD}\) as low as \(3\times 10^{-3}\), as shown in Figure 1c. Moreover, we can also find that the false detection probability of the proposed method remains almost unchanged when the SIR decreases under the same \(E_{b}/N_{0}\). As shown in Figure 1a,c, for the situation of \(E_{b}/N_{0}=-5\) dB, the false detection probability is about \(4\times 10^{-3}\) for \(\text{SIR}=-10\) dB, and as the SIR increases to \(-2\) dB, the \(P_{FD}\) only decreases to \(3\times 10^{-3}\) for the same \(E_{b}/N_{0}\). It can be demonstrated from the abovementioned results that the false detection probability of the proposed method is insensitive to both the SIR and \(E_{b}/N_{0}\) in the given range. From the dashed curves marked with \"+\" in three figures of Figure 1, we can see that the missed detection probability \(P_{MD}\) decreases as the \(E_{b}/N_{0}\) increases under the same SIR. As shown in Figure 1a, the \(P_{MD}\) decreases from \(3\times 10^{-1}\) to \(1\times 10^{-4}\), with the \(E_{b}/N_{0}\) increasing from \(-5\) dB to \(1\) dB. Moreover, we can also see that in the given SIR range, the \(P_{MD}\) of the proposed method remains almost unchanged when the SIR increases under the same \(E_{b}/N_{0}\). As shown in Figure 1a, under the situation of \(E_{b}/N_{0}=-1\) dB, the \(P_{MD}\) of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Variable** & **Value** \\ \hline Channel bandwidth & \(B\) & 250 kHz \\ Symbol period & \(T_{b}\) & 6.25 μs \\ Initial magnitude of signal & \(c_{0}\) & 1 \\ Initial phase of the first symbol & \(q_{0}\) & \(0^{\circ}\) \\ The number of sampling points within a symbol & \(N_{e}\) & 32 \\ Sampling period & \(T_{s}\) & \(T_{b}/N_{s}\) \\ The number of in-band interferences & \(M\) & 3 \\ CFO & \(\Delta f\) & 25.5 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. proposed method is about \(2\times 10^{-3}\) for SIR = \(-10\) dB, whereas with the SIR increasing to \(-2\) dB, the \(P_{MD}\) only decreases to \(1.3\times 10^{-3}\), as shown in Figure 1c. These results indicate that the missed detection probability of the proposed method is sensitive to \(E_{b}/N_{0}\) but insensitive to SIR in the given range. To sum up, the results shown in Figure 1 demonstrate the good performance of the proposed method in signal detection under the situations of low SIR and low \(E_{b}/N_{0}\). In addition, in order to further show the superiority of the proposed method in signal detection, we conduct simulation experiments under the same conditions mentioned before using methods published in [11; 12; 13; 14] and plot their results in Figure 1 together for the convenience of comparison.