Biomedical Engineering Reference
In-Depth Information
|H
(
)|
Ω
IDEAL LOW-PASS FILTER
1
EXAMPLE 8.21
0.8
EXAMPLE 8.22
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Normalized
Ω
FIGURE 11.20
A frequency domain comparison of low-pass filters described in Example Problems 11.23 and
11.24. An ideal low-pass filter with a cutoff frequency at
radians is super-
imposed for comparison. The cutoff frequency of a low-pass filter is usually defined as the frequency at which the
amplitude is equal to 1
p
/4 rads or 0.25 when normalized by
p
p
or approximately 0.71, which matches Example Problem 11.12. Both digital filters have
the same amplitude at f
max
—that is, where normalized
=
O ¼
1.
response to the selected interval do not allow for convergence of the Fourier integral
(Gibbs Phenomena). In the frequency domain, the effects of the truncation can be seen
in the transfer function magnitude of the truncated low-pass filter (Figure 11.21b). The
truncated filter exhibits errors in the passband and stopband regions. These errors are
undesirable because they will distort the signal in the passpand while allowing signals
in the stopband from passing through.
One way to overcome the limitation is to gradually truncate the filter impulse response
with a smooth window function,
). The modified low-pass filter is expressed as the prod-
uct of the ideal filter and the window function:
w
(
t
h
ðÞ¼
h
LP
ðÞ
w
ðÞ¼
W
p
s
inc
ð
W
c
t
Þ
w
ðÞ
ð
11
:
47
Þ
where
) is restricted to the interval -T to T (Figure 11.21c). The windowing procedure is
illustrated in Figure 11.21d, which shows the product of the window with the ideal filter
impulse response. The window allows for a smooth truncation of the impulse response,
thus allowing for convergence of the Fourier integral. As can be seen, the resulting stop-
band and passband errors of the windowed filter (Figure 11.21e) are substantially smaller
than for the truncated ideal filter (Figure 11.21b). Details of the design of window functions
w
(
t
