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Performance of Communication Systems Corrupted by Noise Chap. 7
For the case of a space signal plus noise at the receiver input, we know that the output of the
upper bandpass filter is only Gaussian noise (no signal). Thus, the output of the upper envelope
detector v U is noise having a Rayleigh distribution
v U
(v U |s 2 ) = µ s 2 e - vU>(2s 2 2) , v U Ú 0
(7–62)
0, v U 6 0
where s 2 = N 0 B p . On the other hand, v L has a Rician distribution, since a sinusoid (the space
signal) plus noise appears at the input to the lower envelope detector:
or
v L L+A 2 )>(2s 2 )
f(v L |s 2 ) = µ s 2 e-(v2 I 0 a v LA
s 2 b, v L Ú 0
0, v L 6 0
P e = 1 2 e-A2 >(4s 2 )
P e = 1 2 e-[1>(2TB p)](E b >N 0 )
(7–63)
where s 2 = N 0 B p . Using Eqs. (7–62) and (7–63) in Eq. (7–61), we obtain
q
v L
P e =
L 0 s 2 e-(v L 2 +A 2 )>(2s 2) I 0 a v q
LA
s 2 bc v U U>(2s 2)
Lv L
s 2 e-v2 dv U d dv L
When we evaluate the inner integral, the BER becomes
q
P e = e -A2 >(2s 2 )
v L
(7–64)
L0
s 2 e-v L 2 >s 2 I 0 a v LA
s 2 b dv L
This integral can be evaluated by using the integral table in Appendix A. Thus, for noncoherent
detection of FSK, the BER is
(7–65)
where the average energy per bit is E b = A 2 T/2 and s 2 = N 0 B p .N 0 /2 is the PSD of the input
noise, and B p is the effective bandwidth of each of the bandpass filters. (See Fig. 7–11.)
Comparing Eq. (7–65) with Eq. (7–58), we see that OOK and FSK are equivalent on an E b /N 0
basis. A plot of Eq. (7–65) is given in Fig. 7–14 for the case of the minimum filter bandwidth
allowed, B p = R = 1/T, for no ISI. When comparing the error performance of noncoherently
detected FSK with that of coherently detected FSK, it is seen that noncoherent FSK requires, at
most, only 1 dB more E b /N 0 than that for coherent FSK if P e is 10 - 4 or less. The noncoherent
FSK receiver is considerably easier to build since the coherent reference signals do not have to
be generated. Thus, in practice, almost all of the FSK receivers use noncoherent detection.
Example 7–7 BER FOR FSK SIGNALING WITH NONCOHERENT DETECTION
Evaluate and plot the BER for FSK signaling in the presence of AWGN with noncoherent
detection, where B p = 1/T. Also, calculate and plot the BER for FSK with coherent MF detection.
See Example7_07.m for the solution. Compare these results with those shown in Fig. 7–14.