Graphics Programs Reference
In-Depth Information
∞
∑
1
n
=
(
z
)
nx
()
z
n
=
∞
It follows that
d
X
()
Znx
()
{
}
=
()
z
z
d
In general, a discrete LTI system has a transfer function
H
()
which
describes how the system operates on its input sequence
x
()
in order to pro-
duce the output sequence
y
()
. The output sequence
y
()
x
()
h
()
is computed from
the discrete convolution between the sequences
and
,
∞
∑
y
()
=
x
(
hn m
(
)
(13.115)
m
=
∞
However, since practical systems require that the sequence
x
()
be of finite
length, we can rewrite Eq. (13.115) as
N
∑
y
()
=
x
(
hn m
(
)
(13.116)
m
=
0
where denotes the input sequence length. Taking the Z-transform of Eq.
(13.116) yields
N
Y
()
X
()
H
()
=
(13.117)
and the discrete system transfer function is
Y
()
X
()
H
()
=
-----------
(13.118)
Finally, the transfer function
H
()
can be written as
He
j
ω
He
j
ω
∠
(
)
H
()
z
=
(
)
e
(13.119)
j
ω
=
He
j
ω
He
j
ω
where
(
)
is the amplitude response, and
∠
(
)
is the phase response.
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