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IEEE INTERNET OF THINGS JOURNAL 1

How Friendly are Building Materials as Reﬂectors

to Indoor LOS MIMO Communications?

Yixin Zhang, Student Member, IEEE, Chen Chen, Student Member, IEEE, Songjiang Yang, Student Member,

IEEE, Jiliang Zhang, Senior Member, IEEE, Xiaoli Chu, Senior Member, IEEE, Jie Zhang, Senior Member, IEEE

Abstract—The tremendous popularity of internet of things

(IoT) applications and wireless devices have prompted a mas-

sive increase of indoor wireless trafﬁc. To further explore the

potential of indoor IoT wireless networks, creating constructive

interactions between indoor wireless transmissions and the built

environments becomes necessary. The electromagnetic (EM) wave

propagation indoors would be affected by the EM and physical

properties of the building material, e.g., its relative permittivity

and thickness. In this paper, we construct a new multipath

channel model by characterising wall reﬂection (WR) for an

indoor line-of-sight (LOS) single-user multiple-input multiple-

output (MIMO) system and derive its ergodic capacity in closed-

form. Based on the analytical results, we deﬁne the wireless

friendliness of a building material based on the spatially averaged

indoor capacity and propose a scheme for evaluating the wireless

friendliness of building materials. Monte Carlo simulations val-

idate our analytical results and manifest the signiﬁcant impact

of the relative permittivity and thickness of a building material

on indoor capacity, indicating that the wireless friendliness of

building materials should be considered in the planning and

optimisation of indoor wireless networks. The outcomes of this

paper would enable appropriate selection of wall materials during

building design, thus enhancing the capacity of indoor LOS

MIMO communications.

Index Terms—Building material, wireless friendliness, indoor

communications, EM wave, reﬂection, LOS, MIMO.

I. INTRODUCTION

Internet of things (IoT) connects numerous heterogeneous

devices, and provides infrastructures for smart buildings [1],

[2], smart grids [3], and smart cities [4]. With the roll-

out of 5G systems and the opening horizon of 6G systems,

cellular networks will provide economical, ﬂexible and reliable

wireless connectivities for IoT devices, e.g., by leveraging

5G massive machine type communication (mMTC) [5] and

6G massive broad bandwidth machine type (mBBMT) [6]

technologies.

It is predicted that 80-96% of wireless data trafﬁc will be

consumed indoors in the future [7]. As a result, indoor wireless

trafﬁc required by IoT is growing at an unprecedented rate.

Notably, physical obstacles like walls would affect the indoor

propagation of electromagnetic (EM) waves. Therefore, indoor

Yixin Zhang, Chen Chen, Songjiang Yang, 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.

wireless performance should be one of the indispensable

prerequisites when designing buildings [8], [9].

To meet the high data demand and address the capac-

ity crunch in-building, indoor small base stations (BSs) are

usually equipped with large-scale antenna arrays [10], e.g.

consisting of hundreds of antennas, facilitated by multiple-

input multiple-output (MIMO) technology to achieve spatial

multiplexing/diversity gains [11]–[13]. In order to guarantee

a low spatial correlation, the space intervals among antenna

elements of the MIMO antenna array have to be larger

than half wavelength [14], which therefore will increase the

physical dimension of the indoor small BSs and generate

negative weight and visual consequence on a room. Especially

in industrial environment, deploying BSs in the workspace

may increase the risk of accidents.

To avoid any possible negative impact on the functionality

and appearance of a room, a desirable indoor BS deployment

is to integrate MIMO antenna arrays with interior walls [15],

[16], which however will result in non-negligible coupling

between MIMO antenna arrays and building materials [17]–

[19]. Speciﬁcally, when an EM wave impinges on a wall

surface, the intensity of the wall reﬂected wave can be mea-

sured by the reﬂection coefﬁcient, which depends on the EM

and physical properties of the wall material, i.e., its relative

permittivity and thickness [20]–[25]. The wall reﬂected EM

waves would be superposed with other EM waves, which

may jointly inﬂuence the indoor wireless performance. Hence,

enhancing indoor wireless performance requires a rational

selection and/or design of building materials. In this paper, we

deﬁne the wireless friendliness as a new performance metric

of a building material, which is characterised by its relative

permittivity and thickness. A building material with desirable

wireless friendliness would be beneﬁcial to the performance

of indoor wireless networks.

Metamaterials are known as novel artiﬁcial structures, which

can be customized to build a programmable EM propaga-

tion environment [26]. The meta-atoms in metamaterials are

software-controlled to ﬁrstly capture the signals from the

source and then reﬂect the signal to the destination actively

or passively [27], [28]. Although metamaterials are expected

to contribute to indoor capacity improvement, they are facing

major challenges such as their design complexity growing

exponentially with the number of meta-atoms. Besides, the

energy efﬁciency of metamaterial-aided wireless networking

is no better than massive MIMO [29] and relay-supported

transmissions [30].

By contrast, the wireless friendliness of concrete as a widely

IEEE INTERNET OF THINGS JOURNAL 2

BS

UE

1

d

LOS

H

WR

H

wall

z

2

d

2

Τ

q

Antenna

Linear array

1

D

ε

1

Τ

q

1

R

q

2

R

q

q

D

q

D

BS

UE

1

d

LOS

H

WR

H

wall

z

2

d

2

Τ

q

Antenna

Linear array

1

D

ε

1

Τ

q

1

R

q

2

R

q

q

D

q

D

Fig. 1. The LOS path and the WR path between indoor BS and UE.

used building material has not been sufﬁciently studied. Most

of the existing works on the EM wave propagation loss through

concrete walls were based on measurements [31]–[33]. As far

as we know, a theoretical analysis of indoor multipath capacity

inﬂuenced by the EM and physical properties of concrete is

still missing. As will be shown in the subsequent sections, the

appropriate selection of building materials will increase the

baseline value of indoor capacity. Accordingly, the wireless

performance of building materials should be considered inher-

ently in the design of future smart/green buildings. Hence, it is

of vital importance to build an evaluation scheme to identify

the relationship between the wireless friendliness of a building

material and its EM and physical properties [34].

In this paper, we study the impact of building materials,

especially their relative permittivity and thickness, on the

performance of indoor line-of-sight (LOS) MIMO communi-

cations. To the best of our knowledge, this is the ﬁrst attempt to

study the indoor capacity from the perspective of wall material

design. The main contributions of this paper are summarized

as follows:

•Taking the wall reﬂection (WR) path into account and

based on distance-dependant Rician fading model, a new

indoor LOS MIMO channel model is proposed.

•The marginal probability distribution function (MPDF) of

an unordered squared singular value, the ergodic capacity

of the indoor LOS MIMO channel, and the squared

singular values of its deterministic part are analytically

obtained in closed forms. These analytical expressions

reveal the relationship between the indoor MIMO channel

capacity and the relative permittivity and thickness of the

building material.

•We propose a scheme based on spatially averaged indoor

capacity, which can be used to evaluate the wireless

friendliness of building materials and to guide the design

of a wireless-friendly building.

•The wireless friendliness performance of building mate-

rials is analysed. The optimal values of the permittivity

and thickness of building materials that maximise the

spatially averaged indoor capacity are obtained for both

the omnidirectional and directional BS antenna arrays.

•The analytical results are veriﬁed through Monte Carlo

simulations.

The remainder of this paper is organized as follows. Section

II introduces system model for indoor LOS MIMO downlink

transmissions. In Section III, the analytical MPDF of an

unordered squared singular value, the ergodic capacity of an

indoor LOS MIMO channel, and the squared singular values

of its deterministic part are derived in closed forms. Then

a scheme used for evaluating the wireless friendliness of a

building material is proposed in Section IV. The impact of

directional radiation pattern is discussed in Section V. Monte

Carlo simulation results are provided to verify all analytical

results in Section VI. Finally, Section VII concludes this paper.

II. SY ST EM MO DE L

In this section, we introduce a novel system model for

indoor LOS MIMO communications that incorporates the WR

path and the EM and physical properties of the building

material.

We consider indoor LOS MIMO downlink transmissions, as

shown in Fig. 1. In the considered room, one BS is deployed

close to one of the walls and one user equipment (UE) could be

arbitrarily positioned. Since the strength of a WR path from

a wall other than the wall that is closest to the BS will be

dominated by the distance-dependent path loss, the reﬂected

paths from the other walls would be much weaker than the WR

path from the wall closest to the BS. Hence, the considered

wall refers to the wall closest to the BS hereafter, and the WR

path refers to the wall reﬂected path from the considered wall.

The BS is deployed in parallel with the considered wall with

a small distance of D1from the wall. The BS and a typical

UE are equipped with NTand NRomnidirectional antennas,

respectively, both in linear arrays with inter-antenna spacing

D. The complex frequency-ﬂat linear channel from the BS to

the typical UE is constructed as

y=Hx +n,(1)

where x∈CNT×1,y∈CNR×1denote the transmitted signal

and the received signal, respectively, ndenotes the additive

white Gaussian noise, and His a NR×NTchannel matrix,

[HLOS]nR,nT=µ

4πd1

exp −j2πd1

µ−j2πD

µnR−NR−1

2cos (θR1+ ∆θ)+nT−NT−1

2cos θT1 (2)

[HWR]nR,nT=µΓ

4πd2

exp −j2πd2

µ−j2πD

µnR−NR−1

2cos (θR2+ ∆θ)+nT−NT−1

2cos θT2 (3)

IEEE INTERNET OF THINGS JOURNAL 3

subject to E[Tr{HH†}] = NRNT, where E(·)and Tr{·}

denote the expectation and the trace of a matrix, respectively.

Other than the LOS path, the WR path is taken into account

to capture the impact of building materials on the indoor wire-

less propagation channel. For a certain position in the room,

the LOS path and the WR path are deterministically modelled

by Friis’ formula as NR×NTmatrix HLOS and HWR,

respectively, whose elements are given in (2) and (3) on the

previous page, respectively, in which nT∈ {0,1, ..., NT−1}

and nR∈ {0,1, ..., NR−1}are the indices of transmit and

receive antenna elements, µdenotes the wavelength of EM

waves in the air, d1and d2denote the length of the LOS

path and the WR path, respectively, θT1and θT2denote the

approximated angle of departure (AoD) of the LOS path and

the WR path at the BS array, respectively, while θR1and

θR2denote the approximated angle of arrival (AoA) of the

LOS path and the WR path at the UE array, respectively, ∆θ

denotes the arbitrary orientation angle of the UE array, where

θT1, θT2, θR1, θR2,∆θ∈ {0, π},(θR1+ ∆θ)and (θR2+ ∆θ)

denote equivalent AoA of the LOS path and the WR path

at the UE array, respectively, and Γrepresents the equivalent

reﬂection coefﬁcient of the WR path.

Along the WR path, multiple internal reﬂections are consid-

ered when the EM wave interacts with the building material.

Using plane wave far-ﬁeld approximation, the incident angles

of different order reﬂections are all approximated by α. When

the building material is assumed to be a homogenous dielectric

reﬂector with relative permittivity εand thickness ζ, the

equivalent reﬂection coefﬁcient of the WR path is represented

as [20]

Γ = 1−exp(−j2δ)

1−Γ02exp(−j2δ)Γ0,(4)

where

δ=2πζ

µpε−sin2α, (5)

and the ﬁrst-order reﬂection coefﬁcient Γ0represents the

transverse electric (TE) polarisation ΓTE or the transverse

magnetic (TM) polarisation ΓTM of the incident electric ﬁeld,

respectively, which are given by:

ΓTE =cos α−pε−sin2α

cos α+pε−sin2α,(6)

or

ΓTM =cos α−q(ε−sin2α)/ε2

cos α+q(ε−sin2α)/ε2

.(7)

For simplicity, HLOS and HWR are merged as one matrix

H1, which can be decomposed as

H1=A1hH

β1hα1+A2hH

β2hα2,(8)

A1=µ√NTNR

4πd1

e

−j2πd1

µ,

A2=µΓ√NTNR

4πd2

e

−j2πd2

µ,

αl= 2πD cos θTl/µ,

βl= 2πD cos (θRl+ ∆θ)/µ,

hαl=

e

−j−NT

−1

2αl

√NT

,e

−j1−NT

−1

2αl

√NT

, ..., e

−jNT

−1

2αl

√NT

,

hβl=

ej−NR−1

2βl

√NR

,ej1−NR−1

2βl

√NR

, ..., ejNR−1

2βl

√NR

,

where l∈ {1,2}.

Based on distance-dependant Rician fading model and the

multipath (MP) effect, our channel matrix Hconsists of three

components including the LOS part, the WR part, and the MP

part, which can be presented as

H=s¯

K

1 + ¯

K

¯

H+r1

1 + ¯

KHMP,(9)

where the deterministic matrix ¯

H, including the LOS part and

the WR part, is expressed as

¯

H=H1

kH1kpNRNT,(10)

subject to E[Tr{HH†}] = NRNT, with k·k denoting the F-

norm. The MP components are assumed to be independent and

identically distributed zero mean and unit variance complex

Gaussian random variables arranged in the NR×NTmatrix

HMP.¯

Kis the power ratio between the deterministic part ¯

H

and the random part HMP, which can be obtained through

¯

K=KS, (11)

where Kis the distance-dependant Rician factor as a function

of d1deﬁning the power ratio between the LOS part and the

MP part, given by [35, Eq. (5.22)]

K= 8.7+0.051d1(dB),(12)

and

S=kH1k2

kA1k2=

d2

1

d2

2

Γ2+2Md1

NTNRd2

Γ+1

,(13)

∆β=hH

β2,hH

β1=1

NR

(NR

−1)/2

X

q=−(NR

−1)/2

exp (jq(β1−β2)) = sin (πNRD(cos (θR1+ ∆θ)−cos (θR2+ ∆θ)) /µ)

NRsin (πD (cos (θR1+ ∆θ)−cos (θR2+ ∆θ)) /µ)(16)

∆α=hhα2,hα1i=1

NT

(NT

−1)/2

X

q=−(NT

−1)/2

exp (jq(α1−α2)) = sin (πNTD(cos θT1−cos θT2)/µ)

NTsin (πD(cos θT1−cos θT2)/µ)(17)

IEEE INTERNET OF THINGS JOURNAL 4

in which M=

NT

−1

P

p=0

NR

−1

P

q=0

cos (p(α1−α2) + q(β1−β2)).

III. ANALYSIS OF ERGODIC CAPACITY

In this section, for an arbitrary position in the room, we

derive the two non-zero squared singular values of H1, the

MPDF of an unordered squared singular value and the ergodic

capacity of an indoor LOS MIMO channel Hin closed forms.

A. The distribution of the squared singular value of channel

For notational convenience, we deﬁne m= min{NR, NT}

and n= max{NR, NT}.

Lemma 1: Suppose ϕ1, ϕ2, ..., ϕmare the msquared

singular values of H1, where ϕ1, ϕ2, ..., ϕm−2= 0 and

ϕm−1, ϕm>0. The two non-zero squared singular values

of H1are computed in closed-form as

ϕm−1=kXk2−qkXk4−4|det (X)|2

2,(14)

ϕm=kXk2+qkXk4−4|det (X)|2

2,(15)

X="A1+A2∆β∆α A2∆βkhα∗k

A2∆αkhβ∗kA2khα∗kkhβ∗k#,

hH

β∗=hH

β2−∆βhH

β1,

hα∗=hα2−∆αhα1,

and ∆βand ∆αare given in (16) and (17), respectively [36].

Proof: See Appendix.

Theorem 1: The MPDF of an unordered squared singular

value λof His computed in (18) [37], where the two non-zero

squared singular values of √¯

K¯

Hare given by

φm−1=¯

KNRNTϕm−1

kH1k2=KNRNTϕm−1

kA1k2,(19)

and

φm=¯

KNRNTϕm

kH1k2=KNRNTϕm

kA1k2,(20)

and Di,j is the (i, j)-co-factor of the m×mmatrix Zwhose

(l, k)th entry is given by

(Z)l,k =((n−m+k+l−2)!,1≤l≤m−2,

1F1(n−m+l,n−m+1,φk)

((n−m+l−1)!)−1,otherwise.(21)

The hypergeometric function 0F1w, z2in (18) is deﬁned in

the series form by

0F1w, z2=

∞

X

s=0

(z)2s

s![w]s

,(22)

and the hypergeometric function 1F1(e, o, g)in (21) is given

by

1F1(e, o, g) =

∞

X

s=0

[e]sgs

[o]ss!,(23)

where [r]t=(r+t−1)!

(r−1)! .

Proof: Given the channel model in (8)-(13), the channel

matrix His an NR-by-NTnon-central Wishart matrix with

mean q¯

K

1+ ¯

K¯

H. Hence, the MPDF of an arbitrary squared

value of Hcan be found in [37, (3)], which is derived by the

squared singular values of √¯

K¯

H.

Since H1has only two non-zero squared singular values

given in Lemma 1 and the relationship between H1and ¯

H

is given in (10), the two non-zero squared singular values of

√¯

K¯

Hare given in (19) and (20), respectively, based on (14)

and (15).

Meanwhile, using [37, Lemma 2], we get

lim

τ→0

0F1(n−m+1,(¯

K+1)(φi+τi)λ)Ci,j (φi+τi)

m

Q

k<l

((φl+τl)−(φk+τk))

=fi(λ)Di,j (φi+τi)

m−2

Q

l=1

(l−1)!

m

Q

l=m−1

φm−2

l(φm−φm−1)

,(24)

where τ={τ1, τ2, ..., τm}is an m-dimensional vector whose

elements are distinct,

fi(λ) = (λi−1[n−m+ 1]−1

i−1,1≤i≤m−2,

0F1n−m+ 1,¯

K+ 1φiλ,otherwise,

(25)

Di,j is given in (21) and Ci,j in [37, (3)] is the (i, j)-co-

factor of the m×mmatrix Awhose (i, j)th entry is Ai,j =

(n−m+j−1)!0F1(n−m+j, n −m+ 1, φi). Since the

Hhas only two non-zero squared values, its MPDF can be

derived as (18) by substituting (24) and (25) into [37, (3)].

B. Closed-form ergodic capacity

Theorem 2: The ergodic capacity at a typical position is

given by (26), where the average signal-to-noise-ratio (SNR)

f(λ) = e−φm−φm−1−(¯

K+1)λ

m((n−m)!)2λ

m

X

j=1 (¯

K+ 1)λn−m+j

(φmφm−1)m−2(φm−φm−1)

m−3

Q

l=0

l! (n−m+l)!

· m−2

X

i=1

Di,j ¯

K+ 1λi−1+0F1n−m+ 1,(¯

K+ 1)φm−1λ

(Dm−1,j )−1+0F1n−m+ 1,(¯

K+ 1)φmλ

(Dm,j )−1!

(18)

C(ρ) = κ

m

X

j=1 m−2

X

i=1

(ϑ−1)!

(Di,j )−1

ϑ

X

k=1

Eϑ−k+1 ¯

K+ 1

ρ/NT+

∞

X

p=0 Dm−1,j φp

m−1+Dm,j φp

m

p! (n−m+p)!((τ−1)! (n−m)!)−1

τ

X

k=1

Eτ−k+1 ¯

K+ 1

ρ/NT!

(26)

IEEE INTERNET OF THINGS JOURNAL 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (m)

0

1

2

3

4

5

6

7

8

9

10

y (m)

a

b

c

i

d

BS

e

f

g

h

a: (4.50, 1.25)

b: (3.50, 1.25)

c: (2.50, 1.25)

d: (1.50, 1.25)

e: (0.50, 1.25)

f: (0.50, 3.25)

g: (0.50, 5.25)

h: (0.50, 7.25)

i: (0.50, 9.25)

BS: ( 0, 0.0375)

Fig. 2. Cartesian coordinates for a room, e.g. W=L= 10 (m), where the

BS is close to the considered wall and there is an L-shaped route inside.

at each receiver branch is given by

ρ=¯

KkH1k2ρT

¯

K+ 1NRNT

,(27)

in which ρT=Ekxk2.Eknk2refers to the SNR at

transmitter side,

κ=

exp ¯

K+1

SNR/NT−φm−φm−1

ln2 ((n−m)!)m−1(φmφm−1)m−2(φm−φm−1)

m−3

Q

l=0

l!

,

(28)

ϑ=n−m+j+i−1,τ=n−m+j+p, and EQ(x) =

R∞

1e−xtt−Qdt.

Proof: The ergodic capacity can be derived by taking the

expectation with respect to λas follows [38]

C(ρ) = mElog21 + ρ

NT

λ

=mZ∞

0

log21 + ρ

NT

λf(λ)dλ,

(29)

where f(λ)is given in (18). The integral over

λin (29) is computed by the series expansion of

0F1n−m+ 1,¯

K+ 1φiλin (22) and

Z∞

0

ln (1 + $λ)λη−1e−γ λdλ=(η−1)!

e−γ

$γη

η

X

l=1

Eη−l+1 γ

$

(30)

in [39, Appendix A]. Thus, the ergodic capacity at a typical

UE position is given in (26).

IV. WIRELESS FRIENDLINESS EVALUATION SCHEME FOR

BUILDING MATER IA LS

In this section, we investigate how friendly a wall is to

indoor LOS MIMO transmissions. Aiming to quantify the

wireless friendliness of a building material, a reasonable indi-

cator is the expectation of capacity E(C). However, the value

Algorithm 1: An Wireless Friendliness Evaluation

Scheme for a Building Material

Input: ε0,ζ,K,µ,W,L,X,Y,NT,NR,D,D1,ρT

Output: Cavg

1Calculate the step in xaxis: ∆x=W

X;

2Calculate the step in yaxis: ∆y=L−D1

Y−1;

3Determine the coordinates (x, y)of all sample points :

x=−W/2 + (∆x/2) : ∆x:W/2−(∆x/2);

y=D1: ∆y:L;

4Determine the BS position located at (0, D1);

5for i= 1; i≤Xdo

6for j= 1; j≤Ydo

7Determine the UE location (xi, yj);

8Calculate d1,d2,θT1,θT2,θR1and θR2,α;

9Calculate Γwith (4)-(7);

10 Constructe Hwith (8)-(13);

11 Derive C(xi, yj)with (26)-(28);

12 Calculate Cavg with (31);

13 return Cavg;

of E(C)cannot be calculated straightforwardly. An alternative

solution is to average the capacity values over dense sample

points inside the room, since the limit of the mean capacity

values over sample points equals E(C)as the sampling density

approaches inﬁnity.

Building a two-dimensional Cartesian coordinate system

inside a W×Lrectangular room as shown in Fig. 2, we take

X×Ysample points spatially evenly distributed throughout the

room. For a UE at the location (xi, yj)where i∈ {1,2, ..., X}

and j∈ {1,2, ..., Y }, its downlink ergodic capacity can be

computed by (26)-(28) and denoted as C(xi, yj)in bit/s/Hz.

The capacity spatially averaged over all sample points, used as

an evaluation indicator for measuring the wireless friendliness

of a building material, is given by

Cavg =1

XY

X

X

i=1

Y

X

j=1

C(xi, yj).(31)

According to [20], a simple expression of relative permit-

tivity εis given by ε=ε0−jε1, where the real part and the

imaginary part can be expressed as a function of frequency

f, i.e., ε0=ufvand ε1= 17.98σ/f , where σ=rftis the

conductivity of the building material, and constants u,v,rand

tare compiled in [20, Table III]. In the following, we focus

on the permittivity and thickness of the building material.

Permittivity: The permittivity in this paper refers to the real

part of relative permittivity ε0. The imaginary part is assumed

to be a constant. Note that both εand ε0are unitless.

Thickness: Since the building material in this paper is

assumed to be a homogenous dielectric reﬂector, the building

material’s thickness of ζequals to that of the wall.

The scheme for evaluating the wireless friendliness of a

building material is given in Algorithm 1. The permittivity ε0

and the thickness ζof a wall material are the inputs, and the

output Cavg is computed following (31) as an indicator of its

wireless friendliness. A higher Cavg indicates that a wall made

IEEE INTERNET OF THINGS JOURNAL 6

wall

z

ε

BS

antenna

b

n

m

n

UE

antenna

Main lobe Back lobe

1

d

2

d

1

D

wall

z

ε

BS

antenna

b

n

m

n

UE

antenna

Main lobe Back lobe

1

d

2

d

1

D

Fig. 3. The impact of the radiation pattern of a directional BS antenna on

the EM propagation along the LOS path and the WR path.

of this kind of material would be more friendly to indoor LOS

MIMO communications.

V. IM PACT O F DIRECTIONAL R AD IATI ON PATT ER N

In this section, we consider each transmit element in the BS

linear array as a directional antenna, as shown in Fig. 3. The

main lobe directivity gain and the back lobe directivity gain

are denoted as νmand νb, respectively, where ν2

m+ν2

b= 2

according to the energy conservation law. Due to the dynamic

attitude of UE, we assume an omnidirectional antenna for the

UE antenna for analytical tractability. The deterministic part

of H1should be rewritten accordingly as

H0

1=νmA1hH

β1hα1+νbA2hH

β2hα2.(32)

The two non-zero squared singular values of H0

1are derived

by replacing Xin (14) and (15) with

X0="νmA1+νbA2∆β∆α νbA2∆βkhα∗k

νbA2∆αkhβ∗kνbA2khα∗kkhβ∗k#.(33)

The ¯

Kin channel model (9)-(13), the squared singular

distribution of channel (18)-(20) and the ergodic capacity (26)-

(28) should all be replaced by K0, where K0=KS0, K is

the Rician factor given by (12), and

S0=

H0

1

2

kA1k2=

d2

1ν2

b

d2

2ν2

m

Γ2+2Md1νb

NTNRd2νm

Γ+1

.(34)

Meanwhile, the ergodic capacity C(ρ0)for directional an-

tenna cases should be derived by the average SNR at each

receiver antenna ρ0, which is given by

ρ0=K0kH0

1k2ρT

(K0+ 1) NRNT

.(35)

The wireless friendliness evaluation scheme for directional

BS antenna arrays is similar to Algorithm I by using (32)-(35).

TABLE I

MAI N SIM UL ATION A SS UMP TI ON

Parameter name Parameter value

Frequency f(GHz) 6

Room width W(m) 10

Room length L(m) 10

Inter-antenna spacing D=µ/2(m) 0.025

The distance from BS to wall D1(m) 0.0375

Number of BS antennas NT4

Number of UE antennas NR4

Samples along room xaxis X100

Samples along room yaxis Y100

VI. NUMERICAL RESULTS

In this section, we present and analyse the numerical

results for both omnidirectional and directional BS antenna

arrays to present a comprehensive understanding of the impact

of building materials as reﬂectors on indoor LOS MIMO

communications. Subsection A-C show the results for the

omnidirectional BS antenna array, while Subsection D shows

the results for the directional BS antenna array.

The parameters used in the simulations are given in Table I.

The incident wave is assumed to be TE polarised. The trans-

mission power of the BS is assumed to be equally allocated

to every transmit antenna element. The BS is deployed at

point (0, D1) and its antenna array is deployed parallel to the

considered wall. In the Monte Carlo simulations, the ergodic

capacity at point (xi, yj)is computed by

Csim (xi, yj)=Elog2det I+ρ

NT

HH†.(36)

A. The ergodic capacity for a speciﬁc sample point

In this subsection, we take three points, i.e. (0.3, 0.25), (4.5,

8.0) and (-2.5, 0.55), as examples to verify the correctness of

analytical expression of (18)-(20) and (26)-(28). The Rician

factor Kof the three points is computed by (12) accordingly.

Fig. 4 depicts the MPDF of an unordered squared singular

value of the LOS MIMO channel. It is found that the MPDF

becomes more concentrated as Kis reduced, which reveals

that the squared singular values of matrix Hare more evenly

distributed and thus results in a larger ergodic capacity.

Fig. 5 shows the ergodic capacity versus the transmit

SNR. The ergodic capacity increases when the transmit SNR

increases. Meanwhile, a larger Kleads to a lower ergodic

capacity under the same transmit SNR due to the less con-

centrated MPDF of an unordered squared singular value of

channel matrix H.

From the results for ∆θbeing 0 and π/2in Fig. 4 and Fig.

5, respectively, we can see that ∆θhas a limited impact on the

MPDF of an unordered squared singular value and the ergodic

capacity of indoor LOS MIMO channel. This is because ∆β

in (16) is hardly affected by ∆θ. When ∆θincrease from 0

to π,∆βis always very close to 1. As a result, the squared

singular values of Hderived by (14)-(15) will not change

much with ∆θ. Consequently, the MPDF of an unordered

squared singular value in (18)-(20) and the ergodic capacity

in (26)-(28) of Hwill stay nearly constant for varying ∆θ.

IEEE INTERNET OF THINGS JOURNAL 7

0 5 10 15 20

0

0.05

0.1

0.15

0.2

0.25

0.3

MPDF

13.18 13.2 13.22

0.0404

0.0406

0.0408

Fig. 4. The MPDF of an unordered squared singular value at three points,

for ζ= 0.2,ε= 5.31 −j0.5861f−0.1905 [20, Table III]. Markers represent

simulation values while red solid lines and blue dash lines represent analytical

values when ∆θ= 0 and ∆θ=π/2, respectively.

55 60 65 70 75 80 85 90 95

T (dB)

0

10

20

30

40

50

60

70

Ergodic capacity (bit/s/Hz)

87.67 87.672

31.76

31.765

Fig. 5. Relationship between ergodic capacity and transmit SNR at three

points, for ζ= 0.2,ε= 5.31 −j0.5861f−0.1905. Markers represent

simulation values while red solid lines and blue dash lines represent analytical

values when ∆θ= 0 and ∆θ=π/2, respectively.

Therefore, due to the space limitation, all the numerical results

hereinafter are conducted when ∆θ= 0.

B. The ergodic capacity distribution in a square room

To verify the accuracy of the evaluation indicator Cavg in

(31) and the usefulness of Algorithm 1, the ergodic capacity

at different positions is studied in this subsection.

a b c d e f g h i

Point

0

1

2

3

4

5

6

7

8

9

Ergodic Capacity (bit/s/Hz)

T=50dB No WR

T=60dB No WR

T=50dB WR

T=60dB WR

Fig. 6. Ergodic capacity at the points for L-shaped route, for ζ= 0.2,

ε= 5.31 −j0.5861f−0.1905. Markers represent simulation values while

both the solid and dash lines represent analytical values.

Fig. 7. Ergodic capacity distribution in the 10 m ×10 m square room, for

ρT= 60 dB, ζ= 0.2,ε= 5.31 −j0.5861f−0.1905. Markers represent

simulation values while the lines represent analytical values.

We design a L-shaped route that includes some typical UE

positions in the square room, as shown in Fig. 2. The ergodic

capacities from point a to j along this route, in the presence

or absence of WR path, are shown in Fig. 6. The dash lines

illustrate the results taking into account the WR path. From

point a to e, we observe an increase in capacity as the UE is

approaching the BS except for point d, where the slump in

ergodic capacity is due to the power cancellation caused by

IEEE INTERNET OF THINGS JOURNAL 8

Fig. 8. The ergodic capacity difference between our proposed channel and the

Rician channel, for ρT= 60 dB, ζ= 0.2,ε= 5.31−j0.5861f−0.1905 . The

positive/negative difference indicates constructive/destructive interference.

3.5 4 4.5 5 5.5 6 6.5 7 7.5

Permittivity

2.58

2.6

2.62

2.64

2.66

2.68

2.7

2.72

2.74

2.76

Spatially averaged capacity (bit/s/Hz)

Thickness=0.1

Thickness=0.15

Thickness=0.2

Simulations

Fig. 9. Impact of wall permittivity on spatially averaged capacity for the

omnidirectional BS antenna array for ρT= 60 dB.

the destructive combination of the LOS path and the WR path.

When the UE moves from point e to i, the capacity declines.

This is different from the ergodic capacity under the Rician

fading model without considering the WR path that would

monotonically decrease with an increasing UE-BS distance,

as shown by the solid lines in Fig. 6.

The spatial distribution of the ergodic capacity in a square

room using our proposed model is shown in Fig. 7. It is

observed that the ergodic capacity is not a monotonic function

of the UE-BS distance. This phenomenon can be attributed

0.1 0.15 0.2 0.25 0.3

Thickness (m)

2.6

2.62

2.64

2.66

2.68

2.7

2.72

Spatially averaged capacity (bit/s/Hz)

Permittivity=4.5

Permittivity=5.5

Permittivity=6.5

Simulations

Fig. 10. Impact of wall thickness on spatially averaged capacity for the

omnidirectional BS antenna array for ρT= 60 dB.

Fig. 11. The composite impact of permittivity and thickness on spatially

averaged capacity for the omnidirectional BS antenna array for ρT= 60 dB.

to the constructive and destructive interference between the

EM waves along the LOS path and the WR path. The length

difference in O(λ)leads to the great changes of the amplitude

and phase of the superposed EM wave arriving at the UE.

Fig. 8 plots the ergodic capacity difference between our

proposed channel model based on (8)-(13) and the Rician

fading channel model. The position with a positive/negative

difference corresponds to a location that experiences the

constructive/destructive interference between the EM waves

along the LOS path and the WR path. We can see that the

IEEE INTERNET OF THINGS JOURNAL 9

0 5 10 15 20

0

0.05

0.1

0.15

0.2

0.25

0.3

MPDF

Fig. 12. The MPDF of an unordered squared singular value at three points,

for ζ= 0.2,ε= 5.31 −j0.5861f−0.1905. Markers represent simulation

values while lines represent analytical values.

positions suffering from the destructive interference appear

in certain directions, along which the Fabry-P´

erot resonance

phenomenon of EM waves is observed.

Concluded from Fig. 6-8, the impact of the WR path

that characterises the EM and physical properties of building

materials on indoor ergodic capacity is non-trivial, which

cannot be ignored in indoor LOS MIMO communications.

C. The impact of wall permittivity and thickness on spatially

averaged capacity

In order to identify a wall material with desirable wireless

friendliness, the relationship between the evaluation indicator

named spatially averaged capacity Cavg and the key parame-

ters, i.e., the permittivity ε0and the thickness ζof building

materials is shown in this subsection.

Fig. 9 and Fig. 10 plot the spatially averaged capacity as

a function of the permittivity and the thickness of building

materials, respectively. In Fig. 9, as the permittivity increases

from 3.5 to 7.5, the envelope of each spatially averaged

capacity curve presents a upward trend. The variation in the

spatially averaged capacity becomes more signiﬁcant with the

increase of the permittivity. In Fig. 10, for a given permittivity,

as the wall thickness increases, the spatially averaged capacity

ﬁrst ﬂuctuates with it under a decreasing envelope and grad-

ually converges to a constant value when the thickness goes

beyond 0.25 m. We observe quite severe ﬂuctuations of the

spatially averaged capacity at relatively small wall-thickness

values. That is because the reﬂection coefﬁcient amplitude

ﬂuctuates more severely when the wall is thinner. It should

be highlighted that a tiny lapse in the wall permittivity or the

thickness would bring about changes in the spatially averaged

capacity of up to 0.19 bits/s/Hz.

55 60 65 70 75 80 85 90 95

T (dB)

0

10

20

30

40

50

60

70

Ergodic capacity (bits/s/Hz)

Fig. 13. Relationship between ergodic capacity and transmit SNR at three

points, for ζ= 0.2,ε= 5.31 −j0.5861f−0.1905. Markers represent

simulation values while lines represent analytical values.

a b c d e f g a b

Point

0

1

2

3

4

5

6

7

8

9

Ergodic Capacity (bit/s/Hz)

T=50dB No WR

T=60dB No WR

T=50dB WR

T=60dB WR

Fig. 14. Ergodic capacity at the points for L-shaped route, for ζ= 0.2,

ε= 5.31 −j0.5861f−0.1905. Markers represent simulation values while

both the solid and dash lines represent analytical values.

The composite impact of the building material’s permittivity

and thickness on the spatially averaged capacity is illustrated

in Fig. 11. When the permittivity and the thickness are

conﬁgured in the range from 1.5 to 7.5 and from 0.1 to

0.25 m, respectively, we observe that the optimal parameter

pair of [ε0,ζ] resulting in the highest spatially averaged

capacity of 2.763 bits/s/Hz is [6.92, 0.1], while the worst

pair resulting in the lowest spatially averaged capacity of

2.415 bits/s/Hz is [1.50, 0.1]. We can conclude that certain

combinations of the wall thickness and permittivity values lead

IEEE INTERNET OF THINGS JOURNAL 10

3.5 4 4.5 5 5.5 6 6.5 7 7.5

Permittivity

2.96

2.97

2.98

2.99

3

3.01

3.02

3.03

3.04

3.05

3.06

Spatially averaged capacity (bit/s/Hz)

Thickness=0.1

Thickness=0.15

Thickness=0.2

Simulations

Fig. 15. Impact of wall permittivity on spatially averaged capacity for the

directional BS antenna array for ρT= 60 dB.

to peak values of the spatially averaged capacity, which can

be more than 14.4% higher than the lowest spatially averaged

capacity values associated with some combinations of the wall

thickness and permittivity values that should be avoided during

the selection and/or design of building materials.

D. Analysis for directional BS antenna array

In this subsection, Fig. 12-17 depict the numerical results

for the directional BS antenna array. The main lobe directivity

gain and the back lobe directivity gain are assumed as νm=

p1/3and νb=p5/3, respectively.

Fig. 12 and Fig. 13 are plotted for the same three sample

points shown in Fig. 4 and Fig. 5, i.e. (0.3, 0.25), (4.5, 8.0)

and (-2.5, 0.55). It is found in Fig. 12 that the MPDF of an

unordered squared singular value of the LOS MIMO chan-

nel becomes more concentrated and that the largest squared

singular value of the LOS MIMO channel becomes larger,

when compared with the omnidirectional BS antenna array

in Fig. 4. Moreover, the difference between the MPDF under

different Rician factor Kbecomes less substantial. In Fig. 13,

the ergodic capacity shows an increasing trend with the rise of

transmit SNR. When comparing the red lines in Fig. 5 with the

red lines in Fig. 13, the ergodic capacity under the directional

BS antenna array is shown to be larger than that under the

omnidirectional BS antenna array.

Given the same 9 points along the L-route drawn in Fig. 2,

Fig. 14 shows similar results from Fig. 6. Even though there is

less change in ergodic capacity in the presence or absence of

the WR path compared with the omnidirectional BS antenna

array, it still appears some positions that suffer destructive

interference, such as point d.

With regards to the impact of the wall permittivity and

thickness on spatially averaged capacity, signiﬁcant variation

up to 0.093 bits/s/Hz can be observed from Fig. 15 and Fig.

0.1 0.15 0.2 0.25 0.3

Thickness (m)

2.97

2.98

2.99

3

3.01

3.02

3.03

Spatially averaged capacity (bit/s/Hz)

Permittivity=4.5

Permittivity=5.5

Permittivity=6.5

Simulations

Fig. 16. Impact of wall thickness on spatially averaged capacity for the

directional BS antenna array for ρT= 60 dB.

Fig. 17. The composite impact of permittivity and thickness on spatially

averaged capacity for the directional BS antenna array for ρT= 60 dB.

16. In Fig. 17, it is found that the optimal wall parameter pair

of [ε0,ζ] leading to the highest spatially averaged capacity

of 3.057 bits/s/Hz is [6.84, 0.10], while the worst pair leading

to the lowest spatially averaged capacity of 2.887 bits/s/Hz

is [1.50, 0.10]. The 5.9% difference in the spatially averaged

capacity generated by the certain combinations of the wall

permittivity and thickness is worthy to be considered.

The results above indicate that for directional BS antenna

arrays, the EM and physical properties of building materials

will deﬁnitely exert a substantial inﬂuence on a room’s wire-

less performance.

IEEE INTERNET OF THINGS JOURNAL 11

VII. CONCLUSIONS AND FUT UR E WORKS

In this paper, we ﬁrstly construct a new indoor LOS MIMO

downlink channel model by considering the impact of building

materials on indoor wireless performance. Then, the MPDF

of the squared singular value and the ergodic capacity of the

indoor LOS MIMO channel and the squared singular value

of its deterministic part are obtained in closed forms. On

this basis, a scheme for evaluating the wireless friendliness

of building materials is provided. The analytical results are

veriﬁed through Monte Carlo simulations conducted in the

6 GHz band. Numerical results demonstrate that building

materials as reﬂectors have to be well selected or designed

to avoid the risk of reducing indoor wireless performance,

because a minor discrepancy in the conﬁguration of the relative

permittivity and thickness of the wall material might cause

over 14.4% losses in indoor capacity.

This work is the ﬁrst attempt to investigate how indoor

capacity is inﬂuenced by the EM and physical properties

of building materials, which has laid a solid foundation of

wireless friendliness evaluation of indoor built environments.

In future works, there are some extensive research directions

worthy of in-depth study, including but not limit to: 1) Multi-

layer building materials: It is of great necessity to analyse

both single-layer and multi-layer building materials. One of

the challenges is that the analytical relationship between the

indoor MIMO capacity and the EM and physical properties of

multi-layer building materials is hard to ﬁnd due to the non-

closed-form reﬂection coefﬁcients [20]. Moreover, the optimal

conﬁguration of multi-layer materials would be formulated

into a multi-variate optimisation problem, the complexity of

which would increase with the number of layers. 2) Millimetre

wave (mmWave): MmWave bands have been a promising

candidate for 5G and B5G wireless network, which has been

widely studied in both theoretical and experimental study. The

propagation mechanism for sub-6 GHz bands and mmWave

bands are substantially different due to the orders of magnitude

change in the wavelength [10], [15], [40]. Hence, the work

extending to mmWave bands should start from the accurate

and tractable modelling of indoor mmWave propagation. An-

other scheme is required to evaluate the wireless friendliness

of building materials at mmWave bands. 3) Experimental

validation: It is critical to verify the numerical results of

the proposed channel model and the evaluation scheme with

experimental results. A comprehensive measurement campaign

will be held in future work.

APPENDIX: PRO OF O F LEMMA 1

Since two deterministic components including the LOS path

and the WR path are considered in our model as shown in (8),

it is obvious that H1has two non-zero singular values. The

singular value decomposition of deterministic part H1can be

given as

H1=BHXA,(37)

where

B=hβ1hβ∗/khβ∗kT,

A=hα1hα∗/khα∗kT,

X= 1 ∆β

0khβ∗k! A10

0A2! 1 0

∆αkhα∗k!.

Initially using Gram-Schmidt process, hα1and hH

β1are or-

thonormalized with hα∗/khα∗kand hH

β∗.khβ∗k, respectively,

which means that BHB=Iand AHA=I.

Then Xcan be simpliﬁed as a 2×2matrix, with four

elements given in (14). Assuming Xcould be decomposed as

X=PH √ϕm−10

0√ϕm!Q,(38)

where PHP=Iand QHQ=I, then ϕm−1and ϕmare two

squared singular values of matrix X. Hence we obtain

H1=(PB)H √ϕm−10

0√ϕm!QA.(39)

Here it is interesting to ﬁnd that ϕm−1and ϕmare the two

squared singular values of H1as well, since (PB)HPB =I

and (QA)HQA=I.

Therefore, the two non-zero squared singular values of H1,

i.e. ϕm−1and ϕm, can be easily obtained by conducting

singular value decomposition to low dimensional matrix X,

as given in (14) and (15), respectively.

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Secondquarter 2018.

Yixin Zhang received the B.Eng. degree from Bei-

jing University of Posts and Telecommunications,

Beijing, China, in 2017. She is currently pursuing

the Ph.D. degree in the Department of Electronic and

Electrical Engineering at the University of Shefﬁeld,

Shefﬁeld, UK. Her research interests include electro-

magnetic wave propagation, wireless channel mod-

elling, MIMO antenna conﬁgurations, and indoor

wireless networks.

Chen Chen received the B.Eng. degree from East

China University of Science and Technology, Shang-

hai, China, in 2018. He is currently pursuing his

Ph.D. degree in Wireless Communications at the

University of Shefﬁeld, UK. His current research

interests include millimeter wave networks, green

networks, stochastic geometry and machine learning.

Songjiang Yang received the B.Eng. degree with

class one honours in electronic and communication

engineering from the University of Shefﬁeld, U.K.,

in 2017, where he is currently pursuing the Ph.D. de-

gree. His research interests include millimetre wave

channel modelling and millimetre wave wireless

communications.

Jiliang Zhang (M’15, SM’19) received the B.E.,

M.E., and Ph.D. degrees from the Harbin Institute

of Technology, Harbin, China, in 2007, 2009, and

2014, respectively. He was a Postdoctoral Fellow

with Shenzhen Graduate School, Harbin Institute

of Technology from 2014 to 2016, an Associate

Professor with the School of Information Science

and Engineering, Lanzhou University from 2017 to

2019, and a researcher at the Department of Electri-

cal Engineering, Chalmers University of Technology,

Gothenburg, Sweden from 2017 to 2018. He is now

a Marie Curie Research Fellow at the Department of Electronic and Electrical

Engineering, The University of Shefﬁeld, Shefﬁeld, UK. His current research

interests include, but are not limited to wireless channel modelling, modulation

system, relay system, wireless ranging system, vehicular communications,

ultra-dense small cell networks, neural dynamic, and smart environment

modelling.

IEEE INTERNET OF THINGS JOURNAL 13

Xiaoli Chu is a Professor in the Department of Elec-

tronic and Electrical Engineering at the University

of Shefﬁeld, UK. She received the B.Eng. degree in

Electronic and Information Engineering from Xi’an

Jiao Tong University in 2001 and the Ph.D. degree

in Electrical and Electronic Engineering from the

Hong Kong University of Science and Technology

in 2005. From 2005 to 2012, she was with the

Centre for Telecommunications Research at Kings

College London. Xiaoli has co-authored over 150

peer-reviewed journal and conference papers. She is

co-recipient of the IEEE Communications Society 2017 Young Author Best

Paper Award. She co-authored/co-edited the books “Fog-Enabled Intelligent

IoT Systems”(Springer 2020), “Ultra Dense Networks for 5G and Be-

yond”(Wiley 2019), “Heterogeneous Cellular Networks: Theory, Simulation

and Deployment”(Cambridge University Press 2013), and “4G Femtocells:

Resource Allocation and Interference Management”(Springer 2013). She is

Senior Editor for the IEEE Wireless Communications Letters, Editor for

the IEEE Communications Letters, and received the IEEE Communications

Letters Exemplary Editor Award in 2018. She was Co-Chair of Wireless

Communications Symposium for IEEE ICC 2015, Workshop Co-Chair for

IEEE GreenCom 2013, and has co-organized 8 workshops at IEEE ICC,

GLOBECOM, WCNC, and PIMRC.

Jie Zhang has held the Chair in Wireless

Systems at the Department of Electronic and

Electrical Engineering, University of Shefﬁeld

(www.shefﬁeld.ac.uk) since Jan. 2011.

He is also Founder, Board Chairman and Chief

Scientiﬁc Ofﬁcer (CSO) of Ranplan Wireless

(www.ranplanwireless.com), a public company listed

on Nasdaq OMX. Ranplan Wireless produces a

suite of world leading indoor and the only joint

indoor–outdoor 5G/4G/WiFi network planning and

optimization tools suites including Ranplan Profes-

sional and Collaboration-Hub, which are being used by the world’s largest

mobile operators and network vendors across the globe.

Along with his students and colleagues, he has pioneered research in small

cell and heterogeneous network (HetNet) and published some of the landmark

papers and books on these topics, widely used by both academia and industry.

Prof. Zhang published some of the earliest papers on using social media

network data for proactive network optimisation. Since 2010, he and his

team have developed ground-breaking work in modelling and designing

smart built environments considering both wireless and energy efﬁciency. His

Google scholar citations are in excess of 7000 with an H-index of 35.

Prior to his current appointments, he studied and worked at Imperial

College London, Oxford University, University of Bedfordshire, and East

China University of Science and Technology, reaching the rank of a Lecturer,

Reader and Professor in 2002, 2005 and 2006 respectively.