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Optical Modulators and Modulation Schemes 141

duration of the signal pulse is 30 ps, the duty cycle is 30%. The duty cycle of a NRZ signal can be considered
to be 100%. In the above definition, we have assumed rectangular pulses. For pulses of arbitrary shape, the
duty cycle x can be defined as the ratio of the FWHM of a pulse to the bit interval T ,
b
FWHM
x = . (4.2)
T
b
When rectangular pulses are used, a RZ pulse in a bit interval [−T ∕2, T ∕2] may be written as
b
b
p(t)= 1for |t| < xT ∕2
b
= 0 otherwise. (4.3)

4.4 Power Spectral Density
In this section, we find an expression for the power spectral density (PSD) of various line coders. Let the
message signal be of the form
L
∑
m(t)= A lim a p(t − lT ). (4.4)
0 l b
L→∞
l=−L
Noting that
{p(t − lT )} = ̃p(f)e i2flT b , (4.5)
b
the Fourier transform of m(t) is
L
∑
̃ m(f)= A ̃p(f) lim a e i2flT b . (4.6)
l
0
L→∞
l=−L
The PSD is defined as
2
< | ̃m(f)| >
(f) = lim , (4.7)
m
T→∞ T
where T =(2L + 1)T and < > denotes the ensemble average. From Eq. (4.6), we have
b
2
∗
| ̃m(f)| = ̃m(f) ̃m (f)
L L
∑ ∑
∗
2
= A ̃p(f) lim a e i2flT b ̃ p (f) a e
∗ −i2fkT b
0 l k
L→∞
l=−L k=−L
L L
∑ ∑
2
2
= A |̃p(f)| lim a a e . (4.8)
∗ i2f(l−k)T b
0 L→∞ l k
l=−L k=−L
Using Eq. (4.8) in Eq. (4.7), we obtain
L L
1 ∑ ∑ ∗
2
2
(f)= A |̃p(f)| lim < a a > e i2f(l−k)T b . (4.9)
m 0 L→∞ (2L + 1)T l k
b l=−L k=−L
Let us first consider the case of a polar signal in which a is a random variable that takes values ±1 with equal
l
probability. When k ≠ l,
∗
< a a >= 0. (4.10)
l k
∗
This can be explained as follows. When a a = 1, it corresponds to a = 1 and a = 1, or a =−1 and a =−1;
k
l
l
k
l k
∗
∗
when a a =−1, it corresponds to a =−1 and a = 1, or a = 1 and a =−1. The chance that a a = 1isthe
k
l
l
k
l k l k