172 CHAPTER 4. OPTICAL RECEIVERS
variable, the reduction in the sampled value is also random. The SNR is reduced as a
result of such additional fluctuations, and the receiver performance is degraded. The
SNR can be maintained by increasing the received optical power. This increase is the
power penalty induced by timing jitter.
To simplify the following analysis, let us consider ap–i–nreceiver dominated by
thermal noiseσTand assume a zero extinction ratio. By usingI 0 =0 in Eq. (4.5.11),
the parameterQis given by
Q=
I 1 −〈∆ij〉
(σT^2 +σ^2 j)^1 / 2 +σT, (4.6.12)
where〈∆ij〉is the average value andσjis the RMS value of the current fluctuation∆ij
induced by timing jitter∆t.Ifhout(t)governs the shape of the current pulse,
∆ij=I 1 [hout( 0 )−hout(∆t)], (4.6.13)where the ideal sampling instant is taken to bet=0.
Clearly,σjdepends on the shape of the signal pulse at the decision current. A sim-
ple choice [92] corresponds tohout(t)=cos^2 (πBt/ 2 ), whereBis the bit rate. Here Eq.
(4.3.6) is used as many optical receivers are designed to provide that pulse shape. Since
∆tis likely to be much smaller than the bit periodTB= 1 /B, it can be approximated as
∆ij=( 2 π^2 / 3 − 4 )(B∆t)^2 I 1 (4.6.14)by assuming thatB∆t1. This approximation provides a reasonable estimate of the
power penalty as long as the penalty is not too large [92]. This is expected to be the
case in practice. To calculateσj, the probability density function of the timing jitter∆t
is assumed to be Gaussian, so that
p(∆t)=1
τj√
2 πexp(
−
∆t^2
2 τ^2 j)
, (4.6.15)
whereτjis the RMS value (standard deviation) of∆t. The probability density of∆ij
can be obtained by using Eqs. (4.6.14) and (4.6.15) and noting that∆ijis proportional
to(∆t)^2. The result is
p(∆ij)=1
√
πb∆ijI 1exp(
−
∆ij
bI 1)
, (4.6.16)
where
b=( 4 π^2 / 3 − 8 )(Bτj)^2. (4.6.17)Equation (4.6.16) is used to calculate〈∆ij〉andσj=〈(∆ij)^2 〉^1 /^2. The integration
over∆ijis easily done to obtain
〈∆ij〉=bI 1 / 2 , σj=bI 1 /