563489578934

Sec. 5–11 Minimum-Shift Keying and GMSK 379
where (v 1 - v 2 )T b = 2 p (2∆F)T b and, from Eq. (5–82), h = 2∆FT b . For the continuous-phase
case, u 1 = u 2 ; and Eq. (5–110) is satisfied for a minimum value of h = 0.5, or a peak frequency
deviation of
For discontinuous-phase FSK, u 1 Z
h = 1.0 or
¢F = 1 = 1 R, for MSK
4T b 4
(5–111a)
u 2 ; and the minimum value for orthogonality is
¢F = 1 = 1 R, for discontinuous-phase, FSK
2T b 2
(5–111b)
Now we will demonstrate that the MSK signal (which is h = 0.5 continuous-phase FSK)
is a form of OQPSK signaling with sinusoidal pulse shaping. First, consider the FSK signal
over the signaling interval (0, T b ). When we use Eq. (5–81), the complex envelope is
g(t) = A c e ju(t) = A c e j2p≤F1t 0 m(l)dl
where m(t) = ;1, 0 6 t 6 T b . Using Eq. (5–111), we find that the complex envelope becomes
g(t) = A c e ;jpt>(2T b) = x(t) + jy(t), 0 6 t 6 T b
where the ± signs denote the possible data during the (0, T b ) interval. Thus,
x(t) = A c cos a;1 pt
2T b
b, 0 6 t 6 T b
y(t) = A c sin a;1 pt
2T b
b, 0 6 t 6 T b
(5–112a)
(5–112b)
and the MSK signal is
s(t) = x(t) cos v c t - y(t) sin v c t
(5–112c)
A typical input data waveform m(t) and the resulting x(t) and y (t) quadrature modulation
waveforms are shown in Fig. 5–34. From Eqs. (5–112a) and (5–112b), and realizing that
cos[ ;pt(2T b )] = cos[ pt(2T b )] and sin[ ;pt(2T b )] = ; sin[ pt(2T b )], we see that the ; 1
sign of m(t) during the (0, T b ) interval affects only y(t), and not x(t), over the signaling interval
of (0, 2T b ). We also realize that the sin [ pt(2T b )] pulse of y(t) is 2T b s wide. Similarly, it
can be seen that the ; 1 sign of m(t) over the (T b , 2T b ) interval affects only x(t), and not y(t),
over the interval (T b , 3T b ). In other words, the binary data of m(t) alternately modulate the x(t)
and y(t) components, and the pulse shape for the x(t) and y(t) symbols (which are 2T b wide
instead of T b ) is a sinusoid, as shown in the figure. Thus, MSK is equivalent to OQPSK with
sinusoidal pulse shaping.
The x(t) and y(t) waveforms, as shown in Fig. 5–34, illustrate the so-called Type II
MSK [Bhargava, Haccoun, Matyas, and Nuspl, 1981], in which the basic pulse shape is