Linear Group-Blind Detectors
As before, the basic idea behind group-blind linear detectors
is to suppress the interference from known users based on the spreading
sequences of these users and to suppress the interference from other, unknown
users using subspace-based blind methods. Analogous to the synchronous case, we
have the following three types of linear group-blind detectors. (In this
section, ek denotes an  -vector with
all elements zeros except for the kth, which is
1.)
Definition 3.4: [Group-blind
linear decorrelating detector—multipath CDMA] The weight
vector of the group-blind linear decorrelating detector for user k is given by
the solution to the following constrained optimization problem:
Equation 3.153
Definition 3.5: [Group-blind
linear hybrid detector—multipath CDMA] The weight vector
of the group-blind linear hybrid detector for user k is given by the solution to
the following constrained optimization problem:
Equation 3.154
Definition 3.6: [Group-blind
linear MMSE detector—multipath CDMA] Let 
be the components of the received signal r[i] in (3.148) consisting of signals from known users plus
noise. ( [i] is
the subvector of b[i] containing bits of the
desired users.) The weight vector of the group-blind linear MMSE detector for
user k is given by  , where  
range( ) and range( ) [note that is given in (3.152)], such that
Equation 3.155
Equation 3.156
The following results give expressions for the three
group-blind linear detectors defined above in terms of the known users' channel
matrix and the unknown users' signal subspace components  and
defined in (3.152). The proofs of these
results are similar to those corresponding to the synchronous case.
Proposition 3.7: [Group-blind
linear decorrelating detector (form I)—multipath CDMA] The weight vector of the group-blind linear decorrelating
detector for the kth user is given by
Equation 3.157
Proposition 3.8: [Group-blind
linear hybrid detector (form I)—multipath CDMA] The
weight vector of the group-blind linear hybrid detector for the kth user is given by
Equation 3.158
Proposition 3.9: [Group-blind
linear MMSE detector (form I)—multipath CDMA] The weight
vector of the group-blind linear MMSE detector for the kth user is given by
Equation 3.159
Note that to implement these group-blind linear detectors, the
matrix must be estimated first. The blind channel estimation procedure is
discussed in Section 2.7.3. The channel
estimator discussed there can be used to estimate the channel for each desired
user. Once the desired users' channels are estimated, the matrix can be formed.
As before, the blind channel estimator has an arbitrary phase ambiguity, which
necessitates the use of differential encoding and decoding of the data bits. We
next summarize the group-blind linear hybrid multiuser detection algorithm in
multipath channels.
Algorithm 3.5: [Group-blind
linear hybrid detector (form I)—multipath CDMA]
-
Compute the signal subspace:
Equation 3.160
Equation 3.161
-
Estimate the desired users' channels
(cf. Section
2.7.3):
Equation 3.162
Equation 3.163
Form  using  .
-
Compute the unknown users'
subspace:
Equation 3.164
Equation 3.165
-
Form the detectors:
Equation 3.166
-
Perform differential
detection:
Equation 3.167
Equation 3.168
Note that the group-blind linear decorrelating detector and the
group-blind linear MMSE detector can be implemented similarly. Both require an
estimate of s2, which can be obtained simply as the
mean of the noise subspace eigenvalues  .
Alternatively, the group-blind linear detectors can be
expressed in terms of the signal subspace components Ls and Us of
all users defined in (3.150), as given
by the following three results. The proofs are again similar to their
counterparts in the synchronous case.
Proposition 3.10: [Group-blind
linear decorrelating detector (form II)—multipath CDMA] The group-blind linear decorrelating detector for the kth user is given by
Equation 3.169
Proposition 3.11: [Group-blind
linear hybrid detector (form II)—multipath CDMA] The
group-blind linear hybrid detector for the kth
user is given by
Equation 3.170
Proposition 3.12: [Group-blind
linear MMSE detector (form II)—multipath CDMA] Let the
(rank-deficient) QR factorization of the Pm x r matrix  Us
be
Equation 3.171
where Qs is a  matrix, Rs is
a  nonsingular upper triangular matrix,
and P is a permutation
matrix. The group-blind linear MMSE detector for the kth user is given by
Equation 3.172
Finally, we summarize the form II group-blind linear hybrid
multiuser detection algorithm in multipath channels as follows.
Algorithm 3.6: [Group-blind
linear hybrid detector (form II)—multipath CDMA]
-
Compute the signal subspace:
Equation 3.173
Equation 3.174
-
Estimate the desired users' channels
(cf. Section
2.7.3):
Equation 3.175
Equation 3.176
Form using .
-
Form the detectors:
Equation 3.177
-
Perform difierential
detection:
Equation 3.178
Equation 3.179
It is seen that form I group-blind detectors are based on an
estimate of the signal subspace of the matrix  , whereas
form II group-blind detectors are based on an estimate of the signal subspace of
the matrix Cr. If the signal subspace dimension 
of is less than that of Cr, which is  , form I
implementation in general gives a more accurate estimation of group-blind
detectors. On the other hand, for multipath channels, estimation of the given
users' channels is based on eigendecomposition of Cr.
Hence form II group-blind detectors are more efficient in terms of
implementation since they do not require the eigendecomposition (3.152), which is required by form I
group-blind detectors. If however, the channels are estimated by some other
means not involving the eigendecomposition of Cr,
form I detectors can be computationally less complex than form II detectors,
since the dimension of the estimated signal subspace of the former is less than
that of the latter. (That is, of course, if computationally efficient subspace
tracking algorithms [98] are used instead of the
conventional eigendecomposition.)
Simulation Examples
Next, we provide computer simulation results to demonstrate the
performance of the proposed blind and group-blind linear multiuser detectors
under a number of channel conditions. The simulated system is an asynchronous
CDMA system with processing gain N = 15. Employed
as the user spreading sequences are m-sequences
of length 15 and their shifted versions. The chip pulse is a raised cosine pulse
with roll-off factor 0.5. Each user's channel has L = 3 paths. The delay of each path is uniform on [0,
10Tc]. Hence the maximum delay spread
is one symbol interval (i.e., i = 1). The fading
gain of each path in each user's channel is generated from a complex Gaussian
distribution and fixed for all simulations. The path gains in each user's
channel are normalized so that all users' signals arrive at the receiver with
the same power. The oversampling factor is p = 2.
The smoothing factor is m = 2. Hence this system
can accommodate up to  = 10 users. The number of users in
the simulation is 10, with seven known users (i.e., K = 10 and = 7). The length of each user's signal
frame is M = 200.
In each simulation, an eigendecomposition is performed on the
sample autocorrelation matrix of the received signals. The signal subspace
consists of the eigenvectors corresponding to the largest r eigenvalues. [Recall that  is the
dimension of the signal subspace.] The remaining eigenvectors constitute the
noise subspace. An estimate of the noise variance s2 is given by the average of the Pm – r smallest eigenvalues.
We first compare the performance of four exact detectors (i.e., assuming that H and s2 are known):
-
The linear MMSE detector
-
The linear zero-forcing detector
-
The group-blind linear hybrid detector
-
The group-blind linear MMSE
detector
For each of these detectors, and for each SNR value, the
minimum and maximum bit-error rate (BER) among the seven known users is plotted
in Fig. 3.13. It is seen from this figure
that, as expected, the closer the detector is to the true linear MMSE detector,
the better its performance is.
Next, the performance of the various estimated group-blind
detectors (i.e., the detectors are estimated based on the M received signal vectors) is shown in Fig. 3.14. It is seen that at low SNR, the group-blind
MMSE detectors perform best, whereas at high SNR, the group-blind hybrid
detectors perform best. This is because the hybrid detector zero-forces the
known users' signals and enhances the noise level, whereas the group-blind
linear MMSE detector suppresses both the interference and the noise. At high
SNR, the group-blind hybrid and group-blind MMSE detectors tend to become the
same. However, implementation of the latter requires an estimate of the noise
level. When the noise level is low, this estimate is noisy, which causes the
performance of the group-blind MMSE detector to deteriorate. It is also seen
that the performance of form I detectors is only slightly better than that of
the corresponding form II detectors, at the expense of higher computational
complexity.
Comparing Fig. 3.13 with
Fig. 3.14, it is seen that the
performance of the estimated detectors is substantially diffierent from that of
the corresponding exact detectors for the block size considered here (i.e.,
M = 200). It is known that the subspace detectors
converge to the exact detectors at a rate of  ( ). It is also seen from Fig. 3.14 that the form II hybrid detector
performs very well compared with other forms of group-blind detectors, even
though it has the lowest computational complexity. Hence in subsequent
simulation studies, we compare the performance of the form II hybrid detector
with that of some multiuser detectors proposed previously.
We next compare the performance of the group-blind hybrid
detector with that of the blind detector for the same system. The result is
shown in Fig. 3.15, where the BER curves
for the blind linear MMSE detector, the form II group-blind linear hybrid
detector, and a partial MMSE detector are plotted. The partial MMSE detector
ignores the unknown users and forms the linear MMSE detector for the
known users using the estimated matrix . It is seen that the group-blind
detector significantly outperforms the blind MMSE detector and the partial MMSE
detector. Indeed, the blind MMSE detector exhibits an error floor at high SNR
values. This is due to the finite length of the received signal frame, from
which the detector is estimated. The group-blind hybrid MMSE detector does not
show an error floor in the BER range considered here. Of course, due to the
finite frame length, the group-blind detector also has an error floor. But such
a floor is much lower than that of the blind linear MMSE detector.
Theoretically, both the blind and group-blind detectors
converge to the true linear MMSE detector (at a high signal-to-noise ratio) as
the signal frame size M   . Hence the
asymptotic performance of the two detectors is the same at high signal-to-noise
ratio. However, for a finite frame length M, the
group-blind detector performs significantly better than the blind detector, as
seen from the simulation results above. An intuitive explanation for such
performance improvement is that more information about the multiuser environment
is incorporated in forming the group-blind detector. For example, the
computations for subspace decomposition and channel estimation involved in the
two detectors are exactly the same. However, the blind detector is formed based
solely on the composite channel of the desired user, whereas the group-blind
detector is formed based on the composite channels of all known users. By
incorporating more information about the multiuser channel, the estimated
group-blind detector is more accurate than the estimated blind detector (i.e.,
the former is "closer" than the latter to the exact detector).
It is seen from Fig.
3.13 that when the spreading waveforms and the channels of all users are
known, all three forms of the exact group-blind detectors perform worse than the
linear MMSE detector, which is the exact blind detector. This is because the
zero-forcing and hybrid group-blind detectors zero-force all or some users'
signals and enhance the noise level, whereas the group-blind MMSE detector is
defined in terms of a specific constrained form which in general is different
from the true MMSE detector. However, with imperfect channel information, the
roles are reversed and the group-blind detectors outperform the blind detector.
Of course, both the blind and group-blind detectors are developed based on the
assumption that the multiuser channel is not perfectly known, and a study of the
performance of the exact detectors is only of theoretical interest.
Nevertheless, it is interesting to observe that by changing the assumption on
prior knowledge about the channel, the relative performance of two detectors can
be different.
| 
Sections
Syndication
