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142 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001
Constrained Maximum-Likelihood Detection
in CDMA
Peng Hui Tan, Student Member, IEEE, Lars K. Rasmussen, Member, IEEE, and Teng J. Lim, Member, IEEE
Abstract—The detection strategy usually denoted optimal mul-
tiuser detection is equivalent to the solution of a (0, 1)-constrained
maximum-likelihood (ML) problem, a problem which is known
to be NP-hard. In contrast, the unconstrained ML problem can
be solved quite easily and is known as the decorrelating detector.
In this paper, we consider the constrained ML problem where the
solution vector is restricted to lie within a closed convex set (CCS).
Such a design criterion leads to detector structures which are ML
under the constraint assumption. A close relationship between a
sphere-constrained ML detector and the well-known minimum
mean square error detector is found and verified. An iterative
algorithm for solving a CCS constraint problem is derived based
on results in linear variational inequality theory. Special cases of
this algorithm, subject to a box-constraint, are found to corre-
spond to known, nonlinear successive and parallel interference
cancellation structures, using a clipped soft decision for making
tentative decisions, while a weighted linear parallel interference
canceler with signal-dependent weights arises from the sphere
constraint. Convergence issues are investigated and an efficient
implementation is suggested. The bit-error rate performance is
studied via computer simulations and the expected performance
improvements over unconstrained ML are verified.
Index Terms—Code-division multiple access, interference can-
cellation, multiuser detection.
I. INTRODUCTION
I
N ANY multiple-access system, the available resources
are shared in some way among all active users. As a
consequence, there is a fundamental tradeoff between the
amount of resources available for each user and the corre-
sponding interference encountered due to multiple access. In
code-division multiple-access (CDMA) systems all resources
are in principle available to all users simultaneously. The users
are distinguished from each other by user-specific signature
PaperapprovedbyG. Caire, the Editor for Multiuser Detection and CDMAof
the IEEE CommunicationsSociety.Manuscript received September 1,1999; re-
vised January 6, 2000 and June 9, 2000. This work was supported in part by the
Centre for Wireless Communications (Singapore), the National University of
Singapore, the Swedish Research Council for Engineering Sciences (TFR), and
the Swedish Foundation for StrategicResearch (SSF). This paper was presented
in part at the International Conference on Information, Communication and
Signal Processing, Singapore, December 1999, the 2000 International Zürich
Seminar on Broadband Communication, Zürich, Switzerland, February 2000,
the IEEE Vehicle Technology Conference, Tokyo, Japan, May 2000, and the
IEEE International Symposium on Information Theory, Sorrento, Italy, June
2000.
P. H. Tan and L. K. Rasmussen are with the Telecommunication Theory
Group, Department of Computer Engineering, Chalmers University of
Technology, SE-412 96 Gothenburg, Sweden (email: phtan@ce.chalmers.se;
larsr@ce.chalmers.se).
T. J.Lim is with theCentre for Wireless Communications, Singapore Science
Park II, Singapore 117674 (email: cwclimtj@nus.edu.sg).
Publisher Item Identifier S 0090-6778(01)00257-4.
sequences, modulating the transmitted data symbols using
direct-sequence spread-spectrum techniques. This in turn leads
to a relative high level of multiple-access interference (MAI)
as it is not feasible to maintain low (or zero) cross correlation
among all users in a practical random-access system.
Conventional spread-spectrum detection techniques applied
in CDMA are severely limited in performance by MAI, leading
to both system capacity limitations as well as strict power con-
trol requirements [1]. These limitations are due to the fact that
the traditional matched filter output does not represent a suffi-
cient statistic for detection. A detector working on a true suf-
ficient statistic is generally denoted a multiuser detector, and it
has the potential of alleviating the MAI problems encountered
by conventional techniques.
In order to describe the detection strategies to follow, assume
an asynchronous transmission of
information bits per user
using binary phase-shift keying (BPSK) modulation. The
number of active users is
and the data vector consisting of all
transmitted data symbols for all users is denoted by the column
vector
of dimension . The general maximum-likelihood
(ML) detection problem is equivalent to a constrained quadratic
optimization. The maximally constrained ML detector finds the
ML solution constrained to
where
denotes the set of all binary -tuples represented as column
vectors, i.e., each information symbol estimate must be either
or . This detector has previously been denoted the
optimal multiuser detector [2]. In the area of optimization,
the above ML problem is known as a (0, 1)-constrained (or
Boolean-constrained) quadratic minimization which in turn
represents a combinatorial quadratic minimization. Such a
problem is known to be NP-hard [3] so the (0, 1)-constrained
ML detector is therefore in general too complex for practical
asynchronous DS-CDMA systems, even with a moderate
number of users. For certain special cases of the correlation
matrix
1
it has been shown that (0, 1)-constrained ML detection
can be obtained by successive interference cancellation [4]
or by polynomial-time algorithms [5]–[7]. A class of com-
plexity-limiting (0, 1)-constrained ML detectors was suggested
in [8], assuming a tree-search based detector structure. An iter-
ative structure which is guaranteed to deliver (0, 1)-constrained
ML decisions on some bits was suggested in [9]. The matched
filter outputs are here compared to an iteratively tightened
threshold through which (0, 1)-constrained ML decisions are
made. Decisions on all bits are however not guaranteed. Ap-
proximations to the (0, 1)-constrained ML problem have also
1
The case of all identical cross correlations, i.e., identical off-diagonal ele-
ments of the correlation matrix.
0090–6678/01$10.00 © 2001 IEEE
TAN et al.: CONSTRAINED ML DETECTION IN CDMA 143
been suggested in [10] based on the expectation maximization
algorithm and in [11] based on iterative transformations of the
quadratic minimization problem such that the unconstrained
solution to the transformed problem monotonically approaches
the desired solution. In this paper, however, we will take a more
general approach to complexity-limiting ML detection.
To reduce complexity, the constraints imposed on a feasible
solution can be relaxed. A simple constraint to impose is to
restrict the solution vector to be contained within a closed
convex set (CCS). Examples of CCSs of dimension
are
, an ellipsoid of dimension and a hypercube of
dimension
. The corresponding optimization problem is
known as a CCS constrained quadratic program (CCSQP). The
fully unconstrained ML detector was suggested in [12] and is
denoted the decorrelating detector. Here, a valid solution vector
is found in as each symbol estimate can take on any
real value, i.e., no constraints are imposed. The case is denoted
an unconstrained quadratic program (UQP). This ML solution
is then mapped onto a valid data point through
Sgn .
This is of course a suboptimal mapping and
is not ML [13].
Since the length of the data symbol vector
representing
the transmitted symbols is constant for constant envelope
modulation formats, a simple, sensible constraint to impose on
the quadratic optimization is to confine the solution vector to
lie within a sphere of radius
passing through all possible
data points. This is known in the area of optimization as a
sphere-constrained quadratic program (SQP) [14]. The SQP
problem has been studied intensively in the past and many
results exist [15]. The solution for CDMA, following the
satisfaction of the Karush–Kuhn–Tucker (KKT) conditions, is
a linear detector which is closely related to the MMSE detector.
This case has been considered independently in [16]–[18].
Constraining the data estimate vector to lie within a hyper-
cube described by the data points leads to a problem which
is known as a box-constrained quadratic program (BQP) [14].
Specifically, we consider the case where each element of the
data estimate vector must lie in the range
. Again, this
problem has been considered independently in [16] and [17],
[19]. Such a problem is closely related to the linear comple-
mentarity problem (LCP) [20]. The LCP is equivalent to a BQP
where each element of the data estimate vector is confined to
. Both problems in turn are equivalent to special cases of
the linear variational inequality (VI) problem over a rectangle
[20]. A general iterativealgorithm for solving the LCP was sug-
gested by Ahn in [21]. This algorithm however, is based on the
solutionof the more generalVIproblem which includes the con-
strained quadratic optimization problem under consideration,
i.e., the results in [21] can be extended to include the case of
convex quadratic optimization subject to a CCS constraint.
2
The algorithm can be interpreted as a general interference
cancellation structure where some known successive and par-
allel approaches are recognized as special cases. The tentative
decision function which plays an important role in interference
cancellation is shown to be equivalent to an orthogonal projec-
tionontotheconstraining space. Itisthenshownthat the data es-
2
Algorithms for special cases of the VI problem have also been presented in
[22].
timate vector resulting from the clip-function interference can-
cellation schemes in [23] and [24] is asymptotically equivalent
to the solution of the BQP, i.e., their performance approach that
of the BQP detector as the number of stages tends to infinity.
This interpretation also reveals that the projections for both the
UQP and the SQP correspond to linear (soft) tentative decision
functions, however, the latter case has a gradient which depends
on the received signal. Moreover, we derive conditions for con-
vergence for the general iterative algorithm which in turn pro-
vides new insight into convergence issues for some known non-
linear cancellation structures. Numerical examples show that
only a few stages are needed for practical convergence.
At stated previously, at convergence the clip-function inter-
ference cancellation scheme provides an ML solution which is
constrained to lie within a hypercube defined by the valid data
points. The interference cancellation schemes based on a hy-
perbolic tangent function as a tentative decision suggested in
[25]–[28]alsoprovidea solutionrestrictedtoliewithinthesame
hypercube. This solution is based on MMSE-optimal decision
feedback [29] and is suggested in [27] as an iterative algorithm
tofindthenonlinear MMSE solutionto the detection problem. It
is shownthat the true solution to this problem is a fixedpoint but
convergence of the suggested algorithm is not proven. Simula-
tion results indicate howeverthat it does converge in most cases.
This problem requires a thorough investigation which however,
falls beyond the scope of the present paper.
The paper is organized as follows. In the following section,
the uplink model is described. In Section III we discuss the so-
lution of the constrained ML problem subject to a CCS con-
straint, introduce an iterative algorithm for solving it and con-
sider convergence behavior. The relationship to known cancel-
lation structures are presented in Section IV, and numerical ex-
amples are presented in Section V. Summarizing remarks con-
cludes the paper in Section VI.
Throughout this paper scalars are lowercase, vectors are
boldfaced lowercase, and matrices are boldfaced uppercase.
The symbols
, and are the transposition and inversion
operators, respectively. All vectors are defined as column
vectors with row vectors represented by transposition and
denotes the set of real numbers. The notation denotes
the set of all
-tuples over the set , represented as column
vectors.
II. S
YSTEM MODEL
In this section, we describe the baseband uplink model of
the CDMA communication system used throughout this paper.
The uplink model is based on an asynchronous CDMA system
with single-path channels and the presence of additive white
Gaussian noise (AWGN) with zero mean and variance
. In all our discussions, we assume that users are active
simultaneously.
User
, , in this multiuser CDMA system
transmits a binary information symbolstream
at data rate symbols per second, where
is the symbol interval index and is the length of the data
block. To spread the signal,
is modulated by a spreading
waveform generated from a binary spreading code with a chip
144 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001
rateof chipsper second. The spreading codeused
to modulate the
th bit can be written in a vector as
with
. Binary data and chip formats are as-
sumed for clarity only, and all the later concepts generalize to
-ary formats.
In addition, when the users are allowed random access to the
channel, each user encounters a transmission delay relative to
other users. The delay is measured against an arbitrary reference
selected such that all the transmission delays are constrained
to
. Assume further that all the delays are con-
strained to be integer multiples of the chip interval. The rela-
tive delay normalized to the chip interval is then
. For a delay of the resulting transmitted dis-
crete-time baseband signal due to symbol interval
for user
is then
The dimension of and is .
The continuous-time received signal is down-converted to
baseband, passed through a filter matched to the chip pulse-
shape and sampled. The received baseband signal after chip-
matched filtering is then
(1)
where
is an -dimensional vector of independent, identi-
cally distributed Gaussian random variables of zero mean and
variance
. Here, we have assumed perfect power
control and perfect phase synchronization for clarity and nota-
tionalsimplicity.The resultsdoalsoapplytothegeneralcasesof
no or arbitrary power control as well as to the case of randomly
distributed received signal phases. A more convenient form of
(1) is
where is the matrix of received spreading codes and is the
data symbol vector. This model is discussed in more detail in
[30]. A minimal set of sufficient statistics of dimension
is
obtained through correlation matched to the received spreading
codes of the users. This is to ensure the maximization of the
signal-to-noise ratio [31] and corresponds algebraically to
(2)
where
is the matched filter output vector, is the correla-
tion matrix, and
is a zero mean Gaussian noise vector with
covariance matrix
. In [12], it was shown that is sym-
metric positive definite (SPD) with probability one in an asyn-
chronous system with arbitrary time delays, i.e., a chip asyn-
chronous system. In the chip synchronous model defined here,
there is a nonzero probability that
is semi-positive definite.
As
increases, this probability diminishes. In the rest of the
paper, we assume that
is SPD.
III. CCS-C
ONSTRAINED ML DETECTION
The ML sequence detection criterion is defined as
which incidentally is identical to the MAP criterion assuming
that all data symbols are equally likely. Here,
represents
the set of vectors in which the data estimate vector
is assumed
to exist. Since we are considering an AWGN channel, the nega-
tive log-likelihood function based on the
is described as
The general constrained ML problem for asynchronous CDMA
is then described as
(3)
The so-called optimal ML detector for CDMA [2] is a
(0, 1)-constrained minimization of
where ,
i.e., the ML solution is confined to
. The
complexity in solving this problem is of the order of
, and
thus grows exponentially with the number of users (see, e.g.,
[32] for details).
Here we relax the constraint to a CCS in order to limit the
complexityof the solution algorithm, i.e.,
. As special
cases of
, we consider the real vector space , a sphere
determined by
, and finally a
hypercube (box) described by
.
Here,
is an -vector of all ones and
is a compact notation for for all ,
i.e., each element of
is less than or equal to the corresponding
element in
. We express this set in a compact way as .
A CCS is element-wise separable (ES) if
where and are
appropriate constants. The corresponding constraint for element
is denoted by . Clearly and are ES while is not.
All the cases can be solved by satisfying the KKT conditions
using Lagrange multipliers. The Lagrangian function associated
with the SQP problem is
(4)
A KKT point for the SQP is then described by a pair
[15]
for which
It is well knownthat it is possible to completely detail the global
solution of (3) under a sphere constraint without requiring any
convexity assumption on the objective function.
Proposition 1 (Proposition 2.1 [15]): A point
such that
is a global solution of (3) if and only if there
existsa unique
such that the pair satisfiesthe KKT
conditionin(4),andthematrix
ispositivesemidefinite.
If
is positive definite, then (3) has a unique global
solution.
For a proof, see [15]. In our case, we have assumed that
is symmetric positive definite, and therefore there exists a
TAN et al.: CONSTRAINED ML DETECTION IN CDMA 145
unique solution of the form . This is iden-
tical in form to the MMSE solution which is described as
[33]. The two linear filtersare not identical as the
SQP detector confines the solution to be within a sphere while
no such constraints are imposed on the MMSE solution. It can
be shown however that
, so on average the
MMSE detector does impose the same restriction [34]. Numer-
ical examples have also revealed that
provides, on average, an
accurate estimate of the noise variance
. Furthermore, numer-
ical examples show that the bit-error rate (BER) performance
of the two detectors are virtually identical. With caution, it can
therefore be claimed that there is a close relationship between
the linear MMSE criterion and the sphere-constrained ML crite-
rion. The unconstrained case follows from (4) by setting
.
The result is the well-known decorrelator. No useful interpreta-
tion results from the satisfaction of the KKT conditions for the
box-constraint.
A. Iterative CCSQP Detector
Since
is a strictly convex function as long as is SPD,
the problem of minimizing
over a continuous region can
be solved by a polynomial-time algorithm [35]. One approach
to finding such an algorithm isto consider a VI problem defined
as follows,
Definition1: The VI Problem
isdefinedasfinding
a vector
such that
(5)
where
is a given continuous function from to and
is a given CCS.
Let us further define
We are then ready to establish a close link between CCS-con-
strained quadratic optimization and a special case of the VI
problem.
Proposition 2: If
is a convex function and is a so-
lution to
, then is a solution to the optimization
problem in (3).
Proof: Since
is convex
(6)
But
, since is a solution to .
Therefore, from (6), it can be concluded that
for
all
, i.e., is a minimum point of (3).
The above proposition is also true when is strictly convex.
The proof follows the same arguments. So we can solve (3) by
solving (5) with
.
The existence and uniqueness of a general solution to (5) fol-
lowsdirectly from the proof of Proposition 2 and is summarized
in the following proposition.
Proposition 3: There exists precisely one solution
to (5)
when
if is positive definite.
Proof: If
is positive definite, then is a strictly
convex function. From (3) it follows that
for all and therefore for
all
.
Havingestablished the existenceand uniqueness of a solution
to (5), we move on to the critical question of how to find it.
Before presenting the iterative algorithm, we need to define the
orthogonal projection operation
onto a CCS .
Lemma 1: Let
be a CCS in . Then for each
there is a unique point such that
for all , and is known as the orthogonal projection
of
onto the set with respect to the Euclidean norm, i.e.,
. For an ES CCS, we can
further state this corollary to Lemma 1.
Corollary 1: In case
is ES, then
for all . This is
denoted as an ES projection (ESP). A result based on the
orthogonal projection and which is necessary to prove the final
result is as follows.
Lemma 2: Let
be a CCS in . Then if and
only if
, for all or
, for all .
Proofs for Lemma 1 and Lemma 2 can be found in the liter-
ature, e.g., [20].
A way to generate an iterative solution to (5) is to convert the
given VI problem into a fixed-point formulation [21] using the
following lemma.
Lemma 3 (Lemma 3.1 [21]): Let
be a CCS, and let be
acontinuousfunction.Then
isasolutionto ,
for all
, if and only if
for some or all (7)
Proof: Suppose that
is a solution of , then
multiplying the inequality in (5) by
and adding
to both sides of the resulting inequality, we get
(8)
From Lemma 2, it can be concluded that
.
Conversely, if
for , then (8) directly
and therefore (5) also holds true.
For a CCS constraint and , the solution
then satisfies
for some . For an
ES CCS, we can further state the following corollary to Lemma
3. Here we let
denote element of the vector .
Corollary 2: If
is further ES then is a solution to
,forall ,ifandonlyif
for some or all and any positive diagonal matrix .
Proof: Since the constraining set is ES, the orthog-
onal projection is an ESP and then (7) can be restated as
for some and .
Then according to the one-dimensional case of Lemma 3, we
can claim that for each
(9)
The inequality in (9) can clearly be multiplied with any positive
element
and still be true. Again following Lemma 3,
for all , if and only
if
for some or all ,
and .Using the factthat the projection is ESP,
we collect all the elements in the vector
and the corollary is
proven.
146 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001
It therefore follows that for a ES CCS constraint , the solu-
tion satisfies
(10)
for some
and any positive diagonal matrix .
The following iterative algorithm is proposed in [21] for
solving a linear complementarity problem. However, such
a problem is a special case of the VI problem considered
here [20] and the suggested algorithm is based on the above
results for the general VI problem. The algorithm is thus also
applicable in solving (10) for given
and .
Algorithm 1: For any initial value
, let
(11)
where
, and is the
iteration index. If the orthogonal projection operation can be
decoupled into independent element-wise projections, then
can be either strictly lower triangular, strictly upper triangular
or equal to the null matrix
and is any positive diagonal
matrix. Otherwise,
is equal to the null matrix and .
Algorithm 1 has the form of generalized interference cancel-
lation. In fact, as shown later on for each of the three specific
CCSs considered, special cases of the algorithm correspond to
knownsuccessiveand parallel cancellation structures. The algo-
rithm is serial (successive) in nature when
is strictly upper or
lower triangular. Assuming that
is lower triangular, the itera-
tion of
may be conducted element by element. When ,
the algorithm becomes parallel in nature, as iteration
only de-
pends on estimates from iteration
. This special case of
the algorithm was first suggested in [36].
The choice of
, as well as , and influence the con-
vergence of the algorithm. Some useful sufficient conditions for
convergence of the serial and parallel forms of (11) can however
be found. Consider first the serial case. Let
be partitioned
such that
, where is a diagonal matrix and
and are strictly lower and upper triangular, respectively.
Also, let
.
Theorem 1: If
is either or , the sequence of
Algorithm 1 with
(12)
where
and are the diagonal elements of and ,
respectively, corresponding to user
, symbol interval , con-
verges to the solution of the CCSQP for all realizations of
and .
Proof: The proof is based on showing that for all realiza-
tions of
and , for where denotes
the unique fixed point. The complete proof is included in the
Appendix.
When , the detector becomes a multistage parallel in-
terference canceler, and the conditions for convergence change.
Again, let
to allow for a convergence proof.
Theorem 2: If
, the sequence of Algorithm 1
with
(13)
Fig. 1. An ICU for systematic implementation of IC structures based on (11).
converges to the solution of the CCSQP for all realizations of
and , where and are the maximum eigenvalue of
and the maximum diagonal element of , respectively.
Proof: Theproofis based onasimilar strategyas the proof
of Theorem 1 and is included in the Appendix.
The choice of other triangular forms for leads to an array
of schemes based on block iterations [37], [38]. Previous results
show for the linear case that most restricted and slowest conver-
gence is found for
while least restricted and fastest con-
vergence is found for
or . Choices in between
bridge the gap between the two extreme. It is difficult, if not
impossible, to analytically determine optimal values for the pa-
rameters
, , and . General trends on convergence as these
parameters are varied are studied through computer simulations
in Section V.
IV. R
ELATIONSHIP TO INTERFERENCE CANCELLATION
As mentioned earlier, special cases of Algorithm 1 are equiv-
alent to known successive and parallel interference cancella-
tionstructures.An
-stagesuccessiveinterferencecancellation
(SIC) scheme is described as
(14)
for
where . The relation-
ship between the SIC and the general iterative algorithm is clear
from substituting
, and into
(11). In cases where
, i.e., with relaxed or no power con-
trol, selecting
corresponds to a normalization of
the received amplitudes, adjusting the signal levels to the rele-
vant constraint. The fact that (14) indeed is describing practical
interference cancellation for asynchronous CDMA was demon-
strated for the linear case in [28] and [38].
Similarly, the
-stage weighted parallel interference can-
celer (PIC) [39] is described as
(15)
for
, which can be derivedfrom (11)by
taking
, , and . Again .
Due to the close relationship to interference cancellation,
the general iterative algorithm in (11) can be efficiently imple-
mented when
or . The concept of an interference
cancellation unit (ICU) as a general building block can be ap-
plied, allowing for systematic construction. The corresponding
ICU for user
at stage is shown in Fig. 1. Here,
denotes the residual received vector for user at stage ,
TAN et al.: CONSTRAINED ML DETECTION IN CDMA 147
Fig. 2. An SIC structure systematically constructed using ICU blocks.
Fig. 3. A PIC structure systematically constructed using ICU blocks.
symbol interval and denotes the resulting update
of the residual vector. The residual vector is obtained as
for
for
(16)
with
being a partition of where
contains the columns of associated with symbols already
processed at stage
while contains the remaining
columns of
. Similarly, is a vector containing the
stage
estimates up until user , symbol interval , while
contains stage estimates of the remaining data
symbols. A general successive cancellation structure is then
constructed by inter-connecting ICU blocks as shown in Fig. 2.
The beauty of this representation is that other cancellation
structures can be realized simply by changing the inter-connec-
tions of the ICUs. This is demonstrated by the PIC structure
shown in Fig. 3.
The only difference between the two structures is when and
where the residual vector update is carried out. In this represen-
tation, the inherent detection delay difference between succes-
sive and parallel structures is quite clear. Other combinations of
SIC and PIC as suggested in [40] and [41] can also be imple-
mented following this approach.
A. Tentative Decision Function
The orthogonal projection operation, essential for the algo-
rithm, clearly corresponds to the tentative decision function in
a cancellation structure. For unconstrained ML detection
, the projection is defined by which obvi-
148 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001
Fig. 4. Examples of the tentative decision functions corresponding to the
orthogonal projections. The dashed curve represents the linear function for the
UQP, the dashed and the dotted curves illustrate the functions for the SQP with
and , respectively, while the solid curve is the clipped soft
decision represented by the BQP projection.
ously can be decoupled into independent element-wise projec-
tions,
. This is of course a linear (soft) decision
function with gradient
as illustrated in Fig. 4 by the
dashed line. When the constraining space is a sphere, the ele-
ment-wise (nonseparable) projection is described by
if
if
(17)
where
and . This projection opera-
tion is illustrated in Fig. 5 for a two-dimensional (2-D) example.
It is clear from the above that
is decreasing for increasing
. It is also clear that the overall projection does
not decouple into independent element-wise projections. The
sphere-constrained ML detector can therefore only be imple-
mented as a PIC structure with
and with . Also,
since the projection operation depends on an entire data packet,
an unacceptable detection delay is encountered. This delay to-
gether with other practical problems can be avoided by using an
appropriate block approach [38] in order to perform the orthog-
onal projections. This has been done in the numerical examples.
The corresponding tentative decision function is illustrated in
Fig. 4 for
(the dashed curve) and for (the dotted
curve).
For the box constraint, the orthogonal projection operation
can again be decoupled into independent element-wise orthog-
onal projections. The
th element of the orthogonal projection
vector is in this case
if
if
if
(18)
This projection is illustrated in the 2-D exampleshownin Fig. 6.
The function
is incidentally identical to the clipped
Fig. 5. 2-D example of projection onto a sphere region.
Fig. 6. 2-D example of projection onto a box region.
soft-decision function suggested in [23] and [24] for interfer-
ence cancellation. This tentative decision function is shown in
Fig. 4 as the solid curve. It was shown in [23] that the clip-func-
tion SIC has better BER performance than the linear and hard-
decision SIC, and it is therefore subject to much practical in-
terest [42], [43]. The clip-function weighted PIC also performs
better than the linear and hard-decision weighted PIC [24].
ThegeneralweightedPICscheme corresponds to variouspar-
tial cancellation schemes suggested in the literature for either
the soft (linear) or the clipped soft tentative decision function
[28],[37],[39],[44]–[46].Herewehaveprovidedconditionsfor
convergence for the case of
, i.e., the weights stay fixed
for all stages, but not necessarily the same for each user. For
the linear case it is well-known that
corresponding
to the Jacobi iteration provides faster convergence than
which is usually denoted as the Richardson iteration [47]. The
same is observed through simulations for the nonlinear cases.
For the general case of
, it is difficult to advice any analyt-
ical conditions for convergence. For the linear case conditions
have been presented in [45] and[46], howeverno results are cur-
rently available for the nonlinear cases [48]. In [28], [39], and
[48], some observations based on simulations are presented.
B. Convergence Issues
Aninteresting observationregardingconvergenceis that The-
orems 1 and 2 are independent of the constraining CCS. Hence,
the conditions for convergence of the cancellation schemes are
thesameregardlessof the tentativedecision function. Note how-
ever that different schemes do not experience the same conver-
gence rate and the same resulting BER performance. The results
TAN et al.: CONSTRAINED ML DETECTION IN CDMA 149
Fig. 7. Average BER of the MMSE and SQP detectors. For , iterations were required with , while for , iterations
were required with
.
in Theorems 1 and 2 are known for the linear case [37], how-
ever, an important corollary to Theorem 1 is that the clip-func-
tion SIC represented by (14), where
and
, always converges. Indeed, the good convergence
behavior of the SIC, whether linear, hard-decision or clip-func-
tion, relative to the PIC has been known through simulations
for some time. However, the result presented here represents
the first proof for the guaranteed convergence of the clip-func-
tion SIC. Similar to the SIC case, an important corollary to
Theorem 2 is that the clip-function PIC represented by (15),
where
and , alwaysconverges when
. Again, this result represents the first suf-
ficient condition for convergence of the clip-function PIC.
3
V. N UMERICAL RESULTS
In this section we investigate the BER performance and the
rate of convergence of the SQP and BQP detectors based on nu-
merical examples. Randomly selected long spreading codes are
used and twodifferentscenarios are considered, a lightly loaded
case with
and , as well as a more highly loaded
case where
and . The detectors are designed
with
since we assume perfect power control, i.e., .
The
and of the PIC is selected based on simulation results
such that the fastest possible convergence is achieved. Here we
consider convergence to be reached when the BER for
and
iterations are the same which is not necessarily the same as
having
. The initial data estimate is always chosen
as
.
3
As made clear above, the concept of using a clipped soft-decision function
is quite important. The same function has also been mentioned in [39] and [44]
for PIC.
TABLE I
N
UMBER OF ITERATION REQUIRED FOR CONVERGENCE FOR THE
SQP-PIC STRUCTURE FOR VARYING .DASH DENOTES THAT THE
SCHEME IS NOT CONVERGING
First, we consider the SQP detector. In Fig. 7 the BER of the
PIC is depicted for both cases. The BER of the linearMMSE de-
tector is included for reference. For the lightly loaded case with
, , three iterations were required for conver-
gence. As can be seen in the figure, the performance of the SQP
and the linear MMSE detector are identical. The same behavior
is observed for the more highly loaded case with
. Here,
and 7 iterations are required for convergence. Again
the performance of the SQP and the linear MMSE detector co-
incide.
The influence of the choice of
and has been investigated
for both
and . The number of iterations re-
quired for convergence is obtained while systematically varying
both
and . An important observation is that the rate of con-
vergence for the cases investigated depends only on the product
and not on the individual values of and . It therefore
seems appropriate to select
. The results are summarized
in Table I. Here we can see that fastest convergence is achieved
for
when while when .
In these cases,
should be chosen as close as possible to the
upper limit detailed by (13). It is also clear that the convergence
rate of the PIC structure is quite sensitive to the
ratio.
In Figs. 8 and 9 the BERs of the clipped SIC and the clipped
PIC are depicted. The BER of the MMSE detector and the
single-user bound are used as benchmarks. Fig. 8 shows the
150 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001
Fig. 8. Average BER of the MMSE, clipped SIC and clipped PIC with , .
Fig. 9. Average BER of the MMSE, clipped SIC and clipped PIC with , .
BER of the lightly loaded case, where the clipped SIC needs
three stages to achieve convergence whereas the clipped PIC
(
) requires five stages to converge. For a three-stage
clipped PIC, its BER performance is better than the MMSE
detector and thus also better than the SQP-PIC. It can be seen
that both interference cancellation schemes perform better than
the MMSE detector after convergence but they are still quite
far from the single user bound.
When the system becomes more highly loaded, the conver-
gence rate decreases. This can be seen in Fig. 9, with the clipped
SIC now needing six stages to converge and the clipped PIC
(
) requiring 17 stages. Similarly, a six-stage clipped
PIChas BER performance which isbetter than theSQP-PIC and
MMSE detectors.
Again, the influence of the choice of
and has been in-
vestigated for both
, and for both PIC and
TAN et al.: CONSTRAINED ML DETECTION IN CDMA 151
TABLE II
N
UMBER OF ITERATION REQUIRED FOR CONVERGENCE FOR THE
BQP-PIC STRUCTURE FOR VARYING
.DASH DENOTES THAT THE
SCHEME IS NOT
CONVERGING
TABLE III
T
HE NUMBER OF ITERATION REQUIRED FOR CONVERGENCE FOR THE
BQP-SIC STRUCTURE FOR VARYING .THE DASH DENOTES THAT THE
SCHEME IS NOT CONVERGING
SIC. Again for the cases investigated, only the product of is
of importance. The results are summarized in Tables II and III.
Here we can see that for the PIC, fastest convergenceis achieved
for
when while when .
The same conclusions as for the SQP apply. In the SIC case,
the best choice of
and is for
and for . In these cases, the limit
provided by (12) does not give any indication of the choice for
fastest convergence. Another important observation is that the
SIC structure is only marginally sensitive to the system load as
only one additional iteration is required for
as com-
pared to
. Furthermore, the choice , i.e., no
weighting performed, does not lead to significant performance
degradation.
Comparing the convergence rate characteristics for the PIC
structures of both the SQP and the BQP cases, it is observed that
some disagreement occur. According to Theorem 2, the conver-
gence conditions should be the same for both cases. The reason
for these discrepancies is the approximating window used to ac-
commodate the orthogonal projection for the SQP. This level of
approximation influences the convergence behavior.
VI. C
ONCLUDING REMARKS
In this paper, we have introduced the CCS constrained ML
detector for CDMA as a complexity-limiting alternative to the
(0, 1)-constrained ML detector usually denoted the optimal
detector. Three special cases were used as illustrating examples,
the unconstrained, the sphere-constrained and the box-con-
strained cases. A general iterative algorithm was suggested to
solve the CCS-constrained ML problem and general conditions
for convergence were derived. Known successive and parallel
cancellation schemes were recognized as special cases of the
algorithm. In fact, the algorithm is a general interference can-
cellation structure which can efficiently be implemented using
an interference cancellation unit as a general building block in
a systematic construction. The defining orthogonal projection
onto the constraining space was shown to be equivalent to
the tentative decision function in a cancellation structure.
Specifically, the unconstrained ML detector was shown to lead
to a linear tentative decision function while a sphere-constraint
also lead to such a function. In this case however, the gradient
depends on the length of the projected vector. The box-con-
straint was shown to correspond to a clipped soft-decision
function for making tentative decisions, a function known to
work well in interference cancellation.
The numerical examples show that the number of stages re-
quired to achieve the solution of the constrained ML problems
depends strongly on the
ratio for a PIC structure while
an SIC structure is only slightly influenced by the load. For
both lightly and highly loaded systems, only a few stages of the
clipped SIC are sufficient to achieve the solution of the BQP. A
choice of
(i.e., no additional weighting as compared to
conventional SIC) is always close to the best possible choice for
the SIC examples considered while for the PIC, the best choice
of
decreases with increasing .
A
PPENDIX
PROOFS OF THEOREMS 1 AND
2
Proof:[Theorem1]: Referringtothedefinitionof
in
(3), if
is a decreasing function of , then
for all realizations of and is the CCSQP solution. We prove
this property of
given
as follows:
(19)
We have that
. It is then easy to show that
.
Hence, (19) may be rewritten as
(20)
where
. But it is
clearthateachelement of
is less than or equal to zero, because:
1) if
, ;
2) if
, the corresponding elements of
and haveoppositesigns because .
Therefore, we can conclude that
(21)
since
is symmetric and .
Hence, the sequence
is nonincreasing. In fact, it is a
152 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001
convergent sequence since is continuous on and is thereby
bounded. It follows then that
. Now let be an accumulation point, i.e.,
of the sequence . We show that is in fact a solution of
the CCSQP by noting that
and therefore or
. Hence, we have in the limit as ,
or
, which according to Lemma 3 shows that
is the one and only solution of the CCSQP.
Proof: [Theorem 2]: In this case, the proof is identical
(with
) to the proof above until expression (21). We con-
tinue from there by defining
, and ex-
pressing
in terms of its eigendecomposition as ,we
then have
where , since .
The remainder of the proof follows the proof above, with ob-
vious substitutions.
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Peng Hui Tan (S’00) received the B. Eng. and M.
Eng. degrees in electrical and electronic engineering
from National University of Singapore in 1998
and 1999, respectively. He is currently working
toward the Ph.D. degree within the Telecom-
munication Theory Group in the Department of
Computer Engineering, Chalmers University of
Technology,Gothenburg, Sweden.
Lars K. Rasmussen (M’93) was born on March
8, 1965, in Copenhagen, Denmark. He received the
M.Eng. degree from the Technical University of
Denmark, Lyngby, in 1989, and the Ph.D. degree
from Georgia Institute of Technology, Atlanta, GA,
in 1993.
From 1993 to 1995, he was at the University of
South Australia, Adelaide, Australia. From 1995 to
1998, he was with the Dentre for Wireless Commu-
nications at the National University of Singapore. He
is currently an Associate Professor at Chalmers Uni-
versity of Technology, Gothenburg, Sweden.
Teng J. Lim (M’95) was born in Singapore on May
4, 1967. He received the B.Eng. degree in electrical
engineering from the National University of Singa-
pore in 1992 and the Ph.D. degree from Cambridge
University, Cambridge, U.K., in 1995, in the area of
IIR filtering for acoustic echo cancellation.
SinceSeptember 1995, he has beenwiththe Centre
for Wireless Communications, Singapore, where he
is now a Senior Member of Technical Staff and leads
the Signal Processing Strategic Research Group.