## 4 Rate-Distortion Analysis of Using CS Encoding in WSN
### Rate--Distortion Results  
Figure 2 presents the RD performance of the reconstruction of the temperature signal using L1-magic (in the recovery procedure for the L1-magic algorithm, the \(\mathcal{E}\) parameter used to solve (4) was empirically chosen. We highlight that the RD results to be presented are averaged over 300 runs (using different seeds). For each number of measurements and bit-depth (\(M\),\(B\)), \(\mathcal{E}\) was varied and the \(\mathcal{E}\) value that generates the lowest distortion was employed. ), A*OMP and LASSO. The dimension of the signal block is set to \(N=512\). We consider quantizers bit-depths (B) of \(\{4,6,8,10\}\) and \(M\in\{8,16,32,64,128,256\}\) CS measurements. For each reconstruction algorithm and quantizer bit-depth \(B\) we present the RD curve. One observes that as rate increases (in bits/sample, in this case due to augmenting \(M\)), in general there is an improvement in the reconstruction of the signal (decrease of the normalized MSE) for the three recovery algorithms. This expected behavior is due to the more CS measurements used in the reconstruction. These results also show that smaller reconstruction errors are obtained as the bit-depths of the quantizers increase. L1-magic presented a better RD performance (that is, smaller distortion for the same rate) than both A*OMP and LASSO did. However, L1-magic did not obtain a monotonically decreasing distortion in function of rate. Such behavior has been observed in various simulation settings, and it was also observed to be sensitive on \(\mathcal{E}\) (Equation (4)). The A*OMP and LASSO did not show such non-monotonic behavior. It also can be verified that the LASSO presents a better RD performance than the A*OMP. Consequently, we employ the LASSO algorithm to make a more detailed evaluation of the RD performances of the reconstructions of temperature, humidity, and illumination signals. We generate denser operational RD curves than the ones in Figure 2. The number of CS  
Figure 2: Rate-distortion (RD) results for a temperature signal for several quantization bit-depths using L1-magic, A*orthogonal matching pursuit (OMP), and least absolute shrinkage and selection operator (LASSO). The block length \(N\) is equal to 512. \(M=8\) to \(M=256\), in steps of 2; and the bit-depths (\(B\)) varied within the set [4,5,6,7,8,9,10]. As in the previous results, \(N=512\). Figure 3 shows the RD curves of the reconstruction of the temperature (a), humidity (b), and illumination (c) signals, for each bit-depth (there, we present only the rate-distortion curves for 4, 6, 8, and 10 bits (similar behaviors are observed for quantization using 5, 7, and 9 bits)). The graphs in Figure 3 also present the convex hull [42], which is composed by the points that lead to the best trade-off between rate and distortion, that is, corresponding to the best quantizers for each rate. The RD convex hull (all quantizers were considered in the computation of the convex hull, i.e., \(B\in\{\)4,5,6,7,8,9,10\(\}\)) represents the empirically optimal RD operational curve. Figure 3: Rate-distortion curves and convex hulls for the reconstruction of signals with LASSO for several quantization bit-depths. The block length \(N\) is equal to 512. Considering the same sampling rate, the complexity of CS tends to increase with \(N\) (above we considered \(N=512\)). Measurements are inner products [43] and the quantity of required arithmetic operations per measurement is proportional to the block length \(N\). However, increasing \(N\) may help to better exploit signal sparsity. This motivates investigating the impact of the length of the signal block (\(N\)) in the rate-distortion performance. The number of CS measurements ranges from \(M=8\) to \(M=256\), in steps of 2 and the bit-depths (\(B\)) are within the set {4,5,6,7,8,9,10}. We also consider three different signal block lengths, \(N=\{128,256,512\}\). The reconstruction was obtained with LASSO. Figure 4 presents the results obtained for this experiment.