Group-Blind SISO Multiuser Detector
The heart of the turbo group-blind receiver is the
soft-input/soft-output (SISO) group-blind multiuser detector. The detector
accepts, as inputs, the a priori LLRs for the
code bits of the known users delivered by the SISO MAP channel decoders of these
users, and produces, as outputs, updated LLRs for these code bits. This is
accomplished by soft interference cancellation and MMSE filtering. Specifically,
using the a priori LLRs and knowledge of the
signature sequences and received amplitudes of the known users, the detector
performs a soft-interference cancellation for each user, in which estimates of
the multiuser interference from the other known users and an estimate for the
interference caused by the unknown users are subtracted from the received
signal. Residual interference is suppressed by passing the resulting signal
through an instantaneous MMSE filter. The a
posteriori LLR can then be computed from the MMSE filter output.
The detector first forms soft estimates of the user code bits
as
Equation 6.78
where l2(bk[i]) is the a priori LLR
of the kth user's ith bit delivered by the MAP channel decoder. We denote
hard estimates of the code bits as
Equation 6.79
and denote 
.
In the next step we form an estimate of interference of the
unknown users, I[i], which we denote by 
. We begin
by forming the preliminary estimate
Equation 6.80
where 
and dk[i] is a random variable defined by
Equation 6.81
It will be seen that our ability to form a soft estimate for
dk[i] will allow us
to perform the soft interference cancellation mentioned above. Clearly, dk[i] can take on one of two values, 0 or 2bk[i]. The probability that dk[i] is equal to zero is the probability that the hard
estimate is correct and is given by
Equation 6.82
Recall that for b 
{+1, –1}, the probability that bk[i] = b is related to the
corresponding LLR by [cf. (6.42)]
Equation 6.83
On substituting b = sign {tanh
[½l2(bk)[i]} in (6.83),
we find that
Equation 6.84
Therefore, dk[i] is a random
variable that can be described as
Equation 6.85
We now perform an eigendecomposition on GGT/M where 
1]]. We denote by Uu the
matrix of eigenvectors corresponding to the 
largest
eigenvalues. The span of the columns of Uu
represents an estimate of the subspace of the unknown users (i.e., the
interference subspace). Ideally, that is, when d[i] = 0 in (6.80),
Uu contains the signal subspace spanned by the
unknown interference 
. To refine our estimate of I[i], we project
g[i] onto Uu. The
result is
Equation 6.86
Denote 
and 
. Since,
ideally, 
we have
Equation 6.87
Now we subtract the interference estimate from the received
signal and form a new signal
Equation 6.88
where
Equation 6.89
with
Equation 6.90
For each known user we perform a soft interference cancellation
on z[i] to obtain
Equation 6.91
where
with 
a soft estimate for dk[i], given via (6.85) by
Equation 6.92
Substituting (6.88) into
(6.91), we obtain
Equation 6.93
An instantaneous linear MMSE filter is then applied to rk[i] to obtain
Equation 6.94
The filter 
is chosen to minimize the
mean-square error between the code bit bk[i] and the filter output zk[i]:
Equation 6.95
where the expectation is with respect to the ambient noise and
the interfering users. The solution to (6.95) is given by
Equation 6.96
It is easy to show that
Equation 6.97
where 
. The covariance matrix D[i] has the dimensions 2 x 2
and may be partitioned into four diagonal x
blocks in the following manner:
Equation 6.98
The diagonal elements of D11[i] are
given by
Equation 6.99
Using (6.85), the
diagonal elements of D22[i] are
given by
Equation 6.100
where
Equation 6.101
The diagonal elements of D12[i] and
D21[i] are identical and are given by
Equation 6.102
It is also easy to see that
Equation 6.103
where ek is a -vector whose elements are all zero
except for the kth element, which is 1.
Substituting (6.97) and (6.103) into (6.96), we may write the instantaneous MMSE filter for user
k as
Equation 6.104
As before, we make the assumption that the MMSE filter output
is Gaussian; we may write
Equation 6.105
where mk[i] is the equivalent amplitude of the kth user's signal at the filter output, and hk[i] ~ N (0,
) is a Gaussian noise sample. Using
(6.97) and (6.104), the parameter mk[i] is computed as
Equation 6.106
Equation 6.107
where (6.107) follows
from (6.97), (6.104), and (6.106).
Finally, exactly the same as (6.64), the extrinsic
information, l1(bk[i]), delivered by the SISO multiuser detector is given
by
Equation 6.108
This group-blind SISO multiuser detection algorithm is
summarized as follows.
Algorithm 6.3: [Group-blind
SISO multiuser detector—synchronous CDMA]
-
Given {l2(bk[i])}, form soft and hard estimates of the code bits:
Equation 6.109
Equation 6.110
Denote
-
Let
Equation 6.111
Equation 6.112
Perform an eigendecomposition on
GGT/M,
Equation 6.113
Set Uu
equal to the first columns of U.
-
For i = 0,1, ..., M – 1:
– Refine the estimate of the unknown
interference by projection:
Equation 6.114
– Compute according
to
Equation 6.115
where ak[i] is defined in (6.101). Define
– Subtract 
[i] from r[i] and perform soft interference
cancellation:
Equation 6.116
where 
.
– Calculate D[i], according to (6.99)–(6.102).
– Calculate and apply the MMSE
filters:
Equation 6.117
Equation 6.118
where 
, 
,
and where 
and 
.
– Compute mk[i] according to (6.106).
– Compute the a posteriori LLRs for
code bit bk[i] according to (6.108).
Sections
