## 4 Proposed MP Clutter Suppression Algorithm Based on Joint Optimization
### The Motivation of the Proposed Joint Optimization Algorithm
Correspondingly, the residual vector is updated by \(\hat{\mathbf{e}}_{l}^{\prime}(k,m+1)=\mathbf{x}_{l}-\hat{\mathbf{\Gamma}}(k-1)\hat{\mathbf{S}}_ {l}(k,m)\hat{\mathbf{\kappa}}(k,m)\). The iterative search terminates until \(I(k,m)-I(k,m-1)\) is smaller than the search threshold \(\delta_{2}\), where \(I(k,m)=SINR(k,m)/SNR_{\rm opt}\) denotes the signal-to-interference-plus-noise-ratio (SINR) loss of the \((k,m)\)-th iterative search, \(SINR(k,m)\) and \(SNR_{\rm opt}\) are SINR of the \((k,m)\)-th iterative search and the optimum output signal-to-noise ratio in an interference-free environment, respectively [12]. It also means the profile estimation step of the \(k\)-th external alternation has been completed, and all optimal atoms representing MP clutter have been selected. Consequently, we obtain the profile vector \(\hat{\mathbf{\kappa}}(k)\) in the \(k\)-th external alternation. The corresponding optimal dictionary and modified steering matrix are expressed as \(\hat{\mathbf{S}}_{l}(k)\) and \(\hat{\mathbf{\Phi}}(k)\), respectively. In the ICM-MP error calibration step of the \(k\)-th external alternation, the calibration vector \(\mathbf{h}\) is calculated based on Equation (24)  
\[\hat{\mathbf{h}}(k)=\hat{\mathbf{R}}_{G}^{-1}(k)\Big{(}\hat{\mathbf{r}}_{Ge}(k)-\hat{\mathbf{ \xi}}^{\text{H}}(k)\Big{)}, \tag{30}\]  
where the covariance matrix \(\hat{\mathbf{R}}_{G}(k)=E\Big{[}\hat{\mathbf{G}}_{I}^{\text{H}}(k)\hat{\mathbf{G}}_{I}(k)+ \mathbf{I}_{M}\Big{]}\); the cross-correlation vector \(\hat{\mathbf{r}}_{Ge}(k)=E\Big{[}\hat{\mathbf{G}}_{I}^{\text{H}}(k)\hat{\mathbf{e}}_{I}(k) \Big{]}\); \(\hat{\mathbf{e}}_{i}(k)=\mathbf{x}_{l}-\hat{\mathbf{S}}_{I}(k)\hat{\mathbf{a}}(k)\) and \(\hat{\mathbf{G}}_{I}(k)=diag\big{(}\hat{\mathbf{S}}_{I}(k)\hat{\mathbf{a}}(k)\big{)}(\mathbf{I} _{N\times 1}\otimes\mathbf{I}_{M})\); and \(\hat{\mathbf{\xi}}(k)=\sigma^{2}\big{(}\hat{\mathbf{\Phi}}(k)\hat{\mathbf{a}}(k)\big{)}^{T }(\mathbf{I}_{N\times 1}\otimes\mathbf{I}_{M})\). To efficiently estimate \(\hat{\mathbf{h}}(k)\), let  
\[\hat{\mathbf{B}}(k) =\text{diag}\Big{(}\hat{\mathbf{S}}_{I}^{\ast}(k)\hat{\mathbf{a}}^{\ast} (k)\Big{)}\text{diag}\big{(}\hat{\mathbf{S}}_{I}(k)\hat{\mathbf{a}}(k)\big{)}\] \[=\text{diag}\big{[}\hat{\mathbf{b}}_{1}(k),\hat{\mathbf{b}}_{2}(k),\cdots,\hat{\mathbf{b}}_{N}(k)\big{]} \tag{31}\]  
where the \(1\times M\) vector \(\hat{\mathbf{b}}_{i}(k)\) is composed of \(((i-1)M+1)\)-th to \((iM)\)-th diagonal elements of \(\hat{\mathbf{B}}(k)\). Then, we get:  
\[\hat{\mathbf{G}}_{I}^{\text{H}}(k)\hat{\mathbf{G}}_{I}(k) =(\mathbf{I}_{1\times N}\otimes\mathbf{I}_{M})\hat{\mathbf{B}}(k)(1_{N\times 1 }\otimes\mathbf{I}_{M})\] \[=\big{[}\text{diag}\big{(}\hat{\mathbf{b}}_{1}(k)\big{)},\text{diag} \big{(}\hat{\mathbf{b}}_{2}(k)\big{)},\cdots,\text{diag}\big{(}\hat{\mathbf{b}}_{N}(k )\big{)}\big{]}[\mathbf{I}_{M},\mathbf{I}_{M},\cdots,\mathbf{I}_{M}]^{T}\enspace. \tag{32}\] \[=\text{diag}\bigg{(}\sum\limits_{i=1}^{N}\hat{\mathbf{b}}_{i}(k)\bigg{)}\]  
Thus, \(\hat{\mathbf{G}}_{I}^{\text{H}}(k)\hat{\mathbf{G}}_{I}(k)\) is the diagonal matrix, and the diagonal matrix \(\hat{\mathbf{R}}_{G}(k)=diag\big{(}\big{[}\hat{\mathbf{R}}_{k,1},\hat{\mathbf{R}}_{k,2}, \ldots,\hat{\mathbf{R}}_{k,M}\big{]}\big{)}\). Then, Equation (30) is efficiently calculated by:  
\[\hat{\mathbf{h}}(k)=\big{[}\hat{\mathbf{r}}_{k,1}/\hat{\mathbf{R}}_{k,1},\hat{\mathbf{r}}_{k, 2}/\hat{\mathbf{R}}_{k,2},\ldots,\hat{\mathbf{r}}_{k,M}/\hat{\mathbf{R}}_{k,M}\big{]}^{T}, \tag{33}\]  
where \(\hat{\mathbf{r}}_{k,i}\) is the \(i\)-th element of \(\hat{\mathbf{r}}_{Ge}(k)-\hat{\mathbf{\xi}}^{\text{H}}(k)\). Correspondingly, the update of the ICM-MP fluctuation matrix is given by \(\hat{\mathbf{\Gamma}}(k)=\mathbf{I}_{N}\otimes diag\big{(}\mathbf{I}_{M\times 1}+\hat{\mathbf{h}}(k) \big{)}\). When the gain of the cost function expressed by Equation (17) is less than a predetermined alternation threshold \(\delta_{3}\)[39], the alternating step between Equations (29) and (33) terminates.