Popis metod spektrální analýzy založené na Zolotarevových polynomech

Abstract

The doctoral thesis deals with a new time-frequency transformation based on approximated discrete Zolotarev polynomials (ADZT). Its basis consists of approximated symmetrical Zolotarev polynomials of the rst and the second order. Due the characteristics of these polynomials ADZT achieves a good time and frequency selectivity and it results into the good time localization of non-stationarity parts of the analysed signal without any changes in a frequency resolution. This transform was designed for the purpose of Zolotarev spectra calculation in a doctoral thesis written by Radim Spet k. In that thesis the Zolotarev series was also derived and discrete Zolotarev transform (DZT) was introduced but it has not been implemented yet. The proposed doctoral thesis goes on Radim Spetik's thesis and it adds a missing description of ADZT, systematical analysis of its properties bz using the selected types of signals and veri es the calculation of Zolotarev spectra by the inner product of signal and functions of analysing DZT basis which is composed of symmetrical Zolotarev polynomials. The new relation between approximated and non approximated symmetrical Zolotarev polynomials is found by this veri cation and it is used for the method of DZT basis creation. The main part of this thesis is focused on description of ADZZ behaviour. The several conveniently methods are designed. By using of selected signals the typical properties of ADZT behaviour are described such as the capability of ADZT to e ciently suppress spectral leakage or pulsing of spectral component energy. The thesis also discuss the optimal choice of ADZT parameters including the type and the length of segmentation window, the step of segmentation and the number of leading and trailing zeros added into the signal segment. Based on the proceed analysis the modi cations of ADZT are proposed. The rst modi cation decreases the number of operation of Zolotarev spectra calculation which leads to the lower complexity. The second modi cation improves the time localization of non-stationary parts of analysed signal. The last but not least the comparison of spectral analysis by short-time ADZT to common used spectral methods is done. For this purpose the synthetic signal including the stationary and non-stationary parts is used. The comparison is also illustrated for the real signal such as speech or EKG.