Graphics Programs Reference
In-Depth Information
∞
∑
j
2π
nt
T
s
--------------
x
s
()
=
x
()
P
n
e
(13.99)
n
=
∞
Taking the FT of Eq. (13.99) yields
∞
∑
∞
∑
2π
n
T
s
2π
n
T
s
X
s
()
=
P
n
X
ω
----------
=
P
0
X
()
+
P
n
X
ω
----------
(13.100)
n
=
∞
n
=
∞
n
≠
0
where
X
()
is the FT of
x
()
. Therefore, we conclude that the spectral den-
sity,
X
s
()
, consists of replicas of
X
()
spaced
(
2π
T
s
⁄
)
apart and scaled by
the Fourier series coefficients
P
n
. A Low Pass Filter (LPF) of bandwidth
B
can then be used to recover the original signal
x
()
.
P
0
x
()
x
()
x
s
()
LPF
X
() 0
=
for
ω 2π
B
>
p
()
Figure 13.1. Concept of sampling.
When the sampling rate is increased (i.e., decreases), the replicas of
move farther apart from each other. Alternatively, when the sampling
rate is decreased (i.e., increases), the replicas get closer to one another. The
value of such that the replicas are tangent to one another defines the mini-
mum required sampling rate so that
T
s
X
()
T
s
T
s
x
()
can be recovered from its samples by
using an LPF. It follows that
2π
T
s
1
2
B
------
=
2 π 2()
⇔
T
s
=
-------
(13.101)
The sampling rate defined by Eq. (13.101) is known as the Nyquist sampling
rate. When , the replicas of overlap and, thus, cannot
be recovered cleanly from its samples. This is known as aliasing. In practice,
ideal LPF cannot be implemented; hence, practical systems tend to over-sam-
ple in order to avoid aliasing.
T
s
>
(
12
B
⁄
)
X
()
x
()
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