B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

426 FUNDAMENTALS OF PROBABILITY THEORY
A Rayleigh RV can be derived from two independent Gaussian RVs as follows. Let x and
y be independent Gaussian variables with identical PDFs:
Then
1 2;2 2
Px (x) = -- e-x a
a,Jiii
1 2;2 2
p y
(y) = -- e-y a
a,Jiii
Px y
(x, y) = Px (x)p y (y) = -
2
1
JT(J
2 e- ( x
2 +y
2 )/2 a 2
(8.56)
The joint density appears somewhat like the bell-shaped smface shown in Fig. 8.13. The
points in the (x, y) plane can also be described in polar coordinates as (r, 0), where
(Fig. 8.14a)
r = Jx 2 + y 2 E>=tan- 1 r
X
In Fig. 8.14a, the shaded region represents r < r ::: r + dr and 0 < E> ::: 0 + d0 (where
dr and d0 both --+ 0). Hence, if Pre (r, 0) is the joint PDF of r and 0), then by definition
[Eq. (8.45)), the probability of observing rand 0 in this region is Pre (r, 0) dr d0. But we
also know that this probability is Pxy
(x, y) times the area r dr d 0 of the shaded region.
Hence, [Eq. (8.56)]
and
and [Eq. (8.48a)]
1 ( 2 2 )2 2
-- 2
e-x +y a rdrd0 = Pre(r, 0)drd0
2rra
r ( 2 ,2 i;2 2
P re (r 0) --e- x +y a
' - 2rra 2
= _r _ e- ,2;2 a 2
2rra 2
Pr (r) = 1_: Pre (r, 0) d0
(8.57)
Because E> exists only in the region (0, 2rr),
2
1
rr r 2 2
0 2rra 2
Pr (r) = --e-r f 2 a d0
r 2 1
2
2
= -e-r a u(r)
a 2
Note that r is always greater than 0. In a similar way, we find
(8.58a)
Pe (0) =
l
2rr
1
0
0 ::: E> < 2rr
otherwise
(8.58b)