International Journal of Computational and Applied Mathematics & Computer Science
E-ISSN: 2769-2477
Volume 4, 2024
Synthesis of Singular Systems Walsh and Walsh-like Functions of Arbitrary Order
Authors: , , ,
Abstract: Functionally complete systems of Walsh functions (bases), a particular case of alternating piecewise
constant sequential functions, are widely used in various scientific and technological fields. As applied to the
tasks of spectral analysis of discrete signals, the most interesting are those Walsh bases that deliver linear
coherence of the frequency scales of fast Fourier transform (FFT) processors. By the frequency scales of an
FFT processor, we mean the scale on which the normalized frequencies of the input signal are arranged (input
scale) and the scale on which the signal's spectral components are arranged (output scale). The frequency scales
of the FFT processor are considered linearly coherent if the processor responses with maximum amplitudes and
phases of the same sign are located on the bisector of the Cartesian coordinate system formed by the frequency
scales of the processor. None of the known Walsh bases ordered by Hadamard, Kaczmage, or Paley provide
linear coherence of the frequency scales of the FFT processor. In this study, we develop algorithms to
synthesize two systems, called Walsh-Cooley and Walsh-Tukey systems, which turn out to be the only ones in
the set of classical Walsh systems and sequents Walsh-like systems, respectively, that deliver linear coherence
to the frequency scales of FFT processors.
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Keywords: Walsh function systems, Sequential Walsh-like functions, Linear connectivity of frequency scales
of the DFT processor, Walsh-Cooley and Walsh-Tukey function bases
Pages: 61-80
DOI: 10.37394/232028.2024.4.8