Hindawi MIMO paper.pdf antenna selection

8/10/2019 Hindawi MIMO paper.pdf antenna selection

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Hindawi Publishing CorporationInternational Journal o Antennas and PropagationVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticleA Low Complexity Near-Optimal MIMO Antenna SubsetSelection Algorithm for Capacity Maximisation

Ayyem Pillai Vasudevan1 and R. Sudhakar2

Department of ECE, JCT College of Engineering and Technology, Coimbatore , India Department of ECE, Dr. Mahalingam College of Engineering and Technology, Pollachi , India

Correspondence should be addressed to Ayyem Pillai Vasudevan; ay [email protected]

Received May ; Revised September ; Accepted September

Academic Editor: Christoph F. Mecklenbrauker

Copyright A. P. Vasudevan and R. Sudhakar. Tis is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Multiple input multiple output (MIMO) wireless systems employ a scheme called antenna subset selection or maximising the datarate or reliability or the prevailing channel conditions with the available or affordable number o radio requency (RF) chains. Inthis paper, a low-complexity, and near-optimal perormance ast algorithm is ormulated and the detailed algorithm statementsare stated with the exact complexity involved or capacity-maximising receive-only selection. Te complexities o other receive-only selection comparable algorithms are calculated. Complexities have been stated in terms o both complex-complex ops andreal-real ops. Signicantly, all the algorithms are seen in the perspective o linear increase o capacity with the number o selectedantennas up to one less than the total number o receive antennas. It is shown that our algorithm will be a good choice in terms oboth perormance and complexity or systems, which look or linear increase in capacity with the number o selected antennas upto one less than the total receive antennas. Our algorithm complexity is much less dependent on the number o transmit antennasand is not dependent on the number o selected antennas and it strikes a good tradeoff between perormance and speed, which isvery important or practical implementations.

1. Introduction

Multiple input and multiple output (MIMO) wireless systemscan be used or increasing Shannon capacity, or decreasingbit error rate through, respectively, spatial multiplexing ordiversity. Te more the number o antennas, the more will

be the capacity and diversity order. But, regardless o spatialmultiplexing or diversity concepts, important difficulty inusing a MIMO system is an increased complexity and hencecost dueto the need o increased radio requency (RF) chains,which consist o power ampliers, low noise ampliers,downconverters, upconverters, and so orth.

Tis paper ocuses on maximising the capacity. Becauseo the high cost burden involved in RF chains, it is necessaryto have less number o RF chains, yet maximise the capacity.Tis is done by having a larger number o space links at ourdisposal and selecting the best as many number o links asequal to the number o the RF chains. Selecting the bestsubset links out o a larger number o links is obviously done

by having a larger number o antennas and selecting thebest subset o antennas corresponding to the best links. Teantenna subset selection can be at the transmit side or at thereceive side or at both sides. Tis paper is concentrating onthe selection at the receive side. For a system, which has

total receive antennas and

total transmit antennas, the

optimal way to select a subset oantennas or maximizingcapacity is to carry outdeterminant calculation times asrequired by the capacity ormula given by elatar [] and thenarrive at the highest capacity-giving antenna subset. Suchan exhaustive search method was used in [] or diversity

reception. Similar argument is applicable or transmit side

also. Surely, computations o determinants will becomeprohibitively large. o solve this complexity problem withminimal loss on the capacity perormance, suboptimalalgorithms have been developed.

Various capacity-based single-sided antenna selectionproblems have been discussed in the literature []. In [],


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International Journal o Antennas and Propagation

antenna selection was considered at the transmitter or a lowrank channel. In this paper, there was no mention o RFchain constraint concept with regard to antenna selection;rather, an optimization criterion was built up or maximizingthe capacity and the algorithm o antenna selection was lefas a uture direction. In [], an SNR-based criterion was

developed or antenna selection in transmitter side or spatialmultiplexing systems employing zero orcing-based linearreceivers. In [], the concept o antenna selection with respectto the RF chain was introduced by Molisch et al. Inthatpaper,an analytical bound or the capacity was derived. Te systemwas called hybrid selection MIMO (H-S/MIMO), where inthe case o standard diversity, it is called as hybrid selectionmaximum ratio combining (H-S/MRC). In [], a selectionrule or maximizing the average throughput was given ortransmit antenna selection o spatial multiplexing systems.A norm-based antenna selection algorithm or transmit sidewas suggested by Gore et al. or the Alamouti space timecoding system in []. In this, antennas o largest Euclidean

norms were selected, where it is to be noted that the norm-based algorithm can be very suboptimaly. Norm-based selec-tion can also be used or capacity criterion as suggested in[]. A high-capacity achieving suboptimal antenna selectionalgorithm, whose perormance in terms o capacity, is veryclose to that o the optimal one, and whose computationalcomplexity is not that promising, was given by Gorokhovin []. Tis algorithm, which comes under greedy search,started with the ull set o antennas and one by one deletedthe least capacity contributing antenna. Tis algorithm alsodirectly dealt with elatars exact capacity expression [].Tis expression is shown as () later. Te algorithm changedthe problem o nding determinant into nding an inverse

or the purpose o reducing the computational complexity.However, the inverse lemma used in the algorithm demandsa huge number o ops. Te complexity o this algorithm was

calculated to be(3+ 2) + (22) + (2) +(2) + (2) in [] or receive side selection.It can be noted that or receive side selection, it involvesnot only the total number o receive side antennas but alsothe total transmit antennas. Hence, this algorithm is com-putationally complex. A ollow-up to [] was made in []by Gharavi-Alkhansari and Gershman, who introduced anaddition-based greedy sub-optimal algorithm or receive sideselection. In that paper, the authors ollowed the procedureo starting with empty set o antennas and then added one

by one maximum capacity contributing antenna. Tey usedthe well-known capacity equation given as () later, or theiralgorithm. Te authors o [] also changed the problem onding determinants to nding inverse as done in [], butthe difference is that computation or nding the inverse wasdramatically reduced by using a lemma that nds the inverseby addition and matrix multiplication. A computationallyand constructionally simple algorithm was given by Molischetal.in[]. Tis algorithm does not directly deal with ()andcomes under the classication o ast selection sub-optimalalgorithms group, where, it is understood that this kind oalgorithms does not use directly () and surely there will bea small compromise on capacity. Clearly, this algorithm is

simple, but there is considerable penalty in terms o capacity.In this, correlation between each pair o the rows, where rowscorrespond to receive antennas, is ound and sequentially therst highest correlation antennas are deleted orselecting

antennas. Afer getting a particular high corre-

lation antenna pair, the antenna corresponding to the lower

norm row is deleted. In [], an antenna selection algorithmbased on animoto similarity was proposed. However, thisalgorithm is computationally complex as it involves severalmatrix multiplications. Rate adaptation and single transmitantenna selection were studied in [].

Various articles have been published on the antennaselection at both sides or capacity maximisation. Te paperhave been on decoupled transmit/receive (x/Rx) antennaselection and greedy joint selection. Te concept o both sideselection was rst proposed in []. Te authors suggesteddecoupling or separating selections at transmit and receivesides. Tey suggested optimal search separately on receiveantenna and transmit antenna sides reducing the otherwise

required calculation o

determinants to calcu-lation o + determinants without affecting thecapacity much. Further in [], the decoupling concept was

suggested in terms o greedy algorithm o [] at both sides.A greedy joint algorithm called efficient joint transmit andreceive antenna selection (EJRAS) algorithm was proposedor capacity maximisation in [].

Gharavi-Gershmans receive-side greedy algorithmachieves good capacity perormance, but its complexity isclearly a unction o the number o transmitting antennasand the number o selected antennas. Teir algorithm canbe claimed to be o low complexity only or the case o lowtransmitting antenna number case. Te complexity is o the

order o2, whereas reduction in will surely imposeconstraint on increase in, because i> , linearincrease in capacity is not possible. Tese issues are broughtout quantitatively in Sectionsand. Tough the algorithmproposed in [] is devoid o these constraints, the algorithmis considerably sub-optimal.

In this paper, the authors propose or receive side anaddition-based ast algorithm, which is considerably superiorin perormance to algorithm given in [] with only verysmall percentage increase in complexity as the simulationsand computation calculations show. Our algorithm achievesa capacity, that is, almost equal to greedy algorithm proposedin [] but, it is computationally independent o number

o selected antennas and very importantly, its complexityis much less dependent on the number o transmittingantennas unlike the one in []. Our algorithm is devoid o theconstraint imposed by greedyalgorithm o [] andis superiorin perormance to that in [].

Te paper has been organized as ollows.Section givesthe theoretical model or capacity-based antenna selectionat the receive side. In Section , the algorithm is developed,so that it takes better care o norm than that in [] and theproposed algorithm and its pseudostatements with complex-ity involved are stated. InSection , the computational com-plexities o all the above mentioned algorithms are discussedwith comparison.Section is on simulation considerations,


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InputSpace-

timeencoder

Space-time

decoderOutput

RFswitch

RF chain

RF chain

RF chain

RF chain

RF chain

RF chain

1

2 2

1

2

1

H

MT MRLR

F : A MIMO system with receive side antenna subset selection.

results and discussions. Section gives conclusion.able gives the complexity break-up o Gharavi et al. algorithm.able gives the complexity break-up o Molisch et al.algorithm.

Notations. All the bold and lower case letters reer to matrixvectors, whereas all the bold and upper case letters reer tomatrices. Superscript reers to Hermitian transpose. Teollowing denitions are applicable or the whole paper.

H

Channel matrix o

.

Number o available receive antennas.Number o available transmitting antennas. Symbol energy to noise power ratio.IUnit matrix odimension.

H,: th row vector oH.Te selected antenna subset.Te number o selected antennas.H,: Te sub-channel matrix ormed o rows

corresponding to selected antennas and all columnsoH.

Eigen values oHH

. Angle between two row vectors.h,h Inner product between vectorshand h. Channel capacity o MIMO system.h Euclidean norm o vectorh.2. The System Model

Te capacity o a MIMO system that uses receivingantennas and transmitting antennas ed o equal poweris given by

=log2det I +HH . ()Tis expression can be brought to () by using singularvalue decomposition (SVD). Consider =

=1

log + 1 bps/Hz, ()where is the number o nonzero eigen values. Hence,it suggests that depends on not only the number oindependent columns or the number o independent rows butalso on eigen value distribution. Simple independence is onlya weak requirement as ar as capacity is concerned. Capacity

can be actually maximized by having not just independentcolumns or rows but by having orthogonal columns or rows.Tat is, the eigen values o orthogonal columns or rows willensure that the capacity is maximized.

Antenna selection is an approach, which tries to extractmaximum benet out o the prevailing channel conditionswith the number o available or affordable RF chains. Iavailable RF chains are only, effective receive antennascan be only

in number. Since only

antennas are to be

selected, () will change to

select=log2det I + H,:HH,: . ()Te MIMO system with antenna subset selectionapproach or receive side is depicted inFigure .

3. The Proposed Algorithm

In this section, a new uncorrelation-based algorithm islogically brought or receiver side selection. Te expressionor the capacity with receive antennas can be written withthe application o SVD on it as given in ( ).

In [], Molisch et al. proposed an algorithm, based oncorrelation among the rows, or receive side antenna subset

selection. Te idea was to remove the correlated rows andretain as ar as possible maximally uncorrelated or optimallyorthogonal rows. Molisch et al. algorithm is simple and olow complexity, but the capacity perormance is considerablysuboptimal.

Te reason or the underperormance o Molisch et al.algorithm is that one o the two rows, which have highcorrelation between them and high individual norm valueswill get deleted. Tat is, the norm values are less accountedin that algorithm. But norm values matter as suggested by

product HH in (). Te correlation or nonorthogonalitybetween two rows is measured in terms o the inner producthh

. Te existence o high correlation among any two rows

will mean that the inner product among the rows is high.Let us consider two rows, hand hoH, where

H =[h1

h2h3...

h

] . ()Te rows h1,h2,h3, . . . ,h are o1 . Te inner

product between two rows, hand h, can be written as

hh

=h

h

cos

. ()


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sin

hj

hj

hi

hj

cos

F : Interpretation o inner product between hand interpre-tation o resolution ohalong hand perpendicular toh.

Tat is, correlation = hh cos , where = 1,2,3 , . . . , , = 1, 2, 3, . . . , .Figure describes the inner product.In this,h cos is the projection o h vector on h

vector. Hence, correlation is the norm o vectorhmultipliedby the component o h along h. Te resolution o h isshown inFigure . Te uncorrelation betweenhand h willbethenorm ohmultiplied by the component perpendicular

to h where the perpendicular ish sin . Tat is, theuncorrelation is given byuncorrelation=h h sin . ()Let us dene

hhcorrelation betweenhand hhh uncorrelation betweenhand h.

Ten,

2hh

+ 2hh

=h

2

h

2

cos2

+h

2

h

2sin2

=h2h2 cos2 +sin2=h2h2. ()Hence, the square o uncorrelation is given by2

hh=h2h2 2hh =h2h2 hh2. ()

Molisch et al. algorithm was about nding the correlationand deleting the rows corresponding to high correlation,where correlation between two vectors was dened to bethe inner product between the two row vectors. But theproblem is that high inner products may be due to high

values o norms o the vectors and not necessarily due tohigh values o cos alone, in which case deletion o any onevector will penalize the capacity o the system as the productHH

suggests in (). But, i the same algorithm is steeredto addition concept, such a scenario o deletion o high-norm rows will not happen. Because in addition concept,the uncorrelation between the rows is ound and those rowscorresponding to high values o uncorrelation are retained.Hence, high norm rows will be retained rather than deleted.

We have uncorrelation given by

2hh

=h

2

h

2

h

2

h

2cos2

. ()

In the ollowing, it will be seen that Molisch et al.algorithm does care about correlationaspectwell butdoes notgive due care or norm. For this, we take two pairs o rows, inwhich pair, is more correlated than pair by some percent,but the norm product o pair is more than pair by morepercent. Let us consider

(, )pair o rows and

(, )pair o

rows. Let1be between th and th rows and let 2be betweenth andth rows. Let square oh,hbe percent greaterthan square oh, hand let h2h2 be percent greaterthanh2h2. Tis means that the correlation between h and h is percent greater than the correlation between hand h. Hence, h and h are more correlated than h and

h. We will assume that each oh2 andh2 is greaterthan bothh2 andh2. Tis is a situation o squarenorms multiplication o two rows,andbeing greater thansquare norms multiplication o the other two rows,and,whereas the cos

1is lower than cos

2. Underthe abovestated

assumptions, the uncorrelation becomes

2hh = h2h2 + 0.02h2h2h2h2cos22 0.01h2h2cos22= h2h2 h2h2cos22+ 0.02h2h2 0.01h2h2cos22= 2hh+ 0.02h2h2

0.01

h

2

h

2cos2

2

.

()

Te bracketed part o () will be a +ve value because,

0.02h2h2 will be greater than0.01h2h2cos22.Hence, we see that2hh will be greater than2hh . Hence, iour algorithm is used, we can expect that the higher-normrow o, rows, whose individual norms are greater thanthe individual norms o, rows, will be selected. Due tothis phenomenon, the proposed algorithm will etch morecapacity than that done by Molisch et al. algorithm andwe see in the simulation this is the case. I Molisch et al.algorithm is applied, one amongth andth rows will bedeleted because h, h is higher than h, h. Tis will causea reduction in capacity because, among the rows

and

, one

may be deleted. Te proposed algorithm is stated next, and

its statement version with the complexity involved is statedinable . Consider the ollowing.

() Te channel vector h is dened as theth row oH,withbeing an element o the set= {1,2 , . . . , }.() I= , conclude = ; otherwise, do the steps

to .

() For all (with ), calculate square norm oh,h2. Te square uncorrelation2hh is denedas2

hh= h2h2 2hh ,2hh being the square

correlation. Te square correlation

2hh

is dened as

2hh

= square (

|h

,h

|).


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: Pseudostatements and complexity details o the proposed algorithm.

Sl. no. Algorithm statements Complex ops required Real ops required

= 1,2 ,3 , . . . , 2

8

4

i

<

or

= 1to

2=H,:2

end

i> 1 or = 1to

(2+ 2)(2 )2 (4 0.5)(2 ) or = 1to i > ,= H,: H,: ,=22 2, end

end

end

or = 1to , =argmax, , i H,:2 >H,:2 = ,:= 0 else

= ,:,= 0 end

end

else

=argmax

2

= end = 1,2 , , . . . , end

() I= 1, select(with ), that gives the largesth2, and conclude = {}; otherwise, do steps to.

() For all

and

,

> , compute the square correlation,

2hh = square (|h, h|).() For alland, > , compute the square uncorrela-tion,2hh= h2h2 2hh .

() For Loop, Consider the ollowing.

(a) Choose theand(with, , > ) thatgive the largest2

hh. Ih2 > h2, addto

, otherwise, and addto .(b) Delete

(or

) rom

.

(c) Go to Loop until

indices are in .

4. Complexity Analysis of the Algorithms

In this section, a detailed analysis has been made on thecomplexity o the proposed Molisch et al., and Gharavi-Gershman algorithms. For complexity analysis, the numbers

o multiplications and summations have been accounted.Te multiplications and summations are together called asoating point operations, abbreviated as ops. Te numbero ops demanded by each o the three is taken as thebenchmark or comparison o complexities o the algorithms.Firstly, complex-complex ops and then real-real ops arecalculated. Each complex-complex multiplication involvesour real multiplications and two real additions.complex-complex additions involve2-2real-real additions.

In the case o the proposed algorithm, or single antennaselection, the complexity will be just 2. For morethan , additionally, the inner product among all the rowsand uncorrelation are calculated. Only the lower triangular


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matrix without the diagonal elements o the correlationmatrix is required to be calculated. Te correlation matrix isas ollows:

11

12

1,

... d...

,1,2 ,. ()

Hence, the total complexity o the proposed algorithm is2 ++2 2, where it is to be noted that theexpression is not a unction oand hence the total numbero multiplications and summations is constant with respect to. I= , no selection is required and hence no multi-plications and summations are required. Te correspondingreal number ops or single antenna selection and more thanone antenna selection are, respectively,8 4 and42+ 4 0.52 3.5.

In the case o the algorithm given by Molisch et al.the number o complex-complex ops or selection o both

more than and equal to antenna is the same. Te numbero complex-complex ops is given by2+ (1/2)2(1/2), where the complexity is slightly less thanours. It is to be noted thatthe authors o [] did not makeanyprovision or reduction o complexity or single antenna case.Also, there was no provision to suspend the selection processwhen= .

Te corresponding real-real ops are42+422 2or single and more than one antenna selections.In the case o Gharavi-Gershmans greedy algorithm [], or

calculatingB matrix( 1)times,22complex-complexmultiplications and complex-complex summations have to becalculated

(

1)times. Similarly, a matrix, which consumes

22 + 4 complex-complex ops or one calculation,has to be calculated( 1) times. Further, needs to becalculated (1) times. Hence, total ops corresponding to is(2+ )( 1). Hence, the total complexityo Gharavi-Gershman algorithm is2 + (42+ 4)( 1) + (2+ )( 1).

Te corresponding real-real ops are, respectively, given

by8 4and 8 4 +(8 +142 +4+ 2 4+ 2 2)( 1)or single antennaand more than one antenna cases. Te problem with thisalgorithm is the computational complexity depends on thenumber o, not only receive antennas and transmit antennas,but also selected antennas. Also, the dependence on thenumber o transmit antennas and the number o selectedantennas is large in scale. Tis is to be expected becausethis algorithm depends on calculating the expression o ()or selecting the antennas. Te complex-complex ops andreal-real ops o each o the three algorithms have been,respectively, tabulated as ablesand.

It can be seen rom the complexity analysis discussion asollows.

(i) Te Molisch et al. algorithm complexity dependson only the number o physically present receiveantennas and the transmit antennas. Te dependenceon

is in

(

). Hence it does not vary with

number o selected antennas and the complexity isless dependent on.

(ii) Te complexity o the proposed algorithm dependson only the number o physically present receiveantennas and the transmit antennas. Te dependenceon

is in

(

). Hence, it does not vary with

number o selected antennas and the complexity isless dependent on. Tough the number o mul-tiplications and summations required is very slightlygreater than the Molisch et al. algorithm, this verysmall increase in complexity can be easily disregardedon considering the improvement in perormance interms o capacity given by ouralgorithm over Molischet al. algorithm.

(iii) Te calculation to be done by the algorithm oGharavi-Alkhansari and Gershman [] depends onthe number o selected antennas and the number otransmit antennas. Te dependence is o

(

)and

(2

). Hence, it is more computationally complex.

Te complexity is clearly a unction o and.Tis act clearly imposes a constraint on choice oand.5. Simulation Considerations and Results

It is known that the capacity o a MIMO system is pro-portional to the min(,). In the case o receive-onlyselection, the capacity will be proportional to min(,).For

greater than

, the capacity improvement will not

be on linear order, rather on logarithmic order. Hence, i theantenna selection has to provide a linear increase up to

equal to one less than, it is necessary to keep= .Te simulation carried out in [] assumed that= 4and= 16. Surely, this imposes a constraint on. Such aconstraint will impose discomort in capacity maximisation.Hence, the simulation is carried out or high values o,here in our case or= 16.is assumed to be equal to.

Te ollowing plots have been obtained or the proposedMolisch et al. and Gharavi-Gershman algorithms:

(i) the outage capacity in bits/s/Hz versus,(ii) the outage capacity in bits/s/Hz versus SNR in dB or

two different

,

= and ,

(iii) the cumulative probability density versus Instanta-neous capacity in bits/s/Hz or two different , =6and ,

(iv) the number o ops versus.Because o huge time involvement, the optimal way o

selection simulation or outage capacity versus the numberselected antennasplot has been limited to= = 8 case.Also, the random selection plot has been obtained only orthis case, because the authors elt that it is clear that randomselection is the poorest in terms o capacity perormance andhence it is not necessary to regard urther. Antenna selection


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: Complexity in complex ops or the proposed and other algorithms.

Sl.no.

Te name o algorithm Te complexity in complex ops or more than antenna For single antenna

Proposed 2+ + 2 2 Molisch et al. []

2

+

122

12

Same as theprevious column

Gharavi-Alkhansari and Gershman [] + (42 + 4) 1 + (2+ ) 1 2 : Complexity in real-real ops or the proposed and other algorithms.

Sl.no.

Te name o algorithm Te complexity in real-real ops or more than antenna For single antenna

Proposed 42+ 4 0.52 3.5 8 4 Molisch et al. [] 42+ 4 22 2 Same as theprevious column Gharavi-Alkhansari and Gershman [](16+ 142+ 4+ 2 8+ 2 2) 1 8 4

approach o increasing capacity is applicable or slowadingand quasistatic channels. For such kind o channels, outagecapacity concept is applicable rather than ergodic capacityconcept.

For all the simulations, a -channel average has beenobtained. Simulation was carried out or two types channels.One o the two was o Rayleigh at-ading type, where theelements oH are independent and identically distributed(i.i.d.) complex Gaussian o zero-mean and unitvariance; thatis, real and imaginary parts o H are o . variance It isassumed that the channel elements are uncorrelated; that is,there is no correlation at the transmit side antenna elementsor receive side antenna elements. Te other one type o the

two channels or which simulation was carried out was achannel o some amount o correlation. It was assumed thatthere was no correlation at the base station end and therewas correlation at the mobile station. Tis is based on theact that there can be easibility o maintaining sufficientdecorrelating antenna separation at the base station, whereasit is difficult to maintain the same at the mobile station. TeKronecker model was used or modeling the channel matrix.Te Kronecker model is as ollows:

H

=R

1/2 HR

1/2

, ()

whereR

Receive antenna correlation matrix o

,

R ransmit antenna correlation matrix o , HSpatially white channel.In our case, R is a unit matrix o order. Ris modelled in an exponential way as discussed in []. Temodel is as ollows:

R=

1 4 (1)2 1 d ...4 1 d4... d d d

(1)

2

4

1

, ()

where is the correlation coefficient between the adjacentantennas. It was assumed in our simulation setup that thecorrelation coefficient was .. In such a case, the correlationbetween nonadjacent antennas can be neglected and thecorresponding terms in the correlation matrix can be set to bezero. Consequently, the receive correlation matrix becomes

R= 1 0 0 1 d00 1 d0... d d d0 0 1

, ()where

= 0.2.

Itcan be concluded rom Figure that the outage capacityperormance with respect to o the proposed algorithmis superior to that o Molisch et al. algorithm. It is onlyslightly lower than the optimal one, and Gharavi-Gershmanalgorithm. Te plots have been obtained or the proposedalgorithm, the two other algorithms mentioned earlier, theoptimal one, and random one. Te norm-based algorithm isnot considered, as it is known that it is applicable or only lowSNR condition, though the complexity is very low. Figure describes the variation o outage capacity with the number oselected antennas or= 8,= 8, and SNR= 20 dB. Teplots have been obtained or random and optimal selectionin addition to the three algorithm-based selections. Figure

gives the simulation plot o outage capacity versus number oselected antennas or= 16,= 16, and SNR= 20 dB.Te variation o outage capacity in bits/s/Hz with the numbero selected antennas or= = 16 is tabulatedas able or comparison clarity. It can be seen in the table that thecapacity o the proposed algorithm is almost equal to thato Gharavi-Gershman algorithm and clearly superior to theoriginal Molisch et al. algorithm. Plot as shown inFigure has been obtained or cumulative probability density versusthe instantaneous capacity or two differentvalues, , and with= 16and= 16. It can be seen romFigure that the capacity distribution o our algorithm is superiorto Molisch et al. algorithm and the lowest capacity is only


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1 2 3 4 5 6 7 85

10

1520

25

30

35

40

45

Number of selected antennas

Outageca

pacity(bits/s/Hz)

Proposed

Molisch et al.

Gharavi-Gershman

Optimal

Random

Random

Molisch et al.

Optimal

Proposed

Gharavi-Gershman

Outage capacity versus number of selected antennas withMT = 8

F : Plots showing the variation o outage capacity withnumber o selected antennas or= = 8under i.i.d. channelcondition.

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

80

90


Outagecapacity(bits/s/Hz)

Outage capacity versus number of selected antennas

ProposedMolisch et al.Gharavi-Gershman

F : Plots showing the variation o outage capacity withnumber o selected antennas or= 16and= 16under i.i.d.channel condition.

slightly lower than that o Gharavi-Gershman. Plots as showninFigure have been obtained to see the relation betweenoutage capacity and SNR in dB. Under low SNR condition,the proposed algorithm and Gharavi-Gershman algorithmsperorm almost at the same level. However Molisch et al.

algorithm suffers. Tis suffering is more or high SNRs.Figure shows simulated plots describing variation o

outage capacity withor= = 16under correlatedchannel condition. It may be seen that the capacity reduces ingeneral. It may also be noted that the capacity perormance othe proposed algorithm slightly reduces. Since the proposedalgorithm alls under uncorrelation concept, the correlationslightly affects the perormance. Figure is on variation ocumulative probability density with instantaneous capacityor= = 16 and= 6 and under correlatedchannel condition. Figure is on variation o outage capacitywith SNR in dB or

=

= 16and

= 6and under

correlated channel condition.

35 40 45 50 55 600

0.1

0.2

0.30.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bits/s/Hz)

Cumulative

probabilitydensity

Proposed algorithm

Molisch et al. algorithm

Gharavi-Gershman algorithm

Gharavi-

Molisch et al.

ProposedProposed

Molisch

Gharavi-Gershman

Gershman

Cumulative probability density versus outage capacity forMT = 16

LR = 8

LR = 6

et al.

F : Plots showing the variation o cumulative probabilitydensity versus instantaneous capacity or= 16and= 16or

= 6and under i.i.d. channel condition.

0 2 4 6 8 10 12 14 16 18 205

10

15

20

25

30

35

40

45

50

55

SNR (dB)


Outage capacity versus SNR (dB)

ProposedMolisch et al.Gharavi-Gershman

LR = 8

LR = 6

F: Plots showing the variation o outage capacityversus SNRin dB or= 16,= 16, and= 6and under i.i.d. channelcondition.

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60



Outage capacity (bits/s/Hz) versus number of selected

Proposed algorithm

Molisch et al.

Gharavi-Gershman

antennas under correlation

F : Plots showing variation o outage capacity with numbero selected antennas or

= 16and

= 16under correlated

condition.


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: Variation o outage capacity withor= = 16.No. oant.

Capacity byGhar-Gersh in bps/Hz

Capacity byMolisch in bps/Hz

Capacity byProposed one in bps/Hz

Increase o Ghar-Gersh real-real ops over theproposed as a per cent o the proposed

Complex-Complex Real-Real

. . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . .

: Complexity break-up o Gharavi-Alkhansari and Gershman algorithm [].


= 1,2,3, . . . , B= I or

= 1to

8

4

=H,:

2

end or = 1to =argmax = I < a= 1/ + BH,: (22 + 4)( 1) (62+ 2)( 1) B= B aa (22)( 1) (82 2)( 1) or all

(2

+

)

1 8

4

2 (

1)

=

a

H

,:

2

end

end

end

= 1,2,3, . . . , Figure is on the number o real-real ops versus the

number o selected antennas. As seen in the gure, thenumber o ops demanded by the proposed algorithm isonly very slightly greater than the Molisch et al. algorithm.However, the perormance gain is o a signicant amount asseen inFigure to Figure . As seen inFigure , Gharavi-Gershman algorithm complexity increases almost linearly

with. Columns and oable show complex-complexand real-real op increase witho Gharavi-Gershman asa percent complexity o the proposed algorithm. Practically,real-real ops matter rather than complex-complex ops.Clearly, our algorithm perorms well both in perormanceand complexity. Our algorithm will be a good choice orsystems, which need to have liberty on

up to

1


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: Complexity break-up o Molisch et al. algorithm [].


= 1,2 ,3 , . . . , or

= 1to

2

8

4

2

=H,:

2

end or = 1to

(2 1)(2 )2 (4 2)(2 ) or = 1to i > ,= H,: H,: end

end

end

or = 1to , =argmax

,

, i

H,:

2

>H,:

2

= , := 0 else = , := 0 end

end

=

20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Instantaneous capacity (bits/s/Hz)

Cumulativeprobabilitydensity

Cumulative probability density versus instantaneous

Proposed

Gharavi-Gershman

Molisch et al.

capacity in correlated condition

LR = 8LR = 6

F : Plots showing the variation o cumulative probabilitydensity versus instantaneous capacity or= 16,= 16and= 6and under correlated.with linear increase in capacity with regard to increase in.6. Conclusion

In this paper, a low complexity algorithm was ormulatedand the detailed steps o the algorithm were stated with thecomplexity involvement. Te complexities o other existing

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

2530

35

40

SNR (dB)

Outagecapacity

(bits/s/Hz)

Outage capacity versus SNR (dB) in correlated condition

Proposed

Molisch et al.

Gharavi-Gershman

LR = 8

LR = 6

F: Plots showing the variation o outage capacityversus SNRin dB or,= 16,= 16and= 6and under correlatedcondition.algorithms were calculated. For all the three algorithms,complex-complex ops and real-real ops were calculatedand a comparison was done. Te perormances o all thealgorithms were seen in i.i.d. and correlated channel condi-tions.Our algorithm is balanced in terms o perormance andcomplexity. Our algorithm will be a good choice or systemswhich need to have liberty on

up to

1with linear

increase in capacity with regard to increase in

.


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0 5 10 150

1

23

4

5

6

7

Number of selected antennasRea

l-rea

lopsve

rsusse

lectedantennas Number of real-real ops versus number of selected antennas

Molisch et al.

Gharavi-Gershman

Proposed

MT = 16. Complexity increases

almost linearly beyondLR = 4

MT = 10. Complexity increases

almost linearly beyondL R = 5

104

F : Plots showing thevariation o real-realops as a unctionothe number oselectedantennasor= 16 andand= 16.

Acknowledgments

Te authors are very grateul to the reviewers or the useuland constructive comments they have given afer goingthrough the paper thoroughly. Te comments have urthermotivated the authors towards research and urthered theirresearch aptitude.

References

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[] J. H. Winter and M. Z. Win, Hybrid-selection/opimumcombining, in Proceedings of the IEEE Vehicular TechnologyConference (VTC ), pp. , .

[] D. A. Gore, R. U. Nabar, and A. Paulraj, Selecting an optimalset o transmit antennas or a low rank matrix channel, inProceedings of the IEEE Interntional Conference on Acoustics,Speech, and Signal Processing (ICASSP ), pp. , June.

[] . R. Ramya and S. Bhashyam, Rate adaptation in MIMOantenna selection system with imperect CSI, in Proceedingsof the nd International Conference on Communication Systemsand NETworks (COMSNETS ), January .

[] R. W. Heath Jr., S. Sandhu, and A. Paulraj, Antenna selectionor spatial multiplexing systems with linear receivers, IEEECommunications Letters, vol. , no. , pp. , .

[] A. F. Molisch, M. Z. Win, and J. H. Winters, Capacity oMIMO systems with antenna selection, inProceedings of theInternational Conference on Communications (ICC ), pp. , June .

[] D.A. Gore, R. W. Heath Jr., and A. J.Paulraj, ransmit selectionin spatial multiplexing systems,IEEE Communications Letters,vol. , no. , pp. , .

[] D. Gore and A. Paulraj, Space-time block coding with optimalantenna selection, in Proceedings of the IEEE InterntionalConference on Acoustics, Speech, and Signal Processing (ICASSP), pp. , May .

[] S. Sanayei and A. Nosratinia, Antenna selection in MIMOsystems,IEEE Communications Magazine, vol. , no. , pp., .

[] A. Gorokhov, Antenna selection algorithms or MEA trans-mission systems, in Proceedings of the IEEE InternationalConference on Acoustic, Speech, and Signal Processing (ICASSP), pp. , May .

[] M. Gharavi-Alkhansari and A. B. Gershman, Fast antennasubset selection in MIMO systems, IEEE Transactions on SignalProcessing, vol. , no. , pp. , .

[] A. F. Molisch, M. Z. Win, Y.-S. Choi, and J. H. Winters,Capacity o MIMO systems with antenna selection, IEEETransactions on Wireless Communications, vol. , no. , pp., .

[] Y. Haien and L. Guangjun, Novel antenna selection algorithmbased on animoto similarity,Journal of Systems Engineeringand Electronics, vol. , no. , pp. , .

[] A. Gorokhov, M. Collados, D. Gore, and A. Paulraj, rans-mit/receive MIMO antenna subset selection, in Proceedingsof the IEEE International Conference on Acoustics, Speech, and

Signal Processing (ICASSP ), pp. , May .[] C.-E. Chen, A computationally efficient near-optimal algo-

rithm or capacity- maximization based joint transmit andreceive antenna selection,IEEE Communications Letters, vol., no. , pp. , .

[] G. soulos,MIMO System Technology for Wireless Communica-tions, CRC Press, aylor and Francis Group, .


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