Cryptography Reference
In-Depth Information
sages of size k . Assuming that a message, chosen arbitrarily and equiprobably
among 2 k , is encoded then transmitted through the channel, and assuming that
we use maximum likelihood decoding, then the coding theorem provides a bound
on the mean error probability of decoding the correct codeword. In [3.2], it is
shown that it is possible to limit the PER in the following way, for whatever
value of variable ρ , 0
ρ
1 :
x ) 1 / 1+ ρ 1+ ρ
1) ρ
y
(2 k
PER
Pr( x ) p ( y
|
(3.17)
x
In the case of a channel with equiprobable binary inputs, the probability of each
of the inputs is 1 / 2 and the vectors x and y can be treated independently in
x and y scalar values. Considering that (3.17) is valid for any ρ ,inorderto
obtain the closest upper bound to the PER, we must minimize the right-hand
side of (3.17) as a function of ρ . Introducing the rate R = k/n , it therefore
means minimizing for 0
ρ
1 , the expression :
exp
+exp
d y
k
1
1+ ρ
+
1) 2
2 σ 2 (1 + ρ )
( y +1) 2
2 σ 2 (1 + ρ )
1
2
1
σ 2 π
( y
2 ρR
×
−∞
The explicit value of σ is known for binary inputs (2-PSK and 4-PSK modu-
lations): σ =(2 RE b /N 0 ) 1 / 2 . An exploitable expression of Gallager's upper
bound on the PER of a binary input channel is then:
1+ ρ
d y
cosh y 4 RE b /N 0
1+ ρ
k
+
2 ρR +1 exp −y 2
e −k E N 0
π
min
0 ρ 1
(3.18)
0
This expression links the PER, the rate, the size k of the messages and the signal
to noise ratio E b /N 0 , for a Gaussian binary input channel. It gives an upper
bound of the PER and not an equality. This equation is not very well adapted
to all cases. In particular, simulations show that for a rate close to 1, the bound
is far too lax and does not give really useful results.
If we want to determine the penalty associated with a given packet size, we
can compare the result obtained by evaluating (3.18) with the result obtained
by computing the capacity that considers infinite size packets
3.3.2 Gaussian continuous input channel
In the case of a continuous input channel, we consider the case contrasting with
that of the binary input channel, that is, we will obtain an upper bound on
the practical limits of performance (all the modulations show performance lower
than a continuous input channel). Any modulation used will give performance
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