Modeling and Characterization of MIMO
Mobile-to-Mobile Communication
Channels Using Elliptical Scattering
Geometry
M. Yaqoob Wani, M. Riaz & Noor
M. Khan
Wireless Personal Communications
An International Journal
ISSN 0929-6212
Volume 91
Number 2
Wireless Pers Commun (2016)
91:509-524
DOI 10.1007/s11277-016-3473-8
1 23
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Wireless Pers Commun (2016) 91:509–524
DOI 10.1007/s11277-016-3473-8
Modeling and Characterization of MIMO Mobile-toMobile Communication Channels Using Elliptical
Scattering Geometry
M. Yaqoob Wani1 • M. Riaz1 • Noor M. Khan1
Published online: 4 July 2016
Springer Science+Business Media New York 2016
Abstract In this paper, we develop a two-dimensional (2-D) elliptical geometrical scattering model for multiple-input multiple-output (MIMO) mobile-to-mobile (M2M) communication channels. The elliptical geometry is supposed to be an appropriate approach to
model streets and canyons in M2M communication environment. We assume that both
mobile stations (MSs) are located at the centers of ellipses and are surrounded by uniformly distributed scatterers present on the elliptical loci. The equal spacing between two
consecutive scatterers on the elliptical loci forces the angle-of-arrival (AoA) or angle-ofdeparture (AoD) distribution at either of the MSs to be non-isotropic. We provide an
empirical model for such a non-isotropic AoA and compare its results with the numerical
curves of the elliptical geometry which results in excellent agreement. Utilizing the nonisotropic AoA and the proposed geometrical model, we derive closed-form expressions for
the marginal and joint correlation function of the channel coefficients. We provide various
plots to analyze the correlation among the diffused components of M2M MIMO communication link. Furthermore, a comparison of the correlation curves obtained from the
mathematical expression of the proposed model is carried out with the existing results in
the literature. In order to validate the proposed model, the elliptical geometrical shape is
transformed into circular one. The resulting comparative analysis verifies that the circular
geometrical models are the special cases of our proposed model.
Keywords Correlation function MIMO Channel modeling Elliptical Scattering
& M. Yaqoob Wani
yaqoobwani@gmail.com
M. Riaz
riaz@arwic.com
Noor M. Khan
noor@ieee.org
1
Acme Center for Research in Wireless Communications (ARWiC), Department of Electrical
Engineering, Capital University of Science and Technology, Kahuta Road, Zone-V,
44000 Islamabad, Pakistan
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1 Introduction
Continuous demands for high data rates in all aspect of wireless communication systems
are becoming a constant driving force in today’s research arena. Situations become even
more demanding for mobile-to-mobile (M2M) communication networks where both ends
of the links are surrounded by huge number scattering objects. Such a rich scattering
environment can offer feasible situation for the higher data rate demands using multipleinput and multiple-output (MIMO). A MIMO system composed of multiple antennas at
its both ends can benefit from this rich scattering environment providing spatial multiplexing that can guarantee maximum possible data rates [11, 12, 25]. This is only
possible if we can have good understanding of spatial characteristics of radio fading
channel and can be able to observe correlations among various elements of the multiple
antennas.
Communication channels are modeled in the literature exploiting deterministic,
stochastic and geometry-based stochastic channel modelings [4, 16]. Deterministic channel
modeling approach is employed when locations of the transmitter, receiver and scatterers
are known. In M2M communication environment both the communicating nodes are
mobile and their locations are not fixed, therefore deterministic channel modeling approach
is not appropriate [24]. The stochastic channel modeling approach is based on probability
distribution functions (pdfs) which give insight about the statistical behavior of the channel
and is an empirical approach to analyze the parameters of the channel in a stochastic
manner [7]. Since the vibrant behavior of the M2M wireless channel makes the propagation channel non-stationary, stochastic modeling approach is incapable of modeling the
impact of physical scattering phenomenon in such a dynamic propagation environment.
However, this issue can be resolved through assuming scattering environment quasi-stationary and using geometry-based characteristics of the channel. Exploiting the geometrical channel models, mathematical expressions for various pdfs and statistical parameters
of M2M wireless channel may be derived. Understanding the importance of the geometrical stochastic channel modeling, various geometrical channel models have been proposed
in the literature for M2M communication environment.
Akki and Haber were the first researchers who proposed a fundamental channel model
for M2M communication channels [3]. Using their model, they derived expressions for the
pdfs of the time-correlations in channel coefficients. Using these correlations, the authors
obtained the power spectral density. Patzöld et al. [19] extended the work in [3] and
presented a frequency non-selective circular two-ring model for MIMO M2M communication environment. Assuming infinite scatterers around the mobile stations (MSs), the
authors derived expressions for transmit and receive correlation functions and provided a
framework for the channel capacity. In [26], under the assumption of wide sense stationary
(WSS) scattering, closed-form expressions for the time-autocorrelation function and
Doppler spectrum are provided for M2M communication environment. In [18], a general
analytical solution was provided for the three-dimensional (3D) space–time cross-correlation function (CCF). It was proved that the 3D CCF is the product of two dimensional
(2D) space–time correlation functions of M2M communication channel. Furthermore, the
model was simulated for isotropic and non-isotropic environments. Another concept that
scatterers are not located exactly at the boundary of the circular rings but at varying
distances from the center of the rings, was presented in the form of two-erose-ring model in
[27]. The authors also derived formulas for the complex envelope of the diffused antenna
components. However, the empirical results were presented on the basis of simulation for
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the space–time correlation function assuming an isotropic scattering environment. The
work in [27] was extended by Riaz et al. [23] for the derivation of mathematical
expressions for the time-autocorrelation function and Doppler Spectrum. In [20, 21],
geometrical channel models have been proposed by employing circular disc and circular
strip shapes for M2M scattering environment. The authors derived closed-form expressions
for the pdfs of angle-of-arrival (AoA) and time-of-arrival (ToA) assuming uniform scatterers distribution around MSs. In the previous discussion, two-ring models and two-disc
models for single-input single-output (SISO) and MIMO links in M2M communication
environment are presented. Such channel models can become good foundations for future
research but may not be realistic. The realistic approach to model the streets and canyons
for M2M communication scenario is, however the elliptical shape based channel modeling
proposed in [5, 6].
In [6], an elliptical scattering model for SISO M2M communication environment is
proposed. The authors assumed that scatterers are uniformly distributed within the elliptical regions around the MSs. Using their model, the authors derived expressions for the
pdf of AoA and ToA and analyzed angular spread and delay variations. The same statistical parameters are also analyzed by proposing a more realistic and flexible channel model
in [5], where the authors introduced adjustable scatterer-free regions in the previously
proposed elliptical channel model. These elliptical channel models are better than circular
models. However, these models can be made more beneficial if multiple antenna arrays are
introduced in these models. Furthermore, in the two-ring circular channel models for M2M
communication environment, isotopic scattering environment is considered. However, in
[18], non-isotropic scattering environment is taken into consideration without using any
geometry.
Street canyons in urban areas, highways with road-side vegetation in suburban and rural
areas and deep-cut railway tracks in hilly areas are the demanding candidates for nonisotropic physical channel modeling in MIMO M2M communication links. The layout of
such propagation environments resembles more closely with elliptical shape than circular.
Therefore, such environment can be modeled more realistically with elliptical geometrical
shape. Moreover, elliptical geometrical shapes have been strongly recommended in the
literature [11] to be used in environment-specific vehicular channel modeling. Therefore,
the majority multipath components are expected from roadside scatterers, which benefits a
MIMO system [11, 12, 25]. Utilizing the multiple antenna array at both sides of the mobile
communication link increases system capacity and QoS dramatically [11, 14]. This
motivates us to propose a channel model that addresses all the above-mentioned benefits in
this paper.
We propose two-elliptical eccentricity-based channel model for MIMO M2M communication environment and derive mathematical expressions for the pdf of AOA/AOD
which are further verified through simulation results. Moreover marginal correlation
functions at transmitter and receiver antennas for M2M communication channel are
derived which are then used to develop joint transmit-receive correlation functions.
The rest of the paper is organized as follows: Sect. 2 describes the system model of the
proposed geometrical channel model and pdf of AoA is presented in Sect. 3. Expressions
for transmit and receive correlation functions are provided in Sect. 4. Discussion of the
theoretical and simulation results are presented in Sect. 5. Finally, conclusion of the paper
is given in Sect. 6.
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q
p
St
1
At
bt
at
t
dtp
dpq
Sr
Vr
dqr
Vt
φt
βt
αt
1
Ar
αr
φr
δr
βr
br
2
2
Ar
At
ar
d
Fig. 1 Proposed elliptical channel model for MIMO mobile-to-mobile communication environment
2 System Model
In this section, we present the system architecture of the proposed geometrically-based
two-elliptical channel model for MIMO M2M communication environment as shown in
Fig. 1. Each of the transmitting and receiving MSs is denoted by Mm , where the subscript
m ¼ t or r to denote Mt and Mr , respectively. The MSs are located at the centers of the
ellipses having major axes at and ar and minor axes bt and br with eccentricities et and er,
respectively. The MSs are separated by a distance d such that d ðat þ ar Þ and are
surrounded by uniformly distributed scatterers present on the boundaries of the ellipses.
The ellipses are made rotatable with the directions of motion of the MSs such that their
major axes at and ar make angles at (or ar), respectively, with the x-axis.
Both MSs, being at the same hight are equipped with multiple antennas with configuration Pt Qr , where Pt and Qr are the number of antennas mounted on Mt and Mr ,
respectively. For simplicity, we take Pt = Qr = 2. However, the results can be derived for
ðpÞ
any configuration. The transmit and receive antenna elements are denoted by At and AðqÞ
r
and the separations between antenna elements on Mt and Mr are denoted by dt and dr ,
ðpÞ
respectively. It is assumed that dt bt and dr br . The symbols St and SðqÞ
r in Fig. 1
denote the pth and qth scattering objects around Mt and Mr , respectively.
Moreover, the Mt and Mr are moving with velocities vt and vr making angles at and ar
with the x-axis. Angle-of-departure (AoD) and AoA of the pth and qth multipath signals
are designated as /tðpÞ and /ðqÞ
r , respectively. The tilt angles of the transmit and receive
antennas are denoted by bt and br. The proposed two-elliptical model is based on the
following general assumptions,
1.
2.
3.
4.
5.
Double bounce scattering model is considered between communicating MSs.
Infinite number of scatterers are uniformly distributed on the boundaries of ellipses.
Power is equally reflected from all the scatterers.
Each scatterer behaves as an isotropic antenna.
The scatterers are fixed and MSs are quasi-stationary for a short period of time.
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3 Distribution of AoA/AoD
For the ease of derivation, in most of the geometrical channel modeling approaches,
uniform distribution of scattering objects is assumed thereby relating the scatterer distribution with the AoA. A relationship between arc-lengths and the arriving angles of the
multipath signals at the center of the proposed geometry has been presented in the literature for circular scattering models [10]. However, such relationship between the two
parameters does not exist for the elliptical geometry. In this section, we find a connection
between the arc-lengths with the distribution of AoA of the multipath signals numerically
as well as theoretically for the proposed elliptical geometry where scatterers are located at
its boundary. Researchers usually use different scattering distributions like Gaussian,
Laplacian, Uniform and von Mises distributions for the pdf of AoA/AoD in their proposed
geometric channel models. The physical scattering environment around the MSs can either
be isotropic or non-isotropic like in circular or elliptical models, respectively. The von
Mises probability density function (pdf) is considered matching well in fitting the measured results of azimuth dispersion in mobile radio channels [22]. Abidi et al. in [1]
introduced von Mises distribution for non-isotropic scattering environments which is given
as,
pðaÞ ¼
1
exp½j cosða lÞ;
2pIo ðjÞ
ð1Þ
where Io ð:Þ is the modified Bessel function of first kind of order zero, l 2 ðp; pÞ represents the mean angle and j controls the spread of scatters around the mean l. Above
mathematical expression of the von Mises provided convenience to express closed-form
solutions for correlations functions and other coefficients of the wireless channel. Taking
this advantage, authors in [17, 30] derived closed-form expressions of the correlation
function among different MIMO channel coefficients for non-isotropic scenarios. In order
to obtain the distribution of the AoA of the multipath signals in the proposed scattering
model depicted in Fig. 1. One need to concentrate on the non-isotropicity present in the
proposed system model. This requires establishing a relationship of any of the abovementioned probability distribution models with dimension (shape and size) of the physical
scattering environment that creates non-isotropicity.
For the calculation of AoA distribution, we consider an arbitrary elliptical region
around the Mr shown in Fig. 2. We create some N point-objects as scattering points on
the boundary of the ellipse around either of the MSs of the communication link. The
scattering objects are located at points x_1, y_1, (x_2, y_2, (x_3, y_3,..., (xN, yN) in the
Cartesian coordinates system. Equal spacing between these points is calculated by
dividing the perimeter of the ellipse by N number of scattering points as shown in Fig. 2.
Knowing the arc-length and using the coordinates of the scattering points, straight lines
are drawn from the center of the ellipse to the coordinates of the scattering objects.
Using cosine laws, angle between two consecutive lines (e.g., r1 and r2 ) is measured.
Using the same procedure, angle between rest of the pairs of these lines are measured.
The resultant angles are then used to draw histogram for different values of the
eccentricities er of the ellipse around Mr , as shown in Fig. 3. The figure shows that the
pdf of AoA of multipath signals changes to uniform as the geometry of the scattering
objects changes from an elliptical to a circular shape. The following empirical formula
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15
10
r3
r
5
2
r1
0
r
r
-5
P
P-1
-10
-15
0
5
10
15
20
25
30
35
40
45
50
Fig. 2 Equal distribution of scattering points on the boundary of the ellipse for a specific dimension
Probability Distribution Function
5
x 10
-3
ε=0.92
ε=0.80
ε=0.60
ε=0
4.5
4
3.5
3
2.5
2
1.5
50
100
150
200
250
300
350
Angle-of-Arival [Degrees]
Fig. 3 Rate of occurrence of the AoA using elliptical scattering loci of various eccentricities around Mt and
Mr
for the pdf of AoA is also fitted which is validated by the calculated result as shown in
Fig. 4.
pð/m Þ ¼
1
2
em cos 2/m
2pIo ð2m Þ
where Io ðÞ is the modified bessel function of zeroth order and m ¼
eccentricity of the elliptical loci around Mt or Mr .
123
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 b2m =a2m is the
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Probability Distribution Function
14
x 10
515
-3
Simulation
Theoretical
12
10
8
6
4
2
0
50
100
150
200
250
300
350
Angle-of-Arival [Degrees]
Fig. 4 Comparison of theoretical results of the proposed empirical expression for the pdf of AoA with the
numerical results from the geometry of the system model
4 Derivation of the Correlation Function
In this section, we derive an expression for CCF using the proposed elliptical channel
model for MIMO M2M communication environment. Let the MSs are located at certain
points (i.e., Xo, Yo) in the Cartesian coordinates systems. The generalized equation for each
ellipse, having semi-major am and semi-minor bm axes, can be expressed as,
2
ðxm Xo Þ cos /m þ ðym Yo Þ sin /m
a2m
ð3Þ
2
ðxm Xo Þ sin /m þ ðym Yo Þ cos /m
þ
¼1
b2m
The dynamic radius, r, of the ellipse varies from its minimum to maximum according to
the length of minor and major axes of the ellipse, respectively. An expression for this
dynamic radius is given by,
am bm
r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
am sin /m þ b2m cos2 /m
ð4Þ
The dynamic radius of the ellipse given in (4) is plotted against AoA/AoD shown in Fig. 5.
This dynamic radius can be linked with the distribution of the scattering points lying on the
boundary of the ellipse. From these results, it can be concluded that when scatterers are
uniformly distributed on the boundary of the ellipse, of any arbitrary eccentricity, the AoA
emerges as shown in Fig. 3.
In the proposed geometry depicted in Fig. 1, it can be observed that a signal that is
ðpÞ
ðpÞ
transmitted from the transmit antenna element At , first strikes at the scatterer St of the
scattering ellipse surrounding the transmitter and then travels to the scatterer SrðqÞ of the
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50
Radius of ellipse [meters]
45
40
35
30
25
20
15
10
0
50
100
150
200
250
300
350
φ [in degrees]
Fig. 5 Radius of ellipse w.r.t. AoA
scattering ellipse surrounding the receiver, and finally reaches the receiving antenna element AðqÞ
r . Mathematically, this propagation length of the signal can be expressed as,
D ¼ dtp þ dpq þ dqr
ð5Þ
ðpÞ
where dtp = distance from transmitter Mt to the scattering point St , dpq = distance from
ðpÞ
St
scattering point
to the scattering point SrðqÞ , dqr = distance from scattering point SðqÞ
r to
the receiver Mr :
As each of the MS is moving with velocity vt (or vr ) causing maximum Doppler
ftmax ¼ vt =k (or frmax ¼ vr =k), exploiting the proposed geometrical model, an expression
ð1Þ
for the channel of the communication link from At
to Að1Þ
r can be described as in [18],
P;Q
ðpÞ
ðqÞ
1 X
fpq ej½2pðfT þfr Þtþhr þhpq þho h11 ðtÞ ¼ lim pffiffiffiffiffiffiffi
P;Q!1 PQ
p;q¼1
fpq ¼ xp yq zpq ,
where
ðpÞ
j2p
k ðrt cos /t rr
ðqÞ
cos /r Þ
ðpÞ
ft
ðpÞ
xp ¼ ejpðdt =kÞ cos ð/t
bt Þ
cosð/ðpÞ
t
,
frðqÞ
ð6Þ
ðpÞ
yq ¼ ejpðdr =kÞ cos ð/r
br Þ
,
zpq ¼
cosð/ðqÞ
r
e
,
¼ ftmax
at Þ,
¼ frmax
ar Þ, ho ¼ 2p
k D,
ðr
þ
r
Þ,
where
h
is
the
constant
phase
that
depends
on
specific
orientations
of
hr ¼ 2p
t
r
r
k
the ellipses. Similarly, ho is also a constant phase as interspacing between the MSs is fixed.
Due to the constant behavior of these phases, there will be no effect on the statistics of the
proposed model, therefore these phases can be neglected. Using the proposed geometrical
channel model, one can find the other diffused components
i:e:; h12 ðtÞ;
ð1Þ
h21 ðtÞ and h22 ðtÞ . Space–time correlation function between the transmission links At ð2Þ
ð2Þ
Að1Þ
r and At Ar can be expressed as in [18],
q11;22 ðdT ; dr ; sÞ ¼ Efh11 ðtÞh22 ðt þ sÞg
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where the operator Efg is known as the expectation and is applied on all the random
ðqÞ
phases, AoA and AoD ði:e:; /ðpÞ
t ; /r and hpq Þ in the equation. By substituting the values
of the major axis xp and minor axis yq of the ellipse and their conjugate transpose in (6)
then we get the following expression for the above CCF as,
(
Q
P X
1 X
E xp xpyq yqzpq zpq
q11;22 ðdt ; dr ; sÞ ¼ lim
P;Q!1 PQ
p¼1 q¼1
)
hn oi
ðpÞ
ðpÞ
ðpÞ
ðqÞ
ðqÞ
ðqÞ
exp j 2pðft þ fr ft fr Þt þ hpq hpq ðft þ fr Þs
ð8Þ
ðpÞ
ðpÞ
Using boundary condition i.e., ft þ frðqÞ ¼ ft
in reduced form can be written as,
q11;22 ðdt ; dr ; sÞ ¼
þ frðqÞ
iff p ¼ p and q ¼ q, it follows that (8)
Q
P X
h
i
1 X
ðpÞ
x2p y2q exp j2pðft þ frðqÞ Þs
PQ p¼1 q¼1
ð9Þ
ðpÞ
In the above equation, xp and ft are the functions of AoD, /tðpÞ , while yq and frðqÞ are the
functions of AoA, /ðqÞ
r . It has already been assumed that there are infinite number of
scatterers that reside around the MSs that is P; Q ! 1. In such a case, the discrete random
variables /tðpÞ and /rðqÞ become continuous random variables and take the form /t and /r .
These continuous random variables are represented by certain statistical distributions,
denoted by pð/t Þ and pð/r Þ, respectively.
Power received at the receiver through each diffused component corresponding to
differential angles d/t and d/r is proportional to pð/t Þpð/r Þd/td /r . As the AoA, /r , and
AoD, /t , are statistically independent random variables, therefore, the joint transmitreceive space–time correlation function (CCF) in (9) can be decomposed and that can be
written in the form of the product of transmit and receive correlation functions as,
q11;22 ðdt ; dr ; sÞ ¼ qt ðdt ; sÞqr ðdr ; sÞ
ð10Þ
where the transmit correlation function qt ðdt ; sÞ can be expressed as,
qt ðdt ; sÞ ¼
Zp
x2p ðdt ; /t Þ exp½j2pft ð/t Þspð/t Þd/t
ð11Þ
p
Putting the values of xp ðdt ; /t Þ and ft ð/t Þ from (6) and pð/t Þ from (2) in (11), we get,
1
qt ðdt ; sÞ ¼
2pIo ð2t Þ
Zp
exp½j2pðdt =kÞ cos ð/t bt Þ exp½j2p
p
ð12Þ
2
ftmax cosð/ðpÞ
t at Þs exp½t cos 2/t d/t
Expanding the trigonometric functions in the above equation and rearranging the terms, we
have,
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Zp
o
h n
i
1
2
cosð/
qt ðdt ; sÞ ¼
exp
2p
jðd
=kÞ
cosðb
Þ
jðV
=kÞs
cosða
Þ
þ
Þ
t
t
t
t
t
t
2pIo ð2t Þ
ð13Þ
p
o
i
h n
exp 2p jðdt =kÞ sinðbt Þ jðVt =kÞs sinðat Þ d/t
Substituting, x1 ¼ 2p jðdt =kÞ cos bt jðVt =kÞs cos at þ 2t and x2 ¼ j2pfðdt =kÞ sin bt ðVt =kÞs sin aT g, (13) takes the form,
1
qt ðdt ; sÞ ¼
2pIo ð2t Þ
Zp
exp½x1 cos /t þ x2 sin /t d/t
ð14Þ
p
The above equation is integrated by comparing it with eq. (3.338-4) in [13], we get,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
2
J
qt ðdt ; sÞ ¼
ð15Þ
o 2p x1 þ x2
2pIo ð2t Þ
where Jo ðÞ is the bessel function of zeroth order. Putting back the values of x1 and x2 , (15)
can be expressed in simplified form as,
qt ðdt ; sÞ ¼
1
Jo 2p 4t ðdt =kÞ2 ðsftmax Þ2 22t sftmax cos at
2
Io ðt Þ
!
1=2
dt
2 dt
þ j2t cos bt þ 2s ftmax cos ðat bt Þ
k
k
ð16Þ
Similar equation can be derived for the correlation function at the receiver end. Thus joint
correlation function in (10) can be expressed as,
q11;22 ðdt ; dr ; sÞ ¼
1
Jo
Io ð2t ÞIo ð2r Þ
2p 4t ðdt =kÞ2 ðsftmax Þ2 22t sftmax cos at
1=2
dt
dt
þ j22t cosbt þ 2s ftmax cosðat bt Þ
k
k
2
ðsfrmax Þ
!
Jo 2p 4r ðdr =kÞ2
1=2
dr
dr
22r sfrmax cosar þ j22r cosbr þ 2s frmax cosðar br Þ
k
!
k
ð17Þ
which is a generalized expression for the joint correlation function of MIMO M2M
communication links for more realistic channel models which targets environments like
streets and canyons.
The following remarks can be made about (17):
• If elliptical geometrical shapes are replaced with circular ones, the eccentricities of the
ellipses around transmitter and receiver i.e., t and r , would approach to zero. In such a
case, the scattering environment will become isotropic and the expression for the joint
correlation function in (17) reduces to,
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q11;22 ðdt ; dr ; sÞ ¼ Jo
519
1=2
dt
2p ðdt =kÞ2 ðsftmax Þ2 2s ftmax cos ðat bt Þ
k
Jo
!
1=2
dr
2p ðdr =kÞ2 ðsfrmax Þ2 2s frmax cos ðar br Þ
k
!
ð18Þ
which is the result presented in [8, 19, 28, 29]
• For a SISO case, substituting the spacing between antenna element equal to zero, we
get,
ð19Þ
q11;22 ð0; 0; sÞ ¼ Jo 2pftmax s Jo 2pfrmax s
which is the well known result presented in [2, 3] for SISO M2M communication
environment.
• If the transmitting Mt is supposed to be fixed; the maximum Doppler would occur only
due to the motion of the receiving Mr ; then (19) reduces to,
ð20Þ
qðsÞ ¼ Jo 2pfrmax s
which is the well known CCF of the Jake’s model [9, 15].
5 Results and Description
In this section, description of the obtained theoretical results for the derived correlation
function is presented. From (17), it is clear the correlation function depends on various
parameters like eccentricities of the ellipses, separation between antenna elements,
velocities of the MSs and carrier frequency of the arriving signals. It can be observed that
Transmit Correlation Function
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0
0.1
δt/ λ
0.2
0.3
0.4
0
0.1
0.2
0.3
τ f tmax
0.4
0.5
0.6
0.7
0.8
Fig. 6 Correlation function of the non-isotropically arriving signals using MIMO two-elliptical channel
model (at = 50 m, bt = 20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s)
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3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
σ /λ
2.5
3
0
1
0.5
2
1.5
2.5
3
3.5
4
τ.ftmax
Fig. 7 Correlation function of the non-isotropically arriving signals using MIMO two-ring channel model
(at = 50 m, bt = 20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s)
0.25
ε = 0.92
ε = 0.80
ε = 0.60
ε=0
Cross-Correlations Function
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0
1
2
3
4
5
6
Time [Sec]
Fig. 8 Comparison of correlation function on the basis of different values of eccentricity (at = 50 m, bt =
20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s)
the correlation between antenna elements is maximum at s ¼ 0 and decays with an
increasing value of s. It implies that the MIMO capacity will be minimum at the beginning
as the correlation is maximum there and will thus show an increasing trend with decreasing
values of correlation.
Since in the proposed elliptical model, the scattering objects are assumed to be uniformly spaced on the elliptical loci, the AoA distribution at the receiver, Mr would thus be
non-isotropic as shown in Fig. 3. For such non-isotropic environment where a non-isotropic AoA distribution emerges from the varying distances of the scattering points from
the Mr , expression for the correlation function is derived as given in (17). Various correlation plots with 3-D and 2-D views are generated for the parameters specified in the
captions of the figures. The result presented in Fig. 6 shows excellent agreement with the
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0.25
λ/2
λ/4
λ/8
Transmit Correlations
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0
1
2
3
4
5
6
7
8
9
10
Time [sec]
Fig. 9 Effect of spacing between antenna elements on correlation function, (at = 50 m, bt = 20 m, d =
100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s)
1.1
Transmit Correlation
1
0.9
0.8
0.7
0.6
0.5
vt =100 km/hr
vt =200 km/hr
0.4
vt =300 km/hr
0
10
20
30
40
50
60
70
80
90
100
Temporal Separation, τ [seconds]
Fig. 10 Effect of changing velocity on correlations
simulation results given in [18]. This is basically the elliptical geometrical shape that
forces the correlation functions, qt ðdt ; sÞ, to be having this typical trend. However, the
causes of such trend have not be documented in [18]. If we fix the value of the major axis
of an ellipse and increase its minor from its minimum value to a value equal to its major
axis making eccentricity equal to zero, then the ellipse will be transformed into a circle. In
this case, the correlation function, qt ðdt ; sÞ, shows an isotropic behavior of the arriving
signals as shown in Fig. 7. This result also perfectly matches the theoretical results of the
correlation curves shown in [18].
From the above figures, it is verified that the circular geometrical model is the special
case of our proposed elliptical model. The 2-D curves of the correlation function for
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different values of eccentricities are shown in Fig. 8. It can be seen that the correlation
increases with an increase in eccentricity of the ellipse and becomes maximally uncorrelated when the ellipse is transformed into a circle.
The effect of spacing between antenna elements, dt , on the correlation function is shown
in Fig. 9. The plots are taken for different values of antenna spacing i.e., dt ¼ k=D where,
D ¼ 2; 4; 8. The plots show that the correlation is lesser for larger values of antenna
spacing and vice versa.
It can be seen from (16) and (17) that correlation between antenna elements depends
upon velocity of transmitter and receiver. The effect of motion of transmitter is shown in
Fig. 10 for different velocities i.e., vt = 100, 200, and 300 km/h. From these curves, we
analyze that correlation values are higher for smaller values of velocities. In other words,
we say that correlation decreases with the increase in velocity of the MSs.
6 Conclusion
In this paper, we have developed a 2-D elliptical geometrical scattering model for MIMO
M2M communication channels. The elliptical geometrical shape is supposed to be a more
realistic approach than circular geometry to model the streets and canyons. We have
assumed that both MSs are located at the centers of ellipses and surrounded by uniformly
distributed scatterers present on the elliptical boundaries. Due to being an elliptical
boundaries, the distances of the scattering objects from the MSs are not equal which forces
the AoA distribution at either of the MSs to be non-isotropic. We have provided an
empirical formula for AoA distribution emerged as the result of such non-isotropic arriving
signals. We have compared its results with the numerical curves of the elliptical geometry
that resulted into an acceptable agreement. Utilizing the non-isotropic AoA and the proposed geometrical model, we have derived a closed-form expression for the joint transmitreceive correlation function. Since the AoA and AoD are statistically independent,
therefore, the joint expression for the correlation function can be expressed as a product of
transmit and receive correlation functions. Various plots have been provided to analyze the
correlation functions among the diffused components of the M2M MIMO communication
link. It can be seen from the curves that the correlation depends on spacing between
antenna elements, eccentricity of the ellipses, velocities of the MSs and the frequency of
the arriving signals. Furthermore, the correlation curves obtained from the mathematical
expression of the proposed model have been compared with the existing results in the
literature. In order to validate the proposed model, the elliptical geometrical shape has been
transformed into a circular one. The resulting comparative analysis verified that the circular geometrical models in [8, 19, 28, 29] are the special cases of our proposed model.
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Mohd 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. At present, he pursues his Ph.D. degree in electrical
engineering at Capital University of Science and Technology (CUST)
Islamabad, Pakistan. He is currently with the Acme Center for
Research in Wireless Communications (ARWiC) at CUST. Currently,
he is also working as Assistant Professor with the University of
Lahore, Islamabad, Pakistan. His research interests include channel
modeling and characterization, cellular mobile communication
networks.
Muhammad Riaz was born in Pakistan in 1977. He received his M.Sc.
degree in electronics from Quaid-i-Azam University, Islamabad,
Pakistan, in 2002 and M.S. degree in electronic engineering from
Mohammad Ali Jinnah University (MAJU), Islamabad, Pakistan in
2009. At present, he pursues his Ph.D. degree in electronic engineering
at MAJU, Islamabad, Pakistan. He is currently with the Acme Center
for Research in Wireless Communications (ARWiC) at MAJU. His
research interests include modeling and characterization of mobile-tomobile communication channels and channel equalization.
Noor M. Khan received his B.Sc. degree in electrical engineering
from the University of Engineering and Technology (UET), Lahore,
Pakistan, in 1998 and Ph.D. degree in electrical engineering from the
University of New South Wales (UNSW), Sydney, Australia in 2006.
He held several positions in WorldCall, NISTE, PTCL, UNSW, GIK
Institute of Engineering Sciences and Technology, and Mohammad Ali
Jinnah University, Pakistan from 1998 to 2015. Currently, he is
working as Professor with the Capital University of Science and
Technology (CUST), Islamabad, Pakistan. He has served as Chair and
Co-Chair of the Technical Program Committees of IEEE International
Conference on Emerging Technologies (ICET2012) in 2012 and IEEE
International Multi-topic Conference (INMIC-2009) in 2009, respectively. He has been awarded Research Productivity Award (RPA) by
the Pakistan Council for Science and Technology (PCST) for the years
2011 and 2012. His research interests include channel modeling and
characterization, wireless sensor networks, cellular mobile communication networks and ground-to-air communication systems.
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