TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5499 ~ 55
0
7
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.549
9
5499
Re
cei
v
ed
De
cem
ber 2
5
, 2013; Re
vi
sed
March 18, 20
14; Accepted
April 5, 2014
Spatial Characteristics of Wireless Channel in Tunnel
with Imperfect Walls
Yi Zhang
1
, Zhao Xu
2
, Bo-Ming Song
3
, Yu Huo
*
4
1,2,
3
School of In
formation a
nd
Electrical E
ngi
neer
ing, Ch
in
a Univers
i
t
y
of Minin
g
an
d T
e
chnol
og
y, Xuzh
o
u
,
Jian
gsu 2
210
0
8
, Chin
a,+
86 0
516 8
3
8
840
23
1
Huai
bei Min
i
n
g
(Group) Co
mpan
y Ltd., Huai
bei, An
hui 2
350
25, Ch
ina.
4
T
he Nation
al
and L
o
ca
l Joint
Engin
eeri
ng L
abor
ator
y
of Internet T
e
chnol
o
g
y
on Min
e
, IOT
Perception
Mine R
e
searc
h
Center, Chi
na
Univers
i
t
y
of Minin
g
an
d T
e
chnol
og
y, Xuzh
o
u
, Jiangs
u 22
1
008, Ch
in
a.
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: huo
yu
@al
i
y
u
n
.com
A
b
st
r
a
ct
T
he successful
desig
n an
d a
pplic
atio
n of the physi
c
a
l l
a
ye
r techniq
ues of
the w
i
reless n
e
tw
ork,
inclu
d
i
ng MIM
O, adaptiv
e O
F
DM, and
min
i
ng
ante
nna, t
o
na
me
a few
,
n
eed
the
deta
ile
d kn
ow
led
ge
of th
e
spatia
l ch
aract
e
ristics of t
he
trans
mitted s
i
g
nals. T
h
is p
a
p
e
r prov
id
es
a
mo
da
l a
ppro
a
c
h
to
qua
ntify t
h
e
spatia
l distrib
u
tion of the w
a
v
e
s in tun
nels i
n
detai
l. W
e
devel
op the
mo
dal the
o
ry for prop
agati
on i
n
a
tunne
l w
i
th imperfect w
a
lls, w
h
ich are loss
y, rough,
and
tilted. And the
n
, w
e
discuss the transmiss
i
o
n
pow
er of the
a
n
tenn
a as
a fu
nction
of
the a
ngl
es of d
epart
u
re (AOD). On
this bas
is, w
e
take the
ha
lf-w
ave
dip
o
le
ante
nna
as an ex
a
m
pl
e to an
alyse
b
y
simulati
ons.
T
he theor
etical
result
s show
that the an
gu
l
a
r
pow
er distrib
u
tion i
n
the rect
ang
ular tu
nne
l
follow
s
Gaussian d
i
stributi
o
n
.
T
he
angl
e sp
read (AS) of th
e
w
a
ves can
be
i
n
flue
nce
d
by t
he tu
nne
l w
a
lls
. T
he rou
g
h
nes
s of the w
a
l
l
su
rface is
mo
st
im
po
rta
n
t
i
n
small
tunne
ls
and
at
low
freq
ue
nci
e
s, w
hereas
t
he w
a
l
l
t
ilt
is most i
m
p
o
rtan
t
in larg
e
tu
nn
els and
at hi
g
h
freque
ncies.
Ke
y
w
ords
:
mo
dal the
o
ry, spa
t
ial distri
butio
n, tunnel, i
m
p
e
rfect w
a
lls, w
i
reless netw
o
rk
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Wirel
e
ss Und
e
rg
rou
nd Co
mmuni
cation
Networks (WUCNs) promi
s
e a wid
e
variety of
novel appli
c
a
t
ions, su
ch a
s
enviro
n
men
t
al monitori
ng
, localization,
disa
ster warning in high
way
and
railroad
tunnel
s a
s
we
ll as i
n
u
nde
rgrou
nd
mine
s. The
mai
n
chall
enge
for WUCNs is the
reali
z
ation of
efficient and
reliabl
e und
e
r
groun
d wi
rel
e
ss links. Physical
laye
r tech
niqu
es, such
as MIM
O
, a
daptive OF
DM, and mi
ni
ng ante
nna,
are
re
se
arched to
overcome the
severe
multipath fad
i
ng in
the
s
e
sp
eci
a
l
con
s
train
ed
envi
r
onm
ents [1, 2]. The
spa
t
ial pro
pag
ation
cha
r
a
c
t
e
ri
st
ic
s in t
h
e
wi
re
les
s
environ
ment si
gnificantly infl
uen
ce the pe
rformance of th
ese
physi
cal layer techniq
u
e
s
, so they sh
oul
d be analy
s
e
d
accordingly
[3, 4].
In unde
rgrou
nd, the tunne
l could b
e
viewe
d
as a
h
eavily overm
oded
waveg
u
i
de [5-7].
Con
s
e
quently
, the po
we
r
provide
d
by t
he ant
enna
can
be
effect
ively transmitted only if it
is
effectively co
upled
into th
e allo
we
d p
r
opag
ation m
ode
s. Due t
o
the
co
upli
ng b
e
twe
en
the
antenn
a and
the tunnel wa
veguide
stru
cture an
d the re
flection
s ca
use
d
by the tunnel
wall
s, the
angul
ar po
we
r distri
bution
of the wirele
ss ch
ann
el
is q
u
ite different from the terre
s
trial chan
nel
.
Liena
rd [8]
gi
ven the
map
of the di
re
ctio
ns
of
a
rrival
a
t
a di
stan
ce
o
f
40m i
n
a
tu
nnel
b
y
measurement
s a
nd th
eoretical
cal
c
ul
atio
n. It con
c
lu
de
d that the
spread
angl
e in
crea
se
s
with t
he
tunnel
si
ze. Z
hang
[9] me
a
s
ured
the
re
ceived
sign
al
power versu
s
the
width
o
r
t
he h
e
ight
of t
h
e
tunnel cross se
ction
i
n
two si
ze
tunn
el
s. Th
e
curve
s
sho
w
disto
r
ted d
o
min
a
n
t
-mode
cosi
n
e
function. Nasr [10] and Hu
o [11] calcula
t
ed the di
re
cti
on of arrival
(DOA) in a tu
nnel by the ray
method
in
straig
ht tun
n
e
ls. Su
n [1
2] cal
c
ul
ated
the
sp
atial po
we
r di
st
ribution
of t
he
electroma
gne
tic wave
s tra
n
smi
ssi
on m
ode in a
re
ctangula
r
tunn
el with perfe
ctly cond
ucti
ng
wall
s.
All in all,
the related theore
t
i
cal re
sult
s be
en obtaine
d to date are stil
l too limited to guide
the de
sig
n
a
n
d
ap
plication
of MIMO, a
d
aptive OF
DM
, and
minin
g
antenn
a, etc.
In the
practi
cal
appli
c
ation, the tunnel
wal
l
s are
rou
gh
and may hav
e long ra
nge
tilt variations. In addition, they
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02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5499 – 55
07
5500
are imp
e
rfe
c
t
l
y condu
cting
.
The influen
ce
s of t
hese
factors on t
he spatia
l di
stributio
n of the
wave
s we
re n
o
t con
s
ide
r
ed
in the above work [8-12].
The go
al of
this pap
er i
s
to develo
p
a
theoretica
l
analysi
s
to
quantify the spatial
distrib
u
tion o
f
the wave
s
in tunnel
s in
detail
an
d t
o
provide a
better u
nde
rstandi
ng of t
he
waveg
u
ide e
ffects in tun
nels. In [4], we h
a
ve pro
posed a m
u
l
t
imode op
era
t
ing waveg
u
i
de
model. Its em
pha
sis
wa
s o
n
ly on tunn
el
s with
small
d
egre
e
tilted walls. In this
p
aper,
we exte
nd
the wo
rk in [6
] by consid
eri
ng the ca
se o
f
wall t
ilt, which the tunn
el may also exh
i
bit to a marked
degree. The remaind
e
r of this pap
er is o
r
gani
ze
d as
follows. In Section 2, we d
e
velop the modal
theory for pro
pagatio
n in a tunnel with lo
ssy, rou
gh,
a
nd tilted walls. We discuss
the relation
sh
ip
betwe
en the
angle
s
of de
p
a
rture (A
O
D
)
and the
ord
e
r of each mod
e
in the ray pi
cture. An
d th
en
on this ba
si
s, we get the tra
n
smi
ssi
on po
wer a
s
a
fun
c
tion of AOD. In Section 3, we evalu
a
te the
statistical ch
ara
c
teri
stics
of the angul
ar po
wer
di
stribution an
d
the effects of the dielectric
prop
ertie
s
, ro
ughn
ess an
d tilt of
the tunnel wall
s.
2. Modal Ap
proach
to Angular Dis
t
ribution of
Wa
v
e
s
2.1. The Multimode Model
(1) Strai
ght tunnel mod
e
l
Here
we
con
s
ide
r
th
e g
e
neral
ca
se
of a
radi
o p
r
o
pagatio
n
cha
nnel i
n
a
rectangula
r
tunnel. Sup
p
o
se
that the
recta
ngul
ar tunnel
of
w width and
h height.
L
o
cate
the co
ordi
nate
system in the
centre of the tunnel
cro
ss
se
ction. z axi
s
is defin
ed
a
s
the longitu
d
i
nal dire
ction
of
the tunnel, x
axis a
s
the
wi
dth,
and y axi
s
a
s
the
heig
h
t. Con
s
ide
r
K1 is the
ele
c
trical
paramet
ers
of the mate
ri
al on th
e
side
wall
s, an
d K
2
corre
s
pon
di
ng to the
roof
and flo
o
r. T
h
e ro
ugh
surfa
c
e
of the wall
s i
s
a
s
sumed
to have a
Ga
ussian
di
stri
b
u
tion
with a stand
ard de
ri
vation
equ
al to
2
rough
.
In the recta
n
gular
waveg
u
ide, the ele
c
tri
c
field E is pola
r
i
z
ed
pred
omin
antly in the
hori
z
ontal
an
d vertical di
rection
s
, re
sp
ectively
. The
main field compon
ents
a
r
e the tan
g
e
n
tial
field compo
n
ents i
n
the
tu
nnel [7]. If th
e tunn
el i
s
straight, for ho
ri
zontal
pol
ari
z
ed
(m, n
)
m
o
de
,
they are [6],
0
co
s
c
o
s
ex
p
mn
m
n
x
x
mn
y
m
n
m
n
z
mn
E
E
ik
x
k
y
i
k
z
M
0
1
co
s
c
o
s
ex
p
mn
mn
y
x
mn
y
m
n
m
n
z
mn
mn
E
Hi
k
x
k
y
i
k
z
Z
M
(
1
)
Whe
r
e
0
mn
E
is the mode inten
s
i
t
y on the excitation
plane. We will di
scu
ss them late
r. M
is the
num
ber
of the
allowed m
o
d
e
s th
at pro
pagate
in t
he tun
nel.
Here
we
de
fine
ma
x
0,
1
,
2,
,
mm
;
ma
x
=0
,
1
,
2
,
,
nn
; m and n ca
n not equal to
0 at the sam
e
time [6],
ma
x
2
w
m
;
ma
x
2
h
n
(
2
)
The nu
mbe
r
of the allo
we
d mode
s
whi
c
h
c
oul
d pro
pagate i
n
the
tunnel M i
s
given by
[11, 12]:
ma
x
m
a
x
2
22
1
2
1
1
32
8
Mm
n
wh
wh
(
3
)
mn
Z
is the ch
ara
c
t
e
risti
c
impe
d
ance,
0
22
Z
mn
z
m
n
mn
mn
x
m
n
z
m
n
Ek
H
kk
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Spatial Cha
r
a
c
teri
stics of Wirel
e
ss Cha
nnel in Tun
n
e
l
with Im
perfect Wall
s (Yi Z
hang
)
5501
x
mn
k
,
ymn
k
and
zm
n
k
are defin
ed as [7]:
21
/
2
1
(1
1
)
xm
n
mm
K
ki
ww
K
(
5
)
ym
n
21
/
2
(2
1
)
nn
ki
hh
K
(
6
)
22
2
33
21
2
11
2
1
zm
n
im
K
n
k
wK
h
K
(
7
)
2
22
2
2
0
2
4
xm
n
y
m
n
z
m
n
kk
k
k
(
8
)
Whe
r
e
is the wavele
ngth.
mn
is the loss factor
cau
s
ed b
y
the surfa
c
e
roug
hne
ss [6],
22
33
44
ex
p
(
)
2
ro
u
g
h
mn
z
mn
wh
(
9
)
(2) Tilted tun
nel model
Suppo
se th
at there i
s
lon
g
ran
ge tilt
of
one
ve
rtical
wall
of the tu
nnel. Th
en th
e ray of a
mode en
co
un
ters a p
o
rtion
of the
tilted vertical
wall through a
n
angl
e
1
ti
lt
about the y axis.
Then the ele
c
tromag
netic fi
eld is chan
ge
d from (1
) to:
0
11
cos
c
os
e
x
p
c
o
s
2
s
i
n
2
ti
lt
mn
m
n
x
x
m
n
y
m
n
m
n
z
m
n
ti
lt
ti
lt
E
Ei
k
x
k
y
i
k
z
x
M
0
11
1
c
o
s
c
os
exp
c
os
2
s
i
n
2
t
ilt
mn
m
n
y
x
m
n
y
m
n
m
n
z
m
n
ti
lt
ti
lt
mn
E
Hi
k
x
k
y
i
k
z
x
Z
M
(10
)
The po
we
r co
upling fa
ctor
1
tm
n
g
of the disturb
ed field ba
ck i
n
to the mode
is given by:
2
1
2
2
t
ilt
mn
m
n
tm
n
tilt
mn
mn
E
E
dx
dy
g
E
dx
dy
E
d
x
d
y
(
1
1
)
Then by de
rivation,
22
2
11
1
exp
s
in
2
16
t
m
n
z
m
n
tilt
gk
w
(
1
2
)
Like
wi
se, if there i
s
lon
g
ra
nge tilt of one
hor
i
z
ontal
wall of the tunn
el, the ray of
a mode
encounte
r
s a
portion of the tilted
horizo
n
tal wall thro
ugh an a
ngle
2
tilt
about the x axis. Then
tilting of the floor o
r
roof gi
ves a co
uplin
g factor,
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5499 – 55
07
5502
22
2
22
1
ex
p
s
in
2
16
t
m
n
z
m
n
tilt
gk
h
(
1
3
)
Suppo
se that
(m, n) mod
e
boun
ce
s fro
m
wall
to wa
ll of the strai
ght tunnel m
a
kin
g
a
gra
z
ing a
ngle
1
mn
with the side
walls a
nd
2
mn
with the floor an
d roof. Then [
6
]:
1
0
si
n
2
xm
n
mn
k
m
kw
;
2
0
si
n
2
ym
n
mn
k
n
kh
(
1
4
)
Then in
a tilted wall tunne
l, the numbe
rs of refle
c
tion
s
1
mn
N
and
2
mn
N
experienced by
a ray at the
vertical a
nd
hori
z
ontal
wa
lls of t
he tun
nel, while t
r
a
v
elling a di
stance z
along
the
tunnel, are gi
ven by:
1
1
1
si
n
co
s
2
mn
mn
ti
lt
z
N
w
;
2
2
2
si
n
co
s
2
mn
mn
til
t
z
N
h
(
1
5
)
The loss fa
ctor for a di
stan
ce z i
s
:
12
12
22
2
12
12
sin
2
s
i
n
2
ex
p
+
8
c
os
2
c
os
2
mn
mn
NN
tm
n
t
m
n
t
m
n
tilt
tilt
tilt
tilt
gg
g
mn
z
(
1
6
)
Then field st
rength in a tilted tunnel
can
be
expre
s
se
d by the approximating fun
c
tion.
0
co
s
c
os
R
e
ex
p
mn
mn
x
m
n
y
mn
m
n
m
n
z
m
n
E
E
kx
k
y
g
i
k
z
M
0
1
co
s
c
o
s
R
e
ex
p
mn
m
n
x
m
n
y
mn
mn
mn
z
m
n
mn
E
Hk
x
k
y
g
i
k
z
Z
M
(
1
7
)
2.2. Po
w
e
r
Distribution V
e
rsus
AO
D
Define the e
x
citation plai
n at z=0. Assumi
ng a h
o
rizontally p
o
lari
zed tra
n
smitting
antenn
a that is located at (x0, y0, 0) in
the tunnel. Its surfa
c
e
cu
rre
nt distributio
n
is
K
.
Acco
rdi
ng to
(1), by m
a
tchi
ng the tan
g
e
n
tial
bou
nda
ry conditio
n
s
ov
er the
cross sectio
n
contai
ning th
e antenn
a, for (m, n) mo
de
we obtain:
0
2
co
s
c
os
e
x
p
0
mn
mn
zy
x
m
n
y
m
n
z
m
n
v
Z
E
M
iK
i
k
x
k
y
i
k
z
z
d
V
(
1
8
)
For on
e dire
ction of the tunnel, the lef
t
or
the right
of the anten
na, the tran
smissi
on
power of the (m, n) mode i
s
:
22
,
1
22
mn
mn
mn
mn
mn
xy
EE
Pw
h
Z
Z
(
1
9
)
Here we
cha
r
acteri
ze AO
D of (m, n) m
ode in the ray picture by the azimuth angl
e
mn
and elevatio
n
angle
mn
.
mn
is the an
gle bet
wee
n
the p
r
o
j
ecti
on
of the ray at the h
o
rizontal
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TELKOM
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ISSN:
2302-4
046
Spatial Cha
r
a
c
teri
stics of Wirel
e
ss Cha
nnel in Tun
n
e
l
with Im
perfect Wall
s (Yi Z
hang
)
5503
plane an
d x
axis. Its boundary is
[0
,
2
]
.
mn
is
the angle be
tween the ra
y and y axis. Its
boun
dary i
s
[0
,
]
.
Figure 1
sh
o
w
s the
spatial
paramete
r
s
of
a mo
de in
the ray pictu
r
e
.
They are
[11, 13]:
22
2
π
π
+a
r
c
s
i
n
(
)
2
4
mn
mh
wh
n
;
π
ar
c
s
i
n
(
)
22
mn
n
h
(
2
0
)
Whe
n
the
receivin
g a
n
tenna
is in t
he front of t
he tran
smitting a
n
tenn
a,
0
,
otherwise,
1
.
Then we ca
n get the relatio
n
shi
p
s b
e
twe
en m, n and
AOD:
2
si
n
c
os
w
m
;
2
co
s
h
n
(
2
1
)
Figure 1. Azimuth Angle a
nd Elevation A
ngle of a Wave in Coo
r
di
nal System
On
com
b
inin
g (2)-(9
), (1
6)-(19
)
and
(21),
we
coul
d obtai
n the
tran
smi
ssi
o
n
po
we
r
,
p
of a horizo
n
tally polari
z
ed
antenna de
scrib
ed by AOD. The tran
smissi
on po
wer of a
vertically pola
r
ize
d
anten
na
can be o
b
tai
ned by the sa
me way.
3. Results a
nd Discu
ssi
on
In this sectio
n, we begin
with an initial
exam
ple that illustrate
s the
spatial di
stribution of
the power in
a recta
ngul
ar
tunnel. Then
we an
alyz
e th
e effects of variou
s tunn
el con
d
ition
s
.
Except stu
d
ying the
effect
s of
certai
n p
a
ra
m
e
ters, th
e default tu
n
nel conditio
n
s
a
r
e
set
as follo
ws: the tunnel
cross sectio
n sha
pe is
a re
ct
an
gle with a
hei
ght of 4m an
d a wi
dth of 6
m;
con
s
id
er the gene
ral case
that the walls of t
he tunnel are ma
d
e
of the same material wi
th
electri
c
al p
a
rameters K1=K2=10
-j0.1
8
; the
carri
er
freque
ncy is set to 900MHz; the ro
ugh
s
u
r
f
ac
e ha
s
a
G
a
u
s
s
i
an
dis
t
r
i
b
u
t
ion
w
i
th
a s
t
an
d
a
rd
derivation
eq
ual to
0.08m;
the
sid
e
wall
is
tilted through
1° abo
ut the vertical axis.
In this paper
we take a ho
rizontally pola
r
ized
half
-
wave dipole, whi
c
h is the most typical
linear a
n
tenn
a and ea
sy to transfo
rm int
o
many other
antenn
as, a
s
an example t
o
analyse.
Then the
surf
ace
curre
n
t distributio
n of the tran
smittin
g
antenn
a is:
00
0
0
2
co
s
(
)
44
x
x
Ki
I
y
y
u
x
x
u
x
x
(
2
2
)
It is located i
n
the middle
of the tunnel
cr
o
s
s se
ctio
n. The observation point is 100m
away from th
e transmitter
and in the mi
ddle of the tunnel cro
ss
se
ction.
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TELKOM
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Vol. 12, No. 7, July 201
4: 5499 – 55
07
5504
3.1. Angular
Po
w
e
r Dis
t
ri
bution
[11] comp
ute
d
the ang
ula
r
powe
r
di
strib
u
tion
in a rectangula
r
tunn
el with pe
rfe
c
t walls,
whi
c
h i
s
strai
ght an
d smo
o
th. It con
c
lu
ded th
at the
distrib
u
tion fo
llows G
aussi
an di
strib
u
tio
n
. In
this p
ape
r, we an
alyse
the
distri
bution
i
n
a tu
nnel
wit
h
impe
rfe
c
t walls,
whi
c
h i
s
lossy, rou
gh
and
tilted. Figure
2(a
)
p
r
e
s
en
ts the tran
smissi
on
po
wer ve
rsu
s
A
O
D
dedu
ce
d
from the
m
odal
approa
ch. We normali
ze
the po
wer
of eac
h mod
e
by the maxi
mum po
we
r
among
mod
e
s
.
Gau
ssi
an di
stribution d
e
fin
ed in [14] an
d Lapla
c
ia
n
distrib
u
tion d
e
fined in [15]
are two typical
angul
ar
po
we
r di
strib
u
tion
s for te
rre
stri
al
enviro
n
m
ent
s. We
comp
a
r
e thei
r
powe
r
an
gle
profile
in
Figure 2(b)
and (c). T
o
generat
e
graphs of these 2
probability de
nsity functions, the values of
th
e
me
an
pa
th
AOD
0
,
0
and
th
e an
gle
sp
rea
d
(AS)
,
hav
e be
en
cal
c
ul
ated
so th
at they
dedu
ce
d eith
er from the
Gau
ssi
an di
stribution
or
f
r
om the
actu
al dist
ributio
n
of the field
are
identical. The
y
are given b
y
:
0
E
;
0
E
(
2
3
)
2
2
EE
;
2
2
EE
(
2
4
)
By comp
ari
s
on, we g
e
t a
goo
d a
g
ree
m
ent b
e
twe
e
n
g
r
ap
hs in
Figure 2
(a)
and
(b
),
whe
r
ea
s there is a signifi
cant differe
n
c
e betw
een
figure
s
in Figure 2
(
a
)
an
d (c). It can be
con
c
lu
ded th
at AOD di
stri
bution in the
rectan
gula
r
tu
nnel still follo
ws
Gau
s
sian
distrib
u
tion, e
v
en
con
s
id
er the rough
ne
ss a
n
d
tilt of
the walls.
(a) T
heo
retic
a
l distrib
u
tion
(b) G
a
u
ssi
an
distrib
u
tion
(c) Lapl
aci
an
Distri
bution
Figure 2. Rad
i
ation Field Di
stributio
n
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TELKOM
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ISSN:
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046
Spatial Cha
r
a
c
teri
stics of Wirel
e
ss Cha
nnel in Tun
n
e
l
with Im
perfect Wall
s (Yi Z
hang
)
5505
3.2. Effec
t
s
of Lossy
Walls
The ele
c
tri
c
al
para
m
eters
of the materi
al
on the
wal
l
s K1 an
d K2
are d
e
fined
as [14]:
12
11
2
2
00
;
22
Kj
K
j
f
f
, where
12
0
=8
.
8
5
1
0
F/m is
the permittivity in
vacuum
spa
c
e.
1
and
2
are
the relative pe
rmittivity for the si
de
wall
s or
roof a
nd fl
oor i
n
the
tunnel;
1
and
2
are th
eir
con
ductivity. The
y
are influ
e
n
c
ed
by the h
u
midity, pre
s
sure an
d
temperature
of the
su
rro
u
nding
ro
ck. L
ook up
in [1
6
-
19],
1
and
2
are in th
e
ran
g
e
of 2
~
7
0
;
1
and
2
are in
th
e ra
nge
of 1
0
-
6~1S/m. On
sub
s
tituting i
n
to K1 a
nd K
2
, we
can fin
d
that the
effect of con
ductivity coul
d be complet
e
ly negligibl
e
due to the
small valu
e compa
r
ed to t
he
permittivity. In Figu
re 3
we give the
cu
rves of
AS v
e
rsus the rel
a
tive permitti
vity. Suppose
tha
t
12
and
12
0.009
S
/
m
. It is sh
own t
hat the
influen
ce
of the relative
permittivity is
greate
r
when
its value is small, arou
nd
2-10.
Figure 3. Angle Sprea
d
Versu
s
Rel
a
tive Permittivity
3.3. Effec
t
s
of Rou
gh an
d Tilted Wall
s
To analy
z
e th
e effects
of the wall
rou
ghn
ess an
d wall t
ilt on the spat
ial distri
bution
of the
wave
s, we
compa
r
e 3
tu
nnel
con
d
itio
ns in
this pa
rt: smo
o
th walls a
nd
strai
ght tunnel,
rough
wall
s and
stra
ight tunnel, ro
ugh walls a
n
d
one sid
e
wall
is tilted.
The wall ro
u
ghne
ss and
wall tilt coul
d
influen
ce th
e tran
smi
ssi
o
n
po
wer
distribution in
each dire
ctio
n by the tunnel size
and th
e operating freque
ncy [1, 7].
At first, we analyze the v
a
riation
of AS wi
th the tunnel si
ze fo
r the above 3
tunnel
environ
ment
s. We calculat
e AS with different
widt
h in
a tunnel of 4
m height In
Figure 4(a) a
nd
AS with different height in
a tunnel of 4
m
width in Fi
gure 4
(
b
)
.
It is shown that for straig
ht
tunnels, AS increa
se
s wit
h
the si
ze of the tunnel. Thi
s
re
sult
coh
e
re
s
with
that obtain
e
d
by ray meth
od in
[11]
a
n
d
by m
e
a
s
urements in
[8]. Firstly, in
a
ray
picture of a given mode, the gra
z
ing a
n
g
l
es of the ray decrea
s
e
with the size of the tunnel cro
ss
se
ction [1, 7]. This re
sults i
n
a decrea
s
e
of the sp
rea
d
of angle of the ray relati
ve to the direct
path. Se
con
d
l
y, both of th
e lo
ss o
w
ing
to refle
c
tion
a
nd scatterin
g
by
the wall
s and
th
e
n
u
m
ber
of bou
nces
p
e
r u
n
it len
g
th de
crea
se f
o
r the
g
r
azi
n
g an
gles. It
result
s that th
e tran
smi
s
sio
n
power in
crea
se
s. Thirdly, the large th
e tunnel
si
ze, th
e more the n
u
mbe
r
of p
r
o
pagatio
n mod
e
s
[6]. Conse
q
u
ently, increa
se in tunn
el transve
rs
e dimen
s
ion
s
decrea
s
e
s
th
e first factor but
increa
se
s the
last two facto
r
s. The
n
the net effe
ct is a
n
increa
se of AS with the tunnel
size.
While fo
r the
third
conditio
n
that there is long
rang
e tilt of the tunnel
wall
s, we
ca
n se
e i
n
Figure 4(a) that
decrea
s
es with the
width, and in F
i
gure 4
(
b
)
tha
t
and the p
e
r
ce
ntage
incr
ea
se of
decrease wit
h
the
height. The
reason i
s
t
hat, the reflected mo
de
will be rotated
into other m
ode when t
he tunnel e
x
hibits to
a
marked de
gree tilt. Fro
m
(13
)
–(16
), the
corre
s
p
ondin
g
ord
e
r
of the mode i
s
hi
gher fo
r a
gi
ven dire
ction
in the large
r
tunnel. The
n
the
power
of the
disturbed
field co
uple
d
ba
ck i
n
to the
wave is
small
e
r in the la
rge
r
tunnel fo
r a gi
ven
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TELKOM
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KA
Vol. 12, No. 7, July 201
4: 5499 – 55
07
5506
prop
agatio
n distan
ce. Th
is results
in
larg
er
scat
tering l
o
ss i
n
the l
a
rg
er titled tunn
e
l
.
Con
s
e
quently
, the
wall tilt
contribute
s
to t
he d
e
cre
a
se
of AS with
th
e tunn
el
size.
And th
e l
a
rg
er
the tunnel, the more
role
s the wall tilt pla
y
s.
To
sum
up, t
he
roug
hne
ss is
more im
po
rtant in
sm
all
tunnel
s
while
tilt is mo
re i
m
portant
in large tunnels.
(a) Angl
e sp
read versu
s
wi
dth of the tunnel
(b) Angl
e sp
read versu
s
h
e
ight of the tunnel
Figure 4. Angle spread ve
rsu
s
si
ze
of tunnel cro
ss
se
ction. Ca
se (a)
2
0,
0
ro
ugh
t
i
l
t
m
; Cas
e
(b)
2
0.08
,
0
ro
ugh
ti
lt
m
;
Case
c.
2
0.08
,
1
rough
t
ilt
m
Secon
d
ly, we
analyze the
variation of AS
with the operating frequ
ency for the
above 3
tunnel enviro
n
ments.
In Figure 5 we illustrate the e
ffects of operat
ing frequency on
and
. The above 3
tunnel
co
nditi
ons are
con
s
i
dere
d
. It is
sh
own
that fo
r t
he
straig
ht tu
nnel
s, AS varies
dire
ctly wi
th
the freq
uen
cy
, espe
cially
when it is lowe
r than
500M
Hz. T
h
is
re
su
lt cohe
re
s
with that obtain
ed
by ray metho
d
in [11]. But
for the tunnel
with t
ilted wal
l
s, the AS decre
ases
with the frequ
en
cy a
t
high freq
uen
cies.
Incre
a
se in
freque
ncy d
e
crea
se
s the
gra
z
ing an
gles d
e
fined
by each m
ode an
d
increa
se
s the
pro
pag
ation
mode
s n
u
mb
er. The
n
t
he
mech
ani
sm t
hat the effe
ct of frequ
en
cy on
AS is similar t
o
the mech
an
ism that the effect of
tunnel size o
n
AS.
Beside
s, from
(9) an
d (16
)
, it
is al
so
sh
own
that the roug
hne
ss
i
s
m
o
re impo
rtant a
t
low freque
n
c
ie
s while th
e
wall tilt is mo
re
importa
nt at high frequ
en
ci
es.
Figure 5. Angle Sprea
d
Versu
s
Ca
rri
er Freque
ncy. Ca
se (a)
2
0,
0
ro
ugh
ti
lt
m
; Case (b)
2
0.08
,
0
ro
ugh
ti
lt
m
;
Case
c.
2
0.08
,
1
rough
t
ilt
m
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Spatial Cha
r
a
c
teri
stics of Wirel
e
ss Cha
nnel in Tun
n
e
l
with Im
perfect Wall
s (Yi Z
hang
)
5507
4. Conclusio
n
The stati
s
tica
l characte
ri
stics of the ra
diat
ion field
distrib
u
tion o
f
a half-wave
dipole
antenn
a in the recta
ngul
ar
tunnel are discu
s
s
ed by modal ap
pro
a
ch. The analysis sh
ows that:
(1)
The spatial d
i
stributio
n
of the
wave
s
i
n
the re
ctan
g
u
lar tu
nnel
s f
o
llows G
a
u
s
sian
distrib
u
tion re
gardl
ess of whether t
he tu
nnel walls a
r
e
rough a
nd tilted.
(2)
Angle
sprea
d
in tun
nels coul
d be
affected by
the humi
d
i
t
y, pressure
and
temperature
of the tunne
l walls. Th
e
effect
of condu
ctivity could be
neg
ligible, but the
contri
bution o
f
the relative permittivity of the ma
terial
can n
o
t, espe
cially wh
en its value is lo
w.
(3)
The
rou
ghne
ss of the
wal
l
su
rface i
s
most im
porta
nt in
small tu
nnel
s an
d at
lo
w
freque
nci
e
s. I
t
will cau
s
e t
hat AS incre
a
se
s
with the
tunnel
si
ze
and the
op
erating fre
que
n
c
y.
The
wall tilt is mo
st impo
rtant in larg
e
tunnel
s
an
d
at high frequ
enci
e
s. It will
cau
s
e th
at AS
decrea
s
e
s
wit
h
the tunnel size an
d the o
peratin
g freq
uen
cy.
Ackn
o
w
l
e
dg
ements
This p
ape
r is supp
orte
d b
y
the Fund fo
r the Natio
nal
Scien
c
e an
d
Tech
nolo
g
y Suppo
rt
Program (No.
2012BA
H
12B
00) .
Referen
ces
[1]
Guan K, Z
h
on
g Z
D
, et a
l
.
Measur
ement
of dist
ri
buted
anten
na s
y
st
e
m
s at 2.4GHz
in
a re
al
istic
sub
w
a
y
tu
nn
el
envir
onme
n
t.
IEEE Transactions on Ve
hic
u
lar Technology
. 2012; 6
1
(2): 8
34-8
37.
[2]
Ak
y
i
ldiz
IF
, Sun Z
,
Vuran M
C
. Sign
al
prop
ag
ation tec
h
n
i
qu
e
s
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