Information Technology Reference
In-Depth Information
-
The transmission is carried through a single channel (for example in the
case of cellular communication the channel is the atmosphere), therefore the
message received by the antenna is the sum
u
=
i
u
i
.
The main problem [3] is to extract the individual bits
b
i
from the message
u
.
The bit
b
i
can be estimated by calculating the inner product
ϕ
i
,u
=
i
ϕ
i
,u
j
=
j
b
j
ϕ
i
,ϕ
j
=
b
i
+
j
=
i
b
j
ϕ
i
,ϕ
j
.
The last expression above should be considered as a sum of the information
bit
b
i
and an additional noise caused by the interference of the other messages.
This is the standard scenario also called the
Synchronous
scenario. In practice,
more complicated scenarios appear, e.g.,
asynchronous scenario
- in which each
message
u
i
is allowed to acquire an arbitrary time shift
u
i
(
t
)
u
i
(
t
+
τ
i
),
phase
shift scenario
- in which each message
u
i
is allowed to acquire an arbitrary
phase shift
u
i
(
t
)
→
e
2
π
p
w
i
t
u
i
(
t
) and probably also a combination of the two
where each message
u
i
is allowed to acquire an arbitrary distortion of the form
u
i
(
t
)
→
e
2
π
p
w
i
t
u
i
(
t
+
τ
i
)
.
The previous discussion suggests that what we are seeking for is a large system
→
S
of signals which will enable a reliable extraction of each bit
b
i
for as many
users transmitting through the channel simultaneously.
Definition 1 (Stability conditions).
Two unit signals φ
=
ϕ are called
sta-
bly cross-correlated
if
|
m
ϕ,φ
(
v
)
|
1
for every v
∈
V .Aunitsignalϕ is
called
stably auto-correlated
if
|
A
ϕ
(
v
)
|
1
,forv
=0
.Asystem
S
of signals
is called a
stable
system if every signal ϕ
∈
S
is stably auto-correlated and any
two different signals φ, ϕ
∈
S
are stably cross-correlated.
Formally what we require for CDMA is a stable system
. Let us explain why
this corresponds to a reasonable solution to our problem. At a certain time
t
the
antenna receives a message
S
u
=
i∈J
u
i
,
which is transmitted from a subset of users
J
⊂
I
. Each message
u
i
,
i
∈
J,
is of
the form
u
i
=
b
i
e
2
πi
w
i
t
ϕ
i
(
t
+
τ
i
)=
b
i
π
(
h
i
)
ϕ
i
,
where
h
i
∈
H
. In order to extract
p
the bit
b
i
we compute the matrix coecient
)
o
(1)
,
where
R
h
i
is the operator of right translation
R
h
i
A
ϕ
i
(
h
)=
A
ϕ
i
(
hh
i
)
.
If the cardinality of the set
J
is not too big then by evaluating
m
ϕ
i
,u
at
h
=
h
−
i
we can reconstruct the bit
b
i
. It follows from (1) and (2) that the oscillator
system
m
ϕ
i
,u
=
b
i
R
h
i
A
ϕ
i
+#(
J
−{
i
}
S
O
can support order of
p
3
users, enabling reliable reconstruction when
order of
√
p
users are transmitting simultaneously.
