# Adaptive Wavelet Coding Applied in a Wireless Control System
## 3 System Description
### Implementation of an Adaptive Wavelet Coding  
Adaptive coding and modulation techniques seek to adapt the transmission parameters of a system to the conditions of time-varying channels in order to ensure a more robust and spectrally-efficient transmission [26; 27]. The fundamental parameters to be adapted include coding and modulation, but other parameters can be adjusted, such as power levels. The basic premise of these techniques is to estimate the channel at the receiver and send that estimate back to the transmitter so that the transmission scheme can be adapted according to the channel characteristics. Adaptive coding and modulation for transmitting fading over the channel may lead to an increase of the average code rate, reduction of the required transmission power or reduction of the bit error probability, taking advantage of the favorable channel conditions [28]. This paper proposes the use of adaptive wavelet coding and modulation applied in a control loop over a wireless link to send the feedback and control signals. Thus, when the channel is in a good state,  
Figure 5: Detailed view of the structure of the control loop developed to perform the experimental tests. Figure 6: Control loop structure implemented for performing computer simulations. the code rate will increase, enabling the dispatch of auxiliary data at this time without compromising the main data flow of the control system. In addition, the use of an adaptive encoder enables a low processing overhead and consequently a good energy efficiency. This parameter is very important in a wireless sensor network. The adaptive coding is simulated considering that the transmitter and receiver have perfect information about the current channel state. Taking this information into account, the system can choose the wavelet coding and modulation scheme used in the transmission. The adaptive wavelet encoder used in this work was designed with the assistance of the Markov chain. In the proposed architecture, the input symbols are encoded by the \(2\times 32\) WCM with \(k=1\) when the path gain values of the fading channel are greater than 1.5, and they are encoded by the \(2\times 32\) WCM with \(k=0.5\), when the path gain values are less than or equal to 1.5. Figure 7a illustrates the simulated adaptive wavelet encoder. In this figure, it is possible to verify each encoder operating region according to the proposed architecture. The spectral efficiency estimate of this encoder is equal to 0.55 bits/s/Hz. When compared to conventional channel coding techniques, in a similar environment, the strategy for wavelet coding presents satisfactory gains. Figure 7b shows performance curves obtained through computational simulations of adaptive wavelet coding based on the \(2\times 32\) WCM, previously described, and the wavelet coding systems based on \(2\times 32\) WCM for \(k=1\) and \(k=0.5\), the latter with spectral efficiency equal to 1 bits/s/Hz and 0.5 bits/s/Hz, respectively. For the purpose of comparison, we also present in this figure the performance of an uncoded BPSK system and a system with convolutional coding with the polynomial generator (151, 144), a restriction length of 7 and a code rate of 0.5, associated with a QPSK modulation scheme, and a Viterbi hard receiver. From this figure, it can be observed that the performance gain of the proposed adaptive system, in terms of BER versus \(E_{b}/N_{0}\), can reach two decades when we compare its performance curve with the curve of the system based on the convolutional encoder, in \(E_{b}/N_{0}=25\) dB. As can be seen, the performance of the proposed adaptive wavelet coding has similar performance to the non-adaptive wavelet-coded system based on the \(2\times 32\) WCM with \(k=0.5\), but with a spectral efficiency equal to 0.55 bit/s/Hz, which corresponds to a spectral efficiency gain of approximately 10% in relation to that reference system.