Varieties of Systems of DEF Generated by Isomorphic Transformations

Abstract

Despite more than a century of origin and development, the theory of discrete exponential function (DEF) systems continues to attract the attention of mathematicians and application specialists in various fields of science and technology. One of the most successful applications of the DEF systems is the spectral processing of discrete signals based on fast Fourier transform (FFT) algorithms in the DEF bases. The construction of structural schemes of FFT algorithms is preceded, as a rule, by the factorization of the DEF matrices. The main problem encountered when factorizing DEF matrices is that the elements of such matrices are the degrees of phase multipliers, which are complex-valued quantities. In this connection, the computational complexity of factorization of DEF matrices may be too large, especially when the number of components of the matrix order decomposition is large. In this paper, we propose a relatively simple method of mutually unambiguous transition from complex-valued DEF matrices to matrices whose elements are natural numbers equal to the degree indices of phase multipliers in the canonical DEF matrices. Through this bijective transformation, the factorization of DEF matrices becomes significantly more manageable, streamlining the overall process of factorization.