The ascertainment of scale sampling step for numerical realization adopting even dot-and-grid sampling of the continuous wavelet transform

One of the key discrete approach for the numerical realization of the continuous wavelet transform is to ascertain the scale sampling step, which is the basic requirement for wavelet analytic engineering achievement and research in theory. This article presents the basic theory for the numerical realization of the continuous wavelet transform. According to the highest numeric angle frequency of the signal below π, in this two cases, the article respectively deduces the best results of the scale sampling step of the Morlet mother wave and the Gauss function of each differential coefficient phase of even or odd symmetric mother wave in the numerical realization of the continuous wavelet transform when the even dot-and-grid sampling is adopted. In the aid, it discusses the time shift required in getting the results of odd symmetric mother wave in the numerical realization of the continuous wavelet transform. These useful conclusions solve a fundamental issue of wavelet analyses in engineering practice and research in theory.