BilateralZTransform

BilateralZTransform[expr,n,z]

gives the bilateral Z transform of expr.

BilateralZTransform[expr,{n1,…,nk},{z1,…,zk}]

gives the multidimensional bilateral Z transform of expr.

Details and Options

  • The bilateral Z transform is the discrete analog of the bilateral Laplace transform and plays an important role in digital signal processing and other fields.
  • The bilateral Z transform for a discrete function is given by .
  • The multidimensional bilateral Z transform is given by .
  • The sum is computed using numerical methods if the third argument, z, is given a numerical value.
  • The bilateral Z transform of exists only for complex values of in an annulus given by alpha<TemplateBox[{z}, Abs]<beta. In some cases, the annulus of definition may extend to the exterior or the interior of a disk.
  • The following options can be given:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditions Truewhether to generate answers that involve conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    WorkingPrecision Automaticthe precision used in internal computations

Examples

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Basic Examples  (2)

Define an exponentially decaying sequence:

Compute its bilateral Z transform:

Complex plot of the bilateral Z transform:

Compute the transform at a single point:

Compute the bilateral Z transform of a multivariate function:

Scope  (8)

Bilateral Z transform of the UnitStep function:

Discrete power function:

Combination of power functions:

DiscreteDelta:

Discrete-time, finite support function:

Trigonometric sequence:

Calculate the bilateral Z transform at a single point:

Alternatively, calculate the transform symbolically:

Then evaluate it for a specific value of :

For some functions, the bilateral Z transform can be evaluated only numerically:

Plot the bilateral Z transform using numerical values only:

Options  (3)

Assumptions  (1)

Specify the range for a parameter using Assumptions:

GenerateConditions  (1)

Set GenerateConditions to False to obtain a result without conditions:

WorkingPrecision  (1)

Use WorkingPrecision to obtain a result with arbitrary precision:

Applications  (2)

Define finite duration signals:

Plot the signals in the time domain:

To find the convolution, first calculate the product of the transforms:

Then, perform inversion back to the time domain:

Plot the convolution in the time domain:

Alternatively, find the convolution using DiscreteConvolve:

Define infinite duration signals:

Plot the signals in the time domain:

To find the convolution, first calculate product of the transforms:

Perform the inversion back to the time domain:

Plot the convolution in the time domain:

Alternatively, find the convolution using DiscreteConvolve:

Properties & Relations  (7)

BilateralZTransform and InverseBilateralZTransform are mutual inverses:

BilateralZTransform is closely related to FourierSequenceTransform:

Linearity:

Time shifting:

Scaling in the -domain:

Convolution:

Differentiation in the -domain:

Neat Examples  (1)

Create a table of basic bilateral Z transforms:

Wolfram Research (2021), BilateralZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralZTransform.html.

Text

Wolfram Research (2021), BilateralZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralZTransform.html.

CMS

Wolfram Language. 2021. "BilateralZTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BilateralZTransform.html.

APA

Wolfram Language. (2021). BilateralZTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BilateralZTransform.html

BibTeX

@misc{reference.wolfram_2024_bilateralztransform, author="Wolfram Research", title="{BilateralZTransform}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/BilateralZTransform.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_bilateralztransform, organization={Wolfram Research}, title={BilateralZTransform}, year={2021}, url={https://reference.wolfram.com/language/ref/BilateralZTransform.html}, note=[Accessed: 22-December-2024 ]}