Page 390 - Fiber Optic Communications Fund

Performance Analysis 371

Combining Eqs. (8.267) and (8.273), we obtain

1
P = [P(1|0)+ P(0|1)]
b
2
√
⎧ ⎫
[ DD ( )] ⎛ ( ) ⎞
1 ⎪ 2 √ DD DD 2 ⎪
= exp − 1 + + 1 − Q 1 ⎜ 2 , 1 + ⎟ . (8.274)
2 ⎨ 2 DD ⎜ DD ⎟ ⎬
⎪ ⎪
⎝ ⎠
⎩ ⎭
Note that Eq. (8.274) is the same as Eq. (8.230) obtained for the case of a heterodyne receiver if we replace
het by 2 DD . In this analysis, we have ignored the receiver noise mechanisms such as shot noise and thermal
noise and assumed that the optical filter is a matched filter. Without these approximations and assumptions,
the analysis is quite cumbersome. When the optical filter is not matched to the transmitted signal, analyti-
cal expressions can be obtained using the approaches in Refs. [7]–[9]. In a simplified approach, chi-square
distributions are approximated by Gaussian distributions and the BER can be estimated by calculating the
Q-factor as in Chapter 7. This Gaussian approximation gives reasonably accurate results for OOK, although
it is found to be inaccurate for DPSK signals with direct detection [7].

8.5.2 FSK

For FSK with direct detection, the transmitted signals s (t) and s (t) are the same as those in Section 8.4.3.
0
1
(Eq. (8.166)). Since the energies of signals s (t) and s (t) are equal, we use the matched filters shown in
1
0
Fig. 8.21 (similar to Fig. 8.7). The matched filters can be realized as a bank of band-pass filters. As before,
the matched filters need not be synchronized with the received signal, but can differ by a phase factor .The
signal field u (t) and noise field n (t) at the output of the matched filters are given by
j
F j
∞
1 −it
u (t)= ̃ x()H ()e d, j = 0, 1, (8.275)
j 2 ∫ −∞ j
∞
1 −it
n (t)= ̃ n ()H ()e d, j = 0, 1. (8.276)
Fj 2 ∫ −∞ c j

u 1 (t) + n F1 (t)
t = T b
PD1
H 1 (ω) 2
x(t) = |∙| I (T b )
1
s(t)e *iω c t If I 1 (T b ) > I 0 (T b )
Tx ∑ Comparator select ‘1’,
+
otherwise,
+ t = T b select ‘0’.
H 0 (ω) PD0
n (t)e *iω c t |∙| 2 I 0 (T b )
c
u (t) + n F0 (t)
0
Figure 8.21 Direct detection receiver for FSK.