# Adaptive Wavelet Coding Applied in a Wireless Control System
## 4 Obtained Results
### Control System with Adaptive Wavelet Coding over the Channel Subject to Rayleigh Fading  
For all results presented in this subsection, the variable to be controlled is the level of the second tank (\(L_{2}\)), and the controller gains are \(K_{p}=1.5\), \(k_{i}=0.04\) and \(k_{d}=0.05\). The signals received by the controller and actuator are denoted by \(L_{2}^{\prime}\) and \(v_{p}^{\prime}\), respectively, and the signals sent by the sensor and controller are denoted by \(L_{2}\) and \(v_{p}\), respectively.  
Figure 9a shows the system response of a control loop over a wireless link, but without wavelet coding for a \(E_{b}/N_{0}\) equal to 40 dB. This wireless system uses BPSK modulation and has spectral efficiency equal to 1 bit/s/Hz. From this figure, it can be seen that even at high signal-to-noise ratios, the system is not controllable, since the degenerative effects present in the wireless channel cause an inversion on the bits received by the controller and actuator, distorting the values of those messages. The change of the values received by the controller can be seen in Figure 9b, wherein these messages are distorted in such a way that \(L_{2}\) is degenerated.  
Figures 10a and 11a show the results obtained through computational simulations for the system with the wavelet coding based on \(2\times 32\) WCM for \(k=1\) and \(k=0.5\) in the control loop. It can be seen from this figure that the control system with wavelet coding appears to be controllable from \(E_{b}/N_{0}=15\) dB over a channel with the normalized Doppler in \(f_{D}T_{s}=0.002\), since it has similar performance to the system simulated over wired communication, also shown in the figure. From Figures 9a, 10a and 11a, it can be noticed that the control system with wavelet coding behaves satisfactorily in the considered communication scenarios, but when the control system does not have a channel coding technique equivalent to wavelet coding, the control loop is not controllable, even for high signal-to-noise ratios, which reinforces the importance of wavelet coding in these scenarios. It is also important to note that with the addition of wavelet coding, fewer errors occur, as shown in Figures 10b and 11b.  
Figure 9: (**a**) System response without wavelet coding for various signal-to-noise ratios; (**b**) signal received by the controller on a system without wavelet coding.  
Figure 12a shows the behavior of the control system with the adaptive wavelet coding proposed in this work. In order to evaluate the performance of the proposed control system, a control loop with wired communication between the nodes is also simulated and used as a reference system. From this figure, it can be observed that the system behaves properly when \(E_{b}/N_{0}\geq 15\) dB, since in this case, it has performance similar to the reference system. It is important to note that most wireless systems only operate properly in channels with signal-to-noise equal to 20 dB, indicating that the results obtained from these experiments are satisfactory. Finally, it is important to note that the adaptive wavelet coding has a spectral efficiency of 0.55 and a BER less than \(10^{-6}\) in \(E_{b}/N_{o}\), equal to 25 dB, so the adaptive wavelet coding proposed in this work meets the requirements of flexibility and robustness required in a control loop.  
Figure 11: (**a**) System response with wavelet coding based on \(2\times 32\) WCM for \(k=0.5\); (**b**) signal received by the controller in a system with wavelet coding based on \(2\times 32\) WCM for \(k=0.5\).  
Figure 10: (**a**) System response with based wavelet coding based on \(2\times 32\) WCM for \(k=1\); (**b**) signal received by the controller in a system with wavelet coding based on \(2\times 32\) WCM for \(k=1\).