Local-Search Detection
Clearly, the optimal performance is achieved by the
exhaustive-search detector with the log-likelihood penalty function (i.e., the
ML detector). As will be seen later, the performance of the exhaustive-search
detector with the Huber penalty function is close to that of the ML detector,
while this detector does not require knowledge of the exact noise pdf. However,
the computational complexity of the exhaustive-search detector (4.102) is on the order of 
(2K). We next discuss a local-search approach to
approximating the solution to (4.102),
based on the slowest-descent search method discussed in Section 3.4. The basic idea
is to minimize the cost function C(b;y) over a subset W of
the discrete parameter set {+1, -1}K
that is close to the continuous stationary point b
given by (4.103). More precisely, we
approximate the solution to (4.102) by a
local one:
Equation 4.114
In the slowest-descent-search method, the candidate set W consists of the discrete parameters chosen such that they
are in the neighborhood of Q (Q
K) lines in 
, which are
defined by the stationary point b and the Q eigenvectors of the Hessian matrix 
of C(b;y) at b corresponding to the Q
smallest eigenvalues.
For the three types of penalty functions, the Hessian matrix at
the stationary points are given, respectively, by
Equation 4.115
Equation 4.116
Equation 4.117
where, in (4.115),
Equation 4.118
and in (4.117) the
indicator function I(ya) = 1 if ya and 0 otherwise; hence in this case those rows of
Y with large residual
signals as a possible result of impulsive noise are nullified, whereas other
rows Y of are not
affected.
Denote b* 
sign(b). In general, the slowest-descent-search method
chooses the candidate set W in (4.114) as follows:
Equation 4.119
Hence, {bq,m}m contains the K closest
neighbours of b in {-1, +1}K along the direction of gq Note
that {gq}
represent the Q mutualy orthogonal directions where the cost function
C(b;y) grows
the slowest from the minimum point b.
Finally, we summarize the slowest-descent-search algorithm for
robust multiuser detection in non-Gaussian noise. Given a penalty function r(·), this algorithm solves the discrete optimization problem
(4.114) according to the following
steps.
Algorithm 4.4: [Robust
multiuser detector based on slowest-descent-search—synchronous CDMA]
-
Compute the continuous stationary point
b in (4.103):
Equation 4.120
Equation 4.121
Equation 4.122
Set 
and b*
= sign (
).
-
Compute the Hessian matrix 
given by (4.115) or (4.116)
or (4.117), and its Q smallest
eigenvectors g1, . . . ,
gQ.
-
Solve the discrete optimization problem
defined by (4.114) and (4.119) by an exhaustive search [over (KQ+1)
points].
Simulation Results
For simulations, we consider a synchronous CDMA system with a
processing gain N = 15, number of users K = 6, and no phase offset and equal amplitudes of user
signals (i.e., ak = 1,
k = 1, . . . , K).
The signature sequence s1 of
user 1 is generated randomly and kept fixed throughout simulations. The
signature sequences of other users are generated by circularly shifting the
sequence of user 1.
For each of the three penalty functions, Fig. 4.12 presents the BER performance of the
decorrelative detector, slowest-descent-search detector with two search
directions, and exhaustive-search detector. Searching further slowest-descent
directions does not improve the performance in this case. We observe that for
all three criteria, the performance of the slowest-descent-search detector is
close to that of its respective exhaustive-search version. Moreover, the
LS-based detectors have the worst performance. Note that the detector based on
the Huber penalty function and the slowest-descent search offers significant
performance gain over the robust decorrelator developed in Section 4.4 (Algorithm 4.1).
For example, at the BER of 10-3, it is only less than 1 dB from the
ML detector, whereas the robust decorrelator is about 3 dB from the ML
detector.
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