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Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2008, Article ID 868016, 7 pagesdoi:10.1155/2008/868016

Research ArticleA Semi-Stochastic Propagation Model for the Study ofBeam Tilting in Cellular Systems

Konstantinos B. Baltzis

RadioCommunications Laboratory, Section of Applied and Environmental Physics, Department of Physics,Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

Correspondence should be addressed to Konstantinos B. Baltzis, [email protected]

Received 30 March 2008; Revised 29 June 2008; Accepted 5 August 2008

Recommended by A. Panagopoulos

Base station antenna downtilt mitigates interference and improves the downlink performance of wireless systems. A semi-stochastic propagation model is presented and applied to the study of the impact of the base station beam tilting in cellularcommunications. The two-ray approximation of the proposed model is described analytically. Beam tilting is evaluated in relationto the base station antenna radiation pattern, the antennas height, the propagation environment, the bit error rate, and the signal-to-noise ratio at the receiver front end. Analytically derived expressions for the fading envelopes, the error probability, the optimumtilting angle, and the downlink capacity of a WCDMA system are derived. Theoretical analysis and simulation results are providedto show the characteristics of the model. Comparisons with data in the literature confirm its validity. Furthermore, the effect ofbeam tilting on system downlink performance in terms of bit error rate and capacity is investigated.

Copyright © 2008 Konstantinos B. Baltzis. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

The communication quality in the downlink of a mobilecommunication system is strongly affected by multipathdistortion [1]. Among the proposed techniques, downtiltingof the base station (BS) antenna gives good results providinga form of angle diversity [2, 3]. Maseng [4] analyzed thesystem bit rate in the mobile multipath channel for perfectreflection and omnidirectional BS antennas, showing that themaximum bit rate depends on the propagation environment,the communication range, and the complexity of the mobilestation (MS) receiver. In [5], the effect of beam tiltingwas investigated in view of the multipath distortion underperfect and diffusely scattered reflection. The downlink bitrate as a function of the BS antenna radiation pattern, theantennas height, the sensitivity of the mobile receiver, theexcess distance, and the characteristics of the propagationmedium was examined in [6, 7]. Furthermore, in [8–10],optimum tilting was treated in view of the BS antennacharacteristics, the cells sectorization, the site spacing, thenumber of signal paths, and the traffic volume showing thatoptimum tilting depends on various network parameters.Finally, in [11], an adaptive tilting method was proposedfor interference mitigation and capacity enhancement in

WCDMA systems, revealing the importance of beam tiltingin wireless communications.

In this paper, the effect of beam tilting in the downlinkperformance of a cellular system is investigated. A semi-stochastic propagation model that describes the mobileradio channel is established. The two-ray line-of-sight(LOS) approximation of the model is described extensively.Analytically derived expressions for the fading envelopes,the bit error rate (BER), the optimum tilting angle, andthe downlink capacity of a WCDMA system are obtained.The basic characteristics of the model are explained andits validity is confirmed. Furthermore, the issues of systemperformance and optimum tilting are addressed.

The rest of the paper is organized as follows. In Section 2,the system model is presented. In Section 3, the proposedpropagation models are described and the mathematicalformulation is given. Numerical examples, simulations, anddiscussions are provided in Section 4. Finally in Section 5,concluding remarks are drawn.

2. SYSTEM MODEL

In Figure 1 the model of a mobile system is illustrated. Thehorizontal distance between the BS and the MS is r, the

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2 International Journal of Antennas and Propagation

θ = 0◦

θ

θ0

θ′0r2i

r1i

r

MSBS

h

r1i + r2i = r + xi

Figure 1: Two-dimensional model of a mobile system with tiltedBS wave beam.

difference in height between their antennas (mentioned asBS antenna height for brevity) is h, and the additional pathlength (excess distance) of the ith reflected path relative tothe direct path is xi (it is xi = r1i + r2i− r, where r1i and r2i arethe horizontal distances of the ith path reflection point fromthe BS and the MS, resp.). The tilting angle of the BS antennaradiation pattern defined from an axis parallel to the azimuthplane θ0; θ′0 is the tilting angle defined from the LOS axis, thatis,

θ′0 = θ0 − tan−1

(h

r

). (1)

Without loss of generality, mechanical downtilting isassumed; further improvement can be achieved when BSantenna beam is electrically tilted due to improved interfer-ence suppression [12].

In the examples presented here, the elevation planeradiation pattern of the BS antenna contains csc θ terms andis represented (in natural units) by

G(θ) =

⎧⎪⎪⎨⎪⎪⎩

exp[− k1

(θ − θ0

)2], −π

2≤ θ < θ0 +

HP

2,

k2 csc(θ − θ0

), θ0 +

HP

2≤ θ <

π

2,

(2)

where HP is the half-power beamwidth of the radiationpattern, k1 = 2 ln 2/HP2, and k2 = sin(HP/2)/

√2.

3. MATHEMATICAL FORMULATION

In the wireless channel, the transmitted signal suffersmultiple reflections. It is assumed that there exist N − 1 echosignals with large amplitudes whose appearance influencethe total signal level received by the MS antenna. The echosignals that are sufficiently far away have significantly smalleramplitudes and are left out. Assigning to each ray probabilityPi, the effective BS radiation pattern (i.e., the pattern as seenat the receiver) is given [13] by

F(θ) =N−1∑i=0

(1 +

xir

)−α/2PiG

(θi)e j(2πxi/λ+φi), (3)

where

θi = tan−1(

h

r + xi

). (4)

The φi denotes the phase of the ith ray, and λ is thetransmitted signal wavelength. Variables Pi and φi are inde-pendent and identically distributed (i.i.d.) random variablesdependent on the propagation medium (i = 0 refers to theLOS ray). The path loss exponent α is a terrain-dependentfactor; its values range from 3 to 4.3 in urban areas and areclose to 3.9 and 4.4 in suburban and open (rural) space,respectively, [14]. The ratio

Y ≡ Y(θ) =def

∣∣F(θ)∣∣∣∣G(θ)∣∣ =

∣∣∣∣∣N−1∑i=0

PiXi(θ)e j(2πxi/λ+φi)

∣∣∣∣∣, (5)

where

Xi(θ) =(1 + xi/r

)−α/2G(θi)

G(θ), i = 0, 1, . . . ,N − 1, (6)

is a measure of the fade depth. Using the identity

(∑i

Ii

)2

+

(∑j

Qj

)2

=∑i

∑j

(IiI j +QiQj

)(7)

after some manipulation, (5) becomes

Y = rα/2

G(θ)

√A. (8)

where

A =N−1∑i=0

N−1∑j=0

{PiPj

G(θi)G(θ j)

[r2 +

(xi + xj

)r + xixj

]a/2

× cos(

2πxi − xjλ

+ φi − φj)}

.

(9)

The model’s accuracy increases significantly when asubstantial amount of data regarding terrain and antennagain characteristics is known for every possible ray path.However, a major drawback of the model is its complexity.A common approximation used for the multipath fadingchannel is a two-ray approximation; for example, see [15–20].

A two-ray model which assumes a LOS and a groundreflected propagation path between the transmitter andthe receiver will be presented now. Equal probabilities areassigned to both paths (P0 = P1 = 1). Let us considerX0(θ) = 1, φ0 = 0, x0 = 0 and set X ≡ X1(θ), x ≡ x1 − x0 =x1, and φ ≡ φ1 − φ0 = φ1. Then (8) is simplified into

Y =√

1 + X2 + 2X cos(

2πxλ

+ φ)

, (10)

where φ is uniformly distributed within [−π,π] randomvariable [21, 22].

Konstantinos B. Baltzis 3

The pdf of Y , fY (Y), is calculated (see Appendix A for afew details) by

fY (Y) =

⎧⎪⎪⎨⎪⎪⎩k

Y√4X2 − (1 + X2 − Y 2

)2, Y ∈ [Ymin,Ymax

],

0, otherwise.(11)

In this expression, k is a normalization factor intended toensure unity cumulative distribution function. The termsYmin and Ymax are

Ymin

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1− X , mod(x

λ

)∈[

0,12

],

√1 + X2 + 2X cos

(2πxλ

), mod

(x

λ

)∈(

12

,34

),

√1 + X2 − 2X cos

(2πxλ

), mod

(x

λ

)∈[

34

, 1)

,

Ymax

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

√1 + X2 + 2X cos

(2πxλ

), mod

(x

λ

)∈[

0,14

],

√1 + X2 − 2X cos

(2πxλ

), mod

(x

λ

)∈(

14

,12

),

1 + X , mod(x

λ

)∈[

12

, 1)

,

(12)

where mod(x/λ) is the remainder on dividing x by λ. Aftersome manipulation, we obtain that the expected value of Yis

E{Y} = 2π

(1 + X)[E(π(x

λ+

12

) ∣∣∣∣ 4X

(1 + X)2

)

− E(πx

λ

∣∣∣∣ 4X

(1 + X)2

)],

(13)

where E(φ|m) is the incomplete elliptic integral of the 2ndkind.

Using (10)–(13), approximate expressions for the errorprobability and the optimum tilting angle are derived(see Appendix B). Without loss of generality, the coherentantipodal BPSK modulation scheme is assumed [23]. In theproposed two-ray channel model when the BS beam is tiltedat an angle θ0, the error probability is

Pe|θ0∼= 1

2erfc

(E{Yθ0

}E{Gθ0 (θ)Gθ0=0(θ)

}√γ)

, (14)

where γ is the bit energy-to-noise power spectral densityat the MS receiver front end. The optimum tilting angle isfound by expecting a maximum error probability Pmax

e at anangle θedge = tan−1(h/d), where d is the sector dominancearea (obviously the maximum of the main lobe points at thedesired cell, i.e., θ0 > θedge). It is defined by the BS antenna

radiation pattern, the antennas height, the site spacing, themaximum acceptable bit error rate, the target signal-to-noiseratio at the MS receiver front end, and the propagationchannel characteristics, and is given by the solution of

Gθ0

(θedge

) = erfc−1(2Pmaxe

)E{Y}√γ Gθ0=0

(θedge

), (15)

where erfc−1(x) is the inverse complementary error function.In the case of the antenna radiation pattern of (2), theoptimum tilting angle is given by

θ0, opt∼= tan−1

(h

r

)

+

√√√√ HP2

2 ln 2ln(√

2γ

1 + (r/h)2 ·csc(HP/2)

erfc−1(2Pmaxe

)E{Y}

).

(16)

An approximate expression that describes the effect ofBS antenna tilting on the downlink capacity of a WCDMAcellular network in terms of total deliverable bit rate hasalso been obtained (see Appendix C). The main assumptionsare the equal downlink rate allocation to each of the Kactive users, the fixed channel bandwidth W , and the infiniteBS transmission power. The total deliverable downlink bitrate satisfying a specific bit error rate target is inverselyproportional to the target bit energy-to-noise power spectraldensity Γ at each user’s receiver front end [24–26]. Usersare uniformly distributed within the cells which are approxi-mated as circles with radius d. Each user is allocated a singleunique channelization code which is assumed orthogonalbetween users. To avoid any wastage of power, the bit errorrate is fixed at the maximum that the quality-of-service(QoS) requirements allow. Without loss of generality, thetarget signal-to-noise ratio at each user’s receiver front endand the BER QoS target are assumed equal for all users.Under these assumptions, the ratio between the maximumdeliverable bit rate Rθ0

max when the BS antenna radiationpattern is tilted at a θ0 angle and the corresponding rate withno tilting R0

max is given by

Rθ0max

R0max

∼= 4h4

d4

(∫ π/2cot−1(d/h)

Gθ0 (θ)Gθ0=0(θ)

· cos θ

sin3θdθ)2

, (17)

where [24–26]

R0max =

WK

Γν(K − 1), (18)

where ν is the loss orthogonality factor due to fading,which depends on the propagation environment and themobile’s speed and position [9]. It has to be mentionedthat BS antenna downtilt improves signal level within acell due to better ability to aim the beam towards theintended dominance area, and simultaneously decreasesinterference towards other cells [9]. An optimum downtiltangle is a tradeoff between intercell interference mitigationand coverage threshold. However, under the assumption ofunlimited BS transmission power, the cochannel interference


0 1 2 3 4 5

x (km)

0

0.5

1

1.5

2

Y

Ymax

Ymin

Figure 2: Fading envelopes versus excess distance (slow fading).

from neighbouring cells is negligible [25]. In this case,capacity improvement is caused by the increase in the signallevel only [27].

4. RESULTS AND DISCUSSIONS

In this section, the basic characteristics of the two-raypropagation channel model are examined. The impact ofBS antenna downtilt on system performance and the choiceof optimum tilting are investigated. Finally, the validity ofthe proposed model is verified by comparisons with data inthe literature. In the following examples, the BS is locatedat the center of its cell, consisting of several sectors. TheBS antenna radiation pattern is described by (2). If nototherwise specified, system parameters are r = 2 km, h =50 m, HP = 3◦, θ0 = 3◦, α = 3.4, and λ = 0.33 m.

Figures 2 and 3 depict the fading envelopes Ymax andYmin as a function of the excess distance. In Figure 2,x ranges from zero to 5 km. As x increases, the rangeof the values of Y reduces. In Figure 3, it lies within arange of four signal wavelengths. In this case, the fadingenvelopes are periodic functions repeated every wavelengthof the transmitted signal. In practice, the fading envelopesrepresent the envelopes of the received signal strength due toslow (Figure 2) and fast (Figure 3) fading [1, 23]. It has to benoticed that x depends on various system parameters, such assystem geometry, path loss exponent, BS antenna radiationpattern, and bit rate (e.g., [6, 7, 12]) and is proportionalto the excess delay of the channel. For the system underconsideration, the typical values of x range from half to 2 km.

The dependence of fY (Y) on the system parameters isanalyzed in Figure 4. The f 0

Y curve represents the examplecase previously mentioned (setting the excess distance equalto 1 km); in f 1

Y , f 2Y , and f 3

Y , it is r = 1 km, h = 25 m, andx = 500 m, respectively. Finally in f 4

Y and f 5Y , θ0 and HP

are doubled. The pdfs are almost symmetrical with respectto Y = 1. The pdf spread reduces as r and θ0 decrease, or

0 1 2 3 4

x/λ

0

0.5

1

1.5

2

Y

Ymax

Ymin

Figure 3: Fading envelopes versus normalized excess distance (fastfading).

0 0.5 1 1.5 2

Y

1

2

3

4f Y

(Y)

f 0Y

f 1Y

f 2Y

f 3Y

f 4Y

f 5Y

Figure 4: The pdf of Y for different system configurations.

as h, HP, and x increase, resulting in an improved systemperformance.

Let us consider the MS at the edge of the cell. Theerror probability for BPSK modulation and for various BSantenna downtilting angles (measured from the LOS axis) isshown in Figure 5 (Pe ≡ Pe|θ′0 ). Notice that beam downtiltingimproves system performance up to 3 dB. The optimumtilting angle, θ′0, opt, ranges from one to two degrees (the sameangle measured from an axis parallel to the azimuth plane,θ0, opt, (see Figure 1) lies between two and three degrees).When a Pmax

e = 10−6 is desired, a typical value for data packettransmission [23], and γdB = 10 dB, (16) gives an optimum


0 5 10 15 20

γdB (dB)

10−6

10−5

10−4

10−3

10−2

10−1

100

Pe

θ′0 = 0◦

θ′0 = 1◦

θ′0 = 2◦

θ′0 = 3◦

θ′0 = 4◦

θ′0 = 5◦

θ′0 = 6◦

Figure 5: System performance dependence of the BS beamdowntilting.

tilting angle θ0, opt = 2.68◦. Substituting this value in (1), ris the size of the cell dominance area [28], a θ′0, opt = 1.25◦

is found, in agreement with Figure 5. Notice that for tiltingangles up to a point, a mobile station at the edge of the cellis within the main lobe of the BS radiation pattern. However,when the BS antenna beam is further tilted, its main lobe maynot point to the desired user. As a result of great tilting anglessystem performance degrades.

Equation (16) shows that the optimum tilting angleincreases with BS antenna height and decreases with sitespacing, bit energy-to-noise power spectral density at the MSreceiver front end, and maximum acceptable BER. However,its dependence of the BS antenna half-power beamwidth ismore complicated. The optimum tilting angle increases withHP up to point. Differentiation of (16) with respect to HPshows that the optimum tilting angle is maximized when thehalf-power beamwidth of the BS antenna radiation pattern isthe solution of

HP cot(HP

2

)

− 4 ln(√

2γ

1 + (r/h)2 ·csc(HP/2)

erfc−1(2Pmaxe

)E{Y}

)= 0.

(19)

Further increase in half-power beamwidth reduces theoptimum tilting angle.

The validity of the proposed model was confirmed bycomparisons with data in the literature. In [9], Niemela etal. investigated the impact of mechanical antenna downtilton the performance of a macrocellular WCDMA net-work. In the simulations performed, a six-sectored con-figuration was assumed. The BS antenna heights were{25 m, 35 m, 45 m}. The BS antenna radiation pattern had a

0 2 4 6 8 10

θ0 (deg)

0

1

2

3

4

5

6

7

8

9

Cap

acit

yen

han

cem

ent

(d,h,HP) = (2000, 50, 3)(d,h,HP) = (1000, 50, 3)

(d,h,HP) = (2000, 25, 3)(d,h,HP) = (2000, 50, 6)

Figure 6: Impact of BS antenna downtilting on the capacity of aWCDMA cellular system (distances are measured in metres; anglesare measured in degrees).

half-power beamwidth HP = 6◦ and showed similarities tothat of (2). The target downlink bit energy-to-noise ratio inthe speech services was 8 dB. Their studies have shown thatthe optimum tilting angles were θ0, opt(deg) = {4.9, 5.9, 7.0}and θ0, opt(deg) = {3.8, 4.8, 5.9} for site spacing 1.5 km and2 km, respectively. In order to compare these results withthe ones derived from the proposed model, r was set as halfthe site spacing. An error probability of 10−3 is consideredadequate for speech services [23]. In order to calculateE{Y}, Monte Carlo simulations have been performed. Theoptimum tilting angles predicted by the two-ray model areθ0, opt(deg) = {4.5, 6.1, 7.4} and θ0, opt(deg) = {3.3, 4.8, 6.0}for r = 750 m and r = 1 km, respectively, close to theones reported in [9]. Similar conclusions are obtained fromcomparisons with results in [10]. For example, let us considera three-sectored cell, r = 500 m, and BS antenna half-power beamwidth and height HP = 10.2◦ and h = 50 m,respectively. In the case of speech services a θ0, opt = 12◦ wasreported. Setting γdB = 8 dB and Pmax

e = 10−3, the solutionof (16) gives θ0, opt = 12.4◦.

In Figure 6, the capacity enhancement of a macrocellularWCDMA system in terms of the total downlink bit rateis presented. The users are uniformly distributed withinthe cell and are assigned equal bit rates (the rest of theassumptions are mentioned in the previous section). A sig-nificant improvement in the maximum deliverable downlinkbit rate is observed when BS antenna radiation pattern isdowntilted. Notice that capacity increases when the radiationpattern is tilted up to point. Further tilting degrades systemperformance (a similar behavior has been observed in thestudy of bit error rate). Significant capacity enhancementin sites with smaller dominance area is observed. In thiscase, the optimum tilting angle is higher. Similar results arederived as the BS antenna height increases. Finally, increase


in half-power beamwidth degrades system performance. Thederived conclusions are similar to the ones derived in [9, 28].

5. CONCLUSIONS

In this paper, the impact of BS beam tilting on the downlinkperformance of a cellular system has been investigated.A semi-stochastic propagation model has been proposed.The two-ray model’s approximation has been extensivelydescribed. Approximate expressions for the fading envelopes,the error probability and the optimum tilting angle of acellular system, and the downlink capacity enhancement ofa WCDMA wireless network have been derived. Results haveexhibited the characteristics of the model and treated thechoice of the optimum tilting angle considering the basestation to mobile user posistion, the antennas height, the sitedominance area, the BS antenna radiation pattern, the excessdistance, the signal energy, and the bit error rate. Simulationshave shown the strong impact of tiling in the performanceof a wireless system and the critical role of tilting angle.Comparisons with data in the literature exhibited the validityof the model. The proposed model is a low complexity andadequate tool for wireless networks performance study andoptimization.

APPENDICES

A. THE PDF OF A FUNCTION OF A RANDOM VARIABLE

Suppose that z is a random variable and g(z) is a function ofa real variable z. The expression y = g(z) is a new randomvariable. In order to find the pdf fy(y) for a specific y, wesolve the equation y = g(z). Denoting its N real roots by zn,that is, y = g(z1) = g(z2) = · · · = g(zN ), the pdf fy(y) isgiven [29] by

fy(y) =N∑n=1

fz(zn)

∣∣g′(zn)∣∣ , (A.1)

where fz(z) is the pdf of z and g′(z) is the derivative of g(z)

B. ERROR PROBABILITY AND OPTIMUM TILTINGANGLE EXPRESSIONS

In this appendix, the approximate expressions for the errorprobability and the optimum tilting angle are derived. Theerror probability for coherent antipodal BPSK modulationover an AWGN channel with no fading is [23]

P0e =

12

erfc(√γ) = 1

2erfc

⎛⎝√√√ E0

b

N0

⎞⎠ , (B.1)

where E0b is the bit energy and N0 is the noise power spectral

density at the MS receiver front end. In the proposed two-raychannel model, if the BS beam is not tilted, the bit energy atan angle θ is approximately (see (5))

Eb ∼= E0b·E{Fθ0=0(θ)Gθ0=0(θ)

}2

= E0b·E{Yθ0=0

}2, (B.2)

where E{·} denotes the expected value. When the BS beamis tilted at an angle θ0, using (5) the previous expressionbecomes

Eb ∼= E0b·E{

Fθ0 (θ)Gθ0=0(θ)

}2

= E0b·E{Yθ0Gθ0 (θ)Gθ0=0(θ)

}2

=⇒ Eb ∼= E0b·E{Yθ0

}2·E{Gθ0 (θ)Gθ0=0(θ)

}2

.

(B.3)

Considering that

Pe|θ0= 1

2erfc

(√EbN0

)(B.4)

and using (B.1) and (B.3), (14) is derived.Estimation of the tilting angle that optimizes system

performance is a tradeoff between other cell interferencemitigation and coverage thresholds, and depends on eachnetwork demands. Its value is affected by various factorssuch as the system geometry, the BS antenna radiationpattern, the propagation medium, the link bit rate, and thecells sectorization. Intuitively, the required downtilt angleincreases with the BS antenna height. Correspondingly, a cellwith a small dominance area requires a large downtilt angle.However, the system geometry is not enough to define theoptimum downtilt angle. As it has already been mentioned,in the present proposal the optimum tilting angle is foundby expecting a maximum error probability at the edge ofthe cell. Notice also that the expectation of Y describes thepropagation medium and does not depend on beam tilting.As a result, it holds that E{Yθ0} = E{Y}. Based on theprevious analysis and using (14), it easily comes that theoptimum tilting angle is found from the solution of (15). Inthe case of the antenna radiation pattern of (2), it holds that

Gθ0=0(θedge

) = 1√2

sin(HP

2

)csc θedge,

Gθ0

(θedge

) = exp[− 2 ln 2HP2

(θedge − θ0

)2].

(B.5)

Substituting (B.5) in (15), (16) is easily derived.

C. DOWNLINK THROUGHPUT ENHANCEMENT BYBEAM TILTING

In this appendix, the formulation of the downlink capacityimprovement due to beam tilting in a WCDMA cellularnetwork in terms of total deliverable bit rate under theassumptions already mentioned in Section 3 is presented. Forfixed W and N , the total deliverable bit rate is proportionalto the bit energy at each user’s receiver front end whenthe signal-to-noise ratio QoS target is also fixed [24, 25].Therefore, (B.3) gives that the ratio between Rθ0

max and R0max is

Rθ0max

R0max

∼= E{Gθ0 (θ)Gθ0=0(θ)

}2

(C.1)


(it is reminded that E{Yθ0} = E{Y}). As a result of theuniform distribution of the users within the circular cell, thepdf of their distance ξ from the BS is [30]

fξ(ξ) = 2ξd2

, ξ ∈ [0,d], (C.2)

where ξ = h cot θ, θ ∈ [cot−1(d/h),π/2], (d is the radius ofthe cell).

Using (A.1), the pdf fθ(θ) can be found. After somemanipulation comes that

fθ(θ) = 2h2 cos θ

d2sin3θ. (C.3)

From (C.1) and (C.3), (17) is easily derived.

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