Biomedical Engineering Reference
In-Depth Information
p ()
α
φ
x
(t )
.05
1
t
0
.5
1
α
10
30
50
(a)
(b)
FIGURE 5.1:
(a) PMF and (b) characteristic function magnitude for discrete RV with uniform distri-
bution over 20 points on [0
,
1].
The characteristic function can be found using the sum of a geometric series:
n
−
1
e
jat
n
e
jat
n
e
jhnt
1
−
(
e
jht
)
k
φ
x
(
t
)
=
=
e
jht
.
(5.5)
1
−
=
0
k
Simplifying with the aid of Euler's identity,
sin
b
−
a
2
1
t
exp
j
a
t
n
+
b
n
−
φ
x
(
t
)
=
n
sin
b
−
a
2
1
t
.
(5.6)
2
1
n
−
Figure 5.1 illustrates the PMF and the magnitude of the characteristic function for a discrete RV
which is uniformly distributed over 20 points on [0
,
1]. The characteristic function is plotted
=
φ
x
(
t
) and that
over [0
,π
/
h
], where the span
h
=
1
/
19. Recall from Section 3.3 that
φ
x
(
−
t
)
φ
x
(
t
) is periodic in
t
with period 2
π
/
h
. Thus, Figure 5.1 illustrates one-half period of
|
φ
x
(
·
)
|
.
Definition 5.1.2.
The continuous RV x has a
uniform distribution
on the interval
[
a
,
b
]
if x has
PDF
1
/
(
b
−
a
)
,
a
≤
α
≤
b
f
x
(
α
)
=
(5.7)
0
,
otherwise
.
The mean and variance of a continuous uniform RV are easily computed directly:
b
b
2
a
2
1
−
b
+
a
η
=
α
d
α
=
a
)
=
,
(5.8)
x
b
−
a
2(
b
−
2
a
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