Robust Group-Blind Multiuser Detection in Multipath Channels
We now turn to the group-blind version of the robust multiuser
detector for the multipath channel. As before, we can rewrite (4.150) as
Equation 4.157
Equation 4.158
where  and  [i] contain the data bits in b [i]
corresponding to sets of desired users and undesired users, respectively;  and
contain columns of H corresponding to
desired users and undesired users, respectively. As discussed in Section
2.7.3, based on the knowledge of the spreading waveforms  of the desired
users, by exploiting the orthogonality between the signal subspace and the noise
subspace, we can blindly estimate up to a scale and phase ambiguity for
each user. With such an estimate, we can write
Equation 4.159
where the term  contains the signal carrying the
current bits   of the desired users; and the term
 contains the signal carrying the previous and future bits 
(i.e., the intersymbol interference). Note that in (4.159) the term  represents the estimated channel for
the desired users' current bits, and  is a diagonal matrix containing the
complex scalars of ambiguities; the term  represents
the estimated channel for the desired users' past and future bits, and qI[i] contains the
products of those bits and the complex ambiguities of the corresponding
channels. Following the method outlined in Section 4.7, we first
obtain a robust estimate of z[i] and then translate it into the estimate of 
[i] by again applying Proposition 4.2:
Equation 4.160
Next, we obtain a robust estimate of the sum of the undesired
users' signals based on the relationship
Equation 4.161
Equation 4.162
where  represents the signal subspace obtained
from the eigendecomposition of the autocorrelation matrix of  [i]. Finally, we subtract the estimated undesired users'
signals and the intersymbol interference from r[i] to obtain
Equation 4.163
Equation 4.164
Note that the complex ambiguities in can be
estimated based on the estimate of [i], as
discussed in Section 4.7. Note also that
(4.164) has the same form as (4.141), and
hence similarly to (4.143)–(4.146), the
slowest-descent-search method can then be employed to obtain a robust estimate
of  from (4.164). The algorithm is
summarized below.
Algorithm 4.7: [Robust
group-blind multiuser detector—multipath CDMA]
-
Compute the sample autocorrelation
matrix of the received augmented signal r[i] and its eigendecomposition.
-
Compute the robust estimate of
z[i] following a procedure similar to (4.128)-(4.132).
-
Compute a blind estimate of
according to (3.162)-(3.163).
-
Compute the output of the robust blind
detector according to (4.160).
-
Compute the sum of the undesired users'
signals [i] according to (4.161);
compute the sample autocorrelation matrix of the signal [i] and its
eigendecomposition.
-
Compute the robust estimate of
 [i] in (4.162) following a procedure similar to (4.128)-(4.132).
-
Compute the sum of the desired users'
signals  [i] according to (4.163).
-
Estimate the complex amplitudes of
ambiguities introduced by the blind estimator
based on the robust estimate of [i]
using (3.127)-(3.129) [cf. (3.134)-(3.140)].
-
Form the Huber penalty function and
apply the slowest-descent search of , similar to (4.143)–(4.146).
-
Perform differential
decoding.
Simulation Examples
In the following simulation, the number of users is K = 8 and the spreading gain is N = 15. Each user's channel is assumed to have L = 3 paths and a delay spread of up to one symbol. The
complex gains and the delays of each user's channel are generated randomly. The
chip pulse is a raised cosine pulse with roll-off factor 0.5. The path gains are
normalized so that each user's signal arrives at the receiver with unit power.
The channel is normalized in such a way that the composite of the multipath
channel and the spreading waveform has unit power. The noise parameters are
 = 0.01 and
k = 100. The smoothing factor is m = 2 and the oversampling factor is p = 2. Shown in Fig.
4.14 is the BER performance of the robust blind multiuser detector and that
of the robust group-blind multiuser detector ( = 4). It is
seen that in the presence of both non-Gaussian noise and multipath channel
distortion, the group-blind robust detector substantially improves the
performance of the blind robust detector. Furthermore, most of the performance
gain offered by the slowest-descent search is obtained by searching along only
one direction.
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