Digital Signal Processing Reference
In-Depth Information
Exercise M.2.7
(MATLAB file:
exercise_M_2_7.m
). Generate the uniform random
variables
X
1
and
X
2
having the same mean values
m
1
¼ m
2
¼
3 and variances
s
2
1
=
1/12,
s
2
2
¼
1/48. Estimate the corresponding PDFs.
Solution
From (
2.349
), it follows:
2
2
D
1
=
12
¼
4
D
2
=
12
(2.559)
resulting in:
D
1
¼
2
D
2
¼
1
; D
2
¼
0
:
5
:
(2.560)
The range of the variable
X
1
is:
½
3
0
:
5
;
3
þ
0
:
5
¼½
2
:
5
;
3
:
5
:
(2.561)
Similarly, the random variable
X
2
has the range:
½
3
0
:
25
;
3
þ
0
:
25
¼½
2
:
75
;
3
:
25
:
(2.562)
The generated variables and estimated PDFs are shown in Fig.
2.79
.
a
b
4
2.5
2
3.5
1.5
3
1
2.5
0.5
2
0
100
200 300
400
500
600
700
800
900 1000
2.4
2.6
2.8
3
3.2
3.4
3.6
n
cells
4
3.5
2.5
2
1.5
3
1
2.5
0.5
2
0
100
200 300
400
500
600
700
800
900 1000
2.4
2.6
2.8
3
3.2
3.4
3.6
n
cells
Fig. 2.79
Example of uniform variables with the same mean value and
s
1
2
¼
1/12 and
s
2
2
¼
1/48
.
(
a
) Signals
X
1
and
X
2
.(
b
) Estimated PDFs
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