Digital Signal Processing Reference
In-Depth Information
can fold over into the final useful information band to create aliasing distortion. Therefore, we can use
this frequency as the lower stop frequency edge of the anti-aliasing filter to prevent the aliasing
distortion at the final stage. The upper stopband edge (Nyquist limit) for the anti-image filter at stage 1
is clearly
f
2
, since the filter operates at
f
s
samples per second. So the stopband frequency range at
stage 1 is therefore from
f
s
M
1
f
s
to
f
2
. The aliasing distortion, introduced into the frequency band
2
M
from
f
s
2
M
to
f
s
2
M
1
, will be filtered out after future decimation stage(s).
Similarly, for stage 2, the lower frequency edge of the first image developed after stage 2 down-
sampling is
M
1
M
2
f
s
f
s
f
s
2
M
2
M
¼
As is evident in our two-stage scheme, the stopband frequency range for the second anti-aliasing
filter at stage 2 should be from
f
s
2
M
to
f
s
2
M
1
.
We summarize the specifications for the two-stage decimation as follows:
Filter requirements for stage 1:
Passband frequency range
¼
0to
f
p
Stopband frequency range
¼
f
s
M
1
f
s
to
f
s
2
2
M
Passband ripple
¼ d
p
=
2, where
d
p
is the combined absolute ripple on the passband
Stopband attenuation
¼ d
s
Filter requirements for stage 2:
Passband frequency range
¼
0to
f
p
M
1
M
2
f
s
f
s
to
f
s
2
M
1
Stopband frequency range
¼
2
M
Passband ripple
¼ d
p
=
2, where
d
p
is the combined absolute ripple on the passband
Stopband attenuation
¼ d
s
Example 12.4 illustrates the two-stage decimator design.
EXAMPLE 12.4
Determine the anti-aliasing FIR filter lengths and cutoff frequencies for the two-stage decimator with the following
specifications and the block diagram in
Figure 12.16A
:
Original sampling rate: f
s
¼ 240 kHz
Audio frequency range: 0e3,400 Hz
Passband ripple: d
p
¼ 0:05 (absolute)
Stopband attenuation: d
s
¼ 0:005 (absolute)
FIR filter design using the window method
New sampling rate: f
sM
¼ 8 kHz
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