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．     

多项式剩余类环上循环码新的表示

在编码理论中，多项式剩余类环是非常有意义的，它已经用来构造最优频率希望序列。本文，定义了多项式剩余类环上循环码的离散傅立叶变换及Mattson-Solomon（MS）多项式，证明了多项式剩余类环上的循环码同构于多项式剩余类环的Galois扩张的理想。     

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。     

离散余弦变换的一种推广

离散余弦变换(DCT)在数字信号、图像处理、频谱分析、数据压缩和信息隐藏等领域有着广泛的应用.推广离散余弦变换,给出一个包含三个参数的统一表达式,并证明在许多情形新变换是正交变换.最后给出一种新型离散余弦变换,并证明它是正交变换.     

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…     

图象及数字信号处理中的快速算法研究进展

本文就各种特殊基的FFT算法、互素因子类算法、数论变换、多项式变换、DFT的计算复杂性及FFT的并行算法有关专题,简要地叙述了图象和数字信号处理中的快速算法(离散付里叶交换及卷积计算)的研究概况,并就笔者的观点指出了目前及将来若干进一步研究的主要问题.     

任意长度离散余弦变换的快速算法

§1.引言 离散余弦变换(DCT)有趋于统计最佳交换Kavhunven-Lave变换(KLT)的渐近性质,在通信和信号处理中应用广泛,并在许多方面比离散富里叶变换(DFT)更好。     

有关Clifford矩阵群的几个离散性判别准则

研究了Clifford矩阵变换群的离散性质,给出了几个判别离散群的不等式和定理.     

关于Chrestenson变换的功率谱

一、引言离散 Walsh-Hadamard 变换与离散 Fourier 变换一样,是信号处理的重要工具之一.1955年 N.E.Chrestenson 和 R.G.Selfridge 将 Walsh 函数推广为 p 进情形.对这种广义 Walsh 函数所对应的离散情形的研究亦已获得不少进展,特别是由于它与线性 p     

一个离散晶格方程的N波Darboux变换和无穷守恒律

根据已知离散晶格方程的Lax对,构建了该方程的Ⅳ波Darboux变换和无穷守恒律,通过应用Darboux变换,得到离散晶格方程的范德蒙行列式形式的精确解,通过画图给出了该方程一类特殊的单孤子结构.     

Spectral Decomposition of Cyclic Operators in Discrete Harmonic Analysis

For the operators of the discrete Fourier transform, the discrete Vilenkin–Christenson transform, and all linear transpositions of the discrete Walsh transform, we obtain their spectral decompositions and calculate the dimensions of eigenspaces. For complex operators, namely, the discrete Fourier transform and the Vilenkin–Christenson transform, we obtain real projectors on eigenspaces. For the discrete Walsh transform, we consider in detail the Paley and Walsh orderings and a new ordering in which the matrices of operators are symmetric. For operators of linear transpositions of the discrete Walsh transforms with nonsymmetric matrices, we obtain a spectral decomposition with complex projectors on eigenspaces. We also present the Parseval frame for eigenspaces of the discrete Walsh transform.     

A discrete Hartley transform based on Simpson's rule

The Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform. Copyright © 2015 John Wiley & Sons, Ltd.     

In honor of Professor Dr. Wolfgang Haack on the occasion of his seventieth birthday on April 24, 1972

A discrete transform with a Bessel function kernel is defined, as a finite sum, over the zeros of the Bessel function. The approximate inverse of this transform is derived as another finite sum. This development is in parallel to that of the discrete Fourier transform (DFT) which lead to the fast Fourier transform (FFT) algorithm. The discrete Hankel transform with kernel Jo, the Bessel function of the first kind of order zero, will be used as an illustration for deriving the discrete Hankel transform, its inverse and a number of its basic properties. This includes the convolution product which is necessary for solving boundary problems. Other applications include evaluating Hankel transforms, Bessel series and replacing higher dimension Fourier transforms, with circular symmetry, by a single Hankel transform     

Wavelet transform associated with linear canonical Hankel transform

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.     

3D discrete X-ray transform

The analysis of 3D discrete volumetric data becomes increasingly important as computation power increases. 3D analysis and visualization applications are expected to be especially relevant in areas like medical imaging and nondestructive testing, where elaborated continuous theory exists. However, this theory is not directly applicable to discrete datasets. Therefore, we have to establish theoretical foundations that will replace the existing inexact discretizations, which have been based on the continuous regime. We want to preserve the concepts, properties, and main results of the continuous theory in the discrete case. In this paper, we present a discretization of the continuous X-ray transform for discrete 3D images. Our definition of the discrete X-ray transform is shown to be exact and geometrically faithful as it uses summation along straight geometric lines without arbitrary interpolation schemes. We derive a discrete Fourier slice theorem, which relates our discrete X-ray transform with the Fourier transform of the underlying image, and then use this Fourier slice theorem to derive an algorithm that computes the discrete X-ray transform in O(n⁴logn) operations. Finally, we show that our discrete X-ray transform is invertible.     

The discreteW transform

Four different versions of the discreteW transform (DWT) are introduced. The DWT may be decomposed into the discrete cosine transform (DCT) and the discrete sine transform (DST). Eight versions of both DCT and DST are introduced for the decomposition of the DWT. The relationship among different versions of DWT and their relation with the discrete Fourier transform (DFT) are given. Convolution theorems represented by different versions of the DWT are derived.     

Quadrature formulas for the Laplace and Mellin transforms

A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the integral Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfies given conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. The use of asymptotic formulae yields an algorithm to compute the discrete Laplace transform by using only exponentials.