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

7.9 M-Ary Digital Carrier Modulation 383
Figure 7.37
M-ary PSK
symbols in the
orthogonal
signal space:
(a) M =2;
(b) M =4;
(c) M = 8.
o/2 (t)
i/J2 (t)
i/Jz (t)
M=2 A M=4 A M=8
,fl, (t) i/J1 (t) 'Pl (t)
-A 0 A -A A -A
-A
(a) (b) (c)
M-ary PSK, PAM, and QAM
By making a small modification to Eq. (7.61), PSK signals in general can be written into the
format of
(7.64a)
in which G m = A cos 0 m and b m = -A sin 0 m . In fact, based on the analysis in Sec. 2.6,
J2m cos W e t and J2m sin W e t are orthogonal to each other. Furthermore, they are normalized
over [O, T b ]. As a result, we can represent all PSK symbols in a two-dimensional
signal space with basis functions
such that
o/1 (t) = j2 COS W e t
y
ff.
1/r2 (t) =
sm W e t
<pPSK (t) = G m o/I (t) + b m o/ 2 (t)
(7.64b)
We can geometrically illustrate the relationship of the PSK symbols in the signal space
(Fig. 7.37). Equation (7.64) means that PSK modulations can be represented as QAM signal.
In fact, because the signal is PSK, the signal points must meet a special requirement
that
G 2 + b 2 = A 2 cos 2 0 + (-A) 2 sin 2 0
m m · m m
= A 2 = constant
(7.64c)
In other words, all the signal points must stay on a circle of radius A. In practice, all the signal
points are chosen to be equally spaced in the interest of obtaining the best immunity against
noise. Therefore, for M-ary PSK signaling, the angles are typically chosen uniformly as
2:rr
0 m = 0o + -(m - 1)
M
m = 1, 2, ... , M
The special PSK signaling with M = 4 is an extremely popular and powerful digital
modulation format.* It in fact is a summation of two binary PSK signals, one using the
* QPSK has several effective variations including the offset QPSK.