Calculating algorithm for the signals wavelet-transform using the Chebyshev-Hermit functions

The paper deals with the development of wavelet-transform computation basis which allows to restore the original signal wavelet-coefficients from the coefficients given by decomposition of the original signal with Chebyshev-Hermite functions. To form the basis in question, the wavelet transform of the Chebyshev-Hermite functions is analytically calculated. Derivatives of the Gauss function are used as wavelets. The paper considers the formation of transition bases from the coefficients of signal expansion by Chebyshev-Hermite functions to wavelet transforms using the Gauss functions of the 1st and m-th order as analyzing wavelets. As an example, the basis of the transition from the expansion coefficients of the original signal in terms of the Chebyshev-Hermite functions to the wavelet-transform using the MHAT wavelet is given. In this case, the wavelet transform of the signal is carried out in two stages. At the first stage, the decomposition of the initial signal is obtained in the form of a weighted sum of the Chebyshev-Hermite basis functions. At the second stage, knowing the weight factors of the functions obtained at the first stage, as well as the analytical expression of the continuous wavelet-transform for specific basis functions and the wavelet, it is possible to restore the wavelet-transform of the original signal. The examples of wavelet-transforms of the bases formed are given. Using the obtained formulas for calculating wavelet coefficients, it is possible to construct fast computational processing algorithms. For calculations and graphical representations of the simulation results, the Wolfram Mathematica 11.3 computer algebra system was used.