Linear Group-Blind Multiuser Detection for Synchronous CDMA
We start by considering the following discrete-time signal model for a synchronous CDMA system:
where, as before, K is the total
number of users; Ak, bk[i], and
sk are, respectively, the complex
amplitude, ith transmitted bit, and signature
waveform of the kth user; n[i] ~ Nc(0, s2 IN) 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
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 wk 
for each desired user k, 1 
k , such that the
data bits of these users can be demodulated according to
and
or
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 kth element,
which is 1.
Definition 3.1: [Group-blind linear decorrelating detector—synchronous CDMA] The weight vector dk of the linear decorrelating detector for user k is given by the solution to the following constrained optimization problem:
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
Under the constraint 
, we have 
. It then
follows that for w range(S), ||wH SA||2 is minimized subject to
if and only if wH sl = 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 wk of the group-blind linear hybrid detector for user k is given by the solution to the following constrained optimization problem:
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:
which projects any signal onto the subspace null(
). Recall that the autocorrelation matrix of the received signal in (3.1) is given by
where 
. It is then easily seen that the
matrix 
has an eigenstructure of the form
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 kth 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
Note that in general the linear group-blind MMSE detector mk 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
Proof: Decompose dk as
, where 
and 
.
Substituting this into the constraint 
in (3.7), we have
Hence dk has the form 
for some
. Substituting this into the minimization problem in (3.7), we get
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
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
Proof: Decompose wk as
, where 
range() and
range(). Substituting
this into the constraint 
in (3.9), we have
Hence 
for some 
.
Substituting this into the minimization problem in (3.9), we get
where (3.24) follows
from the fact that 
, and (3.25) follows from (3.12). Hence
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
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
Next we solve 
in (3.14) for some 
. Following the same derivation as
that of (3.25), we obtain
Therefore, we have
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:
where
is given by (3.10). -
Form the detectors:
-
Perform differential detection:
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):
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 Cr in
(3.11) be
The linear group-blind detectors can also be expressed in terms of the signal subspace components Ls and Us 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
Proof: Using the method of Lagrange multipliers to solve the constrained optimization problem (3.7), we obtain
where 
. Substituting (3.39) into the constraint that 
, we
obtain
Hence
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
Proof: Using the method of
Lagrange multipliers to solve the relaxed optimization problem (3.9) over 
, we obtain
where 
is the Lagrange multiplier and 
.
Substituting (3.43) into the constraint
that 
we obtain
Hence
where (3.45) follows
from (3.11), (3.37), and the fact that 
. It is
seen from (3.45) that wk range(Us) =
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 Us, 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 :
where Qs is an N x
matrix, Rs is a 
nonsingular upper triangular matrix,
and P is a permutation
matrix. Then the columns of Qs 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
Proof: Since the columns of
Qs form an orthogonal basis of range(),
following the same derivation as (3.30),
we have
Furthermore, we have
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]
-
Compute the signal subspace:
-
Form the detectors:
-
Perform differential detection:
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.