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

10.1 1 Noncoherent Detection 585
Figure 10.39
Noncoherenl
detection of
binary FSK.
H i(/)
Envelope
detector
r1
Input
Comparator
Decision:
select larger
Ho(/)
Envelope
detector
ro
envelope detectors at t = T b are ro and r1, respectively. The noise components of outputs of
filters Ho (J) and H1 (f ) are the Gaussian RVs no and n1 , respectively, with an 0
= an 1
= a n .
If 1 is transmitted (m = 1), then at the sampling instant, the envelope r1 has the
Ricean PDF*
and ro is the noise envelope with Rayleigh density
The decision is m = 1 if r1 > ro and m = 0 if r1 < ro . Hence, when binary 1 is transmitted,
an error is made if ro > r1 ,
P(E lm = 1) = P(ro > ri)
The event ro > r1 is the same as the joint event "r1 has any positive value t and ro has a value
greater than ri ." This is simply the joint event (0 < r1 < oo, ro > r1). Hence,
P(E lm = 1) = P(O < r1 < oo, ro > r1)
= { oc f 00 Pr 1 r 0
(r1 , ro) dr1 dr o
Because r1 and ro are independent, P r 1 r0
= Pr 1 Pro • Hence,
i cxJ
P(E l m =1) = ('0 r
e-(rf+A)/2 a , I o ( r t p ) f
00
l o
k n
= - e 1 P " I o -- dri
r1 . -(2r 2 +A 2 )/2a 2
( r1 ,4p )
o a; a;
r1
r
e-r5/2 a'/; dr1 dro
Letting x = .Ji ri and a = A p
/ .Ji, we have
* An orthogonal FSK is assumed. This ensures that ro and fJ have Rayleigh and Rice densities, respectively, when 1
is transmitted.
t fJ is the envelope detector and can take only positive values.