Characterization of 3D Elliptical Spatial
Channel Model for MIMO Mobile-toMobile Communication Environment
M. Yaqoob Wani & Noor M. Khan
Wireless Personal Communications
An International Journal
ISSN 0929-6212
Wireless Pers Commun
DOI 10.1007/s11277-017-4479-6
1 23
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1 23
Author's personal copy
Wireless Pers Commun
DOI 10.1007/s11277-017-4479-6
Characterization of 3D Elliptical Spatial Channel Model
for MIMO Mobile-to-Mobile Communication
Environment
M. Yaqoob Wani1 • Noor M. Khan1
Springer Science+Business Media New York 2017
Abstract In this paper, we develop three dimensional (3D) elliptical cylindrical geometrical channel model for multiple-input–multiple-output mobile-to-mobile communication environments. It is assumed that both the mobile nodes are surrounded by uniformly
distributed infinite number of scatterers sprinkled over the surfaces of an elliptical-based
cylindrical shapes. The mobile nodes are located at the centers of the bottom surfaces of
elliptical cylinders and both the mobile nodes are equipped with low-elevated multiple
antenna arrays. The proposed model is designed for urban areas, where mostly the mobile
subscribers reside and are on the move. This model takes into account the effect of multiple
antenna array attributes, roadside infrastructure, the dimensions of the propagation medium, transmit–receiver distance and the velocity of mobile nodes. Using the proposed
channel model, expressions for the joint and marginal cross correlation functions are
derived for non-isotropic scattering environments. The derived expression are simulated
for various parameters to verify their effect on the antenna correlations. The obtained
correlation graph is compared with measured data that confirms a close agreement with it.
Finally, by changing various parameters of the proposed channel model, some existing 2D
and 3D channel models are deduced.
Keywords Mobile-to-mobile channels Correlation function MIMO
channels M2M Wireless channel modeling Elliptical geometry Antenna
correlations
& M. Yaqoob Wani
yaqoobwani@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,
Islamabad 44000, Pakistan
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M. Y. Wani, N. M. Khan
1 Introduction
High capacity wireless links with better quality of service (QoS) are the fundamental
demands of all mobile subscribers in every aspect of the life. Therefore, providing high
capacity data links to mobile subscribers with limited resources is a constant driving force
in the research arena. Mobile-to-mobile (M2M) communication technology give rise to
various innovative advancements over a short span of time in vehicular, railway and
defense sectors to provide audio, data, live video streaming, teleconferencing and other
value added applications without any centralized static base station (BS) [9, 43]. However,
hazardous situation for these high speed, communicating nodes can arise when they are on
the move in scattering-rich urban areas [49]. In such propagation environments, the
channel characteristics are unpredictable and multipath components cause the received
signal in deep fading. Therefore, traditional single-input–single-output (SISO) communication link may not provide the required capacity with QoS in such communication scenarios. Recently, the rich scattering environment has been recognized feasible situation for
the high data-rate applications, if multiple antenna arrays are employed instead of a single
antenna structure at both ends of the wireless link [20]. Multiple-input and multiple-output
(MIMO) systems, therefore, exploit this rich scattering environment in a constructive
manner and provide spatial multiplexing that can guarantee astonishing increase in
throughput [13, 20, 22, 24, 48]. Promised large capacity gain of a MIMO system can be
achieved contingent upon a good understanding of spatial characteristics of the radio
fading channel [24]. In this regard, various techniques are adopted to model the wireless
channel to estimate its statistics on the basis of communication scenario and the distribution of the scattering objects in the propagation environments. Mostly, communication
channels are modeled in the literature using empirical, deterministic, stochastic and
geometry-based stochastic channel modeling (GBSCM) approaches [2, 8, 29]. Empirical
channel models are based on experimental measurements and observations for a particular
communication scenario. Hence, these models are valid only for specific sites under particular circumstances [18]. The deterministic channel modeling approach is applicable
when the transmitter, receiver, scattering objects are static and other channel characteristics
are known. The stochastic channel modeling (SCM) approach is based on the PDFs of
various channel parameters and is also site specific [12]. Since, in Mobile-to-mobile
(M2M) communication environment, both communicating nodes are assumed to be on the
move; therefore, previously discussed channel modeling approaches are incongruous to
model M2M propagation channels. Such non-stationary propagation environments can be
modeled perfectly by GBSCM approaches under the assumption of quasi-stationary
scattering [28]. GBSCM approach is based on fundamental laws of wave propagation and
on the physical geometry of propagation environment [16]. In addition, geometrical
channel models may be utilized for various propagation scenarios just by adjusting the
model input parameters [25]. On the basis of physical dimensions of propagation environments, probability distribution functions of various parameters of the M2M wireless
channel may be derived geometrically. Various 2D and 3D versions of GBSCM’s, have
been proposed in the literature for fixed-to-mobile (F2M) and M2M communication scenarios with both SISO and MIMO antenna structures. In case of F2M framework, mostly
researchers have derived mathematical expressions for PDFs of angle-of-arrival (AoA) and
time-of-arrival (ToA), and space-time correlation functions among MIMO elements
[1, 19, 25, 30, 31, 34, 35]. Besides F2M, Akki and Haber proposed geometrical channel
model for SISO and M2M land communication channels and developed closed-form
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Characterization of 3D Elliptical Spatial Channel Model…
expressions for some statistical characteristics of propagation channel [6, 7]. Authors in
[10, 11, 14, 16, 38–39, 41, 54] have extended precursory research by assuming the scattering objects along circular or elliptical loci with isotropic and non-isotropic distributions
for both SISO and MIMO environments. The authors have derived expressions for different propagation parameters like AoA, ToA, Doppler spread, level crossing rate, average
fade duration and space-time correlation among MIMO antenna elements. In the abovementioned 2D geometrical channel models, signal propagation has been considered only
along the azimuth plane, ignoring the elevation plane. In the previously reported geometrical channel models, the effect of azimuth angle of wavelets were considered for the
channel statistics. Hence, 2D geometrical channel models may be appropriate in some of
the rural areas, but may not decorous for streets, canyons, urban or metropolitan areas.
Because in such areas the source of communication is predominantly with scattered waves
that are diffracted, reflected down the streets or canyons by the edges of the surrounding
infrastructure. Therefore, geometrical channel models that consider the effect of elevation
angle of the wavelets are suitable models for such urban environments. The aspect of
elevation angle of the wavelets is introduced in [15, 32, 33, 52, 55, 57, 58] by presenting
3D geometrical channel models for SISO and MIMO M2M communication systems, where
the authors have assumed that mobile stations are located at the centers of cylinders,
spheres, semi-spheres or ellipsoids. In these proposed 3D channel models, authors have
ignored the propagation-distance traveled by the wavelets from the scatterer located in the
transmitter scattering-region to the scatterer located in the receiver scattering-region. The
authors derived closed-form expressions for Doppler spread, PSD and joint space-time
correlation functions among MIMO coefficients while assuming non-isotropic scattering
environments. Moreover, the authors in [45] have derived the transmit, receive antenna
correlations as separate identity under the context of Kronecker model while ignoring the
effect of distances between transmitter and receiver on the antenna correlations. These
models may be favorable in such urban areas where the mobile nodes reside in close
proximity to each other and the scattering objects surround the mobile nodes in a cylindrical manner. Whereas, in reality the physical layout of streets, canyons, deep-cut railway
tracks and highways are mostly narrow in width and longer in length that have close
resemblance to the elliptical shape than circular ones. High-rise buildings, vegetation and
other infrastructure present along the roadside premises are the sources of multipaths in the
azimuth and elevation planes. Understanding this usefulness of elliptical shape, Riaz
et al. [46, 47] proposed an ellipsoid geometrical channel model for SISO M2M environments, where they derived expressions for the PDFs of ToA and AoA both in azimuth and
elevation planes. The authors assumed that the signals are equally likely from all directions
of the ellipsoid, which contradicts the realistic propagation environment. Because the
probability of AoA of scattered signals from the sky-top of urban areas is almost zero,
Ahmed et al. [5] proposed modified geometrical channel model of [46, 47] with top surface
open and derived PDFs for AoA and AoD. These proposed M2M communications models
can be made more beneficial for high data rate applications if they are equipped with
MIMO systems. This motivates us to propose a geometrical channel model for MIMO
M2M communication environments that can relegate all the above-mentioned impairments
of previously suggested channel models.
In this paper, we propose an eccentricity-based elliptical cylindrical channel model for
MIMO M2M communication environments. We derive expressions for the marginal and
joint correlation functions among the antenna elements. These correlation functions are
further simulated to analyze the impact of different system parameters on the correlations
among antenna-array element. Furthermore, different 2D and 3D existing channel models
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M. Y. Wani, N. M. Khan
are deduced, by changing the parameters of proposed geometry. The correlation curves of
proposed geometry are compared with the results of these previously published geometrical channel models. Finally, simulation result of space-time correlation function is
compared with measurement campaign.
Rest of the paper is organized as follows: Sect. 2 describes the geometry of the proposed eccentricity-based elliptical cylindrical channel model for MIMO M2M. Section 3
presents the derivations of the reference model. Section 4 presents derivation of the closeform expressions of correlation functions for the proposed geometry. Section 5 provides
the comprehensive discussion of the simulation results. Finally, Sect. 6 provides some
concluding remarks.
2 System Model
In this section, we present the system architecture of the proposed elliptical cylindrical
geometrically-based channel model for MIMO M2M communication environment as
shown in Fig. 1. In this proposed channel model, transmitting and receiving mobile stations (MSs) are denoted by MSt and MSr respectively. These MSs are assumed to be
located at the centers bottom surface of the elliptical cylinders, having major axes at and ar
and minor axes bt and br with eccentricities t and r , respectively. The mobile nodes are
moving independently with the velocities of vt and vr making angles at and ar with the xaxis. The center to center distance between the two elliptical cylindrical is represented by d
(such that d at þ ar ) and to avoid the channel may not experience the keyhole behavior
the distance should not be greater than 4Rt Rr Hr =ðkðTr 1ÞðHt 1ÞÞ [21]. The azimuth
plane dimension of the physical propagation channel around the MSs are adjusted in the
system model by eccentricities t and r of the ellipses. Whereas, the surrounding scatter
hight around transmitter and receiver are represented by ht and hr respectively. The scatters
z
AT
z
ST(m)
dpm
(p)
ψT(p)
dmn
AR(q)
βT(m)
ψR(q)
dp∼m
CT
∼
vR
vT
γT
βR(m) dnq∼
CR
∼
AT(p)
AT(p)
Rt
SR(n)
dnq
αT(m)
AR(q)’
∼
AT(p)
AR(q)
γR
Rr C’R
C’T π−θT
αR(n)
∼
AR(q)
π−θR
x
d
y
Fig. 1 Proposed 3D elliptical channel model for MIMO Mobile-to-Mobile channels with Ht ¼ Hr ¼ 2
antenna elements
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Characterization of 3D Elliptical Spatial Channel Model…
present in the close vicinity of mobile nodes are assumed to be uniformly distributed on the
surfaces of elliptical cylinders. Moreover, the elliptical cylinders are rotatable congruous to
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. Mobile nodes are equipped with low hight antennas arrays
with configuration Ht Hr , where Ht and Hr are the number of antennas mounted on MSt
and MSr , respectively. For simplicity, we take Ht = Hr = 2. However, the results can be
derived for any configuration. The transmit and receive antenna array elements are denoted
ðpÞ
by At and ArðqÞ , and the distance between the antenna array elements are denoted by dt and
dr , which are very small as compared to the minor axes of the surrounding ellipses
respectively. The description of the other parameters involved in the proposed geometrical
model are narrated in Table 1.
3 Derivation of the Reference Model
The derivations of the various characteristic of the MIMO M2M fading channel are based
on the reference model depicted in Fig. 1. It can be observed that a signal that is transðpÞ
ðpÞ
mitted from the transmit antenna array element At , first strikes at the scatterer St present
on the surface of elliptical cylindrical surrounding the transmitter node and then travels
Table 1 Definitions of the channel parameters used in the system model
Symbols
Description
d
The distance between center to center of elliptical cylinders surrounding MSt and MSr
Rt ; Rr
The dynamic radius of the transmitter and receiver ellipses, respectively
at ; ar
The major axes of the transmitter and receiver ellipses, respectively
bt ; br
The minor axes of the transmitter and receiver ellipses, respectively
ht ; hr
The hight of the scatterers at transmit and receive elliptical cylinder, respectively
Ht ; Hr
Number of antenna array elements at transmitter and receiver, respectively
t ; r
The eccentricities of the transmitter and receiver ellipses, respectively
dt ðp; p~Þ
The spacing between pth and p~th antenna elements at Tx
dr ðq; q~Þ
The spacing between qth and q~th antenna elements at Tx
ðqÞ
hðpÞ
t ; hr
The azimuth angle of pth transmit and qth receive antenna
element (relative to x-axis), respectively
ðqÞ
wðpÞ
t ; wr
The elevation angle of pth transmit and qth receive antenna
element (relative to x–y plane), respectively
vt ; vr
The velocities of the Tx and Rx, respectively
ct ; cr
The moving directions of the Tx and Rx, respectively
ðmÞ
at ; aðnÞ
r
The azimuth angles of departure (AAoD) and the azimuth
angles of arrival (AAoA), respectively
ðnÞ
bðmÞ
t ; br
The elevation angle of departure (EAoD) and the elevation
angle of arrival (EAoA), respectively
ðpÞ ðmÞ
ðpÞ
~
ðmÞ
ðmÞ
ðqÞ
ðqÞ
~
, d St ; SðnÞ
, d SðnÞ
The distances d At ; St , d At ; St
, and d SðnÞ
r
r ; Ar
r ; Ar
dpm dpm
~ dmn dnq dnq~
pq
dmax
dpm þ dmn þ dnq
p~q~
dmax
dpm
~ þ dmn þ dnq~
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M. Y. Wani, N. M. Khan
towards the scatter SðqÞ
located on the surface of elliptical cylindrical surrounding the
r
receiver node, and then finally reaches the receiving antenna array element AðqÞ
r . During
ðpÞ
ðpÞ
~
~
propagation, the distances covered by the wavelets from At to AðqÞ
to AðrqÞ
, are
r and At
pq
p~q~
denoted by dmax and dmax , respectively. Therefore, the phase change due to the distance
pq
p~q~
traveled by the wavelet can be written as ð2p=kÞdmax
and ð2p=kÞdmax
, where 2p=k is called
wavenumber. Moreover, the joint gain and phase shift due to the collision of the wavelet
pffiffiffiffiffiffiffiffi
ðpÞ
ðqÞ
with St and St cab be expressed as 1= MN and /mn respectively. Other sources of
phase shift in M2M communication
which
is due to
the motion of transmitter and receiver,
ðmÞ
ðnÞ
cos
bðnÞ
can be expressed as 2pfTmax cos at ct cos bðmÞ
t
and
2pf
cos
a
c
Rmax
r
r
t
r t.
Where, fTmax ¼ vt =k and fRmax ¼ vR =k represents the maximum Doppler frequency caused
by the motion of mobile nodes. Moreover, for the ease of derivations of the different
expressions of the proposed channel model, following valid assumptions are considered
1.
2.
3.
4.
5.
Each multipath component of the propagating signal undergoes two bounces while
traveling from the transmitter mobile node to the receiver mobile node.
Infinite number of scatterers are uniformly distributed on the elliptical cylindrical
surfaces with uniformly distributed phases.
Equal distributed power is reflected from all the scatterers.
All the waves reaching at the receiver antenna elements are equal in power.
The scatterers are fixed and MSs are quasi-stationary for a short period of time.
Finally, with the help of above assumptions the diffused components of the transmission
ðpÞ
link from At
to AðqÞ
r can be expressed as,
M X
N
2p
1 X
ej k ðdpm þdmn þdnq Þþj/mn
hpq ðtÞ ¼ lim pffiffiffiffiffiffiffiffi
M;N!1 MN
m¼1 n¼1
e
ðmÞ
ðmÞ
j2ptfTmax cosðat ct Þ cos bt
e
ð1Þ
ðnÞ
ðnÞ
j2ptfRmax cosðar cr Þ cos br
The expressions of the distances dpm , dpm
~ , dnq and dnq~ are obtained by solving the geometry of
the proposed model as shown in Fig. 1. It is observed that these distances are the functions of
random angles of azimuth and elevation of transmit antenna arrays, receive antenna arrays
and scatter. The 3D polar coordinates
of pth
transmit
and qth receive
antenna elements in 3D
space are denoted by
ðpÞ
ðpÞ
ðpÞ
ðpÞ
dTx ; dTy ; dTz
ðpÞ
and
ðqÞ
ðqÞ
ðqÞ
ðpÞ
ðpÞ
dRx ; dRy ; dRz , where dTx ¼ dðAt ;
ðpÞ
ðpÞ
ðqÞ
ðpÞ
ðpÞ
ðpÞ
ðpÞ
Ct Þ cos hðpÞ
t cos wt , dTy ¼ dðAt ; Ct Þ sin ht cos wt , dTz ¼ dðAt ; Ct Þ sin wt , dRx ¼
ðqÞ
ðqÞ
ðqÞ
ðqÞ
ðqÞ
ðqÞ
ðqÞ
ðpÞ
dðAðpÞ
dRy ¼ dðAðpÞ
r ; Cr Þ cos hr cos wr ,
r ; Cr Þ sin hr cos wr ; dRz ¼ dðAr ; Cr Þ sin wr .
Similarly,
the coordinates
in 3D space are denoted by
of mth and nth scatterer
ðm=mÞ
~
ðm=mÞ
~
ðm=mÞ
~
ðn=nÞ
~
ðn=nÞ
~
ðn=nÞ
~
ðm=mÞ
~
ðm=mÞ
~
ðm=mÞ
~
dAx ; dAy ; dAz
¼ Rt cos at
, d Ay
¼
and dAx ; dAy ; dAz
, where dAx
~
ðm=mÞ
Rt sin at
~
ðm=mÞ
, dAz
nÞ
~
Rr tan bðn=
.
t
~
ðn=nÞ
~
mÞ
¼ Rt tan bðm=
, dAx
t
~
ðn=nÞ
nÞ
~
¼ d þ Rr cos aðn=
, dAy
r
~
ðn=nÞ
nÞ
~
¼ Rr sin aðn=
, dRz
r
¼
These polar coordinates depend upon the orientation and configuration of the
antenna arrays. The dynamic radii Rr and Rt of the transmit and receive ellipses can be written
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
in terms of minor ai and major bi axis as, ai bi = a2i sin2 ai þ b2i cos2 ai [44]. Using the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
distance formula, and binomial approximation
ð1 þ xÞ 1 þ x=2 if ðx 1Þ, the
approximated distances can be expressed as,
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Characterization of 3D Elliptical Spatial Channel Model…
dp=p;m
~ Rt
cosbðmÞ
t
ðpÞ
ðpÞ
ðmÞ
ðmÞ
ðpÞ
ðmÞ
ðpÞ
d At ;Ct sinwðpÞ
t sinbt d At ;Ct coswt cosbt cos at ht
ð2Þ
dn;q=q~ Rr
cosbðnÞ
r
ðqÞ
ðnÞ
ðqÞ
ðnÞ
ðqÞ
ðpÞ
ðnÞ
d AðpÞ
r ;Cr sinwr sinbr d Ar ;Cr coswr cosbr cos ar hr
ð3Þ
ðpÞ
ðqÞ
The propagation path length, dmn , from St to St is greater than maxðdpm ;dnq Þ and
contributes significantly in the phase shift to the received signal. It is the function of
ðmÞ
ðnÞ ðnÞ
ðqÞ
ðpÞ
random angles bðmÞ
and dynamic radii of the ellipses and can be
t ;at ;br ar , br , bt
obtained by solving the geometry as shown in Fig. 2. With the help of approximations used
in (2) and (3), the simplified form of dmn can be expressed as,
ðmÞ
ðnÞ
2
2
R
R
cos
a
a
r
t
t
r
R
R
ðmÞ
Rt cos at þ Rr cos aðnÞ
dmn d þ r þ t r
2d 2d
d
ð4Þ
2 2
ðqÞ
ðpÞ
ðqÞ
Rt tan bðpÞ
R
tan
b
r
t
r
Rt tan bt Rr tan br
þ
:
þ
2d
2d
d
ðmÞ
Rr Rt cos aðnÞ
r at
ðmÞ
dmn d Rt cos at þ Rr cos aðnÞ
r
d
ð5Þ
ðpÞ
ðqÞ
2
2 ðpÞ
2
2 ðqÞ
Rt sec bt
Rr sec br
Rt tan bt Rr tan br
þ
:
þ
2d
2d
d
Substituting values of dpm , dnq and dmn in (1), we get,
M X
N
1 X
ap;m bn;q cp;q ej/mn
hpq ðtÞ ¼ lim pffiffiffiffiffiffiffiffi
M;N!1 MN
m¼1 n¼1
ðmÞ
at ct
ej2p½fTmax cosð
Þ
ðmÞ
cos bt þfRmax
cosð
ðnÞ
ar cr
ð6Þ
Þ
ðnÞ
cos br
t :
St(m)
dmn
z
dpm
Sr(n)
x
ht
dqn
y
At(p)
at
βt
Rt
Ar(q)
αt
d
(0,0,0)
ar
bt
hr
βr Rr
(d,0,0)
αr
x
br
Fig. 2 2D view of proposed elliptical channel model for MIMO M2M channels
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M. Y. Wani, N. M. Khan
where
jp
k
d
2þ
ap;m ¼ e
2Rt
ðmÞ
cos b
t
ðmÞ
ðmÞ
2p
j2pd p cos at
ej k ðRt cos at Þ e k Tx
ðmÞ
j2pdT p sin bt þ
e k
jp
k
z
p
j2p
k dT sin at
e
y
ðmÞ
cos bt
ðpÞ
ðRt sec br Þ2
2d
2Rr
d
2þ
ðnÞ
cos br
bn;q ¼ e
ðmÞ
ðmÞ
cos bt
ðnÞ
ðnÞ
2p
j2pd q cos ar
þ ej k ðRr cos ar Þ e k Rx
ðnÞ
ðnÞ
cos br
e
ðnÞ
q
j2p
k dR sin ar cos br
y
2
ðnÞ
q
j2p
k dR sin br þ
e
jp
k
cp;q ¼ e
z
ðRr sec bðqÞ
r Þ
2d
ðpÞ
ðqÞ R R cos aðnÞ aðmÞ
r t
Rt tan b Rr tan br
r
t
t
þ
d
d
ð
Þ
The proposed geometrical can be also extended for Ht Hr antenna arrays structure. The
ðp=qÞ
ðp=qÞ
ðp=qÞ
ðp=qÞ
polar coordinates of antenna elements is denoted by ðdTx =Rx ; dTx =Rx ; dTx =Rx Þ where, dTx =Rx ¼
ðp=qÞ
ðp=qÞ
ðp=qÞ
dðt=rÞ 0:5Hðt=rÞ :5 pðqÞ cos ht=r cos wt=r ; dTx =Rx ¼ dðt=rÞ 0:5Hðt=rÞ :5 pðqÞ
ðp=qÞ
ðp=qÞ
ðp=qÞ
ðp=qÞ
sin ht=r cos wt=r ; dTx =Rx ¼ dðt=rÞ 0:5Hðt=rÞ :5 pðqÞ sin wt=r : The parameter p; p~ 2
f1; . . .; Ht g and q; q~ takes values from f1; . . .; Hr g. The coordinates for p~ element and q~
element can be obtained by replacing dðt=rÞ with dðt=rÞ . Where, dðt=rÞ is the spacing
between two adjacent elements of transmitter (receiver) antenna arrays.
4 Derivation of Space-Time Correlation Function of the Proposed Model
The normalized space-time correlation function between diffused channel coefficients
hpq ðtÞ and hp~q~ðtÞ for the proposed 3D model can be found using the following relation,
E½hpq ðtÞhp~q~ðt þ sÞ
Rpq;p~q~½s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E½jhpq ðtÞj2 þ E½jhp~q~ðtÞj2
ð7Þ
where, E½ is the statistical expectation operator and can be applied only to all random
variables and ðÞ symbolizes as the complex conjugate operation. Using (6) and (7), the
space-time correlation function can be formulated as,
M X
N
h
1 X
E ap;m bn;q cp;q ap;m
~ bn;q~cp~q~
M;N!1 MN
m¼1 n¼1
Rpq;p~q~½s ¼ lim
e
ðmÞ
ðmÞ
ðnÞ
ðnÞ
2jpsðfTmax cosðat ct Þ cos bt þfRmax cosðar cr Þ cos br
ð8Þ
i
Þ ;
It is assumed that infinite number of scattering objects reside around each mobile station,
which implies that the scattering distributions may be transformed from discrete to conðmÞ
tinuous that in turn forces to change the discrete random variables (e.g., at , bðmÞ
t ) into
continuous random variables (at , bt ). Furthermore, we assume that azimuth and elevation
angles are independent of each other, therefore, f ðat ; bt Þ and f ðar ; br Þ can be written in
product form as f ðat Þf ðbt Þ and f ðar Þf ðbr Þ, respectively. Hence, for the continuous time
random variables, the above equation can be written in integration form as,
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Characterization of 3D Elliptical Spatial Channel Model…
Rpq;p~q~ðsÞ Z
Z
bRm
bRm
j2p
ek
bTm
bTm
ðp;pÞ
~
dTx
Z
Z
p
p
p
e2jpsðfTmax cosðat ct Þ cos bt Þ
p
ðp;pÞ
~
cos at cos bt þdTz
sin bt þ
ðRt sec bt Þ2
2d
e2jpsðfRmax cosðar cr Þ cos br Þ e
~
j2p ðp;pÞ
e k dRx
sin ar cos br
j2p
k
~
ðp;pÞ
dTy
~
j2p ðp;pÞ
dTy
ek
sin at cos bt
~
ðq;qÞ
sin at cos bt þdRz
sin br þ
ðRr sec br Þ2
2d
ð9Þ
f ðat Þf ðbt Þf ðar Þf ðbr Þdat dbt dar dbr
where bRm , bTm represents the maximum elevation angles of the scatters present around
~
ðp;pÞ
transmit and receiver mobile nodes and dTx
~
ðp;pÞ
ðpÞ
~
ðpÞ
~
ðq;qÞ
ðqÞ
~
ðqÞ
~
ðq;qÞ
ðqÞ
ðpÞ
~
ðpÞ
~
ðp;pÞ
¼ dTx dTx , dTy
~
ðqÞ
~
ðq;qÞ
ðqÞ
ðpÞ
~
ðpÞ
¼ dTy dTy ,
~
ðqÞ
¼ dTz dTz , dRx ¼ dRx dRx , dRy ¼ dRy dRy , dRz ¼ dRz dRz .
Furthermore, different scattering distribution have been proposed in the literature for
isotropic and non-isotropic environments [30]. In urban areas streets, canyons and highways are more likely non-isotropic environments. So, in this proposed research work, we
have used eccentricity-based (i.e., i ) modified von Mises distribution for azimuth AoA/
AoD [50]. The PDF of azimuth AoA/AoD at each MS can be written as,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
f ðai Þ ¼ 2pI01ð2 Þ ei cos ai ; i ¼ t; r, where, i ¼ 1 b2i =a2i , the azimuth angle ai 2 ðp; pÞ
dTz
i
and I0 ðÞ is the zeroth-order modified Bessel function of the first kind. Similarly, It has
been observed from the experiments that the elevation AoA of incoming signals ranges
from 0
to 20
[51] and the distance between two mobile stations is much larger than each
of their antenna heights; therefore, using the small angle approximation sin bi bi ,
cos bi 1, sec2i b ¼ 1 þ bi =2 and substituting PDF of elevation AoA/AoD, i.e., f ðbi Þ ¼
p
p bi
as proposed in [36], where, the absolute values of elevation angles
4b cos 2 b
im
im
(bim ; i 2 T; R ) lies in the range 00 bim 200 , in Eq. (9) we get,
Z p
Z p
1
exp
½c
cos
a
þ
d
sin
a
da
exp ½cr cos ar þ dr sin ar dar
Rpq;p~q~½s 2
t
t
t
t
t
4p Io ð2t ÞpIo ð2r Þ p
p
2
2
Z
Z
bRm
bRm
bTm
e
j2p
k
ðp;pÞ
~
R
dTz þ 4dt bt
e
j2p
k
ðq;qÞ
~
R
dRz þ 4dr br
bT
m
p
pbt
p
pbr
cos
cos
dbt dbr
4bTm
2bTm 4bRn
2bRn
ð10Þ
where
~
ðp;pÞ
j2pdTx
2jpsfTmax cos ct þ 2t ;
k
ðp;pÞ
~
j2pdTy
2jpsfTmax sin ct ;
dt ¼
k
ðq;qÞ
~
j2pdRz
cr ¼
2jpsfRmax cos cr þ 2r ;
k
~
ðq;qÞ
j2pdRw
dr ¼
2jpsfRmax sin cr :
k
ct ¼
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M. Y. Wani, N. M. Khan
Equation (10) can be further simplified by introducing trigonometric transformation and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rp
the equality p exp ½ci cos ai þ di sin ai dai ¼ I0 ð2p c2i þ di2 Þ [26].
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z
j2p ðp;pÞ~ R2t dTz bt þ 4d bt
Io ð2p c2t þ dt2 Þ bTm p
pbt
k
e
Rpq;p~q~½s cos
dbt
4b
2b
2pIo ð2t Þ
bTm
Tm
Tm
ð11Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z
2
Io ð2p c2r þ dr2 Þ bRm p
pbr j2pk dRzðq;qÞ~ br þR4dr br
e
cos
dbr
2bRn
2pIo ð2r Þ
bRm 4bRn
It is seen that the joint correlation function in (11) is the product of transmit and receive
correlation functions, that can be written as,
Rpq;p~q~½s qð/t ; bt ; dt ; sÞqð/r ; br ; dr ; sÞ
ð12Þ
where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z
j2p ðp;pÞ~ R2t dTz bt þ 4d bt
2pIo ð c2t þ dt2 Þ bTm p
pbt
k
e
qð/t ; bt ; dt ; sÞ cos
dbt
2
2bTm
2pIo ðt Þ
bTm 4bTm
ð13Þ
Integrating (13) with respect to bt , and introduce approximations for the complex error
function as in [4], the simplified expression can be written as,
ðp;pÞ~
1
0
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 k2 cos pð4dTz dþRt ÞbTm
2dk
C
2pIo c2t þ dt2 B
B
C
ð14Þ
qð/t ; bt ; dt ; sÞ @
A
2
ðp;
pÞ
~
2
2
2
2
2
2pIo ðt Þ
d k ðRt þ 4dTz dÞ bTm
0
1
pbTm R2t
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B
ðp;pÞ
~
2p
Io 2p c2t þ dt2 Bcos k bTm dTz þ 2dk C
C
qð/t ; bTm ; dt ; sÞ B
2 C
ðp;pÞ
~
@
A
2pIo ð2t Þ
2
4b d
R b
1 tdkTm þ Tmk Tz
0
1
pbRm R2r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B
~
ðq;qÞ
2p
Io 2p c2r þ dr2 Bcos k bRm dRz þ 2dk C
C
qð/r ; bRm ; dr ; sÞ B
2 C
~
ðq;qÞ
@
A
2pIo ð2r Þ
4bRm dRz
R2r bRm
1
dk þ
k
ð15Þ
ð16Þ
The closed-form expression for the receive space-time correlation function can be obtained
by replacing the index t with index r, and the joint space-time correlation function (12) can
be written as
0
1
2
pb
R
ðq;
qÞ
~
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B
Rm r
2p
Io 2p c2t þ dt2 Io 2p c2r þ dr2 Bcos k bRm dRz þ 2dk C
C
Rpq;p~q~½s B
2 C
ðq;qÞ
~
@
A
2pIo ð2t Þ
2pIo ð2r Þ
4bRm dRz
R2r bRm
1
dk þ
k
0
1
ð17Þ
pbTm R2t
~
ðp;pÞ
2p
Bcos k bTm dTz þ 2dk C
B
C
B
2 C
~
ðp;pÞ
@
A
2
4b d
R b
1 tdkTm þ Tmk Tz
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Characterization of 3D Elliptical Spatial Channel Model…
The existing 3D cylindrical channel model [56] can become the spacial case of our proposed geometrical channel model. By using the assumption that is the radii are much
smaller than the distance d (i.e., maxfRr ; Rt g d) moreover,to avoid keyhole behavior of
the wireless channel the distance d\4Rt Rr Lq =ðkðLp 1ÞðLq 1ÞÞ. Using this assumption
the expression (17) can be reduced to space-time correlation function (34) of [56] for
isotropic propagation environments.
ðq;qÞ
~
ðp;pÞ
~
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2p
cos 2p
b
d
T
Tz
Io 2p c2t þ dt2 Io 2p c2r þ dr2 cos k bRm dRz
m
k
Rpq;p~q~½s 2
2
ð18Þ
ðq;qÞ
~
ðp;pÞ
~
2pIo ð2t Þ
2pIo ð2r Þ
4bRm dRz
4bTm dTz
1
1
k
k
When mobile stations reside on such highways where the low elevated scattering objects
are non isotropically distributed around transmitter and receive. The probability of elevation plane waves cab be negligible. In that case the elevation angles bRm ; bTm can be
illuminated in the proposed space- time correlation functions (17). The resultant expression
is the space-time correlation function given by Wani et al. [50], which is the special case
of the proposed channel model.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Io 2p c2t þ dt2 Io 2p c2r þ dr2
ð19Þ
Rpq;p~q~½s 2pIo ð2t Þ
2pIo ð2r Þ
The expression (19) can be further explored for the space-time correlation functions in
isotropic scattering environments, that is proposed by Pätzold et al. [39] by keeping
eccentricity r ; t ¼ 1.
5 Results and Description
In this section, we describe the theoretical results obtained from the derived correlation
functions (19) and its impact on the MIMO channel capacity. It can be observed that the
derived joint correlation expressions (19) is the function of various parameters that are
linked with physical MIMO system and the propagation channel environment. For the ease
of discussion our focus is mainly on the receive correlation function (16), the receive
correlation can be directly obtained by replacing subscript index t with r. In all simulations,
a normalized sampling period fRmax Ts ¼ :01, is used (where fRmax ; fTmax are the maximum
Doppler frequencies and Ts is the sampling period). Furthermore, the orientation angle of
the antenna arrays, elevation angle of scatterers and other parameters are mentioned on the
respective plots for each simulation. Different 2D and 3D plots of the correlations among
MIMO channel coefficients are obtained for observation and discussion. The simulation
plots of space-time correlation among receive antenna array elements is shown in Figs. 3
and 4. It is observed from the graph that the correlation decays rapidly in both space and
temporal domains by increasing distance (dr ) between antenna array elements and normalized time delay. The temporal correlation is dependent on the velocity vr , AoA/AoD of
multipaths, carrier frequency fc and the speed of light c. Therefore, for the design of MIMO
M2M communication system in the rich scattering environment, antenna spacing, and
velocity of the mobile stations are the constraints that have significant impact on the
system performance.
Similarly, the correlations among receive antenna elements are obtained by varying the
eccentricity (r ) of the receiver ellipse as shown in Fig. 5. It is observed from the graph that
123
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M. Y. Wani, N. M. Khan
[ Correlation ]
1
0.8
0.6
0.4
0.2
0
0
1
τ fRmax
2
3
0
2
Fig. 3 3D space-time correlation function qr ðdr ; sÞ of
ðbr ¼ 20
; ar ¼ 60
; wr ¼ 30
; hr ¼ 60
; ar ¼ 100 m; br ¼ 50 mÞ
8
6
4
σr / λ
22
MIMO
M2M
channel,
1
δr=λ
δr=λ/2
Receive Correlation
0.8
δr=λ/4
δr=λ/5
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
τ fRmax
Fig. 4 Effect of spacing between antenna elements on correlation function. The curves are obtained using
the parameter ðbr ¼ 20
; ar ¼ 60
; wr ¼ 30
; hr ¼ 60
; ar ¼ 100 m; br ¼ 50 mÞ
the correlation has a decreasing trend as of eccentricity is decreased from 0.9 (elliptical
channel model) to 0 (circle model). This indicates that the capacity is degraded when the
mobile units resides in the narrow streets or canyons, similar findings were observed by
Abouda et al. [3]. Therefore, for the design and implementation of MIMO M2M communication system for such propagation environments, the simulation results based on
elliptical channel model are more appropriate than circular channel model. Correlation
among antenna elements is also evaluated for different relative velocities of the mobile
nodes. It is discerned from the simulation results as shown in Fig. 6, that the correlation has
a decreasing trend with the increasing relative velocity of the mobile nodes. Similar
observations were reported in [17, 23]; however, increasing velocity leads to other severe
constraints in the wireless channel that degrade the system throughput.
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Characterization of 3D Elliptical Spatial Channel Model…
Fig. 5 3D space-time correlation function with respect to eccentricity, ðdr ¼ 0:5k; br ¼ 20
; ar ¼ 60
;
wr ¼ 30
; hr ¼ 60
1
Vr = 120km/h
Vr = 100km/h
Vr = 90km/h
Vr = 60km/h
Receive correlation
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
τ fRmax
Fig. 6 2D space-time correlation function for different velocities, ðdr ¼ 0:5k; br ¼ 20
; ar ¼ 60
; wr ¼
30
; hr ¼ 60
; ar ¼ 100 m; br ¼ 50 mÞ
The proposed channel model is transformed into existing 3D circular-based cylindrical
geometrical channel model [56] by adjusting ai ¼ bi where, i ¼ r; t. The comparison of
correlation functions of the deduced 3D geometrical channel model and the proposed
model is depicted in Fig. 7. It is observed from the correlation curves that the space-time
correlations using circular-based channel model is lower as compared to elliptical-based
channel model. Since, the streets and canyons can be modeled more appropriately
exploiting elliptical shape as discussed earlier in introductory section. Therefore, the
theoretical results, based on the elliptical geometry can predict more appropriately the
achievable capacity of MIMO M2M communication systems in streets and canyons.
By adjusting the parameters of the proposed model the correlation curves are validated
by comparing with the measurement results obtained from experimental campaigns carried
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M. Y. Wani, N. M. Khan
1
Circular-based Cylindrical Model
Elliptical-based Cylindrical Model
0.8
Receive Correlation
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
τ fRmax
Fig. 7 Comparison of space-time correlation function of circular-based cylindrical model with the proposed
channel model, ðdr ¼ 0:5k; ar ¼ 60
; wr ¼ 30
; hr ¼ 60
; br ¼ 20
Þ
Space−time correlation
1
Measurement
Proposed
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
λ [m]
Fig. 8 Comparison of space-time correlation function with the measurement results
out for COST2100 channel model in the outdoor communication environment [59]. It is
observed from Fig. 8 that the proposed model is in close agreement with the measurement
data.
Furthermore, the proposed model is more dynamic in nature, i.e. various other existing
geometrical channel models can become the special cases of it. By adjusting the appropriate values for ai ; bi ; wi ; bi ; di ; ði ¼ t; rÞ parameters in Eq. (17) of the proposed channel
model. The 2D and 3D geometrical channel models listed and depicted in Table 2 can be
deduced,
123
Communication
scenarios
MIMO M2M
MIMO M2M
MIMO M2M
MIMO M2M
Scattering model
Proposed
Zajić et al. [56]
Pätzold et al. [39], Stüber et al. [27] and
Zajić et al. [53]
Wani et al. [50]
Table 2 Generalization of proposed geometrical channel model
3D elliptical-based
cylindrical model
3D circular-based cylindrical
model
2D circular model
2D elliptical model
ar ¼ br , at ¼ bt
at ¼ bt , ar ¼ br , bt ¼ br ¼ 0o
at 6¼ bt , ar 6¼ br , bt ¼ br ¼ 0
Respective scattering models
–
Corresponding substitutions
y
AT(p)
Rt
AT(p)
CT
ψT(p)
γT
vT
ST(m)
∼
AT(p)
∼
dp∼m
αT(m)
βT(m)
dpm
C’T π−θTAT(p)
z
y
AT(p)
Rt
AT(p)
CT
ψT(p)
γT
vT
ST(m)
∼
AT(p)
∼
dp∼m
αT(m)
βT(m)
dpm
d
C’T π−θTAT(p)
z
dmn
d
dnq
SR(n)
γR
∼
∼
AR(q)
AR(q)
π−θR
αR(n)
βR(m) dnq∼
z
Rr C’R
vR
CR
ψR(q)
dmn
AR(q)’
AR(q)
x
AR(q)’
AR(q)
SR(n)
γR
∼
∼
AR(q)
AR(q)
π−θR
αR(n)
βR(m) dnq∼
z
Rr C’R
vR
CR
ψR(q)
dnq
x
Geometry of scattering
regions
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Characterization of 3D Elliptical Spatial Channel Model…
123
Communication
scenarios
SISO M2M
SISO M2M
SISO F2M
F2M MIMO
Scattering model
Baltzis et al. [11]
Paul et al. [40]
Baltzis et al. [30]
Abidi et al. [42]
Table 2 continued
Respective scattering models
2D elliptical model
2D circular model
2D model
2D circular model
Corresponding substitutions
at ¼
6 bt , ar 6¼ br , bt ¼ br ¼ 0
,
dr ¼ dt ¼ 0
at ¼ bt , ar ¼ br , dt ¼ dr ¼ bt ¼ br ¼ 0
ar ¼ br , at ¼ bt ¼ 0,
bt ¼ br ¼ dt ¼ dr ¼ fTmax ¼ 0o
at ¼ bt , ar ¼ br ¼ 0, bt ¼ br ¼ 0o
Geometry of scattering
regions
Author's personal copy
M. Y. Wani, N. M. Khan
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Characterization of 3D Elliptical Spatial Channel Model…
6 Conclusion
In this article, 3D eccentricity based elliptical cylindrical geometrical model for MIMO
M2M communication scenario is proposed. The geometry of the proposed model is
rotatable along x–y plane and its dimensions are adjustable in all axes corresponds to the
physical propagation scenario. Based on the proposed model, joint and marginal PDFs of
correlation functions among transmitter and receiver antenna array elements is formulated.
The expression of correlation function is simulated by changing various parameters of the
proposed channel model and obtained correlation results are meticulously described.
Correlation results obtained using proposed model show close agreement with the measurement data. Moreover, the proposed model is versatile in nature, by varying the system
parameters different existing 2D and 3D geometrical channel models are obtained.
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M. Y. Wani, N. M. Khan
Mohd Yaqoob Wani received his B.Sc. degree in electronics from the
University of Kashmir, Srinager, India, in 1991 and MS 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.
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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