WOLFRAM

ZTransform[expr,n,z]

gives the Z transform of expr.

ZTransform[expr,{n1,,nm},{z1,,zm}]

gives the multidimensional Z transform of expr.

Details and Options

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Transform a sequence:

Out[1]=1
Out[2]=2

Transform a multivariate sequence:

Out[1]=1
Out[2]=2

Transform a symbolic sequence:

Out[1]=1

Scope  (25)Survey of the scope of standard use cases

Basic Uses  (7)

Transform a univariate sequence:

Out[1]=1

Transform a multivariate sequence:

Out[2]=2

Compute a typical transform:

Out[1]=1

Plot the magnitude using Plot3D, ContourPlot, or DensityPlot:

Out[2]=2

Plot the complex phase:

Out[3]=3

Generate conditions for the region of convergence:

Out[1]=1

Plot the region for :

Out[2]=2

Evaluate the transform at a point:

Out[1]=1

Plot the spectrum:

Out[2]=2

The phase:

Out[3]=3

Plot both the spectrum and the plot phase using color:

Out[4]=4

Plot the spectrum in the complex plane using ParametricPlot3D:

Out[5]=5

ZTransform will use several properties including linearity:

Out[1]=1

Shifts:

Out[2]=2

Multiplication by exponentials:

Out[3]=3
Out[4]=4

Multiplication by polynomials:

Out[5]=5

Conjugate:

Out[6]=6

ZTransform automatically threads over lists:

Out[1]=1
Out[2]=2

Equations:

Out[3]=3

Rules:

Out[4]=4

TraditionalForm typesetting:

Special Sequences  (13)

Discrete impulses:

Out[1]=1
Out[2]=2

Discrete unit steps:

Out[3]=3
Out[4]=4

Discrete ramps:

Out[5]=5
Out[6]=6

Polynomials result in rational transforms:

Out[1]=1

Factorial polynomials:

Out[2]=2
Out[3]=3

Exponential functions:

Out[1]=1

Exponential polynomials:

Out[2]=2
Out[3]=3

Factorial exponential polynomials:

Out[4]=4
Out[5]=5

Trigonometric functions:

Out[1]=1
Out[2]=2

Trigonometric, exponential and polynomial:

Out[3]=3
Out[4]=4

Combinations of the previous input will also generate rational transforms:

Out[1]=1
Out[2]=2

Different ways of expressing piecewise defined signals:

Out[1]=1
Out[2]=2
Out[3]=3

Rational functions:

Out[1]=1
Out[2]=2
Out[3]=3

Rational exponential functions:

Out[4]=4
Out[5]=5

Hypergeometric term sequences:

Out[1]=1

The DiscreteRatio is rational for all hypergeometric term sequences:

Out[2]=2

Many functions give hypergeometric terms:

Out[4]=4

Any products are hypergeometric terms:

Out[5]=5

Transforms of hypergeometric terms:

Out[6]=6
Out[7]=7
Out[8]=8
Out[9]=9

Holonomic sequences:

Out[1]=1

A holonomic sequence is defined by a linear difference equation:

Out[2]=2

Many special function are holonomic sequences in their index:

Out[3]=3
Out[4]=4

Special sequences:

Out[1]=1
Out[2]=2

Periodic sequences:

Out[1]=1
Out[2]=2

Multivariate transforms:

Out[1]=1
Out[2]=2
Out[3]=3
Out[4]=4
Out[5]=5

Multivariate periodic sequences:

Out[1]=1
Out[2]=2

Special Operators  (5)

Linearity:

Out[1]=1
Out[2]=2

There are several relations to the InverseZTransform:

Out[1]=1

Shifts:

Out[2]=2

Polynomial multiplication:

Out[3]=3
Out[4]=4

Exponential multiplication:

Out[5]=5
Out[6]=6

Differences and shifts:

Out[1]=1
Out[2]=2

Sums:

Out[1]=1
Out[2]=2

Integrals:

Out[1]=1

Options  (4)Common values & functionality for each option

Assumptions  (1)

Without assumptions, typically a general formula will be produced:

Out[1]=1

Use Assumptions to obtain the expression on a given range:

Out[2]=2

GenerateConditions  (1)

Set GenerateConditions to True to get the region of convergence:

Out[1]=1

Method  (1)

Different methods may produce different results:

Out[1]=1
Out[2]=2

VerifyConvergence  (1)

By default, convergence testing is performed:

Out[1]=1

Setting VerifyConvergence->False will avoid the verification step:

Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Solving difference equations:

Out[1]=1
Out[2]=2
Out[3]=3
Out[4]=4

Properties & Relations  (6)Properties of the function, and connections to other functions

ZTransform is closely related to GeneratingFunction:

Out[25]=25

ExponentialGeneratingFunction:

Out[26]=26

FourierSequenceTransform:

Out[19]=19

Use InverseZTransform to get the sequence from its transform:

Out[1]=1
Out[2]=2
Out[3]=3
Out[4]=4

ZTransform effectively computes an infinite sum:

Out[1]=1
Out[2]=2

Linearity:

Out[1]=1

Shifting:

Out[2]=2

Convolution:

Out[3]=3

Derivative:

Out[4]=4

Initial value property:

Out[1]=1
Out[2]=2

Final value property:

Out[1]=1
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

A ZTransform may not converge for all values of parameters:

Out[1]=1
Out[2]=2

Use GenerateConditions to get the region of convergence:

Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Create a gallery of Z transforms:

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).

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 ]}

@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 ]}

@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 ]}