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1
Polarized Rician Fading Models for Performance
Analysis in Cellular Networks
Orestis Georgiou
Toshiba Telecommunications Research Laboratory, 32 Queens Square, Bristol, BS1 4ND, UK
Abstract—Polarization misalignment can significantly affect
wireless communications and should be included in network
models to facilitate performance analysis and design. We there-
fore derive the distribution of the received signal power for the
fading experienced by linearly polarized wireless links which are
misaligned by a fixed angle, and calculate the downlink outage
and coverage probabilities in cellular networks using tools from
stochastic geometry. The derived model is therefore tractable and
can provide insights into how we design cellular systems.
I. INTRODUCTION
Small-scale fading experienced in wireless communications
has a significant effect on the reliability and throughput of
point-to-point links. Consequently, a plethora of fading models
have been developed characterizing the statistical properties
of wireless channels in different propagation environments
[1]. The most popular of these is the Rayleigh fading model
which prescribes an exponential distribution for the channel
gain. The Rayleigh pdf assumes that all multipath received
power is diffusive i.e. there is no specular line-of-sight (LoS)
paths from transmitter to receiver and individual multipath
components have powers which are negligible compared to
the total average power received. Varying the number of
specular and non-specular multipath components, an almost
complete representation of fading models can be analytically
derived [2] encompassing for example Rician fading, and more
exotic ones such as the two-wave with diffuse power (TWDP).
The mathematical tractability of fading distributions has been
essential in network performance analysis and design over the
past decade and thus vital for future deployments [3], [4].
In LoS wireless transmissions it is paramount that the
antennas at both ends of the channel use the same polarization.
Hence, most cellular base stations (BSs) are equipped with
vertically polarized antennas because vertical polarization is
most effective when radiating in all directions such as widely
distributed mobile units (MUs), particularly for rooftop BSs
where horizontal polarization can be 6-10dB weaker than
vertical. In practice, a misalignment of polarization of 45◦
to 90◦degrees in a linearly polarized system can degrade the
signal by 3to 20dB. Equations characterizing the polarization
mismatch loss, also known as polarization loss factor (PLF),
between two spatially separated antennas in free-space were
given as early as 1965 [5] for all combinations of linear,
circular and elliptical polarizations. However, the polarization
of mobile and hand-held units is often random, depending
on how they are held by the user [6]. The same holds in
non-cellular paradigms such as Wi-Fi, body area and sensor
Fig. 1. An unpolarized ray is specularly reflected off a non-metalic surface.
The reflected ray is polarized to a large degree in the plane parallel to the
reflecting surface, especially if the incident ray is at the Brewster angle.
networks, etc. Polarization diversity through MIMO systems
can mitigate polarization mismatch improving SNR by up
to 12dB or 9dB in LoS or non-LoS channels respectively
[7]. This is why many modern BSs transmit two orthogonal
signals at ±45◦polarizations whilst smartphones (tablets) are
equipped with two cellular (Wi-Fi) antennas.
Despite several attempts to propose fading models for
polarized communications, simple fading models amenable to
mathematical tractability are to date not available. This claim
is corroborated by the absence of polarization effects in the
plentiful network analysis, performance, and design literature
[3], [4]. This is particularly surprising given the importance
of polarization mismatch and diversity as explained above. It
is therefore the purpose of this letter to derive such a fading
model from first principles and also to give examples of its
use in cellular networks using tools from stochastic geometry.
II. RICIAN FADING WITH POLARIZATION
For simplicity we consider a single input single output
(SISO) antenna system and modify the textbook derivation of
Rician fading [1] to include polarization effects. This approach
will be extended to more exotic fading distributions such as the
TWDP model [2] and MIMO systems in our future work. A
transmitter sends a simple sinusoidal signal ET(t) =cos 2πft
at a carrier frequency of f. The received signal arising from
the Nmultiple paths of the channel can be expressed as
ER(t) =
N
X
i=0
aicos(2πf t +φi),(1)
where we have assumed a negligible Doppler shift and the
ith multipath signal component has an amplitude aiand a
phase φi. We treat the i= 0 term as the LoS signal1such
that φ0=0. The number of multiple paths is Nwhich can be
1The following analysis still holds if the dominant specular component [8]
i=0 in (1) is a multi-path ray (i.e. non-LoS) with a significantly higher power
than all other multi-path rays i > 0(diffuse). This may occur when the i=0
path undergoes a small number of favourable (e.g. metallic) reflections.
2
treated as equivalent to the number of scattering, reflecting, or
diffracting objects in the channel. Eq. (1) can be rewritten
in terms of inphase and quadrature notation as ER(t) =
(a0+X) cos 2πf t −Ysin 2πft, where X=PN
i=1 aicos φi
and Y=PN
i=1 aisin φi. At this point, the textbook derivation
usually assumes that due to the randomness in the reflecting
objects, the multipath components i > 0will arrive at the
receiver with phases φiuniformly distributed in the range
[0,2π]. Moreover, since Nis large, the central limit theorem
(CLT) will cause Xand Yto converge to a normal distribution
with zero mean and standard deviation σ. Hence, the power of
the received signal is given by (a0+X)2+Y2which follows
a non-central chi-squared distribution if a0>0and σ=1, and
exponential if a0= 0, i.e. the celebrated Rician and Rayleigh
fading models respectively [1].
Instead, we modify the above approach to incorporate
polarization and antenna mismatching effects assuming a
linearly polarized transmitter and receiver which have a LoS
polarization mismatch of θ0which is fixed but unknown (i.e.
a free parameter). We first note that the amplitude of the
received signals aiare attenuated by pathloss and for i > 0
have also suffered absorption and scattering effects when
reflecting off different surfaces. Significantly, the polarization
of the multipath rays at the receiver will typically not be
identical to that of the transmitter. That is, electromagnetic
(EM) radiation will experience changes in its polarization due
to reflections from various surfaces, refraction when travelling
through media of different refractive indices, and scattering off
free charged particles (see Thomson scattering). Ignoring the
latter and focusing on the first two more dominant effects, the
extent to which polarization changes occur can be accurately
described by Fresnel’s equations which depend upon the angle
of incidence. As this is impossible to calculate without a
detailed terrain model and full ray-tracing software, it suffices
to say that reflections off non-metallic surfaces such as asphalt
roadways, buildings, glass, and water reflect EM radiation
predominantly polarized in the plane parallel to the reflecting
surface. This is particularly the case when the incident ray is
at the Brewster angle as illustrated in Fig.1. This is also why
sunglasses are vertically polarized as to filter-out horizontally
polarized light. Hence, in an urban environment it is reasonable
to assume that multipath rays have experienced a number of
reflections off randomly oriented objects (possibly birefringent
causing a polarization rotation) and therefore arrive at the re-
ceiver at a randomly oriented linear polarization. Significantly,
each of these polarized rays (except the LoS one) will have a
mismatch angle θi>0relative to the polarization of the receive
antenna. The attenuation due to polarization mismatch can be
quantified through the polarization loss factor PLF = cos2θ0
describing the power loss when two linearly polarized antennas
in LoS are rotated relative to each other by an angle θ0[5].
Therefore, accounting for the PLF along all multipath rays
(LoS and non-LoS), we re-express the received signal as
ˆ
ER(t) = ˆa0+ˆ
Xcos 2πf t −ˆ
Ysin 2πf t, (2)
where ˆ
X=PN
i=1 ˆaicos φi,ˆ
Y=PN
i=1 ˆaisin φiand ˆai=
ai√cos2θiwith θ0corresponding to the LoS polarization
Fig. 2. The pdf and cdf of equations (3) and (4) using σ=1 and a0=K=2.
angle mismatch between transmitter and receiver antennas,
and θifor i > 0, the non-LoS polarization angle mismatch
between the ith arriving ray and the receiver antenna.
The distribution of received polarization angles θi>0in rich
multipath scattering environments is unknown experimentally,
but much like the received phases φi, it is expected that the
θi>0are uniformly distributed in [0,π
4]and [0,π
2]when there
is polarization diversity (i.e. two ±45◦BS antennas) and there
isn’t polarization diversity, respectively. Hence, we modify the
textbook CLT such that ˆ
X, ˆ
Y∼N(0,ˆσ2)with ˆσ=σand
ˆσ=σ
√2when there is and there isn’t polarization diversity,
respectively. To calculate the pdf of the received power P=
(ˆa0+ˆ
X)2+ˆ
Y2we first define ¯
X, ¯
Y∼N(0,1) and rewrite the
rescaled power as P/ˆσ2=ˆa0
ˆσ+¯
X2+¯
Y2which has a non-
central chi-squared distribution. After a change of variables
we arrive at our main result, a new pdf for the received power
fP(x) = 1
2ˆσexp −x+ ˆa2
0
2ˆσ2I0ˆa0√x
ˆσ2,(3)
for x≥0where I0is the modified Bessel function of the first
kind. Similarly, the cumulative distribution function (cdf) is
FP(x) = 1 −Q1ˆa0
ˆσ,√x
ˆσ,(4)
where Q1(a, b) = R∞
bxe−x2+a2
2I0(ax)dxis the Marcum Q-
function of the first order which can be approximated by
Q1(a, b)≈exp −eν(a)bµ(a)for some polynomial functions
νand µ[9]. The mean and variance of Pare given by ˆa2
0+2ˆσ2
and 4ˆσ2(ˆa2
0+ˆσ2)respectively which have a strong dependence
on θ0. The pdf and cdf are plotted in Fig. 2for different
values of θ0highlighting the qualitative difference in the pdfs
assuming no polarization diversity is employed at the BS.
The Rician factor Kis defined as the ratio of the signal
power of the dominant LoS component over the local-mean
scattered power K=a2
0
2σ2and is usually taken between 3 and
8dB. Here, we define a polarization dependent Rician factor
ˆ
K=ˆa2
0
2ˆσ2with ˆa0=a0√cos2θ0. Note that θ0is a fixed
parameter supported in [0,π
4]and [0,π
2]when there is or there
isn’t polarization diversity at the BS, respectively.
Remark 1: Equations (3)and (4)are the main result of this
letter. The derivation of the modified pdf follows closely that
found in textbooks with only one new parameter, namely, the
polarization mismatch angle θ0. Significantly, the derivation
has been antenna gain diagnostic which can be separately
accounted for in 2D or 3D [10].
3
III. APP LI CATI ON TO CELLULAR NETWOR KS
We will now apply the derived fading model (3) to calculate
the connectivity and coverage in cellular downlink networks.
1) Downlink System Model: We model a cellular network
topology as a two dimensional stationary Poisson point process
(PPP) Φ, with intensity function λ. Each point in Φrepresents
a BS. Let di≥0represent the Euclidean distance from a MU to
the ith nearest BS, such that d1≤d2≤. . . Assuming that this
MU will associate with its nearest BS, the pdf of the distance
d1to the nearest BS is given by fd1(r)=2πλre−λπ r2which
also imposes a constraint on the interference caused from all
other BSs, which, by definition, are located at a distance of at
least d2≥rfrom the MU [3]. In this letter, our main metric
of interest is the signal-to-interference-plus noise-ratio (SINR)
between the associated BS and MU in the downlink. Assuming
that all BSs transmit with equal powers T(no power control)
SINR =TP1g(d1)
N+γI,(5)
where Nis the average background noise power, and I=
Pk>1TPkg(dk)is the interference received at the MU by
other BSs. The received signal power P1from the associated
BS is modelled as a random variable with pdf according to (3)
and therefore depends on the LoS polarization mismatch θ0.
In contrast, the received signal powers Pkfrom the interfering
BSs (k > 1) are modelled as iid exponential random variables
with mean one (i.e. Rayleigh fading). This is reasonable
as it is unlikely that interfering signals from distant non-
associated BSs have a specular component at the receiving
terminal. A similar approach was adopted recently in [11]
where signals from LoS and non-LoS BSs experience different
pathloss attenuations resulting in a counter argument for BS
densification. Instead, we focus on the polarization-modified
fading model (3) and use a common attenuation function [12]
g(di) = δ/(+dη
i), δ, ≥0(6)
which follows from the Friis transmission formula since the
long time average signal-to-noise ratio (SNR) at the receiver
(in the absence of interference) decays with distance like
SNR ∝d−η
i, where η≥2is the pathloss exponent. The
buffer is included in (6) to make the pathloss model non-
singular at zero, while the wavelength dependent constant δ
makes g(di)dimensionless. The factor γ∈[0,1] in (5) serves
as a weight for the interference term and models the frequency
reuse factor (typically at 1
3,1
4,1
7). When γ= 0 transmissions
from all interfering BSs are completely orthogonal to the as-
sociated nearest BS such that there is no inter-cell interference
and SINR=SNR. When γ=1 there is no frequency reuse and
communications are said to be interference-limited.
2) Outage Probability: The outage probability of a wireless
link is a fundamental performance metric of wireless networks
and has been extensively studied in both cellular and mesh
topologies. The connection probability defined as H(d1) =
P[SINR ≥qd1]is the complement of the outage probability
conditioned on the distance d1to the nearest BS and can be
thought of as the probability that at any given instance of time,
the link between a MU and its nearest BS situated at a distance
d1away can achieve a target SINR of at least q. One way of
calculating it is by conditioning the channel realization P1
and writing H(d1) = EP1hPhI ≤ TP1g(d1)−qN
γq P1, d1ii
which requires knowing the distribution of I. The latter can
be obtained via the inverse Laplace transform of the moment
generating function MI(s) = EI[e−sI]of Ias given in [12]:
MI(s) = exp −2πλ Z∞
d11−EPk[e−sTPkg(r)]rdr
= exp −2πλsTδ2F11,1−2
η,2−2
η,−Tδs+
dη
1
dη−2
1(η−2) .
(7)
We can however simplify the computation by rearranging the
order of the integrals required to obtain H(d1)and replacing
sby js in (7) to get the characteristic function of Iwhich we
will then inverse Fourier transform to get a real-valued cdf
H(d1) = 1
2πZ∞
−∞
MI(js)Z∞
0
fP1(x)Z
Txg(d1)−qN
γq
0
ejsz dzdxds
=1
2πZ∞
−∞
MI(js)1
sj+qγe−jNs
γ−ˆa2
0Tsg(d1)
jqγ +Tsg(d1)
jqγ +Tsg(d1)ds,
(8)
where j=√−1,fP1(x)is given by (3) and the polarization
mismatch appears in ˆa0=a0√cos2θ0. Note that the inverse
Fourier operation can be expressed in closed form only in few
cases and for the present one requires numerical integration.
Whilst (8) is an exact calculation of H(d1)(and therefore
the outage probability), it offers little engineering insight. We
may circumvent this by instead conditioning on the interfer-
ence realization I, such that the connection probability can
be expressed as H(d1) = EIhPhP1≥q(N+γI)
Tg(d1)I, d1ii and
make use the cdf of P1given in (4) to calculate
H(d1) = EIhQ1ˆa0
ˆσ,1
ˆσsq(N+γI)
Tg(d1)i
=Q1(A, B)−γB2e−A2+B2
2I0(AB)
2NEI[I] + O(γ2),
(9)
where in the last line of (9) we have Taylor expanded for
small γ1and defined A=ˆa0
ˆσand B=1
ˆσqqN
Tg(d1).
This approximation will be poor in the case of γ= 1 e.g.
where universal frequency reuse is adopted. Note that for noise
limited systems (γ= 0) the connection probability H(d1)is
exactly given by the ccdf of P1(see (4)). Through (9) we
have also obtained a first order approximation to H(d1)for
non-zero γsince it is straight forward to calculate
EI[I] = EPk,dkhX
k>1TPkg(dk)i= 2πTλE[Pk]Z∞
d1
g(r)rdr
= 2πTλδ 2F11,1−2
η,2−2
η,−
dη
1
dη−2
1(η−2) ,
(10)
where 2F1is the Gauss hypergeometric function. For = 0
equation (10) becomes EI[I] = (η−2)dη−2
1−1, while for
η=4 simplifies to EI[I]= 1
2√arctan √
d2
1. Note that equation
4
Fig. 3. Plots of the connection probability H(d1)and the coverage probability
pcfor different values of frequency reuse factor γand LoS polarization
mismatch angle θ0.Left: Marker points are obtained through exact numerical
integration of (8), and solid curves correspond to the first order approximation
given by (9) using q= 0dB. Right: Solid and dashed curves indicate exact
numerical integration of (11) using (8) and (9) to leading order, respectively,
and marker points are obtained through Monte Carlo computer simulations.
(10) diverges for η= 2 thus requiring that η > 2. Second
and higher order correction terms in (9) can be calculated
using the moments EI[In] = dn
dsnMI(−s)s=0 obtainable
via (7). In fact, the Taylor expansion (9) is well behaved if
the random variable Ihas finite moments and hence holds
for more general fading in the interference links Pk>1[13].
Hence, the analytical form of (9) may offer further engineering
insights e.g. can be used compute the diversity and coding
gains in multi-antenna or distributed-antenna-systems (DAS).
3) Coverage Probability: The coverage probability pcgives
the probability that a typical MU is in coverage, i.e. can
achieve an SINR in the downlink greater than the threshold q.
To calculate pcwe de-condition the distance d1to the nearest
BS, however maintain the θ0dependence as we cannot know
the MU antenna polarization and orientation distributions [6]
pc(θ0) = Z∞
0
H(r)fd1(r)dr. (11)
4) Numerical Simulations: Monte Carlo simulations
(MCSs) were employed to validate the network performance
metrics H(d1)and pc(θ0). No polarization diversity is as-
sumed such that ˆσ=σ
√2. An effort was made to use
similar parameters as those presented in [3]: λ=1
8BSs/km2,
T=σ=δ= 1,a0= 10N= 2, and = 0,η= 4.
Fig.3a. compares the exact connection probability (8) to the
asymptotic approximation (9), whilst Fig.3b. compares the
coverage probability (11) to that obtained from MCSs. An
excellent agreement is observed between exact expressions
and the approximate one for H(d1), and the simulated results
for pc(θ0)respectively. Higher order terms in γare however
required for calculating pc(θ0)as approximation errors in (9)
become significant at high SINR thresholds q, as expected.
This is because γand qhave an almost similar effect to H(d1)
and hence the approximation is poor for large γand/or q.
Significantly, Fig.3a. illustrates the severity of polarization
mismatch effect θ0>0on the link-outage probability 1−H(d1)
at almost all length scales d1km. For example, at d1=1.5km
the connection probability is predicted to drop by 5dB (from
0.9to 0.3) due to a θ0=π
2polarization mismatch. Remarkably,
a polarization mismatch has a greater impact on performance
than frequency reuse, at least in the current setting. A similar
effect is observed for the coverage probability (see Fig.3b.),
thus highlighting the importance of polarization diversity.
IV. CONCLUSION
We have derived a simple fading model (see equations (3)
and (4)) for the received signal power experienced by linearly
polarized LoS wireless communications with the modified
Rician fading pdf parametrized by just one new parameter, the
LoS polarization mismatch angle θ0. The model was applied to
a cellular downlink example setting in order to calculate outage
and coverage probabilities using tools from stochastic geome-
try. It was thus shown that a polarization mismatch in the LoS
component can have a greater impact on performance than
frequency reuse. Whilst more involved methods for calculating
these and similar performance metrics exist for more general
fading models [8], [13], it was demonstrated in this letter that
the proposed polarized Rician fading model is mathematically
tractable, and can hence shed light on the impact of not just
channel parameters and physical-layer transmission schemes,
but also on the polarization properties of the antennas and their
effect towards network performance. For example, it would be
interesting to calculate polarization multiplexing and energy
harvesting gains under polarized Rician models in MIMO cel-
lular, distributed antennas systems e.g. CoMP, and future Wi-Fi
paradigms. Therefore, a proper analysis and generalization of
the polarized pdf proposed herein can significantly impact the
design of equalization methods, and diversity schemes used in
interference limited wireless networks.
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