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

5.2 Bandwidth of Angle-Modulated Waves 217
Because a(t) is a periodic signal, Jkta ( t ) is also a periodic signal, which can be expressed as an
exponential Fourier series in the preceding expression. After this, it is relatively straightforward
to write <pFM (t) in terms of the carrier and the sidebands.
Example 5.3
(a) Estimate BFM and BpM for the modulating signal m(t) in Fig. 5.4a for k_r = 2n x 10 5
and k p
= 5n. Assume the essential bandwidth of the periodic m(t) as the frequency of
its third harmonic.
(b) Repeat the problem if the amplitude of m(t) is doubled [if m(t) is multiplied by 2].
(a) The peak amplitude of m(t) is unity. Hence, mp = I. We now determine the
essential bandwidth B of m(t). It is left as an exercise for the reader to show that the
Fourier series for this periodic signal is given by
m(t) = L Cn cos nwot
2n 4
wo = --- =lOn
2 X 10-4
where
n odd
neven
It can be seen that the harmonic amplitudes decrease rapidly with n. The third harmonic
is only 11 % of the fundamental, and the fifth harmonic is only 4% of the fundamental.
This means the third and fifth harmonic powers are 1.21 and 0.16%, respectively, of
the fundamental component power. Hence, we are justified in assuming the essential
bandwidth of m(t) as the frequency of its third harmonic, that is,
For FM:
and
104
B = 3 x - = 15 kHz
2
1 1
t:;.J = -k_rmp = -(2n x 10 5 )(1) = 100
2n 2n
BFM = 2(!:;.f + B) = 230 kHz
Alternatively, the deviation ratio f3 is given by
and
BFM = 2B(f3 + 1) = 30 ( 100 + 1
l5
) = 230 kHz