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

10.2 General Binary Signaling 513
Fi g ure 10.4
Optimum binary
threshold
detection.
t = 1 i,
_ - _ ~ Threshold
Y>-r-->-(7 _ _
_,) d_ ev_ic_ef--Dēc-is-ion-,-;
i, m .,___=_0_ 1
_ f - r (-T-b) _ <_a
(aJ
m = 1 if r(Th) < a 0
0
p,(rll) 1
I ao ! ,.-------.
q/1i,) pof7i,)
(b)
Let Pv (t) and q v (t) be the response of H ( f ) to inputs p(t) and q(t), respectively. From
Eq. (10.7) it follows that
Pv (T h ) = 1_: P(f)H(f)ei 2 HfTh
df
(10.17a)
q v (n) = 1_: Q(f)H(f)ei 2H fTb df
(10.17b)
and a;, the variance, or power, of the noise at the filter output, is
(10.17c)
Without loss of generality, we let P v (T b ) > P v (T b )- Denote n as the noise output at T h . Then
the sampler output r(h) = q v (T b ) + nor p 0 (T b ) + n, depending on whether m = 0 or m = 1,
is received. Hence, r is a Gaussian RV of variance a; with mean q 0 (T b ) or Pv(T b ), depending
on whether m = 0 or 1. Thus, the conditional PDFs of the sampled output r(T b ) are
Optimum Threshold
The two PDFs are shown in Fig. 10.4b. If a v is the optimum threshold of detection, then the
decision rule is
{ 0 ifr < a 0
m - 1 if r > a0
The conditional error probability P(E I m= 0) is the probability of making a wrong decision
when m = 0. This is simply the area Ao under Prim (rlO) from a v to oo. Similarly, P(E Im = 1)