Biomedical Engineering Reference
In-Depth Information
From Digital Filter to Transfer Function
The transfer function for the digital system,
H
(
z
), can be obtained by rearranging the dif-
ference equation (Eq.
(11.23)) and applying Eq.
(11.21).
(
)
is the quotient of
the
H
z
transform of the output,
(
), divided by the
transform of the input,
(
).
z-
Y
z
z-
X
z
y
ð
k
Þþ
a
1
y
ð
k
1
Þþ
a
2
y
ð
k
2
Þ ...þ
a
N
y
ð
k
N
Þ¼
b
0
x
ð
k
Þþ
b
1
x
ð
k
1
Þþ...þ
b
M
x
ð
k
M
Þ
z
1
Y
z
2
Y
ð
z
Þ ...þ
a
N
z
N
Y
ð
z
Þ¼
b
z
1
z
2
X
ð
z
Þ ...: þ
b
M
z
M
X
ð
z
Þ
Y
ð
z
Þþ
a
ð
z
Þþ
a
X
ð
z
Þþ
b
X
ð
z
Þþ
b
1
2
0
1
2
z
1
z
2
...þ
a
N
z
N
Þ¼
X
ð
z
Þð
b
z
1
z
2
...: þ
b
M
z
M
Þ
Y
ð
z
Þð
1
þ
a
þ
a
þ
b
þ
b
1
2
0
1
2
X
ð
z
Þ
¼
b
0
þ
b
1
z
1
þ
b
2
z
1
...þ
b
M
z
M
H
ð
z
Þ¼
Y
ð
z
Þ
þ
a
1
z
1
þ
a
2
z
1
...þ
a
N
z
N
1
ð
11
:
45
Þ
From Transfer Function to Frequency Response
The frequency response (
H
0
(
O
)) of a digital system can be calculated directly from
H
(
z
),
where
is in radians. If the data are samples of an analog signal as previously described,
the relationship between
O
o
and
O
is
O ¼ o
T:
H
0
ðOÞ¼
H
ð
z
Þj
z
¼
e
j
O
ð
11
:
46
Þ
For a linear system, an input sequence of the form
x
ð
k
Þ¼
A sin
ðO
0
k
þ FÞ
will generate an output whose steady-state sequence will fit into the following form:
y
ð
k
Þ¼
B sin
ðO
0
k
þ Þ
Values for B and
can be calculated directly:
¼
AH
0
ðO
B
j
Þ
j
0
ð
H
0
ðO
¼ F þ
angle
ÞÞ
0
EXAMPLE PROBLEM 11.24
The input sequence for the digital filter used in Example Problem 11.23 is
p
2
k
x
ð
k
Þ¼
100 sin
What is the steady-state form of the output?
Solution
1
2
y
ð
k
1
2
x
ð
k
Þ
y
ð
k
Þ
1
Þ¼