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

7.3 Pulse Shaping 347
Fi g ure 7. 13
Vestigial
(raised-cosine)
spectrum.
IP(f)I
R i/2
:+-- fx---- Jx-:
If we choose P(j) to be real-valued and positive then only IP(f)I needs to satisfy Eq. (7.29).
Because IP(J)I is real, Eq. (7.29) implies
lxl < 0.5R b (7.30)
Hence, IP(f) I should be of the form shown in Fig. 7.13. This curve has an odd symmetry about
the set of axes intersecting at point a [the point on IP(l)I curve atf = R b /2]. Note that this
requires that
IP(0.5Rb)I = 0.51P(0) I
The bandwidth, in hertz, of P(j) is 0.5R b + fx, where fx is the bandwidth in excess of
the minimum bandwidth R b /2. Let r be the ratio of the excess bandwidthfx to the theoretical
minimum bandwidth R b /2:
excess bandwidth
r=-----------
theoretical minimum bandwidth
Observe that because fx cannot be larger than R b /2,
fx
0.5R b
= 2fx T b (7.31)
(7.32)
In terms of frequency f, the theoretical minimum bandwidth is R b /2 Hz, and the excess
bandwidth isfx = rR b /2 Hz. Therefore, the bandwidth of P(j) is
R b rR b (1 + r)R b
BT =
2 + - 2
- = --2 - - (7.33)