## 5 Packet Length Control for Sparse Signal Transmission
### Packet Length Effect on Mutual Coherence  
A trade-off exists between the desire to reduce overhead by making packets large and the need to reduce packet error rates in the noisy channel by using small packet length. As shown in Figure 5, supposing a signal consisting of 64 KB of sampled data is transmitted through a wireless medium, with the PRR being 20%, when the payload length increases, once a packet has failed to transmit, more consecutive data will be lost, leading to bursty data loss. Traditionally, bursty data loss is difficult to recover from using interpolation-based methods [15]. Next, we will show the packet length effect on sparse signal transmission. We first introduce mutual coherence as the metric to measure signal transmission quality and then reveal the relationship between packet length and signal recovery performance. The mutual coherence, denoted as \(\mu(A)\),  
\[\mu(\textbf{A})=\max_{i\
eq j,1\leq i,j\leq N}\left\{\frac{|{\textbf{A}_{i}}^{ T}{\textbf{A}_{j}}|}{||\textbf{A}_{i}||.||\textbf{A}_{j}||}\right\} \tag{12}\]  
represents the worst case coherence between any two columns (atoms) of equivalent matrix \(A\) and is one of the most fundamental quantities associated with CS theory. Any \(K\)-sparse signal \(x\) can be exactly recovered from the observation/measurement \(y=Ax\) as long as:  
\[K<\frac{1}{2}[1+\frac{1}{\mu(\textbf{A})}] \tag{13}\]  
Another suitable way to describe \(\mu(A)\), which is also our method in this paper, is to compute the Gram matrix \(G=\tilde{A}^{T}\tilde{A}\), where \(\tilde{A}\) is column-normalized version of \(A\)[36]. Then:  
\[\mu(\textbf{A})=\max_{i\
eq j,1\leq i,j\leq N}|g_{ij}| \tag{14}\]Figure 6 shows the relationship between mutual coherence and packet length with the RCS projection matrix. For convenience, we make the signal dimension \(N\) be divisible by payload length. Because the Fourier basis requires \(N\) to be a power of two, the payload length is chosen to be 1, 8, 32, 64, 128. Though it is not practical using a payload length of 128 bytes, since the maximum packet length in the IEEE 802.15.4 standard is 127 bytes, we only draw it as a reference here by using a 127-byte length to approximate the 128-byte length. Specifically, when the payload length is one byte, it is similar to random sampling [33]. We can see that when the measurement vector is fixed, mutual coherence increases with the growing length of the packet. Furthermore, as the measurement vector number increases, the coherence difference caused by packet length is weakened. However, this is just a clue to the relationship between packet length and signal recovery performance; signal recovery performance is also correlated with other factors, such as signal sparsity. Figure 5: Frequency histogram of consecutive data loss. (**a**) Consecutive loss with the payload length being one byte; (**b**) consecutive loss with the payload length being 16 bytes; (**c**) consecutive loss with payload length being 64 bytes. Figure 6: Packet length effect on relationship between \(\frac{M}{N}\) and mutual coherence using random compressive sensing (RCS) projection. According to [8], the sparsity of typical signals in sensor networks, such as temperature and light, is between 5 and 10. Based on this, we show the relationship between mutual coherence and reconstruction error as follows. Figure 7 shows a noteworthy phenomenon, that as packet length increases, the mutual coherence guaranteeing accurate reconstruction (error being zero) is increasing, as well (except when the payload length is 128 bytes, which is due to its low resolution, making us not find an appropriate value). This figure is drawn through changing the measurement number when different payload lengths are adopted. The reason for the packet length effect is when the measurement number is fixed, a shorter packet length achieves lower mutual coherence.