Page 380 - Fiber Optic Communications Fund

Performance Analysis 361

where n and n dQ are the in-phase and quadrature components of the detector noise. Substituting Eq. (8.195)
dI
in Eq. (8.194), we find
n(t)= n cos ( t +Δ)+ n sin ( t +Δ), (8.196)
IF
Q
IF
I
where
n = n + n ∕(2RA ), (8.197)
I cI dI LO
n = n cQ + n ∕(2RA LO ). (8.198)
Q
dQ
We assume that n(t) is a narrow-band Gaussian noise process with zero mean and it is band-limited to the fre-
quency interval f − B ≤ |f| ≤ f + B. First consider n cos ( t +Δ). After passing through the matched
IF
IF
IF
I
filter, it becomes (see Eq. (8.188))
1 T b n ()s (T + − t){cos [ (t − T )+Δ]+ cos ( (2 + T − t)+Δ)} d. (8.199)
2 ∫ I 1 b IF b IF b
0
As before, the second term on the right-hand side can be ignored. So, it becomes
n cos [ (t − T )+Δ], (8.200)
FI
IF
b
where
1 T b
n FI = 2 ∫ 0 n ()s (T + − t) d. (8.201)
I
1
b
Similarly, the second term of Eq. (8.196) becomes
n sin [ (t − T )+Δ], (8.202)
FQ IF b
where
1 T b
n = n ()s (T + − t) d. (8.203)
FQ 2 ∫ 0 Q 1 b
Combining Eqs. (8.200) and (8.202), the noise output of the matched filter is
n (t)=[n cos ( (t − T )+Δ)+ n FQ sin ( (t − T )+Δ)], (8.204)
b
FI
F
b
IF
IF
het
where n (t) and n FQ (t) are the in-phase and quadrature components of n (t). The PSD of n(t) is N ∕2. From
F
FI
0
Eq. (8.7), we have
2 2
= < n >
F F
N 0 het 1 ∞
2
= |H ()| d (8.205)
2 2 ∫ −∞ I
N 0 het ∞
2
2
= [|̃s ( − )| + |̃s ( + )| d]. (8.206)
1
1
IF
16 ∫ −∞
IF
∗
In Eq. (8.205) we ignore cross-products such as ̃s ( − )s ( + ). This is because ̃s ( − ) and
IF
1
1
IF
IF
1
∗
̃ s ( + ) represent frequency components centered around and − , respectively. If the spectral width
IF
IF
IF
1
of s (t) is smaller than , these frequency components do not overlap. Noting that the contributions from
1
IF
the first and second terms on the right-hand side of Eq. (8.206) are the same, we find
het
N 0 het ∞ N E 1
0
2
2
= |̃s ()| d = . (8.207)
F 8 ∫ −∞ 1 4