Cryptography Reference
In-Depth Information
Under this hypothesis, the error probability Peb ( ρ ) is equal to :
Peb ( ρ )= 1
2 erfc ρ
(2.175)
with:
L
E b
N 0
( α i ) 2
ρ =
n =1
By averaging the probability Peb ( ρ ) on the different realizations of the random
variable ρ , we obtain the bit error probability in presence of diversity of order
L :
Peb =
Peb ( ρ ) p ( ρ )
0
The random variable ρ follows a χ 2 law with probability density:
1
ρ
m
1)! m L ρ L− 1 exp
p ( ρ )=
(2.176)
( L
where m is equal to:
N 0 E ( α j ) 2
After integration, the bit error probability is equal to:
Peb = 1
E b
m =
L L− 1
L
1+ η
2
n
η
1+ n
n
(2.177)
2
n =0
with:
L
=
E b /LN 0
1+ E b /LN 0
( L
1+ n )!
n !( L
1+ n
n
η =
and
1)!
where E b is the average total energy used to transmit an information bit ( E b =
LE b ).
For a high signal to noise ratio, an approximation of the bit error probability
Peb is given by:
2 L
L
E b
LN 0
1
4 E b /LN 0
1
Peb
pour
>> 1
(2.178)
L
In the presence of diversity, the bit error probability Peb decreases following the
inverse of the signal to noise ratio to the power of L .
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