Cryptography Reference
In-Depth Information
Performance on a Rayleigh channel with diversity
Diversity involves repeating the same message (or copies coming from channel
coding) several times by using the carrier frequencies separated by a quantity
higher than or equal to the coherence band
B
c
of the channel. In this case,
we speak of frequency diversity. An alternative to this approach involves trans-
mitting a same message several times on a same carrier but on time intervals
separated by a quantity that is higher than or equal to the coherence time
t
c
of the channel. This is time diversity. Finally, we can transmit a message a
single time and use several sensors spaced typically by a few wavelengths from
the carrier of the modulated signal. In this case, we have space diversity.
Let us assume that we use a 2-PSK modulation to transmit the information
message and a diversity of order
L
. On the time interval
[
iT,
(
i
+1)
T
[
and
considering a coherent reception, after demodulation we have
L
observations of
the form:
r
i
=
α
i
E
b
cos
ϕ
i
+
b
i
,L
(2.171)
where
α
i
is a Rayleigh attenuation,
ϕ
i
the phase (0 or
π
)
carrying the informa-
tion to transmit and
b
i
a white Gaussian noise, with zero mean and variance
equal to
N
0
/
2
.The
L
attenuations
α
i
are mutually independent as well as the
L
terms of noise
b
i
.These
L
attenuations can be seen as
L
independent sub-
channels, also called diversity branches.
E
b
is thus the energy used to transmit
one bit per diversity branch.
To take a decision in the presence of diversity, we construct the variable
Z
i
in the following way:
n
=1
,
2
,
···
L
r
i
·
α
i
Z
i
=
n
=1
The bit error probability
Peb
in presence of diversity is then equal to:
1
2
Pr (
Z
i
>
0
ϕ
i
=
π
)+
1
Peb
=
|
2
Pr (
Z
i
<
0
|
ϕ
i
=0)
(2.172)
Conditionally to
one
realization of the
L
attenuations
α
i
, the decision variable
Z
i
is Gaussian with mean:
=
√
E
b
n
=1
L
(
α
i
)
2
E
{
Z
i
}
if
ϕ
i
=0
(2.173)
−
√
E
b
n
=1
(
α
i
)
2
L
E
{
Z
i
}
=
if
ϕ
i
=
π
and variance:
L
N
0
2
σ
Z
=
(
α
i
)
2
(2.174)
n
=1
