The (non-normalized) DTFT is simply a special case of the zz-transform for the case z=1z=1,
i.e., z=ejωz=ejω for some value ω∈[-π,π]ω∈[-π,π]
X
(
e
j
ω
)
=
∑
n
=
-
∞
∞
x
[
n
]
e
-
j
ω
n
.
X
(
e
j
ω
)
=
∑
n
=
-
∞
∞
x
[
n
]
e
-
j
ω
n
.
(1)The picture you should have in mind is the complex plane. The zz-transform is
defined on the whole plane, and the DTFT is simply the value of the
zz-transform on the unit circle, as illustrated below.
This picture should make it clear why the DTFT is defined only for
ω∈[-π,π]ω∈[-π,π] (or why it is periodic). Using the
normalization above, we also have the inverse DTFT formula:
x
[
n
]
=
1
2
π
∫
-
π
π
X
(
e
j
ω
)
e
j
ω
n
d
ω
.
x
[
n
]
=
1
2
π
∫
-
π
π
X
(
e
j
ω
)
e
j
ω
n
d
ω
.
(2)
![Graph with horizontal axis Re[z] and vertical axis Im[z]. There is a large blue circle centered at the origin on the graph, with a line segment from the origin to a point on the circle in the first quadrant. The acute angle that the line segment makes measured down to the horizontal axis is labeled ω. The point where the line segment intersects the circle is labeled e^j ω.](13_1.png)
