分数维Fourier变换及其快速算法

FRACTIONAL FOURIER TRANSFORM AND ITS FAST ALGORITHM

首先将所有已知的分数维Fourier变换 (DFRT)统一定义在Lagrange多项式插值的框架下 ,从而使人们能够利用简单的计算方法理论分析出各类DFRT逼近到连续分数维Fourier变换 (FRT)的精度 ,同时 ,证明了最近由S .C .Pei,etal.提出的一类DFRT与H .M .Ozakatas提出的DFRT完全等价 .进一步地 ,建立了计算FRT高效的快速算法 ,与已有算法比较 ,新算法具有较少的算术运算量以及分数维阶更广等优点 .

As a first step, a unified framework based on the Lagrange polynomial interpolation for various known discrete fractional Fourier transforms (DFRTs) is developed, and under this framework, the precision of the DFRT which approximates to the continuous fractional Fourier transform (CFRT) can then be theoretically evaluated using simple numerical mathematics. The equivalence between the definition for DFRTs developed by S.C.Pei et al and that by H.M.Ozaktas is proved. An efficient and accurate algorithm for computing the FRT is proposed. Compared with the reported algorithms, the presented algorithm is of less computational complexity and higher fractional order.