ZTransform
✖
ZTransform
Details and Options
- The Z transform for a discrete function is given by .
- The multidimensional Z transform is given by .
- The following options can be given:
-
Assumptions $Assumptions assumptions to make about parameters GenerateConditions False whether to generate answers that involve conditions on parameters Method Automatic method to use VerifyConvergence True whether to verify convergence - In TraditionalForm, ZTransform is output using .
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
https://wolfram.com/xid/0c15ehe-e19h2d
https://wolfram.com/xid/0c15ehe-gkxnrk
Transform a multivariate sequence:
https://wolfram.com/xid/0c15ehe-milsha
https://wolfram.com/xid/0c15ehe-wfnx8
Transform a symbolic sequence:
https://wolfram.com/xid/0c15ehe-fi1sdd
Scope (25)Survey of the scope of standard use cases
Basic Uses (7)
Transform a univariate sequence:
https://wolfram.com/xid/0c15ehe-d09jcj
Transform a multivariate sequence:
https://wolfram.com/xid/0c15ehe-k0pvsr
https://wolfram.com/xid/0c15ehe-hbo84p
Plot the magnitude using Plot3D, ContourPlot, or DensityPlot:
https://wolfram.com/xid/0c15ehe-on2kvh
https://wolfram.com/xid/0c15ehe-ojpkq
Generate conditions for the region of convergence:
https://wolfram.com/xid/0c15ehe-c5cvow
https://wolfram.com/xid/0c15ehe-l96gq9
Evaluate the transform at a point:
https://wolfram.com/xid/0c15ehe-uxedu
https://wolfram.com/xid/0c15ehe-q8h2e2
https://wolfram.com/xid/0c15ehe-b70np3
Plot both the spectrum and the plot phase using color:
https://wolfram.com/xid/0c15ehe-g14qrr
Plot the spectrum in the complex plane using ParametricPlot3D:
https://wolfram.com/xid/0c15ehe-qkoee
ZTransform will use several properties including linearity:
https://wolfram.com/xid/0c15ehe-bwqpke
https://wolfram.com/xid/0c15ehe-dy7uau
Multiplication by exponentials:
https://wolfram.com/xid/0c15ehe-fete5
https://wolfram.com/xid/0c15ehe-bdm1yu
Multiplication by polynomials:
https://wolfram.com/xid/0c15ehe-cv6n4w
https://wolfram.com/xid/0c15ehe-p6cheb
ZTransform automatically threads over lists:
https://wolfram.com/xid/0c15ehe-c7a318
https://wolfram.com/xid/0c15ehe-hejj63
https://wolfram.com/xid/0c15ehe-e2a9sa
https://wolfram.com/xid/0c15ehe-e7gnrl
TraditionalForm typesetting:
https://wolfram.com/xid/0c15ehe-csduj7
Special Sequences (13)
https://wolfram.com/xid/0c15ehe-d7royp
https://wolfram.com/xid/0c15ehe-k9yeeb
https://wolfram.com/xid/0c15ehe-ertwhv
https://wolfram.com/xid/0c15ehe-pe9fc4
https://wolfram.com/xid/0c15ehe-7yya0
https://wolfram.com/xid/0c15ehe-i3vff2
Polynomials result in rational transforms:
https://wolfram.com/xid/0c15ehe-n8rb4v
https://wolfram.com/xid/0c15ehe-uhq3i
https://wolfram.com/xid/0c15ehe-v418j
https://wolfram.com/xid/0c15ehe-hlwbjw
https://wolfram.com/xid/0c15ehe-hcadz0
https://wolfram.com/xid/0c15ehe-op3ewm
Factorial exponential polynomials:
https://wolfram.com/xid/0c15ehe-ji531
https://wolfram.com/xid/0c15ehe-dc817i
https://wolfram.com/xid/0c15ehe-0ut58
https://wolfram.com/xid/0c15ehe-e7zdc2
Trigonometric, exponential and polynomial:
https://wolfram.com/xid/0c15ehe-bmuntv
https://wolfram.com/xid/0c15ehe-esyq3l
Combinations of the previous input will also generate rational transforms:
https://wolfram.com/xid/0c15ehe-evyjyf
https://wolfram.com/xid/0c15ehe-cel9b1
Different ways of expressing piecewise defined signals:
https://wolfram.com/xid/0c15ehe-k9af0
https://wolfram.com/xid/0c15ehe-mi0tg
https://wolfram.com/xid/0c15ehe-fgha9
https://wolfram.com/xid/0c15ehe-gnf42h
https://wolfram.com/xid/0c15ehe-l773h
https://wolfram.com/xid/0c15ehe-n8z9xr
Rational exponential functions:
https://wolfram.com/xid/0c15ehe-ejiqlh
https://wolfram.com/xid/0c15ehe-ihmspl
Hypergeometric term sequences:
https://wolfram.com/xid/0c15ehe-bc43hn
The DiscreteRatio is rational for all hypergeometric term sequences:
https://wolfram.com/xid/0c15ehe-dwrey6
Many functions give hypergeometric terms:
https://wolfram.com/xid/0c15ehe-b53iop
https://wolfram.com/xid/0c15ehe-fhkry
Any products are hypergeometric terms:
https://wolfram.com/xid/0c15ehe-hek40i
Transforms of hypergeometric terms:
https://wolfram.com/xid/0c15ehe-xainz
https://wolfram.com/xid/0c15ehe-gn971i
https://wolfram.com/xid/0c15ehe-mih7p
https://wolfram.com/xid/0c15ehe-bn7z8x
https://wolfram.com/xid/0c15ehe-wcpyw
A holonomic sequence is defined by a linear difference equation:
https://wolfram.com/xid/0c15ehe-cf7dbh
Many special function are holonomic sequences in their index:
https://wolfram.com/xid/0c15ehe-bty6tf
https://wolfram.com/xid/0c15ehe-ggrd1i
https://wolfram.com/xid/0c15ehe-2inbp
https://wolfram.com/xid/0c15ehe-bfx2yq
https://wolfram.com/xid/0c15ehe-pvxl6
https://wolfram.com/xid/0c15ehe-8oniq
https://wolfram.com/xid/0c15ehe-e6c2tq
https://wolfram.com/xid/0c15ehe-covwf
https://wolfram.com/xid/0c15ehe-b8cckx
https://wolfram.com/xid/0c15ehe-h3ezn4
https://wolfram.com/xid/0c15ehe-zeqyd
Multivariate periodic sequences:
https://wolfram.com/xid/0c15ehe-p69ojo
https://wolfram.com/xid/0c15ehe-hg8x9s
Special Operators (5)
https://wolfram.com/xid/0c15ehe-i1qfv
https://wolfram.com/xid/0c15ehe-c2dpja
There are several relations to the InverseZTransform:
https://wolfram.com/xid/0c15ehe-kz0wys
https://wolfram.com/xid/0c15ehe-xhevz
https://wolfram.com/xid/0c15ehe-km3ha7
https://wolfram.com/xid/0c15ehe-qw2nro
https://wolfram.com/xid/0c15ehe-g75o01
https://wolfram.com/xid/0c15ehe-db797d
https://wolfram.com/xid/0c15ehe-gura5a
https://wolfram.com/xid/0c15ehe-cob5xj
https://wolfram.com/xid/0c15ehe-be1wlj
https://wolfram.com/xid/0c15ehe-ns5cp4
https://wolfram.com/xid/0c15ehe-bezs9x
Options (4)Common values & functionality for each option
Assumptions (1)
Without assumptions, typically a general formula will be produced:
https://wolfram.com/xid/0c15ehe-bn9f8z
Use Assumptions to obtain the expression on a given range:
https://wolfram.com/xid/0c15ehe-e1vzyh
GenerateConditions (1)
Set GenerateConditions to True to get the region of convergence:
https://wolfram.com/xid/0c15ehe-dsawsv
Method (1)
VerifyConvergence (1)
By default, convergence testing is performed:
https://wolfram.com/xid/0c15ehe-bz3atu
Setting VerifyConvergence->False will avoid the verification step:
https://wolfram.com/xid/0c15ehe-dcmdxi
Applications (1)Sample problems that can be solved with this function
Properties & Relations (6)Properties of the function, and connections to other functions
ZTransform is closely related to GeneratingFunction:
https://wolfram.com/xid/0c15ehe-jv3pf
ExponentialGeneratingFunction:
https://wolfram.com/xid/0c15ehe-iam0cg
https://wolfram.com/xid/0c15ehe-kt3bs6
Use InverseZTransform to get the sequence from its transform:
https://wolfram.com/xid/0c15ehe-bp78fy
https://wolfram.com/xid/0c15ehe-bi07ea
https://wolfram.com/xid/0c15ehe-gfkaq
https://wolfram.com/xid/0c15ehe-koh7da
ZTransform effectively computes an infinite sum:
https://wolfram.com/xid/0c15ehe-cu285m
https://wolfram.com/xid/0c15ehe-uic2n
https://wolfram.com/xid/0c15ehe-jk87qh
https://wolfram.com/xid/0c15ehe-gfznc
https://wolfram.com/xid/0c15ehe-cqglnw
https://wolfram.com/xid/0c15ehe-hot5il
https://wolfram.com/xid/0c15ehe-g07wsp
https://wolfram.com/xid/0c15ehe-hgovab
https://wolfram.com/xid/0c15ehe-hhakn7
https://wolfram.com/xid/0c15ehe-eyhb57
Possible Issues (1)Common pitfalls and unexpected behavior
A ZTransform may not converge for all values of parameters:
https://wolfram.com/xid/0c15ehe-e91r90
https://wolfram.com/xid/0c15ehe-eswz8t
Use GenerateConditions to get the region of convergence:
https://wolfram.com/xid/0c15ehe-f5pno
Wolfram Research (1999), ZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ZTransform.html (updated 2008).
Text
Wolfram Research (1999), ZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ZTransform.html (updated 2008).
Wolfram Research (1999), ZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ZTransform.html (updated 2008).
CMS
Wolfram Language. 1999. "ZTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/ZTransform.html.
Wolfram Language. 1999. "ZTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/ZTransform.html.
APA
Wolfram Language. (1999). ZTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ZTransform.html
Wolfram Language. (1999). ZTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ZTransform.html
BibTeX
@misc{reference.wolfram_2024_ztransform, author="Wolfram Research", title="{ZTransform}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/ZTransform.html}", note=[Accessed: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_ztransform, organization={Wolfram Research}, title={ZTransform}, year={2008}, url={https://reference.wolfram.com/language/ref/ZTransform.html}, note=[Accessed: 10-January-2025
]}