# Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss
## 3 Double-Layer CS Framework-Based DOA Estimation
### Double-Layer CS Framework  
In this part, we elaborate the double-layer CS framework, which consists of a passive CS process modeling the random packet loss and an active CS process introduced at each array sensor. Furthermore, we present the mutual coherence of the equivalent measurement matrix and the absolute off-diagonal entries' distribution of the corresponding Gram matrix, which account for the satisfactory DOA estimation performance under the double-layer CS framework. For comparison, we first present a single-layer CS framework. Assume \(N\) data samples are acquired at each array sensor, and they are transmitted to the FC to complete one DOA estimate. Under the single-layer CS framework, the raw data samples are directly assembled into packets for transmission. The lossy nature of low-power wireless links is exploited by modeling the random \begin{table}
\begin{tabular}{c c} \hline
**Notations** & **Explanations** \\ \hline \(\mathbf{x}_{h}\) & raw data samples at the \(h\)-th array sensor \\ \(\overline{\mathbf{v}}\) & Fourier sparsifying dictionary \\ \(\mathbf{a}_{h}\) & sparse representation of \(\mathbf{x}_{h}\) under the basis \(\overline{\mathbf{v}}\) \\ \(\mathbf{\phi}_{3}\) & the active CS projection matrix adopted at each array sensor \\ \(\mathbf{\delta}_{h}\) & newly-generated data samples after a projection on \(\mathbf{x}_{h}\) \\ \(\mathbf{\Phi}_{r}\) & the passive CS measurement matrix modeling the packet loss \\ \(\mathbf{y}_{h}\) & received data samples at the FC from the \(h\)-th array sensor \\ \(\mathbf{\tilde{x}}_{h}\) & recovered data samples for the \(h\)-th array sensor at the FC \\ \hline \end{tabular}
\end{table} 
Table 1: Notations and explanations. FC, fusion center. Figure 5: Schematic diagram of the double-layer compressive sensing (CS) framework and the DOA estimation process of DCS-DOA and DCS-direct DOA (DDOA). packet loss during transmission as a passive CS process. Additionally, the passive CS measurement matrix, denoted by \(\mathbf{\Phi}_{r}\in R^{M\times N}\) (\(M\) is the number of received data samples from each array sensor), is constructed as follows:  
\[\mathbf{\Phi}_{r}\left(i,j\right)=\left\{ \begin{array}{ll}1&\text{if}\;\;j=J \left(i\right)\leq N\\ 0&\text{otherwise}\end{array} \right. \tag{6}\]  
where \(i\) is the row index and also the sequence number of the received data samples, \(j\) is the column index and \(J\left(i\right)\) is the corresponding sequence number in the transmitted data vector of received data. The passive CS measurement matrix is determined by the packet reception rate and the packet size (in this paper, we ignore the overhead produced by MHR and MFR and refer to the number of data samples in each packet as the packet size). For example, assume that there are 16 data samples to be transmitted over a lossy link, with the packet reception rate being 50%. When the packet size is one (only one data sample in each packet), the 16 data samples will be assembled into 16 packets for transmission, and eight packets are received with the corresponding packet number being 1, 3, 6, 8, 10, 11, 13, 16; that is, \(J=\left\{1,3,6,8,10,11,13,16\right\}\). Then, the passive CS measurement matrix can be depicted in Figure 5(a). When the packet size is increased to four (four data samples in each packet) for a higher transmission efficiency, the 16 data samples will be assembled into four packets for transmission. Under the same packet reception rate, two packets are received with the corresponding packet number being one and four. In this case, eight data samples are received, but \(J=\left\{1,2,3,4,13,14,15,16\right\}\). Thus, the passive CS measurement matrix is depicted in Figure 5(b). Obviously, when the packet size is larger than one, block data loss will emerge when a packet is missing during transmission. Additionally, this is specifically exhibited in the distribution of the element \"1\" of the passive CS measurement matrix. The \"1\"s will be less dispersive when the packet size is larger.