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

572 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
-[ 1 -Q(a)r
r
- [ 1-Q
u
(10.125a)
(10.125b)
l 0.8.1 Minimum Energy Signal Set
As noted earlier, an infinite number of possible equivalent signal sets exist. Because signal
energy depends on its distance from the origin, however, equivalent sets do not necessarily
have the same average energy. Thus, among the infinite possible equivalent signal sets, the
one in which the signals are closest to the origin, has the minimum average signal energy ( or
transmitted power).
Let m1 , m2, ... , mM be M messages with waveforms s1 (t), s2 (t), . .. , sM (t), represented,
respectively, by points s1, s 2 , ... , SM in the signal space. The mean energy of these
signals is £, given by
M
E = L P(mi) llsill 2
i=l
Translation of this signal set is equivalent to subtracting some vector a from each signal. We
now use this simple operation to yield a minimum mean energy set. We basically wish to find
the vector a such that the new mean energy
M
E' = L P(mi)llsi - all 2 (10.126)
i=l
is minimum. We can show that a must be the center of gravity of M points located at
s1, s 2 , ... , SM with masses P(m1 ), P(m2), ... , P(mM ), respectively,
M
a = L P(m;)si = sj
i=!
To prove this, suppose the mean energy is minimum for some translation b. Then
(10. 127)
M
E' = L P(mi) lls; - bll 2
i=l
M
= L P(mi) ll(si - a) + (a - b) ll 2
i=l
M M M
= L P(m;)llsi - all 2 + 2< (a - b), L P(mi)(si - a)> + L P(m;)lla - bll 2
i=l i=l i=l