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

516 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
The optimum threshold a 0 is obtained by substituting Eqs. (10.17a, b) and (10.22a) into
Eq. (10.19a) and recognizing (via variable substitution) that
1_: P (f)Q (-f) df = 1_: P (-f) Q (j) df = E pq (10.26)
This gives
(10.27)
In deriving the optimum binary receiver, we assumed a certain receiver structure (the threshold
detection receiver in Fig. 10.4). It is not clear yet whether there exists another structure that
may have better performance than that in Fig. 10.4. It will be shown later (in Sec. 10.6) that for
a Gaussian noise, the receiver derived here is the definite optimum. Equation (10.25b) gives P b
for the optimum receiver when the channel noise is white Gaussian. For the case of nonwhite
noise, P b is obtained by substituting f3 max from Eq. (10.21a) into Eq. (10.25a).
Equivalent Optimum Binary Receivers
For the optimum receiver in Fig. 10.4a,
H(f) = P(-f)e - j 2 nfTb _ Q(-f)e - j 2 nfTb
This filter can be realized as a parallel combination of two filters matched to p(t) and q(t),
respectively, as shown in Fig. 10.5a. Yet another equivalent form is shown in Fig. 10.5b.
Because the threshold is (E p
- E q
)/2, we subtract E p
/2 and E q
/2, respectively, from the two
matched filter outputs. This is equivalent to shifting the threshold to 0. In the case of E p = E q
,
we need not subtract E p
/2 and E q
/2 from the two outputs, and the receiver simplifies to that
shown in Fig. 10.5c.
10.2.2 Performance Analysis of General Binary Systems
In this section, we analyze the performance of several typical binary digital communication
systems by applying the techniques derived in the last section for general binary receivers.
Polar Signaling
For the case of polar signaling, q(t) = -p(t). Hence,
and
Substituting these results into Eq. (10.25b) yields
E pq
= -1 00 p 2 (t) dt = -E p
-oo
(10.28)
(10.29)
Also from Eq. (10.22b ),
h(t) = 2p(Tb - t) (10.30)