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

10.8 Equivalent Signal Sets 573
Observe that the second term in the foregoing expression vanishes according to Eq. (10.127)
because
Hence,
M M M
L P(m;)(s; - a) = L P(m;)s; - a L P(m;)
i=l
i=l
i=I
= a - a -1=0
M
E' = 'I:, P(m;) lls; - all 2 + 'I:, P(m;) Ila - h ll 2
i=l
This is minimum when b = a. Note that the rotation of the coordinates does not change the
energy, and, hence, there is no need to rotate the signal set to minimize the energy after the
translation.
1W
i=l
Example 1 0.7 For the binary orthogonal signal set of Fig. 10.32a, determine the minimum energy equivalent
signal set.
Fi g
ure 10.32
Eq uivalent signal
sets.
di-ff"°
d
(a)
t
s (t 'i
d/{r,"_ ) {2/T b k )
7i,
- 1-+-
dtfr,"_
(t)
s'i(IJ
{2/T b k2 /"'\.
-+- C T b
1-+-
(c)
(d)
The minimum energy set for this case is shown in Fig. 10.32b. The origin lies at the center
of gravity of the signals. We have also rotated the signals for convenience. The distances
k1 and k2 must be such that
and
k1 + k2 = d