Page 518 - Fiber Optic Communications Fund

Digital Signal Processing 499

is a Wiener process. If the interval (t − t ) is sufficiently small, the integration can be replaced by the rectan-
0
gular rule. With (t − t )= T samp ,
0
(t)= (t − T samp )− 2f (t − T samp )T samp . (11.13)
i
After discretization, Eq. (11.13) becomes

= l−1 − 2f i,l−1 samp , (11.14)
T
l
where t = lT samp . The phase noise can be interpreted as a one-dimensional random walk [2]. As an example,
consider a drunken man walking randomly on the road. Suppose that at every step there is a 50% chance
that he moves either forward or backward. After two steps, there is a 25% chance that he has moved two
steps forward, a 25% chance that he has moved two steps backward, and 50% chance that he is at his initial
position. After many steps, the mean distance traversed would be close to zero and there would be a large
number of different paths he could have traversed. Since the chance of moving forward or backward at a given
step is independent of the decision at the previous steps, the variance of distance traversed is proportional to
the number of steps. Similarly, in the case of laser phase noise, the phase of the sample n is incremented by
f
−2T samp i,l−1 , where f i,l−1 is a value of instantaneous frequency picked from the Gaussian distribution. From
Eq. (11.14), we have
l−1
∑
≡ (l)− (0)=−2T samp f . (11.15)
i,m
l
m=0
Squaring Eq. (11.15), averaging, and noting that the frequency deviations at each step are independent, we
find
2
2 2
2
< >= 4 T l . (11.16)
l samp f
Note that the phase variance is proportional to l. Solving the laser rate equations with Langevin noise terms,
we find [3]
2
< >= 2ΔlT , (11.17)
l samp
where Δ is the laser linewidth (FWHM). Comparing Eqs. (11.16) and (11.17), we find
2
= Δ . (11.18)
f 2T
samp
Fig. 11.2 shows a few possible evolutions of the laser phase (t) when the linewidth Δ is 5 MHz. Fig. 11.3
shows the evolutions of the phase for two different linewidths. It can be seen that the phase fluctuation is
larger as the linewidth increases.
The phase noise is present in the LO output as well, and the LO output field may be written as

q LO (t)= A LO exp {−i[2f LO t + LO (t)]}, (11.19)

with
2
< >= 2Δ lT , (11.20)
LO,l LO samp
where Δ is the linewidth of LO. The received signal after discretization is given by Eq. (11.9), with
LO
Δ = TX,l + − LO,l , (11.21)
p
l