Wireless Pers Commun DOI 10.1007/s11277-013-1158-0
3D Ellipsoidal Model for Mobile-to-Mobile Radio Propagation Environments Muhammad Riaz · Syed Junaid Nawaz · Noor M. Khan
© Springer Science+Business Media New York 2013
Abstract Geometrical models are mostly used for the study and analysis of the characteristics of radio communication channels. In this paper, a three-dimensional semi-ellipsoidal scattering model is proposed for mobile-to-mobile communication channels, where uniformly distributed scatterers are assumed to be confined within the semi-ellipsoids around mobile stations. The semi-ellipsoidal shape with adjustable dimensions is considered to model the scattering phenomenon in urban streets and canyons. Using the proposed scattering model, a closed-form expression for the joint probability density function of the Angle-of-Arrival in azimuth and elevation planes of the incoming multipath signals is derived at each mobile station. Moreover, various observations are made, which show the impact of scatterers’ elevation and streets’ orientation on the spatial characteristics of mobile-to-mobile communication channel. Keywords Angle-of-arrival · Channel modeling · 3D Semi-ellipsoidal · Geometric modeling · Scattering · Mobile-to-mobile communications
1 Introduction Mobile-to-mobile (M2M) communication is playing pivotal role in most of the emerging wireless communication systems like wireless sensor networks (WSN), vehicular Ad hoc networks (VANET) and intelligent transportation systems (ITS). In M2M environment, both transmitter and receiver are in motion and communicate directly with each other whereas, in fixed-to-mobile communications, mobile stations (MS) communicate with each other via fixed base station. Antenna elevations are low in M2M communication environment and are M. Riaz (B) · N. M. Khan Acme Center for Research in Wireless Communications (ARWiC), Mohammad Ali Jinnah University, Islamabad, Pakistan e-mail:
[email protected] S. J. Nawaz Department of Electrical Engineering, COMSATS Institute of Information Technology, Islamabad, Pakistan
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usually surrounded by scatterers like buildings and trees resulting in a typical non-Line-ofSight (NLoS) environment. Since, in future, a large number of applications are thought to use M2M communications, so in order to ensure a better quality of service, issues regarding data reception over fast fading channels should be pointed out and resolved appropriately. Out of numerous issues, yet one is to model M2M channel accurately. Various channel models have been proposed categorically in the literature from single-input single-output (SISO) to multiple-inputs multiple-outputs (MIMO) [1,9,10,13,15,17,19], one ring to two and three ring scattering models [1,10,13,16–18], and two-dimensional (2D) to three-dimensional (3D) scattering models [3,4,12,14,16,18–20] Nevertheless, still there is a space for a generalized channel model which would be applicable to any realistic M2M propagation environment. In [1], a M2M channel model, applicable to SISO case, is proposed. The closed-form expressions (i.e., analytically representation in relation to finite number of parameters and functions) for the spatio-temporal characteristics of the M2M channel are derived. In [15], time autocorrelation function and Doppler spectrum are derived assuming wide-sense stationary channel in 3D multipath scattering environment. Akki and Haber [1] original work on M2M SISO model was extended in [10] by proposing a two-ring M2M MIMO model. The authors showed that 3D function can be expressed as a product of 2D space-time correlation functions. In [16], correlated double-ring scattering model is proposed where the authors after developing Line-of-Sight (LoS) sum-of-sinusoid model for M2M Ricean fading channel by taking LoS into account between the MSs. The authors also derived relations for level crossing rate and average fade duration. A MIMO M2M Rayleigh fading channel is proposed in [19]. Closed-form expression for joint space-time correlation function and power density spectrum are derived assuming a 2D isotropic scattering environment. A geometrical two-erose-ring model for MIMO M2M communication channels is presented in [17]. The authors assumed that the scatterers are only present not at the boundaries of the circles, rather at various distances from the center of the circles within circular region. They derived the channel diffused components of the link from transmit antenna elements to the receive antenna elements for the proposed model. Taking these diffused channel components, closed-form expressions for the correlation functions and power density spectrum are derived in [13]. Baltzis [2] in presented a 2D single-bounce elliptical model for spatial characteristics of M2M radio channels. They derived closed-form expressions for the probability density function (pdf) of Angle-of-Arrival (AoA) at the MSs by assuming uniform scatterers within adjustable hollow ellipses around the MSs. Time-of-Arrival (ToA) and AoA statistics are investigated in [12] by assuming uniform distribution of scatterers in circular regions around the MSs. In [11], a single-bounce geometrical channel model for M2M communications is considered and pdfs for ToA and AoA are derived assuming uniform distribution of scatterers in an annular strip around each MS. A simplified scattering model for M2M is presented in [3], where scatterers are assumed to confine in elliptical regions. Using this model, closed-form expression for pdf of AoA and integral formula for cumulative distribution function (cdf) of ToA are derived. Three-dimensional (3D) physical models are more realistic than 2D models in M2M communication environment because of low antenna elevations of the both ends of the communication link, compared to the high rise structures around them. Some notable 3D models in the literature are discussed here. A geometrical concentric-cylinders model for time-varying wideband MIMO M2M channel is introduced in [20]. The authors while considering a 3D scattering propagation environment, have showed that many correlation functions are the special cases of the their model. A 3D spherical SISO model for M2M is introduced in [14]. The authors assumed that MSs are located inside scatterer-free spheres and scatterers are only present outside the spheres. Using the spherical model, the authors derived a 3D temporal correlation expression. A 3D spheroid model is presented in [5] for fixed-to-mobile com-
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Fig. 1 Proposed 3D semi-ellipsoidal model for mobile-to-mobile communication environment
munication environment. The authors provided closed-form expressions for pdfs of AoA for azimuth and elevation planes. In [8], a generalized 3D scattering model is proposed for macrocellular communication systems, where the authors derived closed-form expressions for joint and marginal pdfs of the AoA and ToA in the azimuth and elevation planes for a 3D spheroid model with a low MS antenna and an elevated Base Station (BS) antenna. Although 3D modeling is the most realistic to model a low antenna-height M2M communication environment, however, it is more realizable if a flexible geometrical shape is considered in such a way that it can model this low-antenna height M2M link more practically and appropriately with respect to its surrounding high-rise scattering objects. It can thus be said that out of the above mentioned approaches, cylindrical model [20] and spherical model [14] are better techniques than 2D M2M models as far as the realistic 3D M2M scattering environment is concerned; however, still these models seem to be unable to model streets and canyons in urban environment where usually M2M communication takes place. Nevertheless, if the geometry of the streets and canyons is considered, cylindrical shape [20] with scatterers inside or spherical shape [14] with scatterers outside its boundary can not be used to model them; rather a semi-ellipsoid with adjustable dimensions can be utilized to model them more appropriately. In this paper, we propose a geometrically-based 3D semi-ellipsoidal scattering model for mobil-to-mobile communication channels. Scatterers are uniformly distributed in the semiellipsoids. Due to low antenna heights, local scatterers around the MSs will contribute to the AoA in elevation plane as well as in azimuth plane. Exploiting the proposed model, closedform expression for the joint pdf of AoA is derived. The rest of the paper is organized as follows: Sect. 2 describes the proposed 3D semi-ellipsoidal model. In Sect. 3, joint pdf of AoA is derived. Results for the proposed model are discussed in Sect. 4. Finally, conclusion of the paper is provided in Sect. 5.
2 System Model In this section, we present the geometrically-based 3D semi-ellipsoid scattering model for M2M communication channels, depicted in Fig. 1. The scatterers around both the MSs are assumed to be confined within semi-ellipsoids centered at MSs. Both MSs are equipped with omnidirectional antennas residing at an equal height (low height antennas). The distance between both MSi (i = 1, 2 for MS1 and MS2 , respectively) is d. The semi-ellipsoids surrounding the MSs can independently be rotated around their z-axes with a certain angle
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θm i with respect to the LoS component. Some common assumptions are used in [2,5,7,8] to design the proposed model are listed as follows, 1. Each MS is surrounded by uniformly distributed scatterers confined within a semiellipsoid. 2. The power incident on a certain scattering object is reflected with an equal distribution of power in all directions. 3. The signals received at a MS are assumed to arrive with an equal strength from all directions in horizontal and vertical planes. 4. The communication between the MSs is assumed to take place via single isotropic scattering object. 5. All the scatterers have uniform random phases and equal scattering coefficients. Consider that the MSs MS1 and MS2 are located at points (0, 0, 0) and (d, 0, 0), respectively, in the Cartesian coordinates system. The dimensions of both the semi-ellipsoids can independently be set in all the directions. The dimensions of semi-ellipsoid around MSi (i = 1, 2) along with its x, y, and z-axes are denoted by ami , bmi , and cmi , respectively. The angles of the arriving multipath waves in azimuth and elevation planes are shown by φmi and βmi , respectively. The distance of a certain scatterer from MSi is denoted by rmi . The semi-ellipsoid centered at a certain point (X oi , Yoi , Z oi ) in cartesian coordinates system, can be expressed as ((xmi − X oi ) cos θmi + (ymi − Yoi ) sin θmi )2 2 ami +
(− (xmi − X oi ) sin θmi + (ymi − Yoi ) cos θmi )2 2 bmi
+
(z mi − Z oi )2 =1 2 cmi
(1)
Relations for the transformations between cartesian and spherical coordinates systems are, xm1 = rm1 cos φm 1 cos βm 1 , ym1 = rm1 cos βm 1 sin φm 1 , and z m1 = rm1 sin βm 1 xm2 = xm1 + d, ym2 = ym1 , and z m2 = z m1 Volume V of all the scattering region can be obtained by adding the volumes of both the semi-ellipsoids, which can be written in simplified form as, V =
2 π (am1 bm1 cm1 + am2 bm2 cm2 ) 3
(2)
The scattering region contributing towards the arrival of signals at the receiver is divided into two partitions, viz: P1 and P2 , as shown in Fig. 2. The partition P1 consists of those directions in which the scatterers of only one semi-ellipsoid (around MS1 ) contribute towards the arrival of signals; whereas, the P2 consists of those directions in which both the semiellipsoids contribute in the arrival of signals at receiver. The azimuth threshold angles, φt1,m1 and φt2,m1 , shown in Fig. 2, are defined to separate different partitions of the scattering region. These azimuth threshold angles can be obtained in the simplified form as,
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Fig. 2 Scattering regions in the proposed model when observing angle-of-arrival at MS1
⎧ ⎧ ⎨ ⎪ 1 ⎪ ⎪ ⎪ arctan ⎪ ⎪ ⎩ c2 d 2 − Ω a 2 cos2 θ + b2 sin2 θ ⎪ ⎪ m2 m2 m2 m2 m2 ⎪ ⎪ ⎪ ⎪ × Ω a 2 − b2 ⎪ ⎪ cos θm2 sin θm2 + ⎨ m2 m2 φt1,m1 = − 2 2 2 2 ⎪ φt2,m1 ⎪ ⎪ ± bm2 cm2 d Ω cos θm2 ⎪ ⎪ ⎪ 1/2
⎪ c ⎪ 2 d 2 c2 Ω sin2 θ 2 b2 Ω 2 ⎪ ⎪ ; βm1 < arctan m2 +am2 − a m2 ⎪ m2 m2 m2 d ⎪ ⎪ ⎪ ⎩ 0 ; otherwise
(3) where Ω = − Similarly, the elevation threshold angle, βt,m1 , shown in Fig. 2, is computed to separate among the above mentioned two partitions in elevation plane. This threshold angle is computed as a function of azimuth AoA. βt,m1 is the angle between the ground azimuth plane and the the line joining the MS1 and the tangent point (i.e., Tφ ) at the semi-ellipsoid around MS2 . After some algebraic manipulations, the simplified solution for elevation threshold angle βt,m1 can be expressed as follows, ⎧
⎪ 2bφ ⎨ arctan ; φt2,m1 βm1 =0◦ ≤ φm1 ≤ φt1,m1 βm1 =0◦ 2 2 (4) βt,m1 = (ρφ1 +ρφ2 ) −4aφ ⎪ ⎩ 0 ; otherwise 2 cm2
d 2 tan2 βm1 .
where, aφ and bφ are the major and minor axes of the ellipse seen in vertical plane formed within the semi-ellipsoid around MS2 for a certain azimuth angle φm1 (see Fig. 3), which can be derived as, 2 2 2 cos2 (θ 2 2 am2 bm2 bm2 m2 − φm1 ) + am2 sin (φm1 − θm2 ) − d sin φm1 aφ = (5) 2 cos2 (θ 2 2 bm2 m2 − φm1 ) + am2 sin (θm2 − φm1 ) 1 2 2 b2 c2 − b2 c2 Γ 2 cos2 (θ 2 2 2 am2 bφ = m2 + θ R ) − am2 cm2 Γ sin (θm2 + θ R ) m2 m2 m2 m2 am2 bm2 (6) where
Γ =
d2 +
2 1 ρφ1 + ρφ2 − d ρφ1 + ρφ2 cos φm1 4
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Fig. 3 Cross-section view of the scattering semi-ellipsoid region around MS2
and
θ R = arcsin
ρφ1 + ρφ2 sin φm1 2Γ
A horizontal line from MS1 intersects the elliptical base of the semi-ellipsoid around MS2 at points u and z. The distances from MS1 to points u and z are represented by ρφ1 and ρφ2 , respectively. After doing tedious mathematical simplifications, these distances can thus be obtained as, ⎧ −1 φt2,m1 βm1 =0◦ ⎪ ⎪ ; ⎪ ⎪ 2 2 2 2 ⎪ ≤ φm1 ≤ φt1,m1 βm1 =0◦ ⎪ am2 sin (φm1 − θm2 ) + bm2 cos (φm1 − θm2 ) ⎪ ⎪ ⎪ ⎨ + 2 sin(φ 2 ρφ1 × d am2 m1 − θm2 ) sin θm2 − bm2 cos(φm1 − θm2 ) cos θm2 = −
⎪ ρφ2 ⎪ ⎪ ⎪ 2 sin2 (φ −θ )+b2 cos2 (φ −θ )−d 2 sin2 φ ⎪ b a ±a m2 m2 m1 m2 m1 m2 m1 ⎪ m2 m2 ⎪ ⎪ ⎪ ⎩ 0 ; otherwise
(7) The azimuth threshold angles φt1,m1 and φt2,m1 at MS1 , which are functions of βm1 are plotted in Fig. 4a. The elevation threshold angle βt,m1 is a function of φm1 and plotted in Fig. 4b. These threshold angles are used to define the angular limits for the partitions P1 and P2 (see Fig. 2), which are as follows, ⎧ ⎫ ⎨ φt1,m1 ≤ φm1 ≤ φt2,m1 ⎬ or (8) → P1 ⎩ ⎭ βt1,m1 ≤ βm1 ⎧ ⎫ ⎨ φt2,m1 < φm1 < φt1,m1 ⎬ or → P2 (9) ⎩ ⎭ βm1 < βt1,m1
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10
φt2, m1
βt [degrees]
φt [degrees]
12
φt1, m1
10 0 -10 -20 -30
8 6 4 2
0
10
20
30
0
-50
βm1 [degrees]
0
50
φm1 [degrees]
(a)
(b)
Fig. 4 Threshold angles, a azimuth threshold angles, φt,m1 , w.r.t. elevation angle, βm1 , b elevation threshold angle, βt , w.r.t. azimuth angle, φm1 , (am1 = 30 m, bm1 = 20 m, cm1 = 15 m, am2 = 55 m, bm2 = 30 m, cm2 = 20 m, θm2 = 90◦ , and d = 100 m)
where, the symbol “→” implies to a name for a partition based on certain conditions. The upper limit on the distance of scatterers from MS1 in a particular direction can thus be obtained as follows, rm1,max =
; P1 r1 r1 + ρβ2 − ρβ1 ; P2
(10)
The distance of MS1 from a certain scatterer at the boundary of scattering semispheroid around MS1 , can be expressed as r1 =
am1 bm1 cm1 1 2 2 2 2 2 2 2 2 2 2 cm1 cos βm1 am1 +bm1 + bm1 − am1 cos (2θm1 −2φm1 ) +2am1 bm1 sin βm1 (11)
The distances ρβ1 and ρβ2 are the distances of MS1 from the points p and q (Fig. 3), respectively. These distances can be obtained as, ρβ+ 1 = ρβ− 2
⎧ ρ +ρ sec β φ1 φ2 ⎪ sec βm1 − 2 2 m12 ⎪ ⎪ 2 2(bφ +aφ tan βm1 ) ⎪ ⎨
2 (ρ + ρ ) tan2 β 2 b2 4b2 + 4a 2 − (ρ + ρ )2 tan2 β × a ± a ⎪ φ1 φ2 m1 φ1 φ2 m1 φ φ φ φ φ ⎪ ⎪ ⎪ ⎩ 0
; P2
(12) ; P1
3 Spatial Statistics of Radio Links In this section, we derive a mathematical expression for the joint pdf of AoA seen at MS1 through a radial distance rm . The joint pdf of AoA, which is a function of radial distance rm , azimuth angle φm1 , and elevation angle βm1 , can be found as,
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p(rm1 , φm1 , βm1 ) =
f (xm1 , ym1 , z m1 ) |J (xm1 , ym1 , z m1 )| xm 1 = rm 1 cos βm 1 cos φm 1 ym 1 = rm 1 cos βm 1 sin φm 1 z m 1 = rm 1 sin βm 1
The Jacobean transformation in (13) can be written as, −1 ∂x m1 ∂ xm1 ∂ xm1 ∂rm1 ∂φm1 ∂βm1 ∂y 1 m1 ∂ ym1 ∂ ym1 J (xm1 , ym1 , z m1 ) = = 2 ∂rm1 ∂φm1 ∂βm1 rm1 cos βm1 ∂z m1 ∂z m1 ∂z m1 ∂rm1 ∂φm1 ∂βm1
(13)
(14)
It is assumed that scatterers are uniformly distributed within the semi-ellipsoids, and hence scatterer density function for partitions P1 or P2 can be expressed as, 1 ; (xm1 , ym1 , z m1 ) ∈ (P1 or P2 ) f (xm1 , ym1 , z m1 ) = V (15) 0 ; otherwise Substituting J (xm1 , ym1 , z m1 ) from (14) and f (xm1 , ym1 , z m1 ) from (15) in (13), the joint pdf is obtained as, 2 cos β rm1 m1 V Integrating (16) over rm1 for partitions P1 and P2 , we get the joint pdf as, r 3 cos β m1 1 ; P1 3V p(φm1 , βm1 ) = (r +ρ 3 β2 −ρβ1 ) cos βm1 1 ; P2 3V
p(rm1 , φm1 , βm1 ) =
(16)
(17)
Integrating (17) over βm1 , we obtain the marginal pdf of AoA with respect to azimuth plane i.e., p(φm1 ) as, βt,m1
p(φm1 ) =
π
p(φm1 , βm1 ) P2 dβm1 +
2
p(φm1 , βm1 ) P1 dβm1
(18)
βt,m1
0
To get the marginal PDF of AoA with respect to elevation plane i.e., p(βm1 ), integrating (17) over φm1 , we have, φt1,m1
p(βm1 ) =
p(φm1 , βm1 ) P2 dφm1 +
2π −φ t2,m1
p(φm1 , βm1 ) P1 dφm1
φt1,m1
0
2π +
p(φm1 , βm1 ) P2 dφm1
(19)
2π −φt2,m1
Expressions for the angular statistics observed at the other end of communication link (i.e., MS2 ) can also be obtained by following the similar procedure. The derived expressions for the angular statistics observed at MS1 can be used for obtaining the statistics observed at MS2 , by exchanging all the parameters of MS1 with MS2 (e.g., exchanging cm1 with cm2 etc).
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x 10
-4
p(φ
m1
,β
m1
)
4 3 2 1 0 80
100
60 0
40 -100
φm1 [degrees]
20 0
βm1 [degrees]
Fig. 5 Joint probability density function of AoA seen at MS1 , (am1 = 40 m, bm1 = 30 m, cm1 = 20 m, θm1 = 70◦ , am2 = 35m, bm2 = 30 m, cm2 = 25 m, θm2 = 60◦ , and d = 80 m)
4 Results and Discussion In this section, we discuss the obtained theoretical results along with the observations. The results are obtained for the channel parameters shown in the caption of each plot. The joint pdf of AoA with respect to azimuth and elevation planes given in (17) is plotted in 3D as shown in Fig. 5. To elaborate the effects, a 2D plot of joint pdf of AoA is also taken along with contour for different channel parameters is shown in Fig. 6. It is clear that joint pdf of AoA is higher at βm1 = 0◦ and φm1 = 0◦ because in this case the scattering radius is the largest in the LoS direction and a large number of scattering objects get chance to direct the radio signals over LoS or closer to it. The AoA pdf decreases as the semi-ellipsoid is rotated through some angle in azimuth plane. Similarly, AoA pdf decreases with the increase in elevation angle and becomes minimum at maximum elevation angle which is an obvious result that lesser number of signals arrive from large elevation. The signals received at the receiver from the elevation plane are usually observed to be spread over an angle of 20◦ [5], with more than 50 % of energy in the signals arriving from the elevation angles below 16◦ [6]. Marginal pdfs of AoA in azimuth for different values of θm1 is shown in Fig. 7a. It exhibits higher values in the direction of LoS and lower values for NLoS azimuth angles. However, as θm1 is increased from its value equal to zero, a slight decrease in the value of pdf in LoS direction is observed. This increases the rate of occurrence of the NLoS azimuth angles proportionally. Behavior of the marginal pdf of AoA curve in the azimuth for the azimuth angle other than those in the LoS or closer to it depends on the changes in the volume of the scattering region with the rotation of semi-ellipsoid1. It can be observed with the bumps or depressions in the curve on the way from 0◦ to 180◦ . Marginal pdf of AoA in elevation is shown in Fig. 7b. It shows same behavior as that of the pdf of AoA in azimuth for different values of θm1 . However, the curve shows a gradual decrease in its value at 0◦ elevation angle to its minimum value at 90◦ elevation angle. The curve shows no bumps or depressions because of its corresponding change with the change in the volume due to rotation of semi-ellipsoidal. Marginal pdfs of AoA in azimuth for different values of θm2 is shown in Fig. 8a. It can be observed from the figure that pdf of AoA is higher in the direction of LoS and lower for NLoS azimuth angles. As θm2 is increased from its lower value to higher, a significant
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m1
1
, βm )
0.014 0.012
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βm1 = 15 o
15
βm1 = 20 o
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0.01 0.008 0.006
φm1 [degrees]
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p(φ
25
βm1 = 0 o
0.018
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0.004
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100
150
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φm1 [degrees]
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25
30
βm1 [degrees]
(a)
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Fig. 6 Joint pdf of AoA at MS1 in, a azimuth plane for different elevation angles, b (am1 = 40 m, bm1 = 30 m, cm1 = 20 m, θm1 = 70◦ , am2 = 35 m, bm2 = 30 m, cm2 = 25 m, θm2 = 60◦ , and d = 80 m) 0.02 0.018
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θm1 = 60
0.014
θm1 = 0 o θm1 = 90 o
0.12
o
0.1
0.01
m1
0.012
)
θm1 = 90 o
p(β
p ( φm1 )
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θm1 = 0 o
0.008 0.006
0.08 0.06 0.04
0.004 0.02 0.002 0
-150 -100
-50
0
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φm1 [degrees]
(a)
100
150
0
0
20
40
60
80
βm1 [degrees]
(b)
Fig. 7 Marginal pdfs of AoA at MS1 for different values of θm1 in, a azimuth plane, b elevation plane, (am1 = 45 m, bm1 = 35 m, cm1 = 10 m, am2 = 60 m, bm2 = 40 m, cm2 = 20 m, θm2 = 60◦ , and d = 100 m)
decrease in the value of pdf of AoA in LoS direction is observed. The pdf of AoA curve shows a slight decrease for the azimuth angles other than those in the LoS has lower values for higher azimuth angles and vice versa. Marginal pdf of AoA w.r.t. the elevation AoA is plotted in Fig. 8b. It shows similar behavior as that of the pdf of AoA in azimuth for different values of rotational angle θm2 . However, the graph shows a gradual decrease in its value for 0◦ elevation angle to its minimum value at 90◦ elevation angle. Marginal pdf of AoA at MS1 in azimuth and elevation planes for different values of cm1 are plotted in Fig. 9a, b, respectively. The pdf of AoA in azimuth plane in LoS direction
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0.14
θm2 = 0 o θm2 = 30
0.025
o
p ( βm 1)
) m1
p(φ
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0.015
θm2 = 90 o
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θm2 = 60 o 0.02
θm2 = 0 o
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0.01 0.04 0.005
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0.02
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-50
0
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100
0
150
0
20
φm1 [degrees]
40
60
80
βm1 [degrees]
(a)
(b)
Fig. 8 Marginal pdfs of AoA at MS1 for different values of θm2 in, a azimuth plane, b elevation plane,(am1 = 45 m, bm1 = 35 m, cm1 = 10 m, θm1 = 0◦ , am2 = 60 m, bm2 = 40 m, cm2 = 20 m, and d = 100 m) 0.025
0.35 c
m1
=1 m
c
cm1 = 10 m 0.02
=1 m
0.3
cm1 = 20 m cm1 = 30 m
0.015
0.25
p( βm 1 )
p ( φm 1)
m1
cm1 = 30 m
0.01
0.2 0.15 0.1
0.005 0.05 0
0 -150 -100
-50
0
50
φm1 [degrees]
(a)
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150
0
20
40
60
80
βm1 [degrees]
(b)
Fig. 9 Marginal AoA pdf at MS1 for different values of cm1 in a azimuth plane b, elevation plane, (am1 = 45 m, bm1 = 35 m, θm1 = 30◦ , am2 = 60 m, bm2 = 40 m, cm2 = 20 m, θm2 = 20◦ , and d = 100 m)
is higher for low elevations of semi-ellipsoid around MS1 , while it has lower value in the NLoS directions and vice versa. It is because, the scattering objects in the region P1 around MS1 become lesser and lesser in number in the LoS direction with increasing elevations of semi-ellipsoid1 as compared to those present in the whole volume around MS1 . AoA pdf in elevation plane is considerably high for higher values of cm1 because scattering volume P2 in the LoS direction increases with an increase in the whole volume around both the MSs. Marginal pdf of AoA at MS1 in azimuth and elevation planes for different values of cm2 are plotted in Fig. 10a, b, respectively. It is shown that AoA pdf in azimuth plane increases in the LoS direction and decreases in the NLoS directions with an increase in cm2 , but the
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0.35 cm2 = 1 m
cm2 = 1 m
cm2 = 10 m cm2 = 20 m
0.015
cm2 = 30 m
0.25
p ( βm 1 )
p ( φm 1)
cm2 = 30 m
0.3
0.01
0.2 0.15 0.1
0.005
0.05 0
-150 -100
-50
0
50
100
150
0
0
20
40
60
φm1 [degrees]
βm1 [degrees]
(a)
(b)
80
Fig. 10 Marginal AoA pdf at MS1 for different values of cm2 in a azimuth plane, b elevation plane, (am1 = 45 m, bm1 = 35 m, cm1 = 30 m, θm1 = 30◦ , am2 = 60 m, bm2 = 40 m, θm2 = 0◦ , and d = 100 m)
(a)
(b)
Fig. 11 Comparison of marginal pdfs of azimuth AoA observed at MS1 for the proposed 3D model with 2D circular model [12] and 2D elliptical Model [3], a for cm1 = cm2 = 0.3 m, b for cm1 = cm2 = 10 m, (For 3D ellipsoidal and 2D elliptical models, am1 = 30m, bm1 = 20 m, θm1 = 20◦ , am2 = 20 m, bm2 = 15 m, θm2 = 45andd = 100 m), (For 2D circular model, Radii of scattering circular regions around MS1 and MS2 are taken as R1 = 30 m and R2 = 30 m, respectively)
effect of increasing cm2 on the pdf of AoA in elevation is more significant, as shown in Fig. 10b. In Fig. 11a, b, a comparison of the proposed 3D model with a 2D elliptical model [3] is presented, where the effect of increasing the number of scatterers in elevation plane can be observed. As we reduce the the radius of the scattering ellipsoidal in elevation axis (i.e.,
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reducing cm1 and cm2 ), the results for the pdf of azimuth AoA for the proposed 3D model approaches to those obtained in [3]. Moreover, a comparison of the proposed 3D model with a 2D circular model [12] is also presented in Fig. 11a, b. If we substitute equal values for major and minor axes of the scattering semi-ellipsoids along horizontal plane (i.e., a = b) and a very small value for the vertical axis c (i.e., c approaches to zero), the proposed ellipsoidal model deduces to the 2D circular model [12]. In view of the results shown in Figs. 9, 10 and 11, it is concluded that the elevation plane has a prominent role in modeling the spatial characteristics of M2M channels. It is, therefore, necessary to take elevation plane into account in modeling AoA pdf for the performance evaluation of M2M communication links in multipath fading environments.
5 Conclusion In this paper, we have developed a 3D semi-ellipsoidal scattering Model for M2M communication channels with MSs located at the center of each semi-ellipsoid. It has been assumed that scatterers are uniformly distributed around the MSs within the semi-ellipsoids. Using the proposed scattering channel model, a closed-form expression for the joint pdf of AoA has been derived at each MS for azimuth and elevation angles. Marginal pdfs of AoA for azimuth and elevation angles have also been observed for different scenarios of streets’ orientations and scatterers’ elevation. Moreover, theoretical results have been presented that show that the elevation plane has a prominent impact on the spatial characteristics of M2M channels. It is, therefore, necessary to take elevation plane into account in modeling AoA pdf for the performance evaluation of M2M communication links in multipath fading environment.
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Author Biographies Muhammad Riaz was born in Pakistan. He received his M.Sc. from Quaid-i-Azam University, Islamabad, Pakistan, in 2002 and M.S degree in electronic engineering from Mohammad Ali Jinnah University Islamabad, Pakistan in 2009. At present, he persues his Ph.D. degree in electronic engineering at the Mohammad Ali Jinnah University, Islamabad, Pakistan. His research interests include channel modeling, channel equalization, Mobile-to-Mobile Communications.
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3D Ellipsoidal Model Syed Junaid Nawaz received the B.Sc. and M.Sc. degrees in computer engineering from COMSATS Institute of Information Technology (CIIT), Abbottabad, Pakistan, in September 2005 and February 2008, respectively. He received the Ph.D. degree in Electronic Engineering from Mohammad Ali Jinnah University, Islamabad, Pakistan, in February 2012. He pursued a part of his Ph.D. research with Staffordshire University, Stafford, UK. He worked on several research positions with CIIT Abbottabad from September 2005 to January 2008. From February 2008 to August 2012, he worked in the Department of Electrical Engineering, Federal Urdu University of Arts Science and Technology, Islamabad, as an Assistant Professor. He worked as Head of the Electrical Engineering Department from April 2012 to August 2012 at Federal Urdu University of Arts Science and Technology Islamabad. In August 2012, he joined the Department of Electrical Engineering, CIIT, Islamabad as an Assistant Professor. His research interests include physical channel modeling, channel characterization, channel estimation and equalization, adaptive multiuser detection, smart antenna systems, and wireless sensor 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 an Associate Professor with the Mohammad Ali Jinnah University, Islamabad, Pakistan. As recognition of his research contributions, he has been awarded Research Productivity Award (RPA) by the Pakistan Council for Science and Technology (PCST) for the year 2011–2012. 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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