Journal of Shanghai University ( English Edition ) , 2006, 10(1) : 20 - 24

Article ID: 1007-6417(2006)01-0020-05

An antenna selection algorithm for spatial multiplexing systems

in BLAST receiver

FANG Yong ( ~ ~ ) , ZHU Yao- lin ( ~,.I¢~ ~ ) School of Communication and Information Engineering, Shanghai University, Shanghai 200072, P.R . China

Abstract In this paper, the antenna selection problem for spatial multiplexing systems in a BLAST receiver is investigated. In order to

search the optimal antenna subset, a selection criterion is proposed, which is able to obtain the largest minimum post-detection SNR.

The number of required antennas is deduced, which is determined by the system performance requirement for the existing RF chains.

Simulation result shows that the optimal subset in a BLAST receiver has higher minimum sub-channel SNR than that in a ZF receiver.

Computation of system performance can be fitted to a simple function, and it is simpler for the proposed algorithm to compute the

required antenna number.

Key words antenna selection, Bell Labs layered space-time(BLAST), curve fitting.

1 Introduction

Mult ip le- input mul t ip l e -ou tpu t (MIMO) sy s t ems can

offer s ignif icant capac i ty ga ins , c o m p a r e d to single-

input single- ou tpu t (S ISO) s y s t e m s . The ex t r a degrees

of f r eedom of fe red by mul t ip le an tennas can be used

for increas ing bi t ra tes th rough spa t ia l mult i -

p lexing I1-4j . H o w e v e r , mul t ip le an tennas p roces s ing

needs mul t ip le RF cha ins , w h i c h compr i se ampl i f i e r s ,

ana log- to-d ig i ta l conver t e r s , m i x e r s , e t c . , w h i c h are

cos t ly . Fo r MIMO s y s t e m s , one is ab le to c h o o s e opt i -

mal an t ennas to r educe the cos t . Fo r this p u r p o s e , an

opt imal an t enna subse t s e l ec t ion a lgor i thm wi th low-

cos t and l o w - c o m p l e x i t y is r equ i red . Recen t ly , a few

an tennas se lec t ion a lgor i thms have been d e v e l o p e d for

given channel rea l i za t ions . Subse ts of t r ansmi t o r re-

ce ive an tennas are s e l ec t ed by the min imum Shannon

channel capac i ty I~l , and the m a x i m u m Froben ius no rm

of channel ma t r i x [6~ or the m i n i m u m error ra te ET~ . Fa s t

a lgor i thms were p r o p o s e d in Refs . [ 8 ] and [9 ] , and a

pe r fo rmance ana lys i s was given in Ref. [ 10]. However ,

these a lgor i thms are unable to de termine the opt imal

Received Sep. 1, 2005

Project supported by National Natural Science Foundation of

China (Grant No. 60472103 ), Shanghai Excellent Academic

Leader Project (Grant No. 05XP14027 ), and Shanghai Leading

Academic Discipline Project ( Grant No. T0102 )

FANG Yong, Ph. D . , Prof., E-mail: yfang@ staff, shu. edu. cn

an t enna number . In o rde r to obta in the t a rge ted sys tem

per fo rmance with lower cos t , i t is neces sa ry to cons ider

bo th the opt imal an t enna subse t se lec t ion and the an-

t enna number . In this p a p e r , a new an tenna se lec t ion

a lgor i thm is p r o p o s e d , which can bo th sea rch the opt i -

mal an tenna subse t and de te rmine opt imal an tenna

numbe r .

F o r spat ia l mul t ip lexing sys t ems , the ma x imum l ike-

l ihood (ML) rece iver is opt imal when the t ransmi t ted

vec to rs are equally l ike ly . Fo r a high rate sys tem wi th

numerous an tennas , howeve r , l ower complex i ty receiv-

ers such as the l inear r ece ive r o r the Bell Labs l ayered

space- t ime (BLAST) rece ive r are genera l ly p re fe r red .

Because BLAST is an opt imal decoding a lgor i thm in

te rms of t radeoff be tw e e n a lgor i thm complex i ty and de-

coding pe r fo rmance , this p a p e r cons ide r s the receive

an t enna se lec t ion p r o b l e m in a BLAST rece iver .

The r ema inde r of th is p a p e r is o rgan ized as fo l lows .

In Sec t ion 2, the MIMO sys t em m o d e l wi th an t enna se-

l ec t ion is given. The a n t e n n a se l ec t ion c r i te r ion in a

BLAST rece ive r is p r e s e n t e d in Sec t ion 3 , and the a l -

go r i t hm for se lec t ing the an t enna n u m b e r is d e v e l o p e d

to fulfill sys tem p e r f o r m a n c e given R F cha in n u m b e r in

Sec t ion 4. Sect ion 5 a n d 6 give s imu la t ion resul t s and

c onc lu s ions , r e spec t ive ly .

2 Receive antenna selection in a BLAST receiver

Cons ide r rece ive a n t e n n a se l ec t ion for a spa t i a l

Vol. 10 No. 1 Feb. 2006 FANG Y, eta/. : An antenna selection algorithra for spatial multiplexing . . . 21

multiplexing system in a BLAST receiver as shown in

Fig. 1. There are N receive RF chains and NR

(NR > N) receive antennas with M transmit RF

chains. The channel is represented by an NR x M ma-

trix H whose element h~ represents the complex gain

of the channel between the j - t h transmit antenna and

the i - th receive antenna. The subset of N employed

receive antennas is determined by the selection algo-

rithm operating in the receiver which can opt the best

subset p ( p E P ) out of all possible P = C~R subsets of

N receive antennas. Denote Hp as an N x M channel

sub-matrix including rows of H corresponding to the

receive antenna subset p . The corresponding received

signal can be expressed asI~

Y = ~ / E s / M ' H p ' S + W, (1)

where S = [ s l , s2, "" , sM IT is an M × 1 transmitted

signal vector, Y = [ Yl, Y2 , " ' , y~]T an N × 1 received

signal vector, W an N × 1 received noise vector, and

Es the total transmitted signals power independent of

the number of transmit antennas.

NR(inputs) ] NR(outputs)

Fig,1 Receive antenna selection in a BLAST receiver

Assume that the receiver has perfect knowledge of

the channel propagation matrix. The BLAST receiver

performs a QR factorization of the channel matrix. It

then implements two operations: nulling and cancella-

tion [~21 . The channel matrix has the form (recall that

N>~M):

Hp = Q. R, (2)

where Q is an N × N unitary matrix, and R is an N ×

M upper triangular matrix, r ,~ : 0, n > m .

We represent the upper-triangular matrix R as the

sum of a diagonal matrix D and a strictly above-trian-

gular matrix U, i . e . , R = D + U. The nulling and

cancellation are summarized as Eqs. (3) and (4) :

X = Q - 1 Y = ~ / E s / M R S + V, (3)

where the components of V = Q- 1 W are mutually in-

dependent, C N ( 0 , 1 ) . The effect of nulling is to ren-

der the channel matrix in an upper triangular form,

with no amplification of receiver noise.

( Es/M)-V2 D-I[ X - ( Es /M) v2 US]

= S + (Es/M)-I /2D -~ V+ D -1U(S - ~3)

= S + (Es/M)-~/2D-~ V, (4)

where the second equality holds if the estimated trans-

mitted signal ~ = S, i . e . , there are no bit-errors. In-

spection of Eq. (4) discloses that X is subject to inter-

ference from other sub-channels through the off-diago-

nal term U, which can be removed with high proba-

bility through cancellation. Thus nulling and cancel-

lation together produce M independent virtual sub-

channels, and the SNR of the m- th sub-channel is

equal to Esr~,,/MNo. Then

),~c = )%. r 2 ~ , (5)

where )% = Es/MNo.

3 Antenna selection criteria

Performance in spatial multiplexing systems depends

on the receiver types. For an ML receiver, perfor-

mance relies on the minimum Euclidean distance of the

received constellation LT~ . Recall that all components of

S are assumed to utilize the same constellation for the

BLAST receiver. Under this assumption, the m- th

sub-channel with the lowest post-detection SNR will

dominate the error performance of the detection pro-

c e s s [13 '141 . Therefore, the criterion for selecting the

optimal antenna subset is to maximize the minimum

SNR of sub-channels,

H p = m a x t rain ) % ' r ~ m ( H p ) t . (6) p E P m = l , ' " , M

When a ZF receiver is used, the post-processing

SNR of the m- th stream is

yzrm = )%/[ ( H~H, ) 1] m~. (7)

In Ref. [ 7 ] , the selection algorithm based on maxi-

mizing the minimum singular value was presented,

which improved the minimal SNR of ZF receiver. It

first gave the possible lowest limit V zF-r~i" of sub-stream

SNR,

= YOA~n ( H , ) (S)

and then presented the selection criterion,

Hp = max / ) ' 0 " A ~ , ( H , ) } , (9) pEP

where A m~ ( H , ) is the smallest singular value of the

22 Journal of Shanghai University

channel matrix Hp. As a result, the lowest limit of

sub-stream SNR is improved. But it is not the mini-

mum one. To slove this problem, the selection criteri-

on Eq. (9) is changed as

H p = m a x { rain ) ' o / [ ( H ~ H p ) - ~ l m m } . (10) p E P m = l , . . ' , M

In this way, we can obtain the largest minimum

SNR of sub-streams.

4 Computation of required antenna num-

ber

Most antenna selection algorithms focus mainly on

selecting optimal one from all possible antenna sub-

sets. However, when the RF chain number is fixed or

not easy to change, one needs to determine the antenna

number for the requirement of system performance.

Because the minimum sub-channel SNR is not fixed

on some sub-channel and difficult to be represented as

a simple model, we regard the first sub-channel SNR

as a reference system performance. The first sub-

channel SNR can be written as

N

yNC = ¥o " ~-~ [ hp~.l [2, (11) i = 1

where I hp~,~ 12, p i = 1, 2, . . . , NR are i. i. d. chi-

squared variables I151 with probability density function

(p. d. f. ) and cumulative distribution function( c. d. f. )

given respectively by

f ( z ) = f ( l hp,112 = z ) = e -~ , (12)

F ( z ) = P( I h~.~ 12 ~<z) = 1 - e -~ (13)

The average SNR of first channel is as follows,

N

g = el Y~ct = Y0" ~ t l h,~,112t, (14) i = 1

which can be regarded as system performance.

In Eq. (14), the selection algorithm chooses the re-

ceive antennas among the N highest I h.~,~ 12 . For con-

venience, we produce a new set of ordered variables

X K , k = 1 , 2 , " " , N from I h.~,~ 12 , such that XI/> "'"/>

Xk/> "'" >I XN. X~ is the k-th largest of N R random

variables distributed according to Eq. (12) . Then Eq.

(14) is rewritten as follows:

g = ) % e { X I } + "'" + Yoe t X N } . (15)

The k-th highest statistic X~ is smaller than k - 1

variables, and greater than N R - k variables with all

possible combinations. Then based on Ref. [ 16], the

p . d . f . Of Xk is

p ( x ) = C~-R 1 " [ 1 - F ( x ) ] k-1

" C1NR-k+I " f ( x ) ' ~ F ( x ) ~ NR-k . (16)

Therefore, average of the k-th highest statistic X~ is

~/Xk} ~-11 = CNR C~R_ k • l

N R - k -1 1 \~ 1)NR-k- = C'Z, C~R_k+, ~ ( - T C ~ _ ~ ( N R - r ) -2 ,

r = 0

(17)

where F ( x ) NR-k = I1 - [1 - F ( x ) ] l NR-~ = N R - k

\~ Civ,_ k( - 1 ) N ' - ~ - ' [ 1 - F ( x ) ~ NR-k-~ r=O

The system performance g is then obtained:

N

g = Vo --,~ ~ N R ( J N R - k + l k = l

NR -~k

• ~ ( - - 1 ) N R - k - ~ C ~ R _ k ( N R - - r ) - ~ J . (18) r=O

With curvefitting, the Eq. (19) and its inverse can

be represented respectively as

g = G ( N R ) , (19)

N R = ~ V - ' ( g ) 7 , (20)

where )'0 and N are known. G is the fitting function.

F x 7 rounds x to the nearest integer towards infinity

because N R is an integer.

Based on Eq. (20 ) , we can compute the required

antenna number N R according to the system perfor-

mance g for a given RF chain number N .

5 Simulation

In the simulation, the 3 selection criteria were com-

pared. These are maximizing the minimum sub-chan-

nel SNR in a BLAST receiver as shown in Eq. ( 6 ) ,

maximizing the minimum sub-stream SNR in a ZF re-

ceiver as shown in Eq. (10) , and maximizing the pos-

sible lowest limit of sub-stream SNR with ZF receiver

as shown in Eq. (9) . Assume M = 4, N = 5, and N R =

8. Channel realizations are i. i. d. from frame to

frame. Simulation results with 100 channel realizations

are shown in Fig. 2, in which only 25 realizations are

displayed for clarity. From Fig. 2, the minimum sub-

channel SNR in optimal subset with BLAST receiver is

higher than that with ZF receiver, and the minimum

sub-channel SNR in optimal subset with ZF receiver is

higher than the lowest limit of sub-stream SNR in a ZF

Vot. 10 No. 1 Feb. 2006 FANG Y, et ol. : An antenna selection algorithm for spatial multiplexing . . . 23

receiver. The optimal antenna subset chosen in terms

of the minimum sub-channel SNR is different from that

in terms of the lowest sub-channel SNR. Thus, the se-

lection algorithm in Eq. (9) is simple but not exact.

5.0

4.5

4.0

Z 3.5 o,3 E 3.0

.~ 2.5

.~ 2.o 1.5

1.0

0.5 0

J I

L t

I

\ / ' +

5

"~" - - * - - BLAST ', ? ----o---ZF

-- + - - ZF-min

i , $ t

~ g I i I I

10 15 2 0 2 5

Channel realizations Fig.2 Comparison among three selection criteria

In order to get the fitting fnnction G of system per-

formance g , numerical calculation for different RF

chain numbers N, from 4 to 8, is needed, as shown in

Fig. 3. The horizontal axis denotes the antennas num-

ber N R , which is from N to N + 8. The vertical axis

represents system performance. Using CurveExpert

software to fit the 5 curves, we found that these

curves have the same increasing trend and can fit the

logarithmic function very closely with correlation coef-

ficient greater than 0.999 9 and standard deviation less

than 0.021. Then Eq. (18) can be denoted as

g = f l + j2 .1n N R , (21)

where f l and f2 are coefficients of the logarithmic

function. Obviously, they are correlated with the RF

chain number N. In the same way, f l and f2 can be

fitted into a quadratic and linear function respectively

with considerably high correlation according to their

numeric values. We have

8

E

E ~ 9

14

13

12

ll

10

9

8

7

6

5

4 4

' h " " " - - + - - N=6 " + N-7

- - ~'- ,"/=8

I I I I I

6 8 10 12 14

Antenna number N R

I

16

Fig.3 Analytic function of system performance

f l = a + b" N + c" N 2 , (22)

f2 = d + e" N, (23)

where a = 2 . 7 0 2 , b = - 0 . 6 1 0 , c = - 0 . 0 9 0 , d =

0.218 and e = 0.984 are determinate for N = 4 , "'" , 8 .

Substituting Eqs. (22) and (23) into Eq. (21) , we

have

g = G ( N R ) = a + b ' N + c . N 2 + ( d + e . N ) ' l n N R .

(24)

Therefore, the fitting function G is derived which is

a simple, familiar and easy to be inversed. Substitute

Eq. (24) into Eq. (20) , the antenna number selection

can be written as

N R = F e x p ( g - a - b . N - c . N 2 d + e ' N ) 7 . (25)

Finally, we verify reliability of the system perfor-

mance computation as shown in Eq. (18) . The mean

of 400 channel realizations is shown in Fig. 4 for the

case of 5 RF chains. The antenna subset with the max-

imum first sub-channel SNR is used. The simulation

good agreement with the theoretical

j 9.0

= ~ 8.5

E 8.0

7,5 Y*''

7.0

6.5

;>~ 6.0 ult 5.5 / - - * - - Theoretical analysis

¢

5.0 f t t t I t t 1~3 6 7 8 9 10 11 12

A n t e n n a N u m b e r NR

Fig.4 Comparison of simulation result and theoretical analysis

6 Conclusion

In this paper, we explore the antenna selection

problem for spatial multiplexing systems. The sub-

streams can be separated by means of various receiver

algorithms, such as BLAST. According to the sub-

channel post-detection SNR expression, the antenna

selection criterion is given. Besides the selection cri-

terion, we proposed an algorithm for computation of

the required antenna number. The algorithm is able to

determine antenna number for optimal system perfor-

mance when RF chains number fixed. Simulation re-

sult shows that BLAST receiver has better performance

result is in

analysis.

24 Journal of Shanghai University

o v e r ZF r e c e i v e r , a n d t h e p r o p o s e d a l g o r i t h m is s i m p l e

a n d a v a i l a b l e .

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( Editor HONG Ou )