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

5.2 Bandwidth of Angle-Modulated Waves 213
this value. Based on Fig. 5.6c, it is clear that a better FM bandwidth approximation is between
[ 2f, 2f + 4B ]
Therefore, we should readjust our bandwidth estimation. To make this midcourse correction,
we observe that for the case of NBFM, k t
is very small. Hence, given a fixed mp, f is
very small (in comparison to B) for NBFM. In this case, we can ignore the small f term in
Eq. (5.12) with the result
But we showed earlier that for narrowband, the FM bandwidth is approximately 2B Hz. This
indicates that a better bandwidth estimate is
(5.13)
This is precisely the result obtained by Carson, 1 who investigated this problem rigorously
for tone modulation [sinusoidal m(t)]. This formula goes under the name Carson's rule
in the literature. Observe that for a truly wideband case, where f » B, Eq. (5. 13) can be
approximated as
f » B (5.14)
Because w = k_r m p
, this formula is precisely what the pioneers had used for FM bandwidth.
The only mistake was in thinking that this formula will hold for all cases, especially for the
narrowband case, where f « B.
We define a deviation ratio f3 as
f
/3 = ­ B
(5.15)
Carson's rule can be expressed in terms of the deviation ratio as
BFM = 2B(f3 + 1)
(5.16)
The deviation ratio controls the amount of modulation and, consequently, plays a role
similar to the modulation index in AM. Indeed, for the special case of tone-modulated FM, the
deviation ratio f3 is called the modulation index.
Phase Modulation
All the results derived for FM can be directly applied to PM. Thus, for PM, the instantaneous
frequency is given by
Therefore, the peak frequency deviation !).j is given by
f m(t) lmax - [m(t)minl
f = kp
2 ·
2n
(5.17a)