Cryptography Reference
In-Depth Information
To obtain the error binary probability
P
e,bit
, it suces to multiply
P
e,word
by the mean error density
δ
e
defined by (1.13):
exp
Rd
min
E
N
0
−
1
2
δ
e
N
(
d
min
)
P
e,bit
≈
πRd
min
E
N
0
(1.17)
Reading Table 1.1, we note that average number of errors in the 14 competitor
words of weight 4, at the minimum distance from the "all zero" word, is 2.
Equation (1.19) applied to the extended Hamming code is therefore:
exp
exp
E
N
0
2
E
N
0
1
−
2
×
4
×
−
1
2
×
2
4
×
P
e,bit
≈
14
π
2
×
=3
.
5
2
π
E
N
0
E
N
0
4
×
This expression gives
P
e,bit
=2.810
−
5
,1.810
−
6
et 6.2 10
−
8
E
N
0
=7,8
and 9 dB respectively, which corresponds to the results of the simulation of
Figure 1.5. Such agreement between equations and experimentation cannot be
found so clearly for more complex codes. In particular, finding the competitor
codewords at distance
d
min
may not be sucient and we then have to consider
words at distance
d
min
+1,
d
min
+2 etc.
For a same value of
P
e
and
P
e,bit
provided by the relations (1.15) and (1.17)
respectively, the signal to noise ratios
for
NC
C
E
N
0
E
N
0
and
without coding (NC)
and with coding (C) are such that:
⎛
⎝
⎞
⎠
NC
Rd
min
C
−
NC
E
N
0
E
b
N
0
E
b
N
0
C
Rd
min
=log
δ
e
N
(
d
min
)
(1.18)
E
N
0
If
δ
e
N
(
d
min
)
is not too far from unity, this relation can be simplified as:
C
−
NC
≈
E
b
N
0
E
b
N
0
Rd
min
0
NC
E
N
0
The
asymptotic gain
, expressed in dB, provides the gap between
and
C
:
E
N
0
NC
E
N
0
⎛
⎞
E
N
0
⎝
⎠
≈
C
G
a
=10log
10 log (
Rd
min
)
(1.19)
As mentioned above,
Rd
min
appears as a figure of merit which, in a link budget
with a low error rate, fixes the gain that a coding process can provide on a
