Receive Antenna Selection Using ConvexOptimization for MIMO Systems

FengGang SunI,2

ISchool of Information Science and Engineering,Shandong University, 250100

Jinan, Chinasfg1982@mail.sdu.edu.cn

Abstract-In this paper, a receive antenna selection algorithmbased on the theory of convex optimization is proposed toimprove the system performance over Rayleigh fading multiple­input multiple-output (MIMO) channels. The algorithm is basedon the aim of minimizing error rate, and by relaxing the antennaselection variables from discrete to continuous, we arrive at astandard semidefinite convex problem that can be solved veryefficiently. The Monte-Carlo simulations show that the algorithmproposed can provide the performance very close to that of theoptimal selection based on exhaustive search.

Keywords-convex optin,ization; MIMO systems; antennaselection; jading channels.

I. INTRODUCTION

Wireless communication systems using multiple antennas atthe transmitter and the receiver, known as MIMO, can offerimproved system performance, such as the system capacity orbit error rate (BER) in a scattering wireless environment, thushave attracted considerable attention over the past decade [1,2].Along with the improvement of MIMO systems, however,comes the problem of hardware complexity and cost due to thereason that the hardware chain is needed on everytransmitter/receiver. In order to overcome this problem,antenna selection (either at one or at both link ends) scheme isproposed in [3], which is a low-cost and low-complexityalternative that can capture many of the advantages of MIMOsystems. The basic idea of antenna selection is to choose thebest subset of all the antenna combinations at the transmitteror receiver which can achieve the best system performanceaccording to some selection criterion. This method can reducethe number of RF chains and thus lead to a significantsimplification of the MIMO systems. By doing so, theperformance of the MIMO systems increases notably whilewith only negligible hardware complexity.

So far, an intensive research in antenna selection algorithmsfor MIMO systems has been found in a variety of literaturesfor the sake of choosing the appropriate antenna subset. Theoptimal antenna selection algorithm which is based on anexhaustive search of all possible combinations for the one thatgives the best BER or capacity performance exists in [3, 4, 7].However, in such algorithms, the computational complexitybecomes one of the most challenging problems. For MIMOsystems wit NT transmit and NR receive antennas, nT out

This work was supported by National Natural Science Foundation ofChina (60572105), open research fund of National Mobile CommunicationsResearch Laboratory (W200802), the State Key Lab. of Integrated ServicesNetworks (ISN7-02), the Program for New Century Excellent Talents (NCET­0~-0582) in University, and Natural Science Foundation of ShandongProvince (No.Y2007G04).

Ju LiuI, HongJi Xu l and Peng Lan i

2State Key Lab. of Mobile CommunicationsSoutheast University, 210096

Nanjing, Chinajuliu@sdu.edu.cn

ofNT transmit antennas and nR out of NR receive antennas are

selected respectively, which requires the order of(::X:R) computations of determinants, which will become

computationally prohibitive quickly as the number oftransmit/receive antennas grows. Therefore, it is important todevelop low-complexity antenna selection algorithms whichcan provide good performance. A series of selectionalgorithms have been proposed to reduce the computationalcomplexity. The simplest selection algorithm is known as thenorm based selection (NBS) algorithm [5, 6]. This algorithmaims to choose the receive antennas corresponding to the rowsof the channel matrix with the largest Euclidean norm.Although NBS algorithm has a very low complexity, itinduces much system performance loss as compared with theoptimal selection method. The authors in [8] proposed themaximization of the minimum received post-processing SNRas a selection criterion for spatial multiplexing MIMO systemswith linear receivers (Zero Forcing (ZF) or Minimum MeanSquare Error (MMSE)) for the aim of minimizing BER. In [9],the receive antenna selection problem was formulated as anoptimization problem with integer variables which makes ithard to solve. By relaxing the selection variables from discreteto continuous, the problem is then transformed into a convexoptimization problem with continuous constraint. Effectivenumerical methods such as interior point method can then beused to solve it with low complexity. However, [9] onlyconsidered the case of capacity maximization.

In this paper a new receive antenna selection algorithm isproposed. It is based on the criterion of maximizing theminimum singular value of the channel matrix of MIMOsystems. By relaxing the antenna selection variables fromdiscrete to continuous, we arrive at a standard semidefiniteconvex problem which can be solved by some effectivenumerical methods.

The rest of this paper is organized as follows. In Section II,the system model is introduced, while the problem formulationand solution are provided in Section III. Then, Section IVgives some simulation results. Finally, in Section V, weconclude our work.

Notation: Bold uppercase and lowercase letters denotematrices and vectors respectively. Matrix transposition is

denoted by (-)T , whereas the conjugate transposition is

III. RECEIVE ANTENNA SELECTION IN MIMO SYSTEMS

In this Section, we focus on receive antenna selectionalgorithm to improve the system BER performance, thesealgorithms can also be used in transmit antenna selection.

denoted by (-)H . The sets of real numbers, nonnegative real

numbers and complex numbers are IR, IR+ and cerespectively. The set of all complex k x 1 vectors, M x N

matrices are represented by Ck and CMxN respectively. AnN x N identity matrix is denoted as IN. Denote A ~ 0 ifA is a

symmetric matrix and all the eigenvalues ofA are non-positive,and A is known as a positive semidefinite matrix.

(3)~ max EH[(f.11 +~HHH)-I]EIIEI12 =I NT NT

P "H" -I=Amax (f.1I NT +"NH H)T

=~A. (HHH) (4)NT mlO

Equation (4) proves that the performance of MIMO systemswith linear receivers mainly depends on the minimum singular

value of the channel matrix H. Therefore we can choose the

Performance of spatial multiplexing MIMO systems withlinear receivers depends on the minimum received SNRinduced by the particular subset of receive antennas [7].Maximum post processing SNR or maximum minimumsingular value of the channel matrix is the antenna subsetselection criterion proposed for the aim of improving thesystem BER performance. For MIMO systems with linearreceiver such as ZF or MMSE receiver, the post processing

SNR of the k th substream can be expressed as

1SNRk = f.1 (2)

[(f.1I N +NP HHH)-I]k,kT T

where iI E cenRxNT is the channel matrix corresponding to theparticular subset after antenna selection. P is the average

received SNR. The value of f.1 depends on the linear receivers

chosen. f.1 = 0 if ZF receiver is used while f.1 = 1 when MMSE

receiver is used.Let A(A) denotes the eigenvalue of matrix A ,

Amax (A) and Amin (A) represent the maximum and minimum

eigenvalues ofmatrix A , respectively.Due to the scenario that the BER performance mainly

depends on the substream with the worst case, the minimalpost processing SNR is the key parameter that determines thesystem performance. The antenna subset can be selectedsatisfied with the condition that the minimal post processingSNR is maximal.

According to Rayleigh-Ritz theorem, we get

max[(,uINr

+ : HHH)-lk!T

P "H"SNRmin ~Amin(f.1INT +"NH H)-f.1

T

-I P "H"=Amin(f.1IN +-H H)T NT

where ek is the k th column of I NT ' E is the eigenvector

corresponding to the eigenvalue of matrix (f.1IN +~HHH) .T NT

According to the results obtained from (3), the equation of (2)can now be lower bounded as

outputSignal

processingand

decoding

II. SYSTEM MODE

Figure 1. Receive antenna selection model

input

Consider a MIMO system with NT transmit and NR

receive antennas as shown in Fig.I. We suppose that the samenumber of RF chains as the antennas elements are employed atthe transmitter, whereas nRC «NR) RF chains can be used at

the receiver. That is to say that at each transmission epoch, nR

receive antennas are picked for signal reception. The channelis assumed to be flat Rayleigh fading with additive whiteGaussian noise (AWGN) at the receiver. The relationshipbetween input and output for a MIMO system with NT

transmitters and NR receivers can thus be represented as:

r(k)= ~HS(k)+n(k) (1)~N;

where r(k) =['i (k),r2(k), ...,rNR (k)]T ECNR is the received

signal vector that represents the kth sample of the signals

collected at the NR receive antennas,

s(k) = [SI (k),S2 (k), ... ,SNT (k)]T E ceNT is the transmitted signal

vector that represents the k th sample of the transmitted fromthe NT transmit antennas.

n(k) = [nl (k),n2(k), ... ,nNR (k)]T EceNR is zero mean additive

noise with unit energy, p is the average signal-to-noise ratio

(SNR) at each receive antenna. HE CNRXNT is the channelmatrix where hij{i = I, ...,NR ,) = I, ...,NT ) is the path gain

between the }th transmit antenna and the i th receive antenna.

We assume that the perfect channel state information (CSI) isknown at the receiver while performing antenna subsetselection. No CSI is assumed at the transmitter.

The optimization problem (9) is a convex optimizationproblem with linear constraints. However, the problem can notbe directly solved very efficiently. By inducing a newvariable t , the optimization problem (9) can then be simplified.

Let t=Amin (iJ:HiJ:) , then we have tI-iJ:HiJ: ~ 0 that the

optimization problem should follow. Maximizing the

minimum eigenvalue of the matrix iIH iI equals themaximum of t . Then the equivalent problem can be denotedas follows:

maxmize

antenna subset whose corresponding channel matrix iI satisfied

witheS =arg ~Amin {n:Hn:}.This selection algorithm can be solved by exhaustive search

of all possible combinations. When nR out of NR ( NR ~ nR )

receive antennas are selected, this algorithm is effective whilethe number of receive antennas is not very large. However,computational complexity quickly becomes prohibitive whenthe number of receive antennas grows. In order to overcomethis problem, the algorithm is simplified by using convexoptimization [10].

Let hi denote the i th row of the channel matrix H , the

matrix h;h i corresponding to the i th receive antenna can be

denoted as:

NR

subject to tIN - L x.h1!h. ~ 0T i=1 I I I

o~ Xi ~ 1, i =1,2,. ··,NR (10)

The matrix of H H H of all the NR receive antennas can beNR A A

expressed as HHH = L h1!h. . The matrix HHH of thei=1 I I

nR selected receive antenna subset can then be represented as:

IV. SIMULATION RESULTS

In this section, we present the simulation results thatvalidate the simplified selection algorithm of MIMO systemsvia Monte Carlo simulations. In order to compare the BERperformance of the proposed algorithm with other existingalgorithms, we show the performance of various algorithms:the optimal selection using exhaustive search, simplifiedalgorithm using convex optimization, norm based selection(NBS) algorithm and no selection strategy. Assume that thechannel is independent quasi-static Rayleigh flat fading andthe modulation scheme is QPSK. The number of the transmitantennas is 2.

Fig.2 depicts the BER performance comparison between thecase using antenna selection and the case without antennaselection as a function of the received SNR per each antennaunder independent channel. The simulation result shows thatthe BER performance with antenna selection is much betterthan that without antenna selection. At the BER of 10-2

, the

NR

L Xi = nRi=I

We can see that the term on the left hand side of the linearmatrix inequality (LMI) constraint in problem (10) is a non­positive semidefinite matrix, so the problem becomes asemidefinite programming (SDP) problem [10]. However, theLMI constraint is complex, which makes the problemcomplicated. So the transformation of the constraint from acomplex one into a real one is needed. We can transform thecomplex SDP problem into a real one by using the fact [10]that

X ~ 0 ¢::> [mx -3X] ~ 0 (11)3X mx

where X is a complex Hermitian matrix. mx and 3X are thereal and imaginary parts of X, respectively.

Apply (11) into the problem (10), the complex SDPproblem is transformed into a real one. Then the problem canbe solved by efficient methods [10].

From the fractional solution Xi (i = 1, ..., NR) of the problem,

the receive antennas with indices corresponding to thenR largest Xi are selected.

(5)

(9)

(8)

(7)

(6)

o~ Xi ~ 1, i = 1,2,. ··,NR

h~hil Ihi2r h~hiNr

h~rhil h~rhi2 ... IhiNr l2

maximize

subject to

Define xi(i = I, ...,NR ) as:

x. ={ 1, ith

receive antenna selected

I 0, otherwise

HHH = L h1!h.iECs I I

Using (7), Equation (6) can be simplified as:A H A NR H

H H= I x.h. h.i=I I I I

However, the variables Xi (i = 1, ..., NR) are binary valued (0

or 1) integer variables that make the selection problem hard tosolve. We can simply the problem by relaxing the integerconstraints and allowing Xi E [0,1] . Thus the problem of

receive antenna selection for maximizing the minimum

eigenvalue of the matrix HH H is approximated by thefollowing constrained convex relaxation plus rounding scheme:

(

A H A )Amin H H

BER Versus SNR

NBS scheme,SNR=OdB

proposed scheme,SNR=OdB

NBS scheme,SNR=4dB

proposed scheme,SNR=4dB

NBS scheme,SNR=8dBproposed scheme,SNR=8dB

- - - -/ - - - -I - - - -I - - - -__ J _~~]~~~

_===3=::=:::: ===----/---

-I----j---

__ J I _

~~3~~~~1~~~

==::::1====1===- - - --t - -j - - - -I - - - -I - - - -- - - , - - - ~ - - - -I - - - -I - - - -

! 1 1 1

10.1\'

10.2

10.3

C!Wco

10-4

10.5

10.62

REFERENCES

[1] T. L. Marzetta and B. M. Hochwald, "Capacity of a mobile multiple­antenna communication link in Rayleigh flat fading," IEEE Trans.Inform.Theory, vol. 45, 1999, pp. 139-157

[2] G.1. Foschini and M. J. Gans, "On limits of wireless communications ina fading environment when using multiple antennas," Wireless PersonalCommun., vol. 6, 1998, pp. 311-335.

[3] Molisch, A.F., "MIMO systems with antenna selection - an overview,"Radio and Wireless Conference, 2003. RAWCON'03. Proceedings 10­13, Aug. 2003, pp.167 -170.

[4] Sanayei, S.; Nosratinia, A.; "Antenna selection in MIMO systems,"Communications Magazine, IEEE, Volume 42, Issue 10, Oct. 2004 pp.68 -73.

BER Versus Number of selected antennas

V. CONCLUSION

In this paper, a receive antenna selection algorithm isproposed for MIMO systems, which is based on the analysisof the BER performance. By transforming the problem into aconvex optimization problem, it can be solved efficiently.Through the simulation results, we further demonstrate thatthe performance is nearly close to that of the optimal.

Figure 4. BER performance of the proposed scheme and NBS scheme when

NR =10 and nR is selected from 2 to 10

In Fig.4, we plot the performance curves of the proposedantenna selection algorithm and the NBS algorithm versus thenumber of antennas selected at the particular SNR value. Fromthe figure, we can see that the BER performance improvesgreatly when the number of selected antennas increases.However, the increasing speed becomes slower when nR is

becoming larger. It is obvious that when nR is fixed, BER

performance improves greatly as the SNR grows, and thisincreasing speed becomes larger when the SNR increases. Wecan also see that the performance gap between the proposedscheme and the NBS scheme becomes smaller as nR grows

and this gap becomes zero when all antennas are selected. Thisis because when more antennas are selected, the two schemeshave a higher probability to select the same antennas.

4 5 6 7 10the Number of Selected antennas

1614

Random Selection

NBS scheme

optimal algorithm

proposed algorithm

12

Random Selection

NBS scheme

optimal algorithm

proposed algorithm

8 10SNR (dB)

---l.----I I

===E===±====r===i= __===r===r===J___ 1- 1. J_

I I I

===±:===±====±:======±:===±===:±===---t----+-----t------1---1---1--- ----1---1- -1--- -===~===±===~===~===

===E===i===~===3======~===+===~===~=-= ­---I---T---'---~---~-

---T---T---l---~---~--

BER versus SNR with NT =10, N R =15, nR =14

BER versus SNR with NT =2, N R =5, nR =3

---1---

---1---1 1

---1---1

1

I11 1 1

---t----t----+--===;====+:==-+==---'---I---T---::::::::::::L:::::::::::::I:::::::::::::L::::::::::::

___ ~ ~ l _I I I

---r --r---T---I--I I 1 Ir --r---T---I- -1 1 1 II I I I

10.6 L--_-----'---_------L__---"----_----..L__-l-_---L-_-"''--_.

o 8 10 12 14 16SNR (dB)

~wco

Figure 3.

Figure 2.

BER Versus SNR

10·1E~T~~~~~~~~~;r::;;;:::~~;;::::;;;:::~~;;::::;;;:::d

In Fig.3, we repeat the above experiment with large antennanumbers at transmitter and receiver end. The difference is thatin Fig.3 we select 14 out of 15 receive antennas while weselect 3 out of 5 receive antennas in Fig.2. The simulationresult also shows the great improvement in BER performancewhen the proposed antenna selection algorithm is used. Fromthe figure, we can see that when more antennas are installed atthe transmitter or receiver, the system performance can beimproved sharply. The proposed algorithm can obtain muchmore performance gain than the random selection algorithm inthe range of all SNRs while the NBS algorithm can onlyprovide negligible improvement compared to the randomselection algorithm..

NBS scheme and the proposed scheme achieve more SNRgain of2.5dB and 4 dB than the case without applying antennaselection respectively. It is observed that the proposed schemeobtains nearly the same performance as the optimal selectionalgorithm with only negligible performance loss.

[5] Molisch, A.F.; Win, M.Z.; Winters, J.H., "Capacity of MIMO systemswith antenna selection," Communications, 2001. ICC 2001. IEEEInternational Conference on, Volume 2, 11-14 June 2001, pp.570 - 574

[6] Zhuo Chen; Jinhong Yuan; Vucetic, B.; Zhendong Zhou; "Performanceof Alamouti scheme with transmit antenna selection," ElectronicsLetters, Volume 39, Issue 23, 13 Nov. 2003, pp. 1666 - 1668.

[7] R. Heath, S. Sandhu, and A. Paulraj, "Antenna selection for spatialmultiplexing systems with linear receivers," IEEE Commun. Lett., vol. 5,no. 4, Apr. 2001, pp. 142-144

[8] R.W. Heath and A. Paulraj, "Antenna selection for spatial multiplexing

systems based on minimum error rate," IEEE International Conference

on Communications, vol. 7, June 2001, pp. 2276-2280.

[9] A. Dua, K. Medepalli, and A. Paulraj, "Receive antenna selection in

MIMO systems using convex optimization," IEEE Trans. Wireless

Commun., vol. 5, no. 9, Sep. 2006, pp. 2353-2357

[10] S. Boyd and L. Vandenberghe, "Convex Optimization."Cambridge,U.K.: Cambridge Univ. Press, 2004.