# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 3 System Model
### Sparse Jamming Estimation
#### 3.4.2 Bayesian Spare Jamming Estimation Algorithm
The given problem is a form of the so-called input noise model:  
\[\mathbf{\mathrm{b}}=\mathbf{\mathrm{A}}(\mathbf{\mathrm{e}}+\mathbf{\mathrm{w}}). \tag{13}\]  
The closed-form solution to Equation (12) is well known as  
\[\hat{\gamma}_{k}=E_{\mathbf{\mathrm{e}}|\mathbf{\mathrm{b}},\gamma_{k},\sigma^{2}}|e_{ k}|^{2},\ (k=1,2,\ldots,N), \tag{14}\]  
which is a solution to \(\partial Q(\mathbf{\Gamma})/\partial\gamma_{k}=0\) by assuming \(p(\gamma_{k})=1\) and approximating \(Q(\mathbf{\Gamma})\) as  
\[Q(\mathbf{\Gamma})\approx E_{\mathbf{\mathrm{e}}|\mathbf{\mathrm{b}};\gamma,\sigma^{2}} \Bigg{[}\sum_{k}-\frac{1}{2}\log\gamma_{k}-\frac{|e_{k}|^{2}}{\gamma_{k}}+\log p (\gamma_{k})\Bigg{]}. \tag{15}\]  
If we assume a CSCG jamming \(\mathbf{\mathrm{e}}\sim\mathcal{CN}(0,\mathbf{\Gamma})\) and a CSCG noise \(\mathbf{\mathrm{w}}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{\mathrm{I}})\) with given noise variance \(\sigma^{2}\), the conditional PDF \(p(\mathbf{\mathrm{e}}|\mathbf{\mathrm{b}};\mathbf{\Gamma},\sigma^{2})\) becomes CSCG, with mean vector \(\mathbf{\mathrm{\mu}}\) and covariance matrix \(\mathbf{\Sigma}\), where  
\[\hat{\mathbf{\mathrm{e}}}=\underset{\mathbf{\mathrm{e}}}{\text{argmax}}\ p(\mathbf{ \mathrm{e}}|\hat{\mathbf{\Gamma}})p(\hat{\mathbf{\Gamma}})p(\mathbf{\mathrm{b}}|\mathbf{ \mathrm{e}}), \tag{9}\]0  
\[\hat{\mathbf{\mathrm{e}}}=\underset{\mathbf{\mathrm{e}}}{\text{argmax}}\ p(\mathbf{ \mathrm{e}}|\hat{\mathbf{\Gamma}})p(\hat{\mathbf{\Gamma}})p(\mathbf{\mathrm{b}}|\mathbf{ \mathrm{e}}), \tag{9}\]1  
Thus, the ML estimator in Equation (12) becomes  
\[\hat{\mathbf{\mathrm{e}}}=\underset{\mathbf{\mathrm{e}}}{\text{argmax}}\ p(\mathbf{ \mathrm{e}}|\hat{\mathbf{\Gamma}})p(\hat{\mathbf{\Gamma}})p(\mathbf{\mathrm{b}}|\mathbf{ \mathrm{e}}), \tag{9}\]2  
where \(\mu_{k}\) is the \(k\)th element of \(\mathbf{\mathrm{\mu}}\) and \(\Sigma_{kk}\) is the \(k\)th diagonal element of \(\mathbf{\Sigma}\). According to Equations (16)-(18), \(\hat{\mathbf{\Gamma}}\) can be determined by iteratively updating \(\mathbf{\Gamma}^{(t)}\) at the \(t\)th iteration, and by repeating the iteration until convergence. A little is known about the convergence of EM iteration, including the monotonicity property [32]. This property states that the solution converges to at least a saddle point or local optimum. After \(\hat{\mathbf{\Gamma}}\) is converged, the MAP estimation is obtained by  
\[ \begin{array}{ll}\hat{\mathbf{\mathrm{e}}}&=\underset{\mathbf{\mathrm{e}}}{\operatorname {argmax}}\underset{\mathbf{\mathrm{e}}}{\operatorname{argmax}}p(\mathbf{\mathrm{e}} \big{|}\mathbf{\mathrm{b}};\hat{\mathbf{\Gamma}},\sigma^{2})\\ &=\mathbf{\mathrm{\mu}}_{\mathbf{\mathrm{e}}|\mathbf{\mathrm{b}}}(\hat{\mathbf{\Gamma}})\\ &=\hat{\mathbf{\Gamma}}\mathbf{\mathrm{A}}^{H}(\mathbf{\mathrm{A}}\hat{\mathbf{\Gamma}}\mathbf{ \mathrm{A}}^{H}+\sigma^{2}\mathbf{\mathrm{A}}\mathbf{\mathrm{A}}^{H})^{-1}\mathbf{ \mathrm{b}}.\end{array}  \tag{19}\]  
It can be recalled that our goal is to determine the complex-valued codeword \(\mathbf{\mathrm{x}}\), not \(\mathbf{\mathrm{e}}\) itself, which is obtained by the iteration above. To estimate, we must determine not only \(\hat{\mathbf{\mathrm{e}}}\) but also \(\hat{\mathbf{\mathrm{w}}}\); thus, \(\hat{\mathbf{\mathrm{x}}}\) can be calculated by  
\[\hat{\mathbf{\mathrm{x}}}=\mathbf{\mathrm{H}}^{-1}(\hat{\mathbf{\mathrm{e}}}+\hat{\mathbf{ \mathrm{w}}}). \tag{20}\]  
Here, \(\hat{\mathbf{\mathrm{w}}}\) can be obtained during the BSJE iteration using the same method used for determining \(\hat{\mathbf{\mathrm{e}}}\), as \(\hat{\mathbf{\mathrm{w}}}\) also follows a CSCG distribution such that \(\mathcal{CN}(0,\sigma^{2}\mathbf{\mathrm{I}})\).