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

10.8 Equivalent Signal Sets 571
Assuming all symbols to be equiprobable, determine P eM for an AWGN channel with noise
PSD N/2.
Figure 10.31
Analysis of
QPSK.
'
'P 1 ---+- (a)
1E12
(b)
This problem has already been solved in Example 10.4 for a general value of M. Here we
shall solve it for M = 4 to demonstrate the power of the rotation of axes.
Because all the symbols are equiprobable, the decision region boundaries will be
perpendicular bisectors of lines joining various signal points (Fig. 10.3 la). Now
P(Clm1) = P(noise vector originating at s1 remains in R1) (10.124)
This can be found by integrating the joint PDF of components n1 and n2 (originating at
s 1 ) over the region R1. This double integral can be found by using suitable limits, as in
Eq. (10.106). The problem is greatly simplified, however, if we rotate the signal set by
45 ° , as shown in Fig. 10.3 lb. The decision regions are rectangular, and if n1 and n2 are
noise components along ({Ji and ({}2, then Eq. (10.124) can be expressed as
= P ( n1 > -'2) P ( n2 > -'2)
- [ 1 -Q(¼) ]'