Fast convolution methods for hexagonally sampled two‐dimensional signals

Abstract

AbstractFor fast convolution operations on one‐dimensional signal sequences there is the fast cyclic convolution (FCC) method based on sectioning. Also, for a two‐dimensional signal sequence in the convolution operation of a rectangularly sampled signal sequence and a convolving sequence arranged in rectangular blocks, an FCC method based on rectangular sectioning has been reported. On the other hand, two‐dimensional signals band‐limited over a circular region are sampled hexagonally. This arrangement of the convolving sequence in hexagonal blocks to provide it with 12‐fold symmetry is most effective. This paper describes an FCC method based on sectioning for such types of hexagonally sampled signal sequence and convolving sequence arranged in hexagonal blocks, Presupposing the use of a fast Fourier transformation (FFT) of base number 2 in the discrete Fourier transformation in FCC, two types of FCC (hexagonal sectioning and parallelogrammatic sectioning) are proposed and overlap‐and‐save algorithms for these two are given. Also, it is shown that there exists an optimal sectioning block size for which the number of numerical operations becomes minimum in both methods. Next, for both types, the number of numerical operations are calculated and compared. Due to the limited memory capacity useable in FFT calculation, if convolution is not possible with optimal sectioning, the hexagonal sectioning method is advantageous. However, when the memory size is sufficiently large, it is difficult to say which of the two is better.