Ber Calculation
Autor: Mikki • June 27, 2018 • 1,373 Words (6 Pages) • 520 Views
...
Q(z)
1
e z2=2
(12)
zp2
we obtain:
3
Ps
p
e 0:5 s
(13)
2 s
Using Gray coding and assuming that for high signal to noise ratio the errors occur only for the nearest neighbor, Pb can be approximated from Ps by Pb
Ps=2.
2
---------------------------------------------------------------
[pic 2]
Figure 2: MPSK constellation
- BER for MPSK signaling
For MPSK signaling we can calculate easily an approximation of SER using nearest neighbor approximation. Using gure , the symbol error probability can be approximated by:
Ps 2Q
dmin
= 2Q
2A sin
= 2Q
M
p
p
2N0
2N0
This approximation is only good for high SNR.
p
2 s sin( =M)
(14)
- BER for QAM constellation
The SER for a rectangular M-QAM (16-QAM, 64-QAM, 256-QAM etc) with size L = M2 can be calculated by considering two M-PAM on in-phase and quadrature components (see gure 3 for 16-QAM constellation). The error probability of QAM symbol is obtained by the error probability of each branch (M-PAM) and is given by:
2
sqrtM
1)
3
!!
2 (
s
Ps = 1
1
Q
(15)
sqrtM
M
1
r
If we use the nearest neighbor approximation for an M-QAM rectangular con-stellation, there are 4 nearest neighbors with distance dmin. So the SER for high SNR can be approximated by:
c
(16)
In order to calculate the mean energy per transmitted symbol, it can be seen
that
M
Es = 1 A2i (17)
X
M
i=1
3
---------------------------------------------------------------
[pic 3]
Figure 3: 16-QAM constellation
Modulation
Ps( s)
Pb( b)
p
BPSK
Pb = Q
2 b
QPSK
Ps
2Q p
Pb
Q
p
2 b
s
2
MPSK
Ps
2Q p2 s sin
Pb
Q
2 b log2
M sin
M
log2
M
M
p
3
s
4
3
b log2
M
M-QAM
Ps
4Q q
M 1
Pb
log2
M
Q
q
M 1
Table 1: Approximate symbol and bit error probabilities for coherent modula-tion
Using the fact that Ai = (ai + bi) and ai and bi 2 f2i 1 Lg for i = 1; :::; L. After some simple calculations we obtain:
d2
L
2
Es =
min
(2i 1 L)
(18)
2L
=1
...