1840 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
Decentralized Multiuser Detection for Time-Varying
Multipath Channels
Tamer A. Kadous, Student Member, IEEE, and Akbar M. Sayeed, Member, IEEE
Abstract—Multiple-access interference(MAI)and time-varying
multipath effects are the two most significant factors limiting the
performance of code-division multiple-access (CDMA) systems.
While multipath effects are exploited in existing CDMA systems to
combat fading, they are often considered a nuisance to MAI sup-
pression. We propose an integrated framework based on canonical
multipath-Doppler coordinates that exploits channel dispersion
effects forMAI suppression. The canonical coordinates aredefined
by a fixed basis derived from a fundamental characterization of
propagation effects. The basis corresponds to uniformly spaced
multipath delays and Doppler shifts of the signaling waveform that
capture the essential degrees of freedom in the received signal and
eliminate the need for estimating arbitrary delays and Doppler
shifts. The framework builds on the notion of active coordinates
that carry the desired signal energy, facilitate maximal exploitation
of channel diversity, and provide minimum-complexity MAI sup-
pression. Progressively powerful multiuser detectors are obtained
by incorporating additional inactive coordinates carrying only
MAI. Signal space partitioning in terms of active/inactive coordi-
nates provides a direct handle on controlling receiver complexity
to achieve a desired level of performance. System performance is
analyzed for two characteristic time scales relativeto the coherence
time of the channel. Adaptive receiver structures are identified
that are naturally amenable to blind implementations requiring
knowledge of only the spreadingcode of the desired user.
Index Terms—Blind CDMA receivers,channel modeling, dimen-
sion reduction, interference suppression, subspace processing.
I. INTRODUCTION
C
ODE-DIVISION multiple access (CDMA) has emerged
as a promising core wireless technology for meeting the
physical layer challenges of modern communication networks.
Innovative signal processing is playing a key role in the design
of high-performance CDMA receivers. Major signal processing
challenges stem from the following three key factors that have
a significant impact on CDMA system performance: channel
propagation effects manifested as multipath dispersion, multi-
path fading, and temporal variations or Doppler effects; mul-
tiple-access interference (MAI); and complexity of the signal
processing algorithms. Furthermore, these factors affect system
performance in an interrelated fashion and have to be addressed
jointly.
Paper approved by U. Mitra, the Editor for Spread Spectrum/Equalization
of the IEEE Communications Society. Manuscript received August 14, 1999;
revised March 28, 2000. This work was supported in part by the Wisconsin
Alumni Research Foundation and by the National Science Foundation
under Grant CCR-9875805. This paper was presented in part at the IEEE
GLOBECOM’99, Rio de Janeiro, Brazil, December 1999.
The authors are with the Department of Electrial and Computer Engi-
neering, University of Wisconsin-Madison, Madison, WI 53706 USA (e-mail:
akbar@engr.wisc.edu).
Publisher Item Identifier S 0090-6778(00)09891-3.
For signaling waveforms of duration and bandwidth ,
the dimension of the overall signal space is approximately
1
(see, e.g., [2]). In direct-sequence CDMA systems,
is proportional to the spreading gain , where is
the chip duration. Centralized receivers, which have the knowl-
edge of spreading codes of all users, represent the signal space
in terms of symbol-rate sampled outputs of the matched fil-
ters for different users [3]. Decentralized receivers, which have
knowledge of only the spreading code of desired user, represent
the space in terms of
-dimensional chip-rate sampled
2
out-
puts of the matched filter for the desired user. If the number
of (strong) users
is smaller than , it is most advanta-
geous to operate in the lower-dimensional subspace containing
the multiuser signal. While centralized receivers directly ac-
complish this, decentralized receivers rely on the data itself to
adaptively estimate the multiuser subspace (see, e.g., [4], [3],
[5], and [6]). For desired performance of adaptive decentralized
receivers in realistic time-varying scenarios, it is extremely im-
portant to map the received signal to a lower-dimensional sub-
space to enable reliable estimation of requisite statistics and
rapid tracking. However, most existing decentralized receiver
designs operate in the full (
) -dimensional chip-rate sampled
space which can result in unacceptably poor performance in
realistic time-varying scenarios (see, e.g., [7]). Multipath prop-
agation effects distort the signal and make the problem even
more challenging. While there has been considerable recent re-
search on decentralized reception over multipath channels (see,
e.g., [4] and [8]–[11], ), it falls short of jointly addressing the
key issues of propagation effects, MAI suppression, and re-
ceiver complexity, primarily due to the lack of an appropriate
framework relating these aspects of receiver design. In partic-
ular, there is no systematic approach for effecting a judicious
complexity versus performance tradeoff.
We introduce receiver design in canonical multipath-Doppler
coordinates as an integrated framework for combating time-
varying multipath distortion, suppressing MAI, and managing
receiver complexity. The canonical coordinates are derived
from a fundamental characterization of channel propagation
dynamics in terms of uniformly spaced discrete multipath delays
and Doppler shifts of the signaling waveform. These wave-
forms capture the essential degrees of freedom in the received
signal and constitute a canonical fixed basis for representing it.
Consequently, processing in canonical coordinates eliminates
the need for estimating arbitrary delays and Doppler shifts.
While dispersion effects are often considered a nuisance to MAI
1
More precisely,
N
=
(
B
+2
B
)(
T
+
T
)
, where
T
is the multipath
spread and
B
is the Doppler spread of the channel [1]. However, the terms
other than
TB
are relatively small since
T
T
and
B
B
, typically.
2
Or, oversampled outputs to cover all
N
N
dimensions.
0090–6778/00$10.00 © 2000 IEEE
KADOUS AND SAYEED: DECENTRALIZED MULTIUSER DETECTION FOR TIME-VARYING MULTIPATH CHANNELS 1841
suppression, canonical multipath-Doppler coordinates provide
a natural partitioning of the signal space that enables exploita-
tion of propagation effects for MAI suppression and diversity
processing. A keynotion in our framework is that of primary and
secondary coordinates [12]. The primary (active) coordinates
of a desired user depend on its multipath and Doppler spreads
and define a canonical low-dimensional subspace for capturing
its signal energy. The primary coordinates facilitate maximal
exploitation of channel diversity [13] and minimum-complexity
MAI suppression. However, additional degrees of freedom are
needed in general to adequately suppress the MAI that corrupts
the desired signal in the primary coordinates. These additional
degrees of freedom are furnished by the secondary coordinates.
The secondary coordinates may include the active coordinates
of other users (centralized reception) [14], [12] or inactive coor-
dinates of the desired user (decentralized reception) [12] that do
notcontainthedesiredsignal,onlytheMAI.Thegenericreceiver
structure is depicted in Fig. 1(a). The signal space partitioning in
terms of primary/secondary coordinates provides a systematic
approach for tailoring receiver complexity to achieve a desired
levelof performance.
The next section develops the notion of canonical multi-
path-Doppler coordinates. Section III derives a decentralized
minimum-mean-squared-error (MMSE) receiver structure in
terms of primary and secondary coordinates. Performance
analysis in Section IV guides the choice of design parameters.
Examples illustrating various facets of the framework are
presented in Section V. Practical issues related to channel
estimation and adaptive/blind implementations are discussed in
Section VI. Concluding remarks are provided in Section VII.
II. C
ANONICAL MULTIPATH-DOPPLER COORDINATES
This section provides a brief discussion of the concept
of canonical multipath-Doppler coordinates that underlies
our framework [13], [12]. The complex baseband received
waveform
for a single symbol of a single user is given by
(1)
(2)
where
is the information bearing signal, is the user delay,
is complex additive white Gaussian noise (AWGN), and
denotes the spread-spectrum signaling waveform of du-
ration
. Channel propagation is characterized by the multi-
path-Doppler spreading function
that accounts for the
temporal and spectral dispersion produced by the channel [15].
and denote the multipath and Doppler spreads of the
channel, respectively.
3
The key idea behind canonical multipath-Doppler coordi-
nates is that the receiver “sees” only finitely many degrees of
freedom in the signal due to the inherently finite duration
and essentially finite bandwidth of the transmitted waveform
[13], [12], [15]. These essential degrees of freedom are
3
We assume negligible intersymbol interference
T
T
, which is often the
case in CDMA channels. However, the channel is frequency selective (
T >
T
) in most cases, thereby affording multipath diversity.
(a)
(b)
Fig. 1. (a) Generic linear receiver structure based on canonical multipath-
Doppler coordinates. (b) Interpretation of the MMSE receiver in (20) and (21).
The matrix filter
C
suppresses MAI in the primary coordinates
z
by using the
secondary coordinates
z
. The filter
w
further suppresses MAI in the residual
e
and performs diversity combining.
captured by the following fundamental characterization [13],
[12], [15], [16]:
(3)
which corresponds to a uniform sampling of the multi-
path-Doppler plane.
is a time-frequency smoothed
version of
that arises due to the time- and band-limited
nature of
. denotes the representation of the
delay
. The number of degrees of freedom
is determined by the normalized multipath and Doppler spreads,
and
4
and is proportional to the
products
and . We note that the Doppler components
(index
) in (3) capture the temporal channel variations
encountered within a symbol duration. Temporal variations
over symbols are captured by the variations in the channel
coefficients
over symbols.
The channel characterization (3) defines the canonical multi-
path-Dopplercoordinates.Itassertsthatthereceivedsignal
belongs to an -dimensional subspace
spanned by the fixed basis
(4)
,
generated by discretely delayed and Doppler-shifted ver-
sions of the spread-spectrum signaling waveform
[13],
[12]. These basis waveforms define the active coordinates,
, ,
4
b
x
c
(
d
x
e
) denotes the largest (smallest) integer smaller (larger) than
x
.
1842 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
carrying the signal energy. Each user corresponds to unique
active coordinates defined by its spreading waveform
and
the channel spread parameters. The front-end processing for
representing a received symbol in the canonical coordinates
consists of projection (despreading) onto the basis waveformsof
the form (4)
(5)
In (5),
, and are chosen to always include the
active coordinates. However, inactive coordinates,
, corresponding to canonical delays and Doppler
shifts outside the channel spread, may also be used to aid in
MAI suppression. Fig. 2 illustrates the notion of active/inactive
coordinates. Canonical coordinates, taken together for all users
and symbols of interest, along with corresponding channel co-
efficients, constitute sufficient statistics for demodulation—all
signalprocessingcanbeperformedinthecanonicalcoordinates.
An important implication of the canonical representation (3)
is that regardless of the actual physical distribution of multi-
path delays and Doppler shifts, virtually all information is con-
tained in the uniformly spaced canonical coordinates [13], [12].
The main error in (3) is due to the band-limited approximation
and can be made arbitrarily small by sufficient oversampling in
multipath, and by including sufficiently many terms in the sum-
mation (3). In particular, for direct-sequence CDMA,
is in-
versely related to the chip duration
, where is the
spreading gain, and oversampling by a factor
corresponds to
in (3) and (4).
III. R
ECEIVER DESIGN IN THE CANONICAL COORDINATES
Forsimplicity,weillustratemultiuser reception for the case of
synchronized user transmissions and binary phase-shift keying
(BPSK) signaling.
5
The received signal for a single symbol ad-
mits the canonical representation
(6)
(7)
where
denotes the symbol, the canonical multi-
path-Doppler basis waveforms, and
the corresponding
canonical channel coefficients of the
th user. The above signal
representation provides a natural (dictated by channel disper-
sion effects) a priori partitioning of the signal subspace that can
be leveraged for MAI suppression. Centralized receivers rep-
resent the subspace in terms of active coordinates of all users
[14]. Decentralized reception is based on an alternative repre-
sentation in terms of active (primary) and inactive (secondary)
coordinates of the desired user. The focus of this paper is on de-
centralized reception.
5
Asynchronous scenarios can be treated analogously by considering twice as
many interfering users and by processing a block of symbols [3]. The essential
ideas presented here remain unchanged.
Fig. 2. A schematic illustrating active and inactive coordinates. Active
coordinates correspond to multipath-Doppler basis signals that lie within the
channel spread. Inactive coordinates correspond to basis functions outside the
channel spread.
We illustrate the key ideas behind the framework with
decentralized MMSE receiver design and begin by assuming
the knowledge of the channel coefficients of the desired user.
In Section VI, we discuss blind channel estimation issues and
blind implementations of the proposed receivers. The overall
generic receiver structure is shown in Fig. 1(a). The objective
is to choose the primary and secondary filters
and to
yield an MMSE estimate of the bit
of the desired user
(8)
A. Primary Coordinates—Minimal Complexity Reception
The primary coordinates for user 1 of dimension
take the form
6
(9)
where
is the matrix of cross correlations between
the active basis waveforms
and
of users and , respectively,
denotes the vector of channel coefficients
of the th user, and denotes Gaussian noise vector
with correlation matrix
. The powers of different users
are absorbed in the
’s. The first term in (9)
constitutes the signal part in
, while the second term
constitutes the MAI component. Note that all the signal energy
of the desired user is contained in
. In the absence of MAI
(
), only the primary coordinates are needed and the
optimal receiver is the generalized RAKE receiver that exploits
joint multipath-Doppler diversity via maximal-ratio combining
(MRC):
[13]. In the presence of MAI, the fact
that the desired signal belongs to the one-dimensional subspace
spanned by
can be exploited for suppressing in (9). Given
6
For simplicity, we assume the same number of primary coordinates for all
users.
KADOUS AND SAYEED: DECENTRALIZED MULTIUSER DETECTION FOR TIME-VARYING MULTIPATH CHANNELS 1843
, the MMSE receiverbased only on the primary coordinates is
given by
(10)
(11)
Here,
is the correlation matrix of the primary coor-
dinates. This MMSE receiver works in the low-dimensional
primary coordinates to provide minimal-complexity MAI
suppression while maximally exploiting the available (multi-
path-Doppler) diversity to combat fading.
B. Primary and Secondary Coordinates—Enhanced Reception
Secondary (inactive) coordinates
can be progressively in-
corporated into the receiver, via the lower branch in Fig. 1(a), to
improve its MAI suppression capability. The secondary coordi-
nates take the form
(12)
where
is the matrix of cross correlation between
inactive basis waveforms of user 1 and the active basis
waveforms of user
. The Gaussian noise vector has
correlation matrix
, where is the matrix of correlations
between the secondary basis waveforms of user 1. Note that a
particularly attractive feature of the secondary (inactive) coor-
dinates
is that they are signal free—they are only correlated
with the noise and MAI component of
.
7
The MMSE receiver operating on both the primary and sec-
ondary coordinates solves the following problem:
(13)
where
and . The solution to (13) is the
Wiener filter and takes the form
where
(14)
and
(since the secondary coordinates are signal-free).
The submatrices are given by (11) and
(15)
(16)
7
Note that in (12) we are assuming that
~
Q
0
; that is, the active and
inactive basis waveforms are roughly orthogonal. This assumption is based on
the autocorrelation properties of spreading codes. However, our approach can
be readily extended to account for nonzero correlations as well. In particular,
we may use a linearly transformed version of the inactive coordinates that lie in
the orthogonal component of the active subspace.
By using the block matrix inversion formula [17], we can ex-
pand
as
(17)
where
(18)
(19)
Using (14) and (17), we can explicitly characterize
and
as
(20)
(21)
The solution for the primary and secondary filters in (20) and
(21) has an intuitively appealing interpretation as illustrated in
Fig. 1(b). Conditioned on a fixed value of the
, the
matrix
is the linear MMSE estimator of from (
) and is the covariance matrix of the corresponding
estimation error
. Thus, the filter optimally exploits the sec-
ondary coordinates
to suppress MAI in (since is un-
correlated with the signal component
in ). The filter
then optimally suppresses the MAI remaining in the residual
error
by forming an MMSE estimate of from (compare
with the solution in (11) based on the primary coordinates only).
The number of secondary coordinates
can vary between
zero (only primary coordinates) and
(covering the
entire signal space). As we will see, depending on the number
of dominant interfering users, near-optimal performance can be
achieved with significantly low-dimensional (
)
processing.
We note that most of the decentralized receivers proposed in
the literature are based on chip-rate sampled processing. The
received signal is sampled every chip duration and the resultant
-dimensional vector is processed by an -tap filter, which
may be chosen based on the MMSE criterion. In practice, the
-tap MMSE receiver has to be implemented adaptively. For
large
, there are too many degrees of freedom to allow reliable
estimation and tracking of the MMSE filter taps. Thus, it is
necessarytodevelopMMSEdetectors withfewernumberoftaps
(degreesof freedom).Ad hoc lower-dimensional representations
for the filter are often considered (see, for example, [18]). Other
approaches based on subspace tracking are presented in [19]
and, [9]. In our framework, the natural signal space partitioning
in terms of active/inactive coordinates provides a systematic
approach to controlling receiver complexity. The detector
operates in an
-dimensional subspace.
The complexity of the receiver can be progressively increased,
byincreasing
,toachievea desiredlevelof performance.
We also note that the proposed receiver only requires
knowledge of
and . only depends on
the spreading code of the desired user and
can be readily
estimated from data. This solution assumes the knowledge of
of the desired user. In Section VI-B, we will see that the
structure of
can be exploited for blindly estimating .
IV. P
ERFORMANCE ANALYSIS
In this section, we assess the performance of the proposed
MMSE receiver structure under varying conditions. Let
1844 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
be the time scale over which the performance of the receiver
is assessed. Our analysis is based on two distinct time scales
depending on how
compares , the coherence time
of the channel seen by the desired user. Essentially,
is
the time duration over which the channel coefficients
(and
hence
) remain approximately constant. The coherence time
is roughly equal to the reciprocal of the channel Doppler spread
(
) [1]. For , the desired user’s signal ex-
hibits a fixed direction determined by
and the channel is ef-
fectively equivalent to an AWGN channel. Thus, the probability
of error (
) is governed by the instantaneous signal-to-interfer-
ence-plus-noise ratio (SINR)
[20].
8
Over time scales
significantlylonger than
, i.e., , the desired
user’s signal can exhibit up to
de-
grees of freedom and the average
is computed by averaging
over the statistics of . We analyze the receiver over
both time scales and also investigate measures of near–far resis-
tance [3] that are appropriate in the two scenarios.
Another factor that affects system performance is the degrees
of freedom exhibited by the interference. This depends on how
compares to the coherence time of the interference .
9
If , the MAI exhibits up to de-
grees of freedom in the full-dimensional space since each inter-
fering user exhibits a fixed direction within its active subspace
[
in (11), (15), and (16)]. On the other hand,
if
, the MAI can exhibit up to
degrees of freedom, since each interfering user can exhibit up
to
directions in its active subspace due to the time variations
in the channel coefficients.
10
Clearly, in the detector subspace,
spanned by the active and inactive basis vectors,the MAI cannot
exhibit more than
degrees of freedom. How-
ever, unless
, the receiver will not be near–far
resistant. As long as
, we can choose to ensure
. Our analysis will clearly show the dependence
of performance on
and .
To facilitate analysis, we first derive alternate expressions for
the optimum solution. Let
denote the
signal-free component of the canonical coordinates
(22)
The correlation matrix of
is of the form
(23)
8
Note that SINR is really a function of all
h
’s. However, for sufficiently
effective MAI suppression, the residual MAI can be lumped to Gaussain noise
[20].
9
For simplicity, we assume that all interfering users experience the same co-
herence time
T
(which may be different from
T
).
10
Note that rank(
E
[
h h
]
)
D
.
Without loss of generality assume that ; that is,
and .
11
Let
(24)
be the eigendecomposition of
, where contains the eigen-
vectorscorresponding to nonzero eigenvalues(
is the diagonal
matrix of nonzero eigenvalues) and
contains the eigenvec-
tors corresponding to the zero eigenvalues. The second equality
further partitions
and into primary and secondary coor-
dinates. Let
denote the interference subspace
and
denote the orthogonal complement of .
Note that
and
.
admits the eigendecomposition
(25)
where
. Using the block inversion formula (17)
and (25), we get
(26)
(27)
where
denotes the signal-free component of the
residual estimation error
( ) and
. Based on the preceding devel-
opment, the optimum filter solution admits the following repre-
sentations:
(28)
(29)
(30)
where the first equality follows from (23) and the matrix inver-
sion lemma [21], the second one from (26), and the last one
11
Otherwise, we can always prewhiten the canonical coordinates:
z
!
R z
, since
Q
and
~
Q
are known at the receiver.
![Fig. 6. Comparison of analytically computedP [via (38)] versus Monte–Carlo averaging of (37) over allh ’s for different values ofD . Three interfering users. (a) and (b): all users with equal power. (c) and (d): interfering users are three times stronger than desired user. (e) and (f): interfering users are ten times stronger. We note for largeD the receiver performance is close to that of a single-user (MRC) receiver with three-level diversity [1] operating in the absence of MAI.](/figures/fig-6-comparison-of-analytically-computedp-via-38-versus-98hwl0he.webp)
![Fig. 5. Analytically computedP [using (38)] as a function of the relative power of three interfering users (N = 27) for different values ofD .](/figures/fig-5-analytically-computedp-using-38-as-a-function-of-the-3e26f8lu.webp)