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Sec. 7–4 Noncoherent Detection of Bandpass Binary Signals 513
which is a Rician PDF, where
I 0 (z) ! 1 e z cos u du
(7–54)
2p L0
is the modified Bessel function of the first kind of zero order.
The two conditional PDFs, f(r 0 |s 2 ) and f(r 0 |s 1 ), are shown in Fig. 7–10. Actually,
f (r 0 |s 2 ) is a special case of f (r 0 |s 1 ) when A = 0, because, for this condition, we have noise only
at the detector input and Eq. (7–53) becomes Eq. (7–52). Two other plots of Eq. (7–53) are
also given—one for A = 1 and another for A = 4. It is seen that for A/s 1, the mode of the
distribution [i.e., where f (r 0 |s 1 ) is a maximum] occurs at the value r 0 = A. In addition, note
that for A/s 1, f (r 0 |s 1 ) takes on a Gaussian shape (as demonstrated subsequently).
The BER for the noncoherent OOK receiver is obtained by substituting Eqs. (7–52) and
(7–53) into Eq. (7–51):
P e = 1 V T
r 0
(7–55)
2 L0
s 2 e-(r 0 2 +A 2 )>(2s 2 )
I 0 a r 0A
s 2 b dr 0 + 1 q
r 0
2 LV T
s 2 e-r 0 2 >(2s 2 )
dr 0
The optimum threshold level is the value of V T for which P e is a minimum. For A/s 1, the
optimum threshold is close to V T = A/2, so we will use this value to simplify the mathematics. †
The integral involving the Bessel function cannot be evaluated in closed form. However, I 0 (z)
can be approximated by I 0 (z) = e z > 12pz, which is valid for z 1. Then, for A>s 1, the
left integral in Eq. (7–55) becomes
V
1 T
r 0
2 L0
s 2 e-(r 0 2 +A 2 )>(2s 2 )
I 0 a r 0A
s 2 b dr 0 L 1 A/2
r 0
2 L 2ps 2 A e(r 0-A) 2 >(2s 2 )
dr 0
2p
0 C
f
0.75
f(r 0 | s 2 ) (A=0)
0.50
f(r 0 | s 1 ) when A=1
Í=1
0.25
f(r 0 |s 1 ) when A=4
0
Figure 7–10
1 2 3 4 5 6 7
r 0
Conditional PDFs for noncoherent OOK reception.
† Most practical systems operate with A>s 1.