## 4 Proposed MP Clutter Suppression Algorithm Based on Joint Optimization
### The Motivation of the Proposed Joint Optimization Algorithm
Substituting the above equation into Equation (9), we can get:  
\[E\Bigg{\{}\left\|\sum_{p=1}^{N_{T}}\mathbf{x}_{pc,l}-\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{ \alpha}\right\|_{2}^{2}\Bigg{\}} =C+E\Big{\{}\left\|\mathbf{x}_{l}-\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{\alpha} \right\|_{2}^{2}\Big{\}}+2\text{Re}\big{\{}E\big{[}\mathbf{x}_{n}^{\text{H}}\mathbf{ \Gamma}\mathbf{S}_{n}\mathbf{\alpha}\big{]}\big{\}} \tag{14}\] \[=C+E\Big{\{}\left\|\mathbf{x}_{l}-\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{\alpha} \right\|_{2}^{2}\Big{\}}+2\text{Re}\big{\{}E\big{[}\mathbf{x}_{n}^{\text{H}}\mathbf{ \Gamma}diag(\mathbf{x}_{n})\mathbf{\Phi}\mathbf{\alpha}\big{]}\big{\}}\]  
where  
\[E\Big{[}\mathbf{x}_{n}^{\text{H}}\mathbf{\Gamma}diag(\mathbf{x}_{n})\mathbf{\Phi}\mathbf{\alpha} \Big{]}=E\Big{[}\mathbf{x}_{n}^{\text{H}}diag(\mathbf{x}_{n})\Big{]}\mathbf{\Gamma}\mathbf{ \Phi}\mathbf{\alpha}=\sigma^{2}\mathbf{I}_{1\times M\!N}\mathbf{\Gamma}\mathbf{\Phi}\mathbf{\alpha}. \tag{15}\]  
We define \(\mathbf{\psi}=\sigma^{2}\mathbf{I}_{1\times M\!N}\mathbf{\Gamma}\mathbf{\Phi}\), and we can rewrite the Equation (14) as  
\[E\Bigg{\{}\left\|\sum_{p=1}^{N_{T}}\mathbf{x}_{pc,l}-\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{ \alpha}\right\|_{2}^{2}\Bigg{\}}=C+E\Big{\{}\left\|\mathbf{x}_{l}-\mathbf{\Gamma}\bm {S}_{l}\mathbf{\alpha}\right\|_{2}^{2}\Big{\}}+2\text{Re}\{\mathbf{\psi}\mathbf{\alpha}\}. \tag{16}\]  
By minimizing the ICM-MP error calibration matrix and the output power, the optimization problem can be formulated to estimate the profile vector \(\mathbf{\alpha}\) and calibration vector \(\mathbf{h}\)[39]  
\[\big{\{}\mathbf{\hat{\alpha}},\mathbf{\hat{h}}\big{\}}=\underset{\mathbf{\alpha},\mathbf{h}} {\text{argmin}}J(\mathbf{\alpha},\mathbf{h})=\underset{\mathbf{\alpha},\mathbf{h}}{\text{ argmin}}E\Big{\{}\left\|\mathbf{x}_{l}-\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{\alpha}\right\|_{2}^{2} \Big{\}}+2\kappa\|\mathbf{\alpha}\|_{1}+2\text{Re}\{\mathbf{\psi}\mathbf{\alpha}\}+\|\mathbf{ \Lambda}\mathbf{S}_{l}\|_{F}^{2}, \tag{17}\]  
where \(\mathbf{\Gamma}=\mathbf{I}_{N}\otimes diag(\mathbf{I}_{M\times 1}+\mathbf{h})\). Based on the above analyses, the MP clutter suppression problem is equivalent to the estimation problem of the profile vector \(\mathbf{\alpha}\) and calibration vector \(\mathbf{h}\). To overcome this issue, the calibration vector \(\mathbf{h}\) must first be fixed. Thus, the minimization of Equation (17) with a profile \(\mathbf{\alpha}\) becomes:  
\[J_{1}(\mathbf{\alpha})=E\Big{\{}\|\mathbf{x}_{l}-\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{\alpha}\|_{ 2}^{2}\Big{\}}+2\kappa\|\mathbf{\alpha}\|_{1}+2\text{Re}\{\mathbf{\psi}\mathbf{\alpha}\}. \tag{18}\]  
We compute the gradient terms of the above equation with respect to \(\mathbf{\alpha}^{*}\), and let the gradient terms equal zero. Thus, we get:  
\[\mathbf{\alpha}=\mathbf{R}_{S}^{-1}\Big{(}\mathbf{r}_{Sx}-\kappa\text{sign}(\mathbf{\alpha})+ \mathbf{\psi}^{\text{H}}\Big{)}, \tag{19}\]  
where \(\mathbf{R}_{S}=E\Big{[}\mathbf{S}_{l}^{\text{H}}\mathbf{\Gamma}\mathbf{S}_{l}\Big{]};\mathbf{r}_{Sx }=E\Big{[}\mathbf{S}_{l}^{\text{H}}\mathbf{\Gamma}^{\text{H}}\mathbf{x}_{l}\Big{]}\) and \(\text{sign}(\cdot)\) denotes a component-wise function. On the other hand, by fixing the profile vector \(\mathbf{\alpha}\), the estimation of \(\mathbf{h}\) is transformed into the following optimization problem:  
\[\mathbf{x}_{pc,l}=\mathbf{\varepsilon}_{p}{}^{\prime}\mathbf{\Gamma}\mathbf{x}_{l+(l_{p}-l_{ \overline{p}}-l_{\overline{d}})}\odot\mathbf{f}\big{(}\omega_{p}-\omega_{\rm d} \big{)}-\mathbf{\varepsilon}_{p}{}^{\prime}\mathbf{\Gamma}\sum_{\overline{p}=1}^{N_{ \rm T}}\mathbf{\varepsilon}_{\overline{p}}{}^{\prime}\mathbf{x}_{l+(l_{p}+l_{\overline{p}} -2l_{\rm d})}\odot\mathbf{f}\big{(}\overline{\omega}_{p}+\omega_{p}-2\omega_{\rm d} \big{)}. \tag{12}\]0  
and \((k,m,n)(n=1,2,\ldots)\) be the global search and the \(n\)-th local search of the \((k,m)\)-th iterative search, respectively.