Z－变换及其数值逼近

目 录

摘 要 1

Abstract 1

前 言 2

1 Z－变换及其性质 3

1.1 Z－变换式 3

1.2 Z－变换的逆变换 8

1.2.1 留数方法 8

1.2.2 幂级数方法 10

2 Z－变换的应用 12

2.1 求解具有常系数的线性差分方程 12

2.2 脉冲系统的传递函数 14

３ 数值逼近 203.2 Pade逼近的1些定理 23

４ Z－变换的数值逼近 26

结束语 28

致谢词 29

参考文献： 30

附录： 31

摘 要

关键字：Z－变换，数值逼近，拉普拉斯变换。

Abstract

In this dissertation, we deal with the related theory as well as the application of Z-transformation and numerical approximation. First we give an introduction to the basic concepts and properties of Z-transformation; then emphasis is put on some of the applications and its solving methods. The whole paper is divided into three parts: Z-transformation and its properties; the applications of Z-transformation; the numerical approximation of Z-transformation. In the first part, the method to solving inverse transformation is educed on the basis of the definition of Z-transformation and residence theorems; and ten theorems about its properties are also explained. The study here can trace its theoretical foundation from the methods and experience of the research on Laplace transformation. In the second part, we make the study of the application of Z-transformation when we solve linear difference equation bearing constant coefficient and transmission function in pulsing system. Here the educing process of solving differential equation with Laplace methods is practiced. In the last part, we mainly introduce Prony approximation methods of exponential function. We also reveal that the approximation method of Prony exponential function and the corresponding Pade approximation of Z-transformation are of the same pattern.

Keywords: Z-transformation; numerical approximation; Laplace transformation

前 言

随着电子计算机及各种数字技术的不断发展，常常将信号按1定时间采样后再进行传送或处理，这样就产生了离散信号。离散信号作用于线性系统时，这种线性系统就成为离散系统。在研究离散系统的激励与响应的关系以及分析系统的特性时常采用Z－变换，所以Z－变换在实际应用中的作用是非常大的。再者，当今时代是科学技术日新月异地飞速发展的时代，在几乎所有的学科中都有逼近的思想和方法的渗透，这中间既包括自然学科也包括人文学科，也包括理工学科。而在Z－变换的范畴里，由于有些经过Z－变换后得到的式子比较复杂，无法求出精确的解，所以就有运用逼近的思想来解决这1问提的想法，于是就有了Z－变换的数值逼近这1个非常实际的课题。本论文也将在这个问题上做出1定的研究。通过这次的毕业设计，我培养出了刻苦钻研的学习精神和严肃认真的学习态度，这对我以后的学习和工作有很大的益处。