Separation of kernel in the hexagonal discrete fourier transform

For the two‐dimensional signal with circular band‐limiting in frequency space, a square arrangement of sampling points in the real space is usually used. Assuming, however, that the signal is band‐limited in a hexagonal region, being regarded as periodic and adopting the triangular sampling point arrangmeent in real space, the number of sampling points can be reduced by 13.4% that in the usual method. Mersereau has derived a discrete Fourier transform (DFT) for a two‐dimensional signal hexagonal band‐limited both in real and frequency spaces. In his method, however, separation of the Fourier kernel is impossible and Rivard's FFT algorithm is not applicable to the computation of hexagonal DFT. The authors introduce a periodic extension vector system and sampling point generating vector system. By a generalized method, the two‐dimensional DFT is reformulated. It is shown that the kernel can be given a separable expression by suitable choice of coordinates as in the square DFT, actually presenting the method of determining the coordinate. When the kernel of a hexagonal DFT is separable, the computation reduces to that of the one‐dimensional DFT. This permits the application of already developed FFT algorithms, enlarging the range of utilization of the hexagonal DFT.