离散环境中波包系的一些结果

发布时间：2018-06-12 19:44

本文选题：框架算子 + Parseval框架　；参考：《河南大学》2016年硕士论文

【摘要】：框架是在1952年Duffin和Schaeffer研究非调和Fourier分析时提出的.与传统的正交小波相比,框架一般是冗余的,这种冗余性可以导致鲁棒性,意思是说冗余可以使得在低精度下获得的系数却可以在相对高精度下恢复信号.因此框架成为研究的热点之一.它在信号处理、图像处理、采样理论、数字通信等信息学科中是重要的分析工具之一.并且,框架理论在光学、Banach空间理论的研究中也发挥着越来越重要的作用.框架主要分为Gabor框架和小波框架.在实际应用中,Gabor系比较适用于对平稳信号的处理,但小波系更适用于对突变信号的处理.因此,国内外学者们希望找到一类新的函数系来统一 Gabor系和小波系.这一想法与物理学界用于量子力学的一类称为波包的函数系不谋而合.由于在实际应用中输入/输出数据和滤波器都是离散的,所以基于框架的算法实现都是在数字环境中完成的.本文将给出l2(Zd)的一个框架,并给出其生成Parseval框架的充分条件,利用该框架对信号进行分解和重构.进一步,在该框架的基础上给出p-级波包系的概念,对信号进行多层分解.最后,基于框架界比值在滤波器设计中的重要性,本文将通过矩阵互相关性对给定滤波器组框架的框架界进行估计.本文由五部分组成.第一章简要介绍框架及波包的背景知识,以及本文的主要内容和相关结构.第二章引出全文所需要的一些基本概念及性质.第三章从框架算子出发,构造出l2(Zd)上的一个Parseval框架,并依据该框架构造分解算子对信号进行分解,进一步构造重构算子实现对分解信号的完全重建.第四章是本文的重点,将在上一章的基础上构造p-级波包系,对信号进行多层分解,并分析研究了分解的算法及计算量.第五章通过矩阵的互相关性概念,对序列空间中的框架界进行估计。
[Abstract]:The frame was proposed in 1952 when Duffin and Schaeffer studied the nonharmonic Fourier analysis. Compared with the traditional orthogonal wavelet, the frame is generally redundant, and this redundancy can lead to robustness, which means that redundancy can make the coefficients obtained under low precision to recover signals with relatively high accuracy. Therefore, the framework has become one of the hotspots of research. It is one of the important analytical tools in signal processing, image processing, sampling theory, digital communication and other information disciplines. Furthermore, frame theory plays a more and more important role in the study of optical Banach space theory. The frame is mainly divided into Gabor frame and wavelet frame. In practical application, Gabor system is more suitable for stationary signal processing, but wavelet system is more suitable for abrupt signal processing. Therefore, scholars at home and abroad hope to find a new kind of function system to unify Gabor system and wavelet system. This idea coincides with a class of functions called wave packets used in quantum mechanics in physics. Since the input / output data and filters are discrete in practical applications, the frame-based algorithms are implemented in the digital environment. In this paper, we give a frame of l2zd), and give the sufficient conditions for generating Parseval frame, and use this framework to decompose and reconstruct the signal. Furthermore, on the basis of this framework, the concept of p- order wave packet system is given, and the signal is decomposed in multiple layers. Finally, based on the importance of frame bound ratio in filter design, the frame bound of a given filter bank frame is estimated by matrix interrelation. This paper consists of five parts. The first chapter briefly introduces the background of the framework and wave packet, as well as the main content and related structure of this paper. In the second chapter, some basic concepts and properties are introduced. In chapter 3, we construct a Parseval frame based on the frame operator, and decompose the signal according to the frame structure decomposition operator, and construct the reconstruction operator to realize the complete reconstruction of the decomposed signal. The fourth chapter is the focus of this paper. Based on the previous chapter, we construct a p- stage wave packet system, and analyze and study the algorithm and computation of the decomposition. In chapter 5, the frame bounds in sequence space are estimated by the concept of interrelation of matrices.
【学位授予单位】：河南大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O174.2

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