Cryptography Reference
In-Depth Information
The error probability can also be expressed as a function of the relation E b /N 0
where E b is the energy used to transmit a bit with E b = E s / log 2 ( M ) .
We can also try to determine the bit error probability Peb .Allthe M
1
groups of erroneous data appear with the same probability:
Pes
M
(2.89)
1
In a group of erroneous data, we can have k erroneous data among m and this
can occur in m
k
possible ways. Thus, the average number of erroneous data
in a group is:
k m
k
Pes
M
m
2 m− 1
2 m
= m
1 Pes
1
k =1
and finally the bit error probability is equal to:
2 m− 1
2 m
Peb =
1 Pes
(2.90)
where m =log 2 ( M ) .
The error probability for an M-FSK modulation does not have a simple
expression and we have to resort to digital computation to determine this prob-
ability as a function of the relation E b /N 0 . We show that for a given error
probability Peb ,therelation E b /N 0 necessary decreases when M increases. We
also show that probability Pes tends towards a value arbitrarily small when M
tends towards infinity, and for E b /N 0 =4ln2 dB, that is, -1.6 dB.
For a binary transmission ( M =2) , there is an expression of the error probability
Peb .
Let us assume that the signal transmitted is s 1 ( t ) ,wethenhave:
r 1 = E b + b 1
r 2 = b 2
The decision can be taken by comparing z = r 1
r 2 to a threshold fixed to zero.
The error probability Peb 1 conditionally to the emission of s 1 ( t ) ,isequalto:
Peb 1 =Pr
{
z< 0
|
s 1 ( t )
}
Assuming the two signals identically distributed, error probability Peb has the
expression:
Peb = 1
2 ( Peb 1 + Peb 2 )
The noises b 1 and b 2 are non-correlated Gaussian, with zero mean and same
variance equal to N 0 / 2 .Thevariable z , conditionally to the emission of the
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