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Unformatted text preview: PERFORMANCE OF SPATIAL MULTIPLEXING IN THE PRESENCE OF
POLARIZATION DIVERSITY Helmet Bo'lcskeill, Rohit U. Noborll, V. Erceg2), D. Gesbert2), and Arogyaswami J. Pauliof) 1) Information Systems Laboratory, Stanford University
Packard 223, 350 Serra Mall, Stanford, CA 943059510
Phone: (650)7243640, Fax: (650)7238473, email: [email protected]
2) Iospan (formerly Gigabit) Wireless Inc., 3099 North First Street, San Jose, CA 95134 Abstract—In practice large antenna spacings are needed
to achieve high capacity gains in multipleinput multiple
output (MIMO) wireless systems. The use of dualpolarized
antennas is a promising cost effective alternative where
two spatially separated antennas can be replaced by a
single antenna element employing orthogonal polarizations.
This paper investigates the performance of spatial multi
plexing in MIMO wireless systems with dualpolarized an
tennas. We compute estimates of the symbol error rate as
a function of crosspolarization discrimination (XPD) and
spatial fading correlations. Using these estimates, we show
that dualpolarized antennas can signiﬁcantly improve the
performance of spatial multiplexing systems. It is demon—
strated that improvements in terms of symbol error rate of
up to an order of magnitude are possible. We furthermore
ﬁnd that in general for a given SNR there is an optimum
XPD for which the symbol error rate is minimum. Finally,
we present simulation results and we show that our esti
mates closely match the numerical results. 1. INTRODUCTION AND OUTLINE The use of multiple antennas at both ends of a wireless
link has recently been shown to have the potential of drasti
cally increasing capacity through a technique called spatial
multiplexing [1H5]. This capacity gain depends strongly on
transmit and receive antenna spacing. In practice anten
na spacings of several wavelengths are required in order to
achieve signiﬁcant multiplexing gain. Unfortunately, large
antenna spacing increases both size and cost of base stations
and renders the use of multiple antennas in handsets very
difﬁcult. The use of dual—polarized antennas is a promis—
ing cost effective alternative where two spatially separated
antennas can be replaced by a single antenna element em—
ploying orthogonal; polarizations. Contributions. In this paper, we investigate the per—
formance of uncoded spatial multiplexing in systems em
ploying dualpolarized antennas, Although our techniques
are generally applicable, for the sake of simplicity, we con
sider a link with one dualpolarized transmit and one dual
polarized receive antenna. Our contributions are as follows. The work of H. Bolcskei was supported by F‘WF‘ grant .11868
TEC. R. Nabar’s work was supp0rted by the Dr. T. J. Rodgers
Stanford Graduate Fellowship. H. Bdlcskei is on leave from the
Institut fiir Nachrichtentechnik und IIochfrequenztechnik, Tech
nische Universitat VVien, Vienna, Austria. 07803—70414/01/$10.00 ©2001 IEEE 2437 We introduce a channel model for a dualpolarized
singleinput singleoutput link taking into account
spatial fading correlations and crosspolarization dis
crimination (XPD). We propose a method for computing estimates of the
uncoded symbol error rate of spatial multiplexing in
the presence of polarization diversity. We identify the propagation conditions where the use
of polarization diversity is beneﬁcial from an error
probability point of View, and we show that improve
ments in terms of symbol error rate of up to an order
of magnitude are possible. We demonstrate that our symbol error estimates
closely match the simulation results. Our method
can therefore be used to predict performance trends
analytically and helps avoiding timeconsuming com
puter simulations. Organization of the paper. The rest of this paper is
organized as follows. Section 2 introduces the channel mod
el for a dualpolarized singleinput single—output link and
states our assumptions. In Section 3, we derive estimates
for the uncoded symbol error rate of spatial multiplexing
as a function of spatial fading correlations, XPD, and SNR.
Section 4 provides simulation results and demonstrates that
our estimates closely match the simulation results. Finally,
Section 5 contains our conclusions. 2. THE CHANNEL MODEL We consider a system with one dualpolarized transmit
and one dualpolarized receive antenna. The channel is as
sumed to be ﬂat over the frequencyband of interest. The
inputoutput relation is therefore given by1 r=\/E,Hx+n, (1) where x [no $1]T is the 2 X 1 transmit signal vector
whose elements are taken from a ﬁnite (complex) constella
tion chosen such that the average energy of the constellation
elements is 1, r = [7'0 r1]T is the 2 x 1 receive signal vector, n is complexvalued gaussian noise with £{nnH } = (7312, hon her]
hm h1,1 1The superscripts T and H stand for transpose and conjugate
transpose, respectively. H=l is the channel transfer matrix which is also the polarization
matrix, and \/E_,, is a normalization factor. The polariza
tion matrix describes the degree of suppression of individual
co and cross—polarized components, cross correlation, and
cross coupling of energy from one polarization state to the
other polarization state. In practice, the following polar—
izations are generally considered: horizontal, vertical, and
145° slanted polarization. In this paper, we assume that
the transmitter and the receiver employ the same polar
izations, i.e., both of them employ horizontal and vertical
polarization for example. The signals 20 and :51 are trans
mitted on the two different polarizations, and To and 1'1 are
the signals received on the corresponding polarizations. We
emphasize that although we are dealing with one physical
transmit and one physical receive antenna, the underlying
channel is a 2—input 2—output channel, since each polar
ization mode is treated as an independent physical channel.
We assume that the channel is purely Rayleigh fading, i.e.,
the matrix H consists of (in general correlated) complex
gaussian random variables with zero mean. The more gen
eral case taking into account a lineofsight component is
treated in The correlation between the elements of the
matrix H and the variances of the elements depend on the
propagation conditions and the choice of polarizations, re
spectively.
Throughout the paper, we assume2 8{Iho,o[2} = £{lh1,12} = 1
S{lho.1I2} = £{Ih1,oI2} = a, where 0 < or s 1 depends on the XPD. Good XPD yields
small a and vice versa. The case a = 1 can also be in
terpreted as having two physical antennas on each side of
the link employing the same polarization. We furthermore
deﬁne the following correlation coefﬁcients3 Eihophaul = 5{h1,ohi,i} J5 x/E
_ ﬁ_ﬁ. For the sake of simplicity, throughout the paper, we as
sume that S{ho,ohi‘,1} = £{h1,oh3‘1} = 0. Measured values
of XPD and correlation coefficients have been reported for
example in [7H9]. 3. DERIVATION OF SYMBOL ERROR RATE In this section, we ﬁrst discuss the impact of polariza—
tion diversity on spatial multiplexing and then compute an
estimate of the symbol error rate. 3.1. Impact of Polarization Diversity Spatial multiplexing [1]—[5] has the potential to dramat—
ically increase the capacity of wireless radio links with no
additional power or bandwidth consumption. The basic
idea is that if scattering in the multi—antenna channel is rich
enough independent parallel spatial data pipes are created
within the same bandwidth, which ideally yields a linear (in
the number of antennas) capacity increase. Traditionally, 25 stands for the expectation operator.
3The superscript ’ stands for complex conjugate. 243 8 the ability to perform spatial multiplexing has been related
to a rich enough scattering environment. In the present case
virtual multiple antennas are created by employing differ
ent polarizations and the MIMO channel matrix is replaced
by the polarization matrix. It is therefore not clear a pri
ori how spatial multiplexing will perform in the presence
of polarization diversity and how the exact values of XPD
and fading correlations will inﬂuence the performance. We
can, however, quantitatively establish the beneﬁt of polar—
ization diversity. It is well known that the multiplexing gain
is maximized if the condition number of the channel matrix
is 1. Now, it is intuitively clear that for small a the individ
ual realizations of H tend to have lower condition number.
In fact, in the limiting case a = 0 (i.e. perfect XPD) every
realization of H yields orthogonal columns and hence high
multiplexing gain can be expected. 3.2. Error Probability Throughout the paper, we assume that the channel is
unknown in the transmitter, perfectly known in the receiver,
and that maximum—likelihood (ML) decoding is performed.
The receiver computes the ML estimate according to :2 = arg minllr — VEaHxllz,
X where the minimization is performed over the set of all pos—
sible codevectors. Let c and e be two different codevectors of size 2 x 1
and assume that c was transmitted. For a given channel
realization H, the probability that the receiver decides er
roneously in favor of the vector e is given by [10] E3 203, P(c —) ellI) = Q( d’(c,eH)) , (2) where d2(c.eIH) = “H(c  e)2
Upon deﬁning y = H(c — e) we get d2(c,eH) = Ilyll2 and
hence using the Chernoif bound Q(:c) S (5“2/2 it follows
from (2) that E 2
P(c —) elH) g e 4735"” . (3)
Since H was assumed to be gaussian it follows that the
2 X 1 vector y is gaussian as well. The average over all
channel realizations of the right—hand side in (3) is fully
characterized by the eigenvalues of the 2 x 2 covariance
matrix of y [11] deﬁned as Cy = E{yyH}. In particular,
denoting the eigenvalues of C, as z\.(C,,), we get 1 1
1 + AdCﬂﬁé— 1 + A2(Cy)ﬁ§’
where P(c —) e) = £H{P(c ——) eH)} is the pairwise error probability averaged over all channel realizations. Straight
forward manipulations reveal that4 C,, = log — eol2 + ac1 — ell2 + 2R{(co — eo)(c1— el)‘t\/r_x}
r*\/E(co — eol2 +C1 — 61'?) Tx/aﬂco — Col? + [61 r ell?)
alco — col2 +IC1 — 6112 + 2R{(co — eo)(C1 — myth/5} ' The eigenvalues of C3, are given by P(c ——> e) g (4) 472M} stands for the real part of a. a+dzt (a—d)2+4bc
Mo =
2
with a = [co—eo2+acl —e12 + 2R{(co — eo)(c1 — myth/E}
b = n/Eﬂco — 80]2 + Ici — 51'2)
c = rK/Eﬂco — eol2 + I61 — 61'2)
d = aco — eo2 + [c1 — el2 + 2’R{(co — eo)(01 — eﬁ‘tﬁ}. If no polarization diversity is used (Le. 01 = 1) and the
channel matrix is i.i.d., we have )‘1 = A2 = (co — (20]2 +
Ici — ella). In this case the error rate behavior is governed
by error events where only one out of the two symbols is
in error, say (co — 60) 7E 0 with co — eol2 = din1, where
dmin denotes the minimum distance of the scalar constella
tion used. Clearly, the error rate will decay for increasing
(1mm. In the general case, where o < 1 and the individual
entries in H are correlated the error events governing the
performance of spatial multiplexing depend on the channel
geometry induced by the correlation coefﬁcients and the
value of a. In order to avoid having to ﬁnd those error
events for a particular channel geometry, we average over
all possible error events including a weighting which takes
into account that different vector error events cause a dif—
ferent number of scalar symbol error events. In particular,
we want to study the inﬂuence of XPD on the error rate of
spatial multiplexing systems. It is therefore crucial to reveal
how the error probability behaves as a function of XPD for
a given SNR and given t and r. We assume that 4PSK is
employed. This implies that there are 240 error events. The
individual scalar error events (0, — er) can take values from
the set {0, idmimijdmmghdminﬂ + j), :bdm,,,(1 — j)}.
Now, with the relative frequency of an error event 5; where
c:  ei = [(co,. — €O.i) (cm — 81.0]T given by "‘i ‘ 240
with
4, :c = O
w(;1;)= 2, .13 = idmin; ijdmin )
{ 1, idmin(1+j),idmin(1 we estimate the average symbol error rate as 13 = Z n5,P(E,)3(Ei) (5) Here, P(e.;) is (4) evaluated for [(co,, — cod) (01,, — e1,,')]T
and 3(6‘) 2 l 0. In the next section, P is shown to reveal all the relevant
trends and a close match between the exact symbol error
rate and P is found. We note that (4) can be used to study the impact of t, r
and a on P analytically. For example, it follows immediate
ly from (4) and (5) that for t = r = 0 the quantity P will be
minimum for a = 1, i.e., for the case where no polarization
diversity is employed. Indeed, we will ﬁnd in the next sec
tion that polarization diversity improves the multiplexing
gain only in the presence of high correlations. 2.
1 (Cod —eo,.') 95 0, (Cm 81.i) 9'5 0
(60,; — 80,.) = 0 or (cm —ei,.) 3 2439 4. SIMULATION RESULTS In this section, we provide simulation results demon
strating the performance of spatial multiplexing in the p
resence of polarization diversity and spatial fading corre
lation. In particular, we show that P reveals the impact
of polarization diversity on the performance of spatial mul
tiplexing quite accurately. We simulated a system with 1
dual—polarized transmit and l dualpolarized receive anten
na using 4PSK and employing an ML receiver. The signal— to—noiseratio (SNR) was deﬁned as SNR = 10 log Simulation Example The ﬁrst simulation example
serves to demonstrate that P provides an accurate estimate
of the symbol error rate for high SNR. For t = 0.5, 'r = 0.3
and a = 0.4, Fig. 1 shows the symbol error rate obtained
using Monte Carlo simulations along with the estimated
symbol error rate P. It can be seen that especially in the
high S_NR regime the two curves match well. Note, however,
that P is not a strict upper bound on the symbol error rate. 10" ,. ., ID Symbolotrurma 1n‘ _ “1...; 10" i D 10 SNde Fig. 1. Symbol error rate as a function of SNR. Simulation Example 2. The second simulation ex
ample serves to demonstrate the beneﬁt of polarization di
versity. For an SNR_of 15dB, Fig. 2 shows the symbol er
ror rate along with P as a function of a for various values
of t and for 1‘ = 0. It can be seen that for low trans
mit correlation the use of polarization diversity leads to a
performance degradation or equivalently reduced multiplex
ing gain. (Recall that a = 1 can be interpreted as having
two physical antennas on each side of the link all of which
employ the same polarization.) This result conforms with
the investigations in the last paragraph of Sec. 3.2. When
the transmit correlation starts to increase and the condi
tion number of the channel matrix realizations increases or
equivalently the angle between the realizations of the two
columns decreases, polarization diversity yields improved
spatial separation and hence increases the multiplexing
gain. We found that starting at t = 0.85 polarization di
versity, i.e., a < 1 starts improving the multiplexing gain.
We can also see that in the case of fully correlated transmit
antennas, i.e., t = 1, there is an optimum value of a for
which the symbol error rate is minimum. Observe that this
minimum is accurately predicted by P. Fig. 2 furthermore
shows that for high transmit correlation the use of polar
ization diversity can improve the symbol error rate by up to an order of magnitude.  +Slmulaiadl=01 .
.. +srnutaiaai=a7 .. l , @— Simatadl= e EsmedlSUJ + Estmuladqu e Estimad=1 0.9 0.1 Fig. 2. Symbol error rate as a function of a for various
values of the transmit correlation coeﬂicient t. Simulation Example 3. In the last simulation ex
ample, we study the performance of spatial multiplexing
with and without polarization diversity in the presence of
receive correlation only. Fig. 3 shows the symbol error rate
along with P for t = 0 and various values of r. We observe
that for r = 1 the use of polarization diversity increases the
multiplexing gain or equivalently reduces the symbol error
rate. This effect, however, is much less pronounced than in
the case of transmit correlation only. The reason for this is
that in the presence of transmit correlation there are code
vectors which “tend to excite the null space of the channel
matrix” and hence yield very high probability of error. This
does not happen in the case of receive correlation only. A
more detailed description of this observation is provided in [12]. For r = 1, the optimum value of (1 seems to be the
same as in the case of transmit correlation only with t = 1. 10_‘ . .. .l. . —e— slimlaladr:o.1
'+ simulatedr=OJ e— Simulllodr=1
'5 Esmladr=0.1
v * Estimatedr=01
e. Bumled”! y
0.8 [1.7 0.6 Fig. 3. Symbol error rate as a function of a for various
values of the receive correlation coefﬁcient r. 5. CONCLUSION We studied the use of polarization diversity for spatial
multiplexing and found that in the presence of high spa
tial fading correlation dualpolarized antennas can yield a 2440 signiﬁcantly improved multiplexing gain. In particular, we
computed an estimate of the uncoded symbol error rate
which was found to be very accurate in the high SNR regime
and which allows to study the impact of polarization di—
versity on the performance of spatial multiplexing without
having to resort to timeconsuming computer simulations.
We demonstrated that especially in the presence of transmit
correlation the use of polarization diversity can yield signif
icant improvements in terms of symbol error rate. Further—
more, we found that for high spatial fading correlation and
for a given SNR in general there is an optimum value of oz
for which the symbol error rate is minimum. Our symbol
error estimate accurately predicts this point. Finally, we
provided simulation results. 6. REFERENCES [1] A. J. Paulraj and T. Kailath, “Increasing capac
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