Form I Group-Blind Detectors
Define
Equation 3.89
Equation 3.90
and let 
. The following result gives the
asymptotic distribution of the estimated weight vector of the form I linear
group-blind hybrid detector and that of the form I linear group-blind MMSE
detector. The proof is given in the Appendix (Section 3.6.2).
Theorem 3.2: Let
Equation 3.91
be the sample autocorrelation matrix of
the received signals based on M samples. Define
Equation 3.92
Let 
and 
contain,
respectively, the largest 
eigenvalues
of 
and the corresponding eigenvectors.
Let 
be the estimated weight vector of the form I
group-blind linear detector, given by
Equation 3.93
where 
for the group-blind linear hybrid detector and 
for the group-blind linear MMSE detector.
Then
with
Equation 3.94
where
Equation 3.95
Equation 3.96
As before, the SINRs for the form I group-blind detectors can
be expressed in terms of R, s2, and A. However, the closed-form SINR expressions are
too complicated for this case and we therefore do not present them here.
Nevertheless, the SINR of the group-blind linear hybrid detector for orthogonal
signals can be obtained explicitly, as in the following example.
Example 1: Orthogonal Signals
We consider the form I linear hybrid detector. In this case w1 = u =
s1, 
, and 
=
diag 
.
Moreover, 
, so that 
, and 
= 0. Hence 
. Substituting these into (2.119), and
after some manipulation, we get
Equation 3.97
Comparing (3.73) and (3.97), we see that for the orthogonal-signal
case, the form I group-blind hybrid detector always outperforms the form II
group-blind hybrid detector.
In Fig. 3.6 the output
SINR of the two blind detectors and that of the two forms of group-blind hybrid
detectors [given, respectively, by (2.142), (3.73), and (3.97)]
are plotted as functions of the desired user's SNR, f1. It is seen that
in the high-SNR region, the DMI blind detector has the worst performance among
these detectors. In the low-SNR region, however, both the form II group-blind
detector and the subspace blind detector perform worse than the DMI blind
detector. The form I group-blind detector performs the best in this case.
Example 2: Equicorrelated Signals
with Perfect Power Control Although we do not present a closed-form
expression for the output SINR for the form I group-blind detector, we can still
evaluate the SINR for this case as follows. As noted above, the SINR is a
function of the user spreading sequences S
only through the correlation matrix 
. In other words, with the same A and, s2, systems employing different set of
spreading sequences S and S' will have the same SINR as long as 
[even if the spreading sequences take real values rather than the form 
, sj,k 
{+1, –1}]. Hence given R, A, and
s2, we can,
for example, designate S to be of the
form
Equation 3.98
(where 
denotes the Cholesky factor of R) and then use (2.119) and (3.94) to compute the SINR. Note that each column of S in (3.98) has unit norm since the diagonal elements of R are all 1s. Our computation shows that the
performance of the form I group-blind hybrid detector is similar to that of the
form II group-blind hybrid detector, with the exception that the form I detector
behaves similarly to the DMI blind detector in the very low SNR region—namely,
it does not deteriorate as much as do the form I group-blind and subspace blind
detectors. This is shown in Fig. 3.7.
(The performance of the form I group-blind MMSE detector is indistinguishable
from that of the form I group-blind hybrid detector in this case.)
In summary, we have seen that the performance of the subspace
blind detector deteriorates in the low-SNR and high-cross-correlation region,
the form II group-blind detector is resistant to high cross-correlation but not
to low SNR, and the form I group-blind detector is resistant to both high
cross-correlation and low SNR. Although the DMI blind detector is also
insensitive to both high cross-correlation and low SNR, its performance in other
regions is inferior to all the subspace-based blind and group-blind detectors.
Hence we conclude that the form I group-blind detector achieves the best overall
performance among all the detectors considered here.
Finally, we compare the analytical performance expressions
given in this section with the simulation results. The simulated system is the
same as that in Section 2.5 (N = 13, K = 11). The
analytical and simulated SINR performance of the form I and form II group-blind
detectors is shown in Fig. 3.8. 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. 3.9. As
before, the analytical BER performance is based on a Gaussian approximation of
the output of the estimated linear detector. It is seen that as is true for the
DMI blind detector and the subspace detector treated in Section 2.5, the analytical
performance expressions discussed in this section for group-blind detectors
match the simulation results very well. Performance analysis for the group-blind
detectors in 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 group blind multiuser detectors are obtained
in [192, 195]. Moreover,
large-system asymptotic performance of the group-blind multiuser detectors is
given in [604].
Sections
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