Cryptography Reference
In-Depth Information
Figure 11.20 - Standard detector.
We can show that the error probability (before channel decoding) for the
k
-th
user can be written in the form:
⎛
⎞
P
e,k
=
P
b
k
=
b
k
=
+
j
=
k
1
2
K−
1
A
k
σ
b
j
A
j
σ
⎝
⎠
Q
ρ
jk
(11.63)
K−
1
b
−k
∈{−
1
,
+1
}
where
ρ
jk
=
s
j
s
k
measures the intercorrelation between the codes of users
j
and
k
,with
b
−k
=(
b
1
,b
2
,
,b
K
)
.
Assuming that the spreading codes used are such that the intercorrelation
coecients are constant and equal to
0
.
2
, Figure 11.21 gives the performance of
the standard receiver, in terms of error probability of the first user as a function
of the signal to noise ratio, for a number of users varying from 1 to 6. The
messages of all the users are assumed to be received with the same power. We
note of course that the higher the number of users, the worse the performance.
This error probability can even tend towards 1/2, while the signal to noise ratio
increases if the following condition (
Near Far Effect
) is not satisfied:
A
k
>
j
=
k
···
,b
k−
1
,b
k
+1
,
···
A
j
|
ρ
jk
|
.
Optimal joint detection
Optimal joint detection involves maximizing the
a posteriori
probability (prob-
ability of vector
b
conditionally to observation
y
). If we assume that the binary
elements transmitted are equiprobable and given that
y
=
S
T
r
=
RAb
+
S
T
n
with
S
T
n
N
(0
,σ
2
R
)
, we can deduce that the optimal joint detection is given
∼
