Synthesis of Singular Systems Walsh and Walsh-like Functions of Arbitrary Order

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.