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

214 ANGLE MODULATION AND DEMODULATION
If we assume that
mp = [m(t)Jmax = lm(t)nunl
(5. 17b)
then
(5.17c)
Therefore,*
BrM = 2(1).f + B)
= 2 ( k ;:P +B)
(5. 18a)
(5.18b)
One very interesting aspect of FM is that !).w = k f
mp depends only on the peak value of m(t).
It is independent of the spectrum of m(t). On the other hand, in PM, /).w = kpmp depends on the
peak value of m(t). But m(t) depends strongly on the spectral composition of m(t). The presence
of higher frequency components in m(t) implies rapid time variations, resulting in a higher
value of nz p
. Conversely, predominance of lower frequency components will result in a lower
value of ni p
. Hence, whereas the FM signal bandwidth [Eq. (5. 13)] is practically independent
of the spectral shape of m(t), the PM signal bandwidth [Eq. (5. 18)] is strongly affected by the
spectral shape of m(t). For m(t) with a spectrum concentrated at lower frequencies, BpM will
be smaller than when the spectrum of m(t) is concentrated at higher frequencies.
Spectral Analysis of Tone Frequency Modulation
For an FM carrier with a generic message signal m(t), the spectral analysis requires the use of
staircase signal approximation. Tone modulation is a special case for which a precise spectral
analysis is possible: that is, when m(t) is a sinusoid. We use this special case to verify the FM
bandwidth approximation. Let
m(t) = a cos Wmt
From Eq. (5.7), with the assumption that initially a(-oo) = 0, we have
Thus, from Eq. (5.8a), we have
a
a(t) = - sm Wmt
Wm
Moreover
* Equation (5.17a) can be applied only if m(t) is a continuous function of time. If m(t) has jump discontinuities, its
derivative does not exist. In such a case, we should use the direct approach (discussed in Example 5.2) to find
<i' PM (t) and then determine t!.w from <p PM (t).