Page 367 - Fiber Optic Communications Fund

348 Fiber Optic Communications

Using Eq. (8.93) in Eq. (8.39), we obtain
1 √
P = erfc( homo ), (8.96)
b
2
E av
homo = homo . (8.97)
N
0
The parameter homo represents the normalized energy per bit, which serves as a figure of merit in digital
communication.
When homo is much larger than unity, Eq. (8.96) can be approximated as
exp (− homo )
P ≅ . (8.98)
b √
2 homo

8.3.1.1 Relation between Q-factor and BER

The BER and Q-factor are related as follows. From Eq. (7.8), we have
I − I 0
1
Q = . (8.99)
+ 0
1
For a PSK signal, I =−I and = . So,
1
0
0
1
I 1
Q = . (8.100)

1
Suppose that the correlator shown in Fig. 8.8 is used as the matched filter. The mean of bit ‘1’ after the
correlator is (after setting the scaling factor 2RA to unity in Eq. (8.83))
LO
T b
2
I = s (t) dt
1 ∫ 1
0
= E . (8.101)
av
In this case, we have
h(t)= s (T − t), (8.102)
1 b
∗
H()= s () exp (−iT ). (8.103)
1 b
The variance of bit ‘1’ (or bit ‘0’) after the correlator is (see Eq. (8.7))
N homo 1 ∞ N 0 homo T b N homo
0
0
2
2
2
= |H()| d = s (t)dt = E . (8.104)
1 2 ∫ ∫ 1 av
2 −∞ 2 0 2
Here, we have used Parseval’s relations. Substituting Eqs. (8.101) and (8.104) in Eq. (8.100), we find
√
2E av
Q = . (8.105)
N homo
0
From Eqs. (8.97) and (8.96), we have
Q 2
homo
= , (8.106)
2
( )
1 Q
P = erfc √ . (8.107)
b
2 2
Eq. (8.107) holds true even when the matched filter is not used (see Section 8.2.2).