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IEEE AFRICON 2009 23 - 25 September 2009, Nairobi, Kenya

Space-Time-Polarization Diversity in Multiple-InputMultiple-Output Communication Systems

Yakup Kihc, Mutlu Koca and Emin Ananm

Wireless Communication LaboratoryDepartment of Electrical and Electronics Engineering

Bogazici UniversityBebek 34342 Istanbul, Turkey

E-mail: {yakup.kilic.mutlu.koca.anarim}@boun.edu.tr

Abstract-In this work, we consider the use of multiple an­tenna signaling technologies, specifically space-time block coding(ST~C) and spatial multiplexing (SM) schemes, in multiple-inputmultiple-output (MIMO) communication systems employing dualpolarized antennas at both ends. The use of dual-polarizedantennas theoretically double the number of virtual antennasused in the system. On the other hand, the correlation effectsand the cross-polar discrimination (XPD) deteriorates the systemperformance. In our work, we consider these effects and presentperformance analysis of a 2 x 2 system with dual-polarized an­tennas for both STBC and SM cases. We also present simulationresults for both multi-antenna signalling techniques together withhybrid approaches under various XPD and correlation scenarios.Both the theoretical analysis and the simulation results show asignificant performance gain by joint utilization of space, timeand polarization diversity in comparison to uni-polarized systemswith the same number of antennas.

Index Terms-MIMO systems, space-time-block coding, dualpolarization

I. INTRODUCTION

Multiple-input multiple-output (MIMO) transmission tech­niques, such as space-time block coding (8TBC) [1], [2] orspatial multiplexing (8M) [3], are known to achieve significantdiversity or multiplexing gains. However, in MIMO systems,correlations may occur between channel coefficients due toinsufficient antenna spacing and the scattering properties ofthe transmission environment. This may lead to significantdegradation in system performance [4]. In order to have anuncorrelated channel between the transmitter and receiverlarge antenna spacings are required both at the base-stationand the subscriber unit. On the other hand, due to this spacerequirement, deploying multiple antennas may not be feasiblein all communication schemes. For this reason, the use of dual­polarized antennas instead of uni-polarized antennas is a costand space-effective alternative, where two spatially separateduni-polarized antennas are replaced by a single dual-polarizedantenna. Communication with dual-polarized antennas requiretransmitting two independent symbols on the same bandwidthand the same carrier frequency at the same time by using

This work supported by TUBITAK EEEAG under grant number 105E077.

two orthogonal polarizations. However as pointed out in [5],imperfections of transmit and/or receive antennas and XPD arethe results of the two depolarization mechanisms: the use ofimperfect antenna cross-polar isolation (XPI) and the existenceof a cross-polar ratio (XPR) in the propagation channel.These effects degrades the system performance considerably.In [6], a system employing one dual-polarized antenna at thetransmitter and one dual-polarized antenna at the receiver ispresented and the error performance of 2-antenna 8M and8TBC transmission schemes are derived for this virtual MIMOsystem.

Notice that, in [6], a single-input single-output (8180)system is enabled with MIMO capabilities through the use ofdual-polarized antennas. In this work, we present the perfor­mance of MIMO systems employing dual-polarized antennasunder different correlation parameters and XPD factors overcorrelated Rayleigh fading channels. In this regard, not onlythe transmit and receive antenna correlations and the XPDfactor, but also a spatial correlation is included in the systemanalysis. In addition, even when only 2 antennas are used at thetransmitter and at the receiver, a virtual 4 x 4 MIMO system isobtained, where the following four transmission schemes canbe evaluated: 1) 4-antenna 8TBC, 2) 4-antenna 8M, 3) Double2-8TBC (employing two 2-8TBCs from both antennas), 4)2-8TBC+2-8M (employing 2-8TBC at one antenna and 2­8M at the other. The error performance of these schemesare presented with theoretical analysis and simulation results.Notice that this range of transmission alternatives over thesame physical system allow an efficient trade-off between thediversity gain and the multiplexing gain that the overall systemcan achieve. Even though the results can be generalized toany number of transmit/receive antennas, throughout the paperonly a 2 x 2 dual-polarized antenna system is considered whereit is shown to have better performance than the 2x2 uni­polarized antenna systems. The use of dual-polarized antennasleads the way to achieving diversity and multiplexing gainsat a high rate, when combined with the link adaptationalgorithms which are envisioned for next generation wirelesscommunication systems with MIMO capabilities such as thoseproposed with the IEEE 802.11n and 802.16e standards, with

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dual-polarized antenna technology.The rest of this paper is organized as follows : In Section

2, the signal and correlated channel model is presented.In Section 3, we present the error performance analysis ofthe STBC and SM schemes employing two dual polarizedantennas is presented, followed by the discussion of the hybridtransmission schemes in Section 4. In Section 5, we presentthe simulation results and the concluding remarks are drawnout in Section 6.

c Space- Time dSpace - TimeEncoder Decoder

r , ll RIl l

Transmitter Receiver

II. CHANNEL MODEL

In this work the channel model, adopted in [6], is developedfor two dual-polarized antennas at the transmitter and thereceiver as shown in Fig. l. The received signal is definedas,

Verticalpolarization

Horizontal polarization

where x is the transmitted signal vector which is chosenfrom a complex constellation such that the average energyof the constellation element is one, and n is a vector of zeromean i.i.d Gaussian components with variance No/2 for eachelement. The channel matrix for a 2 x 2 dual-polarized systemis defined as,

[K Iv,IV KIh ,lv K 2v,lv K 2h,IV] T

KI v,lh KIh ,lh K 2v,lh K 2h,lh0:/

K 2v Iv = KIh Ih = K 2h Ih = - , (9), , ' a

K 2h,lv = KI v,lh = K 2v,lh = K

K

for the 2 x 1 dual-polarized systems. Since it would be hardto follow long expressions we simply write the equations fora 2 x 1 system and then get the channel correlation matrix fora 2 x 2 system . In the equations, E{.} denotes the expectationoperator and 0 < 0:/ < 1 is defined for the fixed componentof the channel. Also 0: is defined as 0: = X~D in [0,1], thuslarger values of 0: yield good cross-polar discrimination. TheRician K-factor, which denotes the ratio between the power ofthe LOS and the power of the NLOS components, is definedas

Fig. 1. Block diagram of dual-polarized antenna system

(3)

(2)

(I)r=Hx+n

flh- r;;s;;-H= --H+ --HK+I K+I

lhIV,IV hIh,lv h2v,lv h2h,IV]H = hIv,lh hIh,lh h2v,lh h2h,lh

h Iv,2v h Ih,2v h2v,2v h2h,2vh Iv,2h hIh,2h h2v,2h h2h,2h

where subscript kv and kh denotes vertical and horizontalpolarization of the kth antenna element, respectively. Thechannel matrix can be decomposed into the sum of an averageand a variable component as,

where K = 0 represents the pure Rayleigh condition andsuperscript {.}T stands for transpose operation. Normalizedreceive and transmit correlation coefficients are defined as:

In addition, because more than one dual-polarized antennais present, we also consider the spatial correlation betweenantennas. Therefore parameter s, which denotes the spatialcorrelation between antennas, is defined as

S = E{hk,lh:'n,l} (12)

where l = IV,Ih k = IV,Ih and m = 2v ,2h. In order todetermine the effects of transmit, receive and spatial correla-

where the elements of H , denoted as hi,j , represents thefixed components ~f the channel matrix and the elementsof H, denoted as hi ,j, are zero-mean circularly symmetriccomplex Gaussian random variables whose variances dependon the propagation environment and the characteristics ofthe antennas at both link ends . Notice that the fixed andthe variable channel components are assumed to satisfy therelations below in (4)-(8)

E{lhIv, l v I2

} E{lhI h , l h I2

} = E{lh2v,lvI2}- 2

E{lh2h,lhl } = 1, (4)

E{lhl v,l h I2

} E{lhI h , l v I2

} = E{lh2v,lhI2}

- 2E{lh2h,lvl } = 0:/ , (5)

E{lhl v, l v I2

} E{lhIh ,l h I2

} = E{lh2v,lvI2}

- 2E{lh2h,lhl } = 1, (6)

E{lhl v, l h I2

} E{lhI h ,l v I2

} = E{lh2v,lhI2}

- 2E{lh2h,lvl } = 0: , (7)

E{hIh,lhhiv,IV} E{hlv,lhhih,IV} = E{h2h,lhh2v,IV}

E{h2v,lhh2h,IV} = O. (8)

t =

r

E{h1h,lhhiv,lh}

vraE{h2h,lhh2V,lh}

vraE{hIh,lhhih,lV}

vraE{h2h,lhh2h,lV}

vra

E{h1h,lvhiv,lV}

vraE{h2h,lvh2v,lV}

vraE{h1v,lhhiv,lV}

vraE{h2v,lhh2v,lV}

vra

(10)

(II)

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(16)

tions and XPD factor (a), a matrix should be arranged thatcontains all the multiplications of the channel coefficients.Therefore these multiplications can be replaced by t, r, s and aparameters to determine the performance of the polarizationdiversity system according to the different correlations andXPD factors. For this reason, y = vec(H)l is defined anddecomposed into a fixed and a variable component as

y = vee ( J1 : K H ) ,

y = vee ( J1 : K H) ·

Since the channel matrix H is Gaussian, the vector y is alsoGaussian and all the channel realizations can be characterizedby the eigenvalues of the 8 x 8 correlation matrix, which isnonnegative-definite and defined by

Cy = E{yyH} = IIEllH

III. ERROR PERFORMANCES OF SPACE-TIME BLOCKCODING AND SPATIAL MULTIPLEXING

A. Error Performance ofSpace-Time-Block Coding

For the virtual 4-antenna system, we employ the rate-1/2complex G4 code given in [7] as

TXl X2 X3 X4

-X2 Xl -X4 X3

-X3 X4 Xl -X2

x=-X4 -X3 X2 Xl (15)

x* x2 x* X4I 3

-X2 xi -x4 x3

-x3 x4 x* -X21

-X4 -X3 X2 x*1

These symbols are mapped to the horizontal and verticalpolarizations of the dual-polarized antennas. Maximum-ratiocombining is employed at the receiver in order to obtain thedecision metric,

M-I 3

Xk = L L Ihi ,j l2X k + 17j=O i=O

where 17 is a zero-mean Gaussian random variable and M isthe number of receive antennas. In [2] assuming ideal channelstate information (CSI), the probability of transmitting x anddeciding erroneously in favor of e is approximated by

P(x ---+ e I hi,j,i = 1,2, ...,n,j = 1,2, ... ,m)

<exp (-yD(x, e)y"x) (17)

where X = 4EJJo ' and m and n denotes the number of transmitand receive antennas, respectively, and

where E = diag{ '\i}}~l. In the equations, {.}H stands for theconjugate transpose operation. For any length-l vector input,diag{.} operation returns a l x l square matrix whose diagonalelements are the elements of the input vector. The correlationmatrix for a 2 x 1 dual-polarized system is given in (13).Finally, using (13), the correlation matrix for a 2 x 2 dual­polarized system is formed as

D(x,e) =

A(x,e)o

oA(x,e)

oo

(18)

1For a matrix A = [at a2 .. an] where a, is a column vector of A,vee (.) operator returns a vector b = [a[ af . . a;;].

where l s xs refers to a 8 x 8 all-one matrix.

(19)

A(x,e)

Xr- er

oo

Xl- el X~ - e~

is a mn x mn matrix where 0 denotes the n x n all-zero matrix.A(x,e) can be defined as A(x,e) = B(x,e)B(x,e)H whereB (x, e) shows the distance between each symbol on theconstellation defined for l symbol as

xl - el x§ - e§xi - ei x~ - e~

B(x,e) =

(14)SlS X S]C Y 4 x 2

C [CY 4 X 2Y4X4 = sls xs

1 t*va s s r*va 0 s stva a s s 0 r*va s s

s s 1 t*va s s r*va 0

C Y 4 x 2 =S s tva a s s 0 r*va (13)rva 0 s s a t*va s s0 rva s s tva 1 s ss s rva 0 s s a t*vas s 0 rva s s tva 1

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From (17) the probability that the receiver decodes the trans­mitted symbol in error for a given channel realization isapproximated and upper bounded by

P(x ---. elH) :::; N e exp (-Xd~inllyI12) (20)

(28)

ML Decoder

i = arg min " -HiliiicS

R.\-# l ll>~ L- --'

AlamoutiEncoder

SpatialMultiplexer

InformationSymbols

['i x, ]

InformationSymbols

[x3 x4 x5 x6 ]

Fig. 2. Block diagram of proposed hybrid dual-polarized antenna system

where considering BPSK transmission, n(€i) denotes the rel­ative frequency of an error event and is given as

n( .) = W(€i, I v)w(€ i, Ih)w(€ i , 2V)W(€i, 2h)€t 240

with

(21)1

i = O

where r(Cy ) denotes the rank of the correlation matrix.The correlation matrix is orthogonal if and only if

s = r = t = 0 and ex = 1. In this case the channel becomes thei.i.d Rayleigh fading MIMO channel. Thus, the probability oferror is minimized for spatially uncorrelated fading and in theabsence of polarization diversity. Thus a 4 x 4 uni-polarizedsystems has better performance than a 2 x 2 dual-polarizedsystem. In order to get an accurate computation, the expressionin (21) is scaled by an empirically determined constant as in[6].

where N e and d~in are the average number of nearestneighbors and minimum distance of the constellation. For thepure Rayleigh case where K=O, (20) can be written as

r(Cy)-1

P(x ---. elH) :::; N e IT

€ i

The average symbol error rate of SM is then estimated as

(30)

(29)

If error in all orthogonal polarizations,If error in 3 orthogonal polarizations,If error in 2 orthogonal polarizations,If error in any orthogonal polarization.

{2 If €i = 0 (error-free case)

W(€i ,j) = 1 If €i = ±2

and S(€i) represents the number of scalar symbol errors withassociated error event, as follow

IV. HYBRID ApPROACHES

In order to obtain diversity and multiplexing gains to­gether, hybrid approaches combining transmit diversity andSM through the orthogonal polarizations can be employed indual-polarized systems.

Here we propose a hybrid algorithm that employs Alamoutiand SM schemes through the vertical and horizontal compo­nents, respectively, as shown in Fig. 2.

The transmitted signal matrix is as shown below

lX l -X~]X = X 2 X l .

X3 X 4

Xs X6

The channel matrix is defined as

lhlV,IV h2v,lv hlh,lv h2h,IV]H - hlv,2v h2v,2v h lh ,2v h2h,2v

- hlv,lh h2v,lh hlh,lh h2h,lh .h lv,2h h2v,2h hlh,2h h2h,2h

Employing the new H and x, the received signal can be foundaccording to (1) . At the receiver, assuming perfect channelknowledge, maximum-likelihood detection is employed ac­cording to the following decision metric,

(27)

(26)1

i= O

B. Error Performance ofSpatial Multiplexing

In SM scheme, because we are interested in using two dual­polarized antennas both at the transmitter and the receiver,the symbol stream is divided into four substreams and thentransmitted simultaneously from the two orthogonal polar­izations . Assuming perfect channel knowledge, maximum­likelihood detection is employed at the receiver according tothe following equation

x = arg min [r - Hx11 2. (22)z.es

Let € indicates the error between x and X, for a given channelrealization H using the Chemoffbound the probability of erroris given below

P(€IH) :::; exp (-xllzI12) (23)

where X = k and z = H€ which can be decomposed intofixed and variable component as

/K-Z = VK+iH € , (24)

Z = JK ~ 1 H€. (25)

Upon defining Cy = E{zzH} = II:EIIH where the diagonalelements of:E are defined as Ai, the pairwise error probabilityaveraged over all the channel realizations for Rayleigh fadingchannel can be expressed as

r(Cz)-1

P(€) :::; IT

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where x is the detected symbol matrix in similar form to(29). Xi (i = 1, .., 6) are chosen from a constellation set Swhich has complex symbols ofan M -ary phase-shift keying orquadratic-amplitude modulation symbol alphabet. Notice that,in two signalling periods six symbols are transmitted throughthe orthogonal polarizations of antenna elements . Thus thephysical data rate of the system increases by a factor of three.

A double space-time transmit diversity (D-STTD) tech­nique is also proposed in [8] for a 2 x 1 dual-polarizedsystem where the aim is to double the throughput whilehaving the performance of the Alamouti scheme. Here, twoblocks of Alamouti coded signals are sent through verticaland horizontal components separately. However in order toprovide compatibility with other schemes, in our simulationswe implement this scheme with maximum-likelihood detectionwithout zero-forcing interference cancellation at the front end.

V. SIMULATION RE SULTS

In this section simulation results with theoretical analysisare shown for STBC and SM schemes employing dual­polarized 2 x 2 antenna systems. Different channel conditionsare obtained using different receive correlation, transmit cor­relation and spatial correlation coefficients and XPD factors(a) for dual-polarized system under quasi-static Rayleighfading channel conditions while in STBC and SM with uni­polarized antennas the channel is assumed to be uncorrelated.Signal-to-noise ratio (SNR) is defined as 1Olog(2Es /0"; )(dB) for uni-polarized systems with two transmit antennasand IOlog(4Es /0"; ) (dB) for two dual-polarized transmitterantenna systems. In Figs. 3 - 4, the performance of a 2 x 2dual-polarized antenna system employing either STBC orSM are shown for a correlated Rayleigh fading channel fordual-polarized and uni-polarized systems, respectively. In lowcorrelations and low XPD factor case (see Fig. 3) STBC 2 x 2dual-polarized system with QPSK modulation has about 6 dBperformance gain over Alamouti 2 x 2 uni-polarized systemwith BPSK modulation at 10-4 bit-error-rate threshold. Inhigh correlations and high XPD factor case (see Fig. 4) theperformance gain is about 5 dB. Both systems has I bps/Hzthroughput. In addition, STBC scheme has about 10 dB and IIdB performance increase over SM scheme at low correlationand high correlation cases, respectively. Moreover an increasein the correlation coefficients from 0.1 to 0.4 has less than 1dB effect on the performance. However in high SNR regionSM provides 4 bpslHz data rate which is four times the datarate of the STBC scheme.

In Fig. 5, we simulate the different transmission schemesintroduced throughout the paper using QPSK modulationcase. We also simulate the 1 x 1 dual-polarized systememploying Alamouti and SM which is proposed in [6] forcomparison . When considering a 2 x 2 dual-polarized systemSTBC provides about 6 dB gain over D-STTD and 9 dB

24222018

18

16

16

10 12 14S\R(dB)

10 12 14S\R(dB)

-+-Sirllliated 8Malpha =O.2(BPSK)---4- Sirllliated 8Malpha=O.5(BPSK)

'~~ .... -+- Sir1lllated 8Malpha=O.9(BPSK)

..-0" Estimated8Malpha=O.2(BPSK)"S ..-4.. Estimated8Malpha=O.5(BPSK)

·cz::: <,~ ...t> .. Estimated8Malpha=O.9(BPSK)

<,'", ').... -+-EstimatedSTBC alpha=O.2(OPSK)

--*-EstimatedSTBC alpha=O.5(QPSK)~EstimatedSTBC alpha=O.9(QPSK)

."", ...... .. "- ·· "f · · Sir1lllated STBC alpha=O.2(QPSK)

, ~ \;;-. " <, '\:~ ..'\.. .. -*..Estimated8TBC alpha=O.5(OPSK)•. -(I . . Estimated8TBC alpha=O.9(OPSK)

..",::;--s .":,. -e-- /lJarrouli2x2(BPSK)f. " " ~ STBC 4x4(QPSK)

<, ""'~ , '\..

"-5 -, <: '\..

\ . -, -.' .'- .....

.,.1 \ '\. -, ....

-+-Siroo laledSMallila=O.2(BPSK)-+-Siroo laledSMalj:tla=O.5(BPSK)

., ~ = ~.. . ~SiR'kJ laled SM allila=O, 9 (BPSK)

.• 0 " EstimaledSMali=tla=O.2(BPSK)

~."..-<t .,EstimaledSMalj1la=O.5(BPSK)

'( ~.• 1>.,EstimatedSMallila=O.9(BPSK)

" ~""" . :'"'-.___SilllJlatedSIOC allila=O.2(QPSK)

-1-SilllJlatedSISCal~=O_5 (QPSKj

-.-SilllJlatedSISCallila=O.9(QPSK)

.~ ~.<, .•,. ' , EstimatedSTBC allila=O.2(QPSK)

·5 '\. \ <, '<, "..*.,EstimatedSTBC al~=O_5 (QPSK).. (I . , EstimatedSTBC aljila=O.9(QPSK)

, .......~A1amouti 2x2 (BPSK)

( '. <, "--e-STBC 4x4(QPSK)·

·5 \: \ '. -, ,, ""-(

~"\.~ ..

· \: \:. <,

., \ \ \ "oJ -. \

10

10o

10'

10'o

10'

~ 10

10

Fig. 4. Performance of 2x2 dual polarized antennas (r = 0.4, t = 0.4,s = 0.05, a = 0.2,0.5,0.9)

10

10

Fig. 3. Performance of 2x2 dual polarized antennas (r = 0.1, t = 0.1,s = 0.05, a = 0.2,0.5,0.9)

VI. CONCLUSION

In this paper, the performance of multiple dual-polarizedantennas are evaluated for SM and STBC via both theoreticalanalysis and simulation results. In both scenarios, the use of

gain over our proposed hybrid transmission scheme. HoweverSTBC provides 1 bps/Hz data rate while D-STTD and ourproposed hybrid scheme provides 2 bps/Hz and 3 bps/Hzdata rates, respectively. Furthermore, our hybrid scheme has 4dB advantage over SM which provides a transmission rateof 4 bpslHz. Thus a link adaptation algorithm can fullyutilize the channel conditions by switching between those fourtransmission techniques .

(31)x = arg min [r - Hxl12

x , ES

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-+- 1x1Dual-Polarized Alamouti

____ 1x1Dual-Polarized 8M~ -+- 2x2Dual-Polarized D-STTD~::--:~"'" ~ 2x2Dual-Polarized Proposed Hybrid Scheme

~ 2x2Dual-Polarized STBC

"'" :'" -+- 2x2Dual-Polarized 8M<, <,

~ <, <,

....... " "' '-, \ \ '\ "'\. "'-

<, .... .

'\ "' "' " -,\ \ "- <, -, <,

.... .... .. ..

\ \ -, '\ \\ .... .

. . . .. . . . . .. . . ... ,\,\ \

10·

10'

10·

10 12 14 16 18 ~ n ~ ~

SNR(dB)

Fig. 5. Performance of different transmission schemes (r = 0.3, t = 0 .5,8 = 0 .05, a = 0.4)

dual-polarized antennas provide a significant performance im­provement over the use of uni-polarized antennas . In additionwe also propose a new hybrid scheme and provide a simulationtogether with other hybrid structure studied in the literature fordual-polarized systems. This hybrid schemes combined withcorrelation robust STBes, link adaptation algorithms presenta potentially feasible solution for increased data rates.

REFERENCES

[1] S. M. Alamouti , "A simple transmitter diversity scheme for wirelesscommunications," IEEE J. Select. Areas in Commun., vol. 16, no. 8,pp. 1451-1458, Oct. 1998

[2] V. Tarokh, N. Seshadri, and A. R. Calderbank , "Space-time codesfor high data rate wireless communication: Performance criterion andcode construction," IEEE Trans. on In] Theory, vol. 44, no. 2, pp. 744­765, Mar. 1998

[3] G. J. Foschini, "Layered space-time architecture for wireless communi­cation in a fading environment when using multi-element antennas ," BellLabs. Tech. 1., pp. 41-59, Autumn 1996.

[4] J. P. Kennoal , L. Schumacher, K. 1. Pedersen , P. E. Mogensen ,and F. Frederiksen , "A stochastic MIMO radio channel model withexperimental validation," IEEE J. Select. Areas in Commun., vol. 20,no. 6, pp. 1211-1226, Aug. 2002

[5] C. Oestges, B. Clerckx, M. Guilland, and M. Debbah, "Dual polarizedwireless communications: From propagation models to system perfor­mance evaluation," IEEE Trans. on Wireless Comm., vol. 7, no. 10, pp.4019-4031, May 2005.

[6] R. U. Nabar, H. Bolcskei, V. Erceg, D. Gesbert , and A. J. Paulraj,"Performance of multiantenna signaling techniques in the presence ofpolarizat ion diversity," IEEE Trans. on Signal Processing, vol. 50, no.10, pp. 2553-2562 , Oct. 2002.

[7] V. Tarokh, H. Jafarkhani , and A. R. Calderbank , "Space-time blockcoding for wireless communications: Performance results," IEEE J.Select. Areas in Commun. , vol. 17, no. 3 pp.451-460, Mar. 1999.

[8] Y. Deng, A. Burr, and G. White, "Performance of MIMO systems withcombined polarization multiplexing and transmit diversity," In Proc. IEEEVehicular Technology Conference (VTC), May 2005, pp. 869-873.

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