The use of Haar and OB wavelets in the signals analysis

Abstract

In this paper, two mathematical methods of wavelet transform are presented: Haar’s wavelet transform and wavelet transform at oriented basis (OB). Unlike traditional wavelet transforms (like classical Haar’s wavelet) this mathematical approach allows getting more information about the details and behavior of original signal due to more amount of discrete filters that are used for its decomposition. In Haar’s and other wavelet methods there are only two discrete filters are used to decompose initial signal – one low-frequency filter and one high-frequency filter. Low-frequency wavelet coefficients (marked as s-coefficients) give the compressed and approximated version of the initial signal (called trend), and high-frequency wavelet coefficients (marked as d-coefficients) give the high-frequency oscillations around the trend. Such decomposition and calculation of wavelet coefficients is realized at each level of wavelet analysis. While using wavelet transform at oriented basis, there are more than one type of high-frequency wavelet coefficients (marked as d⁽¹⁾-, d⁽²⁾-,…, d^(m)-coefficients) where m is defined by the type of spectral transform at oriented basis (dimension of the matrix of basic function). Number of decomposition levels is defined by the length of initial signal’s interval. In the case of Haar’s wavelet transform this length is determined as N=2ⁿ, and in the case of wavelet transform at oriented basis this length is determined as N=mⁿ. The main principle of wavelet transform lies in the use of scaled and shifted basic functions. The structure and algorithm of multiscale analysis is considered for the cases of Haar’s wavelet where the interval of initial signal is defined as N=2ⁿ, and for OB wavelet with the interval N=mⁿ. The feature of OB wavelet transform is the possibility to operate with more than one high-pass filters that gives more details about the initial signal. In partial case for m=3 basic functions of OB wavelet contains only integer numbers. Moreover, approximately 1/3 of them are zero. Thus, it simplify the calculation significantly. Matrix form of wavelet decomposition is considered for Haar and OB wavelets. Use of matrices generalizes calculation process by combining all decomposition levels in one formula. The matrix method of the calculation of wavelet coefficients simplify the decomposition procedure for initial signal. Thus, it has the advantage against the direct calculation of wavelet coefficients by recurrent formulas. The coefficients of approximation and detailing for the above methods were calculated. It has been proved that wavelet transform at oriented basis has an advantage because it allows to achieve more information about the investigated signal for less amount of decomposition steps and with less calculation losses. As an interesting example, time dependence of discrete function that describes electrical energy consumption in MicroGrid system could be considered as an object for compressing and removing of casual high-frequency oscillations with the help of wavelet analysis.

Ref. 5, fig. 7