Page 99 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.13

where U (X) is the unit step function. Note for a > 0 that only one interval of the real
line exists, where g(x) ≤ y and that is x ∈ [−∞, y/a).

Step 2:
1
⎧ y
⎪ a p X ( ) if a > 0
a
⎨
p Y (y) = δ(y) if a = 0
⎪
⎩ 1 y
− p X ( ) if a < 0.
a a
EXAMPLE 3.13
2
Y = X (the output of a square law or power detector with an input random voltage X).
Step 1:
⎧ √
y
√ p X (β)dβ
− y
⎨ if , y ≥ 0
F Y (y) =
0 if y < 0
⎩
Step 2:
⎧
1 √ 1 √
⎨ √ p X ( y) + √ p X (− y) if y ≥ 0
2 y 2 y
p Y (y) =
0 if y < 0
⎩

EXAMPLE 3.14
Y = XU (X). This example corresponds to the output of an ideal diode with an input
random voltage X.

Step 1:
y

p X (β)dβ if y ≥ 0
−∞
F Y (y) =
0 if y < 0
y
= U (y) p X (β)dβ
−∞
= U (y)F X (y) (3.21)

Step 2:

p Y (y) = U (y) p X (y) + δ(y)F X (0)