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

10.7 General Expression for Error Probability of Optimum Receivers 561
10.7 GENERAL EXPRESSION FOR ERROR
PROBABILITY OF OPTIMUM RECEIVERS
Thus far we have considered rather simple schemes in which the decision regions can be found
easily. The method of computing error probabilities from knowledge of decision regions has
also been discussed. When the number of signal space dimensions grows, it becomes harder
to visualize the decision regions graphically. and as a result the method loses its power. We
now develop an analytical expression for computing error probability for a general M-ary
scheme.
From the structure of the optimum receiver in Fig. 10.18, we observe that if m1 is
transmitted, then the correct decision will be made only if
In other words,
If m 1 is transmitted, then (Fig. 10.18)
(10.1 10)
[ T,11
bk = lo ls1 (t) + n(t)]sk (t) dt + Gk
( 10.lll)
Let
[ TM
Pij = lo s; (t)sj(t) dt i, j =l,2, ... ,M
(10.112)
where the Pij are known as cross-correlations. Thus (if m1 is transmitted),
[TM
bk = Plk + lo
n(t)'k (t) dt + Gk
(10.113a)
N
= Plk + Gk + L Skjnj
j=l
(10.113b)
where nj is the component of n(t) along <pj(t). Note that Plk + Gk is a constant, and variables
nj (j = 1, 2, . .. , N) are independent jointly Gaussian variables, each with zero mean and a
variance of N /2. Thus, variables b k are a linear combination of joint] y Gaussian variables. It
follows that the variables b1, b2, ... , bM are also jointly Gaussian. The probability of making
a correct decision when m1 is transmitted can be computed from Eq. (10.110). Note that b1
can lie anywhere in the range (- oo, oo). More precisely, ifp(b1, b2, . .. , hM lm1) is the joint
conditional PDF of b1, b2, ... , bM , then Eq. (10. 1 10) can be expressed as
(10.114a)