# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 6 Results
### Sparse Jamming Estimation Error  
We evaluated the SJE performance of the GSJE and BSJE algorithms for the proposed system. In general, the BSJE algorithm shows better SJE performance than the GSJE. However, as the two algorithms have advantages and disadvantages, we must select an algorithm for practical applications. * Limitations in BSJE implementation  
In practical implementations, the appropriate maximum number of iterations should be chosen to ensure the completeness of the iterative algorithm. As the maximum number of iterations for BSJE is empirically selected, the output of the algorithm may not sufficiently converge to the solution. In contrast, GSJE has a fixed maximum number of iterations, which is \(N\) in the worst case. Owing to the aforementioned implementation limitations of BSJE, there are several regions in which the BSJE algorithm performs equivalently to, and sometimes worse than, GSJE. * Time complexity  
The BSJE algorithm calculates several matrix products and matrix inversions in a single iteration, whereas the GSJE algorithm requires only a single matrix product and the inversion of a smaller submatrix. In general, GSJE is significantly faster than BSJE, especially when the size of the system becomes larger. * Level of prior knowledge in the GSJE algorithm  
The above two problems discourage the use of the BSJE algorithm. However, the GSJE algorithm requires prior knowledge about the signal to define its termination conditions, and this assumption about prior knowledge might be unrealistic. For example, if the maximum cardinality of the support set is given as the termination condition, a receiver executing the GSJE algorithm must know the support set of the jammer. This assumption that the receiver will know the information a priori is impractical. If this prior information is unknown, the optimum \(K\) is the maximum possible number of jamming coefficients for GSJE. Unfortunately, this value depends on the RIP of the annihilating matrix \(\mathbf{A}\). Since \(\mathbf{A}\) is constructed from the random matrix \(\mathbf{G}\), the \(K\) must be calculated every time the codebook changes. However, evaluating the RIP takes considerable time. The overhead further increases since the codebook must vary over time to guarantee security. Thus calculating the optimum \(K\) for each codebook change is not practicable. Instead, we heuristically evaluated the average RIP for multiple realizations of \(\mathbf{A}\), and roughly set \(K=N/4\) by applying [PERSON]'s bound [51]. Accordingly, we performed a numerical simulation to compare the GSJE and BSJE algorithms. The two algorithms were compared in terms of jamming estimation performance (as measured by MSE) and time complexity (as measured by running time) at various JNRs and levels of prior knowledge for the GSJE algorithm. We set the level of prior knowledge as how much [PERSON] precisely knows about the cardinality of jamming (e.g., sparsity, \(K\)). The measure of prior knowledge is classified into two levels as a function of the true value of \(K\), as listed in Table 2.  
Using the above values, we compared the normalized MSE of the jamming estimation and the corresponding running time. The block length of the complex-valued codeword was set to 1024, and the number of possible message vectors was eight. Thus, the code rate was \(8/2048=1/256\). Among the 1024 complex samples, 128 samples were contaminated by jamming; thus, the jamming rate was \(128/1024=1/8\). We compared the results according to three different JNRs: 10, 5, and 0 dB. The results are listed in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Level of Prior Knowledge** & **Perfect** & **Unknown** \\ \hline
**Maximum iteration of GSJE** & \(\bar{K}=K\) & \(\bar{K}=N/4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definitions for the level of prior knowledge. * MSE performance according to prior knowledge The results reveal the jamming estimation performance of the GSJE and BSJE algorithms.