Sinc函数的非线性逼近及其应用
发布时间：2019-05-21 18:56
【摘要】：Shannon采样定理为信号通信和图像处理奠定了严格的理论基础.根据Shannon采样公式,有限带宽信号可以被精确的恢复.Sinc函数是Shannon采样公式中的插值核.同时Sinc函数还被看作是一个理想的低通滤波器.在信号的实际恢复过程中,通常只涉及到Shannon采样公式中的有限项求和,因此就会产生一个截断误差.如果要得到一个合适的截断误差,就需要很多项求和,因而就带来了很大的计算量.另外,大多数信号都不是严格意义上的有限带宽信号,此时若仍把Sinc函数看作是理想的插值核,则缺乏一个合理的解释.为了解决这些问题,人们便开始从两方面对Shannon采样公式的有限项求和进行改进.一方面,构造一个合适的函数将其加入到Shannon采样公式的有限项求和中,来减小截断误差,此时构造的函数被称为收敛因子；另一方面,构造一个具有紧支集的函数,同时该函数需要满足Sinc函数的一些性质.最后在S hannon采样公式的有限项求和中,用构造的函数来代替Sinc函数.本文将从这两方面来考虑Sinc函数的逼近问题.另外,我们将再次论证当线性多步法达到最高逼近阶时,该差分格式是不稳定的.本文分为五章,具体安排如下：1.第一章,我们介绍了Sinc函数、样条函数、Pade逼近和代数函数逼近的相关内容及研究情况.2.第二章,通过研究Sinc函数的Pade逼近,我们给出了Sinc函数的[2/4]型Pade逼近.然后把[2/4]型Pade逼近看作是一个收敛因子,将其加入到Shannon采样公式的有限项求和中.最后和已有的收敛因子进行了数值实验比较,将[2/4]型Pade逼近作为收敛因子的有限项求和也能得到很好的精度.3.第三章,我们给出了Sinc函数的[2/6]型、[0/2]型、[0/4]型和[0/6]型Pade逼近.然后将[2/6]型Pade逼近和另外三类Pade逼近以及第二章中的三类收敛因子进行数值实验比较,[2/6]型Pade逼近作为收敛因子能得到很好的精度.4.第四章,基于3/1型有理样条函数已有的研究,我们研究了Sinc函数的3/1型有理样条函数逼近,并得到了一类含参数的3/1型有理样条函数.通过分析它的频谱在原点处的泰勒展开式,我们得到：当参数值取2时,该3/1型有理样条函数在低频处有平坦谱.另外,还给出了参数的其它几种合理的取值.最后与已有的几种方法通过图像处理进行比较,我们的方法也能得到很好的图像处理效果.5.第五章,我们从指数函数的代数函数逼近角度,研究了指数函数的[1,n]级代数函数逼近以及与线性多步法的联系.最后我们给出了一个新的证明：当线性多步法达到最高逼近阶时,其差分格式是不稳定的.
[Abstract]:Shannon sampling theorem lays a strict theoretical foundation for signal communication and image processing. According to the Shannon sampling formula, the finite bandwidth signal can be accurately recovered. Inc function is the interpolation kernel in the Shannon sampling formula. At the same time, Sinc function is also regarded as an ideal low-pass filter. In the actual recovery process of the signal, only the finite term summation in the Shannon sampling formula is usually involved, so a truncation error will be produced. If we want to get a suitable truncation error, we need a lot of items to sum up, so it brings a lot of computation. In addition, most of the signals are not limited bandwidth signals in the strict sense, and if the Sinc function is still regarded as an ideal interpolation kernel, there is a lack of a reasonable explanation. In order to solve these problems, people began to improve the finite term sum of Shannon sampling formula from two aspects. On the one hand, a suitable function is constructed to add it to the finite term sum of Shannon sampling formula to reduce the truncation error. At this time, the constructed function is called convergence factor. On the other hand, a function with compact support set is constructed, and the function needs to satisfy some properties of Sinc function. Finally, in the summation of finite terms of S hannon sampling formula, the constructed function is used instead of Sinc function. In this paper, we will consider the approximation of Sinc functions from these two aspects. In addition, we will prove once again that the difference scheme is unstable when the linear multistep method reaches the highest approximation order. This paper is divided into five chapters, the specific arrangements are as follows: 1. In the first chapter, we introduce the related contents and research of Sinc function, Spline function, Pade approximation and Algebra function approximation. 2. In chapter 2, by studying the Pade approximation of Sinc function, we give the [2 鈮，
本文编号：2482318
[Abstract]:Shannon sampling theorem lays a strict theoretical foundation for signal communication and image processing. According to the Shannon sampling formula, the finite bandwidth signal can be accurately recovered. Inc function is the interpolation kernel in the Shannon sampling formula. At the same time, Sinc function is also regarded as an ideal low-pass filter. In the actual recovery process of the signal, only the finite term summation in the Shannon sampling formula is usually involved, so a truncation error will be produced. If we want to get a suitable truncation error, we need a lot of items to sum up, so it brings a lot of computation. In addition, most of the signals are not limited bandwidth signals in the strict sense, and if the Sinc function is still regarded as an ideal interpolation kernel, there is a lack of a reasonable explanation. In order to solve these problems, people began to improve the finite term sum of Shannon sampling formula from two aspects. On the one hand, a suitable function is constructed to add it to the finite term sum of Shannon sampling formula to reduce the truncation error. At this time, the constructed function is called convergence factor. On the other hand, a function with compact support set is constructed, and the function needs to satisfy some properties of Sinc function. Finally, in the summation of finite terms of S hannon sampling formula, the constructed function is used instead of Sinc function. In this paper, we will consider the approximation of Sinc functions from these two aspects. In addition, we will prove once again that the difference scheme is unstable when the linear multistep method reaches the highest approximation order. This paper is divided into five chapters, the specific arrangements are as follows: 1. In the first chapter, we introduce the related contents and research of Sinc function, Spline function, Pade approximation and Algebra function approximation. 2. In chapter 2, by studying the Pade approximation of Sinc function, we give the [2 鈮，
本文编号：2482318
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