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

512 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Figure 10.3
Correlation
detector.
y(t) = ±p(t) + n(t)
p(t)
r(t)
t = T 0
r(T 0 )
Threshold
device
Decision
Hence,
r: r(t) = y(x)p (x + T 0 -
t) dx
(10.16a)
At the decision-making instant t = T 0 , we
r:
have
r(T 0 ) = y(x) p (x) dx
(10.16b)
Because the input y(x) is assumed to start at x = 0 and p (x) = 0 for x > T 0 , we have the
decision variable
{ To
r(T 0 ) = J
o y(x) p (x) dx
(10.16c)
We can implement Eqs. (10. 16) as shown in Fig. 10.3. This type of arrangement, known as the
correlation receiver, is equivalent to the matched tilter receiver.
The right-hand side of Eq. (10. 16c) is the cross-correlation of the received pulse with
p(t). Recall that correlation basically measures the similarity of signals (Sec. 2.7). Thus, the
optimum detector measures the similarity of the received signal with the pulse p(t). Based
on this similarity measure, the sign of the correlation decides whether p(t) or -p(t) was
transmitted.
Thus far we have discussed polar signaling in which only one basic pulse p(t) of opposite
signs is used. Generally, in binary communication, we use two distinct pulses p(t) and q(t) to
represent the two symbols. The optimum receiver for such a case will now be discussed.
10.2 GENERAL BINARY SIGNALING
l 0.2. l Optimum Linear Receiver Analysis
In a binary scheme where symbols are transmitted every T b seconds, the more general transmission
scheme may use two pulses p(t) and q(t) to transmit 1 and 0. The optimum linear
receiver structure under consideration is shown in Fig. 10.4a. The received signal is
Y (t) =
{ p(t) + n(t)
q(t) + n(t)
0 ::: t ::: T b
0::: t::: T b
for data symbol I
for data symbol 0
The incoming signal y(t) is transmitted through a filter H (f), and the output r(t) is sampled
at T b . The decision of whether O or 1 was present at the input depends on whether is or is not
r(Tb) is less than a 0 , where a 0 is the optimum threshold.