Ztransform
Ztransformation
This transform method may be traced back to A. De Moivre [a5] around the year 1730 when he introduced the concept of "generating functions" in probability theory. Closely related to generating functions is the Ztransform, which may be considered as the discrete analogue of the Laplace transform. The Ztransform is widely used in the analysis and design of digital control, and signal processing [a4], [a2], [a3], [a6].
The Ztransform of a sequence , , that is identically zero for negative integers, is defined as
(a1) 
where is a complex number.
By the root test, the series (a1) converges if , where . The number is called the radius of convergence of the series (a1).
Contents
Example 1.
The Ztransform of is given by
Example 2.
The Ztransform of the Kroneckerdelta sequence
is given by
Properties of the Ztransform.
i) Linearity: Let and be the radii of convergence of the sequences and . Then for any ,
ii) Shifting: Let be the radius of convergence of . Then, for ,
a) Rightshifting: , for ;
b) Leftshifting: , for .
iii) Initial and final value.
a) Initial value theorem: ;
b) Final value theorem: .
iv) Convolution: The convolution of two sequences and is defined by
and its Ztransform is given by
Inverse Ztransform.
If , then the inverse Ztransform is defined as . Notice that by Laurent's theorem [a1] (cf. also Laurent series), the inverse Ztransform is unique [a2]. Consider a circle centred at the origin of the plane and enclosing all the poles of . Then, by the Cauchy integral theorem [a1], the inversion formula is given by
and by the residue theorem (cf. also Residue of an analytic function) [a1], .
If in its reduced form, then the poles of are the zeros of .
a) If has simple zeros, then the residue corresponding to the zero is given by
b) If has multiple zeros, then the residue at the zero with multiplicity is given by
The most practical method of finding the inverse Ztransform is the use of partialfractions techniques as illustrated by the following example.
Example.
See also [a2]. Suppose the problem is to solve the difference equation
\begin{equation} x \left( n + 4 \right) + 9 x \left( n + 3 \right) + 30 x \left( n + 2 \right) + 20 x \left( n + 1 \right) + 24 x \left( n \right) = 0 , \end{equation}
where , , , .
Taking the Ztransform yields
Taking the inverse Ztransform of both sides yields
Pairs of Ztransforms.
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References
[a1]  R.V. Churchill, J.W. Brown, "Complex variables and applications" , McGrawHill (1990) 
[a2]  S. Elaydi, "An introduction to difference equations" , Springer (1999) (Edition: Second) 
[a3]  A.J. Jerri, "Linear difference equations with discrete transform methods" , Kluwer Acad. Publ. (1996) 
[a4]  E. Jury, "Theory and application of the ztransform method" , Robert E. Krieger (1964) 
[a5]  A. De Moivre, "Miscellanew, Analytica de Seriebus et Quatratoris" , London (1730) 
[a6]  A.D. Poularikas, "The transforms and applications" , CRC (1996) 
Ztransform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ztransform&oldid=30630