广义M(o)bius变换和算术Fourier变换

离散Fourier变换(DFT)在数字信号处理等许多领域中占有重要地位.近年来,出现一种优于FFT的算术Fourier变换来计算DFT.在广义M(o)bius变换的基础上,本文采用了一种改进的AFT来计算DFT,这种方法可以直接提取DFT的系数,且用数论的方法阐明了这一过程,并展开了进一步的讨论.这也代表了数论方法应用在计算数学领域的一个新的发展方向.     

广义Moebius变换和算术Fourier变换

离散Fourier变换(DFT)在数字信号处理等许多领域中占有重要地位．近年来，出现一种优于FFT的算术Fourier变换来计算DFT．在广义Moebius变换的基础上，本文采用了一种改进的AFT来计算DFT，这种方法可以直接提取DFT的系数，且用数论的方法阐明了这一过程，并展开了进一步的讨论．这也代表了数论方法应用在计算数学领域的一个新的发展方向．     

C-代数上的Fourier变换和C-代数上的Vinogradov不等式

本文给出了C-代数上的Fourier变换，从而推广了经典数论上Fourier变换和任意有限Abel群的Fourier变换．进一步应用这个Fourier变换，给出了C-代数上的Vinogradov不等式．     

广义Mbius变换和算术Fourier变换

离散Fourier变换(DFT)在数字信号处理等许多领域中占有重要地位.近年来,出现一种优于FFT的算术Fourier变换来计算DFT.在广义M     

分数阶Fourier变换的测不准原理

该文参考Fourier变换的性质研究了离散分数阶Fourier变换的测不准原理以及连续分数阶Fourier变换在Lebesgue测度下的测不准原理,使得分数阶Fourier变换的测不准原理性质更一般化.     

一种新的信号处理方法——线调频小波变换

线调频小波变换是处理非平稳信号一种新的方法 .本文分析了线调频小波变换是短时 Fourier变换和小波变换的时频分析的统一形式 ,并能根据信号的特点生成新的时频分析方法 ,说明了线调频小波变换具有传统处理方法无法比拟的优点     

窗口Fourier变换反演公式的级数表示——献给陆善镇教授75华诞

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L^2空间,本文给出重构级数收敛的充分必要条件.     

计算一类数字变换的收缩阵列

本文以DFT的收缩（Systolic）阵列结构为基础，给出了一类数字变换的收缩阵列，这些变换包括离散富里叶变换，离散余弦变换，离散正弦变换，离散Hartley变换，数论变换和多项式变换．     

Fourier变换微积特性定理的普遍公式

采用Fourier变换的极限定义式,给出并证明普遍形式的Fourier主换微积特性定理.采用所得公式计算,可从根本上防止错误的发生,并能避免复杂的极限计算,而直接写出大多数Fourier象函数.     

FGFT的并行算法及其应用

常见的离散Fourier变换(DFT)的推广均定义在一个交换环上。我们在[1]、[2]中给出了DFT在一类非交换环上的推广(FGFT),并将它应用于一些快速线性计算问题。本文将不加证明地列出这些快速算法的并行计算效率。结果表明,这些计算问题亦具有很好的并行性。     

Gauss整环上的三角和与二维Fourier级数

本文在分析了一维AFT（Arithmeric Fourier Transform)推导的基础上,首次将整数环上的三角和推广到Gauss整环上。结合Gauss整环上的Moebius反演公式,推出了计算二维Fourier系数的AFT算法。     

高维广义积分子波变换及其性质

在Soblev空间上定义了一种新的广义积分子波变换,它包括了通常意义下的子波变换,付氏变换,R-变换,同时改进和延伸了朱在文献[1]中引入的广义积分子波变换等.并且对这种新的子波变换的基本性质进行了研究.     

MULTIVARIATE FOURIER TRANSFORM METHODS OVER SIMPLEX AND SUPER-SIMPLEX DOMAINS

In this paper we propose the well-known Fourier method on some non-tensor productdomains in R~d, inclding simplex and so-called super-simplex which consists of (d 1)!simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallelhexagon and a parallel quadrilateral dodecahedron, respectively. We have extended mostof concepts and results of the traditional Fourier methods on multivariate cases, such asFourier basis system, Fourier series, discrete Fourier transform (DFT) and its fast algorithm(FFT) on the super-simplex, as well as generalized sine and cosine transforms (DST, DCT)and related fast algorithms over a simplex. The relationship between the basic orthogonalsystem and eigen-functions of a Laplacian-like operator over these domains is explored.     

模糊Fourier变换算法

本文把普通集合中的离散Fourier变换推广到模糊集合。借助于区间数、模糊数的运算规则及有关性质,给出了模糊离散Fourier变换(FDFT)的定义及算法,而且也讨论了模糊离散Fourier变换中的对应关系以及变换性质的几个定理。     

Faster than the Fast Legendre Transform,the Linear-time Legendre Transform

A new algorithm to compute the Legendre–Fenchel transform is proposed and investigated. The so-called Linear-time Legendre Transform (LLT) improves all previously known Fast Legendre Transform algorithms by reducing their log-linear worst-case time complexity to linear. Since the algorithm amounts to computing several convex hulls and sorting, any convex hull algorithm well-suited for a particular problem gives a corresponding LLT algorithm. After justifying the convergence of the Discrete Legendre Transform to the Legendre–Fenchel transform, an extended computation time complexity analysis is given and confirmed by numerical tests. Finally, the LLT is illustrated with several examples and a LLT MATLAB package is described. This revised version was published online in June 2006 with corrections to the Cover Date.     

一种求循环码对偶码的新方法

本文应用离散傅里叶变换来分析有限域上时域多项式环和它的频域多项式环之间的同构,从另一角度提出了一种由循环码单位元生成它的对偶码的新方法。本文还建立了循环码单位元与它的频域单位元之间的联系,由此可以很方便地生成它的对偶码,理论和实例表明这种方法是行之有效的。     

Detection of the singularities of a complex function by numerical approximations of its Laurent coefficients

For several applications, it is important to know the location of the singularities of a complex function: just for example, the rightmost singularity of a Laplace Transform is related to the exponential order of its inverse function. We discuss a numerical method to approximate, within an input accuracy tolerance, a finite sequence of Laurent coefficients of a function by means of the Discrete Fourier Transform (DFT) of its samples along an input circle. The circle may also enclose some singularities, since the method works with the Laurent expansion. The DFT is computed by the FFT algorithm so that, from a computational point of view, the efficiency is guaranteed. The function samples may be obtained by solving a numerical problem such as, for example, a differential problem. We derive, as consequences of the method, some new outcomes able to detect those singularities which are close to the circle and to discover if the singularities are all external or internal to the circle so that the Laurent expansion reduces to its regular or singular part, respectively. Other singularities may be located by means of a repeated application of the method, as well as an analytic continuation. Some examples and results, obtained by a first implementation, are reported.     

用多项式变换计算多维离散W变换

1．引言多维离散W变换作为多维离散Hartley变换的推广［1－3］，是处理多维问题的一种工具．在计算机视觉、高清晰度电视（HDTV）以及可视电话等领域，经常要对运动图象进行分析和处理，通常称为多帧检测（Multi－WameDetection，简称MFD）［4－5］，这时三维离散w变换是一种可行的方法．由于不需要进行复数运算，比三维离散傅立叶交换（DFT）有优越性．而对运动的三维图象进行处理时，可采用四维离散w变换．对维数更高的多维信号进行处理时，可采用多维离散w变换．对三维以上的w变换，需要的运算量非常大，设计好的快速算法极为重要…     

Root Counting, the DFT and the Linear Complexity of Nonlinear Filtering

The method of root counting is a well established technique in the study of the linear complexity of sequences. Recently, Massey and Serconek [11] have introduced a Discrete Fourier Transform approach to the study of linear complexity. In this paper, we establish the equivalence of these two approaches. The power of the DFT methods are then harnessed to re-derive Rueppel's Root Presence Test, a key result in the theory of filtering of m-sequences, in an elegant and concise way. The application of Rueppel's Test is then extended to give lower bounds on linear complexity for new classes of filtering functions.     

The duality property of the Discrete Fourier Transform based on Simpson's rule

The classical Discrete Fourier Transform (DFT) satisfies a duality property that transforms a discrete time signal to the frequency domain and back to the original domain. In doing so, the original signal is reversed to within a multiplicative factor, namely the dimension of the transformation matrix. In this paper, we prove that the DFT based on Simpson's method satisfies a similar property and illustrate its effect on a real discrete signal. The duality property is particularly useful in determining the components of the transformation matrix as well as components of its positive integral powers. Copyright © 2012 John Wiley & Sons, Ltd.