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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 Z-transform, which may be considered as the discrete analogue of the Laplace transform. The Z-transform is widely used in the analysis and design of digital control, and signal processing [a4], [a2], [a3], [a6].

The Z-transform of a sequence , , that is identically zero for negative integers, is defined as


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).

Example 1.

The Z-transform of is given by

Example 2.

The Z-transform of the Kronecker-delta sequence

is given by

Properties of the Z-transform.

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) Right-shifting: , for ;

b) Left-shifting: , 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 Z-transform is given by

Inverse Z-transform.

If , then the inverse Z-transform is defined as . Notice that by Laurent's theorem [a1] (cf. also Laurent series), the inverse Z-transform 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 Z-transform is the use of partial-fractions techniques as illustrated by the following example.


See also [a2]. Suppose the problem is to solve the difference equation , where , , , .

Taking the Z-transform yields

Taking the inverse Z-transform of both sides yields

Pairs of Z-transforms.


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[a1] R.V. Churchill, J.W. Brown, "Complex variables and applications" , McGraw-Hill (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 z-transform 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)
How to Cite This Entry:
Z-transform. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S. Elaydi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article