On the similarity between discrete harmonic wavelet and discrete orthonormal S transform

Abstract

Two efficient transforms: discrete harmonic wavelet transform (DHWT) and discrete orthonormal S transform (DOST) were developed in the literature. They can be used to analyze, model, and simulate nonstationary signals. Both transforms are efficient and have been used to represent seismic ground motions and wind speeds of high-intensity wind events. The efficiency arises from the fact that they are non-redundant transforms which is similar to Fourier transform but can cope with temporal varying characteristics. DOST originated from the summation of Fourier representation over a frequency band and considers phase shift such that the basis functions are absolutely referenced. DHWT originated from the wavelet concept (i.e., continuous harmonic wavelets). Its basic function is also obtained by considering a frequency band. However, the phase of the basis function is not absolutely referenced. It seems that a detailed discussion of their similarity is never provided in the literature. In fact, papers using DHWT for engineering applications rarely mention DOST and vice versa. In the present study, we provide a detailed comparison of these two transforms in terms of the mathematical derivation of their basis functions, their computer implementation, and their characteristics, including edge effects. In addition, we show their potential use in simulating nonstationary processes. To minimize the edge effects by using DHWT and DOST as well as their variants, we suggest an iterative correction algorithm, which is illustrated by simulating nonstationary processes.