Biomedical Engineering Reference
In-Depth Information
The difference equation is first converted into the
z-
domain:
1
2
Y
ð
z
Þ
z
1
1
2
z
1
1
2
X
ð
z
Þ:
Y
ð
z
Þ
¼
Y
ð
z
Þ
1
¼
Solving for
H(z)
1
2
H
ð
z
Þ¼
Y
ð
z
Þ
X
ð
z
Þ
¼
1
2
1
z
1
gives the filter transfer function. To determine the output, the transfer function is evaluated at the
frequency of the input sinusoid
p
2
ð
z
¼
e
j
Þ
2
2
1
2
1
2
p
H
0
¼
He
j
e
j
0:15p
¼
2
¼
j
¼
0
:
4
j
0
:
2
¼
0
:
45
:
1
2
1
þ
e
j
1
2
1
This transfer function tells us that the output is obtained by scaling the input magnitude by
0.45 and shifting the signal by a phase factor of 0.15p rads. Therefore, the output is
p
2
k
:
y
ð
k
Þ¼
45 sin
15p
Filter design problems begin with identifying the frequencies that are to be kept versus
the frequencies that are to be removed from the signal. For ideal filters, |
H
0
(
O
keep
)|
¼
1
H
0
(
and |
0. The filters in Example Problems 11.23 and 11.24 can both be consid-
ered as low-pass filters. However, their frequency responses (Figure 11.20) show that
neither is a particularly good low-pass filter. An ideal low-pass filter that has a cutoff
frequency of
O
remove
)|
¼
H
0
(
H
0
(
p
/4 with |
O
)|
¼
1 for
O < p
/4 and |
O
)|
¼
0 for
p
/4 and |
O
|
< p
is
superimposed for comparison.
Windowed FIR Filter Design
The ideal filters in Section 11.6.4 provide a general framework from which to build
a variety of filter functions to meet specific design criteria for both analog and
discrete systems. Unfortunately, the ideal low-pass filter is not physically realizable, as
we will see following. Using the ideal low-pass filter impulse response as a start-
ing reference, we will develop a modified filter function that overcomes this limita-
tion. The design of a windowed FIR filter is illustrated following for the case of a
low-pass filter, but the same procedures and concepts apply for high-pass and band-
pass filters.
There are two practical limitations associated with the ideal low-pass filter. First, note
that the impulse response of the ideal low-pass filter has infinite duration, extending from
-
. Thus, implementing an ideal low-pass filter requires an infinite amount of
time. The simplest way to overcome this limitation is to truncate the impulse response
over a finite time interval from -T to T (Figure 11.21a;shownforT
1
to
þ1
0.1 sec). However,
truncation leads to a second undesired effect. The sharp transitions in the impulse
¼
