563489578934

Sec. 5–9 Binary Modulated Bandpass Signaling 363
h = 2¢u
p =¢FT 0 = 2¢F
R
(5–82)
where the bit rate is R = 1T b = 2T 0 . In this application, note that the digital modulation index,
as given by Eq. (5–82), is identical to the FM modulation index defined by Eq. (5–48), since
h = ¢F = ¢F
1/T 0 B
= b f
provided that the bandwidth of m(t) is defined as B = 1T 0 .
The Fourier series for the complex envelope is
g(t) =
n=q
a
n=-q
c n e jnv 0t
(5–83)
where f 0 = 1T 0 = R2,
c n = A T 0/2
c
e ju(t) e -jnv0t dt
T 0 L
-T 0/2
= A T 0 /4
3T 0 /4
c
c e j¢vt-jnv0t dt + e -j¢v(t-(T0/2)) e -jnv0t dt d
T 0 L L
-T 0 /4
T 0 /4
(5–84)
and ∆v = 2p∆F = 2phT 0 . Equation (5–84) reduces to
c n = A c
2
- n)]
casin[(p/2)(h b + (-1) n sin [(p/2)(h + n)]
a
(p/2)(h - n)
(p/2)(h + n)
bd
(5–85)
where the digital modulation index is h = 2∆FR, in which 2∆F is the peak-to-peak frequency
shift and R is the bit rate. Using Eqs. (5–49) and (5–59), we see that the spectrum of this FSK
signal with alternating data is
where
S(f) = 1 2 [G(f - f c) + G * (-f - f c )]
q
q
G(f) = a c n d(f - nf 0 ) = a c n daf - nR 2 b
-q
(5–86a)
(5–86b)
and c n is given by Eq. (5–85).
FSK spectra can be evaluated easily for cases of different frequency shifts ∆F and bit rates
R if a personal computer is used. See Example5_09.m. A summary of three computer runs using
different sets of parameters is shown in Fig. 5–26. Figure 5–26a gives the FSK spectrum for the
Bell 103. For this case, where the Bell 103 parameters are used, the digital modulation index is
h = 0.67 and there are no spectral lines at the mark and space frequencies, f 1 and f 2 , respectively.
Figures 5–26b and 5–26c give the FSK spectra for h = 1.82 and h = 3.33. Note that as the
modulation index is increased, the spectrum concentrates about f 1 and f 2 . This is the same result
predicted by the PSD theorem for wideband FM [Eq. (5–66)], since the PDF of the binary modulation
consists of two delta functions. (See Fig. 5–15.)
-q