Z-Transform

The (unilateral) -transform of a sequence ${a_k}_(k=0)^infty$ is defined as

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

(1)

This definition is implemented in the Wolfram Language as ZTransform[a, n, z]. Similarly, the inverse -transform is implemented as InverseZTransform[A, z, n].

"The" -transform generally refers to the unilateral Z-transform. Unfortunately, there are a number of other conventions. Bracewell (1999) uses the term "-transform" (with a lower case ) to refer to the unilateral -transform. Girling (1987, p. 425) defines the transform in terms of samples of a continuous function. Worse yet, some authors define the -transform as the bilateral Z-transform.

In general, the inverse -transform of a sequence is not unique unless its region of convergence is specified (Zwillinger 1996, p. 545). If the -transform of a function is known analytically, the inverse -transform ${a_n}_(n=0)^infty=Z^(-1)[F(z)](n)$ can be computed using the contour integral

(2)

where is a closed contour surrounding the origin of the complex plane in the domain of analyticity of (Zwillinger 1996, p. 545)

The unilateral transform is important in many applications because the generating function of a sequence of numbers ${a_n}_(n=0)^infty$ is given precisely by $Z[{a_n}_(n=0)^infty](z^(-1))$ , the -transform of ${a_n}$ in the variable (Germundsson 2000). In other words, the inverse -transform of a function gives precisely the sequence of terms in the series expansion of . So, for example, the terms of the series of are given by

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) defines a variant of the unilateral -transform that operates on a continuous function sampled at regular intervals ,

(4)

where is the Laplace transform,

			(5)
			(6)

the one-sided shah function with period is given by

(7)

and is the Kronecker delta, giving

(8)

An alternative equivalent definition is

(9)

where

(10)

This definition is essentially equivalent to the usual one by taking .

The following table summarizes the -transforms for some common functions (Girling 1987, pp. 426-427; Bracewell 1999). Here, is the Kronecker delta, is the Heaviside step function, and is the polylogarithm.

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


1

The -transform of the general power function can be computed analytically as

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

where the are Eulerian numbers and is a polylogarithm. Amazingly, the -transforms of are therefore generators for Euler's number triangle.

The -transform $Z[{a_n}](z)=F(z)$ satisfies a number of important properties, including linearity

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

(14)

translation

$Z[{a_(n-k)}](z)$	$z^(-k)Z[{a_n}](z)$	(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)

scaling

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

(19)

and multiplication by powers of

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

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

The discrete Fourier transform is a special case of the -transform with

(22)

and a -transform with

(23)

for is called a fractional Fourier transform.

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References

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