# Online Sparse DOA Estimation Based on Sub-Aperture Recursive LASSO for TDM-MIMO Radar
## 2 Materials and Methods
### Proposed Method
#### 2.2.2 Proposed Online Strategy  
For TDM-MIMO radars, the sub-aperture data arrive sequentially over time. The \(p\)-th sub-aperture data and the corresponding steering matrix can be expressed as  
\[\mathbf{y}_{p}\triangleq\left[ \begin{array}{c}y(pM_{r}-M_{r}+1)\\ \vdots\\ y(pM_{r})\end{array} \right]\,\ \ \ \ \ \mathbf{A}_{p}\triangleq\left[ \begin{array}{c} \mathbf{b}(pM_{r}-M_{r}+1)\\ \vdots\\ \mathbf{b}(pM_{r})\end{array} \right] \tag{18}\]  
Vector \(\mathbf{y}_{p}\in\mathbb{C}^{M_{r}}\) contains data for the \(p\)-th sub-aperture, and the length is \(M_{r}\), i.e., data blocks 1-4 in Figure 2b (if \(p=1\)). Similarly, the steering matrix \(\mathbf{A}_{p}\in\mathbb{C}^{M_{r}\times K}\) contains the steering vectors of the corresponding subarray elements. The optimization problem (6) can then be rewritten as  
\[\operatorname*{arg\,min}_{\mathbf{s}\in\mathbb{R}^{K}}\left\|\mathbf{y}_{p}- \mathbf{A}_{p}\mathbf{s}\right\|_{2}^{2}+\lambda\|\mathbf{s}\|_{1} \tag{19}\]  
Let \(\overline{\mathbf{s}}\in\mathbb{R}^{K}\) denote the estimated result from the \(p-1\)-th sub-aperture data. Based on the cyclic minimization, we first introduce an intermediate variable \(\mathbf{z}_{p}=\mathbf{y}_{p}-\mathbf{A}_{p}\overline{\mathbf{s}}\) that can be eliminated in subsequent derivations. Then, \(\mathbf{\tilde{y}}_{p,i}=\mathbf{z}_{p}+\mathbf{a}_{p,i}\bar{s}_{i}\in \mathbb{C}^{M_{r}}\), which is substituted into Equation (12)  
\[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\] \[=\left\|\mathbf{z}_{p}+\mathbf{a}_{p,i}\bar{s}_{i}\right\|^{2} \tag{20a}\] \[=\left\|\mathbf{z}_{p}\right\|^{2}+\left\|\mathbf{a}_{p,i}\right\|^{2} \left|\bar{s}_{i}\right|^{2}+2\operatorname{Re}\Bigl{\{}\bar{s}_{i}\mathbf{a} _{p,i}\mathbf{z}_{p}\Bigr{\}}\] \[\beta_{i} =\left\|\mathbf{a}_{p,i}\right\|^{2}\] (20b) \[\gamma_{i} =\left|\mathbf{a}_{p,i}^{*}\mathbf{\tilde{y}}_{i}\right|\] (20c) \[=\left|\mathbf{a}_{p,i}^{*}\left(\mathbf{z}_{p}+\mathbf{a}_{p,i} \bar{s}_{i}\right)\right|\]where \(\bar{s}_{i}\) denotes the current estimate of the \(i\)-th direction, and \(\mathbf{a}_{p,i}\in\mathbb{C}^{M_{r}}\) the steering vector of the \(i\)-th direction for the \(p\)-th sub-aperture, namely, the \(i\)-th column of \(\mathbf{A}_{p}\). The intermediate variable  
\[\boldsymbol{\zeta}_{p}=\mathbf{A}_{p}^{*}\mathbf{z}_{p}\in\mathbb{C}^{K} \tag{21}\]  
and recursive calculation variables are introduced:  
\[\boldsymbol{\Gamma}^{p}\triangleq\mathbf{A}_{p}^{*}\mathbf{A}_{p}= \boldsymbol{\Gamma}^{p-1}+\mathbf{b}_{p}\mathbf{b}_{p}^{*}\in\mathbb{C}^{K \times K} \tag{22}\]  
\[\boldsymbol{\rho}^{p}\triangleq\mathbf{A}_{p}^{*}\mathbf{y}_{p}=\boldsymbol{ \rho}^{p-1}+\mathbf{b}_{p}\mathbf{y}_{p}\in\mathbb{C}^{K} \tag{23}\]  
In minimizing \(r_{i}\), only the calculation of \(\beta_{i}\) and \(\gamma_{i}\) is needed. Considering Equation (22), the expression (20) can be simplified to  
\[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\]0 \[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\]1  
where \(\zeta_{i}\) denotes the \(i\)-th element of \(\boldsymbol{\zeta}_{p}\), and \(\Gamma_{ii}^{p}\) the \(i\)-th diagonal element of matrix \(\boldsymbol{\Gamma}^{p}\). Similarly, Equation (11) can also be represented by recursive variables as  
\[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\]2  
Hence, the \(i\)-th element of the DOA is obtained:  
\[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\]3  
By traversing \(i=1,2,\cdots,K\) for Equations (24)-(26), the target source distribution result \(\\[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\]4\) can be obtained by reiterating (21)-(27) using Equation (21) (which is achieved through the previous result \(\bar{\mathbf{s}}\) by the previous \(p-1\) sub-aperture data) and the new sub-aperture data \(\mathbf{y}_{p}\). The online processing flowchart for the sub-aperture update is shown in \[\alpha_{i} =\left\|\mathbf{\tilde{y}}_{p,i}\right\|^{2}\]5