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

10.6 Optimum Receiver for White Gaussian Noise Channels 555
Figure 10.24
16-ary QAM.
'P2
. . . . . .
• • •
'P1 - Compute
r1s;1 + r2s;2
Comparator
(select
largest)
fh
(a)
(b)
(c)
Let us first calculate the error probability. The first quadrant of the signal space is reproduced
in Fig. 10.24b. Because all the signals are equiprobable, the decision region boundaries
will be perpendicular bisectors joining various signals, as shown in Fig. 10.24b.
From Fig. 10.24b it follows that
P(C lm1) = P(noise vector originating at s1 lies within R1)
=P(n1 > - , n2 > -)
= P ( n1 > - ) P ( n2 > - )
r
= [
1 - Q ( d 2 )
=
[ 1 - Q() r
For convenience, let us define
p = l - Q()
(10. 10 1)
Hence,