Effect of Directional Antennas at Both Ends of the Link on Spatial Characteristics of Cellular

and Mobile Channel Bilal Hasan Qureshi

Dept. of Electronic Engineering Muhammad Ali Jinnah University

Islamabad, Pakistan bilal_oct@yahoo.com

Saeed Iqbal Dept. of Electronic Engineering

Muhammad Ali Jinnah University Islamabad, Pakistan

saeedwattoo@gmail.com

Noor M. khan Dept. of Electronic Engineering

Muhammad Ali Jinnah University Islamabad, Pakistan

noor@ieee.org

Abstract - This paper presents the derivation of the probability density function (PDF) of angle of arrival (AoA) of multipaths at base station (BS) and mobile station (MS) while directional antennas are used at both ends of the communication link. A macrocell environment is modeled by assuming uniform and Gaussian distribution of scatterers around MS. Closed form expressions for PDF of AoA at BS and MS are obtained as a result of using directional antennas at both ends of the link. The behavior of the PDF of AoA at BS and MS is observed and plotted by changing the separation between BS and MS and in case of Gaussian scatter density the effect of changing the standard deviation is shown on the PDF of AoA at BS and MS.

I. INTRODUCTION

It is the need of hour to increase the capacity in cellular and mobile communication systems. To achieve this objective, resources of power and frequency have been utilizedefficiently so far with spectral signal processing techniques in the previous years. But less attention have been given to spatial aspects of the channel. Spectral signal processing techniques cannot meet the increasing demand of capacity. Therefore in recent years, more considerations have been paid to spatial domain parameter for example angle of arrival (AoA) [1]. To exploit the spatial domain parameter efficiently it is essential to have reliable understanding of radio propagation characteristics of transmission path between base station (BS) and mobile station (MS) that leads to the design of effective signal processing techniques [2]. To pursuit this concept, different physical channel models have been proposed in literature. Urban and Rural areas are modeled as microcell and macrocell environments [5], [7]. Indoor regions like shopping centers, big halls and offices are modeled as picocell environment [2].

In macrocells the distance between BS and MS is very large that ranges between 1Km to 10 Km [2]. Moreover the height of BS antenna is greater in macrocells than that of BS antenna in microcells. In macrocells BS antenna is not surrounded by scattering points but MS is. This is common in practice that the multipaths coming from distant scatterers

are less important than form those scatterers which are closer to MS. Due to this reason we assume that in macrocells scatterers are confined in a circle around MS. Apart from circular scattering model many authors proposed elliptical scattering model for physical channel modeling [5], [6]. In both the models either circular or elliptical only those multipaths are considered to carry significant information which experience a single bounce form scatterers because in multiple bounces signal power is attenuated rapidly [6].

To reduce the effect of interference between multipaths adaptive antennas with phase shift mechanism are proposed in literature. However to achieve this goal fixed beam directional antennas are also equally capable in this regard. The PDF of AoA at BS and MS have been found in [3] by using directional antenna at the BS where macrocellenvironment is modeled with the assumption that scatterers are confined uniformly in a circle around MS. A similar kind of work is done in [8] and [9] by using Gaussian scatter density around MS where PDF of AoA is found at BS and MS respectively while directional antenna is used at BS.

In this paper we propose the directional antennas at both ends of the link. We model a macrocell environment with the assumption that scatterers are confined in a circle around MS. Directional antennas are employed at BS and MS. Due to directional antennas only those scatterers are significant which are illuminated by the beamwidth of the BS and MS antennas respectively. Closed form formulas of PDF of AoA of multipaths at MS and BS have been found using uniform and Gaussian scatter density.

The rest of the paper is arranged as follows: System model for directional antennas at both ends of the link is described in section II. The derivation of PDF of AoA of multipaths at MS and BS using uniform scatter density is given in section III. Section IV presents the derivation of PDF of AoA of multipaths at MS and BS using Gaussian scatter density. Results and descriptions are shown in section V. Finally conclusions are made in section VI.

2009 International Conference on Emerging Technologies

978-1-4244-5632-1/09/$26.00 ©2009 IEEE 87

II. SYSTEM MODEL

In this section a macrocell environment is modeled using directional antennas at both ends of the link as shown in Fig. 1. The distance between BS and MS is D. The radius of the circle in which scatterers are confined is R. When a directional antenna of beamwidth α is used only at BS the scatterers present in the region JKEFGO would be illuminated. The length LMS is r and the angles θ1 and θ2

are the same as obtained in [3].

⎪⎪⎩

⎪⎪⎨

⎧

≤<

≤<+

≤<

=

πθθ

θθθαθθ

α

θθ

2

21 tancossin

tan10

;R

;D

;R

r (1)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

−+−= αααθ 2sin2

1cos2sin1cos1 R

D

R

D

(2)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

−−−= αααθ 2sin2

1cos2sin1cos2 R

D

R

D (3)

In addition to the directional antenna at BS, if another directional antenna of beamwidth β is used at MS then the scatterers present inside the region JKLMSNO would be illuminated as shown in Fig. 1

III. PDF OF AOA USING UNIFROM SCATTER DENSITY

The work in this section is presented in two parts. The PDF of AoA at MS is found in part A. Part B presents the PDF of AoA at BS.

A. PDF of AoA at MS

In this part we derive the PDF of AoA at MS if directional antennas are used at both ends of the link. The CDF of AoA at MS using uniform scatter density is given below.

∫+

−=

β

βθ θθ

2

m_unifrom2

1)( dr

AF ; -β < θ < β (4)

Where r is the radius of the circle in which scatterers are confined which is computed by Eq. 1 taking –β < θ < β. The area of the region JKLMSNO is Am_unifrom in which scatterers are illuminated due to directional antennas of beamwidth α and β. This area is actually twice the areas of the sector JKMS and the triangle KLMS shown in the Fig. 1

( )⎭⎬⎫

⎩⎨⎧

−+= 1sin 2

11

22

12m_unifrom θβθ rRRA (5)

Fig 1. Communication system if Directional Antennas are used at both ends of the link using uniform scatter Density.

In the above equation θ1 is computed by Eq. 2 and r is computed by Eq. 1 by taking θ = β because the angle β is such that θ1 < β < θ2. By substituting the values of r and θ1

Am_unifrom can be simplified as,

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛

−+−×

⎭⎬⎫

⎩⎨⎧

++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

−+=

−

−

αααβ

αββ

α

ααα

22

21

22

212

sin1cossincossin

tancossin

tan

sin1cossincosm_unifrom

R

D

R

D

DR

R

D

R

DRA

(6)

The PDF of AoA at MS is then obtained by differentiating Eq. 4 over θ. The parameter Ω used in Eq. 7 below is a normalizing factor such that the area under the curve is unity.

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤<⎭⎬⎫

⎩⎨⎧

+Ω

≤<Ω

=

|| ||; 2

tancossin tan

- ; 2

)(

1m_uniform

2

11m_uniform

2

βθθαθθ

α

θθθ

θθ

A

D

A

R

f

(7)

B. PDF of AoA at BS

The PDF of AoA at BS using uniform scatter density is found by taking the area of the strip of length KL and width ∆θ, whose scatterers are illuminated by the beamwidth of BS antenna with the truncation according to the directional antenna used at MS. The width ∆θ is infinitely small such 88

that the length LL’ and KK’ are more like a straight lines. The length of the strip KL is the difference of x1 and x2 as shown in the Fig. 1. Where x1 is the same as found in [3].

22221 coscos RDDDx +−−= αα (8)

Considering the triangle BMSL x2 can be solved easily as given below.

βcos2222 DrrDx −+= (9)

Area of the strip KK’LL’ is Ab_uniform which is actually area of the rectangular region of length ( x2 - x1 ) and width ∆θ, which can be found as follows,

θαα

θθ

θ

β

dRDDD

DrrDA

⎥⎦

⎤+−+−

∆+

⎢⎣

⎡−+= ∫

222cos2cos

cos222b_uniform

(10)

In the above equation r is computed by Eq. 1. The CDF of the AoA at BS using uniform scatter density is found as under.

∫+

−=

α

αθ θ

πθ 2

b_uniform

2)( d

R

AF ; -α < θ < α (11)

Where πR2 is the area of the circle. Finally PDF of AoA at BS if directional antennas are used at both ends of the link is found by differentiating Eq. 11 over θ.

2

b_uniform

2)(

R

Af

πθθ = ; -α< θ < α (12)

Combining Eq. 10 and Eq. 12 the PDF of AoA at BS issimplified in Eq. 13. The parameter Ω used in Eq. 13 below is a normalizing factor such that the area under the curve is unity.

]2222

2/1

2

22

coscos

cos tancossin

tan 2

tancossin tan

2)(

RDDD

DD

DD

Rf

+−+−

⎭⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

⎢⎢⎣

⎡

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+Ω

=

θθ

βθββ

θ

θββ

θ

πθθ

; -α < θ < α (13)

Fig 2. Communication system if Directional Antennas are used at both ends of the link using Gaussian scatter Density.

IV. PDF OF AOA USING GAUSSIAN SCATTER DENSITY

The work in this section is presented in two parts. The closed form expression for PDF of AoA at MS is found in part A. While part B presents the PDF of AoA at BS.

A. PDF of AoA at MS

The PDF of AoA at MS if directional antennas are used at both ends of the link is found by follow the same derivation of section III part. A, the only difference is that in Eq. 6 we are using r to take the length LMS in the range -β < θ < β. In this section we believe that this length dependents on Gaussian scatter density. Hence PDF of AoA using Gaussian scatter density is found by replacing r with 'r . New length

'r is the effective strength of the length LMS using Gaussian scatter density which is defined as,

dx

rx

'r ∫ ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

−=

0

2

2

1exp

2

14

σπσσ

(14)In Fig. 2 a circle with virtual boundary of radius R is shown around the Gaussian scatterers for simplifications in derivation. The term 4σ is the radius of the circular region in which scatterers are present. The significance of 4σ as the radius of the circle is that 99.9% of the scatterers are present within the circle of radius equal to 4σ. In rest of the equations of this section we would take 4σ as radius of the circle. In Eq. 14 substituting the value of r computed by Eq. 1 'r can be simplified as,

( )( )

⎪⎩

⎪⎨

⎧

≤<⎟⎟⎠

⎞⎜⎜⎝

⎛ +

≤<−

=|||

D'rβθθ

σ

αθασ

θθθσ

1| ; 2

sin cscerf2

11 ; 22 erf2

(15) 89

The effective area of the region JKLMSNO is Am_Gaussian in Gaussian distributed scatterers.

( )⎭⎬⎫

⎩⎨⎧

−+= )(sin')4(21

421

2 12

1m_Gaussian θβσσθ rA (16)

In above equation 'r is computed by Eq. 15 and θ1 is computed by Eq. 2. After substituting these values Am_Gaussian

can be simplified as,

( )

( )

( )

( ) ( )

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞−−

⎪⎩

⎪⎨⎧

⎜⎜

⎝

⎛−

−×

⎭⎬⎫

⎩⎨⎧ +

+

⎪⎭

⎪⎬⎫

−−

⎪⎩

⎪⎨⎧

−=

2

2sin21cos

2sin1cossin

2

sincscerf28

2

2sin21cos

2sin1cos216m_Gaussian

R

D

R

D

D

R

D

R

DA

αα

αβ

σ

αβασ

αα

ασ

(17)

The PDF of AoA at MS using Gaussian scatter density if directional antennas are used at both ends of the link is found as under.

2

m_Gaussain

)'(2

)( rA

fΩ

=θθ ; -β< θ < β (18)

Where 'r is computed by Eq. 15 and Am_Gaussian is computed by Eq. 17. The PDF of AoA is simplified in Eq. 19 which is shown at the bottom of the page. The parameter Ω used in Eq. 18 and Eq. 19 is a normalizing factor such that the area under the curve is unity.

B. PDF of AoA at BS

The PDF of AoA at BS by using Gaussian scatter density is found by follow the same derivation of section III part. Bwith a difference that the area of the strip KK’LL’dependents on Gaussian scatter density. We find r as under.

)12(4 xxr −−= σ (20)

In above equation substituting the values of x1 and x2 from Eq. 8 and Eq. 9 r can be simplified as,

βθββ

θ

θββ

θ

θθσ

cos tancossin

tan2

2

tancossin

tan2

2cos222cos4

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−

+−−+=

DD

DD

DDRDr

; -α < θ < α (21) The effective strength of the strip KL is ''r which is given below.

σσπσ

4

0

2

2

1exp

2

11

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

−−= ∫ dx

rx

''r (22)

( )( )

( ) ( ) ( ) ( )( )

( )

( ) ( )

( ) ( ) ( ) ( )( )

( )

|| || ;

sin 1cos

sin

cossin2

sin cscerf4

sin 1cos

sin cos8

2

sin cscerf

;

sin 1cos

sin

cossin2

sin csc4

sin 1cos

sin cos8

22erf

)(

1

2

22

2

122

22212

2

2

11

2

22

2

122

22212

22

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

≤<

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−−

−⎥⎦

⎤⎢⎣

⎡ ++

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

⎭⎬⎫

⎩⎨⎧

⎟⎠

⎞⎜⎝

⎛ +Ω

≤<−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−−

−⎥⎦

⎤⎢⎣

⎡ ++

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

Ω

=

−−

−−

βθθ

αα

α

βσ

αβασ

αα

ασ

σ

αθασ

θθθ

αα

α

βσ

αβασ

αα

ασ

σ

θθ

R

D

R

D

D

R

D

R

D

D

R

D

R

D

DErf

R

D

R

D

f

(19)

90

Where the value of r is taken from Eq. 21 ''r is simplified as,

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

σσ

2erf

2

114

r''r

; -α < θ < α (23)

The effective area of the strip kk’LL’ is Ab_Gaussian which is actually area of the rectangular region of length ''r and width ∆α which is found as under.

∫∆+

=

θθ

θ

θ b_Gaussian d''rA (24)

Finally the PDF of AoA at BS using Gaussian scatter density if directional antennas are used at both ends of the link can be found as,

2

b_Gaussian

)4(2)(

σπθθ

Af = ; -α < θ < α (25)

Where π(4σ)2 is area of the circle with virtual boundary 4σ. Combining Eq. 24 and Eq. 25 the PDF of AoA at BS using Gaussian scatter density is simplified in Eq. 26 below. The parameter Ω used in Eq. 26 is a normalizing factor such that the area under the curve is unity.

( )

⎥⎥⎦

⎤

⎭⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−

+−−

⎢⎣⎡

+Ω

−Ω

=

2/1

2

2

222

cos tancossin

tan2

tancossin tan

2

1

2

cos

2

cos22erf

168)(

βθββ

θ

θββ

θ

σ

σ

θ

σ

θ

πσπσθθ

DD

DD

DDR

Df

; -α < θ < α (26)

V. RESULTS AND DESCRIPTION

This section presents the results of the derivations in section III and section IV. The PDF of AoA at MS using uniform

and Gaussian scatter densities are given in Eq. 7 and Eq. 19 respectively. Fig. 3 and Fig. 5 shows the PDF of AoA at MS using uniform and Gaussian scatter density. In Fig. 3, α = 5,

β = 80ο, R = 400m, D = 2000m, D = 2500m, D = 3000m, D = 3500m, while in Fig. 5 α = 5, β = 80ο, R = 400m, D = 2000m, σ = 100m, σ = 200m, σ = 300m, σ = 400m are used. The results show that in case of uniform scatter density the PDF of AoA at MS becomes flat as distance between MS and BS increases as shown in Fig. 3 which means that the effect of directional antennas is reducing with an increase in the distance between BS and MS thus tending towardsClark’s model [10] with the truncation according to the beamwidth of directional antenna at MS. The same behavior can also be seen in case of Gaussian scatter density where PDF of AoA at MS becomes more and more flat as σ of Gaussian scatter distribution decreases. It is due to the fact that with decrease in σ of the distribution the scatterers tends towards compactness and hence the effect of directional antenna is negligible.

Fig. 4 and Fig. 6 show the plots of PDF of AoA at BS using uniform and Gaussian scatter densities as given in Eq. 13 and Eq. 26 respectively. In Fig. 4, α = 10 ο, β = 90ο, R = 400m, D = 1000m, D = 1500m, D = 2000m, while in Fig. 6, α = 10 ο, β = 90ο, R = 400m, D = 2000m, σ = 100m, σ = 120m, σ = 140m are used. The result show that the behavior of the PDF of AoA at BS can also be explained in the same manner as explained in the case at MS. The PDF of AoA at BS using uniform scatter density becomes flat as the distance between the MS and BS decreases with a truncation according to α as shown in Fig. 4. A reverse behavior is seen with an increase in the distance between MS and BS where the hump of the PDF curve rises. In case of Gaussian scatter density the PDF of AoA at BS becomes more and more flat as σ of the scatterers distribution increases with the truncation α as shown in Fig. 6.

91

-80 -60 -40 -20 0 20 40 60 802

3

4

5

6

7

8

9

10

11

12x 10

-3

Angle (Degrees)

PD

F o

f A

oA a

t M

S u

sing

Uni

form

Sca

tter

Den

sity

D = 2000

D = 2500 D = 3000

D = 3500

Fig. 3 PDF of AoA at MS for uniform scatter density with α = 5o, β = 80o,R = 400m and D = 2000m, D = 2500m, D = 3000m, D = 3500m

-10 -8 -6 -4 -2 0 2 4 6 8 104

4.5

5

5.5

6

6.5

7x 10

-3

Angle (Degrees)

PD

F o

f A

oA a

t B

S u

sing

Uni

form

Sca

tter

Den

sity

D = 1000

D = 1500D = 2000

Fig. 4 PDF of AoA at BS for uniform scatters density with α = 10o, β = 90o, R = 400m and D = 1000m, D = 1500m, D=2000m

-80 -60 -40 -20 0 20 40 60 800

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Angle (Degrees)

PD

F o

f A

oA a

t M

S u

sing

Gau

ssia

n S

catt

er D

ensi

ty

S.D = 100

S.D = 200S.D = 300

S.D = 400

Fig. 5 PDF of AoA at MS for Gaussian Scatter density with α = 5o, β = 80o, D=2000m,σ (S.D) =100m, σ =200m, σ =300m, σ =400m

-10 -8 -6 -4 -2 0 2 4 6 8 100.03

0.035

0.04

0.045

0.05

0.055

0.06

Angle (Degrees)

PD

F O

f A

oA a

t B

S u

sing

Gau

ssia

n S

catt

er D

ensi

ty S.D = 100

S.D = 120S.D = 140

Fig. 6 PDF of AoA at BS for Gaussian scatter density with α = 10o, β = 90o D = 2000m, σ (S.D) = 100m, σ = 120m, σ = 140m

VI. CONCLUSIONS

In this paper we have derived the close form expression for PDF of AoA of multipaths at BS and MS while directional antennas are used at both ends of the link. We modeled the macrocell mobile environment by assuming uniform and Gaussian distribution of scatters around MS. Four scenarios of the PDF of AoA at BS and MS using uniform and Gaussian scatter density have been explained. The results have been shown by changing the distance between BS and MS in the case of uniform scatter density while in case of Gaussian scatter density the effect of changing the σ has shown in the results.

REFERENCES

[1] A. J. Paulraj, D. A. Gore, R. U. Nabar and H. Bölcskei, “ An overview of MIMO communications: A key to Gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feburary 2004.

[2] N. M. Khan, M. T. Simsim and P. B. Rapajic “A Generalized model for the spatial characteristics of cellular mobile channel,” IEEE Trans. Veh. Technol., vol. 57, no. 1, January 2008.

[3] P. Petrus, J. H. Reed and T. S. Rappaport “Geometrical-based statistical macrocell channel model for mobile environments,” IEEE Trans. Commun., vol. 50, no. 3, pp. 495-502, March 2002.

[4] R. B Ertel and J. H Reed “Angle and time of arrival statistics for circular and elliptical scattering models,” IEEE J. Sel. Areas Commun., vol. 17, no. 11, pp. 1829-1840, November 1999.

[5] R. J. Piechocki, G. V. Tsoulos, and J. P. McGeehan, “Simple general formula for pdf of angle of arrival in large cell operational environments," IEE Electron. Lett., vol. 34, no. 18, pp. 1784-1785, September 1998.

[6] J. C. Liberti, T. S. Rappaport “A Geometrically based model for line of sight multipath radio channels,” in Proc. 46th IEEE Veh. Technol. Conf., pp. 844-848, April–May 1996.

[7] R. Janaswamy “Angle and time of arrival statistics for the Gaussian scatter density model,” IEEE Trans. Wireless Commun., vol. 1, no. 3, pp. 488-497, July 2002.

92

[8] M. P. Lotter and P. V. Rooyen “Modeling Spatial Aspects of Cellular CDMA / SDMA Systems,” IEEE Commun. Lett., vol. no. 3, no. 5, pp. 128-131, May 1999.

[9] C. A. Lopez, D. H. Covarrubias, D. Munoz and M. A. Panduro “Statistical cellular Gaussian scatter density channel model employing a directional antenna for mobile environments,” AEU-Int. J. Elect. and Commun., vol. no. 59, Issue. no. 3, pp.195-199, June 2005.

[10] R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst. Tech. J., vol. 47, no. 6, pp. 957–1000, July–August 1968.

93