Reduced Rank MMSE Detection in Space Time Coded Space Division Multiple Access Systems

REDUCED-RANK MMSE DETECTION IN SPACE-TIME CODED

SPACE-DIVISION MULTIPLE-ACCESS SYSTEMS

Lie-Liang Yang

School ofECS,University ofSouthampton, S017 1BJ,UK

ABSTRACT invokes L array elements or sensors. At the transmitter side Alamouti's STBC scheme based on twotransmit antennas is A multiple-input multiple-output (MIMO) space-division employed for encoding the transmitted symbols, in order to multiple-access(SDMA)systemispresentedandinvestigated,

achieve

adouble-ordertransmit diversity. By contrast, atthe when the space-time block coded (STBC) user signals are receiver side the signals obtained by sampling the receive an-transmitted over Rayleigh fading channels. In the proposed tennaarrays' outputs are linearlyprocessed, in ordertodetect

SDMA systemeachmobileuses twotransmitantennas, while theinformationtransmittedby thedifferentmobileusers. Note

the base-station (BS) receiver employs multiple receive an- that, theanalysis carried outinourforthcoming discourse can tennaarraysand each receiveantennaarrayconsists ofseveral bereadily modified to thecases, whenthe STBC

schemes

us-array elements. The MIMO SDMAsignals at the BS are de- be 2 mit

tennase

ensidered.

tected with the aid ofvarious reduced-rank MMSE_ttwMscems_schemes.

ing

_{In this}T, > 2_contributiontransmitantennas_{we focus on}areconsidered._{the reduced-rank MMSE}

Specifically, three classes of reduced-rank MMSEtechniques detection of the STBC assisted SDMA signals. Specifically, are investigated inconjunction with theproposed SDMA sys- three classes ofreduced-rankMMSEtechniques are extended tem. The threeclasses of reduced-rank MMSE

algorithms

are and

investigated

in

conjunction

with the

proposed

SDMA sys-derived,respectively,basedontheprinciplesof

principal

com- tem, so as to achieve low-complexity MMSE detection. The ponents (PC), cross-spectral metric (CSM) and

Taylor

poly- three classes ofreduced-rankMMSE algorithms are

derived,

nomial approximation (TPA). Our study and simulation re-

respectively,

based on theprinciplesof PC [4] - [7], CSM [6,

8]

sults show that the reduced-rank MMSE detection is

capable

and TPA[9]. Thecharacteristicsofthesereduced-rank MMSE ofachievingasatisfactorytrade-off between theaffordable de-

algorithms

are considered, and their BER

performance

is in-tection complexity and the achievable detection BER perfor-

vestigated by

simulation, when

communicating

over

indepen-mance. dent Rayleigh fading channels. Our study and simulation

re-sults show that the reduced-rank MMSE detection is

capable

I. INTRODUCTION of

achieving

a

satisfactory

traoff between the affordable

de-tection complexity and the achievable detection BER

perfor-In wireless communications MIMO systems equipped with mance.

multipleantennas atboth the transmitter and receiverhold the

promise of attaining substantial spectral efficiency improve- II REPRESENTATIONOF THE RECEIVED SIGNAL mentsrelativetowhat is achievedtoday [1].

Recently,

MIMO

systemshave attracted intense research interests inthecontext Fig. 1shows the schematic ofaSDMAsystem,where each mo-of both MIMOtheoryandapplications. Itis

widely recognized

bile transmitter

employs

twotransmit antennas,while the base that MIMO systemscanbeemployedforachievinga

high

ca- station(BS)receiver has Mantennaarrays,each constituted

by

pacity [1] andahighdiversity order[2], for

mitigating

the ef- Larrayelements. Notethatin

Fig.

1the thinlinesarerelatedto fects of varioustypesofinterferingsignals [3],and forsupport- scalarvariables,while the thick lines

correspond

to vector vari-ing SDMA.A coreprinciplebehind thehighspectral efficiency ables. Forthe kthuser, as showninFig.

1,

at the transmitter achievedby the MIMO systems isthat, when

communicating

sidetwoadjacentdata

symbols

x, and

12

areSTBC

according

over a rich scattering environment providing

nearly indepen-

to Alamouti's space-time transmission scheme. Theresultant dent transmissionpathsfrom each transmitantennatoeachre- STBCsymbolsarethen transmittedintwoconsecutive

symbol

ceive antenna, the multiple antennas

employed by

the trans- periods. Inthe first

symbol period

the

symbol

transmitted from mitter and/or receiver are capable ofproviding extra

degrees

antenna 1 is

Xk1

and the

symbol

transmitted fromantenna 2 of freedom inthespatial-domain, inadditiontothe

degrees

of is Xk2. By contrast, in the second

symbol period

the

symbol

freedom furnished in the more

conventionally

exploited

time- transmitted fromantenna 1 is

-42

and the

symbol

transmit-domain andfrequency-domain. This increased

degrees

of free- ted fromantenna2is

x*,,

where the

superscript

* denotes the domprovide novel ideas forincreasingtheachievable

capacity

complex conjugate.

anddiversityorder ofwirelesssystems, as wellas novel ideas In our studied BS receiver thereareM receiver antenna

ar-forsuppressingthe effects ofintentionalorunintentional inter- rays, as shown in Fig. 1. We assume that the different BS

fering signals. antenna arrays are located sufficiently far apart so that their

In this contribution the application of MIMO principles received signals experience independent fading. As shown in for supporting SDMA is investigated. Specifically, the syn- Fig. 1, each of the M antenna arrays consists of L array ele-chronous multiple-access communications scenario of Fig. 1 is ments, which are linearly correlated elements separated by a considered, where eachtransmitter, such as amobileuser, em- distance of d, whichisassumedto behalfawavelengthin this ploys Tx 2 transmit antennas, while the receiver, which is contribution. We assume that K users are supported by the analogous totheBS, employs M receive antenna arrays. Fur- SDMA system. These users transmit their data synchronously thermore, as seen in

Fig.l1,

each ofthe receive antenna array over flat Rayleigh fading channels. Furthermore, we assume

hkj,lhk,21

m=1, 2,...,M,BS receiveantenna array,where

hk,imn

rep-1 2

z

L resentsthe

fading

envelope, obeying

the

independent

Rayleigh

1 ,+"1 ' Y1 distribution.

Basedon

(1),

letus nowdiscuss the detection of the SDMA \ 2gvk,12,2k,22 L signals. Specifically,wefirst derive thefull-rank MMSE

detec-tor, when

assuming

that the receive

employs

the

knowledge

of

Ag

d 5 S.Space-Time -0 OutpUt

{Dk

} and

{Hk

} associated withall activeusers. Then,various

/ /\uProcessor reduced-rank MMSE detectors are derived.

2 //

\hk,lM,hk,2

III. FULL-RANK MMSE DETECTION

Xk2,

Xhl

|

|In

the context of the full-rank MMSE detection, the received

signal vectory collected at the antenna arrays seen in Fig. 1 is linearlyprocessed by weighting it with thecomplexweight matrix W of size 2ML x 2, in order to yield the estimate

iz

Figure 1: Stylized schematic of the space-time block coding assisted trans- forthetransmitted symbolvectorx astfollows:

mitter,channel model as well as space-time receiver of the SDMA system

us-ing two transmit antennasand M receive antenna arrays, each constituted by I1 =WH (6) Lelements.

where the optimum weight matrixWO is chosen such that the that thecomplex fading envelope remains constant across two mean-squareerror(MSE)between the transmittedsymbol

vec-consecutive STBCsymbol periods. Consequently, the received torx, and the estimatedvector:l isminimized, yielding [4] signalvectorcontaining thesamples collected from M BSan- l

tennaarrays in thecontext oftwo consecutivesymbol periods WO Ry Ryz1 (7)

canbe

expressed

as whereRyis the

auto-correlation matrix

of the

received

signal

K vector y of (1). When assuming that the received power from

y =E

DkHkXk

+n (1) allusersis thesame,

Ry

canbe

expressed

as

k=l

wherey isa2ML-lengthvector,nisa2ML-dimensionalzero- Ry 2 (D1H1HlD

meanAWGNvector,which hasacovariance matrix of K

K

~~~2NO

H] +

1:~~~~~

DkHH

Hj/DjH

+

'42ML

8

Rn

=E

[nnH]

NoI2ML (2) + k F+ ) (8)

where the superscript Hrepresentstheconjugate transpose or 2

Hermitianoperation,

No

is thecomplex AWGN'spower spec- where

Fl

represents the transmitted energy of

symbol

Xk1

or tral

density,

while I2ML is the

identity

matrix of size 2ML. symbol

1k2,

andE2is the

corresponding

covariance matrix

of The otherarguments in

(1)

canbe

expressed

asfollows, the

interference

plus

noise.

In (7)

Ryx1

representsthe

cross-Dk

=

diag

(Dkl,

Dk2,.

..,

Tk

DkMcorrelation

matrix between the received

signal

vectory and the

= lg(k,D2

.,ki)desired

symbol vector

xl,

which is given by

Hk = [HT

HLT

...

HM]T

(3) F1

Xk R[Xkl Xk2]

Ryxi

= 2

DlHl

(9)

where Upon substituting (8) and (9) into (7), the corresponding weight

matrix

WO

isgiven by

dkTn

°

hk,lm hk,2m

4

DkmT

L

[

]d

m

_km

HkmT

=

hk

k,h2m

2m

k-

hk

m_

(4)

(DH,H

H'D

H

+

2) DlHl

(10)

and

dkm

denotes the L-dimensional complex array vector of Upon invoking the matrix inversion lemma [4], we obtain thekthuser and them.threceiveantennaarray, which isgiven W = 1D1H1 (H

DlHE1DiHi

+ 2)1 (11)

dkm

=[i, ejT

sin(h

kr),., ejr(L

l) S( l)] (5)

xro

=(~D'~DH

(HlHD,HE2ll

(H

D ml{DI~y2

HlHDlH

y (2(12)

rFurthermore, it can be shown that the achievable MMSE can sponding to the kth user and the mth receive antenna array bexpseda

Furthermore, we assume that 0 < fkm . 1r. Finally, in (4) eeprsea

hk,im

=hk,im

eJOk,im

represents the CIR in the context of 1 H

il

whereTrace(A) representsthe trace ofA. trix, has received wide research and application [4]. Specif-Above wehave considered the full-rank MMSE detection of ically, the PC-based rank-reduction starts with expressing the the SDMAsignals. As seen in (12), the detectioncomplexityis Hermitian auto-correlation matrix

Ry,

as shownin (8), using dominatedby inverting the matrixE2 of rank2ML,whereM its eigen-decompositionas

is the number of receive antenna arrays, and L is the number

ofarray elementsperreceive antennaarray. Hence, the rank

Ry

= IAzIH (17)

of 2ML mightbe very high, when a SDMA system employs

a high number of antenna arrays and/or each of them has a where 4 is the orthonormal matrix whose columns consist of high number of sensors, forexample,in order tosupport a high theeigenvectors ofRy,i.e.,TXis in the form of

number of users. Consequently,thecomplexity of the MMSE

multiuser detectors might be extreme. Below we investigate

[014((2,*

2MI., (18)

a variety of reduced-rank MMSEdetectors, which invoke the

inversion of matrices having the rank that may be significantly whereI is the eigenvector corresponding to the eigenvalue of

lowerthan2ML. Ai. In(17)A is the diagonal matrix, which isgivenby

IV. REDUCED-RANKMMSE DETECTION A =diag{Al,

A2,

...,A2ML}

(19)

A reduced-rank MMSE detector reduces the number of co- Let us assume that theeigenvalues are ordered asA1 > A2 >

efficients to be estimated or optimized by projecting the re- ... >

A2ML.

Then, for agiven subspace of dimension U, the ceived signal vector onto a lower dimensional subspace [4]. processing matrixPuin (14) in the context of PC is constituted Specifically, letPube the(2MLx U)processingmatrixwith by the firstU columnsof 4X, i.e.,Pu =

[&1,

02,...

,

(U]-its column vectors forming a basis of a U-dimensional

sub-space, where U < 2ML. Then, for a given received 2ML- B. Eigenspace: Cross-Spectral Metric

dimensionalvectory,theU-dimensionalvectorinthesubspace Thecross-spectral metric (CSM) based technique for reduced-canbe

expressed

as rank

filtering

has been

investigated in

[4]-[7].

This

CSM

tech-- _ (pHp)-1pH 14 nique isdesigned withattempting tominimizethe achievable

(Pf

IPu)

14)

MMSE of the reduced-rank MMSE detector. Specifically, with

Su the aid of (17), (18) and (19), we have [4]

where

(PuHPU)

represents a normalization factor. Note N? AE

P",¢

R_1

2?

'iX'

(20) that, an "overbar" is used for indicating the operation in the Y =

Ai

reduced-rank subspace.

Inthesubspace, is the_y vectorinput to the MMSEdetector,

_Upon

_substituting

_{R, into}1

_(I13)

_and

_{remembering ignoring}

_the where it isprocessed by the MMSE filter andoutputestimates factorF1/2,the MMSE of the full-rank MMSE detector can be for the transmitted space-time symbols

{x1

}. Uponfollowing written as

theprinciplesof MMSE[4], itcanbe shown thatwehave

H I~~~~~1 2ML

~HHDH(P.

zl=WfY (15) MMSE1=2I--Trace o ) (21)

where theweight matrix isnowexpressedas

Based on (21), for the CSM-based reduced-rank detection,

WO

= l

(suHRySu)1

SUDlHl

(16)

the

subspace

Pu is constituted

by

the U

eigenvectors

of

Ry,

which correspond with the largest U values of the CSMs de-According to the above analysis, we can know that, for finedas

reduced-rank MMSEdetection, ourmain task istoderivea

U-dimensional detectionsubspacePu,sothat itcanapproximate 1H{D{'( 22

theoriginal 2ML-dimensional space asclosely aspossible or

Ai

1,...

2ML

(22)

that theresulted MMSE isaslowaspossible. Belowarangeof

rank-reduction algorithmsare extended and investigatedasso- The principles behind the CSM-basedreduced-ranktechnique ciated withour SDMAsystems. Notethat,in ourforthcoming can be readily understood by referring to (21), which shows discussion, for simplicity, weignore the factor of

E1/2,

when that, for a given value of U, the U terms having the high-invoking the autocorrelation matrix

Ry

orthecross-correlation est CSMs in the form of (22) must be maintained, in order matrix I? since itis either removed during the derivations for achieving the minimum MSE. Hence, the detection sub-or does not affect the final results at all. This can beshownby space must correspondingly be constituted bythe U eigenvec-(16), where the factor ofF1/2will be removed, since the same tors achievingthese U highest CSMs.

factor is also contained inRy asseenin (8). c alrPlnma prxmto

LetAmax be the maximumeigenvalueof

Ry.

Let pbea con- schemes considered in this contribution. Inoursimulations bi-stantsatisfying0 < p< 1/Amax. Then, the matrix

R`

canbe naryphase shift keying (BPSK) baseband modulation was

as-Taylorexpanded as sumed.

R-1 = p

(pRy)

-l1 p

[I

-(I

-pRy)]-00

pZ

(I-PRy))n

(23) K =

2,M

=

2,L

7,Eb/No

OdB

n=r01

BitErrorRate,

Letthe firstU/2 terms in(23) be usedin orderto

approxi-mateR

1.

Then we have

10-2

U/2-1

Rp

E3

(I-pRY)n

(24)

10o3

aI+

aiRy

+... +

au/2-lRY

(25)

i04

where the coefficients

{an}

are determined by p associated

003060

600

0

with the expansion of (24). Consequently, the estimate tox,

DOA 20

0 DOA22

canbeapproximately expressedas

si

~(aoDiHl

+... +

au/2-lRYu21DlHl)

y

(26)

However, determining the coefficients {an} in(26) mayre-

Figure

2: BER versusDOAs performance oftheSDMA system using

Tx

sultin

high

complexity.

Furthermore, the finite orderapprox- 2

transmit

antennas, M = 2

receive

antennaarrays ofeach

having

L = 7

imations thatresult fromtail-cutting of infinite order approxi- sensors, and supporting K = 2 users, whencommunicating over Rayleigh

mationsgenerally donotleadtothe best fit amongallapproxi- fadingchannels and whenusingthe full-rank MMSE multiuser detection.

mations of thesameorder. Hence,for thepurposeof reduced-rank MMSE detection, we can design the processing matrix

Pu as

PT =2 M=4, L=7, K=10

10

PU

=

[DHl:RyDlH1,

...

Ryu/

1D1H1

(27)

Full...=-rankMMSE

which isa

(2ML

x

U)-dimensional

matrix. Then,the received _ _=== space-time vector yisprojectedontothesubspace determined

_io_2

by

thecolumns of

Pu,

and is

processed by

an

optimum weight

o _ E__

matrixWOderivedinMMSEsense. _10'

The TPA technique has been originally applied for deriv- - X v ing the reduced-rank linear detectors for CDMA systems [9]. 10

In comparison with the other reduced-rank MMSE detec-

-1o-5

tors discussed

previously

in this

section,

the TPA assisted -15 -10 -5 d) 10 15

reduced-rank MMSE detection doesnotdepend ontheeigen- Eb/No (dB) decomposition of the auto-correlation matrix

Ry

According

to[5]andoursimulation resultsinSectionV., theperformance

of thistypeof reduced-rank MMSE detector iscapableofcon-

Figure

3: Principal components: BERversus SNRperbitperformanceof

vergingtofull-rankperformance witharank ofUsignificantly the SDMA system using Tx = 2 transmit antennas, M=4 receive antenna

lower than that of the othertypes of reduced-rank MMSE de- arrays of eachhavingL = 7sensors, andsupportingK = 10users,when

tectorsconsideredinthis contribution. Furthermore,thestudy communicatingoverRayleigh fadingchannels.

in [5] shows that the rank U neededto achievefull-rankper- In order to

gain

an

insight

into the effect of the DOAs on formance does not scale with the system size determined

by

the BER

performance, Fig.

2 shows the BER versusthe DOA the number ofusers,K,and the_{' '} antennaarrayresultant dimen-

_{~~~~~~~~~angles}

_{performance for the SDMA systems,whenthe full-rank}

sions, 2ML. Hence, the rankU of the detection

subspace

can MMSEmultiuser detector of

(12)

employed

the

knowledge

of

usually be significantly lower than the number ofusers sup-

-H}2

2 -

b1,

and

ported by

the SDMA system, when the number ofusers sup-

Dk0k=

and

bDklk=

whcIn

Frg

2 s DOA21=

f21ferens

between user 2's and user 1's signals received, respecively, by *

~~~~~~~~~~~~~~the

first and second antenna arrays. Without loss of

from the DOAs different from that of the desired signals, the detectors and achieves the best BER performance for a multiuserinterference may besubstantially mitigated by there- givenvalue of U. Fromtheresults of Fig. 5we can seethat ceiver with the aid of thebeamforming and MMSE processing. thefull-rankperformance canbeachieved, provided that When the DOAs of both theinterferinguserand the reference the rank of the detection subspace is notlower than 14, user arenearly identical, theBERachieved ishigher than that which issignificantlylower than 20 of the rank of the

sig-achieved, when the interferingsignalsspatially arrive from dif- nal subspace. Furthermore, as we mentionedpreviously

ferent DOAs of the desiredsignals. in SectionC., the TPA scheme doesnotinvokecomplex

eigen-decomposition forforming the detection subspace.

TX=2, M=4, L=7, K=10

= = Full-rank MMSE TX=2, M=4, L=7, K=10

10°

*Full-rank MMSE

bio-3___ ______= °__i_

10-mlo-4 2 = 14 =uL -3

Eb/No (dB) -05U=l2 \ \

-15 -10 -5 0 5 10 15

Eb/No (dB)

Figure 4: Cross-spectral metric: BER versus SNR per bit performance of the SDMA system usingTxc 2 transmit antennas, M =4receive antenna

arrays of each having L =7sensors, andsupporting K =10users, when Figure 5: Taylor polynomial approximation: BER versus SNR per bit communicating over Rayleigh fading channels. performance of the SDMA system usingTxc 2 transmit antennas, M =4 receive antenna arrays of eachhaving L=7sensors, and supporting K=10

users, when communicating over Rayleigh fading channels. Figs. 3 - 5 show the BER performance of the SDMA system,

when using various reduced-rank MMSE detectors. From the

results seen in Figs. 3 - 5, we can have the following observa- ACKNOWLEDGMENT tions.

The author would like toacknowledgewith thanks thefinancial

*The BER performance improves, when increasing the assistance from EPSRC ofUK. rank, U, of the detection subspace,provided that the rank

U is lower than the rank of the signal subspace, or in Fig.5 REFERENCES

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