Performance of Blind Multiuser Detectors
2.5.1 Performance Measures
In previous sections we have discussed two approaches to blind multiuser detection: the direct method and the subspace method. These two approaches are based primarily on two equivalent expressions for the linear MMSE detector [i.e., (2.26) and (2.78)]. When the autocorrelation Cr of the received signals is known exactly, the two approaches have the same performance. However, when Cr is replaced by the corresponding sample autocorrelation, quite interestingly, the performance of these two methods is very different. This is due to the fact that these two approaches exhibit different estimation errors on the estimated detector [193, 194, 197]. In this section we present a performance analysis of the two blind multiuser detectors: the DMI blind detector and the subspace blind detector. For simplicity, we consider only real-valued signals [i.e., in (2.4), Ak > 0, k=1, ..., K and n[i]~N(0, s2IN)].
Suppose that a linear weight vector 
is applied
to the received signal r[i] in (2.5). The output is given by
(2.10). Since
it is assumed that the user bit streams are independent and the noise is
independent of the user bits, the signal-to-interference-plus-noise ratio (SINR)
at the output of the linear detector is given by
The bit-error probability of the linear detector using weight vector w1 is given by
Now suppose that an estimate 
of the weight vector w1 is obtained from the received
signals 
. Denote
Obviously, both 
are random vectors and are functions
of the random quantities 
. In typical adaptive multiuser detection scenarios [183, 549], the estimated detector 
is employed to demodulate future received signals, say r [j], j 
M. Then the output is given by
where the first term in (2.118) represents the output of the true weight vector
w1, which has the same form as
(2.10). The
second term in (2.118) represents an
additional noise term caused by the estimation error Dw1. Hence
from (2.118) the average SINR at the
output of any unbiased estimated linear detector is given
by
with
where 
. Note that in batch processing, on the other hand, the estimated
detector is used to demodulate signals r[i], 0 
i M - 1. Since
Dw1 is
a function of 
, for fixed i, Dw1 and r[i] are in
general correlated. For large M, such correlation
is small. Therefore, in this case we still use (2.119) and (2.120) as the approximate SINR expression.
If we assume further that Dw1 is actually independent of r[i], the average bit-error rate (BER) of this detector is given by
where 
is given by (2.116) and 
denotes the probability density
function (pdf) of the estimated weight vector .
From the discussion above it is seen that to obtain the average
SINR at the output of the estimated linear detector , it
suffices to find its covariance matrix Cw. On
the other hand, the average bit-error rate of the estimated linear detector
depends on its distribution through .
2.5.2 Asymptotic Output SINR
We first present the asymptotic distribution of the two forms of blind linear MMSE detectors for a large number of signal samples, M. Recall that in the direct-matrix-inversion (DMI) method, the blind multiuser detector is estimated according to
In the subspace method, the estimate of the blind detector is given by
where 
and 
contain,
respectively, the largest K eigenvalues and the
corresponding eigenvectors of 
; and where 
contain,
respectively, the remaining eigenvalues and eigenvectors of . The
following result gives the asymptotic distribution of the blind linear MMSE
detectors given by (2.123) and (2.125). The proof is given in the Appendix
(Section
2.8.3).
Theorem 2.1: Let w1 be the true weight vector of the linear MMSE detector given by
and let be the weight vector of the estimated blind linear MMSE
detector given by (2.123) or (2.125). Let the eigendecomposition of the
autocorrelation matrix Cr of the received signal be
Then
with
where
Hence for large M, the
covariance of the blind linear detector, 
, can be
approximated by (2.128). Define, as
before,
The next result gives an expression for the average output SINR, defined by (2.119), of the blind linear detectors. The proof is given in the Appendix (Section 2.8.3).
Corollary 2.1: The average output SINR of the estimated blind linear detector is given by
where
It is seen from (2.132)
that the performance difference between the DMI blind detector and the subspace
blind detector is caused by the single parameter t given by (2.130)—the detector with a smaller t has a higher output SINR.
Let m1, ...,
mK be the
eigenvalues of the matrix R given by (2.131). Denote mmin = min1
k K {mk} and mmax = max1
k K {mk}. Denote also
Amin = min1 k K {Ak} and Amax = max1 k
K {Ak}. The next result gives sufficient
conditions under which one blind detector outperforms the other in terms of the
average output SINR.
Corollary 2.2: If 
,
then 
; and if 
, then 
.
Proof: By rewriting (2.130) as
we obtain the following sufficient condition under which tsubspace < tDMI:
On the other hand, note that
Since the nonzero eigenvalues of SST are the same of those of R = ST S, it follows from (2.138) that
The first part of the corollary then follows by combining (2.137) and (2.139). The second part of the corollary follows a similar proof.
The next result gives an upper and a lower bound on the parameter t in terms of the desired user's amplitude A1, the noise variance s2, and the two extreme eigenvalues of Cr.
Corollary 2.3: The parameter t defined in (2.130) satisfies
Proof: The proof follows from (2.136) and the following fact from Chapter 4 [cf. Proposition 4.2]:
To gain some insight from the result (2.132), we next consider two special cases for which we compare the average output SINRs of the two blind detectors.
Example 1: Orthogonal Signals
In this case, we have uk = sk,
R = IK, and
, k = 1, ..., K. Substituting these into (2.136), we obtain
Substituting (2.141)
into (2.132), and using the fact that in
this case 
, we obtain the following expressions of the average output
SINRs:
where 
is the signal-to-noise ratio (SNR)
of the desired user. It is easily seen that in this case, a necessary and
sufficient condition for the subspace blind detector to outperform the DMI blind
detector is that f1 > 1 (i.e., SNR1 > 0
dB).
Example 2: Equicorrelated Signals
with Perfect Power Control In this case it is assumed that 
, for k
l, 1 k, l K. It is also
assumed that A1= ··· = AK =
A. It is shown in the Appendix (Section 2.8.3) that the
average output SINRs for the two blind detectors are given by
with
and
A necessary and sufficient condition for the subspace blind
detector to outperform the DMI blind detector is 
, which
after some manipulation reduces to
where 
and where 
and 
are the two distinct eigenvalues of R [cf.
the Appendix (Section 2.8.3)]. The
region on the SNR–r
plane where the subspace blind detector outperforms the DMI blind detector is
plotted in Fig. 2.2 for different values
of K. It is seen that in general the subspace
method performs better in the low cross-correlation and high-SNR region.
Figure 2.2. Partition of the SNR–r plane according to the relative performance of two blind detectors. For each K, in the region above the boundary curve, the subspace blind detector performs better, whereas in the region below the boundary curve, the DMI blind detector performs better.
The average output SINR as a function of SNR and r for both blind detectors is shown in Fig. 2.3 It is seen that the performance of the subspace blind detector deteriorates in the high-cross-correlation and low-SNR region; the performance of the DMI blind detector is less sensitive to cross-correlation and SNR in this region. This phenomenon is shown more clearly in Figs. 2.4 and 2.5, where the performance of the two blind detectors is compared as a function of r and SNR, respectively. The performance of the two blind detectors as a function of the number of signal samples M is plotted in Fig. 2.6, where it is seen that for large M, both detectors converge to the true linear MMSE detector, with the subspace blind detector converging much faster than the DMI blind detector; and the performance gain offered by the subspace detector is quite significant for small values of M. Finally, in Fig. 2.7, the performance of the two blind detectors is plotted as a function of the number of users K. As expected from (2.132), the performance gain offered by the subspace detector is significant for smaller values of K, and the gain diminishes as K increases to N. Moreover, it is seen that the performance of the DMI blind detector is insensitive to K.
Figure 2.3. Average output SINR versus SNR and r for two blind detectors. N = 16, K = 6, M = 150. The upper curve in the high-SNR region represents the performance of the subspace blind detector.
Figure 2.4. Average output SINR versus r for two blind detectors. N = 16, K = 6, M = 150, SNR = 15 dB.
Figure 2.5. Average output SINR versus SNR for two blind detectors. N = 16, K = 6, M = 150, r = 0.4.
Figure 2.6. Average output SINR versus the number of signal samples M for two blind detectors. N = 16, K = 6, r = 0.4, SNR = 15 dB.
Figure 2.7. Average output SINR versus the number of users K for two blind detectors. N = 16, M = 150, r = 0.4, SNR = 15 dB.
Simulation Examples
We consider a system with K = 11 users. The users' spreading sequences are randomly generated with processing gain N = 13. All users have the same amplitudes. Figure 2.8 shows both the analytical and simulated SINR performance for the DMI blind detector and the subspace blind detector. For each detector the SINR is plotted as a function of the number of signal samples (M) used for estimating the detector at some fixed SNR. The simulated and analytical BER performance of these estimated detectors is shown in Fig. 2.9. The analytical BER performance is evaluated using the approximation
which effectively treats the output interference plus noise of the estimated detector as having a Gaussian distribution. This can be viewed as a generalization of the results in [386], where it is shown that the output of an exact linear MMSE detector is well approximated with a Gaussian distribution. From Figs. 2.8 and 2.9 it is seen that the agreement between the analytical performance assessment and the simulation results is excellent for both the SINR and BER. The mismatch between analytical and simulation performance occurs for small values of M, which is not surprising since the analytical performance is based on an asymptotic analysis.
Figure 2.8. Output average SINR versus the number of signal samples M for DMI and subspace detectors. N = 13, K = 11. The solid line is the analytical performance, and the dashed line is the simulation performance.
Figure 2.9. BER versus the number of signal samples M for DMI and subspace detectors. N = 13, K = 11. The solid line is the analytical performance, and the dashed line is the simulation performance.
Finally, we note that although in this section we treated only the performance analysis of blind multiuser detection algorithms in simple real-valued synchronous CDMA systems, the analysis for the more realistic complex-valued asynchronous CDMA with multipath channels and blind channel estimation can be found in [196]. Some upper bounds on the achievable performance of various blind multiuser detectors are obtained in [192, 195]. Furthermore, large-system asymptotic performance analysis of blind multiuser detection algorithms is given in [604].
