Blind Multiuser Detection in Correlated Noise
So far in developing the subspace-based linear detectors and the channel estimation methods, the ambient channel noise is assumed to be temporally white. In practice, such an assumption may be violated due to, for example, the interference from some narrowband sources. The techniques developed under the white noise assumption are not applicable to channels with correlated ambient noise. In this section we discuss subspace methods for joint suppression of MAI and ISI in multipath CDMA channels with unknown correlated ambient noise, which were first developed in [551]. The key assumption here is that the signal is received by two antennas well separated so that the noise is spatially uncorrelated.
We start with the received augmented discrete-time signal model
given by (2.169). Assume that the
ambient noise vector n[i] has a covariance matrix 
. Then the
Pm x Pm
autocorrelation matrix Cr of the received signal r[i] is given
by
The linear MMSE detector m1 for
user 1 is given by (2.177) with Cr
replaced by (2.218). As before, we must
first estimate the desired user's composite signature waveform 
given by
(2.198). Notice, however, that when the
ambient noise is correlated, it is not possible to separate the signal subspace
from the noise subspace based solely on the autocorrelation matrix Cr.
To estimate the channel in unknown correlated noise, we make use of two antennas at the receiver. Then the two augmented received signal vectors at the two antennas can be written, respectively, as
where H1 and H2 contain the channel information corresponding to the respective antennas. It is assumed that the two antennas are well separated so that the ambient noise is spatially uncorrelated. In other words, n1[i] and n2[i] are uncorrelated, and their joint covariance is given by
where S1 and S2 are unknown covariance matrices of the noise at the two antennas. The joint autocorrelation matrix of the received signal at the two antennas is then given by
where the submatrices are given by
We next consider two methods for estimating the noise subspaces from the signals received at the two antennas.
Singular Value Decomposition
Assume that both H1 and H2 have full column rank 
;
then the matrix C12 also has
rank r. Consider the singular value decomposition
of the matrix C12,
The Pm x Pm diagonal matrix G has the form G = diag(g1, ..., gr, 0, ...,
0), with g1
· · ·
gr > 0. Now if we partition the matrix Uj as
Uj = [Uj,s
Uj,n]
for j = 1, 2, where Uj,s
and Uj,n contain the first r columns and the last Pm -
r columns of Uj, respectively, the column space of Uj,n is
orthogonal to the column space of Hj:
where 
denotes the orthogonal complement
space of range(Hj). User 1's channel corresponding to antenna
j, fj,1,
can then be estimated from the orthogonality relationship
Canonical Correlation Decomposition
Assume that the matrices C11 and C22 are both positive definite. The canonical correlation decomposition (CCD) of the matrix C12 is given by [18]
or
The Pm x Pm matrix G has the form G = diag(g1, ..., gr, 0, ...,
0), with g1
···
gr > 0. Let
Partition the matrix Lj such that
where Lj,s and Lj,n are the first r columns and the last Pm - r columns of Lj, respectively. The matrix Uj are similarly partitioned into Uj,s and Uj,n. We have [580].
Note, however, that Lj,s does not necessarily span the signal subspace range(Hj) [580].
Now suppose that we have estimated the composite signature
waveform of the desired user 
, using the identified noise subspace
Lj,n. Since 
, we
have
where the second equality in (2.234) follows from (2.231) and the fact that Uj is a
unitary matrix; and the third equality follows from the fact that 
.
Let the estimated weight vectors of the linear MMSE detectors
at the two antennas be 
. To make use of the received signal
at both antennas, we use the following equal-gain differential combining rule
for detecting the differential bit b1[i]:
We next summarize the procedures for computing the linear MMSE
detector 
in unknown correlated noise based on the discussion above.
Let
be the matrix of M received augmented signal sample vectors at antenna j corresponding to one block of data transmission.
Algorithm 2.9: [Blind linear MMSE detector in multipath CDMA with correlated noise—SVD-based method]
-
Compute the auto- and cross-correlation matrices:
-
Perform an SVD on
to get the noise subspace 
. -
Compute the composite signature waveforms
by solving
-
Form the linear MMSE detectors:
Algorithm 2.10: [Blind linear MMSE detector in multipath CDMA with correlated noise—CCD-based method]
-
Perform QR decomposition:
-
Perform an SVD on
:
-
Compute
where
is an upper triangular matrix.
-
Partition
. Compute
the composite signature waveforms 
by solving
-
Form the linear MMSE detectors:
The procedure above is based on the fast algorithm for computing CCD given in [580].
Note that the two methods above operate on the Pm-dimensional signal vectors rj[i], j = 1,2. The same procedures can be applied to the
decimated received signal vectors to operate on the Nm-dimensional signal vectors rj,q[i], j = 1,2, q = 0, ...,
p - 1. As before, such decimation-combining
approach reduces the computational complexity by a factor of 
. It also
reduces the number of users that can be accommodated in the system by a factor
of p.
Simulation Examples
We illustrate the performance of the detectors above via simulation examples. The simulated system is the same as that in Section 2.7.3, except that the ambient noise is temporally correlated. The noise at each antenna j is modeled by a second-order autoregressive (AR) model with coefficients aj = [aj,1, aj,2]; that is, the noise field is generated according to
where nj[i] is the noise at antenna j and sample i, and wj[i] is a complex white Gaussian noise sample with unit variance. The AR coefficients at the two arrays are chosen as a1 = [1, -0.2] and a2 = [1.2, -0.3].
We first consider a five-user system. In Fig. 2.18 the performance of the Pm-dimensional blind linear MMSE detectors is plotted for both SVD- and CCD-based methods. The corresponding performance by the decimation-combining receiver structure is plotted in Fig. 2.19. Next a 10-user system is simulated and the performance of the Pm-dimensional blind linear MMSE detectors is plotted in Fig. 2.20.
Figure 2.18. Performance of Pm-dimensional blind linear MMSE detectors in a five-user system with correlated noise.
Figure 2.19. Performance of decimation-combining blind linear MMSE detectors in a five-user system with correlated noise.
Figure 2.20. Performance of Pm-dimensional blind linear MMSE detectors in a 10-user system with correlated noise.
It is seen from Figs.
2.18–2.20 that CCD-based detectors
are superior in performance to SVD-based detectors. It has been shown that the
CCD has the optimality property of maximizing the correlation between the two
sets of linearly transformed data [18]. Maximizing the correlation of
the two data sets can yield the best estimate of the correlated (i.e., signal)
part of the data. CCD makes use of the information of both 
and 
together with 
and creates the maximum correlation between the two data
sets. On the other hand, SVD uses only the information and does
not create the maximum correlation between the two data sets, and thus yields
inferior performance.