## 4 Proposed MP Clutter Suppression Algorithm Based on Joint Optimization
### Review of the Existing Optimization Problem in SRA and LSA  
In the existing SRA and LSA, the MP clutter is related to the DP clutter snapshot and the modified steering vector in the absence of ICM-MP fluctuation. Thus, the whole temporal plane is discretized uniformly into \(Q\) grid points. The corresponding normalizedDoppler frequency and modified steering vector are \(\omega_{q}\) and \(\mathbf{f}(\omega_{q})=\mathbf{1}_{N\times 1}\otimes\mathbf{v}_{l}\big{(}\omega_{q}\big{)}\), \(q=1,2,\ldots,Q\), respectively. Supposing the MP clutter can be expressed by \(D\) received snapshots, then the sparse model of MP clutter can be approximated by [8]:  
\[\sum_{p=1}^{N_{\rm T}}\mathbf{x}_{pc,l}=\mathbf{S}_{l}\mathbf{\alpha}, \tag{8}\]  
where the range-Doppler dictionary \(\mathbf{S}_{l}=\big{[}\mathbf{S}_{D,l}\odot\mathbf{F}_{1},\mathbf{S}_{D,l}\odot\mathbf{F}_{2}, \ldots,\mathbf{S}_{D,l}\odot\mathbf{F}_{Q}\big{]}\); the \(MN\times D\) matrix \(\mathbf{S}_{D,l}=[\mathbf{x}_{l+1},\mathbf{x}_{l+2},\ldots,\mathbf{x}_{l+D}]\), the \(MN\times D\) matrix \(\mathbf{F}_{q}=1_{1\times D}\otimes\mathbf{f}\big{(}\omega_{q}\big{)}=1_{N\times D} \otimes\mathbf{v}_{l}\big{(}\omega_{q}\big{)}\); \(\mathbf{\alpha}\) indicates the \(DQ\times 1\) profile, and there exists a degree of sparsity in \(\mathbf{\alpha}\)[8].  
By denoting \(\mathbf{S}_{D,n}=\mathbf{x}_{n}\otimes 1_{1\times D}\) and \(\mathbf{S}_{n}=\big{[}\mathbf{S}_{D,n}\odot\mathbf{F}_{1},\mathbf{S}_{D,n}\odot\mathbf{F}_{2}, \ldots,\mathbf{S}_{D,n}\odot\mathbf{F}_{Q}\big{]}\), we obtain [8]:  
\[E\Bigg{\{}\Bigg{\|}\sum_{p=1}^{N_{\rm T}}\mathbf{x}_{pc,l}-\mathbf{S}_{l} \mathbf{\alpha}\Bigg{\|}_{2}^{2}\Bigg{\}} =C+E\Big{\{}\|\mathbf{x}_{l}-\mathbf{S}_{l}\mathbf{\alpha}\|_{2}^{2}\Big{\}} +2\text{Re}\big{\{}E\big{[}\mathbf{x}_{n}^{\rm H}\mathbf{S}_{l}\mathbf{\alpha}\big{]} \big{\}} \tag{9}\] \[=C+E\Big{\{}\|\mathbf{x}_{l}-\mathbf{S}_{l}\mathbf{\alpha}\|_{2}^{2}\Big{\}} +2\text{Re}\big{\{}E\big{[}\mathbf{x}_{n}^{\rm H}\mathbf{S}_{n}\mathbf{\alpha}\big{]} \big{\}}\]  
where \(E\big{[}\mathbf{x}_{n}^{\rm H}\mathbf{S}_{n}\mathbf{\alpha}\big{]}=E\big{[}\mathbf{x}_{n}^{ \rm H}diag(\mathbf{x}_{n})\mathbf{\Phi}\mathbf{\alpha}\big{]}=\mathbf{\zeta}\mathbf{\alpha}\), \(\mathbf{\zeta}=\sigma^{2}1_{1\times MN}\mathbf{\Phi}\), the modified steering matrix \(\mathbf{\Phi}=\big{[}\mathbf{F}_{1},\mathbf{F}_{2},\ldots,\mathbf{F}_{Q}\big{]}\) and \(\sigma^{2}\) is the noise power [8], \(\mathbf{C}\) is a constant independent of parameters \(\mathbf{S}_{l}\) and \(\mathbf{\alpha}\).  
By minimizing the output power, the MP clutter estimation is described as the following optimization problem [8]:  
\[\hat{\mathbf{\alpha}}=\underset{\mathbf{\alpha}}{\text{argmin}}E\Big{\{}\|\mathbf{x}_{l}- \mathbf{S}_{l}\mathbf{\alpha}\|_{2}^{2}\Big{\}}+2\kappa\|\mathbf{\alpha}\|_{1}+2\text{Re} \big{\{}\mathbf{\zeta}\mathbf{\alpha}\big{\}}, \tag{10}\]  
where \(\kappa\) is a sparse constant parameter.  
Based on the above optimization problem, SRA estimates the profile \(\mathbf{\alpha}\) to suppress MP clutter, and LSA modifies the range-doppler dictionary to overcome the off-grid problem. However, the presence of ICM-MP fluctuations significantly affects the MP clutter suppression performance of SRA and LSA.