# A Semi-Stochastic Propagation Model for the Study of  Beam Tilting in Cellular Systems

Authors: Konstantinos B. Baltzis
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

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 cellular communications. The two-ray approximation of the proposed model is described analytically. Beam tilting is evaluated in relation to 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 optimum tilting angle, and the downlink capacity of a WCDMA system are derived. Theoretical analysis and simulation results are provided to show the characteristics of the model. Comparisons with data in the literature confirm its validity. Furthermore, the effect of beam tilting on system downlink performance in terms of bit error rate and capacity is investigated.


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## Body

## 1. Introduction

The
communication quality in the downlink of a mobile communication system is
strongly affected by multipath distortion [1]. Among the proposed techniques, downtilting
of the base station (BS) antenna gives good results providing a form of angle
diversity  [2, 3]. Maseng [4] analyzed the system bit rate in
the mobile multipath channel for perfect reflection and omnidirectional BS
antennas, showing that the maximum bit rate depends on the propagation
environment, the communication range, and the complexity of the mobile station
(MS) receiver. In [5], the effect of beam tilting was investigated in view of
the multipath distortion under perfect and diffusely scattered reflection. The downlink
bit rate as a function of the BS antenna radiation pattern, the antennas height,
the sensitivity of the mobile receiver, the excess distance, and the
characteristics of the propagation medium was examined in [6, 7]. Furthermore,
in [8–10], optimum tilting
was treated in view of the BS antenna characteristics, the cells sectorization,
the site spacing, the number of signal paths, and the traffic volume showing that
optimum tilting depends on various network parameters. Finally, in [11], an adaptive
tilting method was proposed for interference mitigation and capacity enhancement
in WCDMA systems, revealing the importance of beam tilting in wireless
communications.In this paper, the effect of beam tilting in the
downlink performance of a cellular system is investigated. A semi-stochastic
propagation model that describes the mobile radio 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, and the downlink
capacity of a WCDMA system are obtained. The basic characteristics of the model
are explained and its validity is confirmed. Furthermore, the issues of system
performance and optimum tilting are addressed.The rest of the paper is organized as follows. In
Section2, the system model is presented. In Section 3, the proposed
propagation models are described and the mathematical formulation is given.
Numerical examples, simulations, and discussions are provided in Section 4.
Finally in Section 5, concluding remarks are drawn.


## 2. System Model

In
Figure1 the model of a mobile system is illustrated. The horizontal distance between the BS and the MS is r,
the difference in height between their antennas (mentioned as BS antenna height
for brevity) is h,
and the additional path length (excess distance) of the ith reflected path relative to the
direct path is xi (it is xi=r1i+r2i−r, where r1i and r2i are the horizontal distances of the ith path reflection point from the BS
and the MS, resp.). The tilting angle of the BS antenna radiation pattern
defined from an axis parallel to the azimuth plane θ0; θ0′ is the tilting angle defined from the LOS axis,
that is,
(1)θ0'=θ0−tan−1(hr). Without  loss of generality, mechanical downtilting is
assumed; further improvement can be achieved when BS antenna beam
is electrically tilted due to improved interference suppression  [12].Figure 1
Two-dimensional
model of a mobile system with tilted BS wave beam.In  the examples presented here, the elevation
plane radiation pattern of the BS antenna containscscθ terms and is represented (in natural units) by (2)G(θ)={exp[−k1(θ−θ0)2],−π2≤θ<θ0+HP2,k2csc(θ−θ0),θ0+HP2≤θ<π2, where HP is the half-power beamwidth of the radiation pattern, k1=2ln2/HP2, and k2=sin(HP/2)/2.


## 3. Mathematical Formulation

In the wireless channel, the transmitted signal suffers
multiple reflections. It is assumed that there existN−1 echo signals with large amplitudes whose
appearance influence the total signal level received by the MS antenna. The
echo signals that are sufficiently far away have significantly smaller
amplitudes and are left out. Assigning to each ray probability Pi,
the effective BS radiation pattern (i.e., the pattern as seen at the
receiver) is given [13] by (3)F(θ)=∑i=0N−1(1+xir)−α/2PiG(θi)ej(2πxi/λ+ϕi), where (4)θi=tan−1(hr+xi). The ϕi denotes the phase of the ith ray, and λ is the transmitted signal wavelength. Variables Pi and ϕi are independent and identically distributed
(i.i.d.) random variables dependent on the propagation medium (i=0 refers to the LOS ray). The path loss exponent α is a terrain-dependent factor;
its values range from 3 to 4.3 in urban areas and are close to 3.9 and 4.4 in
suburban and open (rural) space, respectively, [14]. The ratio (5)Y≡Y(θ)=def|F(θ)||G(θ)|=|∑i=0N−1PiXi(θ)ej(2πxi/λ+ϕi)|, where (6)Xi(θ)=(1+xi/r)−α/2G(θi)G(θ),i=0,1,…,N−1, is a measure of the fade depth. Using the identity (7)(∑iIi)2+(∑jQj)2=∑i∑j(IiIj+QiQj) after some manipulation, (5)
becomes
(8)Y=rα/2G(θ)A.
where
(9)A=∑i=0N−1∑j=0N−1{PiPjG(θi)G(θj)[r2+(xi+xj)r+xixj]a/2×cos(2πxi−xjλ+ϕi−ϕj)}.The model’s accuracy increases significantly when
a substantial amount of data regarding terrain and antenna gain 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  fading channel is a two-ray approximation; for example,
see [15–20].A two-ray model which assumes a LOS and a ground
reflected propagation path between the transmitter and the receiver will be
presented now. Equal probabilities are assigned to both paths (P0=P1=1). Let us consider X0(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 (10)Y=1+X2+2Xcos(2πxλ+ϕ), where ϕ is uniformly distributed within [−π,π] random variable  [21, 22].The pdf ofY, fY(Y),
is calculated (see Appendix A for a few details) by (11)fY(Y)={kY4X2−(1+X2−Y2)2,Y∈[Ymin,Ymax],0,otherwise. In this expression, k is a normalization factor intended to ensure
unity cumulative distribution function. The terms Ymin and Ymax are (12)Ymin={1−X,mod(xλ)∈[0,12],1+X2+2Xcos(2πxλ),mod(xλ)∈(12,34),1+X2−2Xcos(2πxλ),mod(xλ)∈[34,1),Ymax={1+X2+2Xcos(2πxλ),mod(xλ)∈[0,14],1+X2−2Xcos(2πxλ),mod(xλ)∈(14,12),1+X,mod(xλ)∈[12,1), where mod(x/λ) is the remainder on dividing x by λ.
After some manipulation, we obtain that the expected value of Y is (13)E{Y}=2π(1+X)[E(π(xλ+12)|4X(1+X)2)−E(πxλ|4X(1+X)2)], where E(ϕ|m) is the incomplete elliptic integral of the 2nd kind.Using (10)–(13), approximate expressions
for the error probability and the optimum tilting angle are derived (see
Appendix B). Without
loss of generality, the coherent antipodal BPSK modulation scheme is assumed
[23]. In the proposed two-ray channel model when the BS beam is tilted at an
angle θ0, the error probability is (14)Pe|θ0≅12erfc(E{Yθ0}E{Gθ0(θ)Gθ0=0(θ)}γ), where γ is the bit energy-to-noise power spectral density at the MS
receiver front end. The optimum tilting angle is found by expecting a maximum
error probability Pemax at an angle θedge=tan−1(h/d), where d is the sector dominance area (obviously the
maximum of the main lobe points at the desired cell, i.e., θ0>θedge). It is defined by the BS antenna radiation
pattern, the antennas height, the site spacing, the maximum acceptable bit
error rate, the target signal-to-noise ratio
at the MS receiver front end, and the propagation channel characteristics, and
is given by the solution of (15)Gθ0(θedge)=erfc−1(2Pemax)E{Y}γGθ0=0(θedge), where erfc−1(x) is the inverse complementary error function. In
the case of the antenna radiation pattern of (2),
the optimum tilting angle is given by (16)θ0,opt≅tan−1(hr)+HP22ln2ln(2γ1+(r/h)2⋅csc(HP/2)erfc−1(2Pemax)E{Y}).An approximate expression that describes the
effect of BS antenna tilting on the downlink capacity of a WCDMA cellular
network in terms of total deliverable bit rate has also been obtained (see
AppendixC). The main assumptions are the equal downlink rate allocation to
each of the K active users, the fixed channel bandwidth W,
and the infinite BS transmission power. The total deliverable downlink bit rate
satisfying a specific bit error rate target is inversely proportional to the target
bit energy-to-noise
power spectral density Γ at each user’s receiver front end [24–26]. Users
are uniformly distributed within the cells which are approximated as circles
with radius d.
Each user is allocated a single unique channelization code which is assumed
orthogonal between users. To avoid any wastage of power, the bit error rate is
fixed at the maximum that the quality-of-service (QoS) requirements allow. Without
loss of generality, the target signal-to-noise ratio at each user’s receiver
front end and the BER QoS target are assumed equal for all users. Under these
assumptions, the ratio between the maximum deliverable bit rate Rmaxθ0 when the BS antenna radiation pattern is
tilted at a θ0 angle and the corresponding rate with no
tilting Rmax0 is given by (17)Rmaxθ0Rmax0≅4h4d4(∫cot−1(d/h)π/2Gθ0(θ)Gθ0=0(θ)⋅cosθsin3θdθ)2, where [24–26] (18)Rmax0=WKΓν(K−1), where ν is the loss orthogonality factor due to
fading, which depends on the propagation environment and the mobile’s speed and
position [9]. It has to be mentioned that BS antenna downtilt improves
signal level within a cell due to better ability to aim the beam towards the
intended dominance area, and simultaneously decreases interference towards
other cells [9]. An optimum downtilt angle is a tradeoff between intercell
interference mitigation and coverage threshold. However, under the assumption
of unlimited BS transmission power, the cochannel interference from
neighbouring cells is negligible [25]. In this case, capacity improvement is
caused by the increase in the signal level only [27].


## 4. Results and Discussions

In this section, the basic characteristics of the two-ray propagation
channel model are examined. The impact of BS antenna downtilt on system
performance and the choice of optimum tilting are investigated. Finally, the
validity of the proposed model is verified by comparisons with data in the
literature. In the following examples, the BS is located at the center of its
cell, consisting of several sectors. The BS antenna radiation pattern is
described by (2). If
not otherwise specified, system parameters are r=2 km, h=50 m, HP=3°, θ0=3°, α=3.4,
and λ=0.33 m.Figures2 and 3 depict the fading
envelopes Ymax and Ymin as a function of the excess distance. In
Figure 2, x ranges from zero to 5 km. As x increases, the  range of the values of Y reduces. In Figure 3, it lies within a range
of four signal wavelengths. In this case, the fading
envelopes are periodic functions repeated every wavelength of the transmitted
signal. In practice, the fading envelopes represent the envelopes of the
received signal strength due to slow (Figure 2) and fast (Figure 3) fading [1, 23]. It has to be noticed that x depends on various system parameters,
such as system geometry, path loss exponent, BS antenna radiation pattern, and
bit rate (e.g., [6, 7, 12]) and is proportional to the excess delay of the
channel. For the system under consideration, the typical values of x range from half to 2 km.Figure 2
Fading
envelopes versus excess distance (slow fading).Figure 3
Fading
envelopes versus normalized excess distance (fast fading).The dependence offY(Y) on the system parameters is analyzed in Figure 4. The fY0 curve represents the example case previously
mentioned (setting the excess distance equal to 1 km); in fY1, fY2,
and fY3,
it is r=1 km, h=25 m,
and x=500 m,
respectively. Finally in fY4 and fY5, θ0 and HP are doubled. The pdfs are almost symmetrical with
respect to Y=1.
The pdf spread reduces as r and θ0 decrease, or as h, HP,
and x increase, resulting in an improved system
performance.Figure 4
The
pdf ofY for different system configurations.Let us consider the MS at the edge of the cell.
The error probability for BPSK modulation and  for various BS antenna downtilting angles
(measured from the LOS axis) is shown in Figure5 (Pe≡Pe|θ0'). Notice that beam downtilting improves
system performance up to 3 dB. The optimum tilting angle, θ0,opt′,
ranges from one to two degrees (the same angle measured from an axis parallel
to the azimuth plane, θ0,opt,
(see Figure 1) lies between two and three degrees). When a Pemax=10−6 is desired, a typical value for data packet
transmission  [23], and γdB=10dB,
(16)
gives an optimum tilting angle θ0,opt=2.68°.
Substituting this value in (1), r is the size of the cell dominance area [28],
a θ0,opt′=1.25° is found, in agreement with Figure 5. Notice
that for tilting angles up to a point, a mobile station at the edge of the cell
is within the main lobe of the BS radiation pattern. However, when the BS antenna
beam is further tilted, its main lobe may not point to the desired user. As a
result of great tilting angles system performance degrades.Figure 5
System performance dependence of the BS beam downtilting.Equation (16)
shows that the optimum tilting angle increases with BS antenna height and
decreases with site spacing, bit energy-to-noise power spectral density at the MS receiver front end,
and maximum acceptable BER. However, its dependence of the BS antenna half-power
beamwidth is more complicated. The optimum tilting angle increases with HP up to point. Differentiation of (16)
with respect to HP shows that the optimum tilting angle is
maximized when the half-power beamwidth of the BS antenna radiation pattern is
the solution of (19)HPcot(HP2)−4ln(2γ1+(r/h)2⋅csc(HP/2)erfc−1(2Pemax)E{Y})=0. Further increase in half-power
beamwidth reduces the optimum tilting angle.The validity of the proposed model was confirmed by
comparisons with data in the literature. In [9], Niemelä et al. investigated the
impact of mechanical antenna downtilt on the performance of a macrocellular WCDMA
network. In the simulations performed, a six-sectored
configuration was assumed. The BS antenna heights were {25m,35m,45m}.
The BS antenna radiation pattern had a half-power beamwidth HP=6° and showed similarities to that of (2).
The target downlink bit energy-to-noise ratio in the speech services was 8 dB.
Their studies have shown that the 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 and 2 km, respectively.
In order to compare these results with the ones derived from the proposed model, r was set as half the site spacing. An error
probability of 10−3 is considered adequate for speech services
[23]. In order to calculate E{Y}, Monte
Carlo
simulations have been performed. The optimum tilting
angles predicted by the two-ray model are θ0,opt(deg)={4.5,6.1,7.4} and θ0,opt(deg)={3.3,4.8,6.0} for r=750 m and r=1km, respectively, close to the ones reported in [9].
Similar conclusions are obtained from comparisons with results in [10]. For
example, let us consider a three-sectored cell, r=500m,
and BS antenna half-power beamwidth and height HP=10.2° and h=50m, respectively. In the case of speech services a θ0,opt=12° was reported. Setting γdB=8dB and Pemax=10−3,
the solution of (16)
gives θ0,opt=12.4°.In Figure6, the capacity enhancement of a
macrocellular WCDMA system in terms of the total downlink bit rate is presented.
The users are uniformly distributed within the cell and are assigned equal bit rates
(the rest of the assumptions are mentioned in the previous section). A
significant improvement in the maximum deliverable downlink bit rate is
observed when BS antenna radiation pattern is downtilted. Notice that capacity
increases when the radiation pattern is tilted up to point. Further tilting
degrades system performance (a similar behavior has been observed in the study
of bit error rate). Significant capacity enhancement in sites with smaller
dominance area is observed. In this case, the optimum tilting angle is higher.
Similar results are derived as the BS antenna height increases. Finally,
increase in half-power beamwidth degrades system performance. The derived
conclusions are similar to the ones derived in [9, 28].Figure 6
Impact
of BS antenna downtilting on the capacity of a WCDMA cellular system (distances
are measured in metres; angles are measured in degrees).


## 5. Conclusions

In this paper, the impact of BS beam tilting on the downlink performance
of a cellular system has been investigated. A semi-stochastic propagation model
has been proposed. The two-ray model’s approximation has been extensively described.
Approximate expressions for the fading envelopes, the error probability and the
optimum tilting angle of a cellular system, and the downlink capacity
enhancement of a WCDMA wireless network have been derived. Results have
exhibited the characteristics of the model and treated the choice of the
optimum tilting angle considering the base station to mobile user posistion, the
antennas height, the site dominance area, the BS antenna radiation pattern, the
excess distance, the signal energy, and the bit error rate. Simulations have
shown the strong impact of tiling in the performance of a wireless system and
the critical role of tilting angle. Comparisons with data in the literature
exhibited the validity of the model. The proposed model is a low complexity and
adequate tool for wireless networks performance study and optimization.



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