# 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
Note that the equivalent measurement matrix is \(\mathbf{A}=\mathbf{\Phi}_{r}\mathbf{\Psi}\) under the single-layer CS framework. To account for the hazards of block data loss, we present the histograms of the absolute off-diagonal elements of the corresponding Gram matrix under different packet sizes. As shown in Figure 6(a), when the packet size is increased from 1-64, the number of very correlated columns of the equivalent measurement matrix \(\mathbf{A}\) is increased greatly, so the distribution of entries keeps moving towards the larger side. As a consequence, the DOA estimation performance gradually deteriorates with an increased packet size, which will be discussed in our simulations later. To eliminate the adverse effects of block data loss on DOA estimation in WSAN, besides the passive CS process, we introduce an active CS process at each array sensor to form the double-layer CS framework. At the \(h\)-th array sensor, instead of directly transmitting the raw data vector \(\mathbf{x}_{h}\), we perform a dimension-reduced projection on it before its elements are assembled into packets for transmission, as shown in Figure 8. The projection matrix can be a Gaussian matrix, which makes each of the after-projection data samples a weighted average of the raw data samples. In this way, the transmitted data samples all contain global information of the impinging signals. Besides, a permutation matrix is also competent. By using this matrix, we can rearrange the raw data samples so that the originally adjacent ones will not be assembled into the same data packet. The projection operation using either  
Figure 6: Examples of the passive CS measurement matrix under different packet sizes, with blue grids denoting one and blank grids denoting zero. of the matrices mentioned above can get rid of the hazards of block data loss. The projection matrix is generated in advance and stored in the ROM of the array sensors. For a favorable scalability, an identical projection matrix is chosen for all array sensors. To compensate for the energy consumption generated by the projection operation, we make it dimension-reduced to lower the data volume. Therefore, the active CS projection matrix can be constructed by selecting \(M_{s}\left(M_{s}<N\right)\) random rows from an \(N\times N\) Gaussian or permutation matrix. After the completion of projection at each array sensor, the newly-generated data samples are assembled into data packets according to the packet size and then transmitted to the FC over lossy wireless links. Note that under the double-layer CS framework, the equivalent measurement matrix is denoted by \(\mathbf{A}=\mathbf{\Phi}_{r}\mathbf{\Phi}_{s}\mathbf{\Psi}\). Likewise, we present the histograms of the absolute off-diagonal elements of the corresponding Gram matrix (taking the case where \(\mathbf{\Phi}_{s}\) is a permutation matrix as an example) under different packet sizes. As shown in Figure 7b, the distribution of entries almost stays unchanged since the effect of the projection operation (active CS process) is to reduce the number of very correlated columns in \(\mathbf{A}\). Furthermore, the values of mutual coherence for the single-layer CS and double-layer CS framework under different packet sizes are provided in Table 2. The results are self-explanatory, the values of \(\mu\left(\mathbf{A}\right)\) and \(\mu_{av}\left(\mathbf{A}\right)\) keep rising with the increase of packet size under the single-layer CS framework. However, under the double-layer CS framework, the values remain low regardless of the packet size, which promises a superior DOA estimation performance.