# Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss
## 3 Double-Layer CS Framework-Based DOA Estimation
### DOA Estimation by DCS-DOA
#### 3.2.2 DOA Estimation  
After recovering the sparse vector at each array sensor, DOA estimation can be realized using various algorithms, such as the MUSIC technique and the maximum likelihood estimator. In this paper, a sparse signal reconstruction perspective is adopted for better estimation accuracy and robustness to noise. We denote the recovered sparse vector at the \(h\)-th array sensor by \(\bar{\mathbf{\kappa}}_{h},h=1,...,H\). Now, the array output data are already in the frequency domain, so we can directly perform DOA estimation without the need for DFT conducted in the common wideband DOA estimation process. Assume that \(\bar{\mathbf{\kappa}}_{h}\) contains \(K\left(K\ll N\right)\) nonzero elements; we only need to consider the specific \(K\) nonzero frequency sub-bands. First, we investigate the DOA estimation problem at a single frequency sub-band (denoted by \(f_{k}\), chosen from the \(K\) nonzero frequency components), and the array output vector at \(f_{k}\) can be denoted by \(\bar{\mathbf{\kappa}}\left[f_{k}\right]=\left[\bar{\mathbf{\kappa}}_{1}\left[f_{k} \right],...,\bar{\mathbf{\kappa}}_{H}\left[f_{k}\right]\right]^{T}\). Similarly, the noise vector for the sensor array at the frequency sub-band \(f_{k}\) is denoted by \(\mathbf{n}\left[f_{k}\right]=\left[n_{1}\left[f_{k}\right],...,n_{H}\left[f_{ k}\right]\right]^{T}\). Using a search grid of \(Q\) elements, with each of them indicating a potential source signal at the corresponding incident angle, the array output vector at the frequency sub-band \(f_{k}\) can be formulated as follows:  
\[\bar{\mathbf{\kappa}}\left[f_{k}\right]=\bar{\mathbf{\Lambda}}\left(f_{k},\bar{\mathbf{ \theta}}\right)\bar{\mathbf{\kappa}}\left[f_{k}\right]+\mathbf{n}\left[f_{k}\right] \tag{11}\]  
where \(\bar{\mathbf{\Lambda}}\left(f_{k},\bar{\mathbf{\theta}}\right)=\left[\bar{\mathbf{\alpha }}\left(f_{k},\bar{\theta}_{1}\right),...,\bar{\mathbf{\alpha}}\left(f_{k},\bar{ \theta}_{Q}\right)\right]\) is the array manifold matrix corresponding to the virtual source signal vector \(\bar{\mathbf{\kappa}}\left[f_{k}\right]=\left[\bar{s}[f_{k},\bar{\theta}_{1}], \bar{s}[f_{k},\bar{\theta}_{2}],...,\bar{s}[f_{k},\bar{\theta}_{Q}]\right]^{T}\), and \(\bar{\mathbf{\alpha}}\left(f_{k},\bar{\theta}_{q}\right)\) is the steering vector related to \(\bar{\theta}_{q}\). The sampling grids of all potential directions are denoted by a vector \(\bar{\mathbf{\theta}}=\left[\bar{\theta}_{1},...,\bar{\theta}_{Q}\right]^{T}\). As the sampling grid is sufficiently fine, the number of nonzero elements in \(\bar{\mathbf{\kappa}}\left[f_{k}\right]\) is much smaller than \(Q\). That is to say, \(\bar{\mathbf{\kappa}}\left[f_{k}\right]\) is sparse, and the indexes of the nonzero elements indicate the DOAs of the actual sources. A sparse signal reconstruction perspective for DOA estimation was proposed in [3], and it can be applied to a single frequency sub-band in the wideband case directly, so the aforementioned DOA estimation problem can be resolved by solving the optimization problem below:  
\[ \begin{array}{ll}\min&\left\|\bar{\mathbf{\kappa}}\left[f_{k}\right]\right\|_{1 }\\ \text{subject to }&\left\|\bar{\mathbf{\kappa}}\left[f_{k}\right]-\bar{\mathbf{\Lambda}} \left(f_{k},\bar{\mathbf{\theta}}\right)\bar{\mathbf{\kappa}}\left[f_{k}\right]\right\| _{2}\leq\epsilon_{1}\end{array}  \tag{12}\]  
where \(\epsilon_{1}\) is the user-specified error bound, and the \(l_{1}\)-norm and the \(l_{2}\)-norm represent the sparsity penalty and the residual error, respectively. As the DOAs corresponding to the \(K\left(K\ll N\right)\) nonzero frequency components share the same spatial sparse pattern, we can estimate the DOAs of wideband source signals based on the concept of group-sparsity. Assume that the frequencies of the \(K\) nonzero sub-bands are denoted by \(f_{1},...,f_{K}\).