Wavelet Network Implementation on an Inexpensive Eight Bit Microcontroller

Abstract

The approximation of general continuous functions by nonlinear networks is very useful for system modeling and identification. Such approximation methods can be used, for example, in black-box identification of nonlinear systems, signal processing, control, statistical data analysis, speech recognition, and artificial intelligence. Recently neural networks have been established as a general approximation tool for fitting nonlinear models from input/output data due to their ability of learning rather than complicated process functions (Gao, 2002). Their attractive property is the self-learning ability. A neural network can extract the system features from historical training data using the learning algorithm, requiring little or no a priori knowledge about the process (Patan, 2008). This is why during the past few years the nonlinear dynamic modelling of processes by neural networks has been extensively studied (Narendra & Parthasarathy, 1990; Nerrand et al., 1993; Levin, 1992; Rivals & Personnaz, 1996). In standard neural networks, the nonlinearities are approximated by superposition of sigmoidal functions (Cybenko, 1989). In the other hand, the wavelet theory has found many applications in function approximation, numerical analysis and signal processing. Though this attractive theory has offered efficient algorithms for various purposes, their implementations are usually limited to wavelets of small dimension. The reason is that constructing and storing wavelet basis of large dimension are of prohibitive cost. In order to handle problems of larger dimension, it is necessary to develop algorithms whose implementation is less sensitive to the dimension. And it is known that neural networks are powerful tools for handling problems of large dimension. Due to the similarity between wavelet decomposition and one-hidden-layer neural networks, the idea of combining both wavelets and neural networks has been proposed in various works (Zhang & Benveniste, 1992; Pati & Krishnaprasad, 1993; Hong, 1992; Bakshi & Stephanopoulos, 1993; Tsatsanis & Giannakis, 1993;et al., 1994; Delyon et al., 1995; Saad Saoud & Khellaf, 2009). For example, in (Zhang & Benveniste, 1992) wavelet network is introduced as a class of feedforward networks composed of wavelets, in (Pati & Krishnaprasad, 1993) the discrete wavelet transform is used for analyzing and synthesizing