Z-transform
2020 Mathematics Subject Classification: Primary: 05A15 [MSN][ZBL]
Z-transformation
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
![]() | (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).
Example 1.
The Z-transform of is given by
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Example 2.
The Z-transform of the Kronecker-delta sequence
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is given by
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Properties of the Z-transform.
i) Linearity: Let and
be the radii of convergence of the sequences
and
. Then for any
,
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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
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and its Z-transform is given by
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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
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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
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b) If has multiple zeros, then the residue
at the zero
with multiplicity
is given by
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The most practical method of finding the inverse Z-transform is the use of partial-fractions 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 Z-transform yields
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Taking the inverse Z-transform of both sides yields
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Pairs of Z-transforms.
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|
References
[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) |
Z-transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Z-transform&oldid=34741