The Z-transform of a sequence a0, a1, …, an, … is the function
f(z) = |
|
| . |
The ztrans command takes one or three arguments.
ztrans returns the Z-transform of the sequence.
For example, the Z-transform of the sequence
0, 1, 2, 3, … |
is
f(z) = 0 + 1/z + 2/z2 + 3/z3 + … |
which has closed form
f(z) = z/(z−1)2. |
Input:
Output:
Input:
Output:
Note that
Input:
Output:
since
| 1/xn = 1/(1−1/x) = x/(x−1). |
We also have
Input:
Output:
Note that differentiating both sides of
| 1/zn = z/(z−1) |
gives us
| n/zn−1 = 1/(z−1)2 |
and so, multiplying both sides by z,
| n/zn = z/(z−1)2 = z/(z2 − 2z + 1) |
as indicated above.