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

5.2 Bandwidth of Angle-Modulated Waves 215
and the bandwidth of m(t) is 2nB = W m rad/s. The deviation ratio (or in this case, the
modulation index) is
Hence,
11f 11w akt
/3 = - = - = -
B 2nB W m
(j; FM (t) = A ei ((J)ct+fJ sin (J)mt)
= A ei(J) cf
( eifJ sin (J)mt ) (5.19)
Note that ei fJ sin (J)mt is a periodic signal with period 2n / W m and can be expanded by the
exponential Fourier series, as usual,
where
00
eifJ sin (J)m t = L D n
ei(J)mt
n=-oo
The integral on the right-hand side cannot be evaluated in a closed form but must be integrated
by expanding the integrand in infinite series. This integral has been extensively tabulated and
is denoted by l n (/3), the Bessel function of the first kind and the nth order. These functions are
plotted in Fig. 5.7a as a function of n for various values of {3. Thus,
00
eifJ sin (J)mt
= L l n (f3) ein (J)mt
n=-oo
(5.20)
Substituting Eq. (5.20) into Eq. (5.19), we get
and
00
(j; FM (t) = A L f n (f3)ei((J)c t+n(J)mt)
n=-oo
00
cp FM
(t) =A L l n (f3) cos (w e + nw m )t
n=-oo
The tone-modulated FM signal has a carrier component and an infinite number of sidebands
of frequencies W e ± W m , W e ± 2w m , . .. , W e ± nw m , . .. , as shown in Fig. 5.7b. This is in stark
contrast to the DSB-SC spectrum of only one sideband on either side of the carrier frequency.
The strength of the nth sideband at w = W e +nw m is* l n (/3). From the plots of l n (/3) in Fig. 5.7a,
it can be seen that for a given /3, l n (/3) decreases with n, and there are only a finite number
* Also l- n(fJ) = (-l ) n J n(fJ). Hence, the magnitude of the LSB at (J) = (J)c - n(J)m is the same as that of the USB at
(J) = (J)c +n(J)m,