MITSUBISHI ELECTRIC RESEARCH LABORATORIES
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Capacity of MIMO Systems with Antenna
Selection
Andreas F. Molisch, Moe Z. Win, Yang-seok Choi, Jack H. Winters
TR2005-144 July 2005
Abstract
We consider the capacity of multiple-input multiple-output systems with reduced complexity.
One link-end uses all available antennas, while the other chooses the L out of N antennas that
maximize capacity. We derive an upper bound on the capacity that can be expressed sa sthe sum
of the logarithms of ordered chi-square-distributed variables. This bound is then evaluated ana-
lytically and compared to the results obtained by Monte Carlo simulations. Our results show that
the achieved capacity is close to the capacity of a full-complexity system provided that L is at
least as large as the number of antennas at the other link-end. For example, for L=3, N=8 anten-
nas at the receiver and three antennas at the transmitter, the capacity of the reduced-complexity
scheme is 20 bits/s/Hz compared to 23 bits/s/Hz of a full-complexity scheme. We also present
a suboptimum antenna subset selection algorithm that has a complexity of N2 compared to eht
optimum algorithm with a complexity of (N L).
IEEE Transactions on Wireless Communications
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 1759
Capacity of MIMO Systems With Antenna Selection
Andreas F. Molisch, Fellow, IEEE, Moe Z. Win, Fellow, IEEE,
Yang-Seok Choi, Member, IEEE, and Jack H. Winters, Fellow, IEEE
Abstract—We consider the capacity of multiple-input multiple-
output systems with reduced complexity. One link-end uses all
available antennas, while the other chooses the L out of N an-
tennas that maximize capacity. We derive an upper bound on the
capacity that can be expressed as the sum of the logarithms of
ordered chi-square-distributed variables. This bound is then eval-
uated analytically and compared to the results obtained by Monte
Carlo simulations. Our results show that the achieved capacity is
close to the capacity of a full-complexity system provided that L
is at least as large as the number of antennas at the other link-
end. For example, for L =3, N =8 antennas at the receiver
and three antennas at the transmitter, the capacity of the reduced-
complexity scheme is 20 bits/s/Hz compared to 23 bits/s/Hz of a
full-complexity scheme. We also present a suboptimum antenna
subset selection algorithm that has a complexity of N
2
compared
to the optimum algorithm with a complexity of
N
L
.
Index Terms—Antenna arrays, information rates, MIMO
systems.
I. INTRODUCTION
M
ULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO)
wireless systems are those that have antenna arrays at
both transmitter and receiver. Early simulation studies that re-
vealed the potentially large capacities of those systems were
done in the 1980s [1], and subsequent papers explored the
capacity analytically [2], [3]. Since that time, interest in MIMO
systems has exploded. Layered space–time (ST) receiver struc-
tures [4]–[6] and ST codes [7] make it possible to approach the
capacity limits revealed in [2]. Commercial products based on
such codes are under development [8]. Most importantly, the
standard for third-generation cellular phones [3rd Generation
Manuscript received June 25, 2003; revised February 20, 2004; accepted
April 23, 2004. The editor coordinating the review of this paper and approv-
ing it for publication is G. Leus. This work was supported in part by an
INGVAR grant of the Swedish Strategic Research Fund, a cooperation grant
from the Swedish STINT, the Office of Naval Research Young Investigator
Award N00014-03-1-0489, the National Science Foundation under Grant ANI-
0335256, and the Charles Stark Draper Endowment. Parts of this work were
presented at ICC 2001 and VTC fall 2003.
A. F. Molisch was with AT&T Laboratories-Research, Middletown, NJ
07748 USA. He is now with Mitsubishi Electric Research Laboratory (MERL),
Cambridge, MA 02139 USA and also at the Department of Electroscience,
Lund University, Lund, Sweden (e-mail: Andreas.Molisch@ieee.org).
M. Z. Win was with AT&T Laboratories-Research, Middletown, NJ 07748
USA. He is now with the Laboratory for Information and Decision Systems,
Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:
win@ieee.org; moewin@mit.edu).
Y.-S. Choi was with AT&T Laboratories-Research, Middletown, NJ 07748
USA. He is now with Intel, Inc., Hillsboro, OR 97229 USA (e-mail: yschoi@
ieee.org).
J. H. Winters was with AT&T Labs-Research, Middletown, NJ 07748
USA. He is now with Motia Inc., Middletown, NJ 07748 USA (email: jack.
winters@ieee.org).
Digital Object Identifier 10.1109/TWC.2005.850307
Partnership Project (3GPP)] foresees the use of a simple ST
code [9] with two transmit antennas and one or more receive
antennas for circuit-switched communications and spatial
multiplexing (multiple transmit data streams) for high-speed
downlink packet data access [10].
In an earlier work, it was shown that the incremental gain of
additional receive antennas is negligible if the total number of
receive antennas N
r
is far larger than the number of transmit
antennas N
t
[4].
1
This can be explained by the fact that ad-
ditional antennas do not provide independent communication
channels but just increase the diversity order. This motivates
researchers to explore the possibility of replacing the maximal
ratio diversity that is normally achieved in a such a MIMO
system with selection diversity (SD). Thus, in this paper, we
propose a reduced-complexity MIMO scheme that selects the
L
r
“best” of the available N
r
antennas. Such a scheme can
provide the full number of independent communication chan-
nels, and additionally an SD gain. Compared to the use of all
antennas, the antenna selection has the advantage that only L
r
instead of N
r
receiver RF chains are required. We still require
the full number of antenna elements, but these are usually
inexpensive, as they are patch or dipole antennas that can be
easily produced and placed.
Antenna selection, or more precisely, the principle of using
L out of N antennas, was first studied in the context of antenna
selection at one link-end, while only a single antenna is present
at the other link-end [11]–[14]. This is referred to as “hybrid
selection/maximum ratio combining (MRC)” in the literature.
Therefore, we will employ the term “hybrid selection/MIMO”
(H-S/MIMO) for the more general case studied in this paper,
namely antenna selection at one link-end, and multiple anten-
nas, all of which are used, at the other link-end.
There has been considerable interest in H-S/MIMO in recent
years. The case of antenna selection at the transmitter is treated
in [15] using Monte Carlo simulations; this paper also develops
a criterion for optimal antenna set selection for high signal-to-
noise ratios (SNRs); [16] extended this to the low-SNR case.
It has been shown that antenna selection is beneficial in a low-
rank environment [17] and in interference-limited systems [18].
A selection algorithm for minimizing the bit error probability
of linear MIMO receivers is given in [19]. The use of ST
codes in combination with antenna selection was investigated in
[20] and [21]; the use of antenna selection in transmit–receive
diversity systems with channel knowledge at both link-ends was
1
Under certain circumstances, increasing that number can even lead to
performance degradation, as the channel estimation becomes more difficult and
introduces estimation errors.
1536-1276/$20.00 © 2005 IEEE
1760 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
Fig. 1. Block diagram of the considered system.
treated in [22].
2
A more detailed overview of the literature is
given in [28].
In this paper, we derive analytical bounds for the capacity
distribution function of an H-S/MIMO system at one link-end.
We show that an exact antenna selection algorithm requires
high computational complexity and propose several alternative
methods that have much lower complexity while performing
almost as well as the exact selection criteria. The rest of the
paper is organized as follows: In Section II, we set up the
system model. Analytical bounds for the capacity are derived in
Section III. Next, we present a fast antenna selection algorithm
in Section IV. Section V gives evaluations for the analytical
bounds of H-S/MIMO and compares them to numerical sim-
ulation results. Conclusions and system design considerations
are given in Section VI.
II. S
YSTEM MODEL
We consider the case where the transmitter uses all available
antennas while the receiver uses antenna selection. Fig. 1
exhibits a block diagram. At the transmitter, the data stream
enters an ST encoder, whose outputs are forwarded to the N
t
transmit antennas. The signals are subsequently upconverted
to passband, amplified by a power amplifier, and filtered. For
our model, we omit these stages, as well as their equivalents
at the receiver, which allows us to treat the whole problem
in equivalent baseband. Note, however, that it is exactly these
parts that are most expensive and make the use of reduced-
complexity systems desirable.
From the antennas, the signal is sent through the mobile radio
channel, which is assumed to be flat fading and quasi-static. By
quasi-static, we mean that the coherence time of the channel
is so long that “a large number” of bits can be transmitted
within this time. More specifically, we assume that the data are
encoded with near Shannon limit achieving codes.
3
It has been
shown that LDPC codes with a block length of 10 000 approach
the Shannon limit within less than 1 dB [30]. For a data rate of
10 Mbits/s, such a block can be transmitted within 1 ms, which
is shorter than the typical 10 ms coherence time of wireless
channels. Thus, each channel realization can be associated with
2
Parallel to our work (see also [23] and [24]), an alternative algorithm for the
selection of antenna subsets was presented and a lower bound of the capacity
was derived in [25]–[27]; this algorithm will also be discussed in Section IV.
3
Such a code could be, e.g., the combination of ST processing [6] with a
low-density parity check code [29].
a (Shannon-AWGN) capacity value. The capacity thus becomes
a random variable (RV), rendering the concept of “capacity
cumulative distribution function” and “outage capacity” mean-
ingful performance measures [2].
We denote the N
r
× N
t
matrix of the channel as
H =
h
11
h
12
··· h
1N
t
h
21
h
22
··· h
2N
t
.
.
.
.
.
.
.
.
.
.
.
.
h
N
r
1
h
N
r
2
··· h
N
r
N
t
. (1)
If the channel is Rayleigh fading, the h
ij
are independent
identically distributed (i.i.d.) zero-mean circularly symmetric
complex Gaussian RVs with unit variance, i.e., the real and
imaginary parts have a variance of 1/2 each. Consequently, the
power carried by each transmission channel h
ij
is chi-square
distributed with 2 degrees of freedom. The channel also adds
white Gaussian noise, which is assumed to be independent
among the N
r
receiver antenna elements. Following [2], we
consider the case in which the h
ij
are independently fading, as
this simplifies the theoretical analysis. More involved channel
models are discussed, e.g., in [31]–[33].
The received signal, which is written as
y = Hs + n = x + n (2)
is received by N
r
antenna elements, where s is the transmit
signal vector and n is the noise vector. A control algorithm (to
be discussed in Sections III and IV) selects the best L
r
of the
available N
r
antenna elements and downconverts their signals
for further processing (note that only L
r
receiver chains are
required). ST encoder and decoder are assumed to be ideal so
that the capacity can be achieved. We assume ideal knowledge
of the channel at the receiver so that it is always possible
to select the best antennas. However, we do not assume any
knowledge of the channel at the transmitter. This implies that
no waterfilling can be used and that the available transmitter
power is equally distributed among the transmit antennas.
III. T
HEORY
Let us first explore the scenarios that are suited for
H-S/MIMO. As shown in [2], the capacity is linearly propor-
tional to min(N
r
,N
t
). Any further increase of either N
r
or
N
t
while keeping the other fixed only increases the diversity
MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1761
order and possibly the mean SNR, possibly. Thus, if the num-
ber of antennas at one link-end is limited, e.g., due to space
restrictions, a further increase in the antenna number at the
other link-end does not allow us to add statistically independent
transmission channels (which would imply linear increase in
system capacity), but only provides additional diversity. Since
it is well known that SD has the same diversity order as that of
MRC [34], we can anticipate that a hybrid scheme with N
r
>
L
r
= N
t
will give a good performance. In the next subsections,
we will give a quantitative confirmations of this conjecture.
A. Exact Expression for the Capacity
The capacity of MIMO system using all antenna elements is
given by [2]
C
full
= log
2
det
I
N
r
+
Γ
N
t
HH
†
(3)
where I
N
r
is the N
r
× N
r
identity matrix, Γ is the mean SNR
per receiver branch, and superscript
†
denotes the Hermitian
transpose. The receiver now selects those antennas that allow
a maximization of the capacity, so that
C
select
= max
S(
H)
log
2
det
I
L
r
+
Γ
N
t
H
H
†
(4)
where
H is created by deleting N
r
− L
r
rows from H, and
S(
H) denotes the set of all possible
H, whose cardinality is
N
r
L
r
.
The optimum choice of antennas requires the knowledge of
the complete channel matrix. This may seem to necessitate
the use of N
r
RF chains, which is in contrast with a low-
complexity system. However, in a sufficiently slowly changing
environment, the L
r
RF chains can be cycled through the N
r
antennas during the training bits. In other words, RF chains
are connected to the first L
r
antennas during the first part of
the training sequence, then to the second L
r
antenna during
the next part, and so on. At the end of the training sequence,
we pick the best L
r
antennas. Thus, we only need a few more
training bits instead of additional RF chains and the decrease
in the spectral efficiency due to those additional training bits is
negligible, especially in high-data-rate systems.
B. Capacity Bound for L
r
≤ N
t
An exact analytical solution for C
select
seems difficult. Thus,
we derive analytical bounds in this subsection and verify them
with Monte Carlo simulations in Section V. Our starting point
is the upper capacity bound for the full-complexity system with
N
t
≤ N
r
[2]
C
full
≤
N
t
i=1
log
2
1+
Γ
N
t
γ
i
(5)
where the γ
i
are independent chi-square-distributed RVs with
2N
r
degrees of freedom. The equality applies in the “unrealistic
case when each of the N
t
transmitted components is received
by a separate set of N
r
antennas in a manner where each
signal component is received with no interference from the
others” [2].
In our case, we select the best L
r
out of N
r
receive antennas,
where L
r
≤ N
t
. The upper bound can be obtained similar to
(5), except for exchanging the role of transmitter and receiver,
and selecting those antennas whose instantaneous realizations
of γ
i
are the largest. Since this equation is a crucial starting
point, let us elaborate on its physical interpretation. We consider
a system where each of the N
r
receive antennas has its own
set (of size N
t
) of transmit antennas. Naturally, this case is not
feasible in practice but must result in an upper bound of the
capacity. Each set of transmit antennas corresponding to each
of the N
r
receive antennas can carry one data stream. The max-
imum SNR (which also achieves maximum capacity) for this
data stream can be obtained with maximal ratio transmission,
which in turn results in chi-square-distributed SNR with 2N
t
degrees of freedom at the receiver output. Finally, we select
those L
r
(out of N
r
) receive antennas that give the best SNR,
and thus highest capacity. The capacity bound with antenna
selection is thus
C
bound
=
L
r
i=1
log
2
1+ργ
(i)
(6)
where ρ =
Γ/N
t
, and the γ
(i)
are ordered chi-square-
distributed variables with 2N
t
degrees of freedom, out of a set
of N
r
.
4
The joint statistics of the ordered SNRs γ
(i)
is shown in
(7) at the bottom of the page [14], where Γ(·) is Euler’s Gamma
function [35].
Thus, the characteristic function of the capacity bound is
Φ(jν)=
N
r
!
Γ(N
t
)
N
r
∞
0
dγ
(1)
γ
(1)
0
dγ
(2)
···
γ
(N
r
−1)
0
dγ
(N
r
)
× exp
−jν
L
r
i=1
log
2
1+ργ
(i)
N
r
i=1
γ
N
t
−1
(i)
exp
−γ
(i)
. (8)
4
We use {γ
(i)
} to denote the order set of {γ
i
}, i.e., γ
(1)
>γ
(2)
>
···γ
[N
r
]
. Note that the possibility of at least two equal γ
(i)
’s is excluded as
γ
(i)
= γ
(j)
almost surely for continuous RVs γ
i
.
p
γ
(i)
γ
(1)
,γ
(2)
,...,γ
(N
r
)
=
N
r
!
N
r
i=1
1
Γ(N
t
)
γ
N
t
−1
(i)
exp
−γ
(i)
, for γ
(1)
>γ
(2)
> ···>γ
(N
r
)
0, otherwise
(7)
