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

582 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
instant (Fig. 10.36). The matched filter output is now detected by an envelope detector. The
envelope is sampled at t = Tb for making the decision.
When a 1 is transmitted, the output of the envelope detector at t = Tb is an envelope of a
sine wave of amplitude A p
in a Gaussian noise of variance a;. In this case, the envelope r has
a Ricean density, given by [Eq. (9.86a)]
(10.138a)
Also, when A p
» a n (small-noise case) from Eq. (9 .86c ), we have
y 2n:A p
a;
Pr (rlm = 1) ::::: e -(r-A p ) 2 /2a;
~ __ l _ e -(r-A p ) 2 /2a;
- an.ffir
(10.138b)
(10.138c)
Observe that for small noise, the PDF of r is practically Gaussian, with mean A p
and variance
a;. When O is transmitted, the output of the envelope detector is an envelope of a Gaussian
noise of variance a;. The envelope in this case has a Rayleigh density, given by [Eq. (9.81)]
r
2 1 2 2
Pr (rlm = 0) = - e-r
Both pr (rlm = 1) and Pr (rlm = 0) are shown in Fig. 10.37. Using the argument used earlier
(see Fig. 10.4 ), the optimum threshold is found to be the point where the two densities intersect.
Hence, the optimum threshold a 0 is
or
an
2
a 0
- e -(a 2 o
+A 2 )/2a
P n 2 Io ( Apa o a
-- = -
0
) e -a 2 o ;2an
2
a; a; a;
a n
-A 2 /2a 2 ( Ap ao e P O
Io -- ) = 1
an
2
This equation is satisfied to a close approximation for
A p
ao = - 2
8a 2
1+- n
A 2
p
Figure 1 0.37
Conditional PDFs
in the
noncoherent
detection of ASK
signals.
p/rlm = 0)
r-