# A Capacity Enhancement Method for Frequency-Hopping Anti-Jamming Communication Systems
## 3 Results
### Design of Demodulation Algorithm for FH Capacity Boost Signal
\begin{table}
\begin{tabular}{c c c} \hline
**J** & **Clock-Phase Shift Standard Deviation** & **Maximum Clock-Phase Offset** \\ \hline
1 & 0.023438 & 0.085938 \\
2 & 0.020781 & 0.039063 \\
4 & 0.016406 & 0.054688 \\
8 & 0.0186 & 0.04 \\
16 & 0.0218 & 0.041 \\
32 & 0.023 & 0.04 \\ \hline \end{tabular}
\end{table} 
Table 3: Clock-locked data for Eb/N0 = 14 dB with different J values. * Carrier Synchronization  
The carrier recovery uses the Costas ring [27], which has a block diagram, as shown in Figure 12. It mainly consists of a phase detector, a numerically controlled oscillator, and a loop filter. The phase detector is used to identify the phase, i.e., to track the phase information, which is generally realized with a multiplier. The quadrature frequency converter module multiplies the received IF signal and the local carrier to form I- and Q-way baseband signals. The matched filter module filters out the high harmonics of the baseband signal. The carrier phase error extraction and loop filter cooperate to form the carrier-phase error signal, which controls the carrier signal in the carrier reconstruction module for phase adjustment; the carrier reconstruction module carries out carrier selection and reconstruction according to Information B, as well as the mapping relationship between Information B and the carriers of different frequency-hopping points. The quality of carrier recovery varies for different Eb/N0 values for Information A and different J values for Information B. The quality of carrier recovery is expressed as the average offset difference, \(\sigma_{\mathrm{c}}\), of the carrier-phase error signal, where \(\mathrm{e}_{\mathrm{c}}\) is the error between the carrier recovered from the carrier loop and the carrier at the transmitter, and \(\overline{\mathrm{e}}_{\mathrm{c}}\) is the average value of \(\mathrm{e}_{\mathrm{c}}\). \[\sigma_{\mathrm{c}}=\frac{\sum\limits_{\mathrm{n}}\lvert\mathrm{e}_{\mathrm{c }}(\mathrm{n})-\overline{\mathrm{e}}_{\mathrm{c}}\rvert}{\mathrm{n}} \tag{19}\]  
Figure 11: Maximum shift of clock-phase for different Eb/N0 values and different values of J.  
Figure 10: Standard deviation of clock-phase shift for different Eb/N0 values and different values of J. Figure 13 shows the variation in the standard deviation of the carrier-phase error signal for the J values of 1, 2, 4, 8, 16, and 32 for the Eb/N0 values of Information A of 4, 9, and 14 dB, respectively. In addition, since the error code of Information B creates errors in the reconstructed carrier, which also leads to degradation in the quality of the carrier recovery, the error case of the reconstructed carrier is also presented in Figure 13, denoted as \(\mathrm{cw}_{e}\), where \(\mathrm{cw}_{\mathrm{s}}\) denotes the carrier used for modulation at the transmitter and \(\mathrm{cw}_{\mathrm{r}}\) denotes the carrier recovered from the loop to be used for carrier reconstruction. Figure 13 is a histogram of the error probability of the reconstructed carrier, and it can be seen that the error probability of the reconstructed carrier gradually decreases as the Eb/N0 value of Information A gradually increases with the same J value. At the same Eb/N0 value, the error probability of the reconstructed carrier decreases as the value of J increases. Figure 14 shows the variation in the standard deviation of the carrier phase error signal for the J values of 1, 2, 4, 8, 16, and 32 for the Eb/N0 values of Information A of 4, 9, and 14 dB, respectively. As shown in Figure 14, the carrier recovery quality varies with the Eb/N0 value of Information A. At an Eb/N0 value of 14 dB for Information A, the value of \(\mathrm{\sigma_{c}}\) varies between 0.025 and 0.028. At an Eb/N0 value of 9 dB for Information A, \(\mathrm{\sigma_{c}}\) varies between 0.046 and 0.05, and the carrier jitter worsens by a factor of two, relative to the previous condition. At an Eb/N0 value of 14 dB for Information A, \(\mathrm{\sigma_{c}}\) varies between 0.07 and 0.08, and the carrier jitter deteriorates further.