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

10.1 Optimum Linear Detector for Binary Polar Signaling 507
Figure 10.1
Typical binary
polar signaling
and linear
receiver.
±_p(t) + n(t)
H (f)
h(t)
(a)
Threshold
device
Decision
t -
(b)
t -
noisy, channel, the received signal waveform is
y(t) = ± p(t) + n(t)
where n(t) is a Gaussian channel noise.
10.1 .1 Binary Threshold Detection
0:::: t :S To
(10.1)
Given the received waveform of Eq. (10.1), the binary receiver must decide whether the
transmission was originally a 1 or a 0. Thus, the received signal y(t) must be processed to
produce a decision variable for each symbol. The linear receiver for binary signaling, as
shown in Fig. IO.la, has a general architecture that can be optimum (to be shown later in
Section 10.6). Given the receiver filter H (J) or h(t), its output signal for O :::: t ::: To is simply
y(t) = ± p(t) * h(t) + n(t) * h(t) = ± p 0 (t) + n 0 (t)
(10.2)
.___., .___.,
Po (t) no(t)
The decision variable of this linear binary receiver is the sample of the receiver filter output
at t = tm :
Based on the properties of Gaussian variables in Section 8.6,
n 0 (t) = lo
t
n(r)h(t - r) dr
is Gaussian with zero mean so long as n(t) is a zero mean Gaussian noise. If we define
(10.3)
A p
= Po (tm)
a; = E{n 0 (t m ) 2 }
(10.4a)
(10.4b)
then this binary detection problem is exactly the same as the threshold detection of
Example 8.16. We have shown in Example 8.16 that, if the binary data are equally likely