# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 3 System Model
### Sparse Jamming Estimation
#### 3.4.1 Greedy Sparse Jamming Estimation Algorithm  
To estimate the sparse jamming vector \(\mathbf{e}\) from the measurement \(\mathbf{b}\), one can define a support set \(\mathbb{S}\subset\{1,2,\ldots,N\}\), which is a set of indices \(k\), such that the \(k\)th element of \(\mathbf{e}\) is nonzero. Thus, \(\mathbb{S}=\{k|\ell_{k}\
eq 0,k=1,2,\ldots,N\}\). Let us define \(\mathbf{A}_{\mathbb{S}}\) and \(\mathbf{e}_{\mathbb{S}}\) as submatrices/subvectors of which the columns/elements contain atom \(\mathbf{a}_{k}\) and coefficient \(\ell_{k}\) for \(k\in\mathbb{S}\), respectively. Then, \(\hat{\mathbf{e}}\) can be determined by solving the following optimization:  
\[\hat{\mathbf{e}}=\underset{\mathbf{e}}{\text{argmin}}\|\mathbf{b}-\mathbf{A} \mathbf{e}\|^{2},\text{ subject to }\|\mathbf{e}\|_{0}\leq K, \tag{7}\]  
where \(K\) is the maximum number of nonzero coefficients in \(\mathbf{e}\). Equation (7) is an optimization problem that determines the solution \(\mathbf{e}\) that minimizes the approximation error with the \(l_{0}\) norm constraint. To solve this problem, several greedy algorithms have been proposed. The greedy algorithms iteratively pursue one column of the annihilating matrix \(\mathbf{A}\) at a time, which significantly reduces the approximation error. Thus, the estimate of the support set \(\mathbb{S}\) grows one by one as the iteration progresses, starting from the empty set in the beginning. The matching pursuit (MP) [29], orthogonal MP (OMP) [18; 19] and their variants [30; 31] are most relevant to the derivation presented in this Section. The MP algorithm for SSR was first proposed by [PERSON] and [PERSON] [29]. By applying the MP algorithm to our problem, we can determine one column of the annihilating matrix \(\mathbf{A}\) that maximizes the inner product with a residual vector at each iteration, and then calculate the corresponding nonzero coefficient that minimizes the estimation error. Conversely, the OMP algorithm has an additional procedure that updates all the coefficients in the subvector \(\mathbf{e}_{\mathbb{S}}\) at each iteration using the orthogonal projection of the measurement vector onto the subspace \(\mathbf{A}_{\mathbb{S}}\). In this section, we propose the GSJE algorithm, described as Algorithm 1. The algorithm aims to estimate the sparse jamming vector from the received signal vector \(\mathbf{y}\) in Equation (4). Given \(\mathbf{H}\), \(\mathbf{G}\) and \(\mathbf{y}\), the algorithm calculates \(\mathbf{A}\) using Equation (5). Then, the jamming vector \(\mathbf{e}\) is estimated by approximating the solution of Equation (7) using the greedy iteration. We assume that the channel matrix \(\mathbf{H}\) and codeword space \(\mathbf{G}\) are known to the receiver. The annihilating matrix \(\mathbf{A}\) can then be determined from the SVD of \((\mathbf{H}\mathbf{G})^{H}\), as given in Equation (5), and used to construct the SJE problem. We assume here that the magnitudes of the nonzero coefficients of the jamming vector \(\mathbf{e}\) are significantly larger than those of the noise vector \(\mathbf{w}\), i.e., \(||\mathbf{e}||_{2}\gg\|\mathbf{w}\|^{2}\). This assumption is called the high jamming-to-noise ratio (JNR) assumption. Under this assumption, each iteration of the GSJE will identify a member of the true support set with high probability. In contrast, if the JNR is significantly low (for example, under \(-10\) dB), the accuracy of SJE under the high JNR assumption might be degraded. However, note that the overall system performance, i.e., BER, is no longer affected by the accuracy of the SJE. However, if JNR is in the moderate region, the failure of the GSJE may cause an overall performance degradation. Thus, the model mismatch becomes more significant in the moderate-JNR region. Another challenge is that the GSJE requires certain prior knowledge about jamming. As discussed above, the parameters \(\delta\) and \(K\) that determine the termination condition depend on prior information such as sparsity and energy of jamming.