ListZTransform
ListZTransform[list,z]
gives the Z transform of list as a function of z.
ListZTransform[list,z,k]
places the first element of list at integer time k on the infinite time axis.
ListZTransform[list,{z1,z2,…},{k1,k2,…}]
gives the multidimensional Z transform.
Details
- The one-dimensional Z transform of a list
of length 
is given by 
. - ListZTransform[list,z] is equivalent to ListZTransform[list,z,0].
- ListZTransform[list,…] effectively computes the ZTransform of a sequence given by list.
- ListZTransform[list,z,k] effectively computes ZTransform[f[r],r,z] for a sequence f with f[r-1+k]=list[[r]] for 1<=r<=Length[list] and f[r]=0 otherwise.
Examples
open all close allBasic Examples (2)
Z transform of a constant vector:
H[z_] = ListZTransform[{(1/5), (1/5), (1/5), (1/5), (1/5)}, z]Plot the Z transform on the unit circle:
BodePlot[H[E^I ω], {ω, 0.01, π}, PlotLayout -> "Magnitude", ScalingFunctions -> {"Linear", "Absolute"}]H[u_, v_] = ListZTransform[(1/9)(| | | |
| - | - | - |
| 1 | 1 | 1 |
| 1 | 1 | 1 |
| 1 | 1 | 1 |), {u, v}]Plot3D[Abs[H[E^I u, E^I v]], {u, -π, π}, {v, -π, π}, PlotRange -> {0, 1}]Scope (1)
Z transform of a symbolic sequence:
ListZTransform[{a, b, c}, z]ListZTransform[{a, b, c}, z, -1]Z transform of an offset 3x3 array:
ListZTransform[(1/9)(| | | |
| - | - | - |
| 1 | 1 | 1 |
| 1 | 1 | 1 |
| 1 | 1 | 1 |), {u, v}, {-1, -1}]Applications (1)
Transfer function of a moving-average filter:
H[z_] = ListZTransform[{(1/5), (1/5), (1/5), (1/5), (1/5)}, z]Plot the magnitude of its transfer function:
Plot3D[Abs[H[u + I v]], ...]Plot the magnitude response of the filter:
Plot[Abs[H[E^I ω]], {ω, 0, π}]Properties & Relations (3)
ListZTransform of a list of numbers is equal to the ZTransform of a sum of shifted unit samples:
f[n_] := (1/5)Sum[DiscreteDelta[n - k], {k, 0, 4}]ZTransform[f[n], n, z] == Together@ListZTransform[{(1/5), (1/5), (1/5), (1/5), (1/5)}, z]Inverse of the ListZTransform:
zt = ListZTransform[{(1/5), (1/4), (1/3), (1/2), 1}, z]Table[InverseZTransform[zt, z, n], {n, 0, Length[zt] - 1}]ListZTransform evaluated on a unit circle gives the Fourier transform of the list:
H[z_] = ListZTransform[{(1/5), (1/4), (1/3), (1/2), 1}, z]H[E ^ (I w)] == ListFourierSequenceTransform[{(1/5), (1/4), (1/3), (1/2), 1}, w]Text
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.
APA
Wolfram Language. (2012). ListZTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ListZTransform.html