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

l 0.6 Optimum Receiver for White Gaussian Noise Channels 543
The decision function in Eq. (10.83) now becomes
(10.85)
Note that the decision function is always nonnegative for all values of i. Hence, comparing these
functions is equivalent to comparing their logarithms, because the logarithm is a monotone
function for the positive argument. Hence, for convenience, the decision function will be
chosen as the logarithm of Eq. (10.85). In addition, the factor (rr N) N 1 2 is common for all i
and can be left out. Hence, the decision function to maximize is
1 2
lnP(m;) - N
llq - s; II
(10.86)
Note that I I q - s; I I 2 is the square of the length of the vector q - s; . Hence,
llq - s;11 2 = <q - s;, q- s;>
= llqll 2 + lls;ll 2 - 2<q, s;>
(10.87)
Hence, the decision function in Eq. (10.86) becomes (after multiplying throughout by N /2)
: ln P(m;) - (11qll 2 + 11s;11 2 - 2<q, s;>)
(10.88)
Note that the term I js;I 1 2 is the square of the length of s; and represents E;, the energy of signal
s; (t). The terms Nln P(m;) and E; are constants in the decision function. Let
a; = ½[NlnP(m;) - E;]
(10.89)
Now the decision function in Eq. (10.88) becomes
llqll 2
a· + <q S · > - --
1 , I
The term llq ll 2 /2 is common to all M decision functions and can be omitted for the purpose
of comparison. Thus, the new decision function b; is
2
b; =a; + <q, s;> (10.90)
We compute this function b; for i = I, 2, ... , N, and the receiver decides that m = mk if this
function is the largest for i = k. If the signal q(t) is applied at the input terminals of a system
whose impulse response is h(t), the output at t = TM is given by
1_: q(r)h(TM - r)dr
If we choose a filter matched to s; (t), that is, h(t) = s; (TM - t),
h(TM - r) = s; (r)