# Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss
## 2 Background
### Block Data Loss  
Taking the IEEE 802.15.4 standard as an example, its data packet structure is shown in Figure 2, which consists of the MAC header (MHR), the MAC footer (MFR) and payload. Each transmitted packet has a minimum fixed overhead provided by MHR and MFR. This cost is predetermined and independent of the packet payload size. Intuitively, it is easy to come to the conclusion that using larger packet sizes could achieve a better data transmission efficiency by minimizing the overhead per useful bit transmitted in the payload. However, things may change greatly in practice. Given a bit error rate \(BER\), the packet reception rate (PRR) can be obtained as follows:  
\[PRR=\left(1-BER\right)^{L} \tag{4}\]  
where \(L=L_{payload}+L_{overhead}\) is the whole packet size including two parts: \(L_{payload}\) is the payload size, and \(L_{overhead}\) denotes the size of fixed overhead. Naturally, the relationship between \(PRR\) and payload size under a varying \(BER\) is illustrated in Figure 3a, achieved through simulation. Obviously, \(PRR\) falls as payload size rises, and the downtrend becomes sharper when \(BER\) increases. On the other hand, according to [25], we can define the data transmission efficiency as follows:  
\[E=\frac{L_{payload}\times\left(1-BER\right)^{L_{payload}+L_{overhead}}}{L_{ payload}+L_{overhead}} \tag{5}\]  
where \(E\) represents the normalized data transmission efficiency. Based on Equation (5), the relationship between \(E\) and \(L_{payload}\) under a varied \(BER\) is depicted in Figure 3b.  
Figure 1: The spatial characteristic of wireless links.  
Figure 2: Data packet structure of IEEE 802.15.4. MFR, MAC footer.  
We will take it for granted that the data transmission efficiency will improve with a larger payload size, and this is indeed the case when \(BER\) is sufficiently low, that is wireless links are of high quality. However, in practice, \(BER\) is usually somewhat high; thus, the data transmission efficiency will first rise and then drop with the increase of payload size. This is because data packets with larger sizes are more likely to be lost, exactly as demonstrated in Figure 3a. When a packet with a large size is lost during transmission, all of the data samples in it are missing simultaneously, which is called block data loss. Block data loss does serious harm to the transmission performance, and it becomes more grievous with the increased packet size. For example, assume 65,536 data samples are transmitted over a lossy link, with \(PRR\) being 25%; the histograms of the occurrence number for different sizes of block data loss using different packet sizes are shown in Figure 4. Obviously, the size of block data loss becomes larger when a larger packet size is adopted. Therefore, there is a tradeoff between the desire to improve transmission efficiency by using a large packet size and the need to heighten the \(PRR\) and avoid block data loss by using a small packet size. Briefly, it is a meaningful thing to find some new error-correcting techniques to eliminate the hazards of block data loss, such that the data transmission efficiency is optimized.