## 4 Sparse Signal Transmission Framework
### Problem Formulation  
The sensor node samples the signal and reports data to the sink node for signal recovery periodically. Supposing that the monitoring period includes Ntime slots. \(x_{i}\) denotes the sensed data at time slot \(i\), where \(i=1,2,....N\). Definition 1 Original Data Vector. This is the mathematical presentation of the original signal after signal sampling, which is defined as \(\mathbf{x}=(x_{1},x_{2}...x_{N})\). \(\mathbf{x}\) represents every data sampling in N time slots without missing data elements. Definition 2 Lossy Transmission Matrix. This is also the projection matrix, whose matrix dimensions are \(M\times N\), recording the data loss process caused by a lossy link. \(M\) is the received data number. The matrix has only a single entry of one in each row, and the others zero. The value in projection matrix \(\mathbf{\Phi}\) can be chosen through the following formula:  
\[\mathbf{\Phi}(i,j)=\left\{ \begin{array}{ll}1&ifj{=}\mathrm{J}(i)\leq N\\ 0&otherwise\end{array} \right. \tag{7}\]  
where \(i\) is the row number of \(\mathbf{\Phi}\) and also the received data packets' sequence number. \(\mathrm{J}(i)\) is the received data's original sequence number in \(f\). The effective projection matrix corresponds to a partial Fourier transform projection matrix, which has been shown to meet the restricted isometry property (RIP) and, hence, promises the reconstruction of the sensor signal at the fusion center with a high probability. The feasibility of this approach has been reported earlier in [12; 33; 34]. Figure 3: Process of reliable information transmission. Suppose a signal consisting of 28 sampled data is transmitted through a wireless medium and only nine sampled data are successfully received in the FC, with their sampled sequence number being 1, 4, 9, 12, 14, 17, 21, 25, 28. Then, the corresponding projection matrix is shown in Figure 4, where the blank grid denotes corresponding elements being zero and red being one. Definition 3 Measurement Vector. This is a \(N\times 1\) vector. It is the combination of data packets received after lossy wireless transmissions. Due to the data loss, its elements are either \(x_{i}\) gathered by the sensor node or zero. It can be denoted as the production of \(\Phi\) and \(x\),  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{x} \tag{8}\]  
which conforms to the standard compressive sensing expression. Definition 4 Reconstructed Vector. This is generated by reconstruction algorithms and denoted as \(\hat{x}=(\hat{x}_{1},\hat{x}_{2}...\hat{x}_{N})\). Problem: Original Signal/Information Reconstruction. Given the measurement vector \(y\), this problem is to find an optimal reconstructed vector \(\hat{x}\) that approximates the original data vector \(x\) as closely as possible. _i.e._,  
\[ \begin{array}{l}\mathrm{Objective}\;\;\min||x-\hat{x}||_{2}\\ \mathrm{Subject}\,\mathrm{to}\;\;y\end{array}  \tag{9}\]  
where \(||.||_{2}\) is the \(l_{2}\) norm used to measure the error between vector \(x\) and \(\hat{x}\). In the above problem, the objective is to minimize the absolute error. In order to measure the reconstruction error, we further define the following metric. Definition 5 Reconstruction Error. We define the parameter of the reconstruction error \(\varepsilon\) as follows:  
\[\varepsilon=\frac{\sqrt{\sum\limits_{i}{{{(x_{i}-\hat{x}_{i})}^{2}}}}}{\sqrt {\sum\limits_{i}{{{x_{i}}^{2}}}}} \tag{10}\]  
Reconstruction error reflects the similarity degree of the reconstructed signal and the original one. The smaller the reconstruction error is, the higher the data recovery accuracy of the reconstruction methods will be. Definition 6 Information Acquisition Rate (IAR). Consider that traditional link quality estimation usually adopts packet reception rate, which is based on data acquisition. Aiming to measure the information transmission quality of a certain link, we define a new link performance metric called the  
Figure 4: An example of the projection matrix construction.