## 6 Numerical Results
### Experiment of Data Gathered by Practical Network  
In order to evaluate the performance of the proposed algorithm in dealing with the data acquired from an actual environment, we implemented a WSN consisting of 30 nodes to monitor the luminosity of the environment. The microprocessor of the sensor nodes is CC2530 [23], and the luminosity-observation sensor is BH1750 [24]. The measurement resolution of this sensor is 0.1 (\(lx\)), and the measurement error is \(\pm 5\) (\(lx\)) in the measurement range of 0 to 65,535 (\(lx\)). To test the impact of the collected signals on the proposed method under different experiment environments and topology conditions, we deployed the network into both outdoor and semi-outdoor scenes. The so-called semi-outdoor is as shown in Figure 14B, _i.e._, a part of nodes are placed outdoors, and the rest are placed in the building. In order to obtain the entire environmental luminosity data, the sink node will enquire the sensor readings of all nodes every 5 min. According to our experimental results, our proposed schemes can be directly applied to both uniform and random outdoor deployment experiments; but for the semi-outdoor experimental data, the algorithm is not able to meet the stopping condition within the effective measurement steps. Analysis indicates that, in the semi-outdoor experiment, nodes placed outdoors are exposed to strong light, while their neighbor nodes may be placed in a relatively dimmed building, which will destroy the spatial sparse characteristics of the luminosity data observed by the whole network (for one epoch of the network sensed data (denoted as vector \(x\)) which is shown in Figure 15A, we can see that it is not sparse under wavelet transformation, as shown in Figure 15C). After examining the statistics of the observation data, we find that, although the reading values of the adjacent nodes are very different, for the two consecutive rounds of network sensed data, there are a few differences between the readings of the adjacent nodes. We let \(x_{i}^{t}\) denote the sensor reading of node \(i\) during data collecting epoch \(t\); \(\Delta x_{i}^{t}=x_{i}^{t+1}-x_{i}^{t}\) denotes the difference of node \(i\) between two consecutive periods. Then \(\Delta x_{i}^{t}\), \(\Delta x_{j}^{t}\) of the adjacent nodes should be related (the experimental results are shown in Figure 15B,D; the wavelet coefficient of \(x\) is emanative, but the wavelet transformation  
Figure 14: Network deployment of data gathering WSN. (**A**) Outdoor deployment; (**B**) Semi-outdoor deployment. coefficient of \(\Delta x\) obviously tends to be 0). If we denote \(x_{t+\ 1}\) and \(x_{t}\) as the \((t+1)th\) and the \((t)th\) network sensed data, we can acquire the following two equations:  
\[\Delta x^{t}=x^{t+1}-x^{t} \tag{7}\]  
\[\Delta x^{t}=\Psi U \tag{8}\]  
We assume that the previous rounds of data collection have been completed, and the network sensed data \(x^{t}\) has been worked out at the sink node. Thus, if the sink node has received the measurement results \(y^{t+1}\) at \(t+1\) epoch, where \(y^{t+1}=\Phi^{t+1}x^{t+1}\), from Equation (7), by solving out \(\Delta x^{t}\), we can get \(x^{t+1}\). Actually, \(\Phi^{t+1}\) can be treated as the measurement matrix of \(\Delta x^{t}\), because:  
\[y^{t+1}-y^{t}=\Phi^{t+1}x^{t+1}-\Phi^{t+1}x^{t}=\Phi^{t+1}\Delta x^{t} \tag{9}\]  
Therefore, by Equations (8) and (9), \(\Delta x^{t}\) can be calculated through a typical CS reconstruction method. It should be pointed out that \(y^{t+1}=\Phi^{t+1}x^{t}\
eq y^{t}\) in Equation (9). This is because measurement matrices are randomly generated in different data gathering epochs. With this method, we experiment on two consecutive rounds of data collected under each deployment condition. The experimental results are shown in Table 5. We can see that, with an increasing \(\Delta M\), both the recovery quality of signal and the required number of measurements increase.