B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

348 PRINCIPLES OF DIGITAL DATA TRANSMISSION
Figure 7. 14
Pulses satisfying
Nyquist's first
criterion: solid
curve, ideal
fx = 0 (r = O);
light dashed
curve,
f = Rb/4
(r = 0.5); heavy
dashed curve,
fx = R1,/2
(r = 1).
IP(f)I
0 ,_
p(t )
(a)
(bl
The constant r is called the roll-off factor and is also expressed in terms of percent. For
example, if P(f) is a Nyquist first criterion spectrum with a bandwidth that is 50% higher than
the theoretical minimum, its roll-off factor r = 0.5 or 50%.
A filter having an amplitude response with the same characteristics is required in the
vestigial sideband modulation discussed in Sec. 4.5 [Eq. (4.26)]. For this reason, we shall
refer to the spectrum P(J) in Eqs. (7.29) and (7.30) as a vestigial spectrum. The pulse
p(t) in Eq. (7.23) has zero ISI at the centers of all other pulses transmitted at a rate of
Rb pulses per second. A pulse p(t) that causes zero ISI at the centers of all the remaining
pulses (or signaling instants) is the Nyquist first criterion pulse. We have shown that a pulse
with a vestigial spectrum [Eq. (7.29) or Eq. (7.30)] satisfies the Nyquist's first criterion for
zero ISL
Because O :s: r < 1, the bandwidth of P(J) is restricted to the range Rb/2 to Rb Hz. The
pulse p(t) can be generated as a unit impulse response of a filter with transfer function P(J).
But because P(J) = 0 over a frequency band, it violates the Paley-Wiener criterion and is
therefore unrealizable. However, the vestigial roll-off characteristic is gradual, and it can be
more closely approximated by a practical filter. One family of spectra that satisfies Nyquist's
first criterion is
P(J) =
1,
-
1 . [ 1 - sm ( f-Rb/2 JT ) ]
2 2fx
0,
(7.34)
Figure 7.14a shows three curves from this family, corresponding to fx = 0 (r = 0),
fx = Rb/4 (r = 0.5) and .fx =Rb/2 (r = 1). The respective impulse responses are shown in
Fig. 7.14b. It can be seen that increasingfx (or r) improves p(t); that is, more gradual cutoff
reduces the oscillatory nature of p(t) and causes it to decay more rapidly in time domain. For