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本文讨论了有限阿贝尔群上的希尔伯特变换和张—哈特莱变换的性质。由于引入了群元素排序的概念和定义了函数的奇偶性,从而能够将函数划分出真函数与虚函数两种类别。本文引述了作者和L.Salinas 1986年在国际多值逻辑学术会议论文集上发表文章的主要结果,即实偶函数的奎斯特恩逊谱是实的和偶的;实奇函数的谱是虚的和奇的;取实值的真函数和虚函数两大类型函数的希尔伯特变换同它们的奎斯特恩逊谱的实部与虚部有明确的关系。在这基础上本文详细论证了张—哈特莱变换的性质,并提出了一种快速算法。本文证明了一个函数的奎斯特恩逊功率谱可以直接用张—哈特莱变换计算。因为这种变换是实值的,因此复数运算是可以避免的。
This paper discusses the properties of Hilbert transform and Zhang-Hartley transform over finite Abelians. Due to the introduction of the concept of group element ordering and the definition of the function’s parity, the function can be divided into two categories: true function and virtual function. This paper cites the main results of the article by L. Lalinas and L.Salinas published in Proceedings of the International Multivalued Logic Conference in 1986, that is, the Queston spectrum of real even functions is real and even; the spectrum of real odd functions is Imaginary and odd. The Hilbert transformations of two types of real and imaginary functions of real value have a clear relationship with the real and imaginary parts of their Queston-Son spectrum. On this basis, this paper demonstrates the properties of Zhang - Hadley transform in detail and proposes a fast algorithm. This paper proves that a function of the Questian power spectrum can be directly calculated using the Zhang-Hartley transform. Because this transformation is real, complex operations can be avoided.