## 4 Simulations and Discussion
### FH Signal Tracking  
In this section, the FH signal tracking and dynamic programming performance is analyzed. The simulation results of signal tracking are shown in Figure 5. In the present study, the number of sub-channels is set as \(m=50\) and each sub-channel contains two pairs of sub-bands for signal tracking. The SNR for each signal is 10 dB on the equivalent narrow band.  
Four hops of each source are shown in Figure 5 as examples. It can be seen that the power estimation of sub-channels is effective. Hopping between sub-channels is taken for the TOA estimation, marked by a red line in Figure 6. During a single hop, the power estimation is not stable, because the length of the smoothing filter in the time domain is limited and the noise of all the sub-bands is superimposed.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Sub-band number \(L\) & 195 \\ Periodic sequence frequency \(f_{p}\) & 51.3 MHz \\ Baseband sampling rate \(f_{s}\) & 51.3 MHz \\ Sub-channel number \(m\) & 32, 50 \\ Amplitude of smoothing filter & 1 \\ Carrier frequency & [2, 7 GHz] \\ Hop rate & 10 khop/s \\ Single hop bandwidth \(B\) & 30 MHz \\ Mode type & DQPSK \\ Shaping filter & Raised cosine FIR \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation settings.  
The performance of the TOA estimation is shown in Figure 6, where \(Ls\) represents the length of the smoothing filter. It can be concluded that SNR is the main factor affecting the TOA estimation when it is lower than 10 dB, because the disturbance of noise cannot be removed by the smoothing filter. The TOA estimation is relatively accurate and stable when SNR is higher than 10 dB, and the main factor that affects the estimation is the length of the smoothing filter. The longer the smoothing filter, the better the estimation, because it represents the accumulation of time in the time domain. Based on Equation (12), long-term estimation is closer to real TOA, but it loses real-time performance.  
Figure 5: Results of signals tracking: (**a**) source 1; (**b**) source 2.  
Figure 6: Performance of the TOA estimation.  
In the application, a compromise between accuracy and real-time requirement should be made for a better performance and the filter delay should be eliminated.  
Figure 7 shows the support set estimation by dynamic programming, where \(P_{d}\) is the success rate of the estimation. The length of the smoothing filter for this simulation is \(Ls=256\). The number of sub-channel is set as \(m=32\) and each sub-channel contains three pairs of sub-bands.  
It can be concluded that with increased SNR, the success rate becomes higher. The gain factor has a significant impact on the probability of estimation. When the SNR is higher than 9 dB and the gain rate is higher than 8, the success rate is stable at over 90%. As the power of a single hop is relatively stable and the MWC system spreads all the information of input signals, including noise, to the baseband, the inherent characteristics of the noise limit the sensitivity of estimation. Comparing Figure 7a,b, it can be seen that the signal power limit ratio of the sub-bands \(\gamma\) is not the main factor for the estimation. The lower the ratio, the worse the success rate. For practical applications, the choice of gain factor should consider the channel condition and the requirement of estimation accuracy.