Spatial multiplexing with transmit antenna and constellation selection for correlated MIMO fading channels

Abstract

We consider spatial multiplexing systems in correlated multiple-input multiple-output (MIMO) fading channels with equal power allocated to each transmit antenna. Under this constraint, the number and subset of transmit antennas together with the transmit symbol constellations are determined assuming knowledge of the channel correlation matrices. We first consider a fixed data rate system and vary the number of transmit antennas and constellation such that the minimum margin in the signal-to-noise ratio (SNR) is maximized for linear and Vertical Bell Laboratories Layered Space-Time (V-BLAST) receivers. We also derive transmit antenna and constellation selection criteria for a successive interference cancellation receiver (SCR) with a fixed detection order and a variable number of bits transmitted on each substream. Compared with a system using all available antennas, performance results show significant gains using a subset of transmit antennas, even for independent fading channels. Finally, we select a subset of transmit antennas to maximize data rate given a minimum SNR margin. A lower bound on the maximum outage data rate is derived. The maximum outage data rate of the SCR receiver is seen to be close to the outage channel capacity.

Full text

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 20032829
Spatial Multiplexing With Transmit Antenna
and Constellation Selection for Correlated
MIMO Fading Channels
Ravi Narasimhan, Member, IEEE
Abstract—We consider spatial multiplexing systems in corre-
lated multiple-input multiple-output (MIMO) fading channels
with equal power allocated to each transmit antenna. Under this
constraint, the number and subset of transmit antennas together
with the transmit symbol constellations are determined assuming
knowledge of the channel correlation matrices. We first consider a
fixed data rate system and vary the number of transmit antennas
and constellation such that the minimum margin in the signal-
to-noise ratio (SNR) is maximized for linear and Vertical Bell
Laboratories Layered Space-Time (V-BLAST) receivers. We also
derive transmit antenna and constellation selection criteria for a
successive interference cancellation receiver (SCR) with a fixed
detection order and a variable number of bits transmitted on each
substream. Compared with a system using all available antennas,
performance results show significant gains using a subset of
transmit antennas, even for independent fading channels. Finally,
we select a subset of transmit antennas to maximize data rate
given a minimum SNR margin. A lower bound on the maximum
outage data rate is derived. The maximum outage data rate of the
SCR receiver is seen to be close to the outage channel capacity.
Index Terms—Adaptive modulation, multiple antennas, space-
time processing.
I. INTRODUCTION
M
capacities in wireless communication links [1], [2]. Spatial
multiplexing [3], which is also known as the Vertical Bell
Laboratories Layered Space-Time (V-BLAST) system [4], is
a method to achieve a significant fraction of the theoretical
capacity with reasonable implementation complexity. Spatial
multiplexing achieves the best performance in rich-scattering
channels in which the antenna elements are subject to uncor-
related fading. In practice, however, MIMO channels often
exhibit nonnegligible antenna correlation due to closely spaced
antennas and a finite number of scattering clusters with limited
angular spreads [5]. Previous studies have shown that the per-
formance of spatial multiplexing is quite sensitive to antenna
correlation [6].
Precoding has been proposed as a method to improve the per-
formance of spatial multiplexing in correlated fading channels.
If the channel correlation matrix is known at the transmitter,
ULTIPLE-INPUT multiple-output (MIMO) systems
have emerged as a promising technique to achieve high
ManuscriptreceivedDecember8,2002;revisedApril29,2003.Theassociate
editor coordinatingthe review of this paper and approving it for publication was
Dr. Helmut Bölcskei.
The author is with the Signal Processing Department, Marvel1 Semicon-
ductor, Inc., Sunnyvale, CA 94089 USA (e-mail: nkravi@stanfordalumni.org).
Digital Object Identifier 10.1109/TSP.2003.818205
the optimal precoding consists of transmission on the eigenvec-
tors of the correlation matrix [7], [8]. Furthermore, the bit and
power allocation for each eigenvector is determined by “spa-
tial water-filling” according to the eigenvalues of the correla-
tion matrix. In practice, it is desirable to allocate equal power to
eachtransmit antenna for severalreasons, including limitingthe
overheadinthefeedbackchannel,increasingrobustnesstoinac-
curacies in channel estimates, and reducing the requirements on
the dynamic range of the power amplifiers. Furthermore, equal
bit allocation on each transmit antenna is desirable in many in-
stances for similar reasons. In this context, diagonal precoding
by varying the phases of the transmitted signals has been pro-
posed in [9] for the case of a 2
coding matrix is determined using a numerical search to mini-
mize the error probability for a maximum likelihood (ML) re-
ceiver.Unitary precoding is also proposed in [10] after applying
adiscreteFouriertransform(DFT)tothetransmitdatavector.A
numerical search over unitary precoding matrices is performed
tomaximizeaperformancefunctionrelatedtodiversitygainand
coding gain. Simulation results given in [10] indicate that this
unitary precoding technique performs better than no precoding
for correlated fading channels while providing no apparent gain
for independent fading channels.
Antenna selection has also been proposed for enhanced
performance in correlated fading [11], [12]. Assuming that the
number of RF chains is less than the number of antennas, the
antenna selection algorithms choose the optimum subset of
transmit and receive antennas based on minimum error rate.
The size of theactive subsets of transmit and receive antennasis
fixed by the number of RF chains. Transmit antenna selection is
also discussed in [10], where for a fixed symbol constellation,
simulations indicate a gain in error rate performance by not
transmitting on weak antennas. However, this performance
gain is achieved at the expense of a loss in data rate.
Per-antenna rate control with equal power allocation is ap-
plied to uncorrelated fading channels in [13], where it is shown
that per-antenna rate control at the fading rate nearly achieves
capacity. However, adaptation at the fading rate may be difficult
to achieve in practice due to inaccuracies in channel estimates
and feedback delays. Similarly, in [14], transmit power alloca-
tion for each antenna is determined using instantaneous channel
estimates and, hence, requires adaptation at the fading rate.
In this paper, we propose varying the number of transmit an-
tennasandsymbolconstellationstoimprovetheperformanceof
spatial multiplexing in correlated fading. This technique avoids
a numerical search associated with precoding while providing
2 MIMO system. The pre-
1053-587X/03$17.00 © 2003 IEEE

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2830 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003
performance gains, even for independent fading channels. The
keycontributionofthispaperisasimpleproceduretodetermine
the number of activetransmit antennas (i.e., multiplexing order)
and the corresponding transmit constellations using knowledge
of the channel correlation matrices. Knowledge of correlation
matrices hasbeenusedinseveralpreviousworks (e.g.,[7],[11],
and [12]) since the correlation matrices vary more slowly than
the fading rate and can be estimated more accurately than the
instantaneous channel realization. The algorithms described in
this paper can be used to adapt a generic MIMO design auto-
maticallytoaparticularpropagationenvironmentataratemuch
lower than the fading rate. Equal power allocation for each an-
tennaisassumed.Weconsiderlinear,V-BLAST,andsuccessive
interferencecancellationreceivers.ForthelinearandV-BLAST
receivers, we allow the single transmit constellation to vary, de-
pending on the number of active transmit antennas, to main-
tain a fixed data rate. The V-BLAST receiver algorithm selects
the substream detection order based on the channel realization.
The optimum detection order requires repeated computation of
the pseudo-inverse of the channel matrix. As an alternative ap-
proach, a successive cancellation receiver (SCR) can be used
with a fixed detection order and variable bit allocation across
substreams [15]. The bits are allocated to each substream ac-
cording to its post-processing signal-to-noise ratio (SNR). For
the SCR receiver with fixed detection order, we perform joint
optimization of the bit allocations and the number of active
transmitantennas.Thenumberofactivetransmitantennasis se-
lected to maximize the minimum margin in the post-processing
SNR. The error rate performance of the antenna and constella-
tion selection technique is evaluated for various channel corre-
lation matrices. We then fix the target SNR margin and declare
an outage if the minimum SNR margin is less than the target
value. A lower bound for the maximum outage data rate is de-
rived and evaluated for correlated fading channels.
The paper is organized as follows. In Section II, we briefly
discuss a model for spatial multiplexing in correlated fading
channels.Theselectionofthenumberoftransmitantennas,cor-
respondingantennaelements,andconstellationformarginmax-
imization is discussed in Section III for a fixed data rate. In Sec-
tion IV, we derive a lower bound on the maximum outage data
rate for a given target SNR margin. In Section V, we provide
error rate results to demonstrate the performance of the antenna
and constellation selection algorithm in correlated fading chan-
nels.Wealsocomparethemaximumoutageratewiththeoutage
capacity. Conclusions are given in Section VI.
II. SYSTEM MODEL
We consider a narrowband spatial multiplexing system with
transmit and receive antennas. After antenna selection,
transmit and
and denote the sets of indices of the active transmit and
receive antennas, respectively. For example, if all available an-
tennas are active,
Let
denote the transmitted vector, where
are symbols drawn from possibly different constellations, and
denotethereceivedvector.Wenormalizethe
constellation energy such that
receive antennas are used;
let
and.
, whereis
the total transmit energy. A model for the
matrix
channel
is given by [6], [11], [16]
(1)
where
trices among the active transmit and receive antennas, respec-
tively, and
is a
ments are independent and identically distributed (i.i.d.) com-
plex Gaussian random variables with zero mean and unit vari-
ance.
We assume uniform linear arrays with the distance (in car-
rier wavelengths) between adjacent elements equal to
at the transmitter and receiver, respectively. We also as-
sume that there are
scattering clusters with the angle of de-
parture from the transmitter and angle of arrival at the receiver
of the th cluster denoted by
gles
andare assumed to be Gaussian distributed with
and
angle spreads (small
), the correlation matrices associated
with the th scattering cluster are given by [17]
anddenote the correlation ma-
matrix whose ele-
and
and, respectively. The an-
[6]. For small
(2)
(3)
As noted in [6], (2) and (3) also provide the correct trend of
i.i.d. fading for large angle spreads. Under the narrowband as-
sumption,theoverallcorrelationmatricesaretheweightedaver-
ages of the correlation matrices associated with each scattering
cluster, with the weights equal to the fraction of total power in
each cluster. With the superscript
pose, we also define the eigenvalue decompositions of the cor-
relation matrices as follows:
denoting conjugate trans-
(4)
(5)
where
andare unitary matrices of eigenvectors, and
are diagonal matrices of eigenvalues. In (4) and
(5), the dependence on the active antenna elements has been
suppressedfor notationalconvenience.Now, thereceivedsignal
can be written as
and
(6)
where
vectorwithcovariance
identitymatrix.Inthefollowing,wesuppressthedependenceof
the channel matrix on the active antenna elements.
We assume perfect channel knowledge at the receiver and
perfect knowledge of the channel correlation matrices either
at the transmitter or at the receiver. In the latter case, the re-
ceiverwouldfeedbackthechoiceoftheactivetransmitantennas
and the constellation to the transmitter. We consider detection
is a complex additive white Gaussian noise (AWGN)
,anddenotesthe

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NARASIMHAN: SPATIAL MULTIPLEXING WITH TRANSMIT ANTENNA AND CONSTELLATION SELECTION2831
using linear, V-BLAST, and SCR processing. For the linear re-
ceiver, the transmitted vector is estimated by forming the signal
, which is computed according to
trix is given by
tection and
imum mean squared error (MMSE) detection. For V-BLAST
detection, the transmitted vector is estimated using ordered suc-
cessive interference cancellation with nulling based on the ZF
criterion. Details of the V-BLAST algorithm can be found in
[4]. For SCR detection, successive interference cancellation is
performed using a fixed detection order.
. The receive ma-
for zero forcing (ZF) de-
for min-
III. ANTENNA AND CONSTELLATION SELECTION
FOR FIXED DATA RATE
We now describe a method to select the antennas and symbol
constellations for a fixed data rate. The criterion used for se-
lection is maximization of the minimum SNR margin. Let
denote the desired spectral efficiency [in bits per second per
Hertz (bps/Hz)] of the spatial multiplexing system and SNR
denote the minimum post-processing SNR among all
substreams. We assume either uncoded transmission or that the
coding scheme applied to the transmit bit stream is fixed such
that for a target symbol error probability, the SNR gap from ca-
pacity is
[18]. The gap is nearly constant for most spec-
tral efficiencies encountered in practice. In Section III-A, we
describe the selection criteria for linear and V-BLAST detec-
tion. The selection criteria for SCR detection is derived in Sec-
tion III-B. Since the best performance is achieved when all re-
ceive antennas are active, we set
data
.
A. Linear and V-BLAST Detection
In this subsection,
of the
active transmit antennas. Under this condition, the
minimum SNR margin is given by
bps/Hz are allocated on each
SNR
(7)
A lower bound for
analysis of outage rate in Section IV. In this section, the average
value of the lower bound is used to select the active transmit
antennas and constellations for a fixed data rate. We now obtain
a lower bound for
using the following lower bound for
SNR
.
is obtained in the following for the
Lemma: For a receiver using ZF, MMSE, or V-BLAST de-
tection
SNR
(8)
where
trix
is the minimum eigenvalue of a square ma-
.
Proof: The proof is given in Appendix A.
A lower bound for
Theorem 1: For a receiver using ZF, MMSE, or V-BLAST
detection, we have (9), shown at the bottom of the page.
Proof: The proof is given in Appendix B.
Sinceis random and unknown to the transmitter, we per-
form the maximization using the mean of the minimum eigen-
value
can now be stated as follows.
(10)
The number
tennas are selected such that we have (11), shown at the bottom
of the page. The constellation for transmission is then selected
such that
bps/Hz are sent through each of the
tive transmit antennas.
and element indicesof active transmit an-
ac-
B. SCR Detection
In this subsection, we optimize the substream bit allocations
and number of transmit antennas for SCR detection with fixed
detection order. Let
,
ciencies allocated to each of the
The optimization problem can be stated as follows:
denote the spectral effi-
active transmit antennas.
(12)
where
SNR
(13)
is the SNR margin for the th substream. The optimal bit allo-
cation to satisfy (12) is such that the SNR margins for all sub-
streams are equal, i.e.,
allocation, we obtain the following lower bound.
,. With this bit
(9)
(11)

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2832IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003
Theorem2: FortheSCRsystem,theSNRmargin
(14), shown at the bottom of the page.
Proof: The proof is given in Appendix C.
Comparing (9) and (14), we note that the lower bound for the
minimum SNR margin for the SCR system depends on the geo-
metric mean of the eigenvalues of
the minimum eigenvalues. This result can be interpreted as fol-
lows. In contrast to the linear and V-BLAST systems, the bits
are allocated across transmit antennas for SCR system such that
allsubstreamshavethesameerrorprobability.Underthiscondi-
tionandathighSNR,thebasetwologarithmoftheSNRmargin
can be regarded as the difference between the mean substream
capacity and the mean substream operational spectral efficiency
(assuming
). At high SNR, the mean substream capacity
is approximately the mean of the base two logarithms of the
post-processing substream SNRs or, equivalently, the base two
logarithm of the geometric mean of the post-processing sub-
stream SNRs. The result (14) follows from the fact that the
geometric mean of the post-processing substream SNRs for the
SCR system is proportional to the geometric mean of the eigen-
values of
and .
In order to maximize the lower bound in (14), the selection of
(
, ) is determined by (15), shown at the bottom of the page.
Since
is random and
satisfies
andinstead of
, we use the expected value
(16)
where
(16) can be derived from the results given in [19]. Thus, the pair
(
, ) is chosen such that
is Euler’s constant. The expected value
(17)
Once the pair (
transmit antenna is determined as follows. For a given channel
realization
, the initial bit allocation is obtained assuming
equality in (51) and (58). Thus
, ) is chosen, the bit allocation for each
SNR
(18)
(19)
for
realizations of
the final bit allocation
bit allocations before rounding that are close to an odd multiple
of 0.5 are rounded in the opposite direction until the desired
spectral efficiency
is obtained [18]. The constellation for the
th active transmit antenna is selected such that
transmitted.
. The results are then averaged over several
and rounded to the nearest integer to obtain
,. If, the
bps/Hz are
IV. ANTENNA AND CONSTELLATION SELECTION FOR
MAXIMUM OUTAGE DATA RATE
In this section, we derive a lower bound on the maximum
outage data rate. Recall that outage is declared when
, whereis a given target SNR margin. The
target probability of outage is denoted by
Pr. The cases of linear, V-BLAST, and SCR
detection are discussed separately in the following subsections.
, i.e.,
A. Linear and V-BLAST Detection
Using the lower bound given in (9), we have (20)–(22),
shownatthebottomofthenextpage,where
Pr
lative distribution function (cdf) of
. From (22), we derive a lower bound on the
outage data rate [in bps/Hz], as in (23), shown at the bottom of
the next page, where
.Themaximumoutagedatarate
is obtained by selecting
and
side of (23) for a given
,
is the cumu-
is the inverse cdf of
to maximize the right-hand
and.
(14)
(15)

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NARASIMHAN: SPATIAL MULTIPLEXING WITH TRANSMIT ANTENNA AND CONSTELLATION SELECTION2833
TABLE I
CORRELATED FADING SCENARIOS
B. SCR Detection
For unordered SCR detection,
we have (24)–(26), shown at the bottom of the page, where
Pr
. Therefore, we obtain the following lower bound on the
outage data rate (in bps/Hz):
. Using (14),
(27)
where
isthe
. As in Section IV-A, the
inverse cdfof
maximum outage data rate is obtained by selecting
to maximize the right-hand side of (27).
and
V. SIMULATION RESULTS
We now evaluate the performance of the antenna and constel-
lation selection algorithms in correlated fading channels. In all
the simulations, we use
. Four scenarios of correlated fading are studied, as given in
Table I. We present the results for a fixed data rate followed by
the results for the maximum outage data rate.
and
(20)
(21)
(22)
(23)
(24)
(25)
(26)