Space Time Equalisation Assisted Minimum Bit Error Ratio Multiuser Detection for SDMA Systems

Space-Time Equalisation Assisted Minimum Bit-Error Ratio Multiuser Detection for

SDMA

Systems

S.

Chen, X.C.

Yang

and

L. Hanzo

School of

ECS,

University

of

Southampton, S017

1BJ, U.K. E-mails: {sqc,lh,xy203}@ecs.soton.ac.uk

ABSTRACT

This contribution investigates a space-time equalisation

as-sisted multiuser detection scheme designed for multiple

re-ceiver antenna aided space division multiple access (SDMA)

systems.A novel minimum bit errorratio(MBER) design is invoked for the multiuser detector (MUD), which is shown

to be capable of improving the attainable performance and enhancing system capacity in comparison to that of the standard minimum mean square error (MMSE) design. The

adaptiveMUDcoefficientadjustment procedureof theMBER space-time MUD is implemented using a stochastic gradient

based least bit error rate (LBER) algorithm,which consistently outperforms the classic least mean square (LMS) algorithm,

while maintaining alowercomputational complexitythan the latter.

I. INTRODUCTION

In an effort to further increase the achievable system capacity, antenna arrays can be employed for supportingmultipleusers in a

space division multiple access (SDMA)communications scenario

[1]-[4]. We investigate a space-time equalisation (STE) assisted multiuser detector (MUD) scheme designedfor multiple receiver antenna aided SDMA systems. To gain insight into the multiuser supportingcapability of such an SDMA system, it isusefultodraw

some comparisons with CDMA multiuser systems. In a CDMA system each user is identified by a unique user-specific spreading code. By contrast, an SDMA system differentiates each user by the associateduniqueuser-specific channel impulse response (CIR) encountered at thereceiver antenna. In asimplistic but conceptu-ally appealing interpretation, one may argue that the unique

user-specific CIR plays the role of a user-specific CDMA signature. In this analogy the CIR-signatures are not orthogonal to each other, but this is not a serious limitation, because evenorthogonal spreading codes become non-orthogonal upon convolution by the CIR. However, owing to the non-orthogonal nature of the CIRs, aneffecient multiuser receiver is required for separating the users

in an SDMA system.

The most popular SDMA-receiver design is constituted by the minimum mean square error (MMSE) MUD [3]-[6]. However, as recognised in [7] in a CDMA context, a better strategy is to choose the detector's coefficients so as to directly minimise The financial support of the EPSRC, UK and the EU in the frame-work of the NEWCOM, NEXWAY and PHOENIXprojects isgratefully

acknowledged.

the system's bit error ratio (BER), rather than the mean square error (MSE). We derive the minimum BER (MBER)solution for the STE assisted MUD. It is shown that the MBER design is

moreintelligent in utilizingthesystem's resourcea,resulting inan

enhancedperformance in comparison to the standard MMSE de-sign.Additionally, an adaptive MBER MUD coefficient adjustment algorithm is also considered based on a stochasticgradient learning algorithm referred to asthe leastbiterror rate(LBER) procedure.

It is demonstrated that this adaptive MBER MUD consistently outperforms the leastmeansquare(LMS)designbased MUD and

yet it has a lower computational complexity thanthe latter. Arange of simulation resultsarealsoprovidedinsupportof our theoretical

analysis.

II. SYSTEM MODEL

Consider amultiple antenna aided SDMA system supporting M active users, as depicted in Fig. 1, where eachof the M users is equipped witha single ttansmit antenna and the BS'sreceiver is

assisted by an L-element antenna array. Thesymbol-rate received signal samples xi(k) for 1 <I < L are givenby

M

nc-i

xi(k)

=

E

E

Ci,i,msm(k-i)+nli(k)

=:xi(k)+ni(k), (1) m=l i=O

where ni(k) is a complex-valued Gaussian white noise process associated with

E[Ini(k)12]

= 2,x

i1(k)

denotes the noise-free part oftheIthreceiver antenna'soutput,

sm(k)

is thekthsymbol of user m, while

Cij,m

represents the CIR taps associated with

4'4 y/k)k}fl

r___

nS]-;t S304k)00IMSf v

;

I77

...

Fig. 1. Schematic of an antenna array aided SDMA uplink scenario,

where each of the Musers isequippedwithasingle transmitantennaand the base station's receiveris assistedbyan L-elementantennaarray.

x(k)

O,X)o'

WW2s

XL(k)

w'

I*xw

02

w L 1)l,2m nR) J.2.t

A A ..,

-Fig.2. Space-timeequaliserassisted multiuserdetectorstructure, where Adenotes thesymbol-spaceddelay.

userm and the lthreceiverantenna. For notational

simplicity,

we

assume that each ofthe ML channels has the sameCIR length

of nc.We assumefurthermore that BPSK modulation isusedand hence wehave sm(k) E {±1}. The bank of M STEs shown in Fig. 2constitutes the MUD. The softoutputs oftheM detectors aregiven by

L nF-1

ym(k) ZZ E i, mXi

(k-i),

(2)

1=1 i=O

for1 < m <

M,

wherewl,m =

[WO,i,m

...

WnF-11,,m]T

denotes

the mth detector's weight vector associated with the lth receive

antenna. TheM userdetectors' decisions aredefined by

sm(k-d)=sgn(yR,

(k)),

l<m<M,

(3)

where Sm

(k)

is theestimate of Sm

(k),

while

YRm

(k)

=

[Ym

(k)]

denotes the real part of ym

(k)

and

sgn(e)

is the sign function. Again, for notational simplicity, we assume that each ofthe M detectors has the same decision delay d and all the temporal

filters have the same order nF. Hence we have 0 < d <

nFF+ncC-2.

Letusdefine the variablesWm =

[wTm

...w m]T

xi(k)

[xi(k) xi(k

-

1)

xi(k

-nF +

1)

and

x(k)

[xl(k)

...

xT(k)]T.

Then the output of the mth detectorcan be

written as

L

ym

(k)

=

E

W!'mXl

(k)

=

wmx(k).

1=1

The receivedsignalvector

x(k)

ismodeledby x(k)-

Cs(k)

+

n(k)

=

x(k)

+

n(k),

(4)

with

Cl,m

being the nF x (nF + nc - 1)-dimensional CIR

convolution matrix associated with userm and the Ithreceiver

antenna.Note that the output of the mthdetectorcanbeexpressed

as:

ym(k)-

w-H((k)

+n(k))=

gm(k)

+ em(k), (7)

where em,(k) is Gaussian distributed, having a zero mean and

E[Jem(k)12]

=

2w'WmW

2.

Classically, the mth detector'sweight

vectorwm is given bythefollowing MMSE solution

W(MMSE)m ( C +2cTI)Cn(m-1)(nF+nC-1)+(d+1),

(8)

for1 < m <M,where Idenotes the (LnFxLnF)-dimensional identitymatrix and

Cli

the ith column of C. An adaptive MUD coefficientadjustment algorithmsdesigned for theimplementation

ofthe MMSE solutioncan berealized using the LMSalgorithm. III. MINIMUM BIT ERROR RATE MULTIUSER DETECTION Let us denote the N8 =

2M(nFl+nc--1)

number of possible

transmittedsymbolsequencesofs(k)as

s(q),

1 <q < N8.Denote

furthermorethe

((m-1)(nF+nc-l)+(d+l))th

elementofs(q)

corresponding to the symbol

sm(k

-d), as

s

The noise-free

partof themthdetector input signal x(k)assumesvaluesfromthe signal setdefined as: Xm -{x(q) Cs,1 <q <

N,}.

This

set canbe partitioned intotwosubsets, dependingon thevalueof

sm(k-d),

asfollows:

XW(±){x(q

,) EXm:

sm(k-d)-±1.

Similarly, the noise-free partofthemth detector's output Ym (k)

assumesvalues from the scalarset

Ym -{Ym

=

wmx

X)1

<

q<

N.}.

(9)

Thus

YRn

(k)

=

R[f

m

(k)]

can

only

take the values from theset

YRm A

{YRq

=

R[VmJ[]

1 < q<

NS}I

(10)

andYRm can be divided intothe two subsets conditioned on the valueofsm(k

-d):

{R

-

RE

)

YRm,:

sm(k-d)

=

±1}.

(11)

It isreadilyseen that theconditionalprobabilitydensity function (PDF) ofYRm(k) givensm(k-d) =+1 is:

Nsb 1YRYRm )

Pm-(YRI+l)

2+crnW~MW7, (12)

Nsb q=1

V2irO2WMWm

where -(q+) E

Y)

andN8b= N8/2isthe number of the points in thesetY(+).Thus the BER of the mthdetector associatedwith

thedetector's weightvector wm isgiven by:

(5)

where we have n(k) =

[nl(k)...nL(k)]T

with ni(k)

[nr

(k)ni

(k-1)

...

ni(k-nF+

1)]T,

s(k)

=

[sT

(k)

...s

(k)]T

withSm

(k)

=

[Sm(k)

Sm

(k-1)

... Sm

(k-nF-nC

+

2)]

T, and

C1,2

C2,2

CL,2

* C1,M

*-- C2,M

...

CL,MJ

(6)

Nsb

PE(Wm)

-

>IQ

(g9

(wm))

q=l

where

and

1 00 2

Q(u)- I e 2 dv

9(q+)(W )=

gn(m,d))Rm

gn w

(13)

(14)

(15)

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-

cl'i

C2,1

c

Note that the BER is invariant to a positive scaling of w, Similarly, the BER may be calculated basedonthe other subset,

namely

on

Y".

The MBER solution for the mth detectoris then defined as the weightvector thatminimizes the error probability (13), namely

W(MBER)m

=

argniunPE(wm).

Wm

The gradient of PE(wi) withrespect to wm is given by:

VPE(Wm)=

(16)

N$b (q,s+)

)2

1 N.bq=l

2Nsb

V/27r.

VW

W

q=1

(q,+)W

x sg' (q)

(

YR _ Wm q 1

xsg

(m,dJ)

W Wm -

(17)

Giventhegradient expression (17), the optimization problem (16)

can be solved iteratively by commencing the iterations from an appropriate initialization point, such as theMMSEsolution,using

agradient-based optimization algorithm. The simplified conjugate gradient algorithm [7] provides an efficient means of finding an MBER solution for the optimization problem formulated in (16).

IV. ADAPTIVE MBER MULTIUSER DETECTION The PDF ofYRm(k) can be shown to be explicitly given by:

NP (1YR-b_ )2

P.N(YR)

= 1 W m S

I:

e (18)

NandsteE ofttrcanq=1

and the BER ofthemth detectorcanbecalculated

according

to:

Ns

PE(Wm)

=

NZ

Q

(g(9

(wm))

q=1

with

= sgn(s(q)

)9(q)

Gnm,dWR

(19)

(20) where the summation is carried out over the

N.

number of elements

Y-R4)

e

YRn,.

Inreality, the PDFof

YRm

(k) is unknown. Therefore, some form of PDF estimationisrequired forsupporting theadaptive implementation of the MBER MUD.

Given a block of K training samples

{x(k),

sm(k - d)}

Kk=1,

a Parzen window estimate [8]-[10] of the PDF in (18) is given by:

( K -

(YR-VR,R

(k))

P3m(YR)

=

_{Kv"-7pn E}

Kv.p.

2p,2

P

and the resultant approximate BER formula becomes

K

PE(Wi) = K , Q

(9k(W.)),

k=1

where we have

9k(Wm) = sgn(s8(k -

d))YRm(k)

Pr.

(21)

0

-2

IF

m

0

0) 0

-4

-6

-8

--10

-10 -5 0 5 10 15 20 25

SNR(dB)

Fig. 3. Bit errorratecomparisonof thetheMMSE and MBER multiuser

detectors for the3-user 4-antenna

time-invariant

system.

and

pn

is the chosen kernel width. This approximation is an adequate one, provided thatthe width

pn

ischosen appropriately. In orderto derive a sample-by-sample adaptive MUD coefficient adjustment algorithm for updating

wi,

consider asingle-sample estimateof

Pm

(YR),

namely:

1 (VR-YR-(k))2

Pm

(YR, k)

= -

2p2

Vpn (24)

Conceptually, from this single-sample PDF "estimate", we havea

one-sample

orinstantaneous BER "estimate" PE(W, k). Using the instantaneous stochastic gradient formulaof:

Y2 k

VPE

(wm,v

k) = _sgn

(s8

(k-d))

Rv (k) (25)

2

v~27pn

givesriseto theLBER MUD coefficient adjustment algorithm of: Y2i (k)

wm(k+

1)

=

wm(k)

+psgn(sm(k

- d))e

2-x(k).

(26)

The adaptive gain

it

andkernel widthpn have to be set appropri-ately to ensureadequate performance in terms of the achievable convergence rate and steady-state BER misadjustment. It can be shown that for the BPSKmodulation scheme considered the LBER algorithm is simpler than the LMS algorithm. If the modulation scheme is QPSK, both algorithms have a similar complexity. For higher-order modulation schemes, the LBER algorithm is slightly

morecomplexthan the LMS technique.

V. SIMULATION STUDY

(22)

Time-invariant

system. The system used in our simulations

(22)

supported M =3 userswith the aid of L=4 receiver antennas. All thethree users had an equal power. The 3 * 4 = 12 CIRs are listed in Table I, each having nc =2taps. In the actualsimulation,

all

the 12CIRs were normalized to provide unit channel energy.

(23)

Each temporal filter had a

length

ofnF = 3 and the detector's

decision delaywaschosen to be d= 1. For the setofsimulated CIRs Fig. 3 compares the BER performance of the MMSE and MBER MUDs. It can be seenthat for all three users the MBER

detectors had a better BER performance than the corresponding MMSE detectors. For the specific simulated channel conditions,

the performance gap between the MBER and MMSE detectors

was the smallest for user 3, exhibiting an approximately 1.5 dB

SNRgain atthe BER level of10-4.Atthis BERlevel the MBER

detector of user 2 had theSNRlargestperformance gain over the

corresponding MMSE detector, which wasin excess of6.5dB.

The MMSE and MBER solutions choose the detector's weight

vector very differently. Fig. 4 plots the full conditional PDFs

pm(yl

+ 1), marginal conditional PDFs

Pm(YRJ

+ 1), and the corresponding signal subsets

Y.t)

and

YR+3

for user2at SNR=

2 dB. Since the MMSE detector minimises the MSE term of

E[Ism(k

-

d)

-

ym(k)12],

the

signal

subset Y

has

to be

distributed symmetrically withrespect to bothW[y]and F[y], and

the associated conditional PDF Pm

(yI

+ 1) has to be circular.

However, the BER ofthe detector is mainly determined by the

niinimum distance of the subset

YR+)

fromthedecision threshold

[y]

= 0. By contrast, the MBER detector is not constrained

by the symmetric and circular considerations, it spreads

Ym+)

significantly wider along

![y]

and this leads to a significantly

higher MSE value. However, this also doubles the minimum

distance between

YR+)

and

W[y]

=0, resulting ina substantially

lowerBER,comparedtothe MMSEsolution. Clearly, the MBER design is moreintelligent in utilising the detector's resources. The LMS and LBER MUDs were nextinvestigated for user 2 at SNR= 2 dB.Fig.5 depictsthelearning curvesofthetwoadaptive detectors, averaged over 20 runs and started from two different initial conditions wm(0).In theseinvestigationstheadaptiveMUD

operatedin twodifferentmodes, namely usingpilot-based training,

when the transmitted symbols sm(k - d) were known to the receiverand decision-directed(DD) adaptation, when the detected

symbols 8m(k - d) were used to substitute 8m(k -d). It can be seen that the LBER MUDconsistently outperformed the LMS

MUD.Furthermore, the LBER MUDwascapable of taking the full

advantageof DD adaptation, asdemonstratedin Fig. 5 (b), where

it can also beseen thatthe DD LMS MUDfailed toconverge to the MMSE solution.

Fadingsystem. The system setup and the structureoftheMUDs

were the same as in the time-invariant CIR-based example, but

fading channelswere simulated.Thetransmission framestructure consisted of20training symbols followed by 200 data symbols.

The magnitudes of the CIR taps were i.i.d. Rayleigh processes,

each having the root mean power of

v/_'

+

j,/

05. Idealistic fading conditions were simulated, where theCIR taps were

sub-jected toframe-invariantfading.Hence the CIR taps werefadedat thebeginningof eachtransmission frame at anormalized Doppler frequency of i0-5, generating different channelmagnitudes and phases fordifferentframes, but theywerekeptconstantwithin the

transmissionframe.Fig. 6 compares the bit error rate achieved by the LMS and LBER MUDs of user 1. The performance ofthe

MUDs for the other two users were similar to those shown in Fig. 6.

3-. 3-. -3

0~~ ~ ~ ~~

-0I~~ ~ ~ ~ ~ ~

mM Rely]

(a) MMSE

08

06

04

02

0

-02

-04 -06

-08

.

A

K

.

6

0 2 4

-0

ImLy] Rely]

(b) MBER

Fig. 4. Conditional probability density function

P2(yI

+ 1) (surface), marginal conditional probability density function

P2(YRI

+1) (curve), signal subsets

Y(+)

and

Y(+)

(dots) of the user-2 detector for the3-user

4-antenna time-invariant system at SNR= 2 dB. The detector's weight vector isnormahzed to a unit-length.

VI. CONCLUSIONS

A novel minimum bit error ratedesign has been proposed for a space-time equalisation assisted multiuser detector employed in

multiple antenna aided space division multiple access systems. It has been demonstrated that this MBER design is capable of achieving abetter BER performance and hence ofenhancing the

achievable system capacity, compared to the standard minimum mean square error design. An adaptive implementation of the MBER space-time MUD has also beenderivedbased onthe least biterrorrate algorithm, which was showntoconsistently

outper-form the classicleast meansquarealgorithm, whilemaintaininga

lower computationalcomplexity incomparison to the latter.

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I...

-5

TABLEI

CIRSFOR THE3-USER4-ANTENNATIME-INVARIANT SYSTEM.THEACTUALLY SIMULATEDCIRsWERE

Cj,m(Z)/ICI,m(Z)j

TO PROVIDE UNIT CHANNEL ENERGY.

Ct,m(Z) m= 1 m=2 m=r3

I= 1 (0.5+j0.6) +(0.8-jO.7)z-1 (1.0 +jO.8)+(0.7+jO.4)z-1 (0.6-jO.8)+(-0.5+

jO.3)z-= 2 (0.7 +jlt.0) + (0.2 +jO.5)z-' l (0.4+jO.4)+(0.6+jO.6)z2- (-0.5+jO.3)+(0.9+jO.2)z-1

I=3 (-0.5+jO.7) +(-0.6+jO.7)z-' (-0.3- j0.6)+(0.5-jO.7)z-' (0.7+jO.8)+(-0.2-j0.3)zT

I=4 (0.3-j0.6) +(0.5 -j0.2)z-' (0.8-j0.5)+(0.6+jO.2)z-- (0.2 +jO.1)+(0.9+

j0.2)z-0

-1

0 1000 2000 3000 4000 5000

sample

(a)wm(0) =w(MMSE)m

a)

Cc

_-2

-2 i

m o -3

-4

-5

0 5 10 15

Average SNR (dB)

20 25

Fig.6. Biterrorratecomparisonoftheuser-oneLMSandLBERmultiuser

detectors for the 3-user 4-antenna fadingsystem.

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1e+1

1 e-1

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1 e-2

1 e-3

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Fig. 5. Learningcurvesof theadaptive detector of user-2 for the 3-user

4-antennatime-invariantsystem atSNR=2dB, where DD denotes decision directed adaptation in which sm(k- d) is substituted by its estimate

§m(k-d).

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0-7803-8887-9/05/$20.0002005IEEEl

1 e-1

1 e-2

1e-3

co

Er

-U

co

LMS(1) -LBER (1)

---X~~~~~~~

~~~~~~~~~~~~~~~~~~....

1e-4

1 e-5