## 4 Proposed MP Clutter Suppression Algorithm Based on Joint Optimization
### The Motivation of the Proposed Joint Optimization Algorithm
Initially, we discretize the temporal plane with a relatively large interval \(\Delta\omega\), and build the following range-Doppler dictionary, termed the global dictionary. \[ \begin{array}{ll}\mathbf{S}_{l}&=\big{[}\mathbf{x}_{l+1}\odot\mathbf{f}(\omega_{1}), \ldots,\mathbf{x}_{l+D}\odot\mathbf{f}(\omega_{1}),\ldots,\mathbf{x}_{l+1}\odot\mathbf{f}( \omega_{Q}),\ldots,\mathbf{x}_{l+D}\odot\mathbf{f}(\omega_{Q})\big{]}\\ &=\big{[}\mathbf{g}_{l,1},\ldots,\mathbf{g}_{l,D},\ldots,\mathbf{g}_{l,(Q-1)D+1},\ldots, \mathbf{g}_{l,DQ}\big{]}\end{array} , \tag{25}\]  
where \(\mathbf{g}_{l,\eta}\) (\(\eta=1,2,\ldots,DQ\)) is the \(\eta\)-th global atom. Then, the global index in the \((k,m,0)\)-th global search is picked up by [11]:  
\[\eta(k,m,0)=\arg\max_{\eta\in\{1,2,\ldots,DQ\}}E\Big{[}\mathbf{g}_{l,\eta}^{\rm H} \mathbf{\Gamma}^{\rm H}(k-1)\mathbf{\hat{e}}_{l}^{\prime}(k,m)\Big{]}, \tag{26}\]  
where \(\mathbf{\hat{e}}_{l}^{\prime}(k,m)\) is the residual vector, which is initialized as \(\mathbf{\hat{e}}_{l}^{\prime}(k,1)=\mathbf{x}_{l}\). Based on the selected global index, the range cell and normalized Doppler frequency corresponding to the optimal global atom \(\mathbf{g}_{l,\eta(k,m,0)}\) are denoted by \(\hat{\tau}(k,m)\) and \(\omega(k,m,0)\), respectively. Subsequently, we conducted several local searches. Motivated by the local search technique [11], the local dictionary in the \((k,m,n)\)-th local search is expressed as:  
\[ \begin{array}{ll}\mathbf{S}_{l}(k,m,n)&=\Big{[}\mathbf{x}_{l+\tau(k,m)}\odot\mathbf{f} \Big{(}\omega(k,m,n-1)-\frac{\Delta\omega}{2^{l}}\Big{)},\mathbf{x}_{l+\tau(k,m)} \odot\mathbf{f}(\omega(k,m,n-1)),\mathbf{x}_{l+\tau(k,m)}\odot\mathbf{f}\Big{(}\omega(k,m,n-1)+\frac{\Delta\omega}{2^{l}}\Big{)}\Big{]}\\ &=\big{[}\mathbf{g}_{l,1}(k,m,n),\mathbf{g}_{l,2}(k,m,n),\mathbf{g}_{l,3}(k,m,n)\big{]} \end{array} . \tag{27}\]  
Correspondingly, the local index in the \((k,m,n)\)-th local search is [11]:  
\[\eta(k,m,n)=\arg\max_{\eta\in\{1,2,3\}}a_{\eta}(k,m,n)=\arg\max_{\eta\in\{1,2, 3\}}E\Big{[}\mathbf{g}_{l,\eta}^{\rm H}(k,m,n)\hat{\mathbf{\Gamma}}^{\rm H}(k-1)\mathbf{ \hat{e}}_{l}^{\prime}(k,m)\Big{]}, \tag{28}\]  
where \(a_{\eta}(k,m,n)\) denotes the local response. Thus, the optimal local atom is \(\mathbf{g}_{l,\eta(k,m,n)}\) and the corresponding normalized Doppler frequency is \(\omega(k,m,n)\). When \(\max\big{\{}a_{\eta}(k,m,n)-a_{\eta}(k,m,n-1)\big{\}}\) is less than the local threshold \(\delta_{1}\), the local search terminates, which also means the \((k,m)\)-th iterative search has been completed and an optimal atom is determined. Supposing the optimal normalized Doppler frequency is \(\hat{\omega}(k,m)\), then the optimal atom is \(\hat{\mathbf{g}}_{l}(k_{m})=\mathbf{x}_{l+\hat{\tau}(k,m)}\odot\mathbf{f}(\hat{\omega}(k,m))\). Let \(\hat{\mathbf{S}}_{l}\) and \(\hat{\mathbf{\Phi}}\) be the optimal dictionary and modified steering matrix, respectively. \(\hat{\mathbf{S}}_{l}\) and \(\hat{\mathbf{\Phi}}\) are initialized as \(\hat{\mathbf{S}}_{l}(k,0)=\varnothing\) and \(\hat{\mathbf{\Phi}}_{l}(k,0)=\varnothing\), and updated by \(\hat{\mathbf{S}}_{l}(k,m)=\big{[}\hat{\mathbf{S}}_{l}(k,m-1),\hat{\mathbf{g}}_{l}(k,m) \big{]}\) and \(\hat{\mathbf{\Phi}}(k,m)=\big{[}\hat{\mathbf{\Phi}}(k,m-1),\mathbf{f}(\hat{\omega}(k,m)) \big{]}\), respectively. Therefore, based on Equation (19), the profile vector \(\mathbf{\hat{\alpha}}\) at the \((k,m)\)-th iterative search is calculated by using \(\hat{\mathbf{\Gamma}}(k-1)\). \[\hat{\mathbf{\alpha}}(k,m)=\hat{\mathbf{R}}_{S}^{-1}(k,m)\Big{(}\hat{\mathbf{r}}_{Sx}(k,m)+ \hat{\mathbf{\psi}}^{\rm H}(k,m)\Big{)}, \tag{29}\]  
where \(\hat{\mathbf{R}}_{S}(k,m)=E\Big{[}\hat{\mathbf{S}}_{l}^{\rm H}(k,m)\hat{\mathbf{\Gamma}}^{ \rm H}(k-1)\hat{\mathbf{\Gamma}}(k-1)\hat{\mathbf{S}}_{l}(k,m)\Big{]}\);\(\hat{\mathbf{r}}_{Sx}(k,m)=E\Big{[}\hat{\mathbf{S}}_{l}^{\rm H}(k,m)\hat{\mathbf{ \Gamma}}^{\rm H}(k-1)\mathbf{\pi}_{l}\Big{]}\) and \(\hat{\mathbf{\psi}}(k,m)=\sigma^{2}\mathbf{1}_{1\times MN}\hat{\mathbf{\Gamma}}(k-1)\hat{ \mathbf{\Phi}}(k,m)\).