# Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss
## 3 Double-Layer CS Framework-Based DOA Estimation
### Improved DOA Estimation by DCS-DDOA
#### 3.3.2 Direct DOA Estimation  
Leveraging the joint frequency and spatial domain sparse representation of the sensor array data, DOA estimation can be directly implemented at the FC according to the received data samples. Substituting Equations (16) and (17) into Equation (9), we can have:  
\[\mathbf{y}=\mathbf{\Theta}\mathbf{\alpha}=\mathbf{\Theta}\mathbf{P}\mathbf{\bar{ \mathbf{\alpha}}}=\mathbf{\Theta}\mathbf{P}\left(\mathbf{\bar{\mathbf{B}}} \mathbf{s}+\mathbf{w}\right)=\mathbf{\Gamma}\mathbf{s}+\mathbf{\bar{w}} \tag{18}\]  
where \(\mathbf{\bar{w}}=\mathbf{\Theta}\mathbf{P}\mathbf{w}\) is the subsampled white Gaussian noise at the FC, and \(\mathbf{\Gamma}=\mathbf{\Theta}\mathbf{P}\mathbf{\bar{B}}\) is the joint sparse representation matrix with a dimension of \(HM\times NQ\), combining the frequency sparsity in \(\mathbf{\Theta}\) and the spatial sparsity in \(\mathbf{\bar{B}}\). The structure of \(\mathbf{\Gamma}\) is shown as follows:  
\[\mathbf{\Gamma}=\left[\begin{array}{cccc}\mathbf{\Theta}_{1,1}\mathbf{\bar{ a}}_{1}\left(0\right)&\mathbf{\Theta}_{1,2}\mathbf{\bar{a}}_{1}\left(1 \right)&\cdots&\mathbf{\Theta}_{1,N}\mathbf{\bar{a}}_{1}\left(N-1\right)\\ \mathbf{\Theta}_{2,1}\mathbf{\bar{a}}_{2}\left(0\right)&\mathbf{\Theta}_{2,2} \mathbf{\bar{a}}_{2}\left(1\right)&\cdots&\mathbf{\Theta}_{2,N}\mathbf{\bar{ a}}_{2}\left(N-1\right)\\ \vdots&\vdots&\vdots&\vdots\\ \mathbf{\Theta}_{H,1}\mathbf{\bar{a}}_{H}\left(0\right)&\mathbf{\Theta}_{H,2} \mathbf{\bar{a}}_{H}\left(1\right)&\cdots&\mathbf{\Theta}_{H,N}\mathbf{\bar{ a}}_{H}\left(N-1\right)\end{array}\right] \tag{19}\]  
where \(\mathbf{\Theta}_{h,n}\) is the \(n\)-th column of \(\mathbf{\Theta}_{h}\), i.e., \(\mathbf{\Phi}_{r,h}\mathbf{\Phi}_{s,h}\mathbf{\bar{Y}}\) and \(\mathbf{\bar{a}}_{h}\left(n\right)\) is the \(h\)-th row of \(\mathbf{\bar{A}}\left(n\right)\). Now, we can reconstruct the joint sparse indicator vector \(\mathbf{\bar{s}}\) from the joint measurement vector \(\mathbf{y}\) to obtain the DOA estimation results without the need for recovering the sparse vectors \(\mathbf{\alpha}_{h},1\leq h\leq H\).  
Define a matrix \(\mathbf{\bar{S}}=\left[\mathbf{s}\left[0\right],\mathbf{s}\left[1\right],..., \mathbf{s}\left[N-1\right]\right]\), with \(\mathbf{s}_{q},q=1,...,Q\) denoting the \(q\)-th row vector of \(\mathbf{\bar{S}}\). On the basis of the mentioned representations above and the concept of group sparsity, we can formulate the DOA estimation problem as the following constrained \(l_{1}\)-norm minimization problem:  
\[\begin{array}{ll}\min&\left\|\mathbf{\gamma}\right\|_{1}\\ \text{subject to}&\left\|\mathbf{y}-\mathbf{\Gamma}\mathbf{\bar{s}}\right\|_{2} \leq\varepsilon_{3}\end{array} \tag{20}\]  
where \(\mathbf{\gamma}=\left[\left\|\mathbf{s}_{1}\right\|_{2},\left\|\mathbf{s}_{2} \right\|_{2},...,\left\|\mathbf{s}_{Q}\right\|_{2}\right]^{T}\) is the sparse indicator vector and \(\varepsilon_{3}\) is the user-specified error bound. In the same way, the problem in Equation (20) is solved using the SOCP approach by virtue of the SeDuMi toolbox. Finally, the DOA estimation results of the wideband source signals are spontaneously deduced according to the sparse solution to \(\mathbf{\gamma}\).  
Different from the two-stage DCS-DOA, the DCS-DDOA can desirably exploit the joint sparsity in the frequency and spatial domain. In this integrated approach, it can subtly avoid the error propagation problem existing in the two-stage DCS-DOA. More specifically, the formulated problem is capable of directly obtaining the DOA estimation results, as shown in (20). As a consequence, the DOA estimation performance can be further improved.