Optimal Wales and suspense-based bases disurity transformation of Fourier

Abstract

In the theory and practice of noise-immune encoding and compression of audio and video data, cryptographic information protection, in cellular communication channels and in other fields of science and technology, functionally complete Walsh systems that are a particular case of systems of alternating piecewise constant sequential functions are widely used. As for the problems of spectral analysis of discrete signals of binary-power order (sample size), those Walsh systems used as the bases of the discrete Fourier transform (DFT), which deliver linear coupling to the frequency scales of the DFT processors (and therefore are optimal) are of greatest interest, under which mean the scale of the normalized frequencies of the input signal and the output scale of the frequency channels of the processor. The frequency scales of the DFT processor are considered to be linearly related if the processor responses with the maximum modules and fixed phases (positive or negative but identical for all responses) are located on the bisector of the orthogonal coordinate system formed by the frequency scales. None of the known classical Walsh bases ordered by Hadamard, Kaczmarz or Paley, the required connectivity to the DFT processor scales does not provide. In this study, unique DFT bases have been developed, namely, the Walsh-Coolie basis ( basis) and the alternative Walsh-Tukey basis           ( basis), which are the only one of the many Walsh function systems and sequential function systems, which just cause linear coupling to the frequency scales of the DFT processors. Both bases have the same amplitude-frequency but opposite phase-frequency characteristics in the sense that if the response phase in the Walsh-Cooley base is equal  in some output channel of the point-wise FFT processor, then in the Walsh-Tukey basis . For practical applications, the Walsh-Cooley basis is more preferable compared to the Walsh-Tukey basis, since - the basis is calculated much simpler than  the basis.