调和分析与信道编解码研究

信道编码是通信系统的关键技术之一,其传统的理论工具是代数和有限域GF(p).本文在有噪信道的信道编码数学模型基础上,介绍基于调和分析的新型信道编码理论,分别应用调和分析的重要工具压缩感知与Walsh-Hadamard变换实现信道编解码和信道编码盲识别.本文综述基于调和分析的信道编码的一些基本结果并介绍最新进展,主要包括:Gilbert-Varshamov界、l1解码、RIP(restricted isometry property)条件和Walsh-Hadamard变换等.     

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     

关于从曲面到复 Grassmann流形中的调和映射

本文讨论从曲面到复Grassmann流形Gk,N中的调和映射,给出了调和序列中的基本直射变换与因子分解中的基本旗变换的关系,从而证明了有阶的调和映射与有限的调和映射是一致的.     

UMD空间及其应用

UMD空间是被广泛研究的一类新型的Banach空间，它具有一系列良好的几何性质与分析性质并且与向量值调和分析、随机分析有着广泛深刻的联系。本扼要介绍这类空间的有关问题，主要是以下几个方面：（1）引言（定义与产生背景）；（2）UMD空间的几何特性与分析特征；（3）此类空间的例；（4）在向量值调和分析理论中的应用；（5）关于鞅不等式的最优系数问题。     

奇异积分与分数次积分算子的交换子（英文）

积分和微分算子的交换子是调和分析和偏微分方程中非常重要的工具.最近,本文第三作者给出了一篇综述[Front.Math.China,2011,6(5):821-833],介绍了调和分析领域里一些重要算子的交换子的最新进展并提出了一些公开问题.本文给出了粗糙核以及变量核的奇异积分和分数次积分交换子的一些结果,可看作前一综述的一个补充.     

关于对偶的拟不变测度

<正> 关于局部紧的交换拓扑群上,基于不变测度的富里埃变换理论是大家都知道的.与此相联系的相应于局部紧群上的不变测度,在对偶群上有对偶的不变测度,这两个测度是由 Parseval 等式所联结的.这些都是群上调和分析的基本结果.我们知道在某些相当重     

计算共形几何简介

计算共形几何是数学与计算机科学的交叉领域.本文介绍了计算共形几何中理论基础,计算方法及其在工程领域中的应用等.基本理论概念包括共形映射,单值化定理,共形模,拟共形映射,模空间等.基本计算方法包括调和映照方法,基于黎曼面理论的全纯微分方法,基于几何分析的曲面Ricci流方法.基本应用涵盖了医学图像,机械制造,数字媒体和虚拟/增强现实等领域.     

数学年刊第16卷B辑第2期目次和提要

数学年刊第１６卷Ｂ辑第２期目次和提要Ｒ２到Ｕ（Ｎ）的调和映射的显示结构谷超豪胡和生利用Ｄａｒｂｏｕｘ变换构造Ｒ２到Ｕ（Ｎ）调和映射，并且得到新的调和映射（包括Ｕｎｉｔｏｎ）的显示公式；广义Ｍｏｒａｎ分形的Ｈａｎｓｄｏｒｆｆ维数、测度和Ｆｏｕｒｉｅｒ极...     

关于从曲面到复Grassmann流形中的调和映射

本文讨论从曲面到复Ｇｒａｓｓｍａｎｎ流形Ｇｋ，Ｎ中的调和映射，给出了调和序列中的基本直射变换与 因子分解中的基本旗变换的关系，从而证明了有阶的调和映射与有限的调和映射是一致的．     

一个二进小波变换像空间的再生核函数

1 引言小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为"数学显微镜".基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具[1].     

Channels with arbitrarily varying channel probability functions in the presence of noiseless feedback

In this article we study a channel with arbitrarily varying channel probability functions in the presence of a noiseless feedback channel (a.v.ch.f.). We determine its capacity by proving a coding theorem and its strong converse. Our proof of the coding theorem is constructive; we give explicitly a coding scheme which performs at any rate below the capacity with an arbitrarily small decoding error probability. The proof makes use of a new method ([1]) to prove the coding theorem for discrete memoryless channels with noiseless feedback (d.m.c.f.). It was emphasized in [1] that the method is not based on random coding or maximal coding ideas, and it is this fact that makes it particularly suited for proving coding theorems for certain systems of channels with noiseless feedback.As a consequence of our results we obtain a formula for the zero-error capacity of a d.m.c.f., which was conjectured by Shannon ([8], p. 19).     

Monotonic optimization based decoding for linear codes

New efficient methods are developed for the optimal maximum-likelihood (ML) decoding of an arbitrary binary linear code based on data received from any discrete Gaussian channel. The decoding algorithm is based on monotonic optimization that is minimizing a difference of monotonic (d.m.) objective functions subject to the 0–1 constraints of bit variables. The iterative process converges to the global optimal ML solution after finitely many steps. The proposed algorithm’s computational complexity depends on input sequence length k which is much less than the codeword length n, especially for a codes with small code rate. The viability of the developed is verified through simulations on different coding schemes.     

Calderón–Zygmund operators in the Bessel setting

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.     

Optimal transmission of “television type” gaussian signals through a noiseless feedback channel

Using the methods and results of the theory of conditionally Gaussian filtering of stochastic processes and fields, an optimal scheme of “television type” signal transmission through a noiseless feedback channel is constructed under the usual power conditions, the signals being evolutionary Gaussian fields θ={θ_t(x)),tε[0,T),xεDεRⁿ . Explicit representations for optimal coding and decoding functionals, which are optimal in the sense of a special square criterion, and the expression for the error of signal reproduction are given.     

Upper bounds on the complexity of algebraic cryptanalysis of ciphers with a low multiplicative complexity

Lightweight cipher designs try to minimize the implementation complexity of the cipher while maintaining some specified security level. Using only a small number of AND gates lowers the implementation costs, and enables easier protections against side-channel attacks. In our paper we study the connection between the number of AND gates (multiplicative complexity) and the complexity of algebraic attacks. We model the encryption with multiple right-hand sides (MRHS) equations. The resulting equation system is transformed into a syndrome decoding problem. The complexity of the decoding problem depends on the number of AND gates, and on the relative number of known output bits with respect to the number of unknown key bits. This allows us to apply results from coding theory, and to explicitly connect the complexity of the algebraic cryptanalysis to the multiplicative complexity of the cipher. This means that we can provide asymptotic upper bounds on the complexity of algebraic attacks on selected families of ciphers based on the hardness of the decoding problem.     

基于自适应小波变换和矢量量化的图像压缩编码

运用小波变换进行图像压缩的算法其核心都是小波变换的多分辨率分析以及对不同尺度的小波系数的量化和编码 .本文提出了一种基于能量的自适应小波变换和矢量量化相结合的压缩算法 .即在一定的能量准则下 ,根据子图像的能量大小决定是否进行小波分解 ,然后给出恰当的小波系数量化 .在量化过程中 ,采用一种改进的LBG算法进行码书的训练 .实验表明 ,本算法广泛适用于不同特征的数字图像 ,在取得较高峰值信噪比的同时可以获得较高的重建图像质量 .     

On the complexity of suboptimal decoding for list and decision feedback schemes

We consider bounded distance list decoding, such that the decoder calculates the list of all codewords within a sphere around the received vector. We analyze the performance and the complexity of this suboptimum list decoding scheme for the binary symmetric channel. The reliability function of the list decoding scheme is equivalent to the sphere-packing bound, where the decoding complexity is asymptotically bounded by 2^nR(1-R). Furthermore, we investigate a decision feedback strategy that is based on bounded distance list decoding. Here, any output with zero or many codewords will call for a repeated transmission. In this case the decoding complexity will be of the order 2^nR(1-C), where C denotes the channel capacity. The reliability function is close to Forney's feedback exponent.     

THE FUNDAMENTAL THEOREM OF INFORMATION TRANSMISSION FOR GENERAL MULTI-USER COMMUNICATION NETWORK SYSTEMS

In this paper, we first give a definition of the general multi-user communication network system, and introduce the coding type \[(\mathcal{U},\mathcal{Y},\mathcal{V})\] and \[({\mathcal{U}_0},\mathcal{Y},\mathcal{V})\], for the general network systems (cf Fig. 1, 2). We obtain the following results: 1. For the above \[(\mathcal{U},\mathcal{Y},\mathcal{V})\] type network system, in the general sequence model, the Shannon theorem of information transmission, the coding theorem of multi-user channel and their converse hold water; 2. The converse coding theorem holds water on the region \[D(\mathcal{L}_{L - J}^{({\text{n}})})\] (see(6.4)) for the general sequence model of \[({\mathcal{U}_0},\mathcal{Y},\mathcal{V})\]type network system; 3. On the memoryless multi-user channel for \[({\mathcal{U}_0},\mathcal{Y},\mathcal{V})\] type network system, the D(\[\mathcal{L}_{L - J}^{}\]) is an achievable vetes region For the memoryless multiple-access channel, the \[\mathcal{L}_{L - J}^{}\] is the capacity region, 4. For general \[({\mathcal{U}_0},\mathcal{Y},\mathcal{V})\] type multi-user communication network system, we set up a new decoding method, whick is said to be information threshold distinguishing decoding.     

An operational calculus for the euclidean motion group with applications in robotics and polymer science

In this article we develop analytical and computational tools arising from harmonic analysis on the motion group of three-dimensional Euclidean space. We demonstrate these tools in the context of applications in robotics and polymer science. To this end, we review the theory of unitary representations of the motion group of three dimensional Euclidean space. The matrix elements of the irreducible unitary representations are calculated and the Fourier transform of functions on the motion group is defined. New symmetry and operational properties of the Fourier transform are derived. A technique for the solution of convolution equations arising in robotics is presented and the corresponding regularized problem is solved explicity for particular functions. A partial differential equation from polymer science is shown to be solvable using the operational properties of the Euclidean-group Fourier transform.