Z-變換

序列 ${a_k}_(k=0)^infty$ 的（單邊）-變換定義為

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

(1)

此定義在 Wolfram 語言中實現為ZTransform[a, n, z]。類似地，逆 -變換實現為InverseZTransform[A, z, n]。

“Z-變換”通常指的是單邊 Z-變換。不幸的是，還有許多其他的約定。Bracewell (1999) 使用術語“-變換”（使用小寫）來指代單邊 -變換。Girling (1987, p. 425) 根據連續函式的樣本定義了變換。更糟糕的是，一些作者將 -變換定義為雙邊 Z-變換。

一般來說，序列的逆 -變換不是唯一的，除非指定其收斂區域 (Zwillinger 1996, p. 545)。如果函式的 -變換在分析上已知，則逆 -變換 ${a_n}_(n=0)^infty=Z^(-1)[F(z)](n)$ 可以使用輪廓積分計算

(2)

其中是複平面原點周圍的閉合輪廓，位於的解析域中 (Zwillinger 1996, p. 545)

單邊變換在許多應用中都很重要，因為數字序列 ${a_n}_(n=0)^infty$ 的生成函式正好由 $Z[{a_n}_(n=0)^infty](z^(-1))$ 給出，即 -變換在變數中的 ${a_n}$ (Germundsson 2000)。換句話說，函式的逆 -變換精確地給出了級數展開中的項。例如，的級數項由下式給出

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) 定義了單邊 -變換的一種變體，該變體對以規則間隔取樣的連續函式進行運算，

(4)

其中是拉普拉斯變換，

			(5)
			(6)

週期為的單邊 Shah 函式由下式給出

(7)

並且是克羅內克 delta，給出

(8)

另一種等效定義是

(9)

其中

(10)

透過取，這個定義本質上等同於通常的定義。

下表總結了一些常用函式的 -變換 (Girling 1987, pp. 426-427; Bracewell 1999)。這裡，是克羅內克 delta，是 Heaviside 階躍函式，並且是多對數函式。

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


1

一般冪函式的 -變換可以解析計算為

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

其中是尤拉數，是多對數函式。令人驚訝的是，-變換因此是尤拉數三角形的生成器。

-變換 $Z[{a_n}](z)=F(z)$ 滿足許多重要的性質，包括線性性

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

(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)

縮放

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

(19)

以及乘以的冪

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

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

離散傅立葉變換是 -變換在以下情況下的特例

(22)

並且當

(23)

對於時，-變換被稱為分數傅立葉變換。

另請參閱

雙邊 Z-變換, 離散傅立葉變換, 尤拉數三角形, 尤拉數, 分數傅立葉變換, 生成函式, 拉普拉斯變換, 人口比較, 單邊 Z-變換

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參考文獻

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