快速傅里叶变换FFT的发展现状──纪念FFT发表30周年

THE PRESENT OF FOURIER TRANSFORM ALGORITHMS──THE THIRTIETH ANNIVERSARY OF FFT

博里叶交换快速算法发展已３０年，本文综述了离散变换快速算法的发展，特别是近几年的发展，其中包括传统的基２、基４、基８、分裂基算法的发展以及多维离散傅里叶变换、多维离散余（正）弦变换、多维离散Ｗ变换（哈特莱变换）的快速算法，阐述各种算法是如何将多维变换转换为一维变换的计算，并讨论了在有理数域上计算上述各种变换所需最小实数乘法的次数。

Thirty years have passed since the publication of Cooley-Tukey Fast Fourier Transform in 1965. The recent developments of major discrete transform fast algotithms are reviewed.These algorithms include traditional radix-2, radix-4 radix-8, split-radal djscfcte Fourier transforms, multi-dimensional discrete Fourier transforms, multi-dimensional discrete cosine transforms, multi-dimensional discrete Hartley transforms (discrete W transforms). In this paper, wedemonstrate that the multi-dimensional discrete transforms can be computed byone-dimensional transform and discuss the minimal number of real multiplication required to compute various discrete transforms over the rational number field.