Z-transform of a sequence

13.6.1 Z-transform of a sequence

The Z-transform of a sequence a₀, a₁, …, a_n, … is the function

f(z)=

∞

∑

n=0

a_n

zⁿ

For example, the Z-transform of the sequence 0, 1, 2, 3, … is

f(z)=0+

z²

z³

+⋯

which has closed form

f(z)=

(z−1)²

The ztrans command finds the Z-transform of a sequence.

ztrans takes one mandatory and two optional arguments:
- a_x, 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(a_x ⟨,x,z⟩) returns the Z-transform of the input sequence.

Examples

To find the Z-transform of the identity function:

ztrans(x)

x²−2 x+1

With n as the original variable and z as the transform variable:

ztrans(n,n,z)

z²−2 z+1

Find the Z-transform of the constant function f(x)=1:

ztrans(1)

x−1

Indeed:

∞

∑

n=0

xⁿ

1−

x−1

You also have

ztrans(1,n,z)

z−1

Note that differentiating both sides of

∞

∑

n=0

zⁿ

z−1

gives you

∞

∑

n=0

zⁿ⁻¹

(z−1)²

and so, multiplying both sides by z,

∞

∑

n=0

zⁿ

(z−1)²

z²−2z+1

as indicated in the previous example.