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Sec. 5–6 Phase Modulation and Frequency Modulation 337
which is periodic with period T m = 1f m . Consequently, g(t) could be represented by a Fourier
series that is valid over all time ( q 6 t 6 q); namely,
g(t) =
n=q
a
n=-q
c n e jnv mt
(5–55)
where
c n = A c
T m L
T m /2
-T m /2
(e jb sin v mt )e -jnv mt dt
(5–56)
which reduces to
p
c n = A c c 1 e j(b sin u-nu) du d = A c J n (b)
2p L-p
(5–57)
This integral—known as the Bessel function of the first kind of the nth order, J n (b)—cannot
be evaluated in closed form, but it has been evaluated numerically. Some tabulated values for
J n (b) are given in Table 5–2. Extensive tables are available [Abramowitz and Stegun, 1964]
or can be computed using MATLAB. The Bessel functions are invoked by standard function
calls in mathematical personal computer programs such as MATLAB. Examination of the
integral shows that (by making a change in variable)
J -n (b) = (-1) n J n (b)
(5–58)
A plot of the Bessel functions for various orders n as a function of b is shown in
Fig. 5–10.
Taking the Fourier transform of Eq. (5–55), we obtain
G(f) =
n=q
a
n=-q
c n d(f - nf m )
(5–59)
or
G(f) = A c
n=q
a
n=-q
J n (b)d(f - nf m )
(5–60)
Using this result in Eq. (5–49), we get the spectrum of the angle-modulated signal. See
Example5_06.m. The magnitude spectrum for f 7 0 is shown in Fig. 5–11 for the cases of
b = 0.2, 1.0, 2.0, 5.0, and 8.0. Note that the discrete carrier term (at f = f c ) is proportional to
|J 0 (b)|; consequently, the level (magnitude) of the discrete carrier depends on the modulation
index. It will be zero if J 0 (b) = 0, which occurs if b = 2.40, 5.52, and so on, as shown
in Table 5–3.