ZTransform

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

• 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

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Basic Examples(3)

Transform a sequence:

Transform a multivariate sequence:

Transform a symbolic sequence:

Scope(25)

Basic Uses(7)

Transform a univariate sequence:

Transform a multivariate sequence:

Compute a typical transform:

Plot the magnitude using Plot3D, ContourPlot, or DensityPlot:

Plot the complex phase:

Generate conditions for the region of convergence:

Plot the region for :

Evaluate the transform at a point:

Plot the spectrum:

The phase:

Plot both the spectrum and the plot phase using color:

Plot the spectrum in the complex plane using ParametricPlot3D:

ZTransform will use several properties including linearity:

Shifts:

Multiplication by exponentials:

Multiplication by polynomials:

Conjugate:

Equations:

Rules:

Special Sequences(13)

Discrete impulses:

Discrete unit steps:

Discrete ramps:

Polynomials result in rational transforms:

Factorial polynomials:

Exponential functions:

Exponential polynomials:

Factorial exponential polynomials:

Trigonometric functions:

Trigonometric, exponential and polynomial:

Combinations of the previous input will also generate rational transforms:

Different ways of expressing piecewise defined signals:

Rational functions:

Rational exponential functions:

Hypergeometric term sequences:

The DiscreteRatio is rational for all hypergeometric term sequences:

Many functions give hypergeometric terms:

Any products are hypergeometric terms:

Transforms of hypergeometric terms:

Holonomic sequences:

A holonomic sequence is defined by a linear difference equation:

Many special function are holonomic sequences in their index:

Special sequences:

Periodic sequences:

Multivariate transforms:

Multivariate periodic sequences:

Special Operators(5)

Linearity:

There are several relations to the InverseZTransform:

Shifts:

Polynomial multiplication:

Exponential multiplication:

Differences and shifts:

Sums:

Integrals:

Options(4)

Assumptions(1)

Without assumptions, typically a general formula will be produced:

Use Assumptions to obtain the expression on a given range:

GenerateConditions(1)

Set GenerateConditions to True to get the region of convergence:

Method(1)

Different methods may produce different results:

VerifyConvergence(1)

By default, convergence testing is performed:

Setting will avoid the verification step:

Applications(1)

Solving difference equations:

Properties & Relations(6)

ZTransform is closely related to GeneratingFunction:

Use InverseZTransform to get the sequence from its transform:

ZTransform effectively computes an infinite sum:

Linearity:

Shifting:

Convolution:

Derivative:

Initial value property:

Final value property:

Possible Issues(1)

A ZTransform may not converge for all values of parameters:

Use GenerateConditions to get the region of convergence:

Neat Examples(1)

Create a gallery of Z transforms:

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).

CMS

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

BibTeX

@misc{reference.wolfram_2023_ztransform, author="Wolfram Research", title="{ZTransform}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/ZTransform.html}", note=[Accessed: 20-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_ztransform, organization={Wolfram Research}, title={ZTransform}, year={2008}, url={https://reference.wolfram.com/language/ref/ZTransform.html}, note=[Accessed: 20-April-2024 ]}