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