3D spherical M2M

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3D Mobile-to-Mobile Wireless Channel Model
Prasad T. Samarasinghe, Tharaka A. Lamahewa, Thushara D. Abhayapala and Rodney A. Kennedy
Applied Signal Processing Group,
Research School of Information Sciences and Engineering,
ANU College of Engineering and Computer Science,
The Australian National University,
Canberra ACT 0200, Australia
{prasad.samarasinghe,tharaka.lamahewa,thushara.abhayapala,rodney.kennedy}@anu.edu.au
In this paper we introduce a novel three-dimensional (3D)
M2M channel model based on free-space wave propagation
where the scattering environment surrounding the transmitter
and the receiver is characterized using a random scattering
gain, which is a function of angle of departure (AOD) and
angle of arrival (AOA). When compared with other models
discussed above, this channel model can be used for any
3D scattering environment whereas the other models are
restricted. Based on this channel model we derive the temporal
correlation function for a general 3D scattering environment.
The temporal correlation function is characterized by the
joint angular power distribution at the transmitter and receiver
antennas and the speed of transmitter and receiver antennas.
As a special case, in this paper we consider separable scattering environments, i.e., the angular power distribution at the
receiver is independent of that at the transmitter. In this case,
using the von Mises-Fisher distribution as an example for the
angular power distribution at each end of the channel, we
calculate the temporal correlation function for several fading
scenarios and discuss the usefulness of our proposed model.
The rest of the paper is organized as follows. In Section II,
we present our new 3D M2M channel model. Using this channel model, in Section III we derive the temporal correlation
corresponding to a general scattering environment. In Section
IV we provide numerical examples to show the strength of the
proposed model. Finally, the concluding remarks are given in
Section V.
Notations: Throughout the paper, the following notations
will be used: bold lower letters denote vectors, a unit vector
∗
is represented by x̂ and (·) denotes the complex conjugate
operation. The symbol δ(·) denotes the Dirac delta function
while E {·} denotes the mathematical expectation. The scalar
product
√ between vectors x and y is denoted by x · y and
i = −1.
Abstract—A three-dimensional single-input single-output
Mobile-to-Mobile wireless channel model is developed in this
paper by considering the underlying physics of free space
wave propagation. Based on this channel model, the temporal
correlation function for a general three dimensional scattering
environment is derived. The temporal correlation function is
characterized by the joint angular power distribution at the
transmitter and receiver antennas and the speed of transmitter
and receiver antennas. Using a von Mises-Fisher distribution as
the angular power distribution, the usefulness of the derived
temporal correlation function is discussed.
I. I NTRODUCTION
A communication system in which both the transmitter
and receiver are in motion, can be called as a Mobile-toMobile (M2M) communication system. The main applications
of M2M systems are in mobile ad-hoc wireless networks and
intelligent transport systems. Dedicated Short Range Communication (DSRC) and Wireless Access in a Vehicular Environment (WAVE) are emerging standards for intelligent transport
systems focussed on improving traveler safety, efficiency and
productivity [1]. Understanding of M2M wireless channel and
its behavior will support to improve the above applications.
In the literature, M2M channels have been modeled in
various ways. Akki and Harber [2, 3] were the first to propose a statistical channel model for single-input single-output
(SISO) M2M Rayleigh fading radio channel under non line of
sight conditions. Simulation methods for SISO M2M channels
have been proposed in [4–6]. Also, Pätzold in [7] derived
a two-dimensional(2D) ray-tracing model for multiple-input
multiple-output (MIMO) M2M narrowband channel by considering a geometrical two-ring scattering model. Stüber in [8]
proposed a similar 2D ray-tracing narrowband channel model
for MIMO M2M channel based on ‘double-ring’ geometrical
model. However, since the latter model is independent of the
distances between scatters and antenna elements, it is better
than the former model. In [9] the narrowband channel model
proposed in [10] was extended to a 2D wideband model.
In [11] and [12] these two 2D M2M channels models were
extended to 3D models by assuming scatterers are placed on
cylinders. One shortcoming of these ray-tracing geometrical
based channel models is that they cannot be utilized to
represent general scattering scenarios.
978-1-4244-5434-1/10/$26.00 ©2010 IEEE
II. M OBILE - TO -M OBILE C HANNEL M ODEL
Consider the SISO Mobile-to-Mobile channel model depicted in Fig. 1, where the mobile transmitter is located inside
a scatterer free sphere of radius rT and the mobile receiver
is located inside a scatterer free sphere of radius rR . It is
assumed that scatterers are located outside the two scatterfree transmit and receive regions and are in the far-field of the
receiver and transmitter regions. In Fig. 1 g(φ̂, ϕ̂) represents
30
AusCTW 2010
where we have introduced the shorthand 4 S×S S×S . In
this work we assume that the scattering environment is widesense stationary zero-mean and uncorrelated. Therefore, the
second-order statistics of the scattering gain function g(φ̂, ϕ̂)
can be written as [14]
E g(φ̂, ϕ̂)g ∗ (φ̂ , ϕ̂ ) G(φ̂, ϕ̂)δ(φ̂ − φ̂ )δ(ϕ̂ − ϕ̂ ),
where G(φ̂, ϕ̂) = E |g(φ̂, ϕ̂)|2 characterizes the joint
power spectral density (PSD) or the joint angular power
distribution surrounding the transmitter and receiver regions.
In this case the temporal correlation function in (4) further
simplifies to
G(φ̂, ϕ̂)e−ikuτ û·φ̂ eikvτ v̂·ϕ̂ dφ̂dϕ̂.
(5)
ρ(τ ) =
^
ϕ
^
φ
Scatterers
^ ^
g(φ,ϕ
)
^v
Receiver
u^
(b)
Transmitter
(c)
(a)
Fig. 1.
A general scattering model for a flat fading M2M wireless
communication system
S×S
From (5) above it can be seen that the temporal correlation
function of a mobile-to-mobile fading channel is described
largely by the joint angular power distribution G(φ̂, ϕ̂) and
the speed of transmitter and receiver.
the effective random complex scattering gain function for a
signal leaving the transmitter scatter-free region at a direction
φ̂ (relative to the transmitter region origin) and entering the
receiver scatter-free region from a direction ϕ̂ (relative to the
receiver region origin). Suppose the transmitter is moving at
constant velocity u in the direction of û and the receiver is
moving at constant velocity v in the direction of v̂. Following
the derivation of fixed-to-mobile channel model given in [13],
the signal received at the mobile receiver at time t can be
written as
x(t)g(φ̂, ϕ̂)e−iktuû·φ̂ eiktvv̂·ϕ̂ dφ̂dϕ̂ + n(t),
y(t) =
S×S
III. 3-D S CATTERING E NVIRONMENT
Using the spherical harmonic expansion of plane waves, the
plane wave eikx·ŷ can be expanded in 3D as [15, page 32]
eikx·ŷ =
n=0 m=−n
where x(t) is the baseband transmitted signal, k = 2π/λ is the
wave number with λ being the wavelength, n(t) is the additive
noise at the receiver. The integration in (1) above is over a
unit circle for a two dimensional scattering (or multipath)
environment or over a unit sphere for a three dimensional
scattering environment.
The input-output relationship for the general case then can
be written as
where 0 ≤ θ ≤ π and 0 ≤ ψ ≤ 2π are respectively the
elevation and azimuth angles and Pnm (·) are the associated
Legendre functions of the first kind. By applying the spherical
harmonic expansion (6) in (5), we obtain
(2)
ρ(τ ) =(4π)2
e−inπ/2 jn (kuτ )Ynm (û)
n=0 m=−n
p=0 q=−p
where
The channel temporal correlation, which is the correlation
between the channel gain at time t and time t − τ can then be
found from
p,q
βn,m
=
ρ(τ ) E {h(t)h∗ (t − τ )} .
= E g(φ̂, ϕ̂)g ∗ (φ̂ , ϕ̂ ) eiku(−tû·φ̂+(t−τ )û·φ̂ )
n
∞ p
∞ S×S
×e−ikv(−tv̂·ϕ̂+(t−τ )v̂·ϕ̂ ) dφ̂dϕ̂dφ̂ dϕ̂ ,
(6)
and Ynm (·) are the spherical harmonics, which are defined as,
2n + 1 (n − |m|)! |m|
(7)
Ynm (θ, ψ) P (cos θ)eimψ
4π (n + |m|)! n
where, comparing (1) and (2), the underlying time-varying
fading channel between the receiver and the transmitter can
be written as,
g(φ̂, ϕ̂)e−iktuû·φ̂ eiktvv̂·ϕ̂ dφ̂dϕ̂.
(3)
h(t) =
4
∗
in 4πjn (kx)Ynm (x̂)Ynm
(ŷ),
where jn (r) is the spherical function, which is related to the
Bessel function by
π
J 1 (r)
jn (r) =
2r n+ 2
(1)
y(t) = h(t)x(t) + n(t)
n
∞ S2 ×S2
∗
p,q
eipπ/2 jp (kvτ )Ypq
(v̂)βn,m
,
(8)
∗
G(φ̂, ϕ̂)Yn,m (φ̂)Yp,q
(ϕ̂)dφ̂dϕ̂
(9)
are the scattering environment coefficients which characterize
the 3D scattering environment surrounding the receiver and
transmitter regions.
According to (8), in addition to scattering coefficients, temporal correlation depends on transmitter and receiver velocities (u
(4)
31
and v) and time between two received signals (τ ). Simulation
results in section IV further explain these relationships.
As shown in [16], spherical Bessel functions will exhibit
a high pass nature. Therefore, using this property the infinite
summations in (8) can be approximated as
ρ(τ ) =(4π)2
MT n
proposed in the literature for modeling the scattering distributions GT x (φ̂) and GRx (ϕ̂) at the transmitter and the receiver,
respectively. Few examples are, isotropic model [18], uniform
limited azimuth/elevation model [19] and spherical harmonic
model [19].
In this paper we use the 3D von Mises-Fisher model to
characterize the scattering power distribution at each end
of the channel. In addition, using our temporal correlation
function (8), we derive the temporal correlation function for a
M2M channel in a 3D isotropic scattering environment.
e−inπ/2 jn (kuτ )Ynm (û)
n=0 m=−n
p
MR p=0 q=−p
∗
p,q
eipπ/2 jp (kvτ )Ypq
(v̂)βn,m
,
(10)
IV. E XAMPLES
where NT πerT /λ and NR πerR /λ.
In this section we discuss temporal correlation for two
example scattering distributions and their simulation results.
For both examples we use carrier frequency as 5.9 GHz,
symbol duration Ts = 0.0001 seconds and delay τ = 10Ts .
A. Special Case: Separable Channels
In this section we consider a special case of the joint
angular power distribution G(φ̂, ϕ̂) and derive the scattering
p,q
corresponding to this special
environments coefficients βn,m
case.
When the angle of departure (AOD) φ̂ is independent of the
angle of arrival (AOA) ϕ̂, the joint angular power distribution
G(φ̂, ϕ̂) can be written as
G(φ̂, ϕ̂) = GT x (φ̂)GRx (ϕ̂),
where
A. 3D Isotropic Scattering Distribution
If the waves are transmitting from the transmitter uniformly
to all direction in 3D space, i.e., GT x (φ̂) = 1/2π 2 , the
scattering environment coefficients at the transmitter is given
by
(11)
GT x (φ̂) =
S2
G(φ̂, ϕ̂)dϕ̂,
0,0
βn,m
is the angular power distribution at the transmitter and
GRx (ϕ̂) =
G(φ̂, ϕ̂)dφ̂
is the angular power distribution at the receiver. Fading
channels that satisfy (11) are known as separable channels
or Kronecker channels. As shown in [14], the separability
condition (11) can be assumed when there is a single scattering
cluster in the scattering environment.
In this case, the scattering environment coefficients are
given by
where
0,0
βn,m
p,q
β0,0
√1 ,
4π
0,
n=m=0
otherwise
=
√1 ,
4π
0,
p=q=0
otherwise
The temporal correlation function of the 3D M2M channel, in
this case, becomes
(12)
ρ(τ ) = j0 (kuτ )j0 (kvτ ).
=
=
Similarly, if the waves are impinging on the receiver uniformly
from all direction in 3D space, i.e., GRx (ϕ̂) = 1/2π 2 ,
the corresponding scattering environment coefficients at the
receiver is given by
S2
p,q
0,0 p,q
= βn,m
β0,0 ,
βn,m
S2
GT x (φ̂)Yn,m (φ̂)dφ̂
(13)
(14)
Note that in a 2D scattering environment, as shown in [2],
the temporal correlation function in a 2D isotropic scattering
environment is given by ρ(τ ) = J0 (kuτ )J0 (kvτ ).
The magnitude of the time correlation between two received
signals after 10 symbol period versus magnitude of transmitter
velocity and magnitude of receiver velocity are shown in Fig. 2
and contour plot of the same is shown in Fig. 3, where both
transmitter and receiver scattering distributions are isotropic.
In this simulation, transmitter and receiver are moving with
zero elevation and zero azimuth angles with respect to reference points.The simulation result shows a high degree of
correlation (> 0.3) until both transmitter and receiver reach 20
m/s and reduce to zero temporal correlation when transmitter
and or receiver are at 25 m/s.
are the scattering environment coefficients at the transmitter
and
p,q
∗
β0,0
=
GRx (ϕ̂)Yp,q
(ϕ̂)dϕ̂.
S2
are the scattering environment coefficients at the receiver.
Power distributions are mainly characterized by the mean
angle of arrival ϕ0 (or departure φ0 ) and the angular spread
σ at the receiver (or transmitter). A number of univariate
power distributions in 3D scattering environments1 have been
1 In 2D scattering environments, researchers modeled the scattering environment using azimuthal power distribution models such as von-Mises [17],
Laplacian, uniform-limited, truncated Gaussian, etc.
32
and
1
πκR
1
I 1 (κR )Yp,q (ϕ0 ),
2 sinh κR p+ 2
where κT and κR are the non-isotropy factors at the transmitter
and receiver regions, respectively.
0.8
0.6
0.4
1
0.2
0.9
60
0
0
10
Absolute value of time correlation | ρ(τ) |
Absolute value of time correlation | ρ(τ) |
p,q
=
β0,0
40
20
30
20
40
50
60
0
v (m/s)
u (m/s)
Fig. 2. Time correlation between two receiver signals after τ = 10 symbol
time periods in 3D M2M environment where both transmitter and receiver
scattering distributions are isotropic.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
60
50
0
0
60
40
30
10
20
20
30
40
55
50
10
60
0
v (m/s)
00
50
00 0
0
0
0
00
u (m/s)
45
Fig. 4. Time correlation between two receiver signals after τ = 10 symbol
time periods in 3D M2M environment. Where both transmitter and receiver
scattering distributions are von Mises-Fisher with κ = 1.
35
00
0
30
0
25
0
v (m/s)
40
0
00
00
20
60
0.4
00
00
15
50
0.4
10 0
.8
5
20
30
u (m/s)
40
50
60
40
Fig. 3. Contour plot of time correlation between two receiver signals after
τ = 10 symbol time periods in 3D M2M environment where both transmitter
and receiver scattering distributions are isotropic.
30
0.2
v (m/s)
10
0.2
20 0.4
0.6
10
0.2
The von Mises-Fisher distribution in a 3-D scattering environment defined as [20],
G(θ) =
κ
eκ cos(θ−θ0 ) , |θ − θ0 | ≤ π,
4π sinh κ
0.2
0.4
B. von Mises-Fisher distribution
0.8
0
(15)
0
10
20
30
u (m/s)
40
50
60
Fig. 5. Contour plot of time correlation between two receiver signals after
τ = 10 symbol time periods in 3D M2M environment where both transmitter
and receiver scattering distributions are von Mises-Fisher with κ = 1.
where κ ≥ 0 represents the non-isotropy factor of the distribution and Im (·) is the modified Bessel function of the first
kind. Note that κ = 0 represents isotropic scattering. As shown
p,q
0,0
and β0,0
in [21], the scattering environment coefficients βn,m
corresponding to the von Mises-Fisher distribution are given
by
πκT
1
0,0
∗
βn,m =
I 1 (κT )Yn,m
(φ0 ),
2 sinh κT n+ 2
Assuming both transmitter and receiver sides are of von
Mises-Fisher distribution and separable channels, the magnitude of temporal correlation between two receiver signals after
10 symbol period is shown in Fig. 4 and contour plot of the
same is shown in Fig. 5. In this simulation, transmitter and
receiver are moving with zero elevation and zero azimuth
angles with respect to reference points. In addition, AOA
and AOD also have zero elevation and azimuth angles with
33
respect to reference points. In this situation a high degree of
correlation (> 0.3) is shown until both reach 25 m/s, but
when compared to isotropic case rate of reduction of temporal
correlation is lesser.
The same equation and simulations can be used to understand the impact of the eight angles (elevation and azimuth
angles of transmitter velocity, receiver velocity, AOA and
AOD) to the temporal correlations. One example of different
angles is shown in Fig. 6 where transmitter azimuth and
elevation angles are 300 and other angles remain at zero.
In this situation receiver is reaching to the same temporal
correlation at lesser speeds than the all angles zero example.
Hence the rate of reduction of temporal correlation is higher
with respect to receiver speeds whereas temporal correlation
related to transmitter speeds remains same in both cases.
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45
40
35
v (m/s)
30
25
0.2
20
0.4
0.2
15
10
0.
2
10
0.4
0
0.6
0
0.8
5
20
30
u (m/s)
40
50
60
Fig. 6. Contour plot of time correlation between two receiver signals after
τ = 10 symbol time periods in 3D M2M environment where both transmitter
and receiver scattering distributions are von Mises-Fisher with κ = 1 and
transmitter is moving in azimuth 300 and elevation 300 and other angles
remain at zero.
V. C ONCLUSIONS
In this paper, using underlying physics of wave propagation
in free-space we have proposed a novel 3D model to characterize the fading channel between a moving transmitter and
a moving receiver and also derived 3D generalized temporal
correlation expression. The new model could be used as a
generalized 3D model than any existing model. The flexibility
of our new model is that it can be applied to different scattering
distributions as shown in our simulations, where we have
analyzed temporal correlation between two receiver signals
for isotropic distribution and von Mises-Fisher distribution as
examples.
R EFERENCES
[1] W. Xiang, P. Richardson, and J. Guo, “Introduction and preliminary
experimental results of wireless access for vehicular environments
(wave) systems,” in Mobile and Ubiquitous Systems: Networking and
Services, 2006 Third Annual International Conference on, July 2006,
pp. 1–8.
34
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