Page 366 - Fiber Optic Communications Fund

Performance Analysis 347

where
n (t)
d
n(t)= n (t)+ (8.84)
cr
2RA
LO
is a white noise process with Gaussian distribution. Its PSD is
N homo ASE shot,eff
0
= = + (8.85)
n
2 2 4A 2 R 2
LO
ASE q
= + . (8.86)
2 4R
The factor 1∕2 is introduced in the first term of Eq. (8.85) since the PSD of the real part of n (t) is half of that
c
of n (t). The scaling factor 2RA LO appearing in Eq. (8.83) multiplies both signal and noise and, hence, it is of
c
no consequence in evaluating the performance. Dropping this term, we write the normalized signal used for
decision as
I = s(t)+ n(t). (8.87)
d
8.3.1 PSK: Homodyne Detection

The optical field envelope may be written as
{
s (t) for bit ‘1’
s(t)= 1 (8.88)
s (t)=−s (t) for bit ‘0’.
0 1
We assume that s(t) is real and the filter shown in Fig. 8.6 is matched to s(t). Replacing x(t) by s(t) in Eq. (8.36),
we obtain
T b
2
E = s (t)dt = E , (8.89)
1 ∫ 1 0
0
T b
2
E =− ∫ s (t)dt =−E , (8.90)
1
10
1
0
2 8E 1
max = (E + E − 2E )= . (8.91)
10
1
0
N homo N homo
0 0
Since the bits ‘1’ and ‘0’ are equally probable, the average energy transmitted is
E + E 0
1
E = = E . (8.92)
1
av
2
The average energy forms a basis for comparison of various modulation formats and Eq. (8.91) can be written
as
8E av
max = homo . (8.93)
N
0
The matched filter is given by (Eq. (8.40))
∗ ∗ ∗
H()=[̃s ()− ̃s ()] exp (−iT )= 2̃s () exp (−iT ), (8.94)
1 0 b 1 b
and the threshold r is (Eq. (8.47))
T
E − E 0
1
r = = 0. (8.95)
T
2