Multiscale analysis of discrete functions with a given number of filters

Abstract

The method of discrete wavelet transforms at oriented basis that is constructed by use discrete spectral transform of the functions with modular argument is considered and generalized. 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. While selecting the value m equal to three it gives some advantages in calculation volume and consequently, in the speed of wavelet analysis that could be very useful for the processing of the signals with large interval of definition and non-stationery signals. 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. The use of wavelet transforms with more than two high-frequency filters makes it possible to increase the quantity of data about signal fluctuations and to better localize its characteristic intervals compared with traditional discrete wavelets that operates with one low-frequency and one high-frequency filters. The principle of wavelet transform is based on a multiscale analysis. Basic functions are scaled and shifted along the time axis and by amplitude. A feature of the represented wavelet transform is the using of basic functions of new spectral transforms. These are functions of a symmetric transform on finite intervals and transform at oriented basis. The system of these functions is orthogonal and contains N^p discrete functions of different shapes. One of these functions is a low-pass filter, and all the others are high-pass filters. Sphere of application of wavelet transforms with N basic functions is diagnostics of semiconductor converters, predictive energy-efficient control of energy consumption, analysis of bio-telemetric signals, processing and transmission of images and video signals.

Ref. 7, fig. 1, tabl. 3.