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514
Performance of Communication Systems Corrupted by Noise Chap. 7
Because A>s 1, the integrand is negligible except for values of r 0 in the vicinity of A, so the
lower limit may be extended to -, and 2r can be replaced by 21>(2ps 2 0 >(2ps 2 A)
). Thus,
V
1 T
r 0
(7–56)
2 L0
s 2 e-(r 0 2 +A 2 )>(2s 2 )
I 0 a r 0A
s 2 b dr 0 L 1 V T
1
2 L-q
12ps e-(r 0-A) 2 >(2s 2 )
dr 0
When we substitute this equation into Eq. (7–55), the BER becomes
or
P e = 1 2 L
A/2
-q
1
12ps e-(r 0-A) 2 >(2s 2 )
dr 0 + 1 2 L
q
A>2
r 0
s 2 e-r 0 2 >(2s 2 )
dr 0
P e = 1 2 Qa A 2s b + 1 >(8s 2 )
2 e-A2
Using Q(z) = e -z2 /2> 22pz 2 for z 1, we have
1
P e =
e -A2 >(ds 2 ) + 1 2
22pAA>sB
e -A2 >(ds 2 )
(7–57)
Because A/s 1, the second term on the right dominates over the first term. Finally, we
obtain the approximation for the BER for noncoherent detection of OOK. It is
or
P e = 1 2 e-A2 >(8s 2) , A s 1
P e = 1 2 e-[1>(2TB p)](E b >N 0 ) , E b
N 0
TB p
4
(7–58)
where the average energy per bit is E b = A 2 T/4 and s 2 = N 0 B p .R= 1/T is the bit rate of the
OOK signal, and B p is the equivalent bandwidth of the bandpass filter that precedes the
envelope detector.
Equation (7–58) indicates that the BER depends on the bandwidth of the bandpass
filter and that P e becomes smaller as B p is decreased. Of course, this result is valid only when
the ISI is negligible. Referring to Eq. (3–74), we realize that the minimum bandwidth
allowed (i.e., for no ISI) is obtained when the rolloff factor is r = 0. This implies that the
minimum bandpass bandwidth that is allowed is B p = 2B = R = 1/T. A plot of the BER is
given in Fig. 7–14 for this minimum-bandwidth case of B p = 1/T.
Example 7–6 BER FOR OOK SIGNALING WITH NONCOHERENT DETECTION
Using Eq. (7–57), evaluate and plot the BER for OOK signaling in the presence of AWGN with
noncoherent detection, where B p = 1/T. See Example7_06.m for the solution. Compare this result
with that shown in Fig. 7–14.