Page 186 - Fiber Optic Communications Fund

Optical Modulators and Modulation Schemes 167

1 0 1 1 0 0
1
Differentially
encoded NRZ data
b(t)

–1
0
T b 2T b 3T b 4T b 5T b 6T b
2
Duobinary data
m(t)
–2

Figure 4.31 Input waveform b(t) and duobinary waveform m(t). The data in the interval −T < t < 0isassumedto
b
be ‘1’.
where b is the differentially encoded message data. Using Eq. (4.90), we find
n
m(t)= b + b at nT . (4.93)
n n−1 b
Superposition of pulses p(t) and p(t − T ) leads to a sample value of 2 at T . If both the pulses are −p(t) and
b
b
−p(t − T ), the sample value would be −2at T and if the pulses are p(t) and −p(t − T ), the sample value
b
b
b
would be zero, as illustrated in Fig. 4.31. The Fourier transform of the pulse described by Eq. (4.91) is (see
Example 4.7)
( ) ( ) ( )
2 f f f
̃ p(f)= cos rect exp −i . (4.94)
B B B B
From Fig. 4.32, we see that the bandwidth of the pulse is B∕2 Hz. In contrast, the bandwidth of NRZ-OOK
or RZ-OOK is ≥ B Hz. Eq. (4.94) can be rewritten as
1
̃ p(f)= [1 + exp (i2fT )]rect (f∕B). (4.95)
b
B
The factor exp (i2fT ) corresponds to a time delay of T and, therefore, the pulse p(t) can be generated by
b
b
cascading a delay-and-add filter and an ideal Nyquist filter with the transfer function
{
1for |f| < B∕2
H (f)= , (4.96)
N
0 otherwise
~
|p( f )|

f (Hz)
–B/2 0 B/2

Figure 4.32 Spectrum of the duobinary pulse.