# Antenna Selection for MIMO Systems with Closely Spaced Antennas.

**ABSTRACT** Physical size limitations in user equipment may force multiple antennas to be spaced closely, and this generates a considerable amount of mutual coupling between antenna elements whose effect cannot be neglected. Thus, the design and deployment of antenna selection schemes appropriate for next generation wireless standards such as 3GPP long term evolution (LTE) and LTE advanced needs to take these practical implementation issues into account. In this paper, we consider multiple-input multipleoutput (MIMO) systems where antenna elements are placed side by side in a limited-size linear array, and we examine the performance of some typical antenna selection approaches in such systems and under various scenarios of antenna spacing and mutual coupling. These antenna selection schemes range from the conventional hard selection method where only part of the antennas are active, to some newly proposed methods where all the antennas are used, which are categorized as soft selection. For the cases we consider, our results indicate that, given the presence of mutual coupling, soft selection can always achieve superior performance as compared to hard selection, and the interelement spacing is closely related to the effectiveness of antenna selection. Our work further reveals that, when the effect ofmutual coupling is concerned, it is still possible to achieve better spectral efficiency by placing a few more than necessary antenna elements in user equipment and applying an appropriate antenna selection approach than plainly implementing the conventional MIMO system without antenna selection.

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**ABSTRACT:**MIMO system can offer considerable capacity. But as the antenna’s number increasing, the costs of hardware such as RF chains are very expensive, and limit the using of MIMO system. The traditional antenna selection algorithms based on correlation can give a low complexity but capacity performance loss is large. In this paper, a fast antenna selection algorithm is proposed for MIMO system based on dissimilarity coefficient criteria. The proposed algorithm can be used to calculate the correlation of antennas better, supplies more efficient capacity performance, and has the lower complexity. The simulation results verify the conclusions.01/2011; - SourceAvailable from: repository.lib.ncsu.edu
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**ABSTRACT:**It is well known that the capacity of spatial multiplexing multiple-input multiple-output (SM-MIMO) system employing optimal antenna selection can significantly outperform a system without selection for same number of costly radio frequency chains. However, it requires an exhaustive search for the optimal selection (OS) that grows exponentially with the available number of transmit (u) and receive (m) antennas. In this paper, a novel low complexity receive antenna selection (RAS) technique is proposed for SM-MIMO to maximize the channel capacity over correlated Rayleigh fading environment. It is based on the Euclidean norms of channel matrix rows and the corresponding phase differences due to their direct impact on the capacity. Extensive analysis and simulations have shown near optimal performance for any signal-to-noise-ratio and correlation values with low complexity of ${\mathcal{O} \left({u^{2}m}\right)}$ vector calculations. This technique provides fast RAS to capture most of the capacity gain promised by multiple antenna systems over different channel conditions. Furthermore, it enables efficient spectrum utilization for next generation wireless communications.Wireless Personal Communications 06/2013; 70(4). · 0.43 Impact Factor

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Hindawi Publishing Corporation

EURASIP Journal on Wireless Communications and Networking

Volume 2009, Article ID 739828, 11 pages

doi:10.1155/2009/739828

Research Article

AntennaSelectionforMIMO Systemswith

CloselySpaced Antennas

YangYang,1Rick S.Blum,1and SanaSfar2

1Department of Electrical and Computer Engineering, Lehigh University, 19 Memorial Drive West, Bethlehem, PA 18015, USA

2CTO Office, InterDigital Communications, LLC, 781 Third Avenues, King of Prussia, PA 19406, USA

Correspondence should be addressed to Yang Yang, yay204@lehigh.edu

Received 1 February 2009; Revised 18 May 2009; Accepted 28 June 2009

Recommended by Angel Lozano

Physical size limitations in user equipment may force multiple antennas to be spaced closely, and this generates a considerable

amount of mutual coupling between antenna elements whose effect cannot be neglected. Thus, the design and deployment

of antenna selection schemes appropriate for next generation wireless standards such as 3GPP long term evolution (LTE) and

LTE advanced needs to take these practical implementation issues into account. In this paper, we consider multiple-input

multipleoutput (MIMO) systems where antenna elements are placed side by side in a limited-size linear array, and we examine

the performance of some typical antenna selection approaches in such systems and under various scenarios of antenna spacing

and mutual coupling. These antenna selection schemes range from the conventional hard selection method where only part of the

antennas are active, to some newly proposed methods where all the antennas are used, which are categorized as soft selection. For

the cases we consider, our results indicate that, given the presence of mutual coupling, soft selection can always achieve superior

performanceas comparedtohardselection, andtheinterelementspacing isclosely relatedtotheeffectiveness ofantennaselection.

Ourworkfurtherrevealsthat,whentheeffectofmutualcouplingisconcerned,itisstillpossibletoachievebetterspectralefficiency

byplacingafewmorethannecessaryantennaelementsinuserequipmentandapplyinganappropriateantennaselectionapproach

than plainly implementing the conventional MIMO system without antenna selection.

Copyright © 2009 Yang Yang et al. This is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.Introduction

The multiple-input multiple-output (MIMO) architecture

has been demonstrated to be an effective means to boost

the capacity of wireless communication systems [1], and

has evolved to become an inherent component of various

wireless standards, including the next-generation cellular

systems 3GPP long term evolution (LTE) and LTE advanced.

For example, the use of a MIMO scheme was proposed

in the LTE standard, with possibly up to four antennas at

the mobile side, and four antennas at the cell site [2]. In

MIMO systems, antenna arrays can be exploited in two

different ways, which are [3]: diversity transmission and

spatial multiplexing. However, in either case, one main

problem involved in the implementation of MIMO systems

is the increased complexity, and thus the cost. Even though

the cost for additional antenna elements is minimal, the

radio frequency (RF) elements required by each antenna,

which perform the microwave/baseband frequency transla-

tion, analog-to-digital conversion, and so forth, are usually

costly.

These complexity and cost concerns with MIMO have

motivated the recent popularity of antenna selection (AS)—

an attractive technique which can alleviate the hardware

complexity, and at the same time capture most of the

advantages of MIMO systems. In fact, for its low user

equipment(UE)complexity,AS(transmit)iscurrentlybeing

considered as a baseline of the single-user transmit diversity

techniques in the LTE uplink which is a MIMO single carrier

frequency division multiple access (SC-FDMA) system [4].

Further, when it comes to the RF processing manner, AS can

be categorized into two groups: (1) hard selection, where

only part of the antennas are active and the selection is

implemented in the RF domain by means of a set of switchs

(e.g., [5–7]); (2) soft selection, where all the antennas are

active and a certain form of transformation is performed

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2EURASIP Journal on Wireless Communications and Networking

in the RF domain upon the received signals across all the

antennas (e.g., [8–10]).

A considerable amount of research efforts have been

dedicated to the investigation of AS, and have solidly

demonstrated the theoretical benefits of AS (see [3] for a

tutorial treatment). However, previous works largely ignore

the hardware implementation issues related to AS. For

instance,thephysicalsizeofUEsuchasmobileterminalsand

mobile personal assistants, are usually small and invariable,

and the space allocated for an antenna array is limited. Such

limitationmakestheclosespacingbetweenantennaelements

a necessity, inevitably leading to mutual coupling [11], and

correlated signals. These issues have caught the interest of

some researchers, and the capacity of conventional MIMO

systems (without AS) under the described limitations and

circumstanceswasinvestigated,amongothers,in[12–17].To

give an example, the study in [12] shows that as the number

of receive antenna elements increases in a fixed-length array,

the system capacity firstly increases to saturate shortly after

the mutual coupling reaches a certain level of severeness; and

drops after that.

Form factors of UE limit the performance promised

by MIMO systems, and can further affect the proper

functionality of AS schemes. These practical implementation

issues merit our attention when designing and deploying AS

schemes for the 3GPP LTE and LTE advanced technologies.

There exist some interesting works, such as [18, 19] which

consider AS in size sensitive wireless devices to improve

the system performance. But in general, results, conclusions,

and ideas on the critical implementation aspects of AS

in MIMO systems still remain fragmented. In this paper,

through electromagnetic modeling of the antenna array and

theoreticalanalysis,weproposeacomprehensivestudyofthe

performance of AS, to seek more effective implementation of

ASinsizesensitiveUEemploying MIMOwherebothmutual

coupling and spatial correlation have a strong impact. In this

process, besides the hybrid selection [5–7], a conventional

yet popular hard AS approach, we are particularly interested

in examining the performance potential of some typical

soft AS schemes, including the FFT-based selection [9] and

the phase-shift-based selection [10], that are very appealing

but seem not to have attracted much attention so far. At

the meantime, we also intend to identify the operational

regimes of these representative AS schemes in the compact

antenna array MIMO system. For the cases we consider, we

find that in the presence of mutual coupling, soft AS can

always achieve superior performance as compared to hard

AS. Moreover, effectiveness of these AS schemes is closely

related to the interelement spacing. For example, hard AS

workswellonlywhentheinterelementspacingisnolessthan

a half wavelength.

Additionally, another goal of our study is to address

a simple yet very practical question which deals with the

cost-performance tradeoff in implementation: as far as

mutual coupling is concerned, can we achieve better spectral

efficiency by placing a few more than necessary antenna

elements in size sensitive UE and applying a certain adequate

AS approach than plainly implementing the conventional

MIMO system without AS? Further, if the answer is yes,

l

2r

dr

dr

Lr

···

Figure 1: Dipole elements in a side-by-side configuration (receiver

antenna array as an example).

how would we decide the number of antenna elements for

placement and the AS method for deployment? Our work

will provide answers to the above questions, and it turns out

the solution is closely related to identifying the saturation

point of the spectral efficiency.

This paper is organized as follows. In Section 2, we

introduce the network model for the compact MIMO system

andcharacterizetheinput-outputrelationshipbytakinginto

account the influence of mutual coupling. In Section 3, we

describethehardandsoftASschemesthatwillbeusedinour

study, and also estimate their computational complexity. In

Section 4, we present the simulation results. We discuss our

main findings in Section 5, and finally conclude this paper in

Section 6.

2.Network Model for Compact MIMO

We consider a MIMO system with M transmit and N receive

antennas (M, N > 1). We assume antenna elements are

placed in a side-by-side configuration along a fixed length

at each terminal (transmitter and receiver), as shown in

Figure 1. Other types of antenna configuration are also

possible, for example, circular arrays [11]. But it is noted

that, the side-by-side arrangement exhibits larger mutual

couplingeffectssincetheantennasareplacedinthedirection

of maximum radiation [11, page 474]. Thus, the side-by-side

configuration is more suitable to our study. We define Ltand

Lras the aperture lengths for transmitter and receiver sides,

respectively. In particular, we are more interested in the case

that Lr is fixed and small, which corresponds to the space

limitation of the UE. We denote l as the dipole length, r as

the dipole radius, and dr (dt) as the side-by-side distance

between the adjacent dipoles at the receiver (transmitter)

side. Thus, we have dr= Lr/(N − 1) and dt= Lt/(M −1).

A simplified network model (as compared to [13, 14],

e.g.) for transmitter and receiver sides is depicted in Figure 2.

Figure 3 illustrates a direct conversion receiver that connects

the output signals in Figure 2, where LNA denotes the low-

noise amplifier, LO denotes the local oscillator, and ADC

denotes the analog-to-digital converter. For the ease of the

following analysis, we assume that in the circuit setup, all the

antenna elements at the receiver side are grounded through

the load impedance ZLi, i = 1,...,N (cf. Figure 2), regardless

of whether they will be selected or not. In fact, ZLi,i =

1,...,N constitute a simple matching circuit. Such matching

circuit is necessary as it can enhance the efficiency of power

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EURASIP Journal on Wireless Communications and Networking3

transfer from the generator to the load [20, Chapter 11].

We also assume that the input impedance of each LNA in

Figure 3whichislocatedveryclosetotheantennaelementto

amplify weak received signals, is high enough such that it has

little measurable effect on the receive array’s output voltages.

This assumption is necessary to facilitate the analysis of the

network model. However, it is also very reasonable because

this ensures that the input of the amplifier will neither

overload the source of the signal nor reduce the strength of

the signal by a substantial amount [21].

Let us firstly consider the transmitter side, which can be

regarded as a coupled M port network with M terminals. We

define i = [i1,...,iM]Tand vt= [vt1,...,vtM]Tas the vectors

of terminal currents and voltages, respectively, and they are

related through

vt= ZTi,

(1)

where ZT denotes the impedance matrix at the transmitter

side. The (p,q)-th entry of ZT(p,q), when p/ =q, denotes

the mutual impedance between two antenna elements, and

is given by [20, Chapter 21.2]:

ZT

?p,q?=

jη

4πsin2(kl/2)

?l/2

−l/2F(z) dz,

(2)

where

F(z) =

?e−jkR1

R1

+e−jkR2

R2

−2cos

??

?kl

2

?

·e−jkR0

R0

?

·sin

?

k

?l

2− |z|

.

(3)

In the above expression, η denotes the characteristic

impedance of the propagation medium, and can be calcu-

lated by η =

?

μ/?, where μ and ? denote permittivity and

permeabilityofthemedium,respectively.Likewise,k denotes

the propagation wavenumber of an electromagnetic wave

propagating in a dielectric conducting medium, and can

be computed through k = ω√μ?, where ω is the angular

frequency. Finally R0, R1and R2are defined as

R0=

?

?

?

?

?

?

?

?

?

?p − q?2d2

t

(M −1)2+z2,

?p − q?2d2

R1=

?

?

t

(M −1)2+

?p − q?2d2

?

z −l

2

?2

,

R2=

?

t

(M −1)2+

?

z +l

2

?2

.

(4)

When p = q, ZT(p,q) is the self-impedance of a single

antenna element, and can also be obtained from (2) by

simply redefining R0, R1and R2as follows:

R0=

?r2+z2,

?

?

R1=

r2+

?

?

z −l

2

?2

?2

,

R2=

r2+

z +l

2

.

(5)

Thus, the self-impedance for an antenna element with l =

0.5λ and r = 0.001λ for example, is approximately

?p, p?= 73.08 +42.21jΩ.

Further, let us consider an example that M = 5 antenna

elements of such type are equally spaced over a linear array

of length Lt= 2λ. The impedance matrix ZTis given by

ZT

(6)

ZT=

⎛

⎜

⎜

⎝

⎜

⎜

⎜

⎜

⎜

⎜

⎜

73.08 +42.21j

−12.52 − 29.91j

73.08 +42.21j

4.01 +17.73j

−1.89 −12.30j

4.01 +17.73j

1.08 +9.36j

−12.52 −29.91j

4.01 +17.73j

−12.52 −29.91j

73.08 +42.21j

−1.89 − 12.30j

4.01 +17.73j

−12.52 − 29.91j

4.01 +17.73j

−12.52 −29.91j

73.08 +42.21j

−1.89 −12.30j

1.08 +9.36j

−12.52 −29.91j

4.01 +17.73j

−12.52 −29.91j

73.08 +42.21j

−1.89 −12.30j

−12.52 −29.91j

⎞

⎟

⎟

⎠

⎟

⎟

⎟

⎟

⎟

⎟

⎟

.

(7)

For i = 1,...,M, the terminal voltage vtican be related

to the source voltage xi via the source impedance Zsi by

vti

= xi − Zsiii. Define ZS

x = [x1,...,xM]. Then, from Figure 2, we can obtain the

following results: vt = x − ZSi and vt = ZTi. Therefore,

the relationship between terminal voltages vt and source

voltages x can be written in matrix form as vt = ZT(ZT+

ZS)−1x. Similar to [12], we choose Zsi = Z∗

roughly corresponds to a conjugate match in the presence of

mild coupling. In the case of uncoupling in the transmitter

= diag{Zs1,...,ZsM}, and

T(i,i), which

side, ZT is diagonal, and its diagonal elements are all the

same. Consequently, ZT(ZT+ ZS)−1is also diagonal, and its

diagonalelementcanbedenotedasδT= ZT(1,1)/[ZT(1,1)+

ZS(1,1)]. To accommodate the special case of zero mutual

coupling where vtis equal to x, in our model we modify the

relationship between vtand x into

vt= WTx,

(8)

where WT= δ−1

TZT(ZT+ZS)−1.

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4EURASIP Journal on Wireless Communications and Networking

H

············

x1

x2

Zs1

Zs2

ZsM

i1

i2

iM

vt1

vt2

vtM

vr1

vr2

vrN

ZL2

y1

y2

yN

ZT

ZR

xM

ZL1

ZLN

MIMO

propogation

channel

Overall transmitter side

impedance matrix

Overall receiver side

impedance matrix

Compound MIMO channel

Figure 2: Network model for a (M,N) compact MIMO system.

y1

y2

yN

LNA

LNA

ADC

ADC

I

Q

ADC

ADC

I

Q

90˚

LO

LO

RF chain

RF chain

···

······

90˚

Singal

processing

and

decoding

Figure 3: RF chains at the receiver side.

Denote vr= [vr1,...,vrN] as the vector of open circuited

voltages induced across the receiver side antenna array, and

y = [y1,..., yN] as the voltage vector across the output of

the receive array. Since we assumed high-input impedance of

these LNAs, a similar network analysis can be carried out at

the receiver side and will yield

y = WRvr,

(9)

where WR= δ−1

matrix at the receiver side, and ZL is a diagonal matrix

with its (i,i)th entry given by ZL(i,i) = ZLi = [ZR(i,i)]∗,

i = 1,...,N. δR is given by δR = [ZR(1,1)]∗/{ZR(1,1) +

[ZR(1,1)]∗}. It is noted that the approximate conjugate

match [12] is also assumed at the receiver side, so that the

load impedance matrix ZLis diagonal with its entry given by

Z∗

In frequency-selective fading channels, the effectiveness

of AS is considerably reduced [3], which in turn makes it

RZL(ZR+ ZL)−1. ZRis the mutual impedance

R(i,i), for i = 1,...,N.

difficult to observe the effect of mutual coupling. Therefore,

we focus our attention solely on flat fading MIMO channels.

The radiated signal vt is related to the received signal vr

through

vr= Hvt,

(10)

whereHisaN×M complexGaussianmatrixwithcorrelated

entries. To account for the spatial correlation effect and the

Rayleigh fading, we adopt the Kronecker model [22, 23].

This model uses an assumption that the correlation matrix,

obtained as Ψ = E{vec(H) vec(H)H} with vec(H) being the

operatorstackingthematrixHintoavectorcolumnwise,can

be written as a Kronecker product, that is, Ψ = ΨR⊗ ΨT,

where ΨRand ΨTare respectively, the receive and transmit

correlation matrices, and ⊗ denotes the Kronecker product.

This implies that the joint transmit and receive angle power

spectrum can be written as a product of two independent

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EURASIP Journal on Wireless Communications and Networking5

angle power spectrum at the transmitter and receiver. Thus,

the correlated channel matrix H can be expressed as

H = Ψ1/2

RHwΨ1/2

T,

(11)

where Hwis a N × M matrix whose entries are independent

identically distributed (i.i.d) circular symmetric complex

Gaussian random variables with zero mean and unit vari-

ance. The (i, j)-th entry of ΨRor ΨTis given by J0(2πdij/λ)

[24], where J0is the zeroth order Bessel function of the first

kind, and dijdenotes the distance between the i,j-th antenna

elements.

Therefore, based on (8)–(11), the output signal vector y

at the receiver can be expressed in terms of the input signal x

at the transmitter through

y = WRΨ1/2

= WRΨ1/2

RHwΨ1/2

TWTx +n = Hx +n,

(12)

where H

compound channel matrix which takes into account both the

Rayleighfadinginwirelesschannelsandthemutualcoupling

effect at both transmitter and receiver sides, and n is the

thermal noise. For simplicity, we assume uncorrelated noise

at the receiving antenna element ports. For the case where

correlatednoiseisconsidered,readersarereferredto[16,17].

RHwΨ1/2

TWT can be regarded as a

3.HardandSoft AS for Compact MIMO

We describe here some typical hard and soft AS schemes

that we will investigate, assuming the compact antenna array

MIMO system described in Section 2. For hard AS, we focus

only on the hybrid selection method [5–7]. For soft AS, we

studytwotypicalschemes:theFFT-basedselection[9]which

embeds fast Fourier transform (FFT) operations in the RF

chains, and the phase-shift-based selection [10] which uses

variable phase shifters adapted to the channel coefficients

in the RF chains. For simplicity, we only consider AS at

the receiver side with nRantennas being chosen out of the

N available ones, and we focus on a spatial multiplexing

transmission.

Weassumethatthepropagationchannelisflatfadingand

quasistatic, and is known at the receiver. We also assume that

the power is uniformly allocated across all the M transmit

antennas, that is, E{xxH} = P0IM/M. We denote the noise

power as σ2

ρ = P0/σ2

Shannon limit quite closely are used, the spectral efficiency

(in bits/s/Hz) of this (M,N) full-complexity (FC) compact

MIMO system without AS could be calculated through [1]

?

It is worth noting that the length limits of transmit and

receive arrays, Ltand Lr, enter into the compound channel

matrix H in a very complicated way. It is thus difficult to

find a close-form analytical relationship between CFC(M,N)

and Lt (Lr). Consequently, using Monte Carlo simulations

to evaluate the performance of spectral efficiency becomes a

necessity.

n, and the nominal signal-to-noise ratio (SNR) as

n. Then assuming some codes that approach the

CFC(M,N) = log2

det

?

IM+ρ

MHHH

??

.

(13)

To avoid detailed system configurations and to make

the performance comparison as general and as consistent

as possible, we only use the spectral efficiency as the

performance of interest. Moreover, all these AS schemes we

study here are merely to optimize the spectral efficiency,

not other metrics. Since each channel realization renders a

spectral efficiency value, the ergodic spectral efficiency and

the cumulative distribution function (CDF) of the spectral

efficiency will be both meaningful. We will then consider

them as performance measures for our study.

3.1. Hybrid Selection. This selection scheme belongs to the

conventional hard selection, where nR out of N receive

antennas are chosen by means of a set of switches in the RF

domain (e.g., [5–7]). Figure 4(a) illustrates the architecture

of the hybrid selection at the receiver side. As all the antenna

elementsatthereceiversidearepresumedgroundedthrough

the load impedance ZLi, i = 1,...,N, the mutual coupling

effect will be always present at the receiver side. However,

this can facilitate the channel estimation and allow us to

extract rows from H for subset selection. Otherwise, the

mutual coupling effect will vary with respect to the selected

antenna subsets. For convenience, we define S as the nR× N

selection matrix, which extracts nR rows from H that are

associated with the selected subset of antennas. We further

define S as the collection of all possible selection matrices,

whose cardinality is given by |S| =

with hybrid selection delivers a spectral efficiency of

?N

nR

?

. Thus, the system

CHS= max

S∈S

log2

?

det

?

IM+ρ

M(SH)H(SH)

??

.

(14)

OptimalselectionthatleadstoCHSrequiresanexhaustive

search over all

nR

subsets of S, which is evident by (14).

Note that

?

?N

?

det

IM+ρ

M(SH)H(SH)

?

= det

?

InR+ρ

M(SH)(SH)H?

.

(15)

Then, the matrix multiplication in (14) has a complexity of

O(nRM · min(nR,M)). Calculating the matrix determinant

in (14) requires a complexity of O((min(nR,M))3). Thus, we

can conclude that optimal selection requires about O(|S| ·

nRM · min(nR,M)) complex additions/multiplications. This

estimated complexity for optimal selection can be deemed

as an upper bound of the complexity of any hybrid AS

scheme, since there exist some suboptimal but reduced

complexity algorithms, such as the incremental selection and

the decremental selection algorithms in [7].

3.2. FFT-based Selection. As for this soft selection scheme

(e.g., [9]), a N-point FFT transformation (phase-shift

only) is performed in the RF domain firstly, as shown

in Figure 4(b), where information across all the receive

antennas will be utilized. After that, a hybrid-selection-like

schemeisappliedtoextractnRoutofN informationstreams.

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6EURASIP Journal on Wireless Communications and Networking

RF

switches

1

y1

y2

yN

···

···

···

···

vr1

vr2

vrN

nR

Overall

receiver

side

impedance

matrix

RF chain

RF chain

(a) Hybrid selection

RF

switches

FFT

matrix

F

1

y1

y2

yN

···

···

···

···

···

vr1

vr2

vrN

nR

Overall

receiver

side

impedance

matrix

RF chain

RF chain

(b) FFT-based selection

1

···

···

···

vr1

vr2

vrN

y1

y2

yN

nR

Θ

Overall

receiver

side

impedance

matrix

Phase

shift

matrix

RF chain

RF chain

(c) Phase-shift-based selection

Figure 4: AS at the receiver side for spatial multiplexing transimssions.

We denote F as the N × N unitary FFT matrix with its (k,l)

th entry given by:

F(k,l) =

1

√N

exp

?

−j2π(k −1)(l −1)

N

?

,

∀k,l ∈ [1,N].

(16)

Accordingly, this system delivers a spectral efficiency of

CFFTS= max

S∈Slog2

?

det

?

IM+ρ

M(SFH)H(SFH)

??

.

(17)

The only difference between (14) and (17) is the N-

pointFFTtransformation.SuchFFTtransformationrequires

a computational complexity of O(MN logN). If we assume

N logN

≤

complexity of optimal selection that achieves CFFTScan be

estimated as O(|S| ·nRM ·min(nR,M)), which is the worst-

case complexity.

nR · min(nR,M), then the computational

3.3. Phase-Shift Based Selection. This is another type of soft

selection scheme (e.g., [10]) that we consider throughout

this study. Its architecture is illustrated in Figure 4(c). Let

us denote Θ as one nR × N matrix whose elements are

nonzero and restricted to be pure phase-shifters, that we

will fully define in what follows. There exists some other

work such as [25] that also considers the use of tunable

phaseshifterstoincreasethetotalcapacityofMIMOsystems.

However, in Figure 4(c), the matrix Θ that performs phase-

shift implementation in the RF domain essentially serves

as a N-to-nR switch with nR output streams. Additionally,

unliketheFFTmatrix,Θmightnotbeunitary,andhencethe

resulting noise can be colored. Finally, this system’s spectral

efficiency can be calculated by [10]

CPSS= max

Θ

log2

?

det

?

IM+ρ

M(ΘH)H?

ΘΘH?−1(ΘH)

??

(18)

.

Let us define the singular value decomposition (SVD) of H

as H = UΛVH, where U and V are N × N, M × M unitary

matricesrepresentingtheleftandrightsingularvectorspaces

of H, respectively; Λ is a nonnegative and diagonal matrix,

consisting of all the singular values of H. In particular, we

denote λH,ias the ith largest singular value of H, and uH,ias

the left singular vector of H associated with λH,i. Thus one

solution to the phase shift matrix Θ can be expressed as [10,

Theorem 2]:

Θ = exp

?

j × angle

??uH,1,...,uH,nR

?H??

(19)

where angle{·} gives the phase angles, in radians, of a matrix

with complex elements, exp{·} denotes the element-by-

element exponential of a matrix.

The overall cost for calculating the SVD of H is around

O(MN·min(M,N))[26,Lecture31].Computingthematrix

multiplication in (19) requires a complexity around the

order of O(MNnR). The matrix determinant has an order

of complexity of O((min(nR,M))3). Therefore, the phase-

shift-based selection requires around O(MN · max(nR,M))

complex additions/multiplications.

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EURASIP Journal on Wireless Communications and Networking7

1102030

N

4050 60

0

5

10

15

20

25

30

35

Uncorrelated without mutual coupling

Correlated without mutual coupling

Correlated with mutual coupling

Ergodic spectral efficiency (bits/s/Hz)

Figure 5: Ergodic spectral efficiency of a compact MIMO system

(M = 5) with mutual coupling at both transmitter and receiver

sides.

4.Simulations

Our simulations focus on the case when AS is implemented

only on the receiver side, but mutual coupling and spatial

correlation are accounted for at both terminals. However,

in order to examine the mutual coupling effect on AS at

the receiving antenna array, we further assume M = 5

equally-spaced antennas at the transmitter array, and the

interelement spacing dt is fixed at 10λ. This large spacing

is chosen to make the mutual coupling effect negligible at

the transmitting terminal. For the receiver terminal, we fix

the array length Lr at 2λ. We choose l = 0.5λ and r =

0.001λ for all the dipole elements. Each component in the

impedance matrices ZT and ZR is computed through (2)

which analytically expresses the self and mutual impedance

ofdipoleelementsinaside-by-sideconfiguration.Finally,we

fix the nominal SNR at ρ = 10dB.

As algorithm efficiency is not a focus in this paper, for

both hybrid and FFT-based selection methods, we use the

exhaustive search approach to find the best antenna subset.

For the phase-shift-based selection, we compute the phase

shift matrix Θ through (19) given each H. For each scenario

of interest, we generate 5 ×104random channel realizations,

and study the performance in terms of the ergodic spectral

efficiency and the CDF of the spectral efficiency.

4.1. Ergodic Spectral Efficiency of Compact MIMO. In

Figure 5 we plot the ergodic spectral efficiency of a compact

MIMO system for various N. The solid line in Figure 5

depicts the ergodic spectral efficiency when mutual coupling

and spatial correlation is considered at both terminals. Also

for the purpose of comparison, we include a dashed line

whichdenotestheperformancewhenonlyspatialcorrelation

is considered at both sides, and a dash-dot line which

123

nR

45

3

6

9

12

15

Hybrid selection

FFT based selection

Phase-shift based selection

Reduced full system w/o selection

Ergodic spectral efficiency (bits/s/Hz)

Figure 6: Ergodic spectral efficiency of a compact MIMO system

with AS, where M = 5 and N = 5.

corresponds to the case when only the simplest i.i.d Gaussian

propagation channel is assumed in the system. It is clearly

seen that mutual coupling in the compact MIMO system

seriously decreases the system’s spectral efficiency. Moreover,

in accord with the observation in [12], our results also

indicate that as the number of receive antenna elements

increases, the spectral efficiency will firstly increase, but after

reaching the maximum value (approximately around N = 8

in Figure 5), further increase in N would result in a decrease

of the achieved spectral efficiency. It is also worth noting that

when N = 5, the interelement spacing at the receiver side, dr,

is equal to λ/2, which probably is the most widely adopted

interelement spacing in practice. Thus, results in Figure 5

basically indicate that, by adding a few more elements and

squeezing the interelement spacing down from λ/2, it is

possible to achieve some increase in the spectral efficiency,

even in the presence of mutual coupling. But it is also

observed that such increase is limited and relatively slow as

comparedtothespatial-correlatedonlycase,andthespectral

efficiency will saturate very shortly.

4.2. Ergodic Spectral Efficiency of Compact MIMO with AS.

To study the performance of the ergodic spectral efficiency

with regard to the number of selected antennas nR for a

compact MIMO system using AS, we consider three typical

scenarios, namely, N = 5 in Figure 6, N = 8 in Figure 7, and

N = 12 in Figure 8. In each figure, we plot the performance

of the hybrid selection, FFT-based selection, and phase-

shift-based selection. Additionally, we also depict in each

figure the ergodic spectral efficiency of the reduced full-

complexity (RFC) MIMO, denoted as CRFC(M,nR), where

only nRreceive antennas are distributed in the linear array

and no AS is deployed. In Figure 6, it is observed that

Page 8

8EURASIP Journal on Wireless Communications and Networking

1

23

45678

3

6

9

12

15

Ergodic spectral efficiency (bits/s/Hz)

nR

Hybrid selection

FFT based selection

Phase-shift based selection

Reduced full system w/o selection

Figure 7: Ergodic spectral efficiency of a compact MIMO system

with AS, where M = 5 and N = 8.

CFC(M = 5,N = 5) > CPSS > CFFTS = CHS > CRFC(M =

5,nR), which in particular indicates the following:

(1) Soft AS always performs no worse than hard AS. The

phase-shift-based selection performs strictly better

than the FFT-based selection.

(2) With the same number of RF chains, the system with

AS performs strictly better than the RFC system.

Interestingly, these conclusions that hold for this compact

antenna array case are also generally true for MIMO systems

without considering the mutual coupling effect (e.g., [10]).

But a cross-reference to the results in Figure 5 can help

understand this phenomenon. In Figure 5, it is shown that

when N increases from 1 to 5, the ergodic spectral efficiency

ofthecompactMIMOsystembehavesnearlythesameasthat

of MIMO systems without considering the mutual coupling

effect.Therefore,itappearsnaturalthatwhenASisappliedto

the compact MIMO system with N ≤ 5, similar conclusions

can be obtained. It is also interesting that the FFT-based

selection performs almost exactly the same as the hybrid

selection.

Next we increase the number of placed antenna elements

to N = 8, at which the compact MIMO system achieves the

highest spectral efficiency (cf. Figure 5). We observe some

different results in Figure 7, which are CFC(M = 5,N = 8) >

CPSS> CFFTS> CHS. These results tell the following.

(1) SoftASalwaysoutperformshardAS.Thephase-shift-

based selection delivers the best performance among

all these three AS schemes.

(2) The phase-shift-based selection performs better than

the RFC system when nR

≤ 5. The FFT-based

1 2 3 4 5 6 7 8 9 1011 12

Ergodic spectral efficiency (bits/s/Hz)

3

6

9

12

15

nR

Hybrid selection

FFT based selection

Phase-shift based selection

Reduced full system w/o selection

Figure 8: Ergodic spectral efficiency of a compact MIMO system

with antenna selection, where M = 5 and N = 12.

1 2 3 4 5 6 7 8 9 101112

Ergodic spectral efficiency (bits/s/Hz)

3

6

9

12

15

nR

Phase-shift based selection (N = 5)

Phase-shift based selection (N = 8)

Phase-shift based selection (N = 12)

Reduced full system w/o selection

Figure 9: Ergodic spectral efficiency of a compact MIMO system

with the phase-shift-based selection, where M = 5.

selection performs better than the RFC system when

nR≤ 4. The advantage of using the hybrid selection

is very limited.

We further increase the number of antennas to N =

12. Now the mutual coupling effect becomes more severe,

and different conclusions are demonstrated in Figure 8. It is

observed that CFC(M = 5,N = 12) > CPSS≥ CFFTS> CHS,

indicating the following.

Page 9

EURASIP Journal on Wireless Communications and Networking9

(1) Soft AS performs strictly better than hard AS.

(2) The phase-shift-based selection performs better than

theFFT-basedselectionwhennR< 8.Afterthat,there

is not much performance difference between them.

Also, similar to what we have observed in Figures 6 and 7, in

terms of the ergodic spectral efficiency, none of the systems

with AS outperforms the FC system with N receive antennas

(andthusN RFchains).However,asfortheRFCsystemwith

only nR antennas (and thus nR RF chains), in Figure 8 we

observe the following.

(1) The RFC system always performs better than the

hybrid selection. The hybrid selection seems futile in

this case.

(2) The phase-shift-based selection performs better than

the RFC system when nR< 5. The benefit of the FFT-

basedselectionisverylimited,anditseemsnotworth

implementing.

This indicates that due to the strong impact of mutual cou-

pling in this compact MIMO system, only the phase-shift-

based selection is still effective, but only for a limited range

of numbers of the available RF chains. More specifically,

when nR< 5 it is best to use the phase-shift-based selection,

otherwise the RFC system with nRantennas when 5 ≤ nR<

8. Further increase in the number of RF chains, however, will

notleadtoacorrespondingincreaseinthespectralefficiency,

as demonstrated in Figure 5.

For the purpose of comparison, we also plot the ergodic

spectral efficiency of the phase-shift-based selection scheme

in Figure 9, by extracting the corresponding curves from

Figures 6–8. We find that by placing a few more antenna

elementsinthelimitedspacesothattheinterelementspacing

is less than λ/2, for example, N = 8 in Figure 9, the phase-

shift-based selection approach can help boost the system

spectral efficiency through selecting the best elements. In

fact, the achieved performance is better than that of the

conventional MIMO system without AS. This basically

answers the question we posed in Section 1 that is related to

the cost-performance tradeoff in implementation. However,

further squeezing the interelement spacing will decrease the

performance and bring no performance gain, as can be seen

from the case of N = 12 in Figure 9.

4.3. Spectral Efficiency CDF of Compact MIMO with AS. In

Figure 10, we investigate the CDF of the spectral efficiency

for compact MIMO systems with N = 8 . We consider

the case of nR

= 4 in (Figure 10(a)) and nR

(Figure 10(b)). We use dotted lines to denote the compact

MIMO systems with AS, and dark solid lines for the FC

compact MIMO systems (without AS). We also depict the

spectral efficiency CDF-curves of the RFC systems of N =

4 and N = 6 in Figures 10(a) and 10(b), respectively in

gray solid lines. As can be seen in Figure 10(a), soft AS

schemes, that is, the phase-shift-based and FFT-based AS

methods, perform pretty well as expected, but the hybrid

selection performs even worse than the RFC system with

N = nR= 4 without AS. When we increase nRto 6, as shown

= 6 in

68 10

Spectral efficiency (bits/s/Hz)

1214 16 1820

0

0.2

0.4

0.6

0.8

1

Hybrid

FFT

Phase-shift

FC (N = 8)

RFC (N = 4)

Empirical CDF of

spectral efficiency

(a) (nR= 4)

68 10

Spectral efficiency (bits/s/Hz)

12 14 161820

0

0.2

0.4

0.6

0.8

1

Hybrid

FFT

Phase-shift

FC (N = 8)

RFC (N = 6)

Empirical CDF of

spectral efficiency

(b) (nR= 6)

Figure 10: Empirical CDF of the spectral efficiency of a compact

MIMO system with AS. M = 5 and N = 8.

in Figure 10(b), the performance difference between hard

and soft AS schemes, or between the phase-shift-based and

the FFT-based selection methods, is quite small. But none of

these systems with various AS schemes outperform the RFC

system of N = nR= 6 without AS, which is consistent with

what we have observed in Figure 7.

These results clearly indicate that when the mutual

coupling effect becomes severe, the advantage of using AS

can be greatly reduced, which however, is usually very

pronouncedinMIMOsystemswhereonlyspatialcorrelation

is considered at both terminals, as shown for example in

Figure 11. On the other hand, it is also found that the

spectral efficiency of a RFC system without AS, which is

usually the lower bound spectral efficiency to that of MIMO

systems with AS (as illustrated by an example of Figure 11),

can become even superior to the counterpart when mutual

coupling is taken into account (as shown in Figure 10 for

instance).However,itshouldbenotedthatthisphenomenon

is closely related to the network model that we adopt in

Section 2. In such model, we have assumed that all the

antenna elements are grounded through the impedance

ZLi,i = 1,...,N, regardless of whether they will be selected

ornot.Thus,forMIMOsystemswithN receiveelementsand

with a certain AS scheme, the mutual coupling impact at the

receiversidecomesfromalltheseN elements,andisstronger

than that of a RFC system with only nRreceive elements.

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10EURASIP Journal on Wireless Communications and Networking

68 1012141618202224

Spectral efficiency (bits/s/Hz)

0

0.2

0.4

0.6

0.8

1

Empirical CDF of

spectral efficiency

Hybrid

FFT

Phase-shift

FC (N = 8)

RFC (N = 4)

(a) (nR= 4)

810121416182022 24

Spectral efficiency (bits/s/Hz)

0

0.2

0.4

0.6

0.8

1

Empirical CDF of

spectral efficiency

Hybrid

FFT

Phase-shift

FC (N = 8)

RFC (N = 6)

(b) (nR= 6)

Figure 11: Empirical CDF of the spectral efficiency of a compact

MIMO system with AS. M = 5, N = 8, and mutual coupling is not

considered.

5.Discussions

In our study, we also test different scenarios by varying the

length of linear array Lr, for example, we choose Lr = 3λ,

4λ, and so forth. For brevity, we leave out these simulation

results here, but summarize our main findings as follows.

Suppose the ergodic spectral efficiency of a compact

antenna array MIMO system saturates at Nsat. Our simula-

tion results (e.g., Figure 5) indicate that

Nsat>

?2Lr

λ

+1

?

, (20)

where ?·? rounds the number inside to the nearest integer

less than or equal to it. We also have nR< N for the sake of

deploying AS. Our simulations reveal that the interelement

spacing is closely related to the functionality of AS schemes.

For the cases we study, the conclusion is the following:

(1) Whendr≥ λ/2,bothsoftandhardselectionmethods

are effective, but the selection gains vary with respect

to nR. Particularly, the phase-shift-based selection

delivers the best performance among these tested

schemes.PerformanceoftheFFT-basedselectionand

the hybrid selection appears undistinguishable.

(2) When dr< λ/2, there exist two situations:

(a) When nR ≤ ?2Lr/λ + 1?, the selection gain

of the phase-shift-based selection still appears

pronounced, but tends to become smaller when

nR approaches ?2Lr/λ + 1?. The advantage of

using the FFT-based selection is quite limited.

The hybrid selection seems rather futile.

(b) When ?2Lr/λ + 1? < nR < Nsat, neither soft

nor hard selection seems effective. This suggests

that AS might be unnecessary. Instead, we can

simply use a RFC system with nR RF chains

by equally distributing the elements over the

limited space.

It is noted that in all these cases we examine, soft AS

always has a superior performance over hard selection. This

is because soft selection tends to use all the information

available, while hard selection loses some additional infor-

mation by selecting only a subset of the antenna elements.

Our simulation results also suggest that, if hard selection is

to be used, it is necessary to maintain dr ≥ λ/2. Otherwise,

the strong mutual coupling effect could render this approach

useless. Further, if the best selection gain for system spectral

efficiency is desired, one can place Nsator so elements along

the limited-length linear array, use ?2Lr/λ + 1? or less RF

chains, and apply the phase-shift-based selection method.

Therefore, it becomes crucial to identify the saturation point

Nsat. This in turn requires the electromagnetic modeling of

the antenna array that can take into account the mutual

coupling effect.

6.Conclusion

In this paper, we proposed a study of some typical hard

and soft AS methods for MIMO systems with closely spaced

antennas. We assumed antenna elements are placed linearly

in a side-by-side fashion, and we examined the mutual

coupling effect through electromagnetic modeling of the

antenna array and theoretical analysis. Our results indicate

that, when the interelement spacing is larger or equal to

one half wavelength, selection gains of these tested soft and

hard AS schemes will be very pronounced. However, when

the number of antennas to be placed becomes larger and

the interelement spacing becomes smaller than a half wave-

length, only the phase-shift-based selection remains effective

and this is only true for a limited number of available RF

chains. The same conclusions however, are not observed

for the case of hard selection. Thus it seems necessary to

maintain the interelement spacing no less than one half

wavelength when the hard selection method is desired. On

the other hand, if the best selection gain for system spectral

efficiency is desired, one can employ a certain number of

elements for which the compact MIMO system attains its

maximum ergodic spectral efficiency, use ?2Lr/λ + 1? or

less RF chains, and deploy the phase-shift-based selection

method. This essentially indicates, if the cost-performance

tradeoff in implementation is concerned, by placing a few

more than necessary antenna elements so that the system

spectral efficiency reaches saturation and deploying the

phase-shift-based selection approach, we can achieve better

Page 11

EURASIP Journal on Wireless Communications and Networking11

performance in terms of system spectral efficiency than the

conventional MIMO system without AS. Overall, our study

provides novel insight into the deployment of AS in future

generation wireless systems, including the 3GPP LTE and

LTE advanced technologies.

Acknowledgments

This material is based on research supported by the Air

Force Research Laboratory under agreement FA9550-09-

1-0576, by the National Science Foundation under Grant

CCF-0829958, and by the U.S. Army Research Office under

Grant W911NF-08-1-0449. The authors would like to thank

Dr. Dmitry Chizhik and Dr. Dragan Samardzija of Bell

Laboratories, Alcatel-Lucent for the helpful discussions on

the modeling and implementation issues.

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