6.58.1 The Z-transform of a sequence: ztrans
The Z-transform of a sequence a0, a1, …, an, … is the
function
For example, the Z-transform of the sequence
is
f(z) = 0 + 1/z + 2/z2 + 3/z3 + …
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which has closed form
The ztrans command finds the Z-transform of a sequence.
-
ztrans takes one mandatory and two optional arguments:
-
ax, a formula with a variable for the general
term of a sequence.
- Optionally, x, the variable (by default x).
- Optionally, z, a variable to be used by the resulting
function.
ztrans(ax ⟨ x,z⟩>) returns the
Z-transform of the sequence.
Examples.
-
Input:
ztrans(x)
Output:
- Input:
ztrans(n,n,z)
Output:
- Input:
ztrans(1)
Output:
since
| 1/xn = 1/(1−1/x) = x/(x−1).
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You also have
Input:
ztrans(1,n,z)
Output:
Note that differentiating both sides of
gives you
and so, multiplying both sides by z,
| n/zn = z/(z−1)2 = z/(z2 − 2z + 1)
|
as indicated above.