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

556 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Using similar arguments, we have
P( C lm2) = P( Clm4) = [ 1 - Q ( )] [ 1 - 2Q ( )]
= p(2p - 1)
and
P(Clm3) = (2p - 1) 2
Because of the symmetry of the signals in all four quadrants, we get similar probabilities
for the four signals in each quadrant. Hence, the probability of correct decision is
16
P(C) = L P(Clmi)P(mi)
i=I
1
= 16
[4p 2 + 4p (2p - 1) + 4p (2p - 1) + 4(2p - 1) 2 ]
1
= 4
[9p 2 - 6p + I]
= (3P; I
r
(10.102)
and
P eM = 1 - P ( C) = (p
+ ) ( 1 - p)
In practice, P eM 0 if SNR is high and, hence, P ( C) 1. This means p :::: 1 and p + ½ :::: 1½
[see Eq. (10.102)], and
PeM '.::::'. 3(1 - p) = 3 Q (
) (10.103)
To express this in terms of the received power S;, we determine E, the average energy of the
signal set in Fig. 10.24. Because Ek
, the energy of S k , is the square of the distance of S k from
the origin,
E1 = c : r + c: r = ; d2
Similarly,
E2 = c: r + ( i r = ; d2
and
5 2
E4 = -d
2