Index transforms of multidimensional cyclic convolutions and discrete Fourier transforms

In this correspondence, general equations are described for computing the so-called index transforms which map the m-dimensional discrete Fourier transform (DFT) and cyclic convolution (CC) on n-dimensional DFT or CC, respectively. These transforms are based on mapping indexes of input and output sequences of DFT or CC operators. The general form of these transforms is found, and the necessary and sufficient conditions of their existence are described. Universality of the results is illustrated from the standpoint of practical applications.