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

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参考文献

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