# A Capacity Enhancement Method for Frequency-Hopping Anti-Jamming Communication Systems
## 3 Results
### Design of Demodulation Algorithm for FH Capacity Boost Signal
* Clock Synchronization  
For the Gardner algorithm [26], each code element needs two sampling points, and each code element cycle only needs to calculate one timing error. The Gardner algorithm structure is shown in Figure 9.  
At the receiving end of Figure 2, the I and Q path baseband signals of Information A are sampled at twice the code word rate to generate two paths: the peak point and the over-zero point. After timing error detection is performed, the timing error information is fed into the loop filter and the phase of the resampled clock signal is controlled by the NCO. The NCO output is also used for envelope detection and carrier reconstruction for Information B. When Information B is not transmitted, the system receiver is equivalent to a general BPSK demodulator, and the clock synchronization link is normally locked. In this paper, we focus on the work of the clock loop after the normal transmission of Information A, the moment at which Information B begins to be transmitted (the moment when the carrier begins to hop), and the impact of Information B on the clock loop in the presence of different J values of the error code. The clock loop locking condition is represented by the clock phase offset statistics' mean square difference value and the clock phase maximum offset value. The quality of clock recovery varies for different Eb/N0 values for Information A and different J values for Information B. The quality of clock recovery is expressed as the  
Figure 8: Spectrum after joint multi-symbol wave detection. Figure 9: Gardner structure. average frequency shift difference of the clock-phase error signal and the maximum offset value of the clock phase. The clock recovery quality is expressed in terms of the average frequency shift difference, \(\sigma_{\text{t\_mean}}\), of the clock-phase error signal and the maximum shift value, \(\text{e}_{\text{t\_max}}\), of the clock phase, where \(\text{e}_{\text{t}}\) is the error value between the clock recovered from the timing loop and the standard clock, and \(\overline{\text{e}}_{\text{t}}\) is the average value of \(\text{e}_{\text{t}}\). Figure 5:0  
\(\text{e}_{\text{t\_max}}\) denotes the maximum offset of the clock from the start of transmission of Information B  
Figure 5:1  
Tables 1-3 show the standard deviation values of the clock-phase shift and the maximum shift values of the clock phase within 10 s after the start of frequency hopping, obtained via simulation for different values of J at different SNRs, where the information rate of Information A is 1 kbps, the frequency-hopping points are 64, and the Eb/N0 of Information A is 4 dB, 9 dB, and 14 dB. The offset value is normalized by the hopping period \(\text{T}_{\text{H}}\); a maximum offset value of 0.11 is offset by 0.11 times the hopping period. Figures 10 and 11 present the clock locking for J values of 1, 2, 4, 8, 16, and 32 for the Eb/N0 values of Information A of 4 dB,9 dB, and 14 dB, respectively. Figure 10 shows the standard deviation of the clock-phase offset and Figure 11 shows the maximum clock-phase offset. For the same J value, the clock-locking quality becomes progressively better with the increase in the Eb/N0 values; the effect of the J value on the clock-locking quality is weaker than that of Eb/N0. \begin{table}
\begin{tabular}{c c c} \hline
**J** & **Clock-Phase Shift Standard Deviation** & **Maximum Clock-Phase Offset** \\ \hline
1 & 0.06 & 0.234375 \\
2 & 0.054688 & 0.234375 \\
4 & 0.05 & 0.22 \\
8 & 0.05 & 0.2 \\
16 & 0.04 & 0.193 \\
32 & 0.0351 & 0.185 \\ \hline \end{tabular}
\end{table} 
Table 1: Clock-locked data for Eb/N0 = 4 dB with different J values. \begin{table}
\begin{tabular}{c c c} \hline
**J** & **Clock-Phase Shift Standard Deviation** & **Maximum Clock-Phase Offset** \\ \hline
1 & 0.032656 & 0.103 \\
2 & 0.033563 & 0.109375 \\
4 & 0.034609 & 0.12 \\
8 & 0.036 & 0.12 \\
16 & 0.035 & 0.11 \\
32 & 0.031 & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 2: Clock-locked data for Eb/N0 = 9 dB with different J values.