3.2 Linear
Group
-Blind Multiuser Detection for Synchronous CDMA
We start by considering the following discrete-time signal model for a synchronous CDMA system:
Equation 3.1
Equation 3.2
where, as before,
K
is the total number of users;
A
k
,
b
k
[
i
], and
s
k
are, respectively, the complex amplitude,
i
th transmitted bit, and signature waveform of the
k
th
user
;
n
[
i
] ~
N
c
(0,
s
2
I
N
) is a complex Gaussian noise vector;
;
; and
. In this chapter it is assumed that the receiver has knowledge of the signature waveforms of the first
users (
), whose data bits are to be demodulated, whereas the signature waveforms of the remaining
users are unknown to the receiver. Denote
It is assumed that the users' signature waveforms are linearly independent (i.e.,
S
has full column rank). Hence both
and
also have full column ranks. Then (3.2) can be written as
Equation 3.3
The problem of linear group-blind multiuser detection can be stated as
follows
. Given prior knowledge of the signature waveforms
of the
desired users, find a weight vector
w
k
for each desired user
k
, 1
k
, such that the data bits of these users can be demodulated according to
Equation 3.4
and
Equation 3.5
or
Equation 3.6
The basic idea behind the solution to the problem above is to suppress the interference from known users based on the signature waveforms of these users and to suppress the interference from other unknown users using subspace-based blind
methods
. We first consider the linear
decorrelating
detector, which eliminates the
multiple-access
interference (MAI) completely, at the expense of enhancing the noise level. To facilitate the derivation of its group-blind form, we need the following alternative definition of this detector. In this section we denote
as a
-vector with all elements zeros, except for the
k
th element, which is 1.
Definition 3.1:
[Group-blind linear decorrelating detector ”synchronous CDMA]
The weight vector
d
k
of the linear decorrelating detector for user k is given by the solution to the following constrained optimization problem:
Equation 3.7
This definition is equivalent to the one given in Section 2.2.2. To see this, it suffices to show that
, and
for
l
k
. Since
contains the first
columns
of
S
, then for any
w
we have
Equation 3.8
Under the constraint
, we have
. It then follows that for
w
range
(
S
),
w
H
SA
2
is minimized subject to
if and only if
w
H
s
l
= 0 for
l
=
+ 1, . . .,
K
. Since
rank
(
S
) =
K
, such a
w
range
(
S
) is unique and is indeed the linear decorrelating detector.
The second linear group-blind detector
considered
here is a hybrid detector that zero-forces the interference caused by the
known users and suppresses the interference from unknown users according to the MMSE criterion.
Definition 3.2:
[Group-blind linear hybrid detector ”synchronous CDMA]
The weight vector
w
k
of the group-blind linear hybrid detector for user k is given by the solution to the following constrained optimization problem:
Equation 3.9
Another form of linear group-blind detector is analogous to the linear MMSE detector introduced in Section 2.2.3. It suppresses the interference from the known users and that from the unknown users separately, both in the MMSE sense. First define the following projection matrix:
Equation 3.10
which projects any signal onto the subspace
null
(
). Recall that the autocorrelation matrix of the received signal in (3.1) is given by
Equation 3.11
where
. It is then easily seen that the matrix
has an eigenstructure of the form
Equation 3.12
where
, with
; and the columns of
form an orthogonal basis of the subspace
range
(
S
)
null
(
). We
next
define the linear group-blind MMSE detector. As noted in Chapter 2, any linear detector must lie in the space
. The group-blind linear MMSE detector for the
k
th user has the form
, where
and
, such that
suppresses interference from known users in the MMSE sense, and
suppresses interference from unknown users in the MMSE sense. Formally, we have the following definition.
Definition 3.3:
[Group-blind linear MMSE detector ”synchronous CDMA]
Let
be the
components
of the received signal
r
[
i
]
in (3.3) consisting of the signals from known users plus the noise. The weight vector of the group-blind linear MMSE detector for user k is given by
,
where
range
(
)
and
range
(
)
such that
Equation 3.13
Equation 3.14
Note that in general the linear group-blind MMSE detector
m
k
defined above is different from the linear MMSE detector defined in Section 2.2.3, due to the specific structure that the former imposes.
We next give expressions for the three linear group-blind detectors defined above in terms of the known users' signature waveforms
and the unknown users' signal subspace components
and
defined in (3.12).
Proposition 3.1:
[Group-blind linear decorrelating detector (form I) ”synchronous CDMA]
The weight vector of the group-blind linear decorrelating detector for user k is given by
Equation 3.15
Proof:
Decompose
d
k
as
, where
and
. Substituting this into the constraint
in (3.7), we have
Equation 3.16
Hence
d
k
has the form
for some
. Substituting this into the minimization problem in (3.7), we get
Equation 3.17
Equation 3.18
Equation 3.19
Equation 3.20
where (3.17) follows from (3.11); (3.18) follows from the fact that
; (3.19) follows from (3.12); and (3.20) follows from the fact that
. Hence
Equation 3.21
Proposition 3.2:
[Group-blind linear hybrid detector (form I) ”synchronous CDMA]
The weight vector of the group-blind linear hybrid detector for user k is given by
Equation 3.22
Proof:
Decompose
w
k
as
, where
range
(
) and
range
(
). Substituting this into the constraint
in (3.9), we have
Equation 3.23
Hence
for some
. Substituting this into the minimization problem in (3.9), we get
Equation 3.24
Equation 3.25
where (3.24) follows from the fact that
, and (3.25) follows from (3.12). Hence
Equation 3.26
Proposition 3.3:
[Group-blind linear MMSE detector (formI) ”synchronous CDMA]
The weight vector of the group-blind linear MMSE detector for user k is given by
Equation 3.27
Proof:
We first solve for
in (3.13). Since
range
(
), and
has full column rank
, we can write
for some
. Substituting this into (3.13), we have
Equation 3.28
Next we solve
in (3.14) for some
. Following the same derivation as that of (3.25), we obtain
Equation 3.29
Therefore, we have
Equation 3.30
Based on the results above, we can implement the linear group-blind multiuser detection algorithms based on the received signals
and the signature waveforms
of the desired users. For example, the batch algorithm for the group-blind linear hybrid detector (form I) is summarized as follows.
Algorithm 3.1:
[Group-blind linear hybrid detector (form I) ”synchronous CDMA]
-
Compute the unknown users' signal subspace:
Equation 3.31
Equation 3.32
where
is given by (3.10).
-
Form the detectors:
Equation 3.33
-
Perform differential detection:
Equation 3.34
Equation 3.35
The group-blind linear decorrelating detector and the group-blind linear MMSE detector can be implemented similarly. Note that both of them require an estimate of the noise variance
s
2
. A simple
estimator
of
s
2
is the average of the
N
“
K
eigenvalues in
. Note also that the group-blind linear MMSE detector requires an estimate of the inverse of the energy of the desired users,
, as well. The following result can be found in Section 4.5 (cf. Proposition 4.2):
Equation 3.36
Hence
diag
can be estimated by using (3.36) with the signal subspace parameters
replaced
by their respective sample estimates.
In the results above, the linear group-blind detectors are
expressed
in terms of the known users' signature waveforms
and the unknown users' signal subspace components
and
defined in (3.12). Let the eigendecomposition of the autocorrelation matrix
C
r
in (3.11) be
Equation 3.37
The linear group-blind detectors can also be expressed in terms of the signal subspace components
L
s
and
U
s
of all users' signals defined in (3.37), as given by the following three results.
Proposition 3.4:
[Group-blind linear decorrelating detector (form II) ”synchronous CDMA]
The weight vector of the group-blind linear decorrelating detector for user k is given by
Equation 3.38
Proof:
Using the method of Lagrange multipliers to solve the constrained optimization problem (3.7), we obtain
Equation 3.39
where
. Substituting (3.39) into the constraint that
, we obtain
Equation 3.40
Hence
Equation 3.41
where (3.41) follows from (3.11), (3.37), and the fact that
.
Proposition 3.5:
[Group-blind linear hybrid detector (form II) ”synchronous CDMA]
The weight vector of the group-blind linear hybrid detector for user k is given by
Equation 3.42
Proof:
Using the method of Lagrange multipliers to solve the
relaxed
optimization problem (3.9) over
, we obtain
Equation 3.43
where
is the Lagrange
multiplier
and
. Substituting (3.43) into the constraint that
we obtain
Equation 3.44
Hence
Equation 3.45
where (3.45) follows from (3.11), (3.37), and the fact that
. It is seen from (3.45) that
w
k
range
(
U
s
) =
range
(
S
); therefore, it is the solution to the constrained optimization problem (3.9).
To form the group-blind linear MMSE detector in terms of the signal subspace
U
s
, we first need to find a basis for the subspace
range
(
). Clearly,
range
(
) =
range
(
). Consider the (rank-
deficient
) QR factorization of the
N
x
K
matrix
:
Equation 3.46
where
Q
s
is an
N
x
matrix,
R
s
is a
nonsingular upper
triangular
matrix, and
P
is a permutation matrix. Then the columns of
Q
s
form an orthogonal basis of
range
(
).
Proposition 3.6:
[Group-blind linear MMSE detector (form II) ”synchronous CDMA]
The weight vector of the group-blind linear MMSE detector for user k is given by
Equation 3.47
Proof:
Since the columns of
Q
s
form an orthogonal basis of
range
(
), following the same derivation as (3.30), we have
Equation 3.48
Furthermore, we have
Equation 3.49
Equation 3.50
Equation 3.51
Equation 3.52
where (3.49) follows from
, (3.50) follows from
and (3.51) follows from (3.46). Substituting (3.52) into (3.48), we obtain (3.47).
Based on the results above, we can implement the form II linear group-blind multiuser detection algorithms based on the received signals
and the signature waveforms
of the desired users. For example, the batch algorithm for the linear hybrid group-blind detector (form II) is as follows. (The group-blind linear decorrelating detector and the group-blind linear MMSE detector can be implemented similarly.)
Algorithm 3.2:
[Group-blind linear hybrid detector (form II) ”synchronous CDMA]
In summary, for both the group-blind zero-forcing detector and the group-blind hybrid detector, the interfering signals from known users are nulled out by a projection of the received signal onto the orthogonal subspace of these users' signal subspace. The unknown interfering users' signals are then suppressed by identifying the subspace spanned by these users, followed by a linear transformation in this subspace based on the zero-forcing or MMSE criterion. In the group-blind MMSE detector, the interfering users from the known and unknown users are suppressed separately under the MMSE criterion. The suppression of the unknown users again relies on identification of the signal subspace
spanned
by these users.
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