## 4 Proposed MP Clutter Suppression Algorithm Based on Joint Optimization
### The Motivation of the Proposed Joint Optimization Algorithm  
If we define \(\mathbf{a}_{l_{p},\overline{p}}=\mathbf{\Gamma}\mathbf{x}_{l+(l_{p}+l_{\overline{p}}-2l_ {\alpha})}\odot\mathbf{f}\big{(}\overline{\omega}_{p}+\omega_{p}-2\omega_{\rm d} \big{)}\), then the \((p,\overline{p})\)-th ghost MP clutter in Equation (5) is \(\mathbf{x}_{l_{p},\overline{p}}=\mathbf{\Gamma}\Big{(}\mathbf{\varepsilon}_{p}{}^{\prime} \mathbf{\varepsilon}_{p}{}^{\prime}\mathbf{a}_{l_{p},\overline{p}}\Big{)}=\mathbf{\Gamma} \mathbf{x}_{l_{p},\overline{p}}^{\prime}\). Thus, we get [35]:  
\[E\Big{[}\mathbf{x}_{l_{p},\overline{p}}\mathbf{x}_{l_{p},\overline{p}}^{\rm H}\Big{]}- E\Big{[}\mathbf{x}_{l_{p},\overline{p}}^{\prime}\mathbf{x}_{l_{p},\overline{p}}^{\prime \rm H}\Big{]}=\big{|}\mathbf{\varepsilon}_{p}{}^{\prime}\big{|}^{2}\big{|} \mathbf{\varepsilon}_{\overline{p}}{}^{\prime}\big{|}^{2}\bigg{[}\widetilde{\mathbf{ \Gamma}}-1_{MN\times MN}\bigg{]}\odot\mathbf{R}_{a}, \tag{11}\]  
where \(\mathbf{R}_{a}=E\Big{[}\mathbf{a}_{l_{p},\overline{p}}\mathbf{a}_{l_{p},\overline{p}}^{\rm H }\Big{]}\), \(\widetilde{\mathbf{\Gamma}}=\mathbf{I}_{N\times N}\otimes\text{Toeplitz}(\zeta(0),\zeta(1 ),\ldots,\zeta(M-1))\) and the temporal autocorrelation of the ICM fluctuations is \(\zeta(m)=\exp\big{\{}-8\pi^{2}\sigma_{v}^{2}T_{r}^{2}m^{2}/\lambda^{2}\big{\}}\), \(m=0,1,\ldots,M-1\), \(\lambda\) is wavelength and \(\sigma_{v}\) is the velocity standard deviation [12]. Since \(\big{|}\mathbf{\varepsilon}_{p}{}^{\prime}\big{|}^{2}\ll 1\), \(\big{|}\mathbf{\varepsilon}_{\overline{p}}{}^{\prime}\big{|}^{2}\ll 1\) and each element in the matrix \(\widetilde{\mathbf{\Gamma}}-\mathbf{I}_{MN\times MN}\) is far less than \(1\) in general [12], the effect of the covariance matrix \(E\Big{[}\mathbf{x}_{l_{p},\overline{p}}\mathbf{x}_{l_{p},\overline{p}}^{\rm H}\Big{]}\) in STAP can be equivalent to the effect of \(E\Big{[}\mathbf{x}_{l_{p},\overline{p}}^{\prime}\mathbf{x}_{l_{p},\overline{p}}^{\rm H }\Big{]}\). Thus, Equation (5) can be approximated by:  
\[\mathbf{x}_{pc,l}=\mathbf{\varepsilon}_{p}{}^{\prime}\mathbf{\Gamma}\mathbf{x}_{l+(l_{p}-l_{ \overline{p}}-l_{\overline{d}})}\odot\mathbf{f}\big{(}\omega_{p}-\omega_{\rm d} \big{)}-\mathbf{\varepsilon}_{p}{}^{\prime}\mathbf{\Gamma}\sum_{\overline{p}=1}^{N_{ \rm T}}\mathbf{\varepsilon}_{\overline{p}}{}^{\prime}\mathbf{x}_{l+(l_{p}+l_{\overline{p}} -2l_{\rm d})}\odot\mathbf{f}\big{(}\overline{\omega}_{p}+\omega_{p}-2\omega_{\rm d} \big{)}. \tag{12}\]Therefore, the sparse model of MP clutter in the presence of ICM-MP becomes:  
\[\sum_{p=1}^{N_{T}}\mathbf{x}_{pc,l}=\mathbf{\Gamma}\mathbf{S}_{l}\mathbf{\alpha}=(\mathbf{S}_{l}+\mathbf{ \Lambda}\mathbf{S}_{l})\mathbf{\alpha}, \tag{13}\]  
where \(\mathbf{\Lambda}=\mathbf{\Gamma}-I_{MN}=\mathbf{I}_{N}\otimes diag(\mathbf{h})\) and \(\mathbf{h}=\mathbf{\gamma}_{2}-\mathbf{I}_{M\times 1}\). Here, \(\mathbf{\Lambda}\mathbf{S}_{l}\) and \(\mathbf{h}\), which are the functions of the ICM-MP fluctuation vector \(\mathbf{\gamma}_{2}\), are defined as the ICM-MP errors calibration matrix and the ICM-MP errors calibration vector, respectively.