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ListZTransform
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ListZTransform

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ListZTransform[list,z]

gives the Z transform of list as a function of z.

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ListZTransform[list,z,k]

places the first element of list at integer time k on the infinite time axis.

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ListZTransform[list,{z1,z2,…},{k1,k2,…}]

gives the multidimensional Z transform.

Details

Examples

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Basic Examples  (2)Summary of the most common use cases

Z transform of a constant vector:

In[1]:=1
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https://wolfram.com/xid/0en0n3gj2-hxcway
Out[1]=1

Plot the Z transform on the unit circle:

In[2]:=2
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https://wolfram.com/xid/0en0n3gj2-0fsrfv
Out[2]=2

Two-dimensional Z transform:

In[1]:=1
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https://wolfram.com/xid/0en0n3gj2-d1f7mr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0en0n3gj2-j1c4z9
Out[2]=2

Scope  (1)Survey of the scope of standard use cases

Z transform of a symbolic sequence:

In[1]:=1
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https://wolfram.com/xid/0en0n3gj2-09w14
Out[1]=1

Specify an offset:

In[2]:=2
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https://wolfram.com/xid/0en0n3gj2-rr3z79
Out[2]=2

Properties & Relations  (3)Properties of the function, and connections to other functions

ListZTransform of a list of numbers is equal to the ZTransform of a sum of shifted unit samples:

In[1]:=1
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https://wolfram.com/xid/0en0n3gj2-8b6of
In[2]:=2
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https://wolfram.com/xid/0en0n3gj2-dekd7k
Out[2]=2

Inverse of the ListZTransform:

In[1]:=1
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https://wolfram.com/xid/0en0n3gj2-5b8f8c
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0en0n3gj2-83e38
Out[2]=2

ListZTransform evaluated on a unit circle gives the Fourier transform of the list:

In[1]:=1
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https://wolfram.com/xid/0en0n3gj2-os227
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0en0n3gj2-bss4q6
Out[2]=2
Wolfram Research (2012), ListZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ListZTransform.html.
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Wolfram Research (2012), ListZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ListZTransform.html.

Text

Wolfram Research (2012), ListZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ListZTransform.html.

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Wolfram Research (2012), ListZTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ListZTransform.html.

CMS

Wolfram Language. 2012. "ListZTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ListZTransform.html.

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Wolfram Language. 2012. "ListZTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ListZTransform.html.

APA

Wolfram Language. (2012). ListZTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ListZTransform.html

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Wolfram Language. (2012). ListZTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ListZTransform.html

BibTeX

@misc{reference.wolfram_2025_listztransform, author="Wolfram Research", title="{ListZTransform}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ListZTransform.html}", note=[Accessed: 20-June-2025 ]}

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@misc{reference.wolfram_2025_listztransform, author="Wolfram Research", title="{ListZTransform}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ListZTransform.html}", note=[Accessed: 20-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_listztransform, organization={Wolfram Research}, title={ListZTransform}, year={2012}, url={https://reference.wolfram.com/language/ref/ListZTransform.html}, note=[Accessed: 20-June-2025 ]}

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@online{reference.wolfram_2025_listztransform, organization={Wolfram Research}, title={ListZTransform}, year={2012}, url={https://reference.wolfram.com/language/ref/ListZTransform.html}, note=[Accessed: 20-June-2025 ]}