本文選題：框架算子 + Parseval框架�。� 參考：《河南大學》2016年碩士論文

【摘要】：框架是在1952年Duffin和Schaeffer研究非調(diào)和Fourier分析時提出的.與傳統(tǒng)的正交小波相比,框架一般是冗余的,這種冗余性可以導致魯棒性,意思是說冗余可以使得在低精度下獲得的系數(shù)卻可以在相對高精度下恢復信號.因此框架成為研究的熱點之一.它在信號處理、圖像處理、采樣理論、數(shù)字通信等信息學科中是重要的分析工具之一.并且,框架理論在光學、Banach空間理論的研究中也發(fā)揮著越來越重要的作用.框架主要分為Gabor框架和小波框架.在實際應(yīng)用中,Gabor系比較適用于對平穩(wěn)信號的處理,但小波系更適用于對突變信號的處理.因此,國內(nèi)外學者們希望找到一類新的函數(shù)系來統(tǒng)一 Gabor系和小波系.這一想法與物理學界用于量子力學的一類稱為波包的函數(shù)系不謀而合.由于在實際應(yīng)用中輸入/輸出數(shù)據(jù)和濾波器都是離散的,所以基于框架的算法實現(xiàn)都是在數(shù)字環(huán)境中完成的.本文將給出l2(Zd)的一個框架,并給出其生成Parseval框架的充分條件,利用該框架對信號進行分解和重構(gòu).進一步,在該框架的基礎(chǔ)上給出p-級波包系的概念,對信號進行多層分解.最后,基于框架界比值在濾波器設(shè)計中的重要性,本文將通過矩陣互相關(guān)性對給定濾波器組框架的框架界進行估計.本文由五部分組成.第一章簡要介紹框架及波包的背景知識,以及本文的主要內(nèi)容和相關(guān)結(jié)構(gòu).第二章引出全文所需要的一些基本概念及性質(zhì).第三章從框架算子出發(fā),構(gòu)造出l2(Zd)上的一個Parseval框架,并依據(jù)該框架構(gòu)造分解算子對信號進行分解,進一步構(gòu)造重構(gòu)算子實現(xiàn)對分解信號的完全重建.第四章是本文的重點,將在上一章的基礎(chǔ)上構(gòu)造p-級波包系,對信號進行多層分解,并分析研究了分解的算法及計算量.第五章通過矩陣的互相關(guān)性概念,對序列空間中的框架界進行估計。
[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.
【學位授予單位】：河南大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O174.2

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本文編號：2010854