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

420 FUNDAMENTALS OF PROBABILITY THEORY
Fi g ure 8.1 1
p (x)
Gaussian PDF
, t
with mean m and
variance a 2 .
m
m + 4a
X
Letting (x - m)/a = z,
Therefore,
and
I
f (x-m )/ a 2
Fx (x) =
r,:,-=
e -z l 2 dz
-v 2rr -oo
(x-m)
= 1 -Q - a-
(x-m)
P(x :S x) = I - Q - a
-
(x- m)
P(x > x) = Q - a
-
(8.40a)
(8.40b)
(8.40c)
The Gaussian PDF is perhaps the most important PDF in the field of communications. The
majority of the noise processes observed in practice are Gaussian. The amplitude n of a Gaussian
noise signal is an RV with a Gaussian PDF. This means the probability of observing n in an
interval (n, n + !::,.n) is P n (n) l::,.n, where P n (n) is of the form in Eq. (8.39) [with m = O].
Example 8.16 Threshold Detection
Over a certain binary channel, messages m= 0 and 1 are transmitted with equal probability by
using a positive and a negative pulse, respectively. The received pulse corresponding to 1 is
p(t), shown in Fig. 8.12a, and the received pulse corresponding to 0 is -p(t). Let the peak
amplitude of p(t) be Ap at t = T p
. Because of the channel noise n(t), the received pulses will
be (Fig. 8.12c)
±p(t) + n(t)
To detect the pulses at the receiver, each pulse is sampled at its peak amplitude. In the absence
of noise, the sampler output is either A p
(for m=l) or -A p
(for m=0). Because of the channel
noise, the sampler output is ±A p + n, where n, the noise amplitude at the sampling instant
(Fig. 8.12b), is an RV. For Gaussian noise, the PDF of n is (Fig. 8.12b)
1 2 ;2 2
P n (n) = ---e - n
an
an../iii
(8.41)