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Abstract

Indoor base stations (BSs) equipped with multiple-input multiple-output (MIMO) antenna arrays are commonly deployed in the vicinity of a wall. The wireless friendliness of the wall, determined by the intrinsic electromagnetic (EM) and physical properties of the wall material, significantly influences the indoor wireless performance and thus needs to be thoroughly considered during building design. In this article, for a rectangular room with a BS deployed near one of the walls, by deriving the asymptotic expression of lower-bound indoor wireless capacity of a UE location-specific channel, we reveal that the impact of the BS transmission power and that of the wall material properties on the lower-bound indoor capacity can be decoupled. More specifically, in our derived lower-bound indoor wireless capacity, the properties of the wall material are captured by the logarithmic eigenvalue summation (LES) and logarithmic eigenvalue product (LEP), which are both independent of the BS transmit signal-to-noise ratio (SNR). To simplify the wireless-friendliness evaluation of a wall by leveraging such decoupling, we derive both the LES and LEP in closed forms for a UE location-specific channel, and define the spatially averaged LES, the spatially averaged LEP, and the upper-bound outage probability (all over the room of interest) as new metrics for fast evaluating the wireless friendliness of the wall closest to the BS. Numerical results verify the effectiveness of the three proposed metrics and reveal the crucial impact of room settings and wall materials on the indoor capacity. The proposed approach will enable architects and civil engineers to quickly select building materials according to their wireless friendliness.
IEEE TBC 1
Lower-bound Capacity Based Wireless Friendliness Evaluation for Walls as Reflectors
Yixin Zhang, Graduate Student Member, IEEE, Jiliang Zhang, Senior Member, IEEE,
Xiaoli Chu, Senior Member, IEEE, Jie Zhang, Senior Member, IEEE
Abstract—Indoor base stations (BSs) equipped with multiple-
input multiple-output (MIMO) antenna arrays are commonly
deployed in the vicinity of a wall. The wireless friendliness of
the wall, determined by the intrinsic electromagnetic (EM) and
physical properties of the wall material, significantly influences
the indoor wireless performance and thus needs to be thor-
oughly considered during building design. In this article, for a
rectangular room with a BS deployed near one of the walls,
by deriving the asymptotic expression of lower-bound indoor
wireless capacity of a UE location-specific channel, we reveal that
the impact of the BS transmission power and that of the wall
material properties on the lower-bound indoor capacity can be
decoupled. More specifically, in our derived lower-bound indoor
wireless capacity, the properties of the wall material are captured
by the logarithmic eigenvalue summation (LES) and logarithmic
eigenvalue product (LEP), which are both independent of the BS
transmit signal-to-noise ratio (SNR). To simplify the wireless-
friendliness evaluation of a wall by leveraging such decoupling,
we derive both the LES and LEP in closed forms for a
UE location-specific channel, and define the spatially averaged
LES, the spatially averaged LEP, and the upper-bound outage
probability (all over the room of interest) as new metrics for fast
evaluating the wireless friendliness of the wall closest to the BS.
Numerical results verify the effectiveness of the three proposed
metrics and reveal the crucial impact of room settings and wall
materials on the indoor capacity. The proposed approach will
enable architects and civil engineers to quickly select building
materials according to their wireless friendliness.
Index Terms—Eigenvalue, EM wave, indoor capacity, materi-
als, MIMO, outage, wall reflection, wireless friendliness.
I. INTRODUCTION
It is predicted that mobile video will rise to account for
nearly 80% of all mobile data traffic by 2022, while 70% to
90% of the overall mobile data will be generated indoors [1].
Superior indoor multimedia experience can be supported by
the wireless broadcast services [2]–[4] of mobile networks,
e.g., with the LTE-Broadcast (LTE-B) technology [5]. It is
noteworthy that an indoor mobile network needs to ensure a
high capacity to support high-data-rate services such as live
video streaming [6].
Multiple-input multiple-output (MIMO) technique can be
used to boost indoor wireless performance by enabling mul-
tiple parallel spatial streams between transceivers without
requiring additional bandwidth or higher transmission power
[7]–[9]. To avoid any negative visual impact or inconvenience
on a room, a popular way of deploying indoor base stations
(BSs) equipped with MIMO antenna arrays is to put the BSs
in the edge part of a room, i.e., close to a wall [10], [11].
Yixin Zhang, Jiliang Zhang, and Xiaoli Chu are with the Department of
Electronic and Electrical Engineering, University of Sheffield, S10 2TN, UK.
E-mail: jiliang.zhang@sheffield.ac.uk.
Jie Zhang is with the Department of Electronic and Electrical Engineering,
University of Sheffield, Sheffield, S10 2TN, UK, and also with Ranplan
Wireless Network Design Ltd., Cambridge, CB23 3UY, UK.
This work was supported in part by the European Union’s Horizon 2020
Research and Innovation Programme under Grant 766231 and Grant 752644.
The settings of indoor environments, in terms of room sizes
and aspect ratios, wall material’s thicknesses and relative per-
mittivities, and BS locations and configurations, will affect the
indoor wireless performance [12]–[15]. As such, the materials
of walls have to be carefully selected taking their impacts of
properties on the indoor wireless performance into account in
the building planning and design stage. Generally, the wall
in the vicinity of a BS can be regarded as a lossy dielectric
structure [16]–[18]. An electromagnetic (EM) wave would
suffer reflection loss after hitting on the wall’s surface due to
multiple internal reflections. Measurement results have shown
that the reflection loss is dependent on the incident angles and
polarisation of EM waves, as well as the EM and physical
properties of wall materials [19], [20], which is mathematically
characterised by the Fresnel equations [16], [21]–[23].
The authors in [24], [25] utilized frequency-selective sur-
faces (FSSs) to change the EM properties of the walls in real-
time for the purpose of providing good EM isolation between
rooms. Nevertheless, FSS attached walls were studied only as
active spatial and frequency filters, while the impact of other
EM parameters of walls, such as permittivity and conductivity,
on indoor wireless capacity was not investigated.
In order to predict the impact of a wall that is closest
to the BS on the indoor wireless performance, we proposed
wireless friendliness as a new performance metric of a wall in
[26], where we also developed an approach to evaluating the
wireless friendliness of a wall by adopting the indoor spatially
averaged capacity of a multipath channel as a metric, which
is a function of and is affected by the BS transmission power.
Note that the capacity of the two-ray channel comprising of
the line-of-sight (LOS) path and the wall reflection (WR)
path [18], [27] and the capacity of a multipath channel that
incorporates other multipath components in addition to the
LOS path and the WR path have the same monotonicity, which
can be proven using the results in [28]. Therefore, in this work,
we propose to study a wall’s wireless friendliness based on the
capacity of the two-ray channel. By exploiting the asymptotic
capacity of this two-ray channel, we find that its eigenvalues
can be used to separate the BS transmit signal-to-noise ratio
(SNR) from the interference between the LOS path and the
WR path that determines the wireless friendliness of the wall.
As will be shown in Section II, by leveraging the logarithmic
sum or logarithmic product of the eigenvalues of the two-ray
channel, the influence of the BS transmission power on a wall’s
wireless friendliness can be removed. Consequently, compared
with the metric in [26], the three new metrics proposed
herein no longer require the calculation of the following four
parameters: the BS transmit SNR, the power ratio of the LOS
and WR paths to the other multipath components, and the
two eigenvalues of the two-ray channel, which facilitates a
faster and simpler approach to evaluating a wall’s wireless
friendliness.
IEEE TBC 2
In this article, we investigate how the transverse dimension
(equal to the length or width of a room), longitudinal dimen-
sion (thickness), and dielectric properties (relative permittivity
and conductivity) of the wall closest to the BS affect the
indoor LOS MIMO downlink transmissions in a rectangular
room, which may have various room sizes, aspect ratios, and
distances between the BS and its closest wall. Given that the
wall closest to the BS would exert a much greater impact
on the received signal strength than the other walls due to
the dominating distance-dependent path loss over the much
weaker reflection gain [26], in this article, we focus on the
wireless friendliness evaluation of the wall closest to the BS.
For brevity, the wall closest to the BS in a room is referred
to as the wall hereafter.
The contributions of this work are summarised as follows:
We adopt a two-ray channel model incorporating both the
LOS path and the WR path from the BS’s closest wall,
and derive the lower-bound capacity per unit bandwidth
in the medium and high SNR regimes for an arbitrary
user equipment (UE) location in closed-form.
We show that, while being independent of the BS transmit
SNR, the logarithmic eigenvalue sum (LES) and logarith-
mic eigenvalue product (LEP) in the derived lower-bound
capacity per unit bandwidth of a UE location-specific
channel can jointly characterise the wireless friendliness
of the wall.
Based on the UE location-specific LES and LEP, we
propose the spatially averaged LES, the spatially averaged
LEP, and the upper-bound outage probability of the
room as new metrics for evaluating the wall’s wireless
friendliness. Note that the three proposed metrics for fast
evaluating the wireless friendliness of a wall as a reflector
can be applied to every wall in the considered room when
the channel model takes the WRs from that wall into
consideration.
We also show that the upper-bound outage probability of
a room can be calculated by the cumulative distribution
function (CDF) of the UE location-specific LES and LEP
in the medium and high SNR regimes, respectively.
The effectiveness of the three proposed metrics is nu-
merically evaluated under different room sizes and as-
pect ratios, wall permittivities and conductivities, wall
thicknesses, and BS-wall distances, and is compared with
respect to their advantages, limitations, and applicability.
The rest of the article is organized as follows. In Section
II, we describe the system model for indoor LOS MIMO
downlink transmissions taking the WRs from the BS’s closest
wall into account. In Section III, we derive the lower-bound
channel capacity, LES and LEP for an arbitrary UE location
in closed forms, based on which we propose three metrics for
fast wireless friendliness evaluation of the wall in Section IV.
Numerical results are presented and discussed in Section V.
Finally, we conclude this article in Section VI.
II. SY ST EM MO DE L
As shown in Fig. 1, we consider an indoor LOS MIMO
downlink, where the BS is placed with a small distance of h
BS
( antennas)
UE
( antennas)
1
d
Wall
2
d
2
Τ
q
ε
Τ
q
1
R
q
2
R
q
b
h
D
D
T
N
R
N
Antenna
Linear array
Fig. 1. MIMO channel model with both LOS path and WR
path
m to its closest wall and one UE is arbitrarily positioned in the
room. The BS and UE are each equipped with a uniform linear
array (ULA) with the inter-antenna spacing of Dm, consisting
of NTand NRantennas, respectively. The ULAs at the BS
and the UE are both assumed to be parallel to the wall closest
to the BS. The received signal yCNR×1at an arbitrary UE
location1is given by the transmitted signal xCNT×1, the
channel matrix HCNR×NTand the additive white Gaussian
noise nas
y=Hx +n.(1)
The multiple reflections inside the wall as a homogenous
dielectric reflector can be modelled by the equivalent reflection
coefficient, which is given by [21]
Γ = 1exp(j2δ)
1Γ02exp(j2δ)Γ0,(2)
δ=2πb
µpεsin2α, (3)
where αdenotes the equivalent incident angle of reflections
of all orders under far-field plane wave assumption, µdenotes
the wavelength of the EM waves in the air, bdenotes the
thickness of the wall, εdenotes the relative permittivity of the
wall and is given by ε=ε0j17.98σ/f, where ε0=ufv,
σ=rf tis the conductivity of the wall material, fdenotes the
signal’s frequency, and u,v,rand tare constant parameters
given in [21, Table III], and Γ0denotes the first-order reflection
coefficient, which is given by ΓTE for the transverse electric
(TE) polarisation or ΓTM for the transverse magnetic (TM)
polarisation of the incident electric field, respectively:
ΓTE =cos αpεsin2α
cos α+pεsin2α,(4)
ΓTM =cos αq(εsin2α)2
cos α+q(εsin2α)2
.(5)
1In Sections II and III, we mainly study the lower-bound capacity of the
channel between the BS and an arbitrary UE location. The variables denoting
the UE location are dropped for brevity.
IEEE TBC 3
The wall-reflected EM waves propagating along the WR
path will arrive at the UE receiver in addition to the EM waves
along the LOS path. Based on Friis’ formula, the channel
matrix Hat a specific UE location can be deterministically
modelled as [26]
H=A1hH
β1hα1+A2hH
β2hα2,(6)
αl=2π
µDcos θTl, βl=2π
µDcos θRl,
A1=µ
4πd1
ej2πd1
µ, A2=µΓ
4πd2
ej2πd2
µ,
hαl=ejNT1
2αl, ej1NT
1
2αl, ..., ejNT1
2αl,
hβl=ejNR1
2βl, ej1NR1
2βl, ..., ejNR1
2βl,
where l∈ {1,2},{·}Hrepresents the complex conjugate trans-
pose of a vector or matrix, d1and d2denote the length of the
LOS path and the WR path, respectively, θT1and θT2denote
the equivalent angle of departure (AoD) of the LOS path and
the WR path at the BS antenna array, respectively, θR1and
θR2denote the equivalent angle of arrival (AoA) of the LOS
path and the WR path at the UE antenna array, respectively,
and Γrepresents the equivalent reflection coefficient of the
WR path given in (2)-(3). Since the rank of His 2, matrix
HHHhas only two eigenvalues.
III. LOWE R-BOUND CA PACI TY
The MIMO channel capacity per unit bandwidth at a specific
UE location is given by
C= log2
INR+ρ
NT
HHH
= log21 + ρ
NT
λ11 + ρ
NT
λ2
= log21 + ρ
NT
(λ1+λ2) + ρ2
N2
T
λ1λ2,
(7)
where ρ=PT
N0is the BS transmit SNR, PTand N0denote
the power of the transmitted signal and noise, respectively, λ1
and λ2denote the two eigenvalues of HHH.
Based on (7), in the medium SNR regime, i.e., ρ <
λ1+λ2
λ1λ2NT, the MIMO channel capacity per unit bandwidth
is lower bounded by
Cm= log2ρ
NT
(λ1+λ2)= log2
ρ
NT
+ log2(λ1+λ2),
(8)
where log2(λ1+λ2)is referred to as LES.
In the high SNR regime, i.e., ρλ1+λ2
λ1λ2NT, the MIMO
channel capacity per unit bandwidth is lower bounded by
Ch= log2ρ2
N2
T
λ1λ2= 2log2
ρ
NT
+ log2(λ1λ2),(9)
where log2(λ1λ2)is referred to as LEP.
Since log2
ρ
NTis a constant independent of the room set-
ting or building materials, the LES and LEP can effectively
characterise the wireless friendliness of a wall in the medium
and high SNR regimes, respectively. Following [26, Lemma
1], the LES and LEP at a specific UE location are computed,
respectively, in (10) and (11) on the bottom of this page, where
<(·)denote the real part of a complex,
α=1
NTX
NT1
2
q=NT1
2
exp (jq(α1α2))
=sin (πNTD(cos θT1cos θT2))
NTsin (πD(cos θT1cos θT2)),
(12)
β=1
NRX
NR1
2
q=NR1
2
exp (jq(β1β2))
=sin (πNRD(cos θR1cos θR2))
NRsin (πD(cos θR1cos θR2)).
(13)
Given that
1
NTX
NT1
2
q=NT1
2
cos (q(α1α2)) = ∆α,
1
NRX
NR1
2
q=NR1
2
cos (q(β1β2)) = ∆β,
we obtain a simple expression of the LEP at a specific UE
location as follows
log2(λ1λ2) =2 log2µ2|Γ|NTNR
16π2d1d2+log212
α12
β.
(14)
We note that the inter-antenna spacing D, the wall thickness
b, the LOS path length d1, and the WR path length d2can
typically be given as multiples of the signal wavelength µ.
Thus, according to (10, 12-14), for given EM parameters of a
wall material (i.e., the relative permittivity and conductivity),
the UE-specific LES and UE-specific LEP are not affected by
the signal frequency.
We also find that the number of antennas, i.e., NTand NR,
affects the values of UE-specific LES and UE-specific LEP,
indicating that the MIMO configuration should be considered
when evaluating a wall’s wireless friendliness.
By substituting (10) and (14) into (8) and (9), respectively,
the lower-bound MIMO channel capacity per unit bandwidth
at a specific UE location in the medium and high SNR regimes
are, respectively, given by
Cm=log2µ2ρNR
8π2+log2
1
2d2
1
+
<Γej2πd1d2
µ
(∆αβ)
1d1d2
+|Γ|2
2d2
2
,
(15)
Ch=2log2µ2ρNR
16π2+log2
|Γ|212
α12
β
d2
1d2
2
.
(16)
log2(λ1+λ2) = log2µ2NTNR
16π2d2
1+µ2|Γ|2NTNR
16π2d2
2+µ2NTNR
8π2d1d2αβ<Γej2πd1d2
µ (10)
log2(λ1λ2)= log21+∆2
α2∆α
NTP
NT
1
2
q=NT
1
2
cos (q(α1α2))1+2
β2∆β
NRP
NR
1
2
q=NR
1
2
cos (q(β1β2))µ2|Γ|NTNR
16π2d1d22
(11)
IEEE TBC 4
60 80 100 120 140 160 180
0
10
20
30
40
50
60
70
The spatially averaged capacity (bit/s/Hz)
Capacity (7)
Lower-bound capacity at medium SNR (15)
Lower-bound capacity at high SNR (16)
High SNR
Medium SNR
Fig. 2. The spatially averaged capacity of the room versus
BS transmit SNR.
IV. WIRELESS FRIENDLINESS EVALUATION
FO R WALLS AS REFLECTO RS
In this section, we design three new metrics: the spatially
averaged LES, the spatially averaged LEP, and the upper-
bound outage probability, which are all over the room of
interest, to enable fast evaluation of the wireless friendliness
of the wall closest to the BS.
The wireless friendliness of the wall closest to the BS can be
evaluated based on the lower-bound capacity for all possible
UE locations in a rectangular room [26]. As revealed in (8) and
(9), the impact of the BS transmission power and that of the
wall material properties on the lower-bound indoor capacity
can be decoupled, which enables fast evaluation of the wall’s
wireless friendliness based on the statistics of the LES and
LEP at an arbitrary UE position in the room of interest.
We apply a 2D Cartesian coordinate system inside a rect-
angular room and evenly divide the room area into a X×Y
dense grid. We can calculate the LES and LEP at the centre
of each smallest rectangle (xi, yj), using (10) and (14),
respectively, denoted by LES (xi, yj)and LEP (xi, yj), where
i∈ {1,2, ..., X}and j∈ {1,2, ..., Y }.
A. Spatially averaged LES and spatially averaged LEP of a
room
The impact of room settings on the spatially averaged
capacity over a room can be studied based on the spatially
averaged LES and spatially averaged LEP of that room, which
are computed by
LESavg =1
XY
X
X
i=1
Y
X
j=1
LES (xi, yj).(17)
LEPavg =1
XY
X
X
i=1
Y
X
j=1
LEP (xi, yj).(18)
However, due to the complicated expressions of the UE
location-specific LES and LEP expression in (10) and (14), it
is difficult to derive closed-form expressions for the spatially
averaged LES and spatially averaged LEP of a room. Hence,
-50 0 50 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Outage probability
12 14 16
0.98
1
Fig. 3. The upper-bound outage probability of the room
versus downlink capacity threshold Tat medium SNR of
ρ= 90 dB and high SNR of ρ= 150 dB.
the spatially averaged LES and spatially averaged LEP will be
obtained numerically in Section V.
B. Upper-bound outage probability of a room
To evaluate the percentage of all possible UE locations not
meeting the target downlink capacity, the upper-bound outage
probability of a room is designed as another evaluation metric
of the wireless friendliness of the wall. Under the channel
(6) that considers both the LOS path and the WR path, a
lower value of the upper-bound outage probability of a room
means a better wireless signal coverage inside the room, which
indicates that the wall is more wireless-friendly.
An outage occurs when the capacity at UE location (xi, yj)
falls below a given downlink capacity threshold Tbit/s/Hz.
The upper-bound outage probability of a room is defined to
measure the proportion of UE locations whose capacities are
not greater than Tbit/s/Hz, and is given by
P(T) = Pr C0T
=
Pr LES0Tlog2
ρ
NT, ρ < λ1+λ2
λ1λ2NT,
Pr LEP0T2log2
ρ
NT, ρ λ1+λ2
λ1λ2NT.
(19)
where C0,LES0, and LEP0are the variables denoting the
lower-bound capacity, LES, and LEP at an arbitrary UE
position in the room of interest, respectively. Therefore, the
upper-bound outage probability of a room can be computed
by the CDF of the LES in the medium SNR regime and by
the CDF of the LEP in the high SNR regime. We note that
the downlink capacity threshold Tshould be properly selected
according to the MIMO configuration.
According to the above three proposed evaluation metrics,
the wireless friendliness of the wall will be directly affected by
the wall material’s thickness and relative permittivity, and BS-
wall distance, as they influence the reflection characteristics of
the wall as a reflector. Besides, the size and aspect ratio of a
rectangular room, which determines the transverse dimension
IEEE TBC 5
TABLE I. Material EM parameters under 6 GHz [21]
Material class ε0σ
Concrete 5.31 0.1390
Brick 3.75 0.0380
Plasterboard 2.94 0.0412
Wood 1.99 0.0321
Glass 6.27 0.0364
of the wall closest to the BS, will also influence the wireless
friendliness of the wall.
V. NUMERICAL RES ULTS A ND AN ALYS IS
In this work, the simulation is performed in the 6 GHz band
in a rectangular room with the dimension of W×Lm2. The
BS is deployed on the centreline perpendicular to the Lside.
The incident EM wave is assumed to be TE polarised. The
transmit power is assumed to be equally allocated to each
BS antenna. The inter-antenna spacing of both the UE and
BS antenna array are assumed to be half wavelength. The
UE and BS antenna array each employ a 4-antenna ULA. We
consider five popular wall materials, whose values of relative
permittivity are given in Table. I following [21, Table III].
A. Verification of the analytical derivations
Fig. 2 plots the spatially averaged capacity in the 10×10 m2
room versus the BS transmit SNR. The values of the spatially
averaged capacity are computed by averaging over dense
sample points inside the room with concrete walls of thickness
0.2 m and for the BS-wall distance of 0.05 m. We can see that
the derived lower-bound capacity Cmin (15) and Chin (16)
are very close to the capacity Cin (7) in the medium and high
SNR regimes, respectively.
Fig. 3 plots the upper-bound outage probability versus the
downlink capacity threshold Tat medium SNR of ρ= 90 dB
and high SNR of ρ= 150 dB inside the 10×10 m2room with
concrete walls of thickness 0.2 m and the BS-wall distance of
0.05 m. It is observed that the upper-bound outage probability
increases with the downlink capacity threshold. Meanwhile,
the CDF of the LES and the CDF of LEP tightly match the
actual outage probability Pr(C < T )in the medium and high
SNR regimes, respectively, as given in (19).
B. The impact of room settings on the spatially averaged LES
and spatially averaged LEP of a room
In Fig. 4 and Fig. 5, we discuss the impact of room settings
on the spatially averaged LES and spatially averaged LEP over
the room of interest, respectively.
Fig. 4(a) and Fig. 5(a) plot the spatially averaged LES and
spatially averaged LEP for different room sizes, respectively.
The aspect ratio of a room is defined as r=L/W . The wall
is assumed to be concrete with the thickness of 0.2 m and the
BS-wall distance is 0.05 m. From Fig. 4(a) and Fig. 5(a), given
the same aspect ratio of rooms, a larger room size results in
a smaller spatially averaged LES and spatially averaged LEP
due to the sever path loss caused by the longer LOS path and
WR path.
50 100 150
-19
-18.5
-18
-17.5
-17
-16.5
-16
-15.5
-15
Spatially averaged LES
(a) Under various room sizes
02468
-21
-20
-19
-18
-17
-16
-15
Spatially averaged LES
(b) Under various room aspect ratios
2 4 6 8 10
-17.6
-17.5
-17.4
-17.3
-17.2
-17.1
-17
-16.9
-16.8
-16.7
-16.6
Spatially averaged LES
(c) Under various real part of wall
relative permittivities
0 0.05 0.1 0.15 0.2
-17.5
-17.4
-17.3
-17.2
-17.1
-17
-16.9
-16.8
Spatially averaged LES
(d) Under various wall conductivities
0.1 0.15 0.2 0.25 0.3
-17.4
-17.2
-17
-16.8
-16.6
-16.4
-16.2
-16
Spatially averaged LES
(e) Under various wall thickness
0 0.05 0.1 0.15 0.2
-17.6
-17.4
-17.2
-17
-16.8
-16.6
-16.4
-16.2
-16
-15.8
Spatially averaged LES
0.091 0.093
-16.36
-16.34
(f) Under various BS-wall distances
Fig. 4. The spatially averaged LES of a room in (10) under
different room settings.
Fig. 4(b) and Fig. 5(b) plot the spatially averaged LES and
spatially averaged LEP for different aspect ratios, respectively.
It is seen that, for the same room size, with the increase of r
from 1
8to 8, the spatially averaged LES and spatially averaged
LEP first rapidly increase and then slowly decrease. The peak
values of the spatially averaged LES and spatially averaged
LEP are obtained when rapproaches 2. This phenomenon
can be explained by the path loss towards the UE location
farthest from the BS, where the longest BS-UE distance in
the room is approximately given by d1,max =Wq1 + 1
4r2=
qS
r+Sr
4, where Sis the give area of the rooms with different
r. Obviously, as ris growing above 2, the path loss being
approximately proportional to d2
1,max will increase faster in
growth rate since the derivative of d2
1,max is greater than 0.
IEEE TBC 6
50 100 150
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-64
-62
-60
-58
-56
-54
-52
Spatially averaged LEP
(a) Under various room sizes
02468
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-75
-70
-65
-60
-55
-50
Spatially averaged LEP
(b) Under various room aspect ratios
2 4 6 8 10
-70
-68
-66
-64
-62
-60
-58
-56
Spatially averaged LEP
(c) Under various real part of wall
relative permittivities
0 0.05 0.1 0.15 0.2
-61.5
-61
-60.5
-60
-59.5
-59
-58.5
-58
Spatially averaged LEP
(d) Under various wall conductivities
0.1 0.15 0.2 0.25 0.3
-68
-67
-66
-65
-64
-63
-62
-61
-60
-59
Spatially averaged LEP
(e) Under various wall thickness
0 0.05 0.1 0.15 0.2
-70
-68
-66
-64
-62
-60
-58
-56
-54
-52
-50
Spatially averaged LEP
0.062 0.063
-57.9
-57.85
-57.8
(f) Under various BS-wall distances
Fig. 5. The spatially averaged LEP of a room in (14) under
different room settings.
We arrive at the conclusion that, for the same room size, the
long side wall of a room with an aspect ratio close to 2, which
is deployed closely behind the BS, would be more friendly to
indoor LOS MIMO communications.
The spatially averaged LES and spatially averaged LEP
of a room versus the real part of wall relative permittivity
are depicted in Fig. 4(c) and Fig. 5(c), respectively, while
the spatially averaged LES and spatially averaged LEP of
a room versus the wall conductivity are depicted in Fig.
4(d) and Fig. 5(d), respectively, for a 10 ×10 m2room
with the BS-wall distance of 0.05 m and the wall thickness
of 0.2 m. We can see that the real part of wall relative
permittivity determines the upward or downward trend of the
spatially averaged LES and spatially averaged LEP, while the
conductivity of wall materials affecting the imaginary part
of the relative permittivity strongly influences the fluctuation
magnitude and envelope. Generally, a severer fluctuation arises
out of a smaller conductivity, and the spatially averaged LES
is more prone to this fluctuation than the spatially averaged
LEP. The above results reveal the significant impact of wall
relative permittivity on indoor wireless capacity.
Fig. 4(e) and Fig. 5(e) illustrate the spatially averaged LES
and spatially averaged LEP of a room under different wall
thicknesses, respectively, for a 10×10 m2room with concrete
walls and the BS-wall distance ranging from a quarter of
wavelength (0.0125 m) to wavelength (0.05 m). We observe
that, for a given BS-wall distance, both the spatially averaged
LES and spatially averaged LEP change slightly with the wall
thickness. This is because the wavelength of 0.05 m is not
comparable to the typical wall thickness ranging from 0.1 m to
0.3 m. Meanwhile, the limited variation of δin (3), caused by
the narrow range of typical wall thickness, hardly affects the
equivalent reflection coefficient considering multiple internal
reflections in (2).
Fig. 4(f) and Fig. 5(f) show the spatially averaged LES
and spatially averaged LEP of a room under different BS-
wall distances, respectively, for a 10 ×10 m2room with
concrete walls ranging from 0.1 m to 0.2 m in thickness. It is
found that the curves of spatially averaged LES fluctuate under
a decreasing envelop with the increase of BS-wall distance
from 0.01 m to 0.2 m. Nevertheless, the spatially averaged
LEP present a monotonic increasing tendency with the rise
of BS-wall distance. This is intuitive because, as the BS is
moved away from its closest wall, the difference between
the LOS path and the WR path becomes more substantial.
Consequently, the two paths become more irrelevant, and the
two eigenvalues of the channel becomes closer to each other.
We can conclude that the BS deployment from its closest wall
affects indoor wireless capacity.
C. The impact of room settings on the upper-bound outage
probability of a room
In Fig. 6 and Fig. 7, we discuss the impact of room
settings on the upper-bound outage probability of a room in
the medium and high SNR regimes, respectively.
Fig. 6(a) and Fig. 7(a) present the upper-bound outage
probability of a room at medium and high SNR, respectively,
for different room sizes and aspect ratios with concrete walls
of 0.2 m thickness and 0.05 m BS-wall distance. We can
see that, for the rooms of same size with an aspect ratio
no larger than 2, given a same downlink capacity threshold,
the upper-bound outage probability of the room with a bigger
aspect ratio would be smaller. The room with the aspect ratio
of 4 does not obviously present a lower outage probability
than the room of same size with the aspect ratio of 1.5625.
Meanwhile, for the rooms with the aspect ratio of 1, given
a same downlink capacity threshold, the upper-bound outage
probability is smaller for the room of a smaller size than that
of a bigger size. Therefore, the room size and aspect ratio will
influence the outage performance over a room.
In Fig. 6(b) and Fig. 7(b), the upper-bound outage prob-
ability of a room at medium and high SNR are illustrated,
IEEE TBC 7
-30 -25 -20 -15 -10 -5
log2(1+2)
0
0.2
0.4
0.6
0.8
1
CDF
(a) Under various room sizes
-25 -20 -15 -10
log2(1+2)
0
0.2
0.4
0.6
0.8
1
CDF
Concrete
Brick
Plasterboard
Wood
Glass
(b) Under various wall permittivities
-25 -20 -15 -10 -5
log2(1+2)
0
0.2
0.4
0.6
0.8
1
CDF
-16.88 -16.86
0.581
0.582
0.583
(c) Under various wall thicknesses
-25 -20 -15 -10 -5
log2(1+2)
0
0.2
0.4
0.6
0.8
1
CDF
(d) Under various BS-wall distances
Fig. 6. The upper-bound outage probability of a room in (19) in the medium SNR regime under different room settings.
-100 -90 -80 -70 -60 -50 -40 -30
log2(1 2)
0
0.2
0.4
0.6
0.8
1
CDF
(a) Under various room sizes
-80 -70 -60 -50 -40
log2(1 2)
0
0.2
0.4
0.6
0.8
1
CDF
Glass
Brick
Concrete
Plasterboard
Wood
(b) Under various wall permittivities
-100 -80 -60 -40 -20
log2(1 2)
0
0.2
0.4
0.6
0.8
1
CDF
-61.28 -61.24
0.42
0.425
0.43
(c) Under various wall thicknesses
-100 -80 -60 -40 -20
log2(1 2)
0
0.2
0.4
0.6
0.8
1
CDF
(d) Under various BS-wall distances
Fig. 7. The upper-bound outage probability of a room in (19) in the high SNR regime under different room settings.
respectively, for a 10 ×10 m2room with the BS-wall distance
of 0.05 m under different wall materials with the thickness
of 0.2 m. It is found that, at a high SNR, for a same
downlink capacity threshold, the wall with a larger real part
of relative permittivity would result in smaller upper-bound
outage probability, which is verified by the curves order from
left to right being wood with Re(ε)of 1.99, plasterboard with
Re(ε)of 2.94, brick with Re(ε)of 3.75, concrete with Re(ε)
of 5.31, and glass with Re(ε)of 6.27. The impact of imaginary
part of the relative permittivity affected by the wall material’s
conductivity on the outage performance is much smaller than
that of the real part of relative permittivity. However, the
outage probability at medium SNR is irregular with the relative
permittivity of these five materials due to the higher sensitivity
of LES than LEP. In brief, the EM properties of wall materials
need to be well selected in terms of the outage performance
over a room.
The upper-bound outage probability of a room at medium
and high SNR is shown in Fig. 6(c) and Fig. 7(c), respectively,
under different wall thicknesses for a 10 ×10 m2room with
concrete walls and the BS-wall distance of 0.05 m. We find
that the curves of outage probability under the wall thickness
of 0.1, 0.2, and 0.3 m are with very slight difference, indicating
that the wall thickness in typical range does not substantially
affect the outage performance over a room.
Fig. 6(d) and Fig. 7(d) depict the upper-bound outage
probability of a room at medium and high SNR, respectively,
under different BS-wall distances for a 10 ×10 m2room
with concrete walls of 0.2 m in thickness. It is observed
that, for a same downlink capacity threshold, the upper-bound
outage probability at high SNR becomes smaller as the BS-
wall distance increases from a quarter of wavelength (0.0125
m) to wavelength (0.05 m). Nonetheless, the curves of the
upper-bound outage probability at medium SNR swing left
and right with the change of every quarter of wavelength in
BS-wall distance. This uncertainty of the trend can also be
demonstrated by the fluctuations in Fig. 4(f). Hence, the BS-
wall distance plays a crucial impact on the outage performance
over a room.
D. Comparisons of the three proposed metrics
In this subsection, we compare the three proposed metrics
for evaluating the wireless friendliness of a wall with respect
to their advantages, limitations, and applicability. Note that the
room setting factors mentioned below include the room size
and aspect ratio, wall relative permittivity and thickness, and
the BS-wall distance.
Given the spatially averaged LES and spatially averaged
LEP of a room, respectively, the MIMO channel capacity in
the medium and high SNR regimes can be easily obtained for
a known BS transmit SNR. Meanwhile, according to the trend
of the spatially averaged LES or spatially averaged LEP versus
a room setting factor, the optimal configuration of this factor
under medium and high SNR are thus attained. Moreover, the
spatially averaged LES and spatially averaged LEP over the
UE location are easy to get, which enables fast evaluation of
the wireless friendliness of the wall as a reflector. However, it
IEEE TBC 8
is analytically intractable to derive the expressions of spatially
averaged LES and spatially averaged LEP, and it is uncertain
if the obtained range of the room setting factor will meet the
practical downlink capacity requirement, e.g., not less than the
downlink capacity threshold Tbit/s/Hz.
The upper-bound outage probability of a room, whose ana-
lytical expression is simple based on the CDF of the LES and
LEP in the medium and high SNR regimes, respectively, can
be leveraged to quickly evaluate the indoor capacity at any UE
position against a given downlink capacity threshold and reveal
the percentage of all possible UE locations not achieving the
target downlink capacity. Nevertheless, the trends of upper-
bound outage probability versus each room setting factor is not
intuitive and is affected by the downlink capacity threshold.
Besides, the downlink capacity threshold should be properly
selected for the MIMO configuration of a given size.
In brief, the three proposed metrics have different applicabil-
ity to the wireless friendliness evaluation of walls as reflectors.
When evaluating the impact of a room setting factor on the
indoor wireless capacity and obtaining its optimal configura-
tion, the spatially averaged LES and spatially averaged LES
of a room are preferred in the medium and high SNR regimes,
respectively. When a room setting factor needs to be properly
set to meet the capacity requirement, the upper-bound outage
probability of a room should be adopted.
VI. CONCLUSION
In this work, we have proposed three new metrics for fast
evaluating the wireless friendliness of a wall that is close to
an indoor BS based on how the wall reflections caused by
it affect the lower-bound capacity. We have shown that the
LES and LEP of the channel model that characterises both
the LOS path and the WR path can represent the lower-bound
capacity in the medium and high SNR regimes, respectively,
and derived the LES, LEP, and the lower-bound capacity for
an arbitrary UE location in closed forms. Three new metrics,
i.e., the spatially averaged LES, spatially averaged LEP, and
upper-bound outage probability of a room, have been proposed
and compared with respect to their advantages, limitations, and
applicability. Numerical results validate that the three proposed
metrics are effective indicators of wireless friendliness of the
wall as a reflector under different room sizes and aspect
ratios, wall relative permittivities and thicknesses, and BS-
wall distances. The measurement campaign will be held in the
near future to validate the results herein using the experiment
results.
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