框架理论及其在信号传输中的应用研究

发布时间：2018-05-13 19:48

本文选题：框架 + 融合框架　；参考：《电子科技大学》2017年博士论文

【摘要】：框架理论是泛函分析、非线性逼近理论、算子理论以及信号理论相结合的产物,它是继小波理论之后逐步发展起来的一个新的研究方向。框架理论的发展极大地促进了纯粹数学与工程应用的结合发展,具有十分广阔的应用前景。如今,框架理论已经广泛地应用于图像处理、信号处理、采样理论、数据压缩、系统建模、编码和通信等方面。随着现代信息技术的快速发展和广泛应用,人们更加重视信息资源的开发和利用。尽管框架理论已经得到了较好的发展,但是它作为一个新兴的研究方向仍有许多问题需要进一步研究。本学位论文对框架的基本理论展开研究,并解决框架在信号传输过程中有数据丢失时的重构问题,主要研究内容如下:1.研究基于矩阵的框架设计问题。利用矩阵的奇异值分解得到构造特殊框架的方法,同时,利用酉矩阵得到一些新的紧框架,解决了求解框架算子逆的复杂性问题。该方法操作简单,从而扩大了框架在实际问题中的应用。2.研究融合框架的一些等式和不等式问题。利用有界线性算子的理论和方法,建立了Hilbert空间中的融合框架的等式和不等式。此结论有助于解决融合框架在并行处理和高性能物理实验中的相关问题。3.由于g-框架是框架的广义形式,我们研究g-框架的相关结论。首先通过引入g-框架算子相应的有界线性算子研究g-框架的稳定性。进一步,通过引入最坏情况误差,研究对偶g-框架在有数据丢失情况下的最优对偶g-框架,并讨论规范对偶g-框架是唯一最优对偶g-框架的充分必要条件。最后,利用已知g-框架和有界算子给出逼近对偶g-框架关于局部框架的性质,并证明了两个g-框架是彼此接近时,它们的逼近对偶g-框架也是彼此接近的。4.研究框架理论在信号传输过程中有丢失时的重构问题。基于最优直接法(MOD),提出一种新的搜索最优对偶框架的方法。在信号重构中该方法能够寻找到最优对偶框架,解决对于特殊输入信号不是最优的问题。同时,该方法搜索到的最优对偶框架能够减小重构信号与原始信号的误差,从而在一定程度上解决了信号传输过程中的重构问题。数值实验也验证了新的方法的有效性。
[Abstract]:Frame theory is the product of the combination of functional analysis, nonlinear approximation theory, operator theory and signal theory. It is a new research direction after wavelet theory. The development of frame theory has greatly promoted the combination of pure mathematics and engineering application, and has a very broad application prospect. Nowadays, the framework theory has been widely used in image processing, signal processing, sampling theory, data compression, system modeling, coding and communication. With the rapid development and wide application of modern information technology, people pay more attention to the development and utilization of information resources. Although the frame theory has been well developed, as a new research direction, there are still many problems that need to be further studied. In this dissertation, the basic theory of the frame is studied, and the reconstruction problem of the frame with data loss in the process of signal transmission is solved. The main research contents are as follows: 1. The framework design based on matrix is studied. The method of constructing a special frame is obtained by using singular value decomposition of matrix. At the same time, some new compact frames are obtained by using unitary matrix, and the complexity problem of solving the inverse of frame operator is solved. This method is easy to operate, thus expanding the application of the framework in practical problems. 2. 2. Some equality and inequality problems of fusion frame are studied. By using the theory and method of bounded linear operator, the equality and inequality of fusion frame in Hilbert space are established. This conclusion is helpful to solve the related problems in parallel processing and high performance physics experiments. Since g-frame is a generalized form of frame, we study the relevant conclusions of g-frame. Firstly, the stability of g-frame is studied by introducing the bounded linear operator corresponding to g-frame operator. Furthermore, by introducing the worst-case error, we study the optimal dual g-frame with data loss, and discuss the sufficient and necessary conditions for the canonical dual g-frame to be the only optimal dual g-frame. Finally, by using known g- frames and bounded operators, we give the properties of approximation dual g-frames with respect to local frames, and prove that when two g- frames are close to each other, their approximation dual g-frames are also close to each other. This paper studies the reconstruction of frame theory when it is lost in the process of signal transmission. Based on the optimal direct method, a new method for searching the optimal dual frame is proposed. In signal reconstruction, this method can find the optimal dual frame and solve the problem that the special input signal is not optimal. At the same time, the optimal dual framework searched by this method can reduce the error between the reconstructed signal and the original signal, thus solving the reconstruction problem in the process of signal transmission to a certain extent. Numerical experiments also verify the effectiveness of the new method.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：TN911

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