Signal approximations by shifted Gaussians: a direct approach by finite linear systems

Abstract

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite–rank approximations by linear systems. The main result for this approach is to establish correctness of the finite–rank linear system under consideration. And the main result of the paper is to prove correctness of the finite–rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non–zero.

Generalized Discrete Fourier Transform on the Base of Lagrange and Hermite Interpolation Formulas

“…Description the calculation of direct and inverse GDFT's, associated with the Hermite formula, is possible to produce by Fast Fourier Transform Algorithm in the fft and ifft applications from MatLab according to the following: first, we define the complex amplitudes of the observed signal Y =fft( y ε , n) and the complex amplitudes of the observed signal derivatives Y =fft(y , n), where y, y are vectors signal and its derivatives, second, we get two vectors of complex amplitudes by formulas F = NY − Y, G = Y − (N − 1)Y , thirdly, we will determine of the harmonic components of the signal and its derivatives is carried out by the formulas (26) in the form y = ifft( 1 N (F + εG), n), y = ifft( 1 N (G + (N − 1)εF ), n).…”

“…DFT is widely used [4,6,9,17,20] to perform Fourier analysis in many practical applications. So, for example, DFT is used to effectively solve equations in private derivatives and perform convolution operation, multiplying large whole numbers, coding, filtering the signal analysis [13,16,25,26]. These transforms are also important in transmutation theory [22,23,24].…”

“…The monograph [10] is devoted to the applications of the DFT for digital signal processing,which is characterized by great attention of the authors to the changing educational needs of the reader. The sources [1,5,7,14,17,[22][23][24][25][26][27] set out the theoretical foundations of DFT, FFT and FFT-algorithms and applications to different problems. The Lagrange interpolation formula is written in a baricentric form in the papers [2,16] and it is mostly convenient for our purpose.…”