To the Theory of Discrete Signal Processing

Assume that $G$ is a finite group. A general definition of the $G$-spectrum of a discrete signal, that utilizes irreducible representations of group $G$, is given. If $G$ is an Abelian group, then the $G$-spectrum coincides with the familiar definition. The general definition of $G$-spectrum preserves all the advantages of spectral processing of discrete signals that are inherent in the Abelian case. For lengths that are powers of 2, an infinite sequence $\{G_n\}$ of non-Abelian groups is constructed, for which the $G_n$-spectrum can be calculated 3/4 times more rapidly than the FFT allows in the Abelian case for the same lengths.