Page 355 - Fiber Optic Communications Fund

336 Fiber Optic Communications

AWGN
channel
x(t) + x(t) + n(t) r(t) = u(t) + n F (t) t = T b Threshold Decison
Tx ∑ H(ω) device
r(T b )
+
n(t)

Figure 8.1 A generalized model for the optimum binary receivers.

H() and, therefore, after passing through the filter, the signal component is
{
u (t) when bit ‘1’ is transmitted
1
u(t)= (8.5)
u (t) when bit ‘0’ is transmitted
0
with
∞
1
u (t)= ̃ x ()H() exp (−it) d,
j 2 ∫ j
−∞
j = 0, 1. (8.6)
After passing through the filter, the noise variance is given by
∞
1 N 0 2
2
= |H()| d. (8.7)
2 ∫ −∞ 2
The received signal r(t) can be written as the superposition of the signal and noise at the filter output. The
decision is based on samples of r(t):
r(t)= u(t)+ n (t), (8.8)
F
where n (t) is the noise at the filter output. To determine if the message is bit ‘0’ or bit ‘1’, the received signal
F
r(t) is sampled at intervals of T . Since the noise sample n (T ) is a Gaussian random variable with zero mean
b F b
2
and variance , the received signal sample r(T ) is a Gaussian random variable with mean u(T ) and variance
b b
2
. Its pdf is given by
] }
2
{ [
1 r − u(T )
b
p(r)= √ exp − 2 . (8.9)
2 2
Let r be the threshold. If r(T ) > r , the threshold device decides that the bit ‘1’ is transmitted. Otherwise,
T
T
b
the bit ‘0’ is transmitted. When a bit ‘1’ is transmitted, u(T )= u (T ). In this case, the conditional pdf is
1
b
b
{ [ ] }
2
1 r − u (T )
1
b
1
p(r|‘1’ sent) ≡ p (r)= √ exp − 2 . (8.10)
2 2
Fig. 8.2 shows the conditional pdf p (r). The area of the shaded region in Fig. 8.2 is the chance that the
1
received signal r(T ) < r when bit ‘1’ is transmitted. A bit error is made if the decision device chooses a bit
b T
‘0’ when a bit ‘1’ is transmitted. This happens if r(T ) < r . Therefore, the probability of mistaking a bit ‘1’
b
T
as a bit ‘0’ is the area under the curve p (r) from −∞ to r and is given by
1
T
2
{ [ ] }
1 r T r − u (T )
1
b
P(0|1)= √ ∫ exp − 2 dr. (8.11)
2 −∞ 2