“Unerroric” of multistage discrete Fourier transform of digital signal without arithmetic operations of multiplication

Abstract

There are the issues related to improving the quality of the spectral analysis of a polyharmonic signal by recurrent methods of the multistage discrete Fourier transform (DFT) based on differential digital filtering, which does not require multiplication. The concept of “unerroric” of this DFT of a digital signal is introduced only by addition and shift operations. The philosophical aspect of it is described. The aim of the study is to find a solution to the problem of improving the quality of spectral analysis of polyharmonic signal by recursive methods of multistage discrete Fourier transform based on differential digital filtering without performing arithmetic multiplication operations. Methods of research are the following: directed search and comparative analysis of results of multistage discrete Fourier transform on the basis of difference digital filtering of different orders at integer coefficients of this filtering. The study of the philosophical aspect of "unerroric" of digital methods of spectral analysis of a polyharmonic signal based on recurrence algorithms of multistage discrete Fourier transform with the help of difference digital filters with integer numeric filtration coefficients, which do not require arithmetic operations of multiplication, has shown that further development of the theory and methods of this filtration is necessary. As a result of the conducted research it was revealed that the degree of approximation to the desired level of accuracy of this spectral analysis can be raised by increasing the order of difference digital filtering with integer coefficients, as well as by selecting the order of difference of its algorithm. After philosophical comprehension of the regularities of "unerroric" of digital spectral analysis of polyharmonic signal by multistage discrete Fourier transform recurrence methods on the basis of difference digital filtering only by adding and shifting operations it was suggested to select the element base, technical characteristics of which consider these regularities, for software and hardware implementation of digital algorithms of multistage discrete Fourier transform without performing multiplication operations. It is shown that the use of programmable logic devices (PLD) can be considered as the best variant of the element base for hardware-software implementation of digital algorithms of spectral analysis of a polyharmonic signal by recursive methods of this multistage discrete Fourier transform.