Digital Signal Processing Reference
In-Depth Information
2.3.1 Functions of a Random Variable
Let us examine a few important relationships between functions of random
variables. Most of the noise propagation in linear systems can be best understood
by applying these golden rules partially or in full:
1. Consider a simple relation
z ¼ x þ y
, then the pdf of
z
is obtained by convolving
the
pdf
of
x
and
y
, f
z
ð
z
Þ¼
f
x
ð
x
Þ
f
y
ð
y
Þ
.
2. For the relation
y ¼
m
x þ
c, the
pdf
of
y
is f
y
ð
y
Þ¼
1
x
c
m
Þ
.
3. The central limit theorem says that if
x
i
is a random variable of any distribution,
m
f
x
ð
then the random variable
y
defined as
y ¼
P
i
¼
1
ðx
i
Þ
has a normal distribution,
where N is a large number. As an engineering approximation, N
12 will suffice.
2.3.1.1 Hit Distance of a Projectile
This problem comes under non-linear functions of a random variable. To illustrate
the idea, consider a ballistic object moving under the gravitational field shown in
Figure 2.8 with a gravitational constant g
¼
9
s
2
:
:
81 m
=
v
x
¼
v
cos
and
v
y
¼
v
sin
gt
;
ð
2
:
26
Þ
1
2
gt
2
x
¼
v
t cos
and
y
¼
v
t sin
:
ð
2
:
27
Þ
Figure 2.8 Ballistic motion
Setting y
¼
0 we get t
¼ð
2
v
=
g
Þ
sin
for t
6¼
0
;
then the horizontal distance
where it hits the ground is
2
2
sin
2 sin
cos
sin 2
v
¼
v
2
x
h
¼
v
cos
¼
v
:
g
g
g
Consider
as statistically varying with
v
¼
50 and
v
¼
0
:
2 and
¼ =
3 and
v
¼ =
30 We have derived a closed-form solution (Figure 2.9) for the above pdf in
the interval 85
<
x
<
250:
:
61x
5
10
12
þ
:
73x
4
10
9
:
66x
3
10
6
:
34x
2
10
4
7
5
1
2
f
ð
x
Þ¼
þ
0
:
016x
þ
0
:
4259
:
ð
2
:
28
Þ
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