fast MIMO antenna selection

8/13/2019 fast MIMO antenna selection

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Fast Receive Antenna Selection

for MIMO SystemsJiansong Chen, Xiaoli Yu

Department of Electrical Engineering, University of Southern California, Los Angeles

Abstract- For MIMO systems with large number of transmit

and receive antennas, a major concern is the requirement for

the hardware complexity and computational cost. A promising

solution is to employ antenna subset selection. In this paper, we

focus on antenna subset selection at the receiver where the

antennas are selected to maximize channel capacity. It is shown

that our approach has lower computational complexity than the

existing receive antenna selection techniques while achieving the

similar outage capacity as the optimal selection algorithms. In

particular, the dominant complexity in recursions has been

lowered in two orders of magnitude.

I. INTRODUCTION

Earlier work revealed the potentially large capacities ofMIMO (multiple-input multiple-output) systems provided

from using transmit and receive antenna arrays [1]. This fact

continues to attract attention to develop techniques for

achieving such significant improvement in capacity. In [2]-[3], it has been shown that the extra degrees of freedom

afforded by the multiple antennas can be used to increase the

data rate of MIMO systems through spatial multiplexingtechniques. However, the hardware complexity may be

prohibitive for deploying large number of antennas in MIMO

systems due to the cost of multiple RF chains associated withmultiple antennas. Receive antenna selection is a promising

approach to alleviate the hardware cost while retaining thehigh levels of capacity, where only a small number ofantennas selected from the total available antennas perform at

the receiver. Compared with the systems without any

selection, the systems employing antenna selection has beendemonstrated to suffer a small loss in the capacity if the

receiver chooses an appropriate subset of the available

receive antennas [1].

The optimum solution to antenna selection is by exhaustivesearch. Because it is really time consuming, great effort has

been made to find the simplified approximate solutions.

Previous studies have shown that the norm-based selection

algorithm, where the systems select the receive antennascorresponding to the rows in the channel matrix with largest

possible Euclidean norms [4], has very low complexity andoptimal performance for some special cases. However, insome other circumstances, this method may result in a

significant capacity loss.

A novel antenna selection approach is proposed byGorokhov [5]-[6], which is sub-optimal with respect to the

maximum Shannon capacity criterion. In [5], Gorokhov

suggested a decremental selection algorithm where,beginning with full set of antennas, one antenna is removed

per step so that at each step the capacity loss is minimized.

Further work in [6] proposed an incremental selectionmethod which led to a lower complexity. In [7] Gharavi-

Alkhansari proposed a fast antenna selection algorithm for

incremental selection to maximize channel capacity, whichwas shown to be much faster than Gorokhovs approaches

and have better performance than norm-based selection

method. However, nodecremental selection with the sameorder of complexity was presented in [7]. Evidently, in many

scenarios, decremental selectionis indispensable.In this paper, we propose two novel algorithms for both

decremental andincremental selection to maximize channelcapacity. The proposed algorithms have much lower

computational complexity than that of the Gorokhovs

approaches and can achieve exactly the same capacity as thesub-optimal methods of [5]-[7].

II. MODELOFMIMOCHANNEL

Consider a MIMO system as depicted in Fig.1, which

employs transmit andN Mreceive antennas. The channel isassumed to be flat-fading. Then, the data model has the form

given by [6]

][][][ knkHsEkx s , (1)

where TN ksksksks ][]...,[],[][ 21 (2)

TM kxkxkxkx ][]...,[],[][ 21 (3)

TM knknknkn ][]...,[],[][ 21 (4)are the kth samples of the transmitted signals, the receivedsignals and the additive white Gaussian noise (AWGN) with

energy (2

0N ) and denotes the average signal energy per

antenna.

sE

H is the channel matrix and

represents a scalar channel between the ith receive

and the transmit antennas.

NM),( jiH

jth

We assume that H is a random channel unknown to thetransmitter and perfect channel state information (CSI)

available at the receiver. Then, the capacity of the

MIMO channel specified in (1) is given by

)(HC

HHRNEIHC HsssN 02 /detlog)( (5)where is the

NI NN identity matrix, H

ss ksksER ][][

is the covariance matrix of the transmitted signals

and NRtr ss )( , superscript represents the HermitianH

17770-7803-8622-1/04/$20.00 2004 IEEE


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transpose, and denote trace and determinant of a

matrix, respectively.

tr det

III. FASTRECEIVEANTENNASELECTION

The application of receive antenna selection is shown in

Fig. 1, where M receive antennas are selected fromallM available antennas ( MM ). Then, the capacity

associated to the selected channels is select

H

selectsssNselect HHRNEIHC 02 /detlog)( (6)

where channel matrix is formed by strikingselectH

MM rows fromH. To optimally select the M receiving antennas, the channelcapacity (6) has to be computed over all subsets.

When

M

M

Mis large, the required computational load for findingthe optimal solution to (6) can be high. In order to find

simplified approximated solutions, we propose two

algorithms to lower the computational complexity.

A. Decremental Selection

The decremental selection begins with the full set ofavailable antennas indexed as and then

removes one antenna per step. This process is repeated until

the required number of antennas (

},...,2,1{ Mr

M ) remained. In eachstep, the antenna with the lowest contribution to the system

capacity is removed. After steps, then NnM )( channel

matrix corresponding to the selected )( nM receive

antennas is denoted by . The matrix contains the

rows of

nH nH

)( nM H , which corresponds to the

selected receive antennas. Let)( nM rbe the index set of

the selected antennas at nth step and define as one row

of . By removing from at the step, the

jh

nH jh nH thn( )1

NnM )1( channel matrix is formed and the

corresponding capacity is given by

1nH

11021 /detlog)( nHnsssNn HHRNEIHC jHjnHnsssN hhHHRNEI 02 /detlog (7)

For sake of simplicity, we will assume uncorrelated

transmitters . Applying the Sherman-MorrisonNss IR

formula for determinants of general elementary matrix

(GEM), we obtain that

)()( 1 nn HCHC

Hjn

HnsNjs hHHNEIhNE

1

11002 //1log (8)

According to the last equation, maximization of

w.r.t. given yields)( 1nHC j nH

HjnHnNsjrj hHHINEhp11

0/minarg

(9)

and the subset of available antennas ris updated at the end ofeach step by

}{prr (10)To reduce the computational complexity of (9), let us

define

HHNs HHHINEHG11

00 / (11)

HnHnNsn HHHINEHG11

0/ (12)

and (9) can be then rewritten as

),(minarg jjGp nrj

(13)

where is the diagonal element of ,),( jjGn jth nG p denotes

the index of the largest diagonal element of . After finding

the solution of (13), we remove from at the

nG

ph nH

thn )1( step to obtain the channel matrix

since the receiving antenna is with the lowest

contribution to the system capacity.

NnM )1(

1nH pth

Using matrix inversion lemma, the MM matrix canbe updated in the

nG

thn )1( step by

p

H

p

n

nn gg

ppG

GG

),(1

11

(14)

where is the row of .pg pth nG

Denote our proposed algorithm fordecremental selectionasAlgorithm I, which includes the selection rule (13) and the

recursive updating algorithm (14). The details of Algorithm I

is summarizes in Table I with the right column showing the

complexity corresponding to each part of the algorithm. Note

that the subscript is dropped for simplicity.nNext let us compare the computational complexity of

Algorithm I and Gorokhovs approach. The latter is

MN

Rx.

.

.

.

.

.

M

Tx

RF chain

RF chain

RF

switchH

Coding

RF chain

RF chain

Decoding

Figure1. Receive antenna selection in MIMO

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summarized in Table II. It can be seen that the dominant

complexity in recursions has been lowered from

)'(2 MMMNO to . Considering that thedecremental selection algorithm is often used when

)( MMMO M is

relatively large, the proposed Algorithm I is demonstrated to

have a significantly computational complexity than

Gorokhovs approach.

B. Incremental Selection

As shown in [5] that ifM is small, i.e. )( MM is large,

incremental selection may be more applicable than

decremental selection. Inincremental selection, we start withan empty set of selected antennas and then add one antenna

per step to this set. At each step, the objective is to select one

more antenna, which yields the highest increase of the

capacity. We extend Algorithm I to theincremental selection

and name it Algorithm II.

The corresponding capacity at the step is given bythn )1(

11021 /detlog)( nHnsssNn HHRNEIHC jHjnHnsssN hhHHRNEI 02 /detlog (15)

The updating formulas of Algorithm II are given by

HNs HINEHG 00 / (16)

pHp

nnn gg

ppGGG

),(1

11

(17)

and the selection rule is

),(maxarg jjGp nrj

. (18)

By this selection, is inserted in a proper position in at

the

ph nH

thn )1( step to obtain the Nn )1( channel

matrix .1nH

Algorithm II is summarized in Table I and the dominant

complexity in recursions is . The complexity of the

incremental selection algorithm of [6] is . The

details are described in Table II. Noting that the incrementalselection algorithm is mostly used when

MMO )'( 2NMMO

M is small. By acomparison of Tables I and II, we can see that the proposed

Algorithm II achieves a significant complexity reduction.

ALGORITHEM I

DECREMENTAL SELECTION

Set: HHNs HHHINEHG11

0/:

,

;},...,2,.1{: Mr

for to1n )( MM

compute ,),(minarg: jjGprj

;}{: prr

update

),(1:

ppG

ggGG

p

H

p

;

end

)( 32 NNMO

)( MMMO

)(2 MMMO

ALGORITHEM II

INCREMENTAL SELECTION

Set: , HNs HINEHG 0/:;},...,2,.1{: Mr

for to1n Mcompute ,),(maxarg: jjGp

rj

;}{: prr

update

),(1:

ppG

ggGG

p

H

p

;

end

)( 2NMO

MMO

MMO 2

ALGORITHEM FOR

DECREMENTAL SELECTION

Set: 11

0/:

HHINEA HNs ,

},...,2,.1{: Mr ;

for 1n to )( MM

compute ,Hjj

rjAhhp

minarg:

}{: prr ;

update

;AhAhhAhAAp

H

pp

H

p

1)1(:

end

)( 32 NMNO

)'(2 MMMNO

)(2 MMNO

ALGORITHEM FOR


Set: Ns INEA 0/: ,

},...,2,.1{: Mr ;

for 1n to Mcompute ,H

jjrj

Ahhp

maxarg:

}{: prr ;

update

;AhAhhAhAA pH

pp

H

p

1)1(:

end

)'( 2NMMO

)(2 MMNO

TABLE II

GOROKHOVES APPROACHES [5], [6] TABLE I

PROPOSED APPROACHES

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IV.PERFORMANCE ANALYSIS

The proposed algorithms and Gorokhovs approaches are

summarized in Table I and Table II. As we mentioned in the

previous section, it is shown that the proposed approaches

always have lower computational complexity than that of the

algorithms of [5] and [6].

Table III summarizes Gharavi-Alkhansaris algorithm for

incremental selection and the complexity of this algorithmis . Although the proposed Algorithm II has about

the same order of complexity for incremental selectionwhencompared to Gharavi-Alkhansaris approach, a lack of

capability for decremental selection in Gharavi-Alkhansarisapproach makes our Algorithm I promising, because

decremental selection becomes essential when

MNMO

'M is closertoM . One specific case to illustrate this point is that,if 1' MM , then thedecremental selectionmethod alwaysfinds the optimal solution in the first step. Moreover,

decremental selectionis expected to perform generally betterthan incremental selection. The reason is that antennaincremental rule is based on individual contributions of the

appended antennas while antenna removal rule takes into

account join contributions of all (remaining) antennas [6].

Regarding to memory requirement, our method shows the

highest need in memory among all the algorithms under

comparison. The memory requirement of the proposed

algorithms is since the proposed methods need to

store and update

)(2MO

MM matrix in each step. However, inreal implementation it may not be a big issue.

nG

The motivation of this paper is to lower complexity while

minimizing the channel capacity loss. Since the proposed

algorithms employ the same antenna selection rules as those

of [6] and [7], all methods under comparison should have the

same performance in the sense of capacity. It has been

demonstrated in [7] that the performance difference between

the optimal and all the sub-optimal approaches compared in

this paper is relatively small.

V.CONCLUSION

In this paper, we propose two novel fast receive antenna

selection algorithms for MIMO communication systems.

Comparing with algorithms of [5]-[7] and the optimal

selection method, our approach can achieve the similar

outrage capacity with significant reduction in computational

complexity.

REFERENCES

[1] J. H. Winters, On the capacity of radio communications systems with

diversity in Rayleigh fading environments, IEEE J. Selected AreasComm., 1987.

[2] G. J. Foschini and M. J. Gans, On limits of wireless communications

in a fading environment when using multiple antennas, WirelessPersonal Commun., vol. 6, pp. 311-335, Mar. 1998.

[3] E. Talatar, Capacity of multi-antenna Gaussian channels, Eur. Trans.

Telecommun., vol. 10, pp. 585-595, Nov. 1999.

[4] M. Win and J. Winters, Virtual branch analysis of symbol error

probability for hybrid selection/maximal-ratio combining in Rayleigh

fading,IEEE Trans. Commun., vol. 49, pp. 1926-1934, Nov. 2001.

[5] A. Gorokhov, Antenna Selection Algorithms for MEA Transmission

Systems,Proc. IEEE ICASSP, Orlando, FL, May 2002, pp. 2875-60.[6] A. Gorokhov, D. Gore and A. Paulraj, Receive Antenna Selection for

MIMO Flat-Fading Channels: Theory and Algorithms IEEE Trans.Inform. Theory, vol. 49, pp2687-2696, Oct. 2003

[7] M. Gharavi-Alkhansari and A. B. Gershman, Fast Antenna Subset

Selection in MIMO Systems in IEEE Trans. Signal Processing., vol.52, No.2, Feb. 2004

ALGORITHEM FOR


For to1j M

set H

jjj hh:

end

},...,2,.1{: Mr ;

for to1n Mcompute

jrj

p

maxarg:

}{: prr

)(1

1:

1

1

H

i

H

p

n

i

i

H

p

p

H

n bhbhb

for all rj

update 2||: Hjnjj hb

end

end

)(MNO

MMO

2MNO

MNMO

TABLE III

GHARAVI-ALKHANSARIS APPROACH [7]

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