Digital Signal Processing Reference
In-Depth Information
We choose the Hamming window for this application. The normalized transition band is
f
stop
f
pass
f
sL
¼
3;000 800
18;000
Df ¼
¼ 0:1222
The length of the filter and the cutoff frequency can be determined by
N ¼
3:3
3:3
0:1222
¼ 27
Df
¼
and the cutoff frequency is given by
f
pass
þ
f
stop
2
¼
3;000 þ 800
2
f
c
¼
¼ 1;900 Hz:
12.1.3
Changing the Sampling Rate by a Noninteger Factor L/M
With an understanding of the downsampling and upsampling processes, we now study sampling rate
conversion by a noninteger
L=M
. This can be viewed as two sampling conversion processes. In step 1,
we perform the upsampling process by a factor of integer
L
following application of an interpolation
filter
H
1
ðzÞ
; in step 2, we continue filtering the output from the interpolation filter via an anti-aliasing
filter
H
2
ðzÞ
, and finally execute downsampling. The entire process is illustrated in
Figure 12.8
.
Since the interpolation and anti-aliasing filters are in a cascaded form and operate at the same rate,
we can select one of them. We choose the one with the lower stop frequency edge and choose the most
demanding requirements for passband gain and stopband attenuation for the filter design. A lot of
computational savings can be achieved by using one lowpass filter. We illustrate the procedure via the
following simulation. Let us generate the signal
xðnÞ
by
x
n
¼
5 sin
2
p
1
;
000
n
8
;
000
þ
cos
2
p
2
;
500
n
8
;
000
with a sampling rate of
f
s
¼
8
;
000 Hz and frequencies of 1 kHz and 2.5 kHz. Now we resample
xðnÞ
to
3,000 Hz by a noninteger factor of 0.375, that is,
L
M
3
8
¼
0
:
375
¼
Upsampling is at a factor of
L ¼
3 and the upsampled sequence is filtered by an FIR lowpass filter
designed with a filter length
N ¼
53 and a cutoff frequency of 3,250 Hz at a sampling rate of 3
8,000
¼
24,000 Hz. The spectrum for the upsampled sequence and the spectrum after application of the
interpolation filter are plotted in
Figure 12.9A
.
xn
()
ym
()
Interpolation
filter H1(z)
Anti-aliasing
filter H2(z)
L
M
H
(
z)
FIGURE 12.8
Block diagram for sampling rate conversion.
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