Biomedical Engineering Reference
In-Depth Information
where
x
(
k
) and
y
(
k
) represent the input and output at time
k
,
x
(
k
1) and
y
(
k
1) represent
the input and output one sample into the past, and similarly,
(
k
2) and
(
k
2) correspond
x
x
to the input and output two samples into the past.
Digital systems, like analog systems, can also be defined by their impulse responses,
),
and the convolution sum (Eq. (11.33)). If the response has a finite number of nonzero points,
the filter is called an FIR filter or a “finite impulse response filter.” If the response has an
infinite number of nonzero points, the filter is called an IIR or “infinite impulse response
filter.” One positive quality of digital filters is the ease with which the output for any input
can be calculated.
h
(
k
EXAMPLE PROBLEM 11.22
Find the impulse response for the digital filter
1
2 x ð k Þþ
1
2 y ð k
y ð k Þ¼
1
Þ
Solution
Assume the system is at rest before input begins—that is,
y
(
n
)
¼
0 for
n <
0.
1
2
1
2 y ð
y ð
2
Þ¼
2
Þþ
3
Þ¼
0
þ
0
¼
0
1
2 d
1
2 y ð
y ð
1
Þ¼
ð
1
Þþ
2
Þ¼
0
þ
0
¼
0
1
2
1
2 y ð
1
2 þ
1
2
y ð
0
Þ¼
0
Þþ
1
Þ¼
0
¼
¼
2
1
2 d
1
2 y ð
1
2
1
2
1
2
y ð
1
Þ¼
ð
1
Þþ
0
Þ¼
0
þ
2
3
1
2
1
2 y ð
1
2
1
2
1
2
y ð
2
Þ¼
2
Þþ
1
Þ¼
0
þ
¼
3
4
1
2
1
2 y ð
1
2
1
2
1
2
y ð
3
Þ¼
3
Þþ
2
Þ¼
0
þ
¼
...
.
k þ1
1
2
y ð k Þ¼
u ð k Þ
The impulse response for the filter is an exponential sequence. This is an IIR filter because the
impulse response is of infinite duration.
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