Sliding Window Group-Blind Detector for Asynchronous CDMA
It is not difficult to extend the results of Section 6.4.2 to asynchronous CDMA. The received signal
due to user k(1 
k
K) is given by
Equation 6.119
where tk is the
delay of the kth user's signal, {cj,k}
is a signature sequence of 
1's assigned to the kth user, and y(t) is a normalized
chip waveform of duration Tc = T/N. The total
received signal, given by
Equation 6.120
is match-filtered to the chip waveform and sampled at the chip
rate. The nth matched-filter output during the
ith symbol interval is
Equation 6.121
Substituting (6.119)
into (6.121), we obtain
Equation 6.122
where 
. Then
Equation 6.123
Denote
Equation 6.124
and for j = 0, 1, ..., Ik –1,
Equation 6.125
Then
Equation 6.126
By stacking 
successive received sample vectors,
we define
Equation 6.127
Equation 6.128
where 
. Then we can write the received
signal in matrix form as
Equation 6.129
Define the set of matrices 
such that
is the 
matrix composed of columns jK + 1 through jK + 
of the matrix H. We define the matrix 
.
The size of 
is NI x (2I - 1). We denote by 
the matrix that contains the remaining (2I - 1) columns of H. We define 
[i] and 
[i] by
performing a similar separation of the elements of b[i]. Then we may write (6.129) as
Equation 6.130
This equation is the asynchronous analog to (6.76). We can obtain estimates of 
with
straightforward modifications to Algorithm
6.3.
Simulation Examples
We next present simulation results to demonstrate the
performance of the proposed turbo group-blind multiuser receiver for
asynchronous CDMA. The processing gain of the system is seven and the total
number of users is seven. The number of known users is either two or five, as
noted on the figures. The spreading sequences are randomly generated and the
same sequences are used for all simulations. All users employ the same rate-½,
constraint-length-3 convolutional code (with generators g1 = [110] and g2 = [111]). Each user uses a different
random interleaver, and the same interleavers are used in all simulations. The
block size of information bits for each user is 128. The maximum delay in symbol
intervals is 1. All users use the same transmitted power and the chip pulse
waveform is a raised cosine with roll-off factor 0.5.
Figure 6.11 illustrates
the average bit-error-rate performance of the known users for the group-blind
turbo receiver and the conventional turbo receiver discussed in Section 6.3 for the first
four iterations. The number of known users is five. For the sake of comparison,
we have included plots for the conventional turbo receiver when all of the users are known. The three sets of plots in
this figure are denoted in the legend by "GBMUD," "TMUD," and "ALL KNOWN,"
respectively. Note that the curves for the first iteration are identical for
GBMUD and TMUD. Hence we have suppressed the plot of the first iteration for
TMUD, to improve clarity. Notice that iteration does not significantly improve
the performance of the conventional turbo receiver, whereas the group-blind
receiver provides significant gains through iteration at moderate and high
signal-to-noise ratios. We can also see that the use of more than three
iterations does not provide significant benefits.
In Fig. 6.12, the number
of known users has been changed to two. As we would expect, there is performance
degradation for both conventional and group-blind turbo receivers. In fact, the
conventional receiver gains nothing through iteration for this scenario because
there are now five users whose interference is simply ignored. It is also
apparent that the group-blind turbo receiver will not be able to mitigate all of
the interference of unknown users, even for a large number of iterations. This
is due, in part, to the use of an imperfect interference subspace estimate in
the SISO group-blind multiuser detector.
Sections
