Digital Signal Processing Reference
In-Depth Information
p
(
t
)
x
s
(
t
)
x
(
t
)
X
x
(
t
)
t
0
p(t)
(t)
(t−1)
δ
δ
t
−T
s
T
s
2T
s
0
x
s
(
t
)
t
−T
s
T
s
2T
s
0
x(n)
o
o
o
o
o
o
o
x(2)
o
o
o
o
n
−1
0
1
2
Fig. D.8
Sampling process in the time-domain
through an analog low-pass filter on the condition that the sampling frequency is
more than twice the signal bandwidth B, i.e.,
B
f
s
2
:
The sampling rate f
s
= 2B is called the Nyquist rate
;
which is the minimum rate
for safe sampling. If B [ f
s
/2, replicas of the signal spectrum X(f) that constitute
the DTFT will overlap, and part of the information is damaged. This overlap is
called frequency aliasing. Hence, in practice, we normally bandlimit the signal
before sampling to remove the unimportant frequency content that may cause
aliasing. This can be achieved using a LPF (which is called anti-aliasing filter). If
the sampling frequency is near the Nyquist rate, the anti-aliasing filter should have
a sharp cutoff at f = B, otherwise aliasing may occur. If f
s
2B, we can relax this
condition on the anti-aliasing filter. This is important in hardware implementation
to reduce the LPF complexity.
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