We introduced the zz-transform before as
H
(
z
)
=
∑
k
=
-
∞
∞
h
[
k
]
z
-
k
H
(
z
)
=
∑
k
=
-
∞
∞
h
[
k
]
z
-
k
(1)where zz is a complex number. When H(z)H(z) exists (the sum converges), it can be
interpreted as the “response” of an LSI system with impulse response h[n]h[n]
to the input of znzn. The zz-transform is useful mostly due to its ability
to simplify system analysis via the following result.
If y=h*xy=h*x, then Y(z)=H(z)X(z)Y(z)=H(z)X(z).
First observe that
∑
n
=
-
∞
∞
y
[
n
]
z
-
n
=
∑
n
=
-
∞
∞
∑
k
=
-
∞
∞
x
[
k
]
h
[
n
-
k
]
z
-
n
=
∑
k
=
-
∞
∞
x
[
k
]
∑
n
=
-
∞
∞
h
[
n
-
k
]
z
-
n
∑
n
=
-
∞
∞
y
[
n
]
z
-
n
=
∑
n
=
-
∞
∞
∑
k
=
-
∞
∞
x
[
k
]
h
[
n
-
k
]
z
-
n
=
∑
k
=
-
∞
∞
x
[
k
]
∑
n
=
-
∞
∞
h
[
n
-
k
]
z
-
n
(2)
Let m=n-km=n-k, and note that z-n=z-m·z-kz-n=z-m·z-k. Thus we have
∑
n
=
-
∞
∞
y
[
n
]
z
-
n
=
∑
k
=
-
∞
∞
x
[
k
]
∑
n
=
-
∞
∞
h
[
m
]
z
-
m
z
-
k
=
∑
k
=
-
∞
∞
x
[
k
]
H
(
z
)
z
-
k
=
H
(
z
)
∑
k
=
-
∞
∞
x
[
k
]
z
-
k
=
H
(
z
)
X
(
z
)
∑
n
=
-
∞
∞
y
[
n
]
z
-
n
=
∑
k
=
-
∞
∞
x
[
k
]
∑
n
=
-
∞
∞
h
[
m
]
z
-
m
z
-
k
=
∑
k
=
-
∞
∞
x
[
k
]
H
(
z
)
z
-
k
=
H
(
z
)
∑
k
=
-
∞
∞
x
[
k
]
z
-
k
=
H
(
z
)
X
(
z
)
(3)
This yields the “transfer function”
H
(
z
)
=
Y
(
z
)
X
(
z
)
.
H
(
z
)
=
Y
(
z
)
X
(
z
)
.
(4)
