Digital Signal Processing Reference
In-Depth Information
EXðtfg¼ m ¼
0
;
(6.148)
and that the process is a WS stationary.
The condition necessary for the process to be a mean ergodic (
6.146
) is:
T=
2
ð
T=
ð
2
1
T
1
T
e
2
ljtj
d
t
lim
T!1
R
XX
ðtÞ
d
t ¼
lim
T!1
T=
2
T=
2
2
3
ð
T=
2
ð
0
1
T
4
5
e
2
lt
d
tþ
e
2
lt
d
t
¼
lim
T!1
T=
2
0
1
e
lT
2
lT
1
e
lT
2
lT
e
lT
lT
¼
lim
T!1
þ
¼
lim
T!1
l
e
lT
l
¼
0
¼
lim
T!1
:
(6.149)
From (
6.148
)to(
6.149
), we conclude that the condition for a process to be a
mean ergodic is satisfied. Therefore, we have:
T=
ð
2
1
1
T
lim
T!1
xðtÞ
d
t ¼
xf
X
ðxÞ
d
x ¼ m
X
¼
0
:
(6.150)
1
T=
2
6.7.5 Autocorrelation and Cross-Correlation Ergodic Processes
A time-averaged autocorrelation function is obtained by applying a time average on
a particular realization
x
(
t
) of a stationary process
X
(
t
):
T=
2
ð
1
T
AXðtÞXðt þ tÞ
f
g ¼ R
xx
ðtÞ¼
lim
T!1
xðtÞxðt þ tÞ
d
t:
(6.151)
T=
2
For a particular realization of the process, the obtained result is a deterministic
function of
t
which varies from one realization to another and presents a random
variable.
The condition for ergodicity is that those variations approach zero, i.e., the
variance of the random variable
R
XX
(
t
) must approach zero
.
s
2
lim
T!1
R
XX
¼
0
:
(6.152)
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