Cryptography Reference
In-Depth Information
We obtain the error probability
Peb
by averaging
Peb
(
α
)
over the different
realizations of
α
(
t
)
.
∞
Peb
=
Peb
(
α
)
p
(
α
)
dα
(2.165)
0
where
p
(
α
)
is the probability density of
α
.
Taking into account the fact that, for a given
t
,
α
(
t
)
is a Rayleigh random
variable with probability density
exp
α
2
2
σ
α
p
(
α
)=
α
σ
α
α
≥
0
−
the probabilities
Peb
have the expressions:
1
E
b
/N
0
1+
E
b
/N
0
Peb
=
1
2
2-PSK or 4-PSK modulation
−
(2.166)
1
E
b
/N
0
2+
E
b
/N
0
1
2
2-FSK modulation
Peb
=
−
(2.167)
where
E
b
is the average energy per transmitted bit :
E
b
=
E
α
2
A
2
T
b
2
=
A
2
T
b
σ
α
(2.168)
For high
E
b
/N
0
, the error probabilities can be approximated by:
1
4
E/N
0
2-PSK or 4-PSK
Peb
≈
(2.169)
1
2
E/N
0
2-FSK
Peb
≈
(2.170)
On a Rayleigh fading channel, the performance of the different receivers is
severely degraded compared to those obtained on a Gaussian channel (with
identical
E
b
/N
0
at the input). Indeed, on a Gaussian channel, the error prob-
abilities
Peb
decrease exponentially as a function of the signal to noise ratio
E
b
/N
0
whereas on a Rayleigh fading channel, the decrease in the probability
Peb
is proportional to the inverse of the average signal to noise ratio,
E
b
/N
0
.
To improve the performance on a Rayleigh fading channel, we use two tech-
niques, which we can combine, diversity and, of course, channel coding (which
is, in fact, diversity of information).
