Z-变换

序列 ${a_k}_(k=0)^infty$ 的（单边）-变换定义为

$Z[{a_k}_(k=0)^infty](z)=sum_(k=0)^infty(a_k)/(z^k).$

(1)

此定义在 Wolfram 语言中实现为ZTransform[a, n, z]。类似地，逆 -变换实现为InverseZTransform[A, z, n]。

“Z-变换”通常指的是单边 Z-变换。不幸的是，还有许多其他的约定。Bracewell (1999) 使用术语“-变换”（使用小写）来指代单边 -变换。Girling (1987, p. 425) 根据连续函数的样本定义了变换。更糟糕的是，一些作者将 -变换定义为双边 Z-变换。

一般来说，序列的逆 -变换不是唯一的，除非指定其收敛区域 (Zwillinger 1996, p. 545)。如果函数的 -变换在分析上已知，则逆 -变换 ${a_n}_(n=0)^infty=Z^(-1)[F(z)](n)$ 可以使用轮廓积分计算

(2)

其中是复平面原点周围的闭合轮廓，位于的解析域中 (Zwillinger 1996, p. 545)

单边变换在许多应用中都很重要，因为数字序列 ${a_n}_(n=0)^infty$ 的生成函数正好由 $Z[{a_n}_(n=0)^infty](z^(-1))$ 给出，即 -变换在变量中的 ${a_n}$ (Germundsson 2000)。换句话说，函数的逆 -变换精确地给出了级数展开中的项。例如，的级数项由下式给出

Z^(-1)[y^(-1)(y^(-1)+1)/(y^(-1)-1)^3](n)
=Z^(-1)[-(y(y+1))/((y-1)^3)](n)=n^2.

(3)

Girling (1987) 定义了单边 -变换的一种变体，该变体对以规则间隔采样的连续函数进行运算，

(4)

其中是拉普拉斯变换，

			(5)
			(6)

周期为的单边 Shah 函数由下式给出

(7)

并且是克罗内克 delta，给出

(8)

另一种等效定义是

(9)

其中

(10)

通过取，这个定义本质上等同于通常的定义。

下表总结了一些常用函数的 -变换 (Girling 1987, pp. 426-427; Bracewell 1999)。这里，是克罗内克 delta，是 Heaviside 阶跃函数，并且是多对数函数。

	$Z[{a_n}_(n=0)^infty](z)$
	1


1

一般幂函数的 -变换可以解析计算为

$Z[{n^k}_(n=0)^infty](z)$			(11)
			(12)
			(13)

其中是欧拉数，是多对数函数。令人惊讶的是，-变换因此是欧拉数三角形的生成器。

-变换 $Z[{a_n}](z)=F(z)$ 满足许多重要的性质，包括线性性

$Z[a{a_n}+b{b_n}](z)=aZ[{a_n}](z)+bZ[{b_n}](z),$

(14)

平移

$Z[{a_(n-k)}](z)$	$z^(-k)Z[{a_n}](z)$	(15)
$Z[{a_(n+1)}](z)$	$zZ[{a_n}](z)-za_0$	(16)
$Z[{a_(n+2)}](z)$	$z^2Z[{a_n}](z)-z^2a_0-za_1$	(17)
$Z[{a_(n+k)}](z)$	$z^mZ[{a_n}](z)-sum_(r=0)^(m-1)z^(k-r)a_(rt),$	(18)

缩放

$Z[{b^na_n}](z)=F(z/b),$

(19)

以及乘以的幂

$Z[{n^ka_n}](z)$			(20)
$Z[{n^(-1)a_n}](z)$			(21)

(Girling 1987, p. 425; Zwillinger 1996, p. 544)。

离散傅里叶变换是 -变换在以下情况下的特例

(22)

并且当

(23)

对于时，-变换被称为分数傅里叶变换。

另请参阅

双边 Z-变换, 离散傅里叶变换, 欧拉数三角形, 欧拉数, 分数傅里叶变换, 生成函数, 拉普拉斯变换, 人口比较, 单边 Z-变换

使用探索

更多尝试

参考文献

Arndt, J. "The

-Transform (ZT)." Ch. 3 in "Remarks on FFT Algorithms." http://www.jjj.de/fxt/.Boxer, R. "A Note on Numerical Transform Calculus." Proc. IRE 45, 1401-1406, 1957.Boxer, R. and Thaler, S. "A Simplified Method of Solving Linear and Nonlinear Systems." Proc. IRE 44, 89-101, 1956.Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 257-262, 1999.Balakrishnan, V. K. Schaum's Outline of Combinatorics, including Concepts of Graph Theory. New York: McGraw-Hill, 1995.Brand, L. Differential and Difference Equations. New York: Wiley, 1966.Cadzow, J. A. Discrete-Time Systems: An Introduction with Interdisciplinary Applications. Englewood Cliffs, NJ: Prentice-Hall, 1973.DiStefano, J. J.; Stubberud, A. R.; and Williams, I. J. Schaum's Outline of Feedback and Control Systems, 2nd ed. New York: McGraw-Hill, 1995.Elaydi, S. N. An Introduction to Difference Equations, 2nd ed. New York: Springer, 1999.Germundsson, R. "Mathematica Version 4." Mathematica J. 7, 497-524, 2000.Girling, B. "The Z Transform." In CRC Standard Mathematical Tables, 28th ed (Ed. W. H. Beyer). Boca Raton, FL: CRC Press, pp. 424-428, 1987.Graf, U. Applied Laplace Transforms and z-Transforms for Scientists and Engineers: A Computational Approach using a Mathematica Package. Basel, Switzerland: Birkhäuser, 2004.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Grimaldi, R. P. Discrete and Combinatorial Mathematics: An Applied Introduction, 4th ed. Longman, 1998.Jury, E. I. Theory and Applications of the Z-Transform Method. New York: Wiley, 1964.Kelley, W. G. and Peterson, A. C. Difference Equations: An Introduction with Applications, 2nd ed. New York: Academic Press, 2001.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Levy, H. and Lessman, F. Finite Difference Equations. New York: Dover, 1992.Ljung, L. System Identification: Theory for the User. Prentice-Hall, 1987.Mickens, R. E. Difference Equations, 2nd ed. Princeton, NJ: Van Nostrand Reinhold, 1987.Miller, K. S. Linear Difference Equations. New York: Benjamin, 1968.Ogata, K. Discrete-Time Control Systems. Englewood Cliffs, NJ: Prentice-Hall, 1987.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Sedgewick, R. and Flajolet, P. An Introduction to the Analysis of Algorithms. Reading, MA: Addison-Wesley, 1996.Tsypkin, Ya. Z. Sampling System Theory. New York: Pergamon Press, 1964.Vidyasagar, M. Control System Synthesis: A Factorization Approach. Cambridge, MA: MIT Press, 1985.Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1994.Zwillinger, D. (Ed.). "Generating Functions and