**2. Basic wavelet theory**

Wavelet theory is a natural continuation of Fourier transformation and its modified shortterm Fourier transformation. Over the years, wavelets have been being developed independently in different areas, for example, mathematics, quantum physics, electrical engineering and many other areas and the results can be seen in the increasing application in signal and image processing, turbulence modelling, fluid dynamics, earthquake predictions, etc. Over the last few years, WT has received significant attention in electric power sector since it is more suitable for analysis of different types of transient wavelets when compared to other transformations.

#### **2.1 Development of wavelet theory**

From a historical point of view, wavelet theory development has many origins. In 1822, Fourier (Jean-Baptiste Joseph) developed a theory known as Fourier analysis. The essence of this theory is that a complicated event can be comprehended through its simple constituents. More precisely, the idea is that a certain function can be represented as a sum of sine and cosine waves of different frequencies and amplitudes. It has been proved that every 2π periodic integrible function is a sum of Fourier series 0 *k k* cos sin *<sup>k</sup> a a kx b kx* , for corresponding coefficients *<sup>k</sup> a* i *<sup>k</sup> b* . Today, Fourier analysis is a compulsory course at every technical faculty. Although the contemporary meaning of the term 'wavelet' has been in use only since the 80s', the beginnings of the wavelet theory development go back to the year 1909 and Alfred Haar's dissertation in which he analysed the development of integrable functions in another orthonormal function system. Many papers were published during the 30s'; however, none provided a clear and coherent theory (Daubechies, 1996; Polikar, 1999).

First papers on wavelet theory are the result of research by French geophysicist and engineer, Jean Morlet, whose research focused on different layers of earth, and reflection of acoustic waves from the surface. Without much success, Morlet attempted to resolve the problem using localization technique put forward by Gabor in 1946. This forced him to 'make up' a wavelet. In 1984, Morlet and physicist Alex Grossmann proved stable decomposition and function reconstruction using wavelets coefficients. This is considered to be the first paper in wavelet theory (Teofanov, 2001; Jaffard, 2001).

Grosman made a hypothesis that Morlet's wavelets form a frame for Hilbert's space, and in 1986 this hypothesis was proved accurate by Belgian mathematician Ingrid Daubechies. In 1986, mathematician Ives Meyer construed continuously differentiable wavelet whose only disadvantage was that it did not have a compact support. At the same time, Stephane Mallat, who was dealing with signal processing and who introduced auxiliary function which in a certain way generates wavelet function system, defined the term 'multiresolution analysis' (MRA). Finally, the first stage in the wavelet theory development was concluded with Ingrid Daubechies' spectacular results in 1988 (Graps, 1995).

She created orthonormal wavelet bases of the space of square integrable functions which consists of compactly supported functions with prescribed degree of smoothness. Compact support means that the function is identically equal to zero outside a limited interval, and therefore, for example, corresponding inappropriate integrals come down to certain integrals. Daubechies wavelets reserved their place in special functions family. The most important consequence of wavelet theory development until 1990 was the establishment of a common mathematical language between different disciplines of applied and theoretical mathematics.
