"Introduction". In: Fiber-Optic Communication Systems

6.5. SYSTEM APPLICATIONS 267

random speed changes produce random shifts in the pulse position at the receiver and
are responsible for the timing jitter.
Timing jitter induced by the ASE noise can be calculated using themoment method.
According to this method, changes in the pulse positionqand the frequencyΩalong
the link length are calculated using [125]

q(z)=

1

E

∫∞

−∞

t|A(z,t)|^2 dt, (6.5.20)

Ω(z)=

i

2 E

∫∞

−∞

(

A∗

∂A

∂t

−A

∂A∗

∂t

)

dt, (6.5.21)

whereE=

∫∞

−∞|A|

(^2) dtrepresents the pulse energy.
The NLS equation can be used to find howTandWevolve along the fiber link.
Differentiating Eqs. (6.5.20) and (6.5.21) with respect tozand using Eq. (6.5.31), we
obtain [126]
dΩ
dz

=∑

i

δΩiδ(z−zi), (6.5.22)

dq

dz

=β 2 Ω+∑

i

δqiδ(z−zi), (6.5.23)

whereδΩiandδqiare the random frequency and position changes imparted by noise at
theith amplifier and the sum is over the total numberNAof amplifiers. These equations
show that frequency fluctuations induced by an amplifier become temporal fluctuations
because of GVD; no jitter occurs whenβ 2 =0.
Equations (6.5.22) and (6.5.23) can be integrated in a straightforward manner. For a
cascaded chain ofNAamplifiers with spacingLA, the pulse position at the last amplifier
is given by

qf=

NA

∑

n= 1

δqn+β ̄ 2 LA

NA

∑

n= 1

n− 1

∑

i= 1

δΩi, (6.5.24)

whereβ ̄ 2 is the average value of the GVD. Timing jitter is calculated from this equa-
tion usingσt^2 =〈q^2 f〉−〈qf〉^2 together with〈qf〉=0. The average can be performed by
noting that fluctuations at two different amplifiers are not correlated. However, the tim-
ing jitter depends not only on the variances of position and frequency fluctuations but
also on the cross-correlation function〈δqδΩ〉at the same amplifier. These quantities
depend on the pulse amplitudeA(zi,t)at the amplifier locationzi(see Section 9.5).
Consider a low-power lightwave system employing the CRZ format and assume
that the input pulse is in the form of a chirped Gaussian pulse. As seen in Section 2.4,
the pulse maintains its Gaussian shape on propagation such that

A(z,t)=aexp[iφ−iΩ(t−q)−( 1 +iC)(t−q)^2 / 2 T^2 ], (6.5.25)

where the amplitudea, phaseφ, frequencyΩ, positionq, chirpC, and widthTall are
functions ofz. The variances and cross-correlation ofδqiandΩiat the location of the
ith amplifier are found to be [126]

〈(δΩ)^2 〉=(Ssp/E 0 )[( 1 +Ci^2 )/Ti^2 ], (6.5.26)