## 5 Successive Approximation Using CS-WSNs  
So far, we have used CS to code the signal captured by WSN nodes allowing to configure the RD operating point--\((M,B)\). However, it could be necessary to improve the reconstructed signal at the sink node, even at expenses of more bits. One could retransmit the signal at a higher rate, and this would waste resources. On the other hand, some form of scalability could be pursued. We propose to do that using an incremental transmission framework for CS-Coded WSN-signals. This guarantees good RD performance while economizing resources, and, consequently, saving sensor energy. Extra-bits are transmitted to refine already known quantized CS coefficients or convey unknown coefficients leading to an improve in the signal reconstruction in an incremental fashion. This strategy keeps RD performance as good as possible, since any extra bit transmitted by nodes is used in the reconstruction  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Percentage** & \(\Delta BD\) **(dB)** \\ \hline
0\%–10\% & 0.36 \\
0\%–20\% & 0.64 \\
0\%–30\% & 1.07 \\
0\%–40\% & 1.30 \\
0\%–50\% & 1.73 \\ \hline \hline \end{tabular}
\end{table}
Table 2: \(\Delta BD\) between rate-distortion (RD) curves for each packet loss percentage for the reconstruction of \(N=512\) samples of a temperature signal with LASSO, considering the transmission of 1 CS measurement per packet.  
Figure 6: Rate-distortion convex hulls for each packet loss percentage for the reconstruction of \(N=512\) samples of a temperature signal with LASSO, considering the transmission of 1 CS measurement per packet.  
and not to replace previously known ones. It is straightforward to see that this strategy allows to save energy in sensor nodes.  
We employ a successive approximation scheme such that nodes may incrementally transmit CS measurements. Let a given node to code a set of \(M_{i}\) measurements using \(B_{i}\) bits and transmit them, spending a total of \(R_{i}=M_{i}B_{i}\) bits; this defines the operational point \((M_{i},B_{i})\) in the \(M\times B\) space. Suppose now that the sink node (decoder) needs more information, in order to generate a better version of the reconstructed signal, subject to an increment of \(\Delta R_{i}\) bits. The increment must improve RD performance while remaining as close as possible to the optimal RD points. In any case, \(\Delta R_{i}\) corresponds to the selection of a new operational point \((M_{i}+\Delta M_{i},B_{i}+\Delta B_{i})\). The \(\Delta B_{i}\) and \(\Delta M_{i}\) correspondent increase in the quantity of transmitted bits is  
\[\Delta R_{i}=M_{i}\Delta B_{i}+\Delta M_{i}B_{i}+\Delta M_{i}\Delta B_{i}, \tag{15}\]  
where \(M_{i}\) and \(B_{i}\) are known. \(\Delta R_{i}\) may be seen as the target increment in rate, demanded by the sink, and Equation (15) presents the mapping from the target to the increments in the quantities of bits and measurements leading to a new operational point.  
It is desirable for the sequence of operational points resulting from extra transmissions to provide a path along the optimal RD curve. Therefore, each \((M_{j},B_{j})=(M_{i}+\Delta M_{i},B_{i}+\Delta B_{i})\) should satisfy  
\[(M_{j},B_{j})=\min_{k\in\mathcal{K}|M_{i},B_{k}}\mathcal{D}\left(M_{k},B_{k} \right), \tag{16}\]  
where \(\mathcal{D}(\cdot)\) is the distortion function while \(\mathcal{K}\) is the set of new candidates--Equation (16) means that, from \((M_{i},B_{i})\), one must search for \((M_{j},B_{j})\) leading to the smallest distortion among all \(\mathcal{K}\) candidates. Figure 7 illustrates that idea. It presents an optimal coding path (sequence of encoder setups) on the \(M\times B\) plane, the \((M_{j},B_{j})\) pairs having the best RD trade-of. It depicts the convex hull of the RD of the temperature signal, with the number of measurements (\(M\)) ranging from 8 to 256, in increments of 2, and quantized bit-depths values (\(B\)) varying within the set {4,5,6,7,8,9,10,11,12,13}, for a signal block of length \(N=512\), this considers the NMSE as distortion criterion.