Next we will present various forms of the probability of bit error mathematical equations for some
modulation schemes. The equations will consist of exact, approximate, and bounds in order to give
the reader tools to quickly calculate the probability of error.
First we will provide a summary for the M-ary PSK (MPSK) modulation schemes. The exact
form of the Symbol Error Probability for MPSK is given as
where .
A reasonable approximation exists for M 2 in an AWGN channel given as
Recall for QPSK, the error rate in an AWGN channel is given in the following forms:
Also the QPSK error rate in a flat Rayleigh fading channel is given as
Next we will provide a summary for the M-ary DPSK (MDPSK) modulation schemes. The
Symbol Error Probability for MDPSK in an AWGN channel is given as
PMDPSK
1p
# 3
pp/M
0
expC
r # sin2Qp
MR 1 cosQp
MR# cos(u) S# du
Pb
1
2
# E1 d Eb
No
Eb
No
1U
Pb QaA2Eb
No b
1
2
erfcaAEb
Nob
1
2
# e1 erf aAEb
Nob f
Pb> 1
log2 M
# e1 erf aAEblog2M
No
# sin Sp
MTb f
Pb> 2
log2 M
# QaA2Eb log2M
No
# sin Sp
MTb
r
Eb
No
# log2 M
PMPSK
1p
# 3
pp/M
0
expC
r # sin2Qp
MR sin2(u) S# du
1 Qm(a,b) Q(b,a) expa
a2 b2
2 b # am1
k0 Qb
aRk # Ik(ab)
(m 2) Qm(a,b) Q(a,b) expa
a2 b2
2 b # am1
k1 Qb
aRk
Ik(ab)
Q(a,b) Q(b,a) 1 exp c
a2 b2
2 d # I0(ab)
510 APPENDIX A
Coherent detection of DPSK error rate in an AWGN channel is provided by the following:
For coherent detection of DQPSK, it is given as
The general Bit Error Probability of coherent detection of MDPSK is expressed as
where K 1 for M 2 and K 2 for M 4.
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