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

10.6 Optimum Receiver for White Gaussian Noise Channels 541
the components representing no (t) are independent of the components representing n(t).
Consequently, no(t) is independent of n(t) and contains only irrelevant data.
The received signal r(t) is now reduced to the signal q (t), which contains the desired signal
waveform and the projection of the channel noise on the N -dimensional signal space. Thus,
the signal q(t) can be completely represented in the signal space. Let the vectors representing
n(t) and q(t) be denoted by n and q. Thus,
where s may be any one of vectors s1 , s2, ... , SM .
The random vector n = (n1 , n2, ... , nN) is represented by N independent Gaussian
variables, each with zero mean and variance 0- 0
2
= N /2. The joint PDF of vector n in such a
case has a spherical symmetry, as shown in Eq. (10.69b),
Note that this is actually a compact notation for
10.6.3 (Simplified) Signal Space and Decision Procedure
(10.78a)
(10.78b)
Our problem is now considerably simplified. The irrelevant noise component has been filtered
out. The residual signal q (t) can be represented in an N-dimensional signal space. We proceed
to determine the M decision regions R1 , R2, ... , RM in this space. The regions must be chosen
to minimize the probability of error in making the decision.
Suppose the received vector q = q. Then if the receiver decides m = mk, the conditional
probability of making the correct decision, given that q = q, is
P(Clq = q) = P(mk lq = q) (10.79)
where P(Clq = q) is the conditional probability of making the correct decision given q = q,
and P(mk lq = q) is the conditional probability that mk was transmitted given q = q. The
unconditional probability P(C) is given by
P(C) = l P(Clq = q)p q
(q) dq (10.80)
where the integration is performed over the entire region occupied by q. Note that this is
an N-fold integration with respect to the variables q1 , q2, ... , qN over the signal waveform
duration. Also, because p 4 (q) 2: 0, this integral is maximum when P(Clq = q) is maximum.
From Eq. (10.79) it now follows that if a decision m = mk is made, the error probability is
minimized if the probability
P(C) = l P(Clq = q)p q
(q) dq
is maximized. The probability P(mk lq = q) is called the a posteriori probability of mk . This
is because it represents the probability that mk was transmitted when q was being received.