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  ONSNRGAINOFPOLARIZATIONMATCHING Jyri H¨am¨al¨ainen Olli Piirainen Risto Wichman Nokia Networks Nokia Networks Helsinki University of TechnologyP.O. Box 319, 90651 Oulu, Finland P.O. Box 319, 90651 Oulu, Finland P.O. Box 3000, 02015 HUT, Finland ABSTRACT Recently, results demonstrating the feasibility of polariza-tion matching within mobile environments have been pre-sented. In this paper, the SNR gain from polarization match-ing is analyzed assuming a simple Rayleigh fading channelmodel, and it is observed that the SNR gain from polariza-tion matching is moderate even when the mobile polariza-tion is perfectly known in the base station. However, if thecross–polarization coupling ratio is high, large losses causedby polarization mismatch can be avoided in individual radiolinks when using polarization matching. 1 INTRODUCTION Several multi–antenna techniques, such as beamforming andtransmit diversity, increasing the uplink and downlinkcapac-ityandcoveragehavebeenpresentedforboth2Gand3Gsys-tems. Given the strict complexityrequirementsof handsets itis ofgreatimportancetofindmulti–antennasolutionsthatarebased on algorithms which are implemented in base stationrather than in mobile station. Beamforming is a classical ex-ample of this kind of solution providing average link perfor-mance improvement. In beamforming,the goal is to increasethe average strength of the electric field nearby the mobilestation by transmitting the same linearly preprocessed signalfrom strongly correlated antennas in such a way that signalsadd up constructively in the direction of the mobile. Whilebeamforming is based on the estimated direction of mobilestation, it is also possible to increase link performance if thepolarizationofthetransmitterandreceiverarematchedprop-erly. Inthispaper,westudytheso–calledpolarizationmatch-ing between transmit and receive antennas.Recently, it has been demonstrated [1, 2] that polarizationmatching can be feasible in mobile environments. Roughly,the polarization matching is a method where base stationemploys the information concerning the mobile polarizationwhen selecting downlink transmit weights. The mobile po-larization can be known either from uplink measurements orit can be estimated basedon the feedbacksignalingfrommo-bile station.The paper is structured as follows: Section 2 presents thesystem model, and Section 3 analyzes the gain achievablefrom polarization matching. The paper is concluded in Sec-tion 4. 2 SYSTEM MODEL Consider a system utilizing two dual–polarized transmit an-tennas in the base station and a single receive antenna in themobile station. The polarization of the mobile station an-tennaisdenotedby u , where u is atwo-dimensionalcomplexvector such that   u   = 1 . For linear vertical polarization wehave u V    = (1 , 0) T  while the linear horizontal polarizationisof the form u H   = (0 , 1) T  . We assume that the base station isequipped with vertically and horizontally polarized transmitantenna branches. In the following we adopt the model usedin [3]. In a singlepath fadingenvironmentthemobilestationreceives the signal(1)  r  = u ∗ ·  u V   H  V    + u H  H  H   s + n, where  s  is the transmitted symbol,  n  is zero–mean Gaussiannoise, H  V    =  w V   h VV    + w H  h HV    , H  H   =  w H  h HH  + w V   h VH  and  w  = ( w V    ,w H  ) T  is the transmit weight vector. Weremark that  h XY    is the impulse response corresponding tothe signal transmitted using  X  –polarization and received us-ing  Y  –polarization. In the analysis we assume that coeffi-cients  h XY    are uncorrelated. This assumption is well validin the broadside of the antenna system. It is emphasizedthat a single antenna receiver can receive only part of thetwo–dimensional signal while using dual–polarized anten-nas the mobile would be able to receive all the available sig-nal energy. All coefficients  h XY    are assumed to be complexzero-mean Gaussian variables and we denote  E {| h XY    | 2 }  =2 σ 2 XY    . Usingthenotation a = ( u ∗ · u V   , u ∗ · u H  )  thereceivedsignal can be expressed in the form r  =  aCw  s + n,  C =  h VV    h HV   h VH   h HH   . 3 POLARIZATIONMATCHING Let us study the selection of the optimal transmit weight vec-tor w . We denote by  u X  the projection of mobile polariza-tion to the  X  –polarization,  i.e. ,  u X  = u ∗ · u X . The channel  autocorrelation matrix is given by R = E { C ∗ R  p C } ,  R  p  = a ∗ a =  | u V    | 2 u ∗ V   u H  u ∗ H  u V    | u H  | 2  , where the expectation has been taken elementwise. Theproblem of polarization matching is the following: Findweight vector  w  such that  w ∗ Rw  is maximized. The so-lution to this problem is obvious. If mobile polarization is u , then we select w such that u  =  w V   u V    +  w H  u H  . Thus,we match the polarization of the transmitted signal with thereceiver polarization. Since u V    and u H   are orthogonal unitvectors, the above matching operation is always possible.In order to illustrate the benefits given by the polarizationmatching we study first the single antenna transmission uti-lizing a vertically polarized antenna, hence we take  w V    = 1 , w H   = 0 . Assume that mobile polarization is linear. Thenwe can write u = (sin θ, cos θ ) T  and the received signal be-comes r  =  H  ( θ ) s  + n  = ( h VV    sin θ  + h VH   cos θ ) s  + n. For a given  θ  the received SNR is γ  ( θ ) = E {| H  ( θ ) | 2 | θ }  = 2 σ 2 VV    sin 2 θ  + 2 σ 2 VH   cos 2 θ. If receiver polarization is vertical we have  θ  =  π/ 2  while θ  = 0  for horizontal polarization. The ratio XPR VH   =  σ 2 VV   σ 2 VH  between received vertical power and horizontal power is re-ferred to as cross–polarization coupling ratio. The usage of the subscript emphasizes that  XPR VH   presents a leakeageof power from vertical to horizontal polarization. This no-tation is adopted since the leakeage from horizontal to ver-tical polarization is not necessarily the same as  XPR VH  .Moreover,  XPR VH   depends on the environment and val-ues  0  −  15  dB have been given in literature. It is seen thatif   XPR VH   is high, then the mismatch between transmitterand receiver polarizations can cause large losses in receivedSNR.Let us compute the expected SNR when  θ  is uniformlydistributedon  ( − π/ 2 ,π/ 2) , i.e. , theorientationofthemobilepolarizationis random. Since  E { sin 2 θ }  = E { cos 2 θ }  = 1 / 2 we find that  γ   = E { γ  ( θ ) }  =  σ 2 VV    +  σ 2 VH  . In the sequelwe scale the transmit powers such that  σ 2 VV    +  σ 2 VH   = 1 .Then  γ   = 1  for the single antenna case. Similarly, we set σ 2 HH   + σ 2 HV    = 1 .Consider next the random matching. This corresponds tothe system where  w V    and  w H   are random but   w   = 1 .Now H  ( θ ) =  w V    ( h VV    sin θ  +  h VH   cos θ )+ w H  ( h HH   cos θ  + h HV    sin θ ) and after some elementary operations we find that  γ   = 1  asin the case of single antenna transmission. However, thereis a major difference between random matching and a singleantenna system since the transmit weights can be changedcontinuously in random matching. This implies that polar-ization mismatch — as well as perfect match — occurs onlyoccasionally. This provides interleaving gain when channelcoding is employed.Consider the polarization matching assuming that mobilepolarization is perfectly known in the transmitter. Now wetake  w V    = sin θ ,  w H   = cos θ  and the received signal in themobile is given by H  ( θ ) =  h VV    sin 2 θ  + h HH   cos 2 θ + ( h VH   + h HV    )sin θ cos θ. Since  E { sin 4 θ }  = E { cos 4 θ }  = 3 / 8  and  E { sin 2 2 θ }  = 1 / 2 we obtain γ   = 34( σ 2 VV    + σ 2 HH  ) + 14( σ 2 VH   + σ 2 HV    ) . We remark that if   XPR VH  , XPR HV    → ∞ , then theachieved SNR gain against single antenna case and randommatching is  1 . 76  dB which is the same as in the case of fast(within channel coherence time) mobile directed antenna se-lection with two transmit antennas. Contrary to transmit di-versity designed to mitigate fast fading, polarization match-ing does not provide diversity gain against fast fading butit gives polarization matching gain against polarization mis-match. The difference is essential: while signal attenuationcaused by fast fading may last only a short time dependingon the mobile speed, the polarization mismatch can last verylong time depending on the mobile antenna orientation. Fi-nally, we note that if   XPR VH   =  XPR HV    = 0 dB,  γ   = 1 and polarization matching does not provide any gain. 4 CONCLUSIONS The polarization matching proposed in [1, 2] was studied,and it was found out that the SNR gain from polarizationmatching is moderate even when the mobile polarization isperfectly known in the base station. However, if the cross–polarizationcouplingratio is high,large losses caused by po-larization mismatch can be avoided in individual radio linkswhen using polarization matching. This results as an im-proved network performance since the co-channel interfer-ence is attenuated. It is also expected that the practical gainfrom polarization matching is higher than indicated by ouranalysis since there is usually a correlation between transmitantenna branches when using cross-polarized antennas. References [1] S. Miller, J. Shapira: ”Transmission Considerations forPolarization-SmartAntennas”,in  Proc. IEEE Vehicular Technology Conference , May 2001.[2] S. Miller, J. Shapira: ”A Novel Polarization Smart An-tennas”, in  Proc. IEEE Vehicular Technology Confer-ence , May 2001.[3] W. C. Jakes (ed),  Microwave Mobile Communications ,IEEE Press, 1974.
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