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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

1

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 y∈CNR×1at an arbitrary UE

location1is given by the transmitted signal x∈CNT×1, the

channel matrix H∈CNR×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 = 1−exp(−j2δ)

1−Γ02exp(−j2δ)Γ0,(2)

δ=2πb

µpε−sin2α, (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=ε0−j17.98σ/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 α−pε−sin2α

cos α+pε−sin2α,(4)

ΓTM =cos α−q(ε−sin2α)/ε2

cos α+q(ε−sin2α)/ε2

.(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

e−j2πd1

µ, A2=µΓ

4πd2

e−j2πd2

µ,

hαl=e−j−NT−1

2αl, e−j1−NT

−1

2αl, ..., e−jNT−1

2αl,

hβl=ej−NR−1

2βl, ej1−NR−1

2βl, ..., ejNR−1

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

= log21 + 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., ρ≥λ1+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

NT−1

2

q=−NT−1

2

exp (jq(α1−α2))

=sin (πNTD(cos θT1−cos θT2)/µ)

NTsin (πD(cos θT1−cos θT2)/µ),

(12)

∆β=1

NRX

NR−1

2

q=−NR−1

2

exp (jq(β1−β2))

=sin (πNRD(cos θR1−cos θR2)/µ)

NRsin (πD(cos θR1−cos θR2)/µ).

(13)

Given that

1

NTX

NT−1

2

q=−NT−1

2

cos (q(α1−α2)) = ∆α,

1

NRX

NR−1

2

q=−NR−1

2

cos (q(β1−β2)) = ∆β,

we obtain a simple expression of the LEP at a speciﬁc UE

location as follows

log2(λ1λ2) =2 log2µ2|Γ|NTNR

16π2d1d2+log21−∆2

α1−∆2

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

+

<Γej2πd1−d2

µ

(∆α∆β)

−1d1d2

+|Γ|2

2d2

2

,

(15)

Ch=2log2µ2ρNR

16π2+log2

|Γ|21−∆2

α1−∆2

β

d2

1d2

2

.

(16)

log2(λ1+λ2) = log2µ2NTNR

16π2d2

1+µ2|Γ|2NTNR

16π2d2

2+µ2NTNR

8π2d1d2∆α∆β<Γej2πd1−d2

µ (10)

log2(λ1λ2)= log21+∆2

α−2∆α

NTP

NT

−1

2

q=−NT

−1

2

cos (q(α1−α2))1+∆2

β−2∆β

NRP

NR

−1

2

q=−NR

−1

2

cos (q(β1−β2))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 C0≤T

=

Pr LES0≤T−log2

ρ

NT, ρ < λ1+λ2

λ1λ2NT,

Pr LEP0≤T−2log2

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

-66

-64

-62

-60

-58

-56

-54

-52

Spatially averaged LEP

(a) Under various room sizes

02468

-80

-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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