# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 3 System Model
### Sparse Jamming Estimation  
Let us recall the received vector specified in Equation (4). There always exists a nonzero annihilating matrix \(\mathbf{A}\) such that \(\mathbf{A}\mathbf{H}\mathbf{x}=0\), unless \(\mathbf{H}\mathbf{x}\) spans \(\mathbb{C}^{N}\). Let a matrix \(\mathbf{G}\) be a generator matrix or codebook matrix of which the columns span the range space of the codewords. Here, note that the generator or codebook matrix \(\mathbf{G}\) is constructed by a random Gaussian distribution, as discussed in Section 2. Because the codeword space is a subspace of \(\mathbb{C}^{N}\), singular-value decomposition (SVD) can be applied to determine the annihilating matrix \(\mathbf{A}\). Consider an SVD of \(\left(\mathbf{HG}\right)^{H}\), i.e.,  
\[\left(\mathbf{HG}\right)^{H}=\mathbf{U}\big{[} \begin{array}{cc}\mathbf{D}& 0\end{array} \big{]}\bigg{[} \begin{array}{c}\mathbf{V}_{1}^{H}\\ \mathbf{A}\end{array} \bigg{]}. \tag{5}\]  
It can be determined that \(\mathbf{A}\) satisfies \(\mathbf{A}\mathbf{H}\mathbf{x}=0\) for all possible values of \(\mathbf{x}\). By multiplying \(\mathbf{A}\) to both sides of Equation (4), [PERSON] obtains the measurement vector \(\mathbf{b}\),  
\[ \begin{array}{ll}\mathbf{b}&=\mathbf{A}(\mathbf{e}+\mathbf{w})\\ &=\mathbf{A}\widetilde{\mathbf{e}},\end{array}  \tag{6}\]  
where \(\widetilde{\mathbf{e}}:=\mathbf{e}+\mathbf{w}\) is the jamming-plus-noise vector. Because \(\mathbf{b}\) and \(\mathbf{A}\) are known to [PERSON], \(\mathbf{e}\) can be obtained by solving for the SJE problem of Equation (6), which is in the form of a LIP. Unfortunately, the SJE problem is underdetermined because the annihilating matrix \(\mathbf{A}\) is the null space of \(\mathbf{HG}\). However, let us recall the assumption that \(\mathbf{e}\) is a sparse vector. It is known that a sparse vector can be recovered using SSR techniques even if the LIP is underdetermined [7; 8; 16; 17; 18; 24]. There are two conditions related to the successful reconstruction of the jamming vector \(\mathbf{e}\) from Equation (6) using SSR techniques. The first condition is that the LIP should have good properties. The goodness of LIP is often measured using a condition referred to as RIP [7] of the sensing matrix. In the proposed SJE problem, the sensing matrix corresponds to the annihilating matrix \(\mathbf{A}\). Several studies have shown that a sensing matrix has a good RIP if the matrix is created from an i.i.d. zero-mean Gaussian. However, the annihilating matrix \(\mathbf{A}\) in the SJE problem is not created from such a distribution, but is created from the null space of \(\mathbf{HG}\). Several studies have discussed the RIP of the null-space matrix. [PERSON] et al. [7] and [PERSON] et al. [25] argued that the null-space matrix of an i.i.d. Gaussian random matrix demonstrates a good property because the null space can consist of an i.i.d. Gaussian random basis. [PERSON] et al. [26] showed that the property of the sensing matrix that is created from the null space of a Toeplitz matrix is good for SSR. Note that \(\mathbf{HG}\) is a Gaussian matrix because the OFDM channel matrix \(\mathbf{H}\) is a Gaussian matrix and \(\mathbf{G}\) is also a Gaussian matrix (refer to Section 2 for more details). Further, \(\mathbf{HG}\) can be an i.i.d. Gaussian matrix or Gaussian Toeplitz matrix when the channel is a frequency nonselective channel. Thus,these results suggest that the SJE problem can be solved successfully, depending on the channel. The second condition for successful SSR is that the jamming with noise vector \(\widehat{\mathbf{e}}\) should be close to the sparse vector. If the energy of a sparse vector is much larger than that of a non-sparse vector, the sum of the two vectors is said to satisfy soft sparsity. [PERSON] et al. [27] discussed the convergence of the SSR under the soft sparsity model. [PERSON] and [PERSON] [28] studied the premeasurement noise model, which is similar to the soft sparsity model.