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 jyri.k.hamalainen@nokia.com olli.piirainen@nokia.com risto.wichman@hut.ﬁ
ABSTRACT
Recently, results demonstrating the feasibility of polarization matching within mobile environments have been presented. In this paper, the SNR gain from polarization matching is analyzed assuming a simple Rayleigh fading channelmodel, and it is observed that the SNR gain from polarization matching is moderate even when the mobile polarization 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 downlinkcapacityandcoveragehavebeenpresentedforboth2Gand3Gsystems. Given the strict complexityrequirementsof handsets itis ofgreatimportancetoﬁndmulti–antennasolutionsthatarebased on algorithms which are implemented in base stationrather than in mobile station. Beamforming is a classical example of this kind of solution providing average link performance 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 thepolarizationofthetransmitterandreceiverarematchedproperly. Inthispaper,westudytheso–calledpolarizationmatching 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 polarization can be known either from uplink measurements orit can be estimated basedon the feedbacksignalingfrommobile 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 Section 4.
2 SYSTEM MODEL
Consider a system utilizing two dual–polarized transmit antennas in the base station and a single receive antenna in themobile station. The polarization of the mobile station antennaisdenotedby
u
, where
u
is atwodimensionalcomplexvector 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 using
Y
–polarization. In the analysis we assume that coefﬁcients
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 antennas the mobile would be able to receive all the available signal energy. All coefﬁcients
h
XY
are assumed to be complexzeromean 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 vector
w
. We denote by
u
X
the projection of mobile polarization 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 solution 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 utilizing 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 becomes
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 referred 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 notation is adopted since the leakeage from horizontal to vertical polarization is not necessarily the same as
XPR
VH
.Moreover,
XPR
VH
depends on the environment and values
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 polarization 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 selection with two transmit antennas. Contrary to transmit diversity designed to mitigate fast fading, polarization matching does not provide diversity gain against fast fading butit gives polarization matching gain against polarization mismatch. 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. Finally, 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 polarization mismatch can be avoided in individual radio linkswhen using polarization matching. This results as an improved network performance since the cochannel interference 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 crosspolarized antennas.
References
[1] S. Miller, J. Shapira: ”Transmission Considerations forPolarizationSmartAntennas”,in
Proc. IEEE Vehicular Technology Conference
, May 2001.[2] S. Miller, J. Shapira: ”A Novel Polarization Smart Antennas”, in
Proc. IEEE Vehicular Technology Conference
, May 2001.[3] W. C. Jakes (ed),
Microwave Mobile Communications
,IEEE Press, 1974.