Digital Signal Processing Reference
In-Depth Information
The mean value is zero and, thus constant. Therefore, the first condition (
6.32
)is
satisfied.
Next we must verify whether or not the autocorrelation function depends only on
t
,
R
XX
ðt; t þ tÞ¼XðtÞXðt þ tÞ
¼ ðX
1
cos
o
0
t þ X
2
sin
o
0
tÞ X
1
cos
o
0
ðt þ tÞþX
2
sin
o
0
ðt þ tÞ
ð
Þ
¼ X
1
cos
o
0
t
cos
o
0
ðt þ tÞþX
2
sin
o
0
t
sin
o
0
ðt þ tÞ
þ X
1
X
2
cos
o
0
t
sin
o
0
ðt þ tÞþX
1
X
2
sin
o
0
t
cos
o
0
ðt þ tÞ:
(6.37)
Knowing that variables
X
1
and
X
2
are uncorrelated, and using (
6.35
) we get:
X
1
X
2
¼ X
1
X
2
¼
0
:
(6.38)
From (
6.35
), we have:
X
1
¼ s
1
¼ s
2
;
(6.39)
X
2
¼ s
2
¼ s
2
:
Finally, from (
6.37
)to(
6.39
), we arrive at:
R
XX
ðt; t þ tÞ¼s
2
cos
o
0
t
cos
o
0
ðt þ tÞþ
sin
o
0
t
sin
o
0
ðt þ tÞ
½
¼ s
2
cos
o
0
t:
(6.40)
The obtained result shows that the second condition (
6.33
) is satisfied as well,
indicating that the process is a WS stationary process.
6.5.3 What Does Autocorrelation Function Tell Us
and Why Do We Need It?
From the definition of an autocorrelation function, we know that an autocorrelation
function is a statistical characteristic of a process that is obtained by observing
the process in two points, the same as a joint density function of a second order.
The question which arises is: “why do we need an autocorrelation function if we
already have the second order joint density function?”
In order to answer this question, the following discussion is in order.
Consider two processes
X
(
t
) and
Y
(
t
), shown in Fig.
6.7
.
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