Joint antenna selection and link adaptation for MIMO systems

Abstract

Multi-input multi-output (MIMO) systems, with multiple antennas at both the transmitter and the receiver, are anticipated to be widely employed in future wireless networks due to their predicted tremendous system capacity. To protect the transmitted data against random channel impairment, it is desirable to consider link adaptation, such as rate adaptation and power control, to improve the system performance and guarantee certain quality of service. Based on the observation that link adaptation and antenna selection problems are often coupled, we propose a joint antenna subset selection and link adaptation study for MIMO systems. After the formulation of the multidimensional joint optimization problem, the main contribution of this paper lies in the design of efficient algorithms approaching the optimal solution for both uncorrelated and correlated MIMO channels. Specifically, we propose one simplified antenna selection and link adaptation rule based on the expected optimal number of active antennas for uncorrelated MIMO with Rayleigh fading and one for correlated MIMO channels only based on the slowly varying channel correlation information. Our proposed algorithms are verified through numerical results, demonstrating significant gains over traditional MIMO signaling, while feasible for practical implementation.

Full text

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Abstract—Multi-input multi-output (MIMO) systems, with
multiple antennas at both the transmitter and receiver, are
anticipated to be widely employed in future wireless networks due
to their predicted tremendous system capacity. In order to protect
the transmitted data against random channel impairment, it is
desirable to consider link adaptation, such as rate adaptation and
power control to improve the system performance and guarantee
certain quality of service. Based on the observation that link
adaptation and antenna selection problems are often coupled, we
propose a joint antenna subset selection and link adaptation study
for MIMO systems. After
multi-dimensional joint optimization problem, the main
contribution of this paper lies in the design of efficient algorithms
approaching the optimal solution for both uncorrelated and
correlated MIMO channels. Specifically, we propose one
simplified antenna selection and link adaptation rule based on the
expected optimal number of active antennas for uncorrelated
MIMO with Rayleigh fading, and one for correlated MIMO
channels only based on the slowly varying channel correlation
information. Our proposed algorithms are verified through
numerical results, demonstrating
traditional MIMO signaling while feasible for practical
implementation.

Index Terms— Antenna
Cholesky-decomposition, MIMO systems

the formulation of the
significant gains over
selection, QR-decomposition,
I. INTRODUCTION
HE use of multiple antennas at both the transmitter and
receiver side, so as to form a multiple-input
multiple-output (MIMO) antenna system, is an emerging
technology that makes building high data rate wireless
networks a reality [12][13]. Transmitting independent data
streams simultaneously from different antennas through spatial
multiplexing (see, e.g., [1]) effectively realizes the high
spectral efficiency promised by MIMO systems, but leaves the
transmitted data unprotected from random channel impairment.
Therefore, it is often desirable to consider link adaptation, such
as rate adaptation and power control to improve the system
performance and guarantee certain quality of service
[5][8][21][24].

Manuscript received June 28, 2004; revised April 1,2005 and June 28, 2005.
The associate editor coordinating the review of this paper and approving it for
publication was Dr. Chengshan Xiao.
The authors are with the Department of Electrical and Computer
Engineering, NC State University, Raleigh, NC 27606 USA (phone:
919-513-0299; fax: 919-515-2285;
Huaiyu_Dai@ncsu.edu).
e-mail: qzhou@ncsu.edu;
One of the drawbacks with an MIMO system is the increased
complexity and hardware cost due to the expensive RF chains
required by each active antenna. It is of increasing research
interest recently to find a good antenna selection scheme that
can significantly reduce such cost while incurring little
performance loss. Generally, there are two goals for antenna
subset selection in MIMO systems: one aims to maximize the
channel capacity [19][25], the other aims to minimize the bit
error rate for spatial multiplexing systems when some practical
signaling schemes are used [6][7][14][16].
It is interesting to notice that link adaptation and antenna
selection problems are actually coupled for MIMO systems,
when practical signal processing techniques such as
zero-forcing successive interference cancellation (ZF-SIC) (as
used in V-BLAST) are employed at the receiver for data
decoupling and detection. This is because the decoupled
subchannel gains (post-detection signal-to-noise ratio (SNR))
are determined by the active antenna subset, while some weak
subchannels are naturally dropped during link adaptation
process. Motivated by this fact, we propose a joint antenna
subset selection and link adaptation study for MIMO systems.
In a real propagation environment, the capacity of a MIMO
system may be lower than what is predicted with rich scattering
assumption due to fading correlation [10][26]. Meanwhile, link
adaptation and antenna selection are expected to achieve more
gains in correlated MIMO channels due to more prominent
subchannel discrepancies. Furthermore, fading correlation
information varies much more slowly, hence it is feasible and
advantageous to implement antenna selection and link
adaptation only based on the correlation information rather than
on the instantaneous channel information. The author in [9]also
proposed some simplified rules for joint antenna selection and
link adaptation based on the channel correlation information,
aiming to maximize some lower bounds of the minimum post
SNR. Therefore the performance of these rules depends on how
tight the lower bounds would be. Furthermore, the exhaustive
search entailed there might make these rules still complex in
implementation.
In this paper, we consider the problem of joint antenna
selection and link adaptation for an uncoded spatial
multiplexing system with a ZF-SIC receiver, for both
uncorrelated and correlated MIMO channels. Our goal is to
minimize the bit error rate given a throughput and power
constraint. We allow all the available resources, including the
number of active transmit antennas, symbol constellation size
and transmit power dynamically adapted to the channel
Joint Antenna Selection and Link Adaptation for
MIMO Systems
Quan Zhou, Student Member, IEEE, and Huaiyu Dai, Member, IEEE
T

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conditions. The main contributions of this paper are
summarized below:
1).Both incremental and decremental antenna selection rules
with link adaptation are proposed for uncorrelated MIMO
systems. Both rules are realized with recursive algorithms, thus
greatly reducing the computational complexities and feasible
for practical implementation. Rigorously speaking, neither rule
provides the optimum solution, but the performance loss is
negligible
2).For uncorrelated MIMO channels with independent and
identically distributed (i.i.d) Rayleigh fading, we propose an
antenna selection rule based on the expectation of the optimal
number of active antennas. Based on this rule, the
computational complexities can be further reduced while little
performance degradation would be incurred. Such computation
reduction is especially prominent for large MIMO systems.
3).For correlated MIMO channels, we propose an
incremental antenna selection rule with link adaptation based
on the slowly varying channel covariance information, which is
also implemented in a recursive way to avoid the computational
complexity of exhaustive search.
This paper is organized as follows. In Section II, we
introduce the MIMO system model with transmit antenna
selection, and formulate the problem of joint antenna subset
selection and link adaptation. In Section III, we develop
incremental and decremental antenna selection rules with link
adaptation for uncorrelated MIMO channels. We also propose a
simplified rule based on the expected optimal number of active
antennas to further reduce complexity. In Section IV, we
develop an antenna selection rule with link adaptation for
correlated MIMO channels only based on the slowly varying
channel correlation information. Simulation results are given
and analyzed in Section V. Finally, in Section VI, we make
conclusions and propose some future work.
II. PROBLEM FORMULATION
A. MIMO System with Transmit Antenna Selection
Without loss of generality, we assume a narrowband MIMO
system with total
t
K transmit and
the channel between
t
K transmit and
denoted by H . In our study, the antenna selection is only
carried out at the transmitter side, and it is easily shown that the
best performance is achieved when all receive antennas are
active [22]. With
t
N out of
t
K transmit antennas to be chosen,
we denote the selected subset of transmit antennas by p and
the channel matrix between the selected
and
r
N
receive antennas by
correspond to the selected antennas. The received signals are
then given by

( ) p
=+
yH x n, (1)
( ,,...,)
t
N
x xx
x
( ,,...,)
r
N
y yy
y
r
N receive antennas, with
N receive antennas
r
t
N transmit antennas
, whose columns
( ) p
H
where
12
T
=
is the transmitted signal vector,
12
T
=
is the received signal vector, and
12
( ,
n n
,...,)
t
T
N
n
=
n
is assumed to be i.i.d Gaussian with zero
mean and variance of
the index p in (1) in the following discussion when no
ambiguity incurs. All through this paper we assume
2
n
σ . For ease of description, we will drop
rt
NN
≥
.
B. ZF-SIC with QR Decomposition Interpretation
The zero-forcing successive interference cancellation,
widely used in MIMO detection, can be simply interpreted by
matrix QR decomposition. With
unitary matrix and R is an upper triangular matrix, we can
apply
Q to the received vector to obtain
detailed as
...
t
N
rrr
y
yrr
⎜⎟
⎜⎟
=
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
from which the transmitted symbols
=
HQR , where Q is a
HH
==+
y
?
Q yRx n ?
,
1,1
0
1,2 1,
111
2 2,2
:
0
2,22
,
...
:::
:
0
:
..
:
t
ttt
tt
N
NNN
N N
r
x
x
n
n
?
y
?
xn
?
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎟
⎛
⎜
⎜
⎜
⎜
⎜
⎝
⎞ ⎛
⎟ ⎜
⎟ ⎜
+
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎠
?
?
?
, (2)
11
, ,...,
tt
NN
xxx
−
can be
detected successively. Assume no error propagation during
interference cancellation process1, it is clear that QR
decomposition decomposes an
rt
NN
×
MIMO channel matrix
H into
t N subchannels with
subchannel.
, i i r
being the gain for the i th
−

C. Joint Antenna and Link Adaptation
As mentioned in the introduction, the link adaptation
problem and the antenna selection problem are often coupled
for a MIMO system. Furthermore, it is often beneficial to use
only a good subset of antennas in MIMO communications to
reduce hardware complexity and energy consumption. The
induced performance loss is often negligible when judicious
antenna selection is made and link adaptation is employed. To
this end, we propose to jointly consider the antenna selection
and link adaptation for wireless MIMO communications.
Antenna selection and link adaptation can be realized either at
the transmitter or at the receiver, depending on the availability
of channel state information. In the latter case, the receiver will
only feed back the selected active antenna subset and
corresponding communication modes to the transmitter.
In this paper we assume QAM modulation for illustration
purpose. For square M-ary QAM with average powerγ , the
minimum Euclidean distance d is
6
1
M
−
which is also a good approximation for energy efficient
“non-square” QAM in a large range of interest [2].
Assume there are

d
γ
=
, (3)
t N active antennas in use. For the
subchannel with gain, the square of the minimum
i th
−
, i i r

1 This assumption is reasonable at sufficiently high SNR regimes and
commonly adopted in relevant study to simplify analysis. Our simulation results
validate its effectiveness.

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Euclidean distance of the output constellation is given as
2
6
i ii
i out
i
M
−
where
i
M are the power and constellation size
allocated to the i th
−
substream. As with many other
multi-channel communications, the performance of a spatial
multiplexing system is usually limited by the weakest link.
Thus the optimization problem can be sensibly formulated as:
{
(, , , ):,
tiiiTiT
i
i
=
where
2
log
ii
bM
=
is the number of bits allocated to the i th
subchannel,
constraints imposed on the system.
In (5) we want to find an optimal antenna subset together
with its optimal bit and power allocation, subject to the total
throughput and power constraints. The number of active
antennas
complicating the problem. To our best knowledge, the global
optimal solution is open and often a thorough search has to be
resorted to, which is typically infeasible for practical
implementations. Therefore we take some effective steps to
decouple the original problem into some suboptimal ones,
which will be shown to yield excellent performance
nonetheless.
First, assuming the set of active antennas and associated bit
allocation are given, as the system performance is limited by
the worst subchannel, to maximize the aggregate performance
we would like to allocate power so as to achieve the same
output minimum Euclidean distances for all subchannels, i.e.,
222
1,, ( )
...
t
outN oute
ddd
, given as
γ
=
×
−
Thus our optimization goal is simplified as

,
2
,
1
r
d
γ
=
,
1,2,...,
t
iN
=
, (4)
iγ and

}
1
1
2
,
i out
{1,...
∈
}
max
N
=
∑
min
Nt
∑
t
t
iN
N p b bb
d
γγγ
==
, (5)
−

Tb and
T γ are the total throughput and power
t N can also be an optimization parameter, thus further
===

2
( )
e
22
1
,,
11
6
6
()( 1)
1
tt
TT
NN
j jj jj
jj
j
d
rrM
M
γ
−
−
==
=
×−
∑∑
. (6)
2
,
1
min( 1)min(),()
t
N
j jjtt
j
rMNN
−
=
×−=
∑
gm
,1
tt
NK
≤≤
, (7)
subject to
21222
log () log () ... log (
++
)
t
TN
bMMM
=+
,
where
2
2
1,1
r
,
()( ,.....,)
tt
T
tN N
rN
−
−
=
g
is named the antenna gain
vector, while
1
()(1,...,1)
t
T
tN
NMM
=−−
m
is named the bit
allocation vector, and i denotes the inner product between
them. Our target is to find an optimal pair of (
for a given
1
tt
NK
≤≤
, when the number of active antennas is not given
beforehand.
Given
through a thorough search in principle, which is still not an easy
)
(),()
tt
NN
gm

t N , and further choose the best pair among
t N , the optimal pair of ()
(),()
tt
NN
gm
can be found
task when
antenna selection and bit allocation problems by exploiting the
discrete and finite-alphabet nature of the bit allocation vector
()
t
N
m
.When the total throughput and the modulation set are
given, the possible choices of the bit allocation vector can be
determined in advance by a lookup table. Furthermore, by
lemma 1 given in the appendix, in order to minimize (7), only
one permutation (decreasing order) of the elements in the bit
allocation vector needs to be considered for each possible
combination. With this decoupling, the optimization problem is
finally approximated as an antenna selection problem to find a
suitable ()
t
N
g
followed by table lookup to find a matching
()
t
N
m
. Some simple recursive algorithms are proposed in the
next section to avoid exhaustive search while incurring little
performance degradation.
Finally, note that our proposed algorithms can be readily
extended to other modulation schemes. For example, when
PSK is employed, the minimum Euclidean distance of M-ary
t
N and
t
K are large. We further decouple the
PSK with power γ
is given by
2
2sin ()
d
M
π
γ
=
.
Correspondingly, (7) becomes
2
2
,
1
mincsc ( )
t
N
j j
j
j
r
M
π
−
=
×
∑

subject to the same constraint.
III. JOINT ANTENNA SELECTION AND LINK ADAPTATION FOR
UNCORRELATED MIMO CHANNELS
We first consider the uncorrelated MIMO channels where
the channel matrix H can be modeled with i.i.d. complex
Gaussian entries. Two basic recursive algorithms are proposed
to choose the desired antenna gain vector (
selection means the “desired” antennas are recursively added to
an initially empty active antenna set while decremental
selection means “undesired” antennas are recursively removed
from an initially full antenna set2. When
the incremental selection rule described in III.A, while we can
use the decremental rule in III.B when
general link adaptation problem where
advance, we can search over all possible 1
the optimal one. To reduce complexity, we propose an adaptive
selection rule based on estimation of
)
t
N
g
: incremental
tt
NK
<<
, we can use
t
N is close to
N is unknown in
N
≤≤
t
K . In a
t
tt
K
to find
t
N in III.C.
A. Incremental Selection Rule with Link Adaptation
Intuitively, we want
1,1
r
,
2,2
r
, …,
,
tt
N N
r
as large as
possible. Our incremental recursive rule works as follows:
starting from a column of H (
rt
NK
×
) which results in
maximum
1,1
r
(corresponding to the largest vector norm), we
successively choose from the remaining columns of H such
that the next subchannel gain is maximized. The subchannel

2 Same notations are used in [19]and other literature with different problem
settings.

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gain of the newly added antenna can be obtained in a
closed-form solution, which is described by the following
lemma.
Lemma2.a Assume the QR decomposition of a matrix
with k independent columns is
( ) k
H

( )
k
( ) ( )
k
R
⎤
⎦
h
, the first k diagonal
R
k
=
HQ
. Then for
the enhanced matrix
( 1) ( )
kk
+
⎡
=⎣
k
HH
with QR
decomposition
elements of
(1)
( 1) (
R
1)
k
k
+
=++
H
k +
Q
( 1)
R
keep the same with those of
( ) k
, while
the ( 1)
k th
+−
one is given by
( ) ( )
k
Q
HHH
k
−
k +
one is given by
h h h Qh ;
similarly, the first k column vectors of
with those of
( ) k
Q
(1)
Q
th
keep the same
, while the (1)
k
+−
1
(:, 1)(:, ) (:, )
k
H
l
kll
=
+=−∑
QhQ hQ

Proof: see the appendix.
Based on Lemma 2.a, assume in the k
the k selected columns of H and the QR decomposition of
( ) k
H
is
( ) ( )
kk
QR
, then in the (
the column vector h from
H H
remaining columns of H
th
−
step,
( ) k
H
stores
1)
k th
+−
step, we choose
( )
k
\
) in a way such that
(which represents the
1,1
( )
k
( )
k
HHH
kk
r
Furthermore, it can also be shown as follows that the
successively generated antenna gains are already ordered.
Lemma2.b In the above incremental selection rule for
uncorrelated MIMO,
1,1 2,2
||
rr
≥
Proof: see the appendix.
Lemma 2.b shows that the elements in the selected antenna
2
()(,
t
Nrr
=
g
an increasing order. Thus we only need to arrange the elements
of candidate bit allocation vectors
in the lookup table according to lemma 1, which saves storage
space and increases the matching speed for (7). We further
assume
)(
ˆ
t
N
m
is the optimal bit allocation vector that
minimizes
( ),()
tt
NN
gm
for a given (
The incremental selection rule with link adaptation for
uncorrelated MIMO is summarized in the table I.
Two points are noteworthy for the above algorithm. First,
due to Lemma 2.a and 2.b, the antenna selection and link
adaptation process is significantly expedited. Secondly, due to
the recursive nature of the algorithm, searching an optimal
for a general link adaptation problem does not mean
of effort (as calculation for
t
N + is just one step further based
on calculation for
t
N ), but rather the worst-case effort where
all
t
K transmit antennas must be deployed. Nonetheless, in
case nearly all the
t
K transmit antennas would be deployed, we
provide a decremental selection rule for link adaptation, which
is described in the following subsection.
++=−
h h h QQh
is maximized.
,,
.....
tt
k k
r
N N
r
≥ ≥≥≥
.
gain vector
2
2
1,1 2,2,
,.....,)
tt
T
N N
r
−
−−
are already in
()
t
N
m
in a deceasing order
)
t
N
g
.
t
N
t
K times
1

TABLE I
INCREMENTAL ANTENNA SELECTION RULE WITH LINK ADAPTATION FOR
UNCORRELATED MIMO


Set
set)
for
{1,2,3,..,}
t K
=
I
and
= Φ
p
(empty set),
= Φ
g
(empty set),
= Φ
Q
(empty
1
t
i to K
=


2
(:, )
i
i
α = H
;
end

p
1
(1) argmax
t
i
i Kα
≤ ≤
;
=
,
2
1,1
r
1max
i K
≤ ≤
t
i
α=
,
2
1,1
(1) 1/r
=
g
,
(:,1) (:, (1))/
p
(:, (1) ;
p
=
QHH


\ (1)
I p
=
=
I

2
t
for k to N

update
2
(:, )(:, 1)
H
ii
ik
αα
=−−
HQ
, for all i∈I ;
( )
k
argmax
i
i
α
∈
=
I
;
p
;
2
,
max
i
∈
=
k k
r
i
α=
I
2
,
( ) 1/
k
k k
r
g
;
1
1
(:, ) (:, ( ))
p
(:, )(:, ( ) (:, )
k
p
k
H
l
kkll
−
=
=−∑
QHQHQ
;
(:, )(:, )/
k
(:, ) ;
kk
=
QQQ

( ) k
= −
IIp
;
end
assume
()
ˆ (
m
)argmin
m
(),()
t
)
ttt
N
NNN
=
gm
;


(1:)
t
returnN
p
and ˆ (
t N
m
;

B. Decremental Selection Rule with Link Adaptation
Our proposed decremental selection rule is related to the
V-BLAST ordering rule first proposed in [20], which
successively chooses the antenna (among those not already
chosen) that maximizes post-detection SNR under the
assumption of perfect feedback. Accordingly, we can
successively discard the antenna (among those not already
chosen) that minimizes the post-detection SNR under the
assumption of perfect feedback. Usually repeated matrix
inversion will be involved during the process of discarding,
which may introduce much computation complexity and
numerical instability. Thanks to the work in [4], we can avoid
computing the inversion of the deflated channel matrix by
means of a recursive square-root algorithm.
The decremental selection rule with link adaptation for
uncorrelated MIMO is summarized in the table II.

TABLE II
DECREMENTAL ANTENNA SELECTION RULE WITH LINK ADAPTATION FOR
UNCORRELATED MIMO


Set
HH ;
Using [4] to find square root of (
(
()
t
K=
)
1
()()
()
tt
KK
H
−
HH
, assume
)
1/2
−
()()()
()
ttt
KKK
H
=
PHH
;

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5
Assume
( )
,:
l
()
1
ˆ
l
argmin
t
t
K
l K
≤ ≤
=
P
(
( )
,:
l
P
means the length of the l th
−

row), discard the ˆlth
−
column of
()
t
K
H
, and assume the deflated matrix to be
( 1)
t
K −
H
;
k
for
1(1)
tt
toKN
=−+

Using [4] to find square root of
()()
1
()
H
tt
KkKk
−−
−
HH
, based on
( 1)
t
Kk
− +
P
;
assume
Discard the ˆl
()()()
1/2
−
()
H
ttt
KkKkKk
−−−
=
PHH
;
th
−
column of
()
t
Kk
−
H
, where
( )
,:
l
( )
k
1
ˆ
l
argmin
l k
≤ ≤
=
P
,
assume the deflated matrix to be
end
Compute the sorted antenna gain vector (
( 1)
t
Kk
− −
H
;
)
t
N
g
for
()
t
N
H
;
assume
()
ˆ (
m
) argmin
m
( ),()
t
ttt
N
NNN
=
gm
;
()
t
N
return H
and ˆ (
)
t N
m
.
C. Simplified Link Adaptation for Uncorrelated Rayleigh
MIMO Channels
In a general link adaptation problem where
in advance, we need to test all possibilities 1
the optimal one using either the incremental or decremental
selection rule. In this subsection, we propose a simplified
selection rule based on the estimation of the optimal number of
active transmit antennas to further reduce the complexity. With
t
N is not fixed
NK
≤
tt
≤
to find
i.i.d. complex Gaussian channel matrix,
2
, i i r
+ −
in (7) is a
2
χ distributed random variable with 2 (
freedom [3], where the probability density function of the
2
χ distribution with v degrees of freedom is given as:
1)
r
Ni
×
degrees of

( 2)/2/2
/2
( | )
f x v
2
the gamma function defined as
( /2)
v
vx
v
xe
−−
=
Γ
, (8)
where
( ) x
Γ
is
1
0
( )
x
xt
t e dt
∞
−−
Γ=∫
.
The expected value of
2
,
1/
i i r
can be obtained as

0
1/(
⎧
⎨
⎩
2)2
2
1
x
( | )
f x v dx
v
+∞
for
for
v
v
+∞
∫
−>
=
=
, (9)
Replacing
2
,
1/
i i r
In (7) with their expected values, we have
()
1
1
min( ),()min(1),
2()
1
subject to
b
=
≥
Therefore, we can estimate the optimal number of active
antennas in a pre-processing stage as follows: for all possible
bit allocation vectors
( )'
t
Ns
m
that minimizes (10), denoted as
?
, which is our estimate of the
,
t
N
ttj
j
r
tt
ENNM
Nj
NK
=
=×−
−
≤≤
∑
gm
(10)

21
≥
222
12
log () log (
+
) ... log (
++
),
t
t
TN
N
MMMand
MMM
?
. (11)
that satisfy (11), find the one
ˆ (
)
t
N
m
; then find
1
ˆ
argmin
N
≤
( ),()
tt
ttt
K
NENN
≤
=
gm
optimal number of active antennas in i.i.d Rayleigh fading
MIMO channels. Thus we can decide to use either the
incremental or decremental selection rule for joint antenna
selection and link adaptation for different system settings based
on the value of
t
search optimal
t
N only in the range around
reduce the computational complexity. Simulations results show
that searching
t
N in the range of
only three bit allocation vectors ˆ (
ˆ (
1)
t
N +
m
in the bit allocation lookup table incur little
performance loss (See Section V).
N?. Furthermore, we can restrict ourselves to
t
N? to further
1,1
tt
N
?
N
?
⎡
⎣
⎤
⎦
, ˆ (
m
−+
and storing
1)
t
N −
m
)
t
N
and
IV. JOINT ANTENNA SLECTION AND LINK ADAPTATION FOR
CORRELATED MIMO CHANNELS
A. Correlated MIMO Channels
In this section, we extend the study of joint antenna selection
and link adaptation to correlated MIMO channels. We assume
correlation only exists at the transmitter side, as described by
the “one-ring” model in [10]. This model is feasible, e.g., for
the outdoor macrocell situation where the transmitter at the
base station is elevated high above the local scattering
environment, while there are sufficient local scatters around the
mobile receivers. For an
r
NK
×
can be modeled as
wT
=
HH A with
is an
rt
NK
×
matrix containing i.i.d. complex Gaussian
random variables and
T
R
semi-positive definite matrix representing the covariance
matrix for each row of H .
Again, we assume
t
N out of
As before, the channel matrix between the
antennas and
r
N receive antennas can be described as
( )( ) ( )
wT
ppp
=
HHA
, where p contains the indices of the
selected antennas, and
T
A
corresponding submatrix of
T
R .
We assume uniform linear arrays at both the transmitter and
receiver, with antenna spacing
wavelength). We also assume there are L clusters of scatterers
in the environment and the angle of departure for the l
path cluster is Gaussian distributed as
( , )
i jth
−
entry of the transmit covariance matrix contributed by
the lth
−
scattering cluster can be shown to be approximated
as [9][16]-[18]:
t
MIMO system, the channel
=
AAR , where
H
*
H
TTT
w
H
is a
tt
KK
×
Hermitian
t
K antennas are to be selected.
t
N transmit
H
( )
p
( )
p
( )
p
H
TT
=
AR
is the
T
Δ (relative to the carrier
th
−

2
l
~ ( ,
θ σ
)
ll
N
θ
. Then the

2
1(2 (
2
)sin())
2 (
π
) cos()
,
,
Tll
Tl
ij
jij
T l
i j
ee
π θ σ
θ
−−Δ
−−Δ
⎡
⎣
⎤
⎦
≈
R
. (12)
For a narrowband system, the net correlation matrix can be
obtained by summing the covariance matrices contributed by
the L clusters weighted by the fraction of power in the
corresponding cluster. As a counterpart to (1), the received
signal in correlated MIMO can be written as: