Digital Signal Processing Reference
In-Depth Information
The Hamming window is selected, since it provides 0.019 dB ripple and 53 dB stopband attenuation. The
normalized transition band is given by
f
stop
f
pass
f
s
¼
1;000 800
6;000
Df ¼
¼ 0:033
The length of the filter and the cutoff frequency can be determined by
N ¼
3:3
3:3
0:033
¼ 100
Df
¼
We choose an odd number; that is, N ¼ 101, and
f
pass
þ
f
stop
2
¼
800 þ 1;000
2
f
c
¼
¼ 900 Hz
12.1.2
Sampling Rate Increase by an Integer Factor
Increasing a sampling rate is a process of upsampling by an integer factor of
L
. This process is
described as follows:
m
L
8
<
x
m ¼ nL
y
m
¼
(12.9)
:
otherwise
0
where
n ¼
0
;
1
;
2
;
/
; xðnÞ
is the sequence to be upsampled by a factor of
L
, and
yðmÞ
is the upsampled
sequence. As an example, suppose that the data sequence is given as follows:
xðnÞ
:
884
5
6
.
After upsampling the data sequence
xðnÞ
by a factor of 3 (adding
L-
1 zeros for each sample), we have
the upsampled data sequence
wðmÞ
as
wðmÞ
:
800 800 400
500
600
.
The next step is to smooth the upsampled data sequence via an interpolation filter. The process is
illustrated in
Figure 12.5A
.
Similar to the downsampling case, assuming that the data sequence has the current sampling period
of
T
, the Nyquist frequency is given by
f
max
¼ f
s
=
2. After usampling by a factor of
L
, the new
sampling period becomes
T=L
, thus the new sampling frequency is changed to be
f
sL
¼ Lf
s
(12.10)
This indicates that after upsampling, the spectral replicas originally centered at
f
s
,
2
f
s
,
.
are
included in the frequency range from 0 Hz to the new Nyquist limit
Lf
s
=
2 Hz, as shown in
Figure 12.5B
.
To remove those included spectral replicas, an interpolation filter with a stop frequency
edge of
f
s
=
2 in Hz must be attached, and the normalized stop frequency edge is given by
f
2
T
L
¼
p
L
U
stop
¼
2
p
radians
(12.11)
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