# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 6 Results
### Bit Error Rate under Jamming  
We evaluated the BER at the receiver under jamming attacks in multiple scenarios. The results presented in Figures 7 and 8 illustrate the BER performance of the proposed system in the AWGN channel under a jamming attack with a JNR of 0 dB. The block length of the complex-valued codeword was set to 512 and the number of possible message vectors was four. Thus, the code rate was \(4/1024=1/256\). At the same time, we evaluated the BER of an uncoded DSSS that had same rate in order to use this as a baseline. Among the 512 complex-valued samples, 64 samples were contaminated by jamming; thus, the jamming rate was \(64/512=1/8\). We evaluated the BER under jamming over the \(E_{b}/N_{0}\) region from 0 dB to 16 dB, corresponding to SNRs from \(-24\) dB to \(-8\) dB, as SNR is calculated by \((E_{b}/N_{0})\cdot(\text{code rate})\approx(E_{b}/N_{0})-24\text{ dB}\). Note that the SNR is far below the noise floor. As illustrated in Figure 8, the BER of the proposed method using the BSJE and GSJE algorithms with perfect prior knowledge approaches the BER of jamming-free QPSK as \(E_{b}/N_{0}\) grows. In contrast, the BER of the GSJE without any prior knowledge cannot achieve  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**JNR** & \multicolumn{2}{c}{**10 dB**} & \multicolumn{2}{c}{**5 dB**} & \multicolumn{2}{c}{**0 dB**} \\ \hline
**Prior Knowledge** & **Perfect** & **Unknown** & **Perfect** & **Unknown** & **Perfect** & **Unknown** \\ \hline MSE (BSJE) & 0.0525 & 0.0523 & 0.166 & 0.166 & 0.523 & 0.523 \\ \hline MSE (GSJE) & 0.0446 & 0.0880 & 0.193 & 0.295 & 0.754 & 0.992 \\ \hline Run time (BSJE) & 11,300 & 11,500 & 10,900 & 11,800 & 12,600 & 11,600 \\ \hline Run time (GSJE) & 977 & 2530 & 900 & 2510 & 896 & 2440 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mean squared error (MSE) comparison results between BSJE and GSJE algorithms, according to jamming-to-noise ratio (JNR).  
anti-jamming performance. Furthermore, if the proposed codeword is drawn using the codebook method, the BER of the proposed method with the BSJE algorithm outperforms that of the jamming-free QPSK, as illustrated in Figure 9. Thus, the proposed method achieves a coding gain while simultaneously having an anti-jamming property.  
Figures 9 and 10 illustrate the BER performance of the proposed system over a frequency-selective fading channel. The simulation was performed over a frequency-selective fading channel with frequency-domain channel coefficients following an i.i.d. CSCG random distribution \(\mathcal{CN}(0,\mathbf{I})\), with the other parameters being identical to those in the AWGN channel simulation. Because several subbands have significantly low gain, the BER of the uncoded QPSK signal is significantly more degraded than that in the AWGN channel simulation. In contrast, the BER degradation of the proposed method is tolerable when compared to that of the uncoded QPSK. The method using BSJE demonstrates the highest performance, and the method using GSJE shows the lowest performance among the proposed methods. Consistent with the AWGN channel simulation, the performance of the codebook method illustrated in Figure 11 is superior to that of the linear Gaussian method illustrated in Figure 10.  
Figure 10: BER performance of the proposed system over a frequency-selective channel at JNR = 0 dB for the linear Gaussian method.  
Figure 9: BER performance of the proposed system in an AWGN channel at JNR = 0 dB for the codebook method.