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

548 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Thus, the decision rule is
m1
Decision (q) = { m2
randomly m1 or m2
if d 2 - d 2 < C
I 2
if df - d} > C
if df - d} = c
The boundary of the decision regions is given by df - d} = c. We now show that such a
boundary is given by a straight line perpendicular to line s 1-s2 and passing through s 1-s2
at a distance µ, from s 1 , where
(10.92)
where d is the distance between s 1 and s2. To prove this, we redraw the pertinent part of
Fig. 10.20b as Fig. 10.20c, from which it is evident that
Hence,
Therefore,
df = a 2 + µ, 2
di = a 2 + (d - µ,) 2
df - d} = 2dµ, - d 2 = c
c+d 2
µ, =
2.d
This is the desired result. Thus, along the decision boundary df -di is constant and equal to c.
The boundaries of the decision regions for M > 2 may be determined via similar argument.
The decision regions for the case of three equiprobable two-dimensional signals are shown
in Fig. 10.21. The boundaries of the decision regions are perpendicular bisectors of the lines
joining the original transmitted signals. If the signals are not equiprobable, then the boundaries
will be shifted away from the signals with larger probabilities of occurrence.
For signals in N-dimensional space, the decision regions will be N-dimensional hypercones.
If there are M messages m1 , m2, ... , mM with decision regions R1 , R2, ... , RM ,
respectively, then P(Clm;), the probability of a correct decision when m; is transmitted, is
given by
and P(C), the probability of a correct decision, is given by
(10.93)
M
P( C) = L P(m;)P( Clm;)
i=l
(10.94)
and P e M, the probability of error, is given by
P e M = 1 -P(C)
(10.95)