Basics of Synthesis of Walsh Systems and Walsh-like Functions

In this study, we develop various systems of discrete Walsh-like (0,1)-sequent functions (bases). Walsh-like in the image space, we will refer to functions in which the number of zeros and ones in each half of the definition interval is not necessarily the same, as it takes place in the images of functions of classical Walsh systems. The choice of the Discrete Fourier Transform (DFT) basis function systems is determined by the requirements of computational convenience and, ultimately, by the labor intensity of the algorithms for realizing the desired transformation. Based on these considerations, using real bases based on Walsh function systems and their extensions - Walsh-like systems seem to be relevant and promising for a variety of applications. In this paper, we show that subsets of both classical and Walsh-like systems contain unique systems, called Walsh-Cooley and Walsh-Tukey systems, whose bases deliver linear coherence to the frequency scales of DFT processors. None of the canonical Walsh systems, which include Walsh systems ordered by Hadamard, Kaczmarz, or Paley, possesses the above property. We discuss the extension of the power of the set of Walsh-like bases by synchronous permutation of rows and columns of arbitrary Walsh-like systems. An algorithm is developed to determine the permutation of the signal sample numbers at the input of the DFT processor, using which the formation of the signal spectrum on the required Walsh basis is achieved.