Page 483 - Fiber Optic Communications Fund

464 Fiber Optic Communications

where I is the mean current given by
0
{ ( )}
∞
√ ∑
I = P Re 1 + u + 2 u (10.286)
0 0 000 0mm
m=−∞
and
∞
∑ ∑
I = b u lin,n + Re(u lmn )b b b , (10.287)
n
l m n
n=−∞,n≠0 l+m−n=0,l≠0,m≠0
where
√ 2 2 2
u lin,n = P exp (−n T ∕2T ). (10.288)
0
s
0
For BPSK, we have
⟨b ⟩ = 0, (10.289)
n
⟨b b ⟩ = , (10.290)
n m nm
where is the Kronecker delta function. To calculate the variance, Eq. (10.287) is rewritten as
nm
I = I + I + I , (10.291)
lin IFWM,d IFWM,nd
where I , I IFWM,d, and I IFWM,nd represent random currents due to linear ISI, degenerate IFWM, and
lin
non-degenerate IFWM, respectively. An IFWM triplet is degenerate if l = m. From Eq. (10.287), we have
∞
∑
I lin = b u lin,n , (10.292)
n
n=−∞,n≠0
∑
I IFWM,d = Re{u }b , (10.293)
n
lln
l+m−n=0,l=m
∑
I IFWM,nd = 2 Re{u lmn }b b b . (10.294)
l m n
l+m−n=0,l<m,l≠m≠n
The factor 2 is introduced to account for the fact that the summation is carried out only over the region of
l < m. In Eqs. (10.293) and (10.294), the terms corresponding to intra-pulse SPM and IXPM are excluded.
The variance of I lin is ∑ ∑
<I 2 >= < b b >u lin,m u lin,n , (10.295)
m n
lin
m≠0 n≠0
Using Eq. (10.290), Eq. (10.295) simplifies to
( )
2 2
∑ 2 ∑ −m T s
2
<I >= u = P exp , (10.296)
lin lin,m 0 2
m≠0 m≠0 T 0
∑ ∑
2
2
<I > = Re[u ]Re[u ′ l ′ n ′] < b b ′ >
IFWM,d lln l n n
l+m−n=0,l=m l ′ +m ′ −n ′ =0,l ′ =m ′
∑
2 2
= (Re[u ]) . (10.297)
lln
l+m−n=0,l=m
′
′
In Eq. (10.297), we have used Eq. (10.290) and when n = n , l has to be equal to l to satisfy l + m − n = 0
′
′
′
and l + m − n = 0.