Page 402 - Fiber Optic Communications Fund

Performance Analysis 383

1
2 2 2
⟨|̃n ()| ⟩ = {⟨|̃n ( + )| ⟩ + ⟨|̃n ( − )| ⟩
1
IF
IF
cI
cI
4
∗
+⟨̃n ( + )̃n ( − )+ c.c.⟩}. (8.344)
IF
IF
cI
cI
Typically, an optical filter of bandwidth B is introduced before the photodetectors. B is greater than the signal
o
o
bandwidth, but much smaller than ∕2. In this case, n (t) and n (t) are band-limited to B ∕2 and the
IF
o
cI
cQ
spectra due to ̃n ( + ) and ̃n ( − ) do not overlap. Hence, the last two terms on the right-hand side
IF
IF
cI
cI
2
of Eq. (8.344) are zero. Since the PSD of n (t) is proportional to ⟨|̃n ()| ⟩, from Eq. (8.344) and Fig. 8.31(b),
1
1
we find
ASE
(f)= for |f − f | < B ∕2
o
IF
8
n 1
ASE
= for |f + f | < B ∕2
o
IF
8
= 0 elsewhere. (8.345)
Let
n (t)= n sin ( t +Δ). (8.346)
2 cQ IF
Proceeding as before, we find
ASE
(f)= for |f − f | < B ∕2
IF
o
8
n 2
ASE
= for |f + f | < B ∕2
IF
o
8
= 0 elsewhere. (8.347)
ρ
n cI
ρ ASE /2
f
B o
(a)
ρ n 1
ρ n cI ( f + f IF )/4 ρ n cI ( f * f IF )/4
ρ ASE /8
* f IF f IF f
B o B o
(b)
Figure 8.31 ASE PSD: (a) base band; (b) after multiplication by cos( t).
IF