Geometrical Modeling of Scattering Environment for Highways in Umbrella Cell Based MIMO Communication Systems

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Geometrical Modeling of Scattering
Environment for Highways in Umbrella
Cell Based MIMO Communication Systems
M. Yaqoob Wani, M. Riaz & Noor
M. Khan
Wireless Personal Communications
An International Journal
ISSN 0929-6212
Wireless Pers Commun
DOI 10.1007/s11277-018-5666-9
1 23
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Wireless Pers Commun
https://doi.org/10.1007/s11277-018-5666-9
Geometrical Modeling of Scattering Environment
for Highways in Umbrella Cell Based MIMO
Communication Systems
M. Yaqoob Wani1 • M. Riaz2 • Noor M. Khan3
Ó Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this paper, we develop a three-dimensional (3D) eccentricity-based cylindrical geometrical channel model for nonisotropic multiple-input-multiple-output (MIMO)
communication systems under umbrella macrocellular environment. We use elliptic
cylindrical geometry to model the scattering phenomenon in streets, canyons and highways. The scattering objects like, high-rise building, trees and vegetation that lie along the
roadside premises are modeled by the height of an elliptical cylinder. The proposed
channel model targets fast moving vehicles on the highways in an umbrella-cell of cellular
communication networks. We assume that both ends of the communication link are
equipped with multiple antenna arrays, where, mobile-station antenna height is lower than
base-station antenna. Utilizing the proposed MIMO communication channel model, we
obtain closed-form expressions for the space–time correlation function among the MIMO
antenna elements. The obtained theoretical expressions are plotted and analyzed for different values of channel parameters. Finally, we compare the proposed model with the
existing models in the literature and prove that our model can be deduced to the existing
two-dimensional and 3D channel models.
& M. Yaqoob Wani
[email protected]
M. Riaz
[email protected]
Noor M. Khan
[email protected]
1
Department of Computer Science and Information Technology (CS&IT), The University of Lahore,
Islamabad Campus, Islamabad, Pakistan
2
Department of Electronics and Electrical Systems, The University of Lahore, Islamabad Campus,
6.7 Japan Road, Zone-V, Islamabad 44000, Pakistan
3
Department of Electrical Engineering, Acme Center for Research in Wireless Communications
(ARWiC), Capital University of Science and Technology, Kahuta Road, Zone-V, Islamabad 44000,
Pakistan
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Keywords Umbrella cell Correlation function MIMO capacity Channel
modeling Cylinder
1 Introduction
Wireless communications has shown tremendous growth in the last few decades due to its
potential for facilitating communication links between machines, robots, aircrafts, ships and
automobiles, etc. One of the major fields of wireless communication, which expanded
exponentially in the last two decades, is cellular mobile communications. It is one of the
apogee applications of wireless communication systems, where a stagnant BS with a high rise
antenna mounted on the top of a structure communicates with the MSs having low elevated
antennas located within its vicinity [32]. Because of the prodigious applications of cellular
mobile communication, there has been seen a surge expansion in terms of number of subscribers in the last two decade. This exponential increase in the number of mobile subscribers
demanding discriminate features like multimedia applications, live video streaming, internet
access and other data-hungry value-added applications, forced the system to increase its
spectral efficiency, data rate and link performance. This triggered a gigantic need to increase
the throughput with improved quality of service (QoS) and large cellular coverage. In cellular
mobile communication systems, high capacity with limited frequency spectrum can be
achieved by sub-dividing larger cells in to the smaller ones like microcells, picocells, or
femtocells [33, 43]. In such a communication system, mobile subscribers access the system
using various multiple access techniques like frequency-division multiple access (FDMA),
time-division multiple access (TDMA), code-division multiple access (CDMA) and
orthogonal frequency-division multiple access (OFDMA) [43, 45]. A mobile subscriber when
crosses the boundary of its serving cell and enters into an neighboring cell during progressing
call, hand-off takes place [4, 23, 43]. The smaller size of the cell architecture creates most
crucial dilemma in the cellular mobile communication, i.e, it increases the occurrence rate of
handoff [48]. These frequent handoffs become an imbroglio situation for speedy vehicles on
highways across the microcells [34]. To overcome this frequent problem, fast moving mobile
vehicles are handed over to an umbrella cell as shown in Fig. 1.
In [10, 11, 13, 14, 24, 31, 36, 47], numerous approaches have been presented by the
theorists and researchers to handover the fast-moving vehicles to an umbrella cell. In an
umbrella cell, a dedicated BS with high powered antenna mounted on high rise structure is
used to cover the large area along the highway to serve these high speed mobile users.
Therefore, for the beneficial design of umbrella cell, depth knowledge of the scattering
Fig. 1 Umbrella-based cellular communication system
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channel between the high rise BS and speedy mobile vehicle is extremely important. In this
regard, various approaches like deterministic, stochastic and geometrical-based channel
modeling approaches have been published in the literature to analyze the statistics of
propagation channel [6, 46]. Geometrically-based channel modeling approach is used to
model an umbrella cell for the fast-moving vehicles on highways. In this regard, various
regular shaped geometrical channel models for SISO F2M environments have been provided in [5, 7, 15, 25, 28, 38, 41]. These traditional SISO systems are venerable to
multipath fading effects therefore, they can not meet the QoS and high date-rate
requirements in the rich scattering environments. In contrast to it, theoretically and
empirically MIMO antenna architecture have been proved to be the promising candidate
that can provide huge throughput in F2M and mobile-to-mobile (M2M) communication
environments [20, 22]. However, the high correlation among the antenna array elements is
an important practical factor that can degrade the performance of MIMO systems [8, 20].
Therefore, a geometrical channel model is required that can be used to predict the performance of MIMO F2M communication systems in such propagation environments.
Various 2D [1, 9, 18] and 3D [26, 37] geometrical channels models have been recommended for F2M communication environments utilizing multiple antenna arrays at each
end of the communication link. The authors analyzed the models for various channel
characteristic like angle-of-arrival (AoA), time-of-arrival (ToA), Doppler spread and crosscorrelations among the MIMO links. These channels may be applicable in some scenario of
mobile communication, however, these models are not appropriate to model umbrella cell
scattering environment along highways. An elliptical geometrical shape is more suitable to
model the scattering environment along the highways. To accommodate the high-rise
scattering objects along the highways like buildings, mountains, and trees, are modeled
using the elliptical-based cylindrical shape. The elevated walls of the cylinder represent
these elevated scatterers. In this research article, we suggest an eccentricity-based geometrical channel model that represent an umbrella cell in macrocellular environment. We
assume that signal propagation takes place in azimuth as well as in elevation planes. To
attain higher data rate, multiple antenna array system is installed on each end of the
communication link. Using the proposed model, we derive expressions for joint and
marginal correlation functions among MIMO antenna elements.
The rest of the article is as follows: Sect. 2, describes the methodology to obtain the
mathematical expressions of correlations functions among the MIMO links. Section 3,
system model of the proposed umbrella cell is described. The derivation of system model is
given in Sect. 4. Expressions space–time correlation functions is provided in Sect. 5. The
discussion of the theoretical results are presented in Sect. 6, and lastly, outcome of the
research article is given in Sect. 7.
2 Research Methodology
In this research article, we present and derive the mathematical expressions of the space–
time correlation functions among multiple antenna elements installed on the BS and MS.
The proposed 3D elliptical-based cylindrical channel model portrays an umbrella cell of
the macro-cellular mobile communication environment that accommodates the fast moving
vehicles on highways. It is assumed the MS is surrounded by scattering objects lying in
azimuth and elevation planes in an elliptical-based cylindrical manner. Signals originated
from any antenna array element of the BS propagate towards MS in both horizontal and
vertical planes. The wavelets are reflected, diffracted and scattered from the scattering
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objects located in the vicinity of the MS in haphazard manner resulting in multipath
propagation. The multipath signals with different amplitudes and phases superimposed
constructively or destructively at the receive antenna array elements that result in formation of complex envelope. The statistics of received signal at each antenna element
depends on the distribution of scattering objects, transmit and receive correlations among
array elements [17].
Using the geometry of the proposed system model depicted in Fig. 2, expression of the
diffused components (5) is obtained with the help of some valid assumptions and binomial
approximations. The diffused components are further used for the formulation of the joint
space–time correlation function (12). Finally, the obtained correlation expressions for
transmit, receive and joint correlation functions are simulated and plots are generated in
Matlab for various channel parameters.
3 System Model
This section presents a detailed description of the system model of the proposed fixed-tomobile (F2M) MIMO elliptical-based cylindrical geometrical channel model for umbrellacell as shown in Fig. 2. In this proposed, channel model the mobile subscriber is supposed
At(p)
z
ψt(p)
dnp
z
δt
x
y
At(p)
dnp
dqn
Ar(q)
βr(n)
ψr(q)
Sr(n)
ht
dqn
δr
Ar(q)
hr
At(p)
vr
Ar(p)
ar
γr
br π−θr
(q)
αr(n)
x
π−θt(p)
Ar(p)
D
At(p)
y
Fig. 2 The proposed elliptical-based cylindrical channel model for umbrella cell in macrocellular
communication environment
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to be located at the center of elliptical cylinder holding low elevated antenna array and the
base BS is located on the top of tower of height ht , fixed with multiple antenna array
structure. The eccentricity r models the azimuth dimensions of the elliptical scattering
region around the MS; whereas, the height hr of the elliptical cylinder models its elevation.
The BS and MS are equipped with P and Q number of antenna array elements respectively,
for the sake of simplicity, we take P ¼ Q ¼ 2, however, the results can be derived for any
ðpÞ
configuration. The antenna array elements of BS and MS are symbolized as At and AðqÞ
r ,
and spacings between them are denoted by dt and dr , respectively. The MS is assumed to
be located at (0, 0, 0) and BS (D, 0, 0), at an arbitrary time instant, in the Cartesian
coordinate system, and the azimuth distance between the center of cylinder to base-station
is denoted by D. Moreover it is also assumed that the MS is moving with the speed of vr in
the direction of cr . The description of other variables used in the proposed geometry are
listed in Table 1.
4 Derivation of Channel Parameters
The derivation of the space–time correlation among MIMO antenna elements is based on
~
ðp;pÞ
the reference system model depicted in Fig. 2. The multipath signals are arrived at At
~
qÞ
from Aðq;
via striking at the scattering point SrðnÞ that is situated at the surface of elliptical
r
ðpÞ
cylinder. The propagation distance covered by the transmitted signal from At
to AðpÞ
r and
~
ðqÞ
At
~
pq
p~q~
to AðrqÞ
, are denoted by dmax
and dmax
, respectively. The phase change due to the
pq
p~q~
distance traveled by the signal can be written as ð2p=kÞdmax
and ð2p=kÞdmax
, where 2p=k
is called wavenumber. The gain and phase shift due to the collision of the signals with SðnÞ
r
Table 1 Notations of various channel parameters used in the system model
Symbols
Description
Rr
The dynamic radius of the receiver ellipses
ðpÞ
At , AðqÞ
r
Antenna array elements at BS and MS, respectively
SðnÞ
r
Represents the n-th scatter located on the surface of elliptic cylinder
dt
The spacing between p-th and p~-th antenna elements at Tx
dr
The spacing between q-th and q~-th antenna elements at Rx
ðpÞ
hðqÞ
r ; ht
The azimuth angle of q-th receive and p-th transmit antenna element with respect to x-axis,
respectively
ðpÞ
wðqÞ
r ; wt
The elevation angle of q-th receive and p-th transmit antenna element with respect to x–y plane,
respectively
vr
Velocities of the MS
cr
The moving direction of the MS
ðnÞ
aðnÞ
r ; br
pq
dmax
p~q~
dmax
AoA of multipath in azimuth and elevation plane respectively
ar ; br
Represents the major and minor axes of ellipse respectively
r
Represents the eccentricity of the ellipse surrounding the MS
The distance between p-th antenna element to q-th antenna element, dqn ? dnp
The distance between p~-th antenna element to q~-th antenna element, dqn
~ ? dnp~
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M. Y. Wani et al.
pffiffiffiffi
can be written as 1= N and xn , respectively. Phase change also occur due to the motion of
ðnÞ
MS and can be expressed as 2pfRmax cosðaðnÞ
r cr Þ cos br t, where, fRmax ¼ vr =k represents
the maximum Doppler frequency caused by the motion of receiver mobile node. It is also
assumed that both antenna element spacing dt ; dr are much smaller than br . The AoA in
azimuth plane and elevation plane are assumed to be random variable and independent of
each other [3]. Moreover, xn is also supposed to be a random variable that is uniformly
distributed over ½p; p and is also independent from AoA. Furthermore, for the ease of
derivation of the space–time correlation among MIMO antenna arrays of the proposed
channel model, the following valid assumptions are considered:
1.
2.
The proposed eccentricity-based channel model is assumed to be a single-bounced.
Infinity number of scattering points are uniformly distributed on the surface of the
elliptical cylindrical geometry.
Uniform power is reflected from all scatterers.
The MS is surrounded by stationary scattering points.
Each scatterer behaves as an omni-directional source.
3.
4.
5.
Finally, with the help of above assumptions, the diffused components of the transmission
ðpÞ
link from At
to AðqÞ
r can be expressed as in [2],
N
1 X
2p pq
ej k dmax þjxn
hpq ðtÞ ¼ lim pffiffiffiffi
N!1 N
n¼1
e
ðnÞ
ð1Þ
ðnÞ
j2ptvkr cosðar cr Þ cos br
ðpÞ
pq
¼ dqn þ dnp , is the total distance traveled by the wave-vector from At to AðqÞ
The dmax
r .
The distances dqn and dnp are obtained by solving the geometry of the proposed model as
shown in Fig. 2. The 3D polar coordinates
of qth
and pth transmit
receive
antenna element
ðqÞ
ðqÞ
ðqÞ
antenna element are denoted by dRx ; dRy ; dRz
ðqÞ
ðpÞ
ðpÞ
ðpÞ
and dTx ; dTy ; dTz , respectively. Where,
ðqÞ
ðqÞ
ðpÞ
ðqÞ
ðqÞ
ðqÞ
ðqÞ
dRx ¼ dr cos hðqÞ
r cos wr ; dRy ¼ dr sin hr cos wr ; dRz ¼ dr sin wr ; dTx ¼ dt
ðpÞ
cos hðpÞ
t
ðpÞ
ðpÞ
ðpÞ
ðpÞ
cos wðpÞ
.
t ; dTy ¼ dt sin ht cos wt ; dTz ¼ dt sin wt
ðn or nÞ
~
ðn or nÞ
~
; dRy
;
Similarly, the coordinates of nth scatterer in 3D space is denoted by dRx
~
ðn or nÞ
dRz
~
or nÞ
or nÞ
~
or nÞ
~
Þ, where dRðn or nÞ~ ¼ Rr cos aðn
; dRðn or nÞ~ ¼ Rr sin aðn
; dRðn or nÞ~ ¼ Rr tan bðn
.
t
r
r
x
z
y
These polar coordinates depend upon the orientation and configuration of the antenna
arrays. Moreover, the dynamic radius Rr of the ellipse around the receiver can be written in
terms of minor br and major ar axes is given in [42] as,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
Rr ¼ ar br = a2r sin2 ar þ b2r cos2 ar
Using the distance formula and binomial approximation
the approximated distances can be expressed as,
dqn Rr
cos bðnÞ
r
cos aðnÞ
r
123
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 þ xÞ 1 þ x=2 if ðx 1Þ,
ðnÞ
ðqÞ
ðqÞ
dr sin wðqÞ
r sin br dr cos hr cos wr
cos bðnÞ
r
dr sin hðqÞ
r
cos wðqÞ
r
sin aðnÞ
r
cos bðnÞ
r
ð3Þ
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d2t
h2 R2 sec2 bðnÞ
dt Rr tan bðnÞ
r
r sin wt
þ t þ r
2D 2D
2D
D
ðpÞ
ðpÞ
dt Rr cos aðnÞ
ðpÞ
ðnÞ
r cos ht cos wt
þ dt cos wðpÞ
t cos ht Rr cos ar D
ðpÞ
ðpÞ
ðpÞ
ðnÞ
ðnÞ
ht Rr sin hðpÞ
cos
w
h
R
sin
a
d
R
sin
a
t r
t r
r
r sin ht cos wt
t
t
þ
D
D
D
dnp D þ
ð4Þ
Analogously, the distances dqn
~ and dnp~ can be also obtained using the same geometry.
Moreover, the constant phase shift caused by the time independent terms D; d2t =2D; h2t =2D
can be set to zero without loss of generality. Furthermore, for the ease of understanding and
avoiding complexity, Eq. (1) can be rewritten as,
N
ðnÞ
ðnÞ
1 X
Grt ej2ptfRmax cosðaR cr Þ cos br
hpq ðtÞ lim pffiffiffiffi
N!1 N
n¼1
ð5Þ
2p
where Grt ¼ ej k ðdqn þdnp Þ .
5 Derivation of the Correlation function
The normalized space–time correlation function between diffused channel coefficients
hpq ðtÞ and hp~q~ðtÞ for the proposed 3D MIMO F2M non-isotropic scattering propagation
environment can be obtained using the following relation given in [2, 51],
Efhpq ðtÞhp~q~ðt þ sÞg
Rpq;p~q~ðsÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Efjhpq ðtÞjg2 þ Efjhp~q~ðtÞjg2
ð6Þ
where Efg is the statistical expectation operator and can be applied to all random variables, ðÞ symbolizes the complex conjugate operation and p; p~ 2 {1,...,P}, q; q~ 2
{1,...,Q}. Using Eqs. 5 and 6 the expression for the space–time correlation function can be
written as,
N
ðnÞ
ðnÞ
1X
Grt Grt ej2psfRmax cosðaR cr Þ cos br
N!1 N
n¼1
Rpq;p~q~ðdr ; dt ; sÞ ¼ lim
ð7Þ
In F2M multipath propagation environments, the scattering objects present along the
roadside premises are the sources of the energy efficient multipath. Therefore, the distribution of the AoA of multipath depends on the distribution of scattering objects that are
lying in the close vicinity of MSs. For the ease of derivation of mathematical expression of
space–time correlation function, it is assumed that infinity scattering objects are present
around the MS, that implies the discrete scattering distributions may be transformed to a
continuous scattering distributions that in turn forces us to change the discrete random
ðnÞ
variables aðnÞ
r and br into continuous random variables ar and br , respectively. Furthermore, we assume that azimuth and elevation angles are independent of each other;
therefore, the joint PDF f ðar ; br Þ of AoA at MS can be written in product form as
f ðar Þf ðbr Þ, where, f ðar Þ and f ðbr Þ are the PDF’s of AoA in azimuth plane and elevation
plane respectively. Therefore, the Eq. (7) can be written in integration form as,
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Rpq;p~q~ðdr ; dt ; sÞ Z
Z
bRm
bRm
j2p
k
e
p
e2jpsðfTmax cosðar cr Þ cos br Þ
p
ðq;qÞ
~
dRx cos ar cos br þ
ðp;pÞ
~
Ty
D
Rr d
~
ðp;pÞ
Rr d
Tx
D
cos ar
e
j2p
k
~
ðp;pÞ
sin ar þj2p
k
~
ðq;qÞ
dRz
sin br þ
Rr
dTz
ðq;qÞ
~
dRy sin ar cos br
tan br
D
ð8Þ
!
R2 sec2 b
þ r 2D r
e
f ðar Þf ðbr Þdar dbr
~
ðq;qÞ
where
dRx
~
ðp;pÞ
ðpÞ
ðqÞ
~
ðqÞ
~
ðq;qÞ
ðqÞ
¼ dRx dRx ; dRy
~
ðpÞ
~
ðp;pÞ
~
ðqÞ
~
ðq;qÞ
¼ dRy dRy ; dRz
ðpÞ
ðqÞ
~
ðqÞ
~
ðp;pÞ
¼ dRz dRz ; dTx
ðpÞ
~
ðpÞ
¼ dTx dTx ;
~
ðpÞ
dTy ¼ dTy dTy ; dTz ¼ dTz dTz ; and bRm is the maximum elevation AoA of the
scatterers causes at the MS. To describe isotropic and non-isotropic scattering environments, various distributions have been proposed in the literature for AoA/AoD [2, 29, 44].
In urban or suburban areas the physical layouts of the streets, canyons and highways in the
azimuth plane resemble elliptical shapes. Thus the roadside scatterers can be modeled nonisotropically distributed. Hence, eccentricity-based modified von Mises distribution proposed in [49], is more appropriate for such nonisotropic scattering environments to model
azimuth AoA/AoD. The expression of PDF of azimuth AoA/AoD at MS is,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
pðar Þ ¼ 2pI01ð2 Þ er cos ar , where, r ¼ 1 b2r =a2r , the azimuth angle ar 2 ½p; p and I0 ðÞ
r
is the zeroth-order modified Bessel function of the first kind. Similarly, different scattering
distributions have been also proposed in the literature for elevation AoA [2, 12]. In
addition, Parsons et al. in [39] proposed PDF of elevation AoA f ðbr Þ ¼ 4bp cos
rm
0
p br
2 brm
,
0
where, the absolute values of elevation angles brm lie in the range 0 brm 20 [50].
Z bR Z p
m
1
p
p br
Rpq;p~q~ðdr ; dt ; sÞ ¼
cos
2pI0 ð2r Þ bRm p 4bRm
2 bRm
2
er cos ar e2jpsðfRmax ðcos ar cos cr cos br þsin ar sin cr cos br ÞÞ
ðp;pÞ
~
e
e
e
j2p
k
~
ðq;qÞ
dRx cos ar cos br þ
Rr d
j2p
k
~
ðq;qÞ
dRy sin ar cos br þ
ðq;qÞ
~
dRz
ðp;pÞ
~
Ty
D
cos ar
sin br þ
ð9Þ
Rr d
j2p
k
Tx
D
sin ar
ðp;pÞ
~
Rr d
tan br R2 sec2 bðnÞ
Tz
þ r 2D r
D
dar dbr
The integration in (9) lacks the closed-form solution for the space–time correlation
functions among the MIMO antenna elements. Hence, we integrate this equation numerically by using small angle approximation, i.e., sin br br , cos br 1, sec2 br ¼ 1 þ br =2.
This assumption of small elevation angles is valid for the proposed model because the
distance between the MS and BS is much larger than the antenna heights. By incorporating
these approximations, we get the following expression,
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Rpq;p~q~ðdr ; dt ; sÞ 1
2pI0 ð2r Þ
j2p
k
e
bRm
Z
p
e2jpsðfTmax ðcos ar cos cr þsin ar sin cr ÞÞ
p
bRm
ðq;qÞ
~
dRx
j2p
k
e
Z
~
ðp;pÞ
Rr d
cos ar þ DTx
ðp;pÞ
~
Rr d
br R2 b
ðq;qÞ
~
r r
dRz br þ TDz
þ 4D
cos ar
~
ðq;qÞ
j2p
k
dRy sin ar þ
e
ðp;pÞ
~
Rr d
Ty
D
sin ar
ð10Þ
p
p br
dar dbr
cos
4bRm
2 bRm
2
er cos ar
The above integrations for variables ar and br can be written separately as,
Z p
1
e2jpsðfRmax ðcos ar cos cr þsin ar sin cr ÞÞ
Rpq;p~q~ðdr ; dt ; sÞ 2pI0 ð2r Þ p
~
ðp;pÞ
Rr d
~
ðp;pÞ
Ty
ðq;qÞ
~
j2p
e
j2p
k
ðq;qÞ
~
dRx þ
Rr d
Tx
D
þ2r
cos ar
~
ðp;pÞ
R2
Tz
þ4Dr
D
e
k
dRy þ
D
sin ar
dar
ð11Þ
p
p br
dbr
cos
4bRm
2 bRm
bRm
~
ðp;pÞ
ðp;pÞ
~
Rr dTy
Rr dTx
ðq;qÞ
~
ðq;qÞ
~
j2p
j2p
2
2jpsfRmax cos cr þ r ; x2 ¼ k dRy þ D
By substituting, x1 ¼ k dRx þ D
ðp;pÞ
~
Rr dTz
~
ðq;qÞ
R2r
j2p
2jpsfRmax sin cr ; x3 ¼ k dRz þ D þ 4D in (11), we get the following expression,
Z
j2pbr
k
bRm
e
~
ðq;qÞ
dRz þ
Rr d
Z p
1
Rpq;p~q~ðdr ; dt ; sÞ ex1 cos ar þx2 sin ar dar
2pI0 ð2r Þ p
Z bR
m
p br
x3 br p
dbr
e
cos
4bRm
2 bRm
bRm
ð12Þ
Equation (12) can be written in simplified form by using Bessel function notations given in
[21],
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Io
x21 þ x22 cosð2p b x3 Þ
k Rm
h
i
ð13Þ
Rpq;p~q~ðdr ; dt ; sÞ I0 ð2r Þ
1 ð4bRm x3 Þ2
k
6 Result Description
In this section, we present the theoretical results of the space–time correlation function and
elaborate the impact of various channel parameters on the correlations among MIMO
antenna elements. It is observed that the derived expression (13) is the function of various
model parameters and physical dimensions of the propagation channel. The values of the
parameters used in plotting different curves of space–time correlations are mentioned in
the caption of each plot. A normalized sampling period (step size) fRmax Ts ¼ 0:01, is used
for plotting the curves. Various 2D and 3D plots of the correlations among MIMO antenna
elements are presented for discussion and observation.
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M. Y. Wani et al.
Receive Correlation
1500
1000
500
0
200
150
100
Eleva
tion a
50
ngle
of tra
nsmit
0
-50
anten
na, ψ
T
-100
-150
[in de
gree]
-200
-200
-150
-100
-50
0
50
nna, ψ R
ive ante
of rece
ion angle
100
150
200
e]
[in degre
Elevat
Fig. 3 3D space time correlation function with respect elevation angles of transmit and receive antenna
arrays
The correlations among MIMO antenna elements shown in Figs. 3, 4, 5 and 6 are
plotted against the orientation of antenna arrays in the azimuth and elevation planes. The
plot depicted in Fig. 3, shows clearly that the correlation is maximum when the elevation
angles of any antennas is set at 0
or 180
, and the correlations show a decreasing trend
with the increase in elevation angle of any the antenna from 0
to 90
. The plot is generated
using channel parameters dr ¼ dt ¼ k=2; br ¼ 20
; ar ¼ p=4; wr ¼ 30
; hr ¼ ht ¼ 2p=3; ar
¼ 500 m; br ¼ 200 m. Therefore, for the design of an umbrella cell in cellular mobile
communication, antenna orientation has significant impact on the system performance.
The proposed elliptical cylindrical channel model is generic in nature and can be
reduced to cylindrical model proposed by Feng et al. in [16] by keeping eccentricity r ¼ 0
and distance D = 5 km and other channel parameters are dr ¼ dt ¼ k=2; br ¼ 20
;
ar ¼ p=4; hr ¼ ht ¼ 2p=3; ar ¼ br ¼ 100 m. The joint correlation graph is plotted against
elevation angles of transmit and receive antenna arrays, shown in Fig. 4. Resultant plot is
well matched with the correlation plot given in [16]. This implies that the channel model
given in [16] is the special case of our proposed channel model.
Similarly, 3D plot shown in Fig. 5, of joint space–time correlation function is taken
against azimuth AoA for the channel parameters dr ¼ dt ¼ k=2; br ¼ 20
; wr ¼
30
; hr ¼ ht ¼ 2p=3; ar ¼ br ¼ 100 m, and its 2D slices are plotted in Fig. 6a, b. It is
observed from the plots that the correlation among the MIMO antenna array elements are
dependent on the PDF of AoA (ar ) in azimuth plane of the nonisotropic scattering environments; a similar observation was reported in [30, 40].
The antenna spacing is another most important system parameter that has significant
effect on the correlations among the MIMO antenna elements [19, 27]. The graph shown in
Fig. 7, demonstrates the effect of antenna spacing on antenna correlations. It is seen as the
spacing is increased, the correlation shows a decreasing trend and vice versa. Hence,
keeping the antenna elements at appropriate spacing, maximum throughput can be
achieved from the MIMO systems.
Moreover, the proposed 3D eccentricity based channel model is transformed into 3D
circular based geometrical channel model given in [35], by making eccentricity equal to
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Geometrical Modeling of Scattering Environment for Umbrella Cell...
90
80
Receive Correlation
70
60
50
40
30
20
200
10
100
0
0
-200
-150
-100
-50
0
-100
50
Elevation angle of receive antenna, ψR [in degrees]
100
150
it an
nsm
f tra
le o
g
n
on a
tenn
a, ψ T
es]
egre
[in d
-200
200
ti
a
Elev
Fig. 4 Receive correlation with respect to elevation angles of transmit and receive antenna arrays [16]
x 10
-3
3
Receive Correlation
2.5
2
1.5
1
0.5
0
1
0.8
0.6
τf
0.4
Rmax
0.2
0
0
50
100
150
250
200
300
350
400
Azimuth angle, α [in degrees]
r
Fig. 5 Joint space–time correlation function of azimuth AoA and normalized time delay
zero. It is observed from Fig. 8 that elliptical geometry shows increased correlation as
compared to circular geometry. It thus confirms the statement that elliptical geometry is
more appropriate shape to model scattering environments of streets, canyons or highways
in umbrella cellular environment; therefore, the results of our proposed channel model are
more justified than circular ones to design and analyze MIMO M2M communication links.
These curves are plotted for the following channel parameters dt ¼ k=2; br ¼
20
; hr ¼ p=6; wr ¼ 30
; hr ¼ ht ¼ 2p=3; ar ¼ br ¼ 200 m.
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M. Y. Wani et al.
3
x 10
-3
3
x 10
-3
τ fRmax = 0
α r = 0o
αr = 30
αr = 60
o
αr = 90o
2
1.5
τ fRmax = 0.8
1.5
1
0.5
0.5
0.2
0.4
0.6
0.8
τ fRmax = 0.5
2
1
0
0
τ fRmax = 0.1
2.5
Receive Correlation
Receive Correlation
2.5
o
0
0
1
τ fRmax
50
100
150
200
250
300
350
Azimuth angle [in degrees]
(b)
(a)
Fig. 6 2D space time correlation function with respect azimuth AoA
0.12
δr = 0.3λ
δr = 0.5 λ
0.1
δr = λ
Receive Correlation
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized time delay, τ fRmax
Fig. 7 Effect of antenna spacing on space–time correlation function
7 Conclusion
In this research article, a single bounce 3D eccentricity-based cylindrical geometrical
channel model for MIMO F2M propagation environment has been presented. The proposed
elliptic cylindrical geometry is rotatable along the horizontal plane about vertical axis and
its dimensions are adjustable corresponding to the physical propagation environment.
Based on the proposed model, mathematical expressions of joint and marginal space–time
correlation functions among the transmitter and receiver antenna array elements have been
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Geometrical Modeling of Scattering Environment for Umbrella Cell...
0.12
Elliptical-Based Cylindrical Model
Circular-Based Cylindrical Model
0.1
Receive Correlation
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
0
0.5
1
1.5
τ fRmax
Fig. 8 Comparison of space–time correlation function of proposed elliptical based cylindrical channel
models with the circular based cylindrical channel models [16]
formulated. Various curves are generated using the obtained theoretical expressions for the
joint and marginal correlation functions for different values of channel parameters. The
resultant correlation plots are thoroughly elaborate over the system performance. Moreover, the proposed F2M MIMO channel model is dynamic in nature, by changing the some
parameters of the its geometry, pre-proposed 2D and 3D F2M channel models can be
extracted.
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Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
M. Yaqoob Wani received his B.S. degree in electronics from the
University of Kashmir, Srinager, India, in 1991 and M.S. Telecommunication and Networking from the Iqra University, Islamabad
Pakistan in 2008. He received his Ph.D. degree in electrical engineering from Capital University of Science and Technology (CUST),
Islamabad, Pakistan in 2017. He is with the Acme Center for Research
in Wireless Communications (ARWiC) at CUST. Currently, he is also
working as Associate Professor/Head of Department Software Engineering with the University of Lahore, Islamabad, Pakistan. His
research interests include channel modeling and characterization,
cellular mobile communication networks and vehicle-to-vehicle
communications.
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M. Riaz was born in Pakistan in 1977. He received his M.Sc. degree in
electronics from Quaid-i-Azam University, Islamabad, Pakistan, in
2002. He did his M.S and Ph.D. degree in electronic engineering from
Mohammad Ali Jinnah University (MAJU), Islamabad, Pakistan in
2009 and 2015, respectively. At present, he is with the University of
Lahore, Islamabad Campus as Assistant Professor/Head of Department
of Electronics and Electrical Systems. He is also with the Acme Center
for Research in Wireless Communications (ARWiC) at Capital
University of Science and Technology, Islamabad. His research
interests include modeling and characterization of cellular and mobileto-mobile communication channels, localization of wireless sensor
network, MIMO communication systems, channel equalization and
estimation, and vehicle-to-vehicle communications.
Noor M. Khan was born in Pakistan in 1973. He received the B.Sc.
degree in electrical engineering from the University of Engineering
and Technology, Lahore, Pakistan, in 1998 and the Ph.D. degree from
the University of New South Wales, Sydney, Australia, in 2007. From
2002 to 2007, he was a casual academic with the University of New
South Wales. He is currently Professor with the Capital University of
Science and Technology (CUST), Islamabad, Pakistan. His research
interests include smart antenna systems, adaptive multiuser detection,
mobile-to-mobile communications, wireless sensor networks, channel
characterization and estimation, and physical channel modeling for
mobile communications.
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