Digital Signal Processing Reference
In-Depth Information
Stage1
Stage2
x
()
y
()
Anti-aliasing
filter H1(z)
Anti-aliasing
filter H2(z)
M
1
M
2
f
s
f
MM
f
M
s
s
f
M
s
12
1
FIGURE 12.14
Multistage decimation.
Anti-aliasing filter at
stage 1 operating at
the sampling rate of f
s
Removed by H1(z)
Image after
decimation at
stage 1
Image before
decimation at
stage 1
Removed in
future stage(s)
Prevented
by H1(z)
f
M
f
s
/2
s
2
Final stop band
frequency edge
f
M
f
M
f
M
s
s
s
f
s
2
1
1
Stop band frequency
edge at stage 1
FIGURE 12.15
Stopband frequency edge for the anti-aliasing filter at stage 1 for two-stage decimation.
2
M
after final down-
f
s
2
M
sampling. So our useful information bandwidth should stop at the frequency edge of
. Next, we
need to determine the stop frequency edge for the anti-aliasing lowpass filter at stage 1 before the first
decimation process begins. This stop frequency edge is actually the lower frequency edge of the first
f
s
M
1
image replica centered at the sampling frequency of
after the stage 1 decimation. This lower
frequency edge of the first image replica is then determined by
f
s
M
1
f
s
2
M
After downsampling, we expect that some frequency components from
f
s
2
M
1
to
f
s
M
1
f
s
to be folded
2
M
over to the frequency band between
f
s
2
M
and
f
s
2
M
1
. However, these aliased frequency components do not
affect the final useful band between 0 Hz to
f
s
2
M
and will be removed by the anti-aliasing filter(s) in the
M
1
f
s
2
M
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