# Adaptive Wavelet Coding Applied in a Wireless Control System
## 1 Introduction  
The traditional communication architectures employed in industrial control and manufacturing systems are generally based on wired systems. According to [1], this type of architecture is generally centralized and, thus, does not provide some desired requirements for modern industrial systems such as: modularity, decentralized control, ease of diagnostics and low cost. An alternative to overcome the difficulties imposed by wired networks is the use of wireless sensors in industrial plants [2]. Flexibility, low cost and scalability are some of the advantages of wireless sensor networks. However, the use of wireless sensors in industrial environments should ensure low demand for memory and processing resources, low bit error rate, low power consumption, dynamic topology and security [3; 4]. Wireless sensors use a communication susceptible to distortions caused by flat fading, produced by multiple propagation paths of the signal, and by shading, as a function of the obstacles present. In an industrial environment in addition to multiple paths and obstacles, there are other sources of noise, such as welding equipment, microwave ovens and wireless communication instruments. Therefore, there is a high level of fading on a large and small scale, according to [5]. Thereby, when transmitted in industrial environments, the signals are subject to distortions and errors caused by the propagation channel, which can cause problems in the controlled and/or monitored equipment. These communication problems are more critical in industry than in telecommunications systems designed for voice transmission. The use of wireless sensors in a control loop has other challenges to overcome such as packet loss, packet error, routing optimization, topology, latency and jitter, among others. There are several protocols and techniques in the literature that address these challenges, some of which are analyzed by [6; 7; 8]. The packet loss can be addressed using control techniques, such as the use of a model-based controller, as analyzed by [9]. Another approach used to combat the degenerative effects present in the wireless communications scenarios, and applied in control systems, is the use of filtering, diversity and coding techniques, some of which are analyzed by [10; 11; 12]. Diversity techniques consist of generating redundancy (replicas) of the transmitted signal at the receiver. They are then transmitted on independent channels, being affected differently in an uncorrelated way by the channel [13]. Mitigation of transmission errors favors the control system, although it can lead to a reduction in the system spectral efficiency. However, wavelet coding can be used to generate time diversity without compromising spectral efficiency [14]. It was initially proposed by [PERSON] et al. in [15] as an alternative channel coding technique to minimize fading effects, based on the orthogonality properties that exist between the rows of the wavelet coefficient matrix (WCM). It is also important to note that wavelets are also used in source coding techniques or even in systems with joint source-channel coding [16; 17; 18]. However, this work proposes the use of wavelets in an unusual way. Specifically, we use filters based on wavelet coefficients as channel encoders in order to increase the robustness of the communication system in relation to the degenerative effects present in the wireless communication. In the encoding process considered in this work, each data bit sent to the wavelet encoder is successfully multiplied by the coefficients of a given row in the WCM, thus disseminating information in multiple coded symbols. The resulting encoded symbols are not equiprobable and present multiple levels, being transmitted in different time intervals. Due to the WCM orthogonality properties, the spread information of each bit may be collected in the decoder by proper correlation with the WCM rows used for encoding.