1240 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 6, JUNE 2001
Adaptive Minimum-BER Linear Multiuser Detection
for DS-CDMA Signals in Multipath Channels
Sheng Chen, Senior Member, IEEE, Ahmad K. Samingan, Bernard Mulgrew, Member, IEEE, and Lajos Hanzo
Abstract—The problem of constructing adaptive minimum bit
error rate (MBER) linear multiuser detectors is considered for di-
rect-sequence code division multiple access (DS-CDMA) signals
transmitted through multipath channels. Based on the approach
of kernel density estimation for approximating the bit error rate
(BER) from training data, a least mean squares (LMS) style sto-
chastic gradient adaptive algorithm is developed fortraining linear
multiuser detectors. Computer simulation is used to study the con-
vergence speed and steady-state BER misadjustment of this adap-
tive MBER linear multiuser detector, and the results show that it
outperformsanexisting LMS-style adaptiveMBER algorithm first
presented at Globecom’98 by Yeh et al.
Index Terms—Adaptive algorithms, linear multiuser detectors,
minimum bit error rate, minimum mean square error, stochastic
gradient algorithms.
I. INTRODUCTION
W
ITHIN the class of linear multiuser detectors for
DS-CDMA signals, the minimum mean square error
(MMSE) detector [1]–[5] is the most popular one as it provides
good performance and can readily be implemented using stan-
dard adaptive filter techniques such as the LMS and recursive
least squares algorithms [3], [5]. However, it is well known that
the MMSE solution is not always optimal in this application,
and the BER of the MMSE linear multiuser detector can, in
certain situations, be distinctly inferior to the optimal MBER
solution [6]–[9]. Gradient optimization for obtaining the
theoretical MBER linear multiuser detector is considered in
[6]
1
for narrowband Gaussian CDMA channels [i.e., channels
that do not introduce intersymbol interference (ISI)]. Gradient
optimization to achieve the theoretical MBER solution is also
addressed in [7], where constraints are added to ensure a global
solution by gradient-based algorithms at the expense that the
solution obtained may not be the true MBER solution. There
are two stochastic gradient adaptive algorithms for realizing
the MBER linear multiuser detector in the literature [8], [9].
The adaptive algorithm given in [8] uses a difference ap-
proximation to estimate the gradient of one-sample error prob-
Manuscript receivedApril 27, 2000; revised February 21, 2001. The associate
editor coordinating the review of this paper and approving it for publication was
Prof. Dimitrios Hatzinakos.
S. Chen, A. K. Samingan, and L. Hanzo are with the Department of Elec-
tronics and Computer Science, the University of Southampton, Southampton,
U.K. (e-mail: sqc@ecs.soton.ac.uk).
B. Mulgrew is with the Department of Electronics and Electrical Engineering,
the University of Edinburgh, Edinburgh, U.K.
Publisher Item Identifier S 1053-587X(01)03884-3.
1
The “adaptive” algorithm in [6] is, in fact, nonadaptive as it requires the
received signal minus the noise component. This is the same to an off-line opti-
mization with the perfect channel knowledge and all user bits.
ability and moves the detector weights in the negative direction
of the estimated stochastic gradient. The algorithm only adjusts
the detector weight vector when the detector makes an error.
2
The main drawback of this algorithm is therefore a very slow
convergence, particularly when the error rate is very low. For
the sake of distinguishing this stochastic adaptive MBER algo-
rithm from others, it will be called the difference approxima-
tion adaptive MBER (DMBER) algorithm. The adaptive algo-
rithm reported in [9], which is called the approximate MBER
(AMBER) detector, is appealing due to its computational sim-
plicity. It is a stochastic gradient algorithm that is identical to the
signed-error LMS algorithm [10], except in the vicinity of the
decision boundary, where it is modified to continue updating the
weights when the signed-error LMS algorithm would not. The
AMBER algorithm therefore can continuously update when the
detector weight vector has reached the regions of very low error
rate.
Adaptive MBER linear equalizers have been investigated for
a longer time [11]–[16]. In particular, the adaptive MBER equal-
izer presented in [15] and [16] is a LMS-style stochastic gra-
dient algorithm and has been shown to outperform the approxi-
mate MBER linear equalizer first reported in [17], which is the
counterpart of the AMBER linear multiuser detector of [9]. In
this study, we extend the adaptive MBER algorithm of [15] and
[16] to multiuser detection for DS-CDMA channels and develop
a new adaptive MBER linear multiuser detector. For the pur-
pose of distinguishing it from the two above-mentioned adap-
tive algorithms, this new LMS-style stochastic gradient adap-
tive algorithm will be referred to as the least BER (LBER) al-
gorithm. Our investigation involving simulation shows that this
new LBER linear multiuser detector is superior in performance
over the AMBER linear multiuser detector of [9].
The paper is organized as follows. Section II presents the
DS-CDMA system model used and provides the necessary no-
tations and definitions. Section III is devoted to formulating the
MBER solution for the linear multiuser detector and developing
a gradient search algorithm. In Section IV,the proposed adaptive
MBER multiuser detector is derived. Kernel density estimation
is employed to approximate the BER as a smooth function of
training data, and this leads to the formulation of a LMS-style
stochastic gradient adaptive algorithm called the LBER. Com-
parison with the two existing stochastic adaptive algorithms (the
DMBER and the AMBER) is also given in this section. Sec-
tion V gives some computer simulation results, and the paper
concludes in Section VI.
2
More precisely, when the difference of the detector decisions corresponding
to the weights perturbed in opposite directions is nonzero.
1053–587X/01$10.00 © 2001 IEEE