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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Capacity Limits of MIMO Channels
Andrea Goldsmith, Senior Member, IEEE, Syed Ali Jafar, Student Member, IEEE, Nihar Jindal, Student Member, IEEE,
and Sriram Vishwanath, Student Member, IEEE
Invited Paper
Abstract—We provide an overview of the extensive recent
results on the Shannon capacity of single-user and multiuser
multiple-input multiple-output (MIMO) channels. Although
enormous capacity gains have been predicted for such channels,
these predictions are based on somewhat unrealistic assumptions
about the underlying time-varying channel model and how well
it can be tracked at the receiver, as well as at the transmitter.
More realistic assumptions can dramatically impact the potential
capacity gains of MIMO techniques. For time-varying MIMO
channels there are multiple Shannon theoretic capacity definitions
and, for each definition, different correlation models and channel
information assumptions that we consider. We first provide a
comprehensive summary of ergodic and capacity versus outage
results for single-user MIMO channels. These results indicate that
the capacity gain obtained from multiple antennas heavily depends
on the available channel information at either the receiver or
transmitter, the channel signal-to-noise ratio, and the correlation
between the channel gains on each antenna element. We then focus
attention on the capacity region of the multiple-access channels
(MACs) and the largest known achievable rate region for the
broadcast channel. In contrast to single-user MIMO channels,
capacity results for these multiuser MIMO channels are quite
difficult to obtain, even for constant channels. We summarize
results for the MIMO broadcast and MAC for channels that are
either constant or fading with perfect instantaneous knowledge
of the antenna gains at both transmitter(s) and receiver(s). We
show that the capacity region of the MIMO multiple access and
the largest known achievable rate region (called the dirty-paper
region) for the MIMO broadcast channel are intimately related
via a duality transformation. This transformation facilitates
finding the transmission strategies that achieve a point on the
boundary of the MIMO MAC capacity region in terms of the
transmission strategies of the MIMO broadcast dirty-paper region
and vice-versa. Finally, we discuss capacity results for multicell
MIMO channels with base station cooperation. The base stations
then act as a spatially diverse antenna array and transmission
strategies that exploit this structure exhibit significant capacity
gains. This section also provides a brief discussion of system level
issues associated with MIMO cellular. Open problems in this field
abound and are discussed throughout the paper.
Index Terms—Antenna correlation, beamforming, broadcast
channels (BCs), channel distribution information (CDI), channel
state information (CSI), multicell systems, multiple-access channels (MACs), multiple-input multiple-output (MIMO) channels,
multiuser systems, Shannon capacity.
Manuscript received November 8, 2002; revised January 31, 2003. This work
was supported in part by the Office of Naval Research (ONR) under Grants
N00014-99-1-0578 and N00014-02-1-0003. The work of S. Vishwanath was
supported by a Stanford Graduate Fellowship.
The authors are with the Department of Electrical Engineering, Stanford
University, Stanford, CA 94305 USA (e-mail: andrea@wsl.stanford.edu;
syed@wsl.stanford.edu; njindal@wsl.stanford.edu; sriram@wsl.stanford.edu).
Digital Object Identifier 10.1109/JSAC.2003.810294
I. INTRODUCTION
IRELESS systems continue to strive for ever higher
data rates. This goal is particularly challenging for
systems that are power, bandwidth, and complexity limited.
However, another domain can be exploited to significantly
increase channel capacity: the use of multiple transmit and
receive antennas. Pioneering work by Winters [81], Foschini
[20], and Telatar [69] ignited much interest in this area by
predicting remarkable spectral efficiencies for wireless systems
with multiple antennas when the channel exhibits rich scattering and its variations can be accurately tracked. This initial
promise of exceptional spectral efficiency almost “for free”
resulted in an explosion of research activity to characterize the
theoretical and practical issues associated with multiple-input
multiple-output (MIMO) wireless channels and to extend these
concepts to multiuser systems. This tutorial summarizes the
segment of this recent work focused on the capacity of MIMO
systems for both single-users and multiple users under different
assumptions about spatial correlation and channel information
available at the transmitter and receiver.
The large spectral efficiencies associated with MIMO channels are based on the premise that a rich scattering environment
provides independent transmission paths from each transmit antenna to each receive antenna. Therefore, for single-user systems, a transmission and reception strategy that exploits this
sepastructure achieves capacity on approximately
is the number of transmit antennas and
rate channels, where
is the number of receive antennas. Thus, capacity scales linrelative to a system with just one transmit
early with
and one receive antenna. This capacity increase requires a scattering environment such that the matrix of channel gains between transmit and receive antenna pairs has full rank and independent entries and that perfect estimates of these gains are
available at the receiver. Perfect estimates of these gains at both
the transmitter and receiver provides an increase in the constant
multiplier associated with the linear scaling. Much subsequent
work has been aimed at characterizing MIMO channel capacity
under more realistic assumptions about the underlying channel
model and the channel estimates available at the transmitter and
receiver. The main question from both a theoretical and practical standpoint is whether the enormous capacity gains initially
predicted by Winters, Foschini, and Telatar can be obtained in
more realistic operating scenarios and what specific gains result
from adding more antennas and/or a feedback link to feed receiver channel information back to the transmitter.
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GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
MIMO channel capacity depends heavily on the statistical properties and antenna element correlations of the
channel. Recent work has developed both analytical and
measurement-based MIMO channel models along with the corresponding capacity calculations for typical indoor and outdoor
environments [26]. Antenna correlation varies drastically as a
function of the scattering environment, the distance between
transmitter and receiver, the antenna configurations, and the
Doppler spread [1], [65]. As we shall see, the effect of channel
correlation on capacity depends on what is known about the
channel at the transmitter and receiver: correlation sometimes
increases capacity and sometimes reduces it [16]. Moreover,
channels with very low correlation between antennas can still
exhibit a “keyhole” effect where the rank of the channel gain
matrix is very small, leading to limited capacity gains [12].
Fortunately, this effect is not prevalent in most environments.
The impact of channel statistics in the low-power (wideband)
regime has interesting properties as well: recent results in this
area can be found in [71].
We focus on MIMO channel capacity in the Shannon
theoretic sense. The Shannon capacity of a single-user time-invariant channel is defined as the maximum mutual information
between the channel input and output. This maximum mutual
information is shown by Shannon’s capacity theorem to be the
maximum data rate that can be transmitted over the channel
with arbitrarily small error probability. When the channel
is time-varying channel capacity has multiple definitions,
depending on what is known about the channel state or its
distribution at the transmitter and/or receiver and whether
capacity is measured based on averaging the rate over all
channel states/distributions or maintaining a constant fixed or
minimum rate. Specifically, when the instantaneous channel
gains, called the channel state information (CSI), are known
perfectly at both transmitter and receiver, the transmitter can
adapt its transmission strategy relative to the instantaneous
channel state. In this case, the Shannon (ergodic) capacity is
the maximum mutual information averaged over all channel
states. This ergodic capacity is typically achieved using an
adaptive transmission policy where the power and data rate
vary relative to the channel state variations. Other capacity
definitions for time-varying channels with perfect transmitter
and receiver CSI include outage capacity and minimum-rate
capacity. These capacities require a fixed or minimum data rate
in all nonoutage channel states, which is needed for applications with delay-constrained data where the data rate cannot
depend on channel variations (except in outage states, where
no data is transmitted). The average rate associated with outage
or minimum rate capacity is typically smaller than ergodic
capacity due to the additional constraints associated with these
definitions. This tutorial will focus on ergodic capacity in the
case of perfect transmitter and receiver CSI.
When only the channel distribution is known at the transmitter (receiver) the transmission (reception) strategy is based
on the channel distribution instead of the instantaneous channel
state. The channel coefficients are typically assumed to be
jointly Gaussian, so the channel distribution is specified by
the channel mean and covariance matrices. We will refer to
knowledge of the channel distribution as channel distribution
685
information (CDI). We assume throughout the paper that CDI is
always perfect, so there is no mismatch between the CDI at the
transmitter or receiver and the true channel distribution. When
only the receiver has perfect CSI the transmitter must maintain
a fixed-rate transmission strategy optimized with respect to its
CDI. In this case, ergodic capacity defines the rate that can
be achieved based on averaging over all channel states [69].
Alternatively, the transmitter can send at a rate that cannot be
supported by all channel states: in these poor channel states the
receiver declares an outage and the transmitted data is lost. In
this scenario, each transmission rate has an outage probability
associated with it and capacity is measured relative to outage
probability1 (capacity CDF) [20]. An excellent tutorial on
fading channel capacity for single antenna channels can be
found in [4]. For single-user MIMO channels with perfect
transmitter and receiver CSI the ergodic and outage capacity
calculations are straightforward since the capacity is known for
every channel state. Thus, for single-user MIMO systems the
tutorial will focus on capacity results assuming perfect CDI at
the transmitter and perfect CSI or CDI at the receiver. Although
there has been much recent progress in this area, many open
problems remain.
In multiuser channels, capacity becomes a -dimensional region defining the set of all rate vectors (
) simultaneously achievable by all users. The multiple capacity definitions for time-varying channels under different transmitter and
receiver CSI and CDI assumptions extend to the capacity region
of the multiple-access channel (MAC) and broadcast channel
(BC) in the obvious way [28], [48], [49], [70]. However, these
MIMO multiuser capacity regions, even for time-invariant channels, are difficult to find. Few capacity results exist for timevarying multiuser MIMO channels, especially under the realistic assumption that the transmitter(s) and/or receiver(s) have
CDI only. Therefore, the tutorial focus for MIMO multiuser systems will be on ergodic capacity under perfect CSI at the transmitter and receiver, with a brief discussion of the known results
and open problems for other capacity definitions and CSI/CDI
assumptions.
Note that the MIMO techniques described herein are applicable to any channel described by a matrix. Matrix channels
describe not only multiantenna systems but also channels with
crosstalk [85] and wideband channels [72]. While the focus
of this tutorial is on memoryless channels (flat-fading), the results can also be extended to channels with memory (ISI) using
well-known methods for incorporating the channel delay spread
into the channel matrix [59], as will be discussed in the next
section.
Many practical MIMO techniques have been developed
to capitalize on the theoretical capacity gains predicted by
Shannon theory. A major focus of such work is space-time
coding: recent work in this area is summarized in [21]. Other
techniques for MIMO systems include space–time modulation
[30], [33], adaptive modulation and coding [10], space–time
1Note that an outage under perfect CSI at the receiver only is different than
an outage when both transmitter and receiver have perfect CSI. Under receiver
CSI only an outage occurs when the transmitted data cannot be reliably decoded
at the receiver, so that data is lost. When both the transmitter and receiver have
perfect CSI the channel is not used during outage (no service), so no data is lost.
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
TABLE I
TABLE OF ABBREVIATIONS
Fig. 1.
equalization [2], [51], space–time signal processing [3],
space–time CDMA [14], [34], and space–time OFDM [50],
[52], [82]. An overview of the recent advances in these areas
and other practical techniques along with their performance
can be found in [25].
The remainder of this paper is organized as follows. In
Section II, we discuss the capacity of single-user MIMO
systems under different assumptions about channel state and
distribution information at the transmitter and receiver. This
section also describes the optimality of beamforming and
training issues. Section III describes the capacity region of the
MIMO MAC and the “dirty-paper” achievable region of the
MIMO BC, along with a duality connection between these
regions. The capacity of multicell systems under dirty paper
coding (DPC) and opportunistic beamforming is discussed in
Section IV, as well as tradeoffs between capacity, diversity, and
sectorization. Section V summarizes these capacity results and
describes some remaining open problems and design questions
associated with MIMO systems.
A note on notation: We use boldface to denote matrices and
for expectation.
denotes the determinant and
vectors and
the inverse of a square matrix . For any general matrix
,
denotes the conjugate transpose and Tr
denotes the
denotes a diagtrace. denotes the identity matrix and diag
onal matrix with the ( ) entry equal to . For symmetric maimplies that is positive semidefinite.
trices the notation
A table of abbreviations used throughout the paper is given
in Table I.
II. SINGLE-USER MIMO
In this section, we focus on the capacity of single-user MIMO
channels. While most wireless systems today support multiple
users, single-user results are still of much interest for the insight they provide and their application to channelized systems,
where users are allocated orthogonal resources (time, frequency
bands, etc.). MIMO channel capacity is also much easier to derive for single users than for multiple users. Indeed, single-user
MIMO channel with perfect CSIR and distribution feedback.
MIMO capacity results are known for many cases, where the
corresponding multiuser problems remain unsolved. In particular, very little is known about multiuser capacity without the assumption of perfect channel state information at the transmitter
(CSIT) and at the receiver (CSIR). While there remain many
open problems in obtaining the single-user capacity under general assumptions of CSI and CDI, for several interesting cases
the solution is known. This section will give an overview of
known results for single-user MIMO channels with particular
focus on special cases of CDI at the transmitter, as well as the
receiver. We begin with a description of the channel model and
the different CSI and CDI models we consider, along with their
motivation.
A. Channel Model
transmit antennas and a reConsider a transmitter with
ceiver with receive antennas. The channel can be represented
by the
matrix . The
received signal is equal
to
(1)
transmitted vector and is the
addiwhere is the
tive white circularly symmetric complex Gaussian noise vector,
normalized so that its covariance matrix is the identity matrix.
The normalization of any nonsingular noise covariance matrix
to fit the above model is as straightforward as multiplying
to yield the effective channel
the received vector with
and a white noise vector.
The CSI is the channel matrix . Thus, with perfect CSIT or
CSIR, the channel matrix is assumed to be known perfectly
and instantaneously at the transmitter or receiver, respectively.
When the transmitter or receiver knows the channel state perfectly, we also assume that it knows the distribution of this state
perfectly, since the distribution can be obtained from the state
observations.
1) Perfect CSIR and CDIT: The perfect CSIR and CDIT
model is motivated by the scenario where the channel state can
be accurately tracked at the receiver and the statistical channel
model at the transmitter is based on CDI fed back from the receiver. This distribution model is typically based on receiver estimates of the channel state and the uncertainty associated with
these estimates. Fig. 1 illustrates the underlying communication
model in this scenario, where denotes the complex Gaussian
distribution.
The salient features of the model are as follows.
• Conditioned on the parameter that defines the channel
at different time
distribution, the channel realizations
instants are independent identically distributed (i.i.d.).
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
Fig. 2.
MIMO channel with perfect CSIR and CDIT ( fixed).
• In a wireless system the channel statistics change over
time due to mobility of the transmitter, receiver, and the
scattering environment. Thus, is time-varying.
• The statistical model depends on the time scale of interest.
For example, in the short term, the channel coefficients
may have a nonzero mean and one set of correlations
reflecting the geometry of the particular propagation
environment. However, over a long term the channel coefficients may be described as zero-mean and uncorrelated
due to the averaging over several propagation environments. For this reason, uncorrelated, zero-mean channel
coefficients is a common assumption for the channel
distribution in the absence of distribution feedback or
when it is not possible to adapt to the short-term channel
statistics. However, if the transmitter receives frequent
and it can adapt to these time-varying
updates of
short-term channel statistics then capacity is increased
relative to the transmission strategy associated with just
the long-term channel statistics. In other words, adapting
the transmission strategy to the short-term channel statistics increases capacity. In the literature adaptation to
the short-term channel statistics (the feedback model of
Fig. 1) is referred to by many names including mean and
covariance feedback, imperfect feedback and partial CSI
[38], [40], [42], [45], [46], [56], [66], [76].
• The feedback channel is assumed to be free from noise.
This makes the CDIT a deterministic function of the CDIR
and allows optimal codes to be constructed directly over
the input alphabet [8].
• For each realization of the conditional average transmit
.
power is constrained as
• The ergodic capacity of the system in Fig. 1 is the caaveraged over the different realizations
pacity
where
is the ergodic capacity of the channel shown
in Fig. 2. This figure represents a MIMO channel with perfect CSI at the receiver and only CDI about the constant
distribution at the transmitter. Channel capacity calculations generally implicitly assume CDI at both the transmitter and receiver except for special channel classes, such
as the compound channel or arbitrarily varying channel.
This implicit knowledge of is justified by the fact that the
channel coefficients are typically modeled based on their
long-term average distribution. Alternatively, can be obtained by the feedback model of Fig. 1. Thus, motivated
by the distribution feedback model of Fig. 1, we will provide capacity results for the system model of Fig. 2 under
different distribution ( ) models. For clarity, we explicitly state when CDI is available at either the transmitter
or receiver, to contrast with the case where CSI is also
available.
687
Computation of
for general
is a hard problem.
Almost all research in this area has focused on three special
cases for this distribution: zero-mean spatially white channels,
spatially white channels with nonzero mean, and zero-mean
channels with nonwhite channel covariance. In all three
cases, the channel coefficients are modeled as complex jointly
Gaussian random variables. Under the zero-mean spatially
white (ZMSW) model, the channel mean is zero and the
channel covariance is modeled as white, i.e., the channel
elements are assumed to be i.i.d. random variables. This
model typically captures the long-term average distribution of
the channel coefficients averaged over multiple propagation
environments. Under the channel mean information (CMI)
model, the mean of the channel distribution is nonzero while
the covariance is modeled as white with a constant scale factor.
This model is motivated by a system where the channel state
is measured imperfectly at the transmitter, so the CMI reflects
this measurement and the constant factor reflects the estimation
error. Under the channel covariance information (CCI) model,
the channel is assumed to be varying too rapidly to track its
mean, so the mean is set to zero and the information regarding
the relative geometry of the propagation paths is captured by a
nonwhite covariance matrix. Based on the underlying system
model shown in Fig. 1, in the literature the CMI model is
also called mean feedback and the CCI model is also called
covariance feedback. Mathematically, the three distribution
models for can be described as follows:
Zero-Mean Spatially White (ZMSW):
;
Channel Mean Information (CMI):
;
Channel Covariance Information (CCI):
.
is an
matrix of i.i.d. zero mean, unit variance
Here,
complex circularly symmetric Gaussian random variables. The
channel mean and are constants that may be interpreted as
the channel estimate based on the feedback and the variance of
and
are called the rethe estimation error, respectively.
ceive and transmit fade covariance matrices. Although not completely general, this simple correlation model has been validated
through recent field measurements as a sufficiently accurate representation of the fade correlations seen in actual cellular sysand the variance
tems [13]. Under CMI the channel mean
of the estimation error are assumed known and under CCI
and
are asthe transmit and receive covariance matrices
sumed known.
2) CDIT and CDIR: In highly mobile channels, the assumption of perfect CSI at the receiver can be unrealistic.
Thus, we now consider a model where both transmitter and
receiver only have information about the channel distribution.
Even for a rapidly fluctuating channel where reliable channel
estimation is not possible, it might be possible for the receiver
to track the short-term distribution of the channel fades, as
the channel distribution changes much more slowly than the
channel itself. The estimated distribution can be made available
to the transmitter through a feedback channel. Fig. 3 illustrates
the underlying communication model.
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Instead, due to the changing propagation environment wireless
channels vary over time, assuming values over a continuum. The
capacity of fading channels is investigated next.
C. Fading MIMO Channel Capacity
Fig. 3.
MIMO channel with CDIR and distribution feedback.
Fig. 4.
MIMO channel with CDIT and CDIR ( fixed).
Note that the estimation of the channel statistics at the
receiver is captured in the model as a genie that provides the
receiver with the correct channel distribution. The feedback
channel represents the same information being made available
to the transmitter simultaneously. This model is slightly optimistic because in practice the receiver estimates only from the
received signal and therefore will not have a perfect estimate.
As in the previous section, the ergodic capacity turns out to
be the expected value (expectation over ) of the ergodic ca, where
is the ergodic capacity of the channel
pacity
in Fig. 4. In this figure, is constant and known at both the
transmitter and receiver (CDIT and CDIR). As in the previous
is difficult for general , so we
section, the computation of
restrict ourselves to the same three channel distribution models
described in the previous subsection: the ZMSW, CMI, and CCI
models.
Next, we summarize the single-user MIMO capacity results
under various assumptions on CSI and CDI.
B. Constant MIMO Channel Capacity
When the channel is constant and known perfectly at the
transmitter and the receiver, the capacity is
Tr
(2)
where is the input covariance matrix. Telatar [69] showed that
the MIMO channel can be converted to parallel, noninterfering
single-input single-output (SISO) channels through a singular
value decomposition (SVD) of the channel matrix. The SVD
parallel channels with gains corresponding
yields
of . Waterfilling the transmit power
to the singular values
over these parallel channels leads to the power allocation
With slow fading, the channel may remain approximately
constant long enough to allow reliable estimation of the channel
state at the receiver (perfect CSIR) and timely feedback of this
state information to the transmitter (perfect CSIT). However,
in systems with moderate to high user mobility, the system
designer is inevitably faced with channels that change rapidly.
Fading models where only the channel distribution is available
to the receiver (CDIR) and/or transmitter (CDIT) are more
applicable to such systems. Capacity results under various
assumptions regarding CSI and CDI are summarized in this
section.
1) Capacity With Perfect CSIT and Perfect CSIR: Perfect
CSIT and perfect CSIR model a fading channel that changes
slow enough to be reliably measured by the receiver and fed
back to the transmitter without significant delay. The ergodic
capacity of a flat-fading channel with perfect CSIT and CSIR is
simply the average of the capacities achieved with each channel
realization. The capacity for each channel realization is given
by the constant channel capacity expression in the previous section. Thus, the fading MIMO channel capacity assuming perfect
channel knowledge at both transmitter and receiver is
2) Capacity With Perfect CSIR and CDIT: ZMSW
Model: Seminal work by Foschini and Gans [22] and Telatar
[69] addressed the case of perfect CSIR and a ZMSW channel
distribution at the transmitter. Recall that in this case, the
is assumed to have i.i.d. complex Gaussian
channel matrix
). As described in the introduction, the
entries (i.e.,
two relevant capacity definitions in this case are capacity versus
outage (capacity CDF) and ergodic capacity. For any given
input covariance matrix the input distribution that achieves the
ergodic capacity is shown in [22] and [69] to be complex vector
Gaussian, mainly because the vector Gaussian distribution
maximizes the entropy for a given covariance matrix. This
leads to the transmitter optimization problem—i.e., finding the
optimum input covariance matrix to maximize ergodic capacity
subject to a transmit power (trace of the input covariance matrix) constraint. Mathematically, the problem is to characterize
the optimum to maximize
(3)
is the waterfill level,
is the power in the th
where
is defined as
. The
eigenmode of the channel and
channel capacity is shown to be
(4)
Although the constant channel model is relatively easy to analyze, wireless channels in practice are not fixed or constant.
(5)
Tr
Tr
(6)
where
(7)
is the mutual information with the input covariance matrix
and the expectation is with respect to the channel
. The mutual information
is achieved by
matrix
transmitting independent complex circular Gaussian symbols
along the eigenvectors of . The powers allocated to each
eigenvector are given by the eigenvalues of .
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
689
It is shown in [22] and [69] that the optimum input covariance
matrix that maximizes ergodic capacity is the scaled identity
matrix, i.e., the transmit power is divided equally among all the
transmit antennas. Thus, the ergodic capacity is given by
(8)
An integral form of this expectation involving Laguerre polyand simultaneously become
nomials is derived in [69]. If
. Exlarge, capacity is seen to grow linearly with
pressions for the growth rate constant can be found in [32] and
[69].
Telatar [69] conjectures that the optimal input covariance
matrix that maximizes capacity versus outage is a diagonal
matrix with the power equally distributed among a subset of the
transmit antennas. The principal observation is that as the capacity CDF becomes steeper, capacity versus outage increases
for low outage probabilities and decreases for high outage
probabilities. This is reflected in the fact that the higher the
outage probability, the smaller the number of transmit antennas
that should be used. As the transmit power is shared equally
increases (so
between more antennas the expectation of
the ergodic capacity increases) but the tails of its distribution
decay faster. While this improves capacity versus outage for
low outage probabilities, the capacity versus outage for high
outages is decreased. Usually, we are interested in low outage
probabilities2 and, therefore, the usual intuition for outage
capacity is that it increases as the diversity order of the channel
increases, i.e., as the capacity CDF becomes steeper. Foschini
and Gans [22] also propose a layered architecture to achieve
these capacities with scalar codes. This architecture, called Bell
Labs Layered Space–Time (BLAST), shows enormous capacity
gains over single antenna systems. For example, at 1% outage,
12 dB signal-to-noise ratio (SNR) and with 12 antennas, the
spectral efficiency is shown to be 32 b/s/Hz as opposed to the
spectral efficiencies of around 1 b/s/Hz achieved in present day
single antenna systems. While the channel models in [22] and
[69] assume uncorrelated and frequency flat fading, practical
channels exhibit both correlated fading, as well as frequency
selectivity. The need to estimate the capacity gains of BLAST
for practical systems in the presence of channel fade correlations and frequency selective fading sparked off measurement
campaigns reported in [24] and [55]. The measured capacities
are found to be about 30% smaller than would be anticipated
from an idealized model. However, the capacity gains over
single antenna systems are still overwhelming.
3) Capacity With Perfect CSIR and CDIT: CMI and CCI
Models: Recent results indicate that for MIMO channels
the capacity improvement resulting from some knowledge
of the short-term channel statistics at the transmitter can be
substantial. These results have ignited much interest in the
capacity of MIMO channels with perfect CSIR and CDIT
under general distribution models. In this section, we focus on
the cases of CMI and CCI channel distributions, corresponding
to distribution feedback of the channel mean or covariance
2The capacity for high outage probabilities becomes relevant for schemes that
transmit only to the best user. For such schemes, it is shown in [6] that increasing
the number of transmit antennas reduces the average sum capacity.
matrix. Key results on the capacity of such channels have been
recently obtained by several authors including Madhow and
Visotsky [76], Trott and Narula [58], [57], Jafar and Goldsmith
[42], [40], [38], Jorsweick and Boche [45], [46], and Simon
and Moustakas [56], [66].
Mathematically the problem is defined by (6) and (7), with
the distribution on determined by the CMI or CCI. The optimum input covariance matrix in general can be a full rank matrix which implies either vector coding across the antenna array
or transmission of several scalar codes in parallel with successive interference cancellation at the receiver. Limiting the rank
of the input covariance matrix to unity, called beamforming, essentially leads to a scalar coded system which has a significantly
lower complexity for typical array sizes.
The complexity versus capacity tradeoff is an interesting
aspect of capacity results under CDIT. The ability to use scalar
codes to achieve capacity under CDIT for different channel
distribution models, also called optimality of beamforming,
captures this tradeoff and has been the topic of much research
in itself. Note that vector coding refers to fully unconstrained
signaling schemes for the memoryless MIMO Gaussian
channel. Every symbol period, a channel use corresponds to the
transmission of a vector symbol comprised of the inputs to each
transmit antenna. Ideally, while decoding vector codewords the
receiver needs to take into account the dependencies in both
space and time dimensions and therefore the complexity of
vector decoding grows exponentially in the number of transmit
antennas. A lower complexity implementation of the vector
coding strategy is also possible in the form of several scalar
codewords being transmitted in parallel. It is shown in [38] that
without loss of capacity, any input covariance matrix, regardless
of its rank, can be treated as several scalar codewords encoded
independently at the transmitter and decoded successively at
the receiver by subtracting out the contribution from previously
decoded codewords at each stage. However, well-known
problems associated with successive decoding and interference
subtraction, e.g., error propagation, render this approach
unsuitable for use in practical systems. It is in this context that
the question of optimality of beamforming becomes important.
Beamforming transforms the MIMO channel into a single-input
single-output (SISO) channel. Thus, well established scalar
codec technology can be used to approach capacity and since
there is only one beam, interference cancellation is not needed.
In the summary given below, we include the results on both the
transmitter optimization problem, as well as the optimality of
beamforming.
Multiple-Input Single-Output (MISO) Channels: We first
consider systems that use a single receive antenna and multiple
transmit antennas. The channel matrix is rank one. With perfect CSIT and CSIR, for every channel matrix realization it is
possible to identify the only nonzero eigenmode of the channel
accurately and beamform along that mode. On the other hand,
with perfect CSIR and CDIT under the ZMSW model, it was
shown by Foschini and Gans [22] and Telatar [69] that the optimal input covariance matrix is a multiple of the identity matrix. Thus, the inability of the transmitter to identify the nonzero
channel eigenmode forces a strategy, where the power is equally
distributed in all directions.
690
Fig. 5.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Plot of necessary and sufficient conditions (9). <Author: Fig. 5 not cited in text>
For a system using a single receive antenna and multiple
transmit antennas, the transmitter optimization problem under
CSIR and CDIT is solved by Visotsky and Madhow in [76] for
the distribution models of CMI and CCI. For the CMI model
) the principal eigenvector of the optimal input
(
is found to be along the channel mean
covariance matrix
vector and the eigenvalues corresponding to the remaining
eigenvectors are shown to be equal. When beamforming is
optimal, all power is allocated to the principal eigenvector.
) the eigenvectors of the
For the CCI model (
are shown to be along
optimal input covariance matrix
the eigenvectors of the transmit fade covariance matrix and
the eigenvalues are in the same order as the corresponding
eigenvalues of the transmit fade covariance matrix. Moreover, Visotsky and Madhow’s numerical results indicate that
beamforming is close to the optimal strategy when the quality
of feedback improves, i.e., when the channel uncertainty
decreases under CMI or when a stronger channel mode can be
identified under CCI. We will discuss quality of feedback in
more detail below. Under CMI, Narula and Trott [58] point out
that there are cases where the capacity is actually achieved via
beamforming. While they do not obtain fully general necessary
and sufficient conditions for when beamforming is a capacity
achieving strategy, they develop partial answers to the problem
for two transmit antennas.
A general condition that is both necessary and sufficient for
optimality of beamforming is obtained by Jafar and Goldsmith
in [40] for both the CMI and CCI models. The result can be
stated as follows.
The ergodic capacity can be achieved with a unit rank matrix
if and only if the following condition is true:
(9)
where for the CCI model
are the two largest eigenvalues of the channel
1)
;
fade covariance matrix
is exponential distributed with unit mean, i.e.,
2)
;
and for the CMI model
1)
;
has a noncentral chi-squared distribution. More
2)
where
precisely,
is the zeroth-order modified Bessel function of the
first kind.
Further, for the CCI model the expectation can be evaluated
to express (9) explicitly in closed form as
(10)
The optimality conditions are plotted in Fig. 5. For the CCI
model the optimality of beamforming depends on the two largest
of the transmit fade covariance matrix and
eigenvalues
the transmit power . Beamforming is found to be optimal
when the two largest eigenvalues of the transmit covariance matrix are sufficiently disparate or the transmit power is sufficiently low. Since beamforming corresponds to using the principal eigenmode alone, this is reminiscent of waterpouring solutions where only the deepest level gets all the water when it
is sufficiently deeper than the next deepest level and when the
quantity of water is small enough. For the CMI model the optimality of beamforming is found to depend on transmit power
and the quality of feedback associated with the mean informaof the
tion, which is defined mathematically as the ratio
norm squared of the channel mean vector and the channel uncertainty . As the transmit power is decreased or the quality
of feedback improves beamforming becomes optimal. As menso quality
tioned earlier, for perfect CSIT (uncertainty
) the optimal input strategy is beamforming,
of feedback
while in the absence of mean feedback (quality of feedback
so the CMI model becomes the ZMSW model), as shown by
Telatar [69], the optimal input covariance has full rank, i.e.,
beamforming is necessarily suboptimal. Note that [40], [57],
[58], and [76] assume a single receive antenna. Next, we summarize the analogous capacity results for MIMO channels.
MIMO Channels: With multiple transmit and receive antennas, capacity with CSIR and CDIT under the CCI model with
) is obtained by
spatially white fading at the receiver (
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
Jafar and Goldsmith in [42]. Like the single receive antenna
case the capacity achieving input covariance matrix is found
to have the eigenvectors of the transmit fade covariance matrix
and the eigenvalues are in the same order as the corresponding
eigenvalues of the transmit fade covariance matrix. Jafar and
Goldsmith also presented in closed form a mathematical condition that is both necessary and sufficient for optimality of beamforming in this case. The same necessary and sufficient condition is also derived independently by Jorsweick and Boche in
[45] and Simon and Moustakas in [66]. In [46], Jorsweick and
Boche extend these results to incorporate fade correlations at
the receiver as well. Their results show that while the receive
fade correlation matrix does not affect the eigenvectors of the
optimal input covariance matrix, it does affect the eigenvalues.
The general condition for optimality of beamforming found by
Jorsweick and Boche depends upon the two largest eigenvalues
of the transmit covariance matrix and all the eigenvalues of the
receive covariance matrix.
Capacity under the CMI model with multiple transmit and
receive antennas is solved by Jafar and Goldsmith in [38] when
the channel mean has rank one and is extended to general
channel means by Moustakas and Simon in [67]. Similar to
the MISO case, the principal eigenvector of the optimal input
covariance matrix and of the channel mean are the same and
the eigenvalues of the remaining eigenvectors are equal. For
the case where the channel mean has unit rank, a necessary
and sufficient condition for optimality of beamforming is also
determined in [38].
These results summarize our discussion of channel capacity
with CDIT and perfect CSIR under different channel distribution models. From these results we notice that the benefits
of adapting to distribution information regarding CMI or CCI
fed back from the receiver to the transmitter are twofold. Not
only does the capacity increase with more information about
the channel distribution, but this feedback also allows the
transmitter to identify the stronger channel modes and achieve
this higher capacity with simple scalar codewords.
We conclude this section with a discussion on the growth of
capacity with number of antennas. With perfect CSIR and CDIT
under the ZMSW channel distribution, it was shown by Foschini
and Gans [22] and by Telatar [69] that the channel capacity
. This linear increase occurs
grows linearly with
whether the transmitter knows the channel perfectly (perfect
CSIT) or only knows its distribution (CDIT). The proportionality constant of this linear increase, called the rate of growth,
has also been characterized in [15], [31], [68], [69]. Chuah et al.
[15] show that with perfect CSIR and CSIT, the rate of growth of
is reduced by channel fading correlacapacity with
tions at high SNR but is increased at low SNR. They also show
that the mutual information under CSIR increases linearly with
even when a spatially white transmission strategy
is used on a correlated fading channel, although the slope is reduced relative to the uncorrelated fading channel. As we will
see in the next section, the assumption of perfect CSIR is crucial for the linear growth behavior of capacity with the number
of antennas.
In the next section, we explore the capacity when only CDI
is available at the transmitter and the receiver.
691
4) Capacity With CDIT and CDIR: ZMSW Model: We saw
in the last section that with perfect CSIR, channel capacity
grows linearly with the minimum of the number of transmit and
receive antennas. However, reliable channel estimation may
not be possible for a mobile receiver that experiences rapid
fluctuations of the channel coefficients. Since user mobility
is the principal driving force for wireless communication
systems, the capacity behavior with CDIT and CDIR under the
ZMSW distribution model (i.e.,
is distributed as
with
at either the receiver or transmitter) is of
no knowledge of
particular interest. In this section, we summarize some MIMO
capacity results in this area.
One of the first papers to address the MIMO capacity with
CDIR and CDIT under the ZMSW model is [53] by Marzetta
and Hochwald. They model the channel matrix components as
i.i.d. complex Gaussian random variables that remain constant
for a coherence interval of symbol periods after which they
change to another independent realization. Capacity is achieved
transmitted signal matrix is equal to the
when the
product of two statistically independent matrices: a
isotropically distributed unitary matrix times a certain
random matrix that is diagonal, real, and nonnegative. This
result enables them to determine capacity for many interesting
cases. Marzetta and Hochwald show that, for a fixed number
of antennas, as the length of the coherence interval increases,
the capacity approaches the capacity obtained as if the receiver
knew the propagation coefficients. However, perhaps the most
surprising result in [53] is the following: In contrast to the linear
under the perfect CSIR
growth of capacity with
assumption, [53] showed that in the absence of CSIT and CSIR,
capacity does not increase at all as the number of transmit antennas is increased beyond the length of the coherence interval
. The MIMO capacity for this model was further explored by
Zheng and Tse in [89]. They show that at high SNRs capacity
is achieved using no more than
transmit antennas. In particular, having more transmit antennas
than receive antennas does not provide any capacity increase
at high SNR. Zheng and Tse also find that for each 3-dB SNR
.
increase, the capacity gain is
Notice that [53], [89] assume block fading models, i.e., the
channel fade coefficients are assumed to be constant for a block
symbol durations. Hochwald and Marzetta extend their
of
results to continuous fading in [54] where, within each independent -symbol block, the fading coefficients have an arbitrary
time correlation. If the correlation vanishes beyond some lag
, called the correlation time of the fading, then it is shown in
[54] that increasing the number of transmit antennas beyond
antennas does not increase capacity. Lapidoth and
Moser [47] explored the channel capacity of this CDIT/CDIR
model for the ZMSW distribution at high SNR without the
block fading assumption. In contrast to the results of Zheng and
Tse for block fading, Lapidoth and Moser show that without
the block fading assumption, the channel capacity grows only
double logarithmically in SNR. This result is shown to hold
under very general conditions, even allowing for memory and
partial receiver side information.
5) Capacity With CDIR and CDIR: CCI Model: The results
in [53] and [89] seem to leave little hope of achieving the high
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
capacity gains predicted for MIMO systems when the channel
cannot be accurately estimated at the receiver and the channel
distribution follows the ZMSW model. However, before resigning ourselves to these less-than-optimistic results we note
that these results assume a somewhat pessimistic model for
the channel distribution. That is because most channels when
averaged over a relatively small area have either a nonzero
mean or a nonwhite covariance. Thus, if these distribution
parameters can be tracked, the channel distribution corresponds
to either the CMI or CCI model.
Recent work by Jafar and Goldsmith [37] addresses the
MIMO channel capacity with CDIT and CDIR under the
CCI distribution model. The channel matrix components are
modeled as spatially correlated complex Gaussian random
variables that remain constant for a coherence interval of
symbol periods after which they change to another independent
realization based on the spatial correlation model. The channel
correlations are assumed to be known at the transmitter and
receiver. As in the case of spatially white fading (ZMSW
model), Jafar and Goldsmith show that with the CCI model the
transmitted signal matrix
capacity is achieved when the
isotropically distributed
is equal to the product of a
random
unitary matrix, a statistically independent
matrix that is diagonal, real and nonnegative and the matrix
.
of the eigenvectors of the transmit fade covariance matrix
It is shown in [37] that the channel capacity is independent
eigenvalues of the transmit fade
of the smallest
covariance matrix, as well as the eigenvectors of the transmit
and
. Also, in
and receive fade covariance matrices
contrast to the results for the spatially white fading model
where adding more transmit antennas beyond the coherence
) does not increase capacity, [37] shows
interval length (
that additional transmit antennas always increase capacity as
long as their channel fading coefficients are spatially correlated.
Thus, in contrast to the results in favor of independent fades
with perfect CSIR, these results indicate that with CCI at the
transmitter and the receiver, transmit fade correlations can be
beneficial, making the case for minimizing the spacing between
transmit antennas when dealing with highly mobile, fast fading
channels that cannot be accurately measured. Mathematically,
), capacity
[37] proves that for fast fading channels (
is a Schur-concave function of the vector of eigenvalues of
the transmit fade correlation matrix. The maximum possible
capacity gain due to transmitter fade correlations is shown to
db.
be 10
6) Frequency Selective Fading Channels: While flat fading
is a realistic assumption for narrowband systems where
the signal bandwidth is smaller than the channel coherence
bandwidth, broadband communications involve channels that
experience frequency selective fading. Research on the capacity
of MIMO systems with frequency selective fading typically
takes the approach of dividing the channel bandwidth into
parallel flat fading channels and constructing an overall block
diagonal channel matrix with the diagonal blocks given by the
channel matrices corresponding to each of these subchannels.
Under perfect CSIR and CSIT, the total power constraint then
leads to the usual closed-form waterfilling solution. Note
that the waterfill is done simultaneously over both space and
frequency. Even SISO frequency selective fading channels can
be represented by the MIMO system model (1) in this manner
[59]. For MIMO systems, the matrix channel model is derived
by Bolcskei, Gesbert and Paulraj in [5] based on an analysis
of the capacity behavior of OFDM-based MIMO channels
in broadband fading environments. Under the assumption of
perfect CSIR and CDIT for the ZMSW model, their results
show that in the MIMO case, unlike the SISO case, frequency
selective fading channels may provide advantages over flat
fading channels not only in terms of ergodic capacity but also
in terms of capacity versus outage. In other words, MIMO
frequency selective fading channels are shown to provide both
higher diversity gain and higher multiplexing gain than MIMO
flat-fading channels. The measurements in [55] show that
frequency selectivity makes the CDF of the capacity steeper
and, thus, increases the capacity for a given outage as compared
with the flat-frequency case, but the influence on the ergodic
capacity is small.
7) Training for Multiple-Antenna Systems: The results
summarized in the previous sections indicate that CSI plays a
crucial role in the capacity of MIMO systems. In particular,
the capacity results in the absence of CSIR are strikingly
different and often quite pessimistic compared with those that
assume perfect CSIR. To recapitulate, with perfect CSIR and
CDIT MIMO channel capacity is known to increase linearly
when the CDIT assumes the ZMSW or CCI
with
distribution models. However, in fast fading when the channel
changes so rapidly that it cannot be estimated reliably at the
receiver (CDIR only) the capacity does not increase with the
where is the
number of transmit antennas at all for
channel decorrelation time. Also at high SNR under the ZMSW
distribution model, capacity with perfect CSIR and CDIT
increases logarithmically with SNR, while the capacity with
CDIR and CDIT increases only double logarithmically with
SNR. Thus, CSIR is critical for obtaining the high capacity
benefits of multiple-antenna wireless links. CSIR is often
obtained by sending known training symbols to the receiver.
However, with too little training the channel estimates are
poor, whereas with too much training there is no time for data
transmission before the channel changes. So the key question
to ask is how much training is needed in multiple-antenna
wireless links. This question itself is the title of the paper
[29] by Hassibi and Hochwald where they compute a lower
bound on the capacity of a channel that is learned by training
and maximize the bound as a function of the receive SNR,
fading coherence time, and number of transmitter antennas.
When the training and data powers are allowed to vary, the
optimal number of training symbols is shown to be equal to
the number of transmit antennas—which is also the smallest
training interval length that guarantees meaningful estimates
of the channel matrix. When the training and data powers are
instead required to be equal, the optimal training duration may
be longer than the number of antennas. Hassibi and Hochwald
also show that training-based schemes can be optimal at high
SNR, but are suboptimal at low SNR.
D. Open Problems in Single-User MIMO
The results summarized in this section form the basis of our
understanding of channel capacity under different CSI and CDI
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
693
the noise vector where
is circularly
symmetric complex Gaussian with identity covariance. The received signal at the base station is then equal to
..
.
Fig. 6. System models of the (left) MIMO BC and the (right) MIMO MAC
channels.
assumptions. These results serve as useful indicators for the benefits of incorporating training and feedback schemes in a MIMO
wireless link to obtain CSIR/CDIT and CSIT/CDIT, respectively. However, our knowledge of MIMO capacity with CDI
only is still far from complete, even for single-user systems. We
conclude this section by pointing out some of the many open
problems.
1) Combined CCI and CMI: Capacity under CDIT and perfect CSIR is unsolved under a combined CCI and CMI
distribution model even with a single receive antenna.
2) CCI: With perfect CSIR and CDIT capacity is not known
under the CCI model for completely general correlations.
3) CDIR: Almost all cases with only CDIR are open problems.
4) Outage capacity: Most results for CDI only at either the
transmitter or receiver are for ergodic capacity. Capacity
versus outage has proven to be less analytically tractable
than ergodic capacity and contains an abundance of open
problems.
III. MULTIUSER MIMO
In this section, we consider the two basic multiuser MIMO
channel models: the MIMO MAC and the MIMO BC. Since
the capacity region of a general MAC has been known for quite
a while, there are many results on the MIMO MAC for both
constant channels and fading channels with different CSI and
CDI assumptions at the transmitters and receivers. The MIMO
BC, however, is a relatively new problem for which capacity
results have only recently been found. As a result, the field is
much less developed, but we summarize the recent results in the
area. Interestingly, the MIMO MAC and MIMO BC have been
shown to be duals, as we will discuss in Section III-C2.
A. System Model
To describe the MAC and BC models, we consider a celantennas and
lular-type system in which the base station has
mobiles has
antennas. The downlink of this
each of the
system is a MIMO BC and the uplink is a MIMO MAC. We
to denote the downlink channel matrix from the base
will use
station to user . Assuming that the same channel is used on the
. A picuplink and downlink, the uplink matrix of user is
ture of the system model is shown in Fig. 6.
be the transmitted signal of user
In the MAC, let
denote the received signal and
(i.e., mobile) . Let
where
In the MAC, each user (i.e., mobile) is subject to an individual
power constraint of . The transmit covariance matrix of user
is defined to be
. The power constraint implies
for
.
Tr
denote the transmitted vector signal
In the BC, let
be the received signal
(from the base station) and let
at receiver (i.e., mobile) . The noise at receiver is represented
and is assumed to be circularly symmetric comby
). The received signal of
plex Gaussian noise (
User is equal to
(11)
The transmit covariance matrix of the input signal is
. The base station is subject to an average power con.
straint , which implies Tr
B. MIMO Multiple-Access Channel
In this section, we summarize capacity results on the multiple-antenna MAC. We first analyze the constant channel scenario and then consider the fading channel. Since the capacity
region of a general MAC is known, the expressions for the capacity of a constant MAC are quite straightforward. For the
fading case, one must consider different assumptions about the
CSI and CDI available at the transmitter and receiver. We consider three cases: perfect CSIR and CSIT, perfect CSIR and
CDIT, and CDIT and CDIR. As above, under CDI, we consider
three different distribution models: the ZMSW, CMI, and CCI
models.
1) Constant Channel: The capacity of any MAC can be
written as the convex closure of the union of rate regions corresponding to every product input distribution
satisfying the user-by-user power constraints [18]. For the
Gaussian MIMO MAC, however, it has been shown that it is
sufficient to consider only Gaussian inputs and that the convex
hull operation is not needed [11], [86]. For any set of powers
, the capacity of the MIMO MAC is shown
in (12), at the bottom of the next page. The th user transmits a
. Each
zero-mean Gaussian with spatial covariance matrix
) corresponds to a
set of covariance matrices (
-dimensional polyhedron (i.e.)
and the capacity region is equal to the union (over all
covariance matrices satisfying the trace constraints) of
694
Fig. 7.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Capacity region of MIMO MAC for
N = 1.
all such polyhedrons. The corner points of each polyhedron can be achieved by successive decoding, in which
users’ signals are successively decoded and subtracted
out of the received signal. For the two-user case, each set
of covariance matrices corresponds to a pentagon, similar in form to the capacity region of the scalar Gaussian
MAC. The corner point where
and
corresponds to decoding
user 2 first (i.e., in the presence of interference from user 1)
and decoding user 1 last (without interference from user 2).
Successive decoding can reduce a complex multiuser detection
problem into a series of single-user detection steps [27].
The capacity region of a MIMO MAC for the single transmit
) is shown in Fig. 7. When
, the coantenna case (
variance matrix of each transmitter is a scalar equal to the transmitted power. Clearly, each user should transmit at full power.
is the
Thus, the capacity region for a -user MAC for
) satisfying
set of all rate vectors (
(13)
For the two-user case, this reduces to the simple pentagon seen
in Fig. 7.
, however, a union must be taken over all coWhen
variance matrices. Intuitively, the set of covariance matrices that
are different from the set of covariance matrices
maximize
that maximize the sum rate. In Fig. 8, a MAC capacity region
is shown. Notice that the region is equal to the union
for
of pentagons (each pentagon corresponding to a different set of
transmit covariance matrices), a few of which are shown with
dashed lines in the figure. The boundary of the capacity region is in general curved, except at the sum rate point, where
the boundary is a straight line [86]. Each point on the curved
portion of the boundary is achieved by a different set of covariance matrices. At point A, user 1 is decoded last and achieves
as a water-fill of the
his single-user capacity by choosing
(independent of
or
). User 2 is decoded first,
channel
is chosen as
in the presence of interference from user 1, so
Fig. 8. Capacity region of MIMO MAC for
N > 1.
and the interference from user 1.
a waterfill of the channel
The sum-rate corner points B and C are the two corner points of
the pentagon corresponding to the sum-rate optimal covariance
and
. At point B user 1 is decoded last,
matrices
whereas at point C user 2 is decoded last. Thus, points B and
C are achieved using the same covariance matrices but different
decoding orders.
Next, we focus on characterizing the optimal covariance
) that achieve different points on the
matrices (
boundary of the MIMO MAC capacity region. Since the MAC
capacity region is convex, it is well known from convex theory
that the boundary of the capacity region can be fully characterover
ized by maximizing the function
all rate vectors in the capacity region and for all nonnegative
) such that
. For a fixed
priorities (
), this is equivalent to finding the
set of priorities (
point on the capacity region boundary that is tangent to a line
whose slope is defined by the priorities. See the tangent line
in Fig. 8 for an example. The structure of the MAC capacity
region implies that all boundary points of the capacity region
are corner points of polyhedrons corresponding to different sets
of covariance matrices. Furthermore, the corner point should
correspond to successive decoding in order of increasing
priority, i.e., the user with the highest priority should be
decoded last and, therefore, sees no interference [70], [73].
Thus, the problem of finding the boundary point on the capacity
assumed to be
region associated with priorities
in descending order (users can be arbitrarily re-numbered to
satisfy this condition) can be written as
subject to power constraints on the trace of each of the covariance matrices. Note that the covariances that maximize the func-
(12)
Tr
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
tion above are the optimal covariances. The most interesting
and useful feature of the optimization problem above is that
the objective function is concave in the covariance matrices.
Thus, efficient convex optimization tools exist that solve this
problem numerically [7]. A more efficient numerical technique
) covarito find the sum-rate maximizing (i.e.,
ance matrices, called iterative waterfilling, was developed by Yu
et al. [86]. This technique is based on the Karush Kuhn Tucker
(KKT) optimality conditions for the sum-rate maximizing covariance matrices. These conditions indicate that the sum-rate
maximizing covariance matrix of any user in the system should
be the single-user water-filling covariance matrix of its own
channel with noise equal to the actual noise plus the interfertransmitters.
ence from the other
2) Fading Channels: As in the single-user case, the capacity
of the MIMO MAC where the channel is time-varying depends
on the definition of capacity and the availability of CSI and CDI
at the transmitters and the receiver. The capacity with perfect
CSIT and CSIR is very well studied, as is the capacity with
perfect CSIR and CDIT under the ZMSW distribution model.
However, little is known about the capacity of the MIMO MAC
with CDIT at either the transmitter or receiver under the CMI or
CCI distribution models. Some results on the optimum distribution for the single antenna case with CDIT and CDIR under the
ZMSW distribution can be found in [62].
With perfect CSIR and CSIT the system can be viewed as a set
of parallel non interfering MIMO MACs (one for each fading
state) sharing a common power constraint. Thus, the ergodic
capacity region can be obtained as an average of these parallel
MIMO MAC capacity regions [87], where the averaging is done
with respect to the channel statistics. The iterative waterfilling
algorithm of [86] easily extends to this case, with joint space
and time waterfilling.
The capacity region of a MAC with perfect CSIR and CDIT
under the ZMSW distribution model was found in [23] and [63].
In this case, Gaussian inputs are optimal and the ergodic capacity region is equal to the time average of the capacity obtained at each fading instant with a constant transmit policy (i.e.,
a constant covariance matrix for each user). Thus, the ergodic
capacity region is given by
Tr
If the channel matrices
have i.i.d. complex Gaussian entries
and each user has the same power constraint, then the optimal
covariances are scaled versions of the identity matrix [69].
There has also been some work on capacity with perfect
CSIR and CDIT under the CCI distribution model [41]. In
this paper, Jafar and Goldsmith determine the optimal transmit
covariance matrices when there is transmit antenna correlation
that is known at the transmitters. This topic has yet to be fully
investigated.
695
Asymptotic results on the sum capacity of MIMO MAC
channels with the number of receive antennas and the number
of transmitters increasing to infinity were obtained by Telatar
[69] and by Viswanath et al. [80]. MIMO MAC sum capacity
with perfect CSIR and CDIT under the ZMSW distribution
model (i.e., each transmitter’s channel is distributed as
)
[69]. Thus, for
is found to grow linearly with
systems with large numbers of users, increasing the number
of receive antennas at the base station ( ) while keeping the
number of mobile antennas ( ) constant can lead to linear
growth. Sum capacity with perfect CSIR and CSIT also scales
, but perfect CSIT is of decreasing
linearly with
value as the number of receive antennas increases [32], [80].
Furthermore, the limiting distribution of the sum capacity with
perfect CSIR and CSIT was found to be Gaussian by Hochwald
and Vishwanath [32].
C. MIMO Broadcast Channel
In this section, we summarize capacity results on the
multiple-antenna BC. When the transmitter has only one
antenna, the Gaussian broadcast channel is a degraded broadcast channel (i.e., the users can be absolutely ranked by their
channel strength), for which the capacity region is known [18].
However, when the transmitter has more than one antenna, the
Gaussian broadcast channel is generally nondegraded.3 The
capacity region of general nondegraded broadcast channels is
unknown, but the seminal work of Caire and Shamai [9] and
subsequent research on this problem have shed a great deal
of light on this channel and the sum capacity of the MIMO
BC has been found. In subsequent sections, we focus mainly
on the constant channel, but we do briefly discuss the fading
channel as well which is still an open problem. Note that the
antennas and each
BC transmitter (i.e., the base station) has
receiver has antennas, as described in Section III-A.
1) Dirty Paper Coding (DPC) Achievable Rate Region: An
achievable region for the MIMO BC was first obtained for the
case by Caire and Shamai [9] and later extended to the
multiple-receive antenna case by Yu and Cioffi [83] using the
idea of DPC [17]. The basic premise of DPC is as follows. If the
transmitter (but not the receiver) has perfect, noncausal knowledge of additive Gaussian interference in the channel, then the
capacity of the channel is the same as if there was no additive
interference, or equivalently as if the receiver also had knowledge of the interference. DPC is a technique that allows noncausally known interference to be “presubtracted” at the transmitter, but in such a way that the transmit power is not increased.
A more practical (and more general) technique to perform this
presubtraction is the cancelling for known interference technique found by Erez et al. in [19].
In the MIMO BC, DPC can be applied at the transmitter when
choosing codewords for different receivers. The transmitter first
picks a codeword (i.e., ) for receiver 1. The transmitter then
chooses a codeword for receiver 2 (i.e., ) with full (noncausal)
knowledge of the codeword intended for receiver 1. Therefore,
3The multiple-antenna broadcast channel is nondegraded because users receive different strength signals from different transmit antennas. See [18] for a
precise definition of degradedness.
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Fig. 9.
Dirty paper rate region,
H
= [1 0:5],
H
= [0:5 1], P = 10.
the codeword of user 1 can be presubtracted such that receiver
2 does not see the codeword intended for receiver 1 as interference. Similarly, the codeword for receiver 3 is chosen such that
receiver 3 does not see the signals intended for receivers 1 and 2
) as interference. This process continues for all
(i.e
receivers. If user
is encoded first, followed by user
,
etc., the following is an achievable rate vector:
(14)
is defined as the convex
The dirty paper region
hull of the union of all such rates vectors over all positive
such that
semi-definite covariance matrices
Tr
and over all permutations
Tr
:
One important feature to notice about the dirty paper rate
equations in (14) is that the rate equations are neither a concave
nor convex function of the covariance matrices. This makes numerically finding the dirty paper region very difficult, because
generally a brute force search over the entire space of covariance
matrices that meet the power constraint must be conducted. The
dirty paper rate region for a two-user channel with
and
is shown in Fig. 9.
Note that DPC and successive decoding (i.e., interference
cancellation by the receiver instead of the transmitter) are
completely equivalent capacity-wise for scalar channels, but
this equivalence does not hold for MIMO channels. It has been
shown [36] that the achievable region with successive decoding
is contained within the DPC region.
2) MAC-BC Duality: In [74], Vishwanath, Jindal, and
Goldsmith showed that the dirty paper rate region of the
multiantenna BC with power constraint is equal to the union
of capacity regions of the dual MAC, where the union is taken
over all individual power constraints that sum to
(16)
(15)
where
form
.
is given by (14). The transmitted signal is
and the input covariance matrices are of the
. From the dirty paper result we find that
are uncorrelated, which implies
This is the multiple-antenna extension of the previously established duality between the scalar Gaussian broadcast and multiple-access channels [44]. In addition to the relationship between the two rate regions, for any set of covariance matrices in
the MAC/BC (and the corresponding rate vector), [74] provides
an explicit set of transformations to find covariance matrices in
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
697
the BC/MAC that achieve the same rates. The union of MAC
capacity regions in (16) is easily seen to be the same expression
Tr
instead of
as in (12) but with the constraint
(i.e., a sum constraint instead of individual
Tr
constraints).
The MAC-BC duality is very useful from a numerical
standpoint because the dirty paper region leads to nonconcave
rate functions of the covariances, whereas the rates in the dual
MAC are concave functions of the covariance matrices. Thus,
the optimal MAC covariances can be found using standard
convex optimization techniques and then transformed to the
corresponding optimal BC covariances using the MAC-BC
transformations given in [74]. A specialized algorithm to
find the optimal MAC covariances can be found in [35]. An
algorithm based on the iterative waterfilling algorithm [86] that
finds the sum rate optimal covariances is given in [43].
The dirty paper rate region is shown in Fig. 9 for a channel
and
. Notice that the dirty paper
with two-users,
rate region shown in Fig. 9 is actually a union of MAC regions,
where each MAC region corresponds to a different set of in, each of the MAC redividual power constraints. Since
gions is a pentagon, as discussed in Section III-B1. Similar to the
MAC capacity region, the boundary of the DPC region is curved,
except at the sum-rate maximizing portion of the boundary. For
case, duality also indicates that rank-one covarithe
ance matrices (i.e., beamforming) are optimal for DPC. This
fact is not obvious from the dirty paper rate equations, but follows from the transformations of [74] which find BC covariances that achieve the same rates as a set of MAC covariance
case).
matrices (which are scalars in the
Duality also allows the MIMO MAC capacity region to be
expressed as an intersection of the dual dirty paper BC rate regions [74, Corollary 1]
Asymptotic results for the sum-rate capacity of the MIMO
BC for
under the ZSMW model can be obtained by combining the asymptotic results for the sum-rate capacity of the
MIMO MAC with duality [32]. Thus, the role of transmitter side
information reduces with the growth in the number of transmit
antennas and, hence, the sum capacity of the MIMO BC with
users and
transmit antennas tends to the sum capacity of
a single-user system with only receiver CSI and
receive antennas and transmit antennas, which is given by
. Thus, the asymptotic growth under CSIR and CSIT
and
or CDIT under the ZMSW model is linear as
the growth rate constant can be found in [32]. As seen for
the MIMO MAC, for systems with large numbers of users, increasing the number of transmit antennas at the base station ( )
while keeping the number of mobile antennas ( ) constant can
lead to linear growth.
D. Open Problems in Multiuser MIMO
Multiuser MIMO has been the primary focus of research in
recent years, mainly due to the large number of open problems
in this area. Some of these are as follows.
1) BC with perfect CSIR and CDIT: The broadcast channel
capacity is only known when both the transmitter and the
receivers have perfect knowledge of the channel.
2) CDIT and CDIR: Since perfect CSI is rarely possible, a
study of capacity with CDI at both the transmitter(s) and
receiver(s) for both MAC and BC is of great practical
relevance.
3) Non-DPC techniques for BC: DPC is a very powerful capacity-achieving scheme, but it appears quite difficult to
implement in practice. Thus, non-DPC multiuser transmissions schemes for the downlink (such as downlink
beamforming [60]) are also of practical relevance.
IV. MULTICELL MIMO
(17)
3) Optimality of DPC: DPC was first shown to achieve the
,
sum-rate capacity of the MIMO BC for the two-user,
channel by Caire and Shamai [9]. This was shown by
proving that the Sato upper bound [61] on the broadcast channel
sum-rate capacity is achievable using DPC. The sum-rate optimality of DPC was extended to the multiuser channel with
by Viswanath and Tse [79] and to the more general
case by Vishwanath et al. [74] and Yu and Cioffi [84].
It has also recently been conjectured that the DPC rate region
is the actual capacity region of the multiple-antenna broadcast
channel. Significant progress toward proving this conjecture is
made in [75] and [77].
4) Fading Channels: Most of the capacity problems for
fading MIMO BCs are still open, with the exception of sum-rate
capacity with perfect CSIR and CSIT. In this case, as for the
MIMO MAC, the MIMO BC can be split into parallel channels
with an overall power constraint (see Li and Goldsmith [48] for
a treatment of the scalar case).
The MAC and the BC are information theoretic abstractions
of the uplink and the downlink of a single cell in a cellular
system. However, a cellular system, by definition, consists of
many cells. Due to the fundamental nature of wireless propagation, transmissions in a cell are not limited to within that cell.
Users and base stations in adjacent cells experience interference from each other. Also, since the base stations are typically
not mobile themselves there is the possibility for the base stations to communicate through a high-speed reliable connection,
possibly consisting of optical fiber links capable of very high
data rates. This opens up the opportunity for base stations to cooperate in the way they process different users’ signals. Analysis of the capacity of the cellular network, explicitly taking
into account the presence of multiple cells, multiple users and
multiple antennas, and the possibilities of cooperation between
base stations is inevitably a hard problem and runs into several long-standing unsolved problems in network information
theory. However, such an analysis is also of utmost importance
because it defines a common benchmark that can be used to
gauge the efficiency of any practical scheme, in the same way
that the capacity of a single-user link serves as a measure of the
performance of practical schemes. There has been some recent
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
research in this area that extends the single-cell MAC and BC
results to multiple cells. In this section, we summarize some of
these results.
The key to the extension of single-cell results to multiple-cell
systems is the assumption of perfect cooperation between base
stations. Conceptually, this allows the multiple base stations
to be treated as physically distributed antennas of one comcoorposite base station. Specifically, consider a group of
antennas and mobiles, each with
dinated cells, each with
antennas. If we define
to be the downlink
channel of user from base station , then the composite downand the composite
link channel of user is
uplink channel is . The received signal of user can then be
, where
is the composite transwritten as
mitted signal defined as
. Here, we let
represent the transmit signal from base .
First, let us consider the uplink. As pointed out by Jafar
et al. [36] the single-cell MIMO MAC capacity region results
apply to this system in a straightforward way. Thus, by assuming perfect data cooperation between the base stations,
the multiple-cell uplink is easily seen to be equal to the
MAC capacity region of the composite channel, defined as
in (12), where the power
constraints of the th mobile is .
On the downlink, since the base stations can cooperate perfectly, DPC can be used over the entire transmitted signal (i.e.,
across base stations) in a straightforward manner. The application of DPC to a multiple-cell environment with cooperation between base stations is pioneered in recent work by Shamai and
Zaidel [64]. For one antenna at each user and each base station, they show that a relatively simple application of DPC can
enhance the capacity of the cellular downlink. While capacity
computations are not the focus of [64], they do show that their
scheme is asymptotically optimal at high SNRs.
The MIMO downlink capacity is explored by Jafar and
Goldsmith in [39]. Note that the multicell downlink can be
solved in a similar way as the uplink. But this requires perfect
data and power cooperation between the base stations. If we
represent the transmit vector for User from base
let
station , the composite transmit vector intended for User
is
. Thus, the composite covariance of
. The covariance matrix
user is defined as
. Assuming
of the entire transmitted signal is
perfect data cooperation between the base stations, DPC can be
applied to the composite vectors intended for different users.
Thus, the dirty paper region described in Section III-C1, (15),
can be achieved in the multicell downlink.
While data cooperation is a justifiable assumption for capacity computations in the sense that it captures the possibility
of base stations cooperating among themselves as described
earlier in this section, in practice each base station has its
own power constraint. The per-base power constraint can be
, where
is the power constraint
expressed as
at base . Thus power cooperation, or pooling the transmit
power for all the base stations to have one overall transmit
power constraint, is not realistic. Note that on the uplink
the base stations are only receiving signals and, therefore,
Fig. 10.
Optimal sum rate relative to HDR.
no power cooperation is required. The per-base power constraints restrict consideration to covariance matrices such that
Tr
. This is equivalent
on the sum of the first
diagonal entries
to a constraint of
, a constraint of
on the sum on the next
diagonal
of
, etc. These constraints are considerably stricter
entries of
than a constraint on the trace of
as in the single-cell case.
Though DPC yields an achievable region, it has not been
shown to achieve the capacity region or even the sum-rate
capacity with per-base power constraints. Additionally, the
MAC-BC duality (Section III-C2) which greatly simplified
calculation of the dirty paper region does not apply under
per-base power constraints. Thus, even generating numerical
results for the multicell downlink is quite challenging.
However, data and power cooperation does give a simple
upper bound on the capacity of the network. Based on numerical comparisons between this upper bound and a lower bound
on capacity derived in [39], Jafar and Goldsmith find that the
simple upper bound with power and data cooperation is also a
good measure of the capacity with data cooperation alone.
Note that current wireless systems use the high data rate
(HDR) protocol and transmit to only one user at a time on
the downlink, where this best user is chosen to maximize the
average system data rate. In contrast, DPC allows the base
station to transmit to many users simultaneously. This is particularly advantageous when the number of transmit antennas
at the base station is much larger than the number of receive
antennas at each user—a common scenario in current cellular
systems. To illustrate the advantages of DPC over HDR, even
for a single cell, the relative gains of optimal DPC over a
strategy that serves only the best user at any time are shown
in Fig. 10. Note that this single-cell model is equivalent to the
multicell system with no cooperation between base stations so
that the interference from other cells is treated as noise. With
cooperation between base stations the gains are expected to be
even more significant as DPC reduces the overall interference
by making some users invisible to others.
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
The capacity results described in this section address just a
few out of many interesting questions in the design of a cellular
system with multiple antennas. Multiple antennas can be used
not only to enhance the capacity of the system but also to drive
down the probability of error through diversity combining. Recent work by Zheng and Tse [88] unravels a fundamental diversity versus multiplexing tradeoff in MIMO systems. Also, instead of using isotropic transmit antennas on the downlink and
transmitting to many users, it may be simpler to use directional
antennas to divide the cell into sectors and transmit to one user
within each sector. The relative impact of CDIT and/or CDIR on
each of these schemes is not fully understood. Although in this
paper we focus on the physical layer, smart schemes to handle
CDIT can also be found at higher layers. An interesting example
is the idea of opportunistic beamforming [78]. In the absence
of CSIT, the transmitter randomly chooses the beamforming
weights. With enough users in the system, it becomes very likely
that these weights will be nearly optimal for one of the users. In
other words, a random beam selected by the transmitter is very
likely to be pointed toward a user if there are enough users in
the system. Instead of feeding back the channel coefficients to
the transmitter the users simply feed back the SNRs they see
with the current choice of beamforming weights. This significantly reduces the amount of feedback required. By randomly
changing the weights frequently, the scheme also treats all users
fairly.
V. CONCLUSION
We have summarized recent results on the capacity of MIMO
channels for both single-user and multiuser systems. The great
capacity gains predicted for such systems can be realized in
some cases, but realistic assumptions about channel knowledge
and the underlying channel model can significantly mitigate
these gains. For single-user systems the capacity under perfect
CSI at the transmitter and receiver is relatively straightforward
and predicts that capacity grows linearly with the number of antennas. Backing off from the perfect CSI assumption makes the
capacity calculation much more difficult and the capacity gains
are highly dependent on the nature of the CSI/CDI, the channel
SNR, and the antenna element correlations. Specifically, assuming perfect CSIR, CSIT provides significant capacity gain at
low SNRs but not much at high SNRs. The insight here is that
at low SNRs it is important to put power into the appropriate
eigenmodes of the system. Interestingly, with perfect CSIR and
CSIT, antenna correlations are found to increase capacity at low
SNRs and decrease capacity at high SNRs. Finally, under CDIT
and CDIR for a zero-mean spatially white channel, at high SNRs
capacity grows relative to only the double log of the SNR with
the number of antennas as a constant additive term. This rather
poor capacity gain would not typically justify adding more antennas. However, at moderate SNRs the growth relative to the
number of antennas is less pessimistic.
We also examined the capacity of MIMO broadcast and multiple-access channels. The capacity region of the MIMO MAC
is well-known and can be characterized as a convex optimization problem. Duality allows the DPC achievable region for
the MIMO BC, a nonconvex region, to be computed from the
699
MIMO MAC capacity region. These capacity and achievable
regions are only known for ergodic capacity under perfect CSIT
and CSIR. Relatively little is known about the MIMO MAC
and BC regions under more realistic CSI assumptions. A multicell system with base station cooperation can be modeled as
a MIMO BC (downlink) or MIMO MAC (uplink), where the
antennas associated with each base station are pooled by the
system. Exploiting this antenna structure leads to significant capacity gains over HDR transmission strategies.
There are many open problems in this area. For single-user
systems the problems are mainly associated with CDI only
at either the transmitter or receiver. Most capacity regions
associated with multiuser MIMO channels remain unsolved,
especially ergodic capacity and capacity versus outage for the
MIMO BC under perfect receiver CSI only. There are very few
existing results for CDI at either the transmitter or receiver for
any multiuser MIMO channel. Finally, the capacity of cellular
systems with multiple antennas remains a relatively open area,
in part because the single-cell problem is mostly unsolved and
in part because the Shannon capacity of a cellular system is
not well-defined and depends heavily on frequency assumptions and propagation models. Other fundamental tradeoffs
in MIMO cellular designs such as whether antennas should
be used for sectorization, capacity gain, or diversity are not
well understood. In short, we have only scratched the surface
in understanding the fundamental capacity limits of systems
with multiple transmitter and receiver antennas, as well as
the implications of these limits for practical system designs.
This area of research is likely to remain timely, important, and
fruitful for many years to come.
REFERENCES
[1] A. Abdi and M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” IEEE J. Select. Areas
Commun., vol. 20, pp. 550–561, Apr. 2002.
[2] N. Al-Dhahir, “Overview and comparison of equalization schemes for
space-time-coded signals with application to EDGE,” IEEE Trans.
Signal Processing, vol. 50, pp. 2477–2488, Oct. 2002.
[3] N. Al-Dhahir, C. Fragouli, A. Stamoulis, W. Younis, and R. Calderbank,
“Space-time processing for broadband wireless access,” IEEE Commun.
Mag., vol. 40, pp. 136–142, Sept. 2002.
[4] E. Biglieri, J. Proakis, and S. S. Shitz, “Fading channels: Information
theoretic and communication aspects,” IEEE Trans. Inform. Theory, vol.
44, pp. 2619–2692, Oct. 1998.
[5] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, pp.
225–234, Feb. 2002.
[6] S. Borst and P. Whiting, “The use of diversity antennas in high-speed
wireless systems: Capacity gains, fairness issues, multi-user scheduling,” Bell Labs Tech. Mem., 2001.
[7] S. Boyd and L. Vandenberghe. (2001) Introduction to Convex
Optimization With Engineering Applications. [Online]. Available:
www.stanford.edu/~boyd/cvxbook.html
[8] G. Caire and S. Shamai, “On the capacity of some channels with channel
state information,” IEEE Trans. Inform. Theory, vol. 45, pp. 2007–2019,
Sept. 1999.
[9]
, “On achievable rates in a multi-antenna broadcast downlink,” in
Proc. 38th Annual Allerton Conf. Commununications, Control, Computing, Oct. 2000, pp. 1188–1193.
[10] S. Catreux, V. Erceg, D. Gesbert, and R. W. Heath, “Adaptive modulation and MIMO coding for broadband wireless data networks,” IEEE
Commun. Mag., vol. 40, pp. 108–115, June 2002.
[11] R. Cheng and S. Verdu, “Gaussian multiaccess channels with ISI: Capacity region and multiuser water-filling,” IEEE Trans. Inform. Theory,
vol. 39, pp. 773–785, May 1993.
700
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
[12] D. Chizhik, G. Foschini, M. Gans, and R. Valenzuela, “Keyholes, correlations and capacities of multielement transmit and receive antennas,”
IEEE Trans. Wireless Commun., vol. 1, pp. 361–368, Apr. 2002.
[13] D. Chizhik, J. Ling, P. Wolniansky, R. Valenzuela, N. Costa, and K.
Huber, “Multiple input multiple output measurements and modeling
in Manhattan,” in Proc. IEEE Vehicular Technology Conf., 2002, pp.
107–110.
[14] Chong and L. Milstein, “The performance of a space-time spreading
CDMA system with channel estimation errors,” in Proc. Int. Communications Conf., Apr. 2002, pp. 1793–1797.
[15] C. Chuah, D. Tse, J. Kahn, and R. Valenzuela, “Capacity scaling in
MIMO wireless systems under correlated fading,” IEEE Trans. Inform.
Theory, vol. 48, pp. 637–650, Mar. 2002.
[16] C.-N. Chuah, D. N. Tse, J. Kahn, and R. A. Valenzuela, “Capacity
scaling in MIMO wireless systems under correlated fading,” IEEE
Trans. Inform. Theory, vol. 48, pp. 637–650, Mar. 2002.
[17] M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol.
29, pp. 439–441, May 1983.
[18] T. M. Cover and J. A. Thomas, Elements of Information Theory. New
York: Wiley, 1991.
[19] U. Erez, S. Shamai, and R. Zamir, “Capacity and lattice strategies for
cancelling known interference,” in Proc. Int. Symp. Information Theory
Applications, Nov. 2000, pp. 681–684.
[20] G. J. Foschini, “Layered space-time architecture for wireless communication in fading environments when using multi-element antennas,” Bell
Labs Tech. J., pp. 41–59, 1996.
[21] G. J. Foschini, D. Chizhik, M. Gans, C. Papadias, and R. A. Valenzuela, “Analysis and performance of some basic spacetime architectures,” IEEE J. Select. Areas Commun., Special Issue on MIMO Systems,
pt. I, vol. 21, pp. 303–320, Apr. 2003.
[22] G. J. Foschini and M. J. Gans, “On limits of wireless communications in
a fading environment when using multiple antennas,” Wireless Personal
Commun.: Kluwer Academic Press, no. 6, pp. 311–335, 1998.
[23] R. G. Gallager, “An inequality on the capacity region of multiaccess
fading channels,” in Communication and Cryptography—Two Sides of
One Tapestry. Boston, MA: Kluwer, 1994, pp. 129–139.
[24] M. J. Gans, N. Amitay, Y. S. Yeh, H. Xu, T. Damen, R. A. Valenzuela,
T. Sizer, R. Storz, D. Taylor, W. M. MacDonald, C. Tran, and A.
Adamiecki, “Outdoor BLAST measurement system at 2.44 GHz:
Calibration and initial results,” IEEE J. Select. Areas Commun., vol.
20, pp. 570–581, Apr. 2002.
[25] D. Gesbert, M. Shafi, D. S. Shiu, P. Smith, and A. Naguib, “From theory
to practice: An overview of MIMO space-time coded wireless systems,”
IEEE J. Select. Areas Commun. Special Issue on MIMO Systems, pt. I,
vol. 21, pp. 281–302, Apr. 2003.
[26] L. Greenstein, J. Andersen, H. Bertoni, S. Kozono, D. Michelson,
and W. Tranter, “Channel and propagation models for wireless system
design I and II,” IEEE J. Select. Areas Commun., vol. 20, Apr./Aug.
2002.
[27] T. Guess and M. K. Varanasi, “Multiuser decision-feedback receivers for
the general Gaussian multiple-access channel,” in Proc. Allerton Conf.
Communications, Control, Computing, Monticello, IL, Oct. 1996, pp.
190–199.
[28] S. Hanly and D. Tse, “Multiaccess fading channels-Part II: Delay-limited capacities,” IEEE Trans. Inform. Theory, vol. 44, pp. 2816–2831,
Nov. 1998.
[29] B. Hassibi and B. Hochwald, “How much training is needed in multiple-antenna wireless links?,” IEEE Trans. Inform. Theory, vol. 49, pp.
951–963, Apr. 2003.
, “Cayley differential unitary space-time codes,” IEEE Trans. In[30]
form. Theory, vol. 48, pp. 1485–1503, June 2002.
[31] B. Hochwald, T. L. Marzetta, and V. Tarokh, “Multi-antenna channelhardening and its implications for rate feedback and scheduling,” IEEE
Trans. Inform. Theory, 2002, submitted for publication.
[32] B. Hochwald and S. Vishwanath, “Space-time multiple access: Linear
growth in sum rate,” in Proc. 40th Allerton Conf. Communications, Control, Computing, Monticello, IL, Oct. 2002.
[33] M. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens, and R. Urbanke, “Systematic design of unitary space-time constellations,” IEEE
Trans. Inform. Theory, vol. 46, pp. 1962–1973, Sept. 2000.
[34] H. Huang, H. Viswanathan, and G. J. Foschini, “Multiple antennas in cellular CDMA systems: Transmission, detection and spectral efficiency,”
IEEE Trans. Wireless Commun., vol. 1, pp. 383–392, July 2002.
[35] H. C. Huang, S. Venkatesan, and H. Viswanathan, “Downlink capacity
evaluation of cellular networks with known interference cancellation,”
in Proc. DIMACS Workshop on Signal Processing Wireless Communications, DIMACS Center, Rutgers Univ., Oct. 7–9, 2002.
[36] S. Jafar, G. Foschini, and A. Goldsmith, “Phantomnet: Exploring optimal multicellular multiple antenna systems,” in Proc. Vehicular Technology Conf., 2002, pp. 261–265.
[37] S. Jafar and A. Goldsmith. Multiple-Antenna Capacity in Correlated
Rayleigh Fading With no Side Information. [Online]. Available:
http://wsl.stanford.edu/publications.html.
, “Transmitter optimization and optimality of beamforming for
[38]
multiple antenna systems with imperfect feedback,” IEEE Trans.
Wireless Commun., submitted for publication.
[39] S. A. Jafar and A. Goldsmith, “Transmitter optimization for multiple
antenna cellular systems,” in Proc. Int. Symp. Information Theory, June
2002, p. 50.
[40] S. A. Jafar and A. J. Goldsmith, “On optimality of beamforming for
multiple antenna systems with imperfect feedback,” in Proc. Int. Symp.
Information Theory, June 2001, p. 321.
, “Vector mac capacity region with covariance feedback,” in Proc.
[41]
Int. Symp. Information Theory, June 2001, p. 321.
[42] S. A. Jafar, S. Vishwanath, and A. J. Goldsmith, “Channel capacity and
beamforming for multiple transmit and receive antennas with covariance feedback,” in Proc. Int. Conf. Communications, vol. 7, 2001, pp.
2266–2270.
[43] N. Jindal, S. Jafar, S. Vishwanath, and A. Goldsmith, “Sum power iterative water-filling for multi-antenna Gaussian broadcast channels,” in
Proc. Asilomar Conf. Signals, Systems, Computers, Pacific Grove, CA,
Nov. 3–6, 2002.
[44] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of Gaussian
multiple-access and broadcast channels,” in Proc. Int. Symp. Inform.
Theory, June 2002, p. 500.
[45] E. Jorswieck and H. Boche, “Channel capacity and capacity-range of
beamforming in MIMO wireless systems under correlated fading with
covariance feedback,” IEEE J. Select. Areas Commun., submitted for
publication.
, “Optimal transmission with imperfect channel state information
[46]
at the transmit antenna array,” Wireless Personal Commun., submitted
for publication.
[47] A. Lapidoth and S. M. Moser, “Capacity bounds via duality with applications to multi-antenna systems on flat fading channels,” IEEE Trans.
Inform. Theory, submitted for publication.
[48] L. Li and A. Goldsmith, “Capacity and optimal resource allocation for
fading broadcast channels-Part I: Ergodic capacity,” IEEE Trans. Inform. Theory, vol. 47, pp. 1083–1102, Mar. 2001.
, “Capacity and optimal resource allocation for fading broadcast
[49]
channels-Part II: Outage capacity,” IEEE Trans. Inform. Theory, vol. 47,
pp. 1103–1127, Mar. 2001.
[50] Y. Li, J. Winters, and N. Sollenberger, “MIMO-OFDM for wireless communication: Signal detection with enhanced channel estimation,” IEEE
Trans. Commun., pp. 1471–1477, Sept. 2002.
[51] A. Lozano and C. Papadias, “Layered space-time receivers for frequency-selective wireless channels,” IEEE Trans. Commun., vol. 50,
pp. 65–73, Jan. 2002.
[52] B. Lu, X. Wang, and Y. Li, “Iterative receivers for space-time blockcoded OFDM systems in dispersive fading channels,” IEEE Trans. Wireless Commun., vol. 1, pp. 213–225, Apr. 2002.
[53] T. Marzetta and B. Hochwald, “Capacity of a mobile multiple-antenna
communication link in Rayleigh flat fading,” IEEE Trans. Inform.
Theory, vol. 45, pp. 139–157, Jan. 1999.
, “Unitary space-time modulation for multiple-antenna communi[54]
cations in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 46,
pp. 543–564, Mar. 2000.
[55] A. F. Molisch, M. Stienbauer, M. Toeltsch, E. Bonek, and R. S. Thoma,
“Capacity of MIMO systems based on measured wireless channels,”
IEEE J. Select. Areas Commun., vol. 20, pp. 561–569, Apr. 2002.
[56] A. Moustakas and S. Simon. Optimizing Multi-Transmitter Single-Receiver (MISO) Antenna Systems With Partial Channel Knowledge. [Online]. Available: http://mars.bell-labs.com.
[57] A. Narula, M. Trott, and G. Wornel, “Performance limits of coded diversity methods for transmitter antenna arrays,” IEEE Trans. Inform.
Theory, vol. 45, pp. 2418–2433, Nov. 1999.
[58] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use
of side information in multiple antenna data transmission over fading
channels,” IEEE J. Select. Areas Commun., vol. 16, pp. 1423–1436, Oct.
1998.
[59] G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, pp. 357–366, Mar. 1998.
[60] F. Rashid-Farrokhi, K. R. Liu, and L. Tassiulas, “Transit beamforming
and power control for cellular wireless systems,” IEEE J. Select. Areas
Commun., vol. 16, pp. 1437–1450, Oct. 1998.
GOLDSMITH et al.: CAPACITY LIMITS OF MIMO CHANNELS
[61] H. Sato, “An outer bound on the capacity region of the broadcast
channel,” IEEE Trans. Inform. Theory, vol. 24, pp. 374–377, May 1978.
[62] S. Shamai and T. L. Marzetta, “Multiuser capacity in block fading with
no channel state information,” IEEE Trans. Inform. Theory, vol. 48, pp.
938–942, Apr. 2002.
[63] S. Shamai and A. D. Wyner, “Information-theoretic considerations for
symmetric, cellular, multiple-access fading channels,” IEEE Trans. Inform. Theory, vol. 43, pp. 1877–1991, Nov. 1997.
[64] S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity
via co-processing at the transmitting end,” in Proc. IEEE Vehicular Technology Conf., May 2001, pp. 1745–1749.
[65] D. Shiu, G. Foschini, M. Gans, and J. Kahn, “Fading correlation and its
effect on the capacity of multi-element antenna systems,” IEEE Trans.
Commun., vol. 48, pp. 502–513, Mar. 2000.
[66] S. Simon and A. Moustakas, “Optimizing MIMO antenna systems with
channel covariance feedback,” IEEE J. Select. Areas Commun., vol. 21,
pp. 406–417, Apr. 2003.
, “Optimality of beamforming in multiple transmitter multiple
[67]
receiver communication systems with partial channel knowledge,” in
Proc. DIMACS Workshop Signal Proessing Wireless Communications,
DIMACS Center, Rutgers Univ., Oct. 7–9, 2002.
[68] P. J. Smith and M. Shafi, “On a Gaussian approximation to the capacity
of wireless MIMO systems,” in Proc. Int. Conf. Communications, Apr.
2002, pp. 406–410.
[69] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans.
Telecomm. ETT, vol. 10, no. 6, pp. 585–596, Nov. 1999.
[70] D. Tse and S. Hanly, “Multiaccess fading channels-Part I: Polymatroid
structure, optimal resource allocation and throughput capacities,” IEEE
Trans. Inform. Theory, vol. 44, pp. 2796–2815, Nov. 1998.
[71] A. Tulino, A. Lozano, and S. Verdu, “Capacity of multi-antenna channels in the low power regime,” in Proc. IEEE Information Theory Workshop, Oct. 2002, pp. 192–195.
[72] S. Verdu, “Spectral efficiency in the wideband regime,” IEEE Trans.
Inform. Theory, vol. 48, pp. 1319–1343, June 2002.
[73] S. Vishwanath, S. Jafar, and A. Goldsmith, “Optimum power and rate
allocation strategies for multiple access fading channels,” in Proc. Vehicular Technology Conf., May 2000, pp. 2888–2892.
[74] S. Vishwanath, N. Jindal, and A. Goldsmith, “On the capacity of multiple
input multiple output broadcast channels,” in Proc. Int. Conf. Communications, Apr. 2002, pp. 1444–1450.
[75] S. Vishwanath, G. Kramer, S. Shamai(Shitz), S. A. Jafar, and A. Goldsmith, “Outer bounds for multi-antenna broadcast channels,” in Proc.
DIMACS Workshop on Signal Processing Wireless Communications, DIMACS Center, Rutgers Univ., Oct. 7–9, 2002.
[76] E. Visotsky and U. Madhow, “Space-time transmit precoding with imperfect feedback,” IEEE Trans. Inform. Theory, vol. 47, pp. 2632–2639,
Sept. 2001.
[77] P. Viswanath and D. Tse, “On the capacity of the multi-antenna broadcast channel,” in Proc. DIMACS workshop on Signal Processing Wireless Communications, DIMACS Center, Rutgers Univ., Oct. 7–9, 2002.
[78] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using
dumb antennas,” IEEE Trans. Inform. Theory, vol. 48, pp. 1277–1294,
June 2002.
[79] P. Viswanath and D. N. Tse, “Sum capacity of the multiple antenna
Gaussian broadcast channel,” in Proc. Int. Symp. Information Theory,
June 2002, p. 497.
[80] P. Viswanath, D. N. Tse, and V. Anantharam, “Asymptotically optimal
water-filling in vector multiple-access channels,” IEEE Trans. Inform.
Theory, vol. 47, pp. 241–267, Jan. 2001.
[81] J. Winters, “On the capacity of radio communication systems with diversity in a Rayleigh fading environment,” IEEE J. Select. Areas Commun.,
vol. 5, pp. 871–878, June 1987.
[82] Y. Xin and G. Giannakis, “High-rate space-time layered OFDM,” IEEE
Commun. Lett., pp. 187–189, May 2002.
[83] W. Yu and J. Cioffi, “Trellis precoding for the broadcast channel,” in
Proc.Global Communications Conf., Oct. 2001, pp. 1344–1348.
[84] W. Yu and J. M. Cioffi, “Sum capacity of a Gaussian vector broadcast
channel,” in Proc. Int. Symp. Information Theory, June 2002, p. 498.
[85] W. Yu, G. Ginis, and J. Cioffi, “An adaptive multiuser power control algorithm for VDSL,” in Proc. Global Communications Conf., Oct. 2001,
pp. 394–398.
[86] W. Yu, W. Rhee, S. Boyd, and J. Cioffi, “Iterative water-filling for vector
multiple access channels,” in Proc. IEEE Int. Symp. Information Theory,
2001, p. 322.
701
[87] W. Yu, W. Rhee, and J. Cioffi, “Optimal power control in multiple access
fading channels with multiple antennas,” in Proc. Int. Conf. Communications, 2001, pp. 575–579.
[88] L. Zheng and D. Tse, “Optimal diversity-multiplexing tradeoff in multiple antenna channels,” in Proc. Allerton Conf. Communications, Control, Computing, Monticello, IL, Oct. 2001, pp. 835–844.
[89] L. Zheng and D. N. Tse, “Packing spheres in the Grassmann manifold:
A geometric approach to the noncoherent multi-antenna channel,” IEEE
Trans. Inform. Theory, vol. 48, pp. 359–383, Feb. 2002.
Andrea Goldsmith (S’90–M’93–SM’99) received
the B.S., M.S., and Ph.D. degrees in electrical
engineering from University of California, Berkeley,
in 1986, 1991, and 1994, respectively.
She was an Assistant Professor in the Department
of Electrical Engineering, California Institute of
Technology (Caltech), Pasadena, from 1994 to
1999. In 1999, she joined the Electrical Engineering
Department, Stanford University, Stanford, CA,
where she is currently an Associate Professor.
Her industry experience includes affiliation with
Maxim Technologies, Santa Clara, CA, from 1986 to 1990, where she worked
on packet radio and satellite communication systems and with AT&T Bell
Laboratories, Holmdel, NJ, from 1991 to 1992, where she worked on microcell
modeling and channel estimation. Her research includes work in capacity of
wireless channels and networks, wireless information and communication
theory, multiantenna systems, joint source and channel coding, cross-layer
wireless network design, communications for distributed control and adaptive
resource allocation for cellular systems and ad-hoc wireless networks.
Dr. Goldsmith is a Terman Faculty Fellow at Stanford University and a recipient of the Alfred P. Sloan Fellowship, the National Academy of Engineering
Gilbreth Lectureship, a National Science Foundation CAREER Development
Award, the Office of Naval Research Young Investigator Award, a National
Semiconductor Faculty Development Award, an Okawa Foundation Award,
and the David Griep Memorial Prize from University of California, Berkeley.
She was an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS from
1995 to 2002 and has been an Editor for the IEEE WIRELESS COMMUNICATIONS
MAGAZINE since 1995. She is also an Elected Member of Stanford’s Faculty
Senate and the Board of Governors for the IEEE Information Theory Society.
Syed Ali Jafar (S’99) received the B.Tech. degree
in electrical engineering from the Indian Institute of
Technology (IIT), Delhi, in 1997 and the M.S. degree in electrical engineering from California Institute of Technology (Caltech), Pasadena, in 1999. He
is a Graduate Research Assistant in the Wireless Systems Lab, Stanford University, Stanford, CA, and is
currently working toward the Ph.D. degree in electrical engineering.
He was a Summer Intern in the Wireless Communications Group of Lucent Bell Laboratories,
Holmdel, NJ, in 2001 and has two pending patents resulting from that work.
He was also an Engineer in the satellite networks division of Hughes Software
Systems, India, from 1997 to 1998. His research interests include spread-spectrum systems, multiple antenna systems, and multiuser information theory.
Nihar Jindal (S’99) received the B.S. degree in
electrical engineering and computer science from
University of California, Berkeley, in 1999 and
the M.S. degree in electrical engineering from
Stanford University, Stanford, CA, in 2001, and is
currently working toward the Ph.D. degree at the
same university.
His industry experience includes summer internships at Intel Corporation, Santa Clara, CA, in 2000
and at Lucent Bell Labs, Holmdel, NJ, in 2002.
His research interests include multiple-antenna
channels and multiuser information theory and their applications to wireless
communication.
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003
Sriram Vishwanath (S’99) received the B.Tech. degree in electrical engineering from the Indian Institute of Technology (IIT), Madras, in 1998 and the
M.S. degree in electrical engineering from the California Institute of Technology (Caltech), Pasadena,
in 1999. He is a graduate fellow currently working
toward the Ph.D. degree in electrical engineering at
Stanford University, Stanford, CA.
His research interests include information and
coding theory, with a focus on multiple antenna
systems. His industry experience includes work at
National Semiconductor Corporation, Santa Clara, CA, in the Summer of 2000
and at the Lucent Bell Labs, Murray Hill, NJ, during the Summer of 2002.