Wavelets and the Sampling Theorem
2.1 Introduction
Processing continuous -time (CT) signals is a major step and requirement in different applications, including communication systems, control systems, power systems, power electronics, image processing, and biomedical applications. Furthermore, the search for proper techniques is one of the main challenges for accurate and realistic implementation of CT processing. There are many working techniques, including Fourier transforms, Laplace transform, and Z -transform. Such CT processing techniques work well for CT signals, which satisfy the conditions of fi nite energy, periodicity, stationarity, and fi nite number of discontinuities. In processing any signal, it is desirable and often necessary to apply a frequency -time based analysis technique in an attempt to extract all the frequency components, including the transient components present in the processed signal, and to preserve their location in time. This constraint has made it hard to process the large class of CT signals that do not satisfy the conditions of classical signal processing techniques.
In general, processing a CT signal, which involves sampling and reconstructing the CT signal, can be represented as an N -dimensional approximation case. Such a representation has been used to interpret the sampling theorem in the context of the wavelet -based multiresolution analysis (MRA). There are different types of wavelet basis functions that are capable of constructing MRAs, which can support sampling structures. However, such constructed MRAs are capable of supporting uniform sampling structures. The different types of available wavelet basis functions, along with the main characteristics of their associated MRAs, are provided in this chapter. Furthermore, this chapter presents a new type of wavelet basis function, capable of supporting a nonuniform recurrent sampling -reconstruction structure.
One of the most effective methods for decomposing CT signals is through an MRA that is constructed by orthogonal basis functions. In this type of signal processing, a CT signal is broken into orthogonal time -localized frequency channels (scales). The required orthogonal basis functions are generated by integer -indexed