利用本原单位根构造正交变换

Using primitive unit roots to construct orthogonal transforms

在整环R上,基于本原单位根提出一种新的正交变换,它概括了整数剩余类环、有限域、复数域上的正交变换及多项式变换,包括离散傅里叶变换、阿达玛变换、Chrestenson变换,可导出多实值对称变换、离散W变换和离散正弦型变换.证明了离散正弦型变换的正交性.所有这些变换基于与离散傅里叶变换的联系都有快速算法.该变换对系统掌握常用的变换和构造新的变换大有帮助.

A new orthogonal transform based on the primitive unit root in the integral ring is proposed,which summarizes the orthogonal transform in the residue class rings of integers,finite fields and the complex field as well as the polynomial transform,including Discrete Fourier Transform,Hadamard Transform and Chrestenson Transform.The symmetric Transform of Multiple Real Values,Discrete W Transform and Discrete Sine-type Transform can be derived from it.The orthogonality of the Discrete Sine-type Transform has been proved.All transforms possess a fast algorithm based on related of to the DFT.This paper is very beneficial to grasping typical transforms systematically and constructing a new transform.