Graphics Programs Reference
In-Depth Information
EX
()
[
]
=
X
(13.93)
ℜ
X
()
EX
()
Xt
τ
=
[
(
+
)
]
If the time average and time correlation functions are equal to the statistical
average and statistical correlation functions, the random process is referred to
as an ergodic random process. The following is true for an ergodic process:
T
⁄
∫
2
1
---
lim
x
()
t
d
=
EX
()
[
]
=
X
(13.94)
T
→
∞
T
⁄
2
T
⁄
∫
2
1
---
x
∗
()
xt
τ
lim
(
+
)
t
d
=
ℜ
X
()
(13.95)
T
→
∞
T
⁄
2
The covariance of two random processes
X
()
and
Y
()
is defined by
C
XY
(
tt
τ
,
+
)
=
EX
()
EX
()
[
{
[
]
}
Yt
τ
{
(
+
)
EYt
τ
[
(
+
)
]
}
]
(13.96)
which can be written as
C
XY
tt
τ
(
,
+
)
XY
()
XY
=
(13.97)
13.9. Sampling Theorem
Most modern communication and radar systems are designed to process dis-
crete samples of signals bearing information. In general, we would like to
determine the necessary condition such that a signal can be fully reconstructed
from its samples by filtering, or data processing in general. The answer to this
question lies in the sampling theorem which may be stated as follows: let the
signal be real-valued and band-limited with bandwidth ; this signal can
be fully reconstructed from its samples if the time interval between samples is
no greater than
x
()
B
12()
⁄
.
Fig. 13.1
illustrates the sampling process concept. The sampling signal
p
()
is periodic with period
T
s
p
()
, which is called the sampling interval. The Fourier
series expansion of
is
∞
∑
j
2π
nt
T
s
--------------
p
()
=
P
n
e
(13.98)
n
=
∞
The sampled signal
x
s
()
is then given by
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