utility fair optimization of antenna tilt angles in lte networks

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE/ACM TRANSACTIONS ON NETWORKING 1

Utility Fair Optimization of Antenna Tilt Angles inLTE Networks

Bahar Partov, Douglas J. Leith, Senior Member, IEEE, and Rouzbeh Razavi, Senior Member, IEEE

Abstract—We formulate adaptation of antenna tilt angle as autility fair optimization task. This optimization problem is non-convex, but in this paper we show that, under reasonable condi-tions, it can be reformulated as a convex optimization. Using thisinsight, we develop a lightweight method for finding the optimalantenna tilt angles, making use of measurements that are alreadyavailable at base stations, and suited to distributed implementa-tion.

Index Terms—Antenna tilt angle, Long Term Evolution (LTE),proportional fairness, maximizing capacity, optimization.

I. INTRODUCTION

T HE ANTENNA tilt angle of wireless base stations isknown to be a key factor in determining cell coverage and

to play a significant role in interference management [1], [2].While traditionally adjustment of tilt angle has largely beencarried out manually, modern base stations increasingly allowautomated adjustment. This creates the potential for moredynamic adaptation of tilt angle, for example to better matchcell coverage to the distribution of user equipment and traffic,to reduce coverage holes created by failures in neighboringstations, to better manage interference from the user deploy-ment of femtocells, etc. The benefits of self-configuration andself-optimization are already recognized in LTE release 9 [3],and automated adaptation of tilt angle in particular has been thesubject of recent interest.In this paper, we formulate adaptation of antenna tilt angle

as a utility fair optimization task. Namely, the objective is tojointly adjust antenna tilt angles within the cellular network soas to maximize user utility, subject to network constraints. Ad-justments at base stations must be carried out jointly in a co-ordinated manner in order to manage interference. This opti-mization problem is nonconvex, but in this paper we show that,under certain conditions, it can be reformulated as a convex opti-mization. Specifically, we show that: 1) in the high signal-to-in-terference-plus-noise ratio (SINR) operating regime and withan appropriate choice of decision variables, the optimization isconvex for any concave utility function; and 2) in any SINRregime, the optimization can be formulated in a convex manner

Manuscript received February 28, 2013; revised September 18, 2013;accepted November 25, 2013; approved by IEEE/ACM TRANSACTIONS ONNETWORKING Editor S. Puthenpura. This material is based upon work sup-ported by the Science Foundation Ireland under Grant No. 11/PI/1177 and BellLabs Ireland.B. Partov and D. J. Leith are with the Hamilton Institute, NUI Maynooth,

Maynooth, Ireland (e-mail: [email protected]).R. Razavi is with the Bell Laboratories, Alcatel-Lucent, Dublin, Ireland.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNET.2013.2294965

when the objective is a proportional fair rate allocation. Sincethe optimization is not well suited to solution using standarddual methods, we develop a primal-dual method for findingthe optimal antenna tilt angles. This approach is lightweight,making use of measurements that are already available at basestations, and suited to distributed implementation.The rest of the paper is organized as follows. In Section II, we

summarize the existing work in the area. In Section III, we in-troduce our network model, which is based on 3GPP standard,and in Section IV, we analyze its convexity properties in thehigh-SINR regime. In Section V, we extend the analysis to gen-eral SINR regimes. In Section VI, we carry out a performanceevaluation of a realistic setup, and finally, in Section VIII, wesummarize our conclusions.

II. RELATED WORK

The analysis and modeling of the impact of the antenna tiltangle on cell performance has been well studied; see, for ex-ample, [4], [5], and references therein. Recently, self-optimiza-tion of tilt angle has started to attract attention, but most ofthis work makes use of heuristic approaches. In [6], a heuristicmethod is proposed for adjusting tilt to maximize average spec-tral efficiency within the network, while [7] proposes a combi-nation of fuzzy and reinforcement learning. In [8], simulated an-nealing is considered for joint self-configuration of antenna tiltangle and power, and in [9], a noncooperative game approachbetween neighboring base stations is studied. Offline planningof tilt angle is considered, for example, in [10], using a heuristicsearch method combined with a mixed integer local search. Inthe present paper, we take a more formal, rigorous approach andshow that tilt angle optimization can, in fact, be formulated asa convex problem. Building on this result, we then introducea lightweight distributed algorithm based on primal-dual sub-gradient updates and show that this algorithm is guaranteed toconverge arbitrarily closely to the network optimum.

III. NETWORK MODEL

A. Network Architecture

The network consists of a set of base stations and a set ofuser equipment (UE), with UE receiving downlink traffictransmitted from base station . For base stations withsectoral antennas, we define a separate element in for eachantenna. We denote the geographical coordinates of basestation by and of user equipment by . Thedistance between user and base station is therefore given by

(1)

1063-6692 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


2 IEEE/ACM TRANSACTIONS ON NETWORKING

Fig. 1. Schematic illustrating relationship between antenna main lobe and tiltangle.

Fig. 2. Comparison of antenna main lobe vertical gain model (4) (dashed line)and measured antenna gain (solid line) for a Kathrein 742215 antenna,.

B. Antenna Gain and Path Loss

The received power on subcarrier from base stationat user is given by , where isthe base station antenna gain, is the path loss between andand is the base station transmit power for subcarrier .

For simplicity, shadowing and fast fading are not considered inthe equations. We model path loss, as recommended in [11], by

(2)

with fixed path-loss factor , path-loss exponent , and dis-tance in kilometers. For a given antenna type, the antennagain can be determined given the relative positions ofand , the antenna tilt angle , and the azimuth angle .

With regard to the latter, changing the tilt and/or azimuth angleschanges the direction of the antenna’s main lobe; see Fig. 1. Wewill assume that the azimuth angle is held fixed, but allow theantenna tilt angle to be adjusted within the interval . Fol-lowing [11], the antenna gain can then be modeled by

(3)

where is the maximum gain of the antenna

(4)

is the antenna vertical attenuation, isthe height difference between the base station and UE (which,for simplicity, we assume is the same for all base stations andusers), and is the vertical half-power beamwidth of theantenna. Fig. 2 illustrates the ability of (4) to accurately modelthe main lobe of an antenna that is popular in cellular networks.

Fig. 3. Illustrating linear approximation to.

It will prove useful to use the quantity. It will also prove useful to consider the

following linear approximation to antenna gainexponent about tilt angle :

(5)

This linear approximation is illustrated by the solid line in Fig. 3.It is reasonably accurate provided adaptation of the antennaangle does not cause to change sign (in which casethe side of the main antenna lobe facing the user changes, andso the slope of the linear approximation changes sign). This isassumed to be the case for the antennas of base stations otherthan that to which the UE is associated, which is only a mild as-sumption since otherwise interference from these base stationscan be expected to be excessive.

C. User Throughput

The downlink throughput of the user equipment asso-ciated with base station is given by

(6)

where is the vector of tilt angles, is the maximumachievable throughput (limited by the available modulation andcoding schemes), and

(7)

Here, is the number of subcarriers, the channel band-width, a loss factor capturing nonideal coding, etc., and

SINR on subcarrier for user

(8)

where is the received power frombase station by user is thereceived power from base station by user , and


PARTOV et al.: UTILITY FAIR OPTIMIZATION OF ANTENNA TILT ANGLES IN LTE NETWORKS 3

is the channel noise for user on subcarrier . Observe that inwe make use of linear approximation .

IV. HIGH-SINR REGIME

In the high-SINR regime, the downlink throughput (6) can beaccurately approximated by

(9)

where

(10)

A. Utility Fair Optimization of Tilt Angle

Under the assumption of high-SINR operation, we can for-mulate the selection of antenna tilt angles as the following opti-mization problem :

(11)

s.t. (12)

(13)

where is a concave increasing utility function and. Constraint (12) captures restrictions on the range of feasibleantenna tilt angles, while (13) ensures that each user receives aspecified minimum throughput (which is expected to mainly beimportant for users at the edge of a cell who might otherwise beassigned too low a throughput).

B. Convexity Properties

Lemma 1: is strictly concave in .Proof: We have

Now , and are constants, so we onlyneed to consider concavity with respect to . It can be verifiedthat .

Lemma 2: is convex.Proof: Rewrite as

. This can beexpressed as with

. By [12, p. 74], the log ofa sum of exponentials is convex. The additive term acts asa translation, and by [12, p. 79], convexity is preserved undertranslation. By approximation (5), the function isaffine in , and by [12, p. 79], when composed with the log ofa sum of exponentials, the resulting function remains convex.It follows from Lemmas 1 and 2 that we have the following

theorem.

Theorem 1: is concave in .Proof: Recalling from (8) and (10)

(14)

By Lemmas 1 and 2, is concave in . Since the func-tion is concave and nondecreasing, it follows that

is also concave in .Note that is not strictly concave in since

is only strictly concave in but not inthe other elements of . Nevertheless, under mild conditions,

is strictly concave in :Theorem 2: Suppose , i.e., every base stationhas at least one associated UE . Then,

is strictly concave in (and so the solution to problem isunique).

Proof: We have

Recall is strictly concave in (by Lemma 1).The sum is therefore strictly con-cave in every . It is thereforestrictly concave in (in more detail, for any , and

, we have

).The result then follows from the fact that the sum of a strictlyconcave function and a concave function is strictly concave.We have the following corollary.Corollary 1: When each base station has at least one user

with throughput less that , then is strictly con-cave in .

Proof: Let denote the set of users withthroughput less than . When each base station has at leastone user with throughput less that , then .Now is strictly concavein by Theorem 2. It then follows immediately that

is strictlyconcave in .

C. Convex Optimization

The objective function in optimization problem is con-cave in (since is concave increasing and is con-cave by Theorem 1, then is concave), and constraints(12) and (13) are linear (and so convex). Hence, optimizationproblem is convex. It follows immediately that a solution



exists. The Slater condition is satisfied, and so strong dualityholds.

D. Difficulty of Using Conventional Dual Algorithms

The Lagrangian is

(15)

where denotes the set of multiplers. The dual function is , where

. The main KKT conditions are. That is

(16)

Given , we can use (16) to find . The optimal vector ofmultipliers is . Since is concave, astandard dual-function approach is to find using subgradientascent techniques, and then find the optimal tilt angle .However, solving (16) to obtain the primal variables is tricky ingeneral since it imposes complex, implicit dual constraints for asolution to exist. Consequently, the dual subgradient approachis unattractive for solving problem .

E. Distributed Algorithm for Finding Optimal Solution

We consider the following primal-dual algorithm.

Algorithm 1: High SINR

Initialize: , step sizedo

(17)

(18)

(19)

loop

where, in Algorithm 1, projection equals when ,and 0 otherwise

(20)

(21)

(22)

(23)

and denotes any subgradient of withrespect to .

Observe that each iteration (17)–(19) of Algorithm 1 simulta-neously updates both the primal variable and the multipliers

. It possesses the following convergence property.Lemma 3: For Algorithm 1, suppose is bounded

for all . Then, there exists constant such that

where is a solution to optimization problem ,, and denotes the

usual Euclidean norm.Proof: Optimization problem is convex, the objec-

tive and constraint functions are differentiable, and the Slatercondition is satisfied. The result now follows by direct applica-tion of Lemma 8 in the Appendix.Since as , Lemma 3 tells us

that update (17)–(19) converges to a ball around an op-timum , the size of the ball decreasing with stepsize . The size of the ball is measured in terms of metric

, and recall that by complementaryslackness .

F. Message Passing and Implementation

Algorithm 1 can be implemented in a distributed manner.Namely, each base station carries out local tilt angle up-dates according to (17) and (19), and also carries out update (18)for each user associated with base station . For this, each basestation needs to evaluate (20)–(23). Evidently, (21)–(23) canbe evaluated using locally available information (the tilt angleof base station and the current downlink throughput of userassociated with base station ). In contrast, evaluating (20) re-quires information sharing between base stations. Specifically,it is necessary to evaluate

(24)

The first term in (24) is the sensitivity of the throughput of usersassociated to base station to changes in its tilt angle . Thiscan either be directly measured by base station (by perturbingthe tilt angle), or calculated using

(25)

where

(26)

This calculation requires knowledge of the pointing anglebetween base station and user . This pointing angle

can be determined from knowledge of the location of users,information that is usually available to modern base stationssince location-based services (LBS) are of high importance formobile network providers. For example, in the US, carriers arerequired by the FCC to provide location-based information ofthe mobile users for E911 services and to within a specified



Fig. 4. Example network topology. Base stations are indicated by solid squareslabeled 1–3, UEs by dots.

accuracy [13]. Within Release 9 of 3GPP, a set of enhancedpositioning methods is standardized for LTE [14].The second term in (24) is the sensitivity of the throughput of

users associated to base stations other than to changes in tiltangle . This can be calculated as

(27)

This requires user received power from base station ,user received power from the base station to which itis associated, the user SINR , and the pointing angle .All of this information is available to the base station to whichthe user is associated (via user equipment received power andSINR reports), but not to neighboring base stations, and so mustbe communicated to them.We note that antenna tilt angle updates are likely to occur on

a relatively long timescale in practice. Capturing hourly basedtraffic patterns of the mobile users may therefore also providerelatively reliable traffic distribution information, which mightalso be used.

G. Example

We illustrate the application of the foregoing high-SINR anal-ysis to the scenario shown in Fig. 4. We use a simple scenariohere to help gain insight, with a more realistic setup consideredin detail in Section VI. The scenario consists of regularly spacedbase stations each with three sector antennas. The base stationradio parameters are detailed in Table I based on 3GPP stan-dard [11]. The users are primarily located in two clusters, asindicated in Fig. 4. One cluster of 16 users is associated withthe first sector of base station 1, and the other cluster of 16 userswith the third sector of base station 2. Clustering of users cre-ates a challenging tilt angle assignment task since a poor choiceof tilt angles will have a strong effect on network performance.Additionally, two users are located close to the midpoint be-tween these base stations. Ensuring adequate coverage at celledges is commonly an issue for network operators, and so weexpect a performance tradeoff between serving these edge usersand serving users located in the clusters. For concreteness, weselect utility function , so that optimization problem

Fig. 5. Tilt angle versus iteration number with Algorithm 1.

Fig. 6. Normalized network sum-throughput versus iteration number withAlgorithm 1. Here, the network sum-throughput is normalized by dividing bythe number of users in the network.

TABLE ISIMULATION PARAMETERS

corresponds to maximizing the network sum-throughput,subject to every user obtaining aminimum throughput of 64 kb/sand to physical constraints that the allowable tilt angles must liein the interval [5 , 20 ].Fig. 5 shows tilt angle time histories for the two base sta-

tions when using Algorithm 1. It can be seen that the tilt an-gles converge to the optimum in less than 600 iterations. Fig. 6shows the corresponding network sum-throughput versus time.



Fig. 7. Normalized network sum-throughput as minimum throughput con-straint is varied. The network sum-throughput is normalized by dividing bythe number of users in the network.

Also shown is the network sum-throughput for fixed antenna an-gles of 8 . Optimizing the tilt angles increases the network sumthroughput by almost a factor of 18 compared to the use of fixedangles. As already noted, the improvement is expected to beparticularly pronounced in this simple example since the usersare grouped into clusters, and so angling the antennas to pointtoward their respective clusters both greatly increases receivedpower and decreases interference. We note that significant per-formance gains are, however, also observed in the more realisticscenario studied in Section VI, and this reflects the fundamentalimportance of antenna tilt angle to network performance.Fig. 7 illustrates the impact of the minimum throughput con-

straint on network sum-throughput. It can be seen that as isincreased from zero to 2 Mb/s, the network sum-throughput de-creases, but that the impact is minor. Note that as is increasedbeyond 2 Mb/s, the optimization becomes infeasible as the sta-tions at the cell edge are unable to support such high rates.As discussed in Section IV-F, UE location information is used

when calculating (20) in Algorithm 1. In practice, this locationinformation will be approximate in nature. Fig. 8 plots the op-timized network sum-throughput versus the standard deviationof the location error when zero-mean Gaussian noise is addedto the true user locations. It can be seen that, as might be ex-pected, the optimized sum-throughput falls as the noise level isincreased. However, the decrease is small (less than 5%) evenfor relatively large location errors.

V. ANY SINR: PROPORTIONAL FAIR RATE ALLOCATION

In this section, we relax the assumption of operation inthe high-SINR regime. However, this comes at the cost ofrestricting attention to proportional fair rate allocations. Weconsider the following utility fair optimization problem :

(28)

s.t. (29)

(30)

where is given by (6).

Fig. 8. Optimized network sum-throughput versus magnitude of location error.Error bars indicate the standard deviation for 100 runs of simulation for eachpoint. The network sum-throughput is normalized by dividing by the number ofusers in the network.

A. Convexity Properties

We recall the following.Lemma 4 [15]: is concave and

nondecreasing in .Turning now to , we begin by observing the following.Lemma 5: is concave in .Proof: From (7), we have

(31)

(32)

where . That is, the mapping fromvector to is the vector composition of inLemma 5 and . By Lemmas 1 and 2, is concavein . By [12, p. 86], the vector composition is a nondecreasingconcave function and a concave function is concave.Theorem 3: is concave in .Proof: From (6), we have

(33)

(34)

where follows from the fact that the function is monoton-ically increasing. By Lemma 5, is concave. Since the

function is concave nondecreasing, when composed with, it is concave i.e., is concave in .

B. Convex Optimization

It follows from Theorem 3 that the objective of optimiza-tion problem is concave. Constraints (29) are linear (soconvex). The right-hand side (RHS) of constraint (30) is con-cave, again by Theorem 3, and so this constraint is convex. Itfollows that optimization problem is convex and a solu-tion exists.



Fig. 9. (a) Comparing normalized sum-throughput from Fig. 6 and whensolving proportional fair allocation problem . As expected, the sumthroughput is lower for the proportional fair allocation. (b) Comparing nor-malized sum-log-throughput. As expected, the sum-log-throughput is higherfor the proportional fair allocation. In all cases, the network throughput isnormalized by dividing by the number of users in the network.

C. Distributed Algorithm

The Slater condition is satisfied, and strong duality holds. Wecan therefore apply a similar approach as in Section IV-E todevelop a distributed algorithm for finding the optimal antennatilt angles.The Lagrangian is

(35)

We can now apply Algorithm 1 to solve provided we usethe appropriate gradients

(36)

(37)

(38)

(39)

with (40), shown at the bottom of the page.

D. Example

We revisit the example in Section IV-G. Fig. 9 comparesthe results for optimization problems and . Fig. 9(a)shows the sum-throughput from Fig. 6 and when solvingproportional fair allocation problem . As expected, the

Fig. 10. User throughput assignments for (a) sum-throughput maximization inthe high-SINR regime and (b) proportional fair rate allocation.

sum throughput is lower for the proportional fair allocation.Fig. 9(b) compares the sum-log-throughput. As expected, thesum-log-throughput is higher for the proportional fair alloca-tion problem . Fig. 10 shows detail of the throughputsassigned to individual users to maximize sum-throughput andfor proportional fairness. It can be seen that the throughputassignments are broadly similar in both cases, with the primarydifference being the throughputs assigned to the two userslocated at the cell edge (numbered 33 and 34 in Fig. 10).The proportional fair allocation assigns significantly higherrate to these edge stations than does the max sum-throughputallocation.

VI. PERFORMANCE EVALUATION

In this section, we consider a realistic example based on datafrom the cellular network covering Grafton Street and DawsonStreet in downtown Dublin, Ireland; see Fig 11. These are majorshopping streets close to the center of Dublin city, with a largenumber of cellular users. We consider a section of the networkwith 21 sectors in a 1500 1500-m area and with an inter-site distance of 800 m. Environmental characteristics are de-rived from experimental measurement data with a combina-tion of non-line-of-sight and line-of-sight paths. Path loss andlog-normal shadow fading parameters are derived from [11] formacro urban scenarios and detailed in Table II. There are 1350users, with locations as shown in Fig. 11(b). We focus on theperformance experienced by the 388 users associated with thecenter base station [indicated by BS1 in Fig. 11(b)]. Fig. 12

and

(40)



Fig. 11. Dublin, Ireland example. (a) Street map. (b) User and base-stationlocations.

Fig. 12. Proportional fair rate allocation, Dublin example. For comparison, re-sults are also shown when a fixed tilt angle of 8 is used (indicated by dashedlines). (a) Normalized sum-log-throughput. (b) CDF of user throughputs.

TABLE IIDUBLIN SCENARIO SIMULATION PARAMETERS

shows the proportional fair rate allocation. For comparison, re-sults are also shown when a fixed tilt angle of 8 is used. It canbe seen from Fig. 12(a) that the sum-log-throughput objectivefunction is improved by 22% by tilt angle optimization, and thatAlgorithm 1 converges rapidly to the optimal allocation. Fromthe cumulative distribution function (CDF) in Fig. 12(b), it canbe seen the user throughputs are also significantly increased,with the median throughput increased by almost a factor of 4compared to use of fixed angles.

VII. LTE SIMO LINKS AND MMSE POST-PROCESSING

In this section, we extend the performance evaluation to con-sider LTE single-input–multiple-output (SIMO) links with onetransmit antenna on the BS and two receive antennas at theUE. The presence of two antennas at the receiver allows theUE to cancel one interferer. Hence, if interference is dominatedby a single transmitter, then we expect the use of SIMO linkswill allow intercell interference to be significantly reduced. Our

interest here is in the impact that this interference cancella-tion has on the size of throughput gain achievable by tilt angleadjustment.We consider an SIMO link with linear minimummean square

error (LMMSE) post-processing applied to the received signalto mitigate neighboring cell interference. Defining channelvector , the channel gain for user is

(41)

where is a zero mean Gaussian random variable representingslow fading effects, is a Rayleigh flat fading vector, andis the power of the transmitted signal assuming all base stationstransmit at . We can consider the elements ofto be independent complex random Gaussian processes corre-sponding to the channels of base station and user , providedthat the antenna elements are sufficiently separated (typically onthe order of half a wavelength apart). We identify the intercellinterference vector for user by the strongestinterferer

(42)

The remaining intercell interference is modeled as spatiallywhite Gaussian noise [16], which comprises the noise vector

where and are independent Gaussianvariables

(43)Hence, the received signal is given by

(44)

with . The linear MMSE combining vectoris given by

(45)

where is the autocorrelation of interference vector

(46)

By applying theMMSEweights on the received signal, the post-processing SINR is calculated as

(47)

We average the post-processing SINRs over the multipathfading samples. Using the averaged post-processing SINRs,user throughputs with and without tilt optimization can becalculated using (6).Fig. 13 shows the CDFs of the user throughputs for SIMO

links with MMSE detection, with and without flat fading.



Fig. 13. User throughput CDFs with SIMO links and MMSE detection, Dublinexample. Fast fading is modeled by generating 300 samples using the 3GPPtypical urban channel model, where the speeds of the mobile user and carrierfrequency are 3 km/h and 2 GHz, respectively. (a) With fast fading. (b) Withoutfast fading.

Fig. 14. Contribution of the strongest interference to the total interference.(a) Cumulative distributed function of in the central BS coverage area. (b) Dis-tribution of relative to the user positions.

As expected, the use of MMSE detection yields significantimprovements in the user throughputs. The throughput gainsachieved by tilt optimization can be compared for SISO linksand for SIMO links with an optimal LMMSE detector by com-paring Figs. 12(b) and 13. The gain in the mean user throughputachieved by tilt optimization is decreased from 83.07% to67.42% when MMSE detection is employed. However, thegain in the log-sum-rate (which is the objective function ofoptimization ) only changes from 22.29% to 22.00%. Thatis, while MMSE detection enhances intercell interference miti-gation, tilt optimization can still yield significant improvementsin network capacity.We can investigate this behavior in more detail as follows.

Let

(48)

be the ratio of the largest interferer to the total interferenceexperienced by a user . The CDF of for the Dublin exampleis shown in Fig. 14(a). It can be seen that approximately 40%of users have values less than 0.5, i.e., for 40% of users, thestrongest interferer power is less than the sum of the power ofthe other interferers. Fig. 14(b) shows the corresponding spatialdistribution of . It can be seen that the strongest interferer isdominant at the edge of antenna sectors and along the nullsof the sector antennas. However, the intensity of the strongestinterferer decreases along the edges of the base station cov-erage area and alongside the antennas. Table III details thethroughput gains achieved by tilt angle optimization for both

TABLE IIIUSER THROUGHPUT GAINS DUE TO TILT ANGLE OPTIMIZATION FOR BOTH

SISO AND SIMO LINKS AND VERSUS , DUBLIN EXAMPLE

single-input–single-output (SISO) and SIMO links and forusers with different ratios. It can be seen that the throughputgain obtained by tilt angle optimization for users withis reduced when SIMO links are used. However, the gain issimilar for both SISO and SIMO links for users with ,once MMSE post-processing is applied, and as noted abovethis consists of approximately 40% of users.In summary, although the mean user throughput is improved

for both fixed and optimal tilt angles for SIMO links withMMSE, tilt optimization can still yield considerable perfor-mance gains.

VIII. CONCLUSION AND FUTURE WORK

In this paper, we formulate adaptation of antenna tilt angleas a utility fair optimization task. Namely, the objective is tojointly adjust antenna tilt angles within the cellular networkso as to maximize user utility, subject to network constraints.Adjustments at base stations must be carried out jointly ina coordinated manner in order to manage interference. Thisoptimization problem is nonconvex, but we show that undercertain conditions, it can be reformulated as a convex op-timization. Specifically, we show that: 1) in the high-SINRoperating regime and with an appropriate choice of variables,the optimization is convex for any concave utility function; and2) in any SINR regime, the optimization can be formulated ina convex manner when the objective is a proportional fair rateallocation. Since the optimization is not well suited to use ofstandard dual methods, we develop a primal-dual method forfinding the optimal antenna tilt angles. This approach is light-weight, making use of measurements that are already availableat base stations, and suited to distributed implementation. Theeffectiveness of the proposed approach is demonstrated using anumber of simulation examples, including a realistic examplebased on the cellular network in Dublin, Ireland, and is foundto yield considerable performance gains.

APPENDIXOPTIMIZATION PROBLEM

Consider the optimization problem

s.t.

with convex,convex. For simplicity, we will assume that

are differentiable, but this could be relaxed. The optimiza-tion problem is convex, and so at least one solution exists; let

denote the set of solutions. Assuming the Slater condition



is satisfied, then strong duality holds, and the KKT conditionsare necessary and sufficient for optimality. The Lagrangian is

where is the multiplier associated with constraintand . At an optimum , the multi-pliers must lie in set .

A. Gradient Algorithm

We consider the following primal-dual update:

(49)

(50)

where step size and aresubgradients of with respect to and , respec-tively. We have and

with a subgradient ofwith respect to and a subgradient of with re-spect to . Projection when , 0 otherwise.

B. Fixed Points

Lemma 6 (Fixed Points): withis a fixed point of the dynamics (49) and (50).

Proof: From the KKT conditions, andso is a fixed point of (49). Since is feasible,

. We need to consider two cases:(i) , in which case is a fixed pointof (50); and (ii) , in which case by comple-mentary slackness , and this is also a fixed point of (50).Hence, every is a fixed point of the dynamics (49) and(50).

C. Convergence

Let. Observe that: 1) ; and 2)

if and only if and .Lemma 7: Under update (49) and (50)

where and

.Proof: From (49), for any , we have

(51)

where follows from the fact that(from the definition of a subgra-

dient). From (50), we have for any that

(52)

where follows from the observation that

Then, from (51) and (52)

Lemma 8: Under update (49) and (50), when(e.g., this holds when

is bounded and are continuous), wehave

where and .



Proof: By Lemma 7

Hence

For , recalland . Hence,

and . Therefore,. Substituting for

then yields the result.

ACKNOWLEDGMENT

The authors would like to thank Dr. H. Claussen andDr. D. Lopez-Perez of Bell Labs, Dublin, Ireland, for theirvaluable comments and suggestions.

REFERENCES[1] I. Forkel, A. Kemper, R. Pabst, and R. Hermans, “The effect of elec-

trical and mechanical antenna down-tilting in UMTS networks,” inProc. 3rd Int. Conf. 3G Mobile Commun. Technol., 2002, pp. 86–90,Conf. Publ. No. 489.

[2] F. Athley and M. Johansson, “Impact of electrical and mechanical an-tenna tilt on LTE downlink system performance,” in Proc. 71st IEEEVTC-Spring, 2010, pp. 1–5.

[3] LTE; Evolved Universal Terrestrial Radio Access (E-UTRA); Self-Con-figurating, Self-Optimizating Network (SON); Use Cases and Solu-tions, 3GPPTR 36.902 (Release 9), 2010.

[4] E. Benner and A. Sesay, “Effects of antenna height, antenna gain,and pattern downtilting for cellular mobile radio,” IEEE Trans. Veh.Technol., vol. 45, no. 2, pp. 217–224, May 1996.

[5] O. Yilmaz, S. Hamalainen, and J. Hamalainen, “Analysis of antennaparameter optimization space for 3GPP LTE,” inProc. 70th IEEE VTC-Fall, 2009, pp. 1–5.

[6] H. Eckhardt, S. Klein, and M. Gruber, “Vertical antenna tilt optimiza-tion for LTE base stations.,” in Proc. IEEE VTC-Spring, 2011, pp. 1–5.

[7] R. Razavi, S. Klein, and H. Claussen, “A fuzzy reinforcement learningapproach for self optimization of coverage in LTE networks,” Bell Lab.Tech. J., vol. 15, pp. 153–175, Dec. 2010.

[8] A. Temesváry, “Self-configuration of antenna tilt and power for plug& play deployed cellular networks,” in Proc. IEEE WCNC, 2009, pp.1–6.

[9] G. Calcev and M. Dillon, “Antenna tilt control in CDMA networks,”in Proc. 2nd Annu. ACM Int. Workshop Wireless Internet, 2006, p. 25.

[10] A. Eisenblatter and H. Geerdes, “Capacity optimization for UMTS:Bounds and benchmarks for interference reduction,” in Proc. 19thIEEE PIMRC, 2008, pp. 1–6.

[11] 3rd Generation Partnership Project; Technical Specification GroupRadio Access Network; Evolved Universal Terrestrial Radio Access(E-UTRA); Further Advancements for E-UTRA Physical Layer As-pects, (Release 9), 2010.

[12] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,U.K.: Cambridge Univ. Press, 2004.

[13] FCC, Washington, DC, USA, “Wireless E911 location accuracy re-quirements,” 2011 [Online]. Available: http://www.fcc.gov/document/wireless-e911-location-accuracy-requirements-1/

[14] Evolved Universal Terrestrial Radio Access (E-UTRA); LTE Posi-tioning Protocol (LPP), 3 GPP TS 36.355 (Release 9), 2010–02.

[15] J. Papandriopoulos, S. Dey, and J. Evans, “Optimal and distributedprotocols for cross-layer design of physical and transport layers inmanets,” IEEE/ACM Trans. Netw., vol. 16, no. 6, pp. 1392–1405, Dec.2008.

[16] Universal Mobile Telecommunications System (UMTS); SpatialChannel Model for Multiple Input Multiple Output (MIMO) Simula-tions, (Release 11), 2012.

Bahar Partov received the undergraduate degree inelectrical engineering from the University of Tabriz,Tabriz, Iran, in 2007, and the master’s degree intelecommunication and information systems fromthe University of Essex, Colchester, U.K., in 2009,and is currently pursuing the Ph.D. degree in networkmathematics at the Hamilton Institute, Maynooth,Ireland, together with Bell Labs Alcatel-LucentIreland.Her current research interests are distributed algo-

rithms for self-organized networks.

Douglas J. Leith (M’02–SM’09) graduated fromthe University of Glasgow, Glasgow, U.K., in 1986,where he received the Ph.D. degree in engineeringin 1989.He was appointed Senior Lecturer with the Uni-

versity of Strathclyde, Glasgow, U.K., In 1995, andas a Full Professor with the National University ofIreland, Maynooth, Ireland, in 2001, where he wasinstrumental in establishing the Hamilton Institute(http://www.hamilton.ie), of which he is currentlyDirector. His current research interests include

optimization, network congestion control, resource allocation in wirelessnetworks, coding theory, and data privacy.

Rouzbeh Razavi (M’09–SM’13)e received a masterdegree with distinction in telecommunication and in-formation system and a Ph.D. in wireless communi-cation networks both from the University of Essex,UK.He is a Member of Technical Staff with the

Department of Autonomous Networks and SystemsResearch, Bell Laboratories, Alcatel-Lucent, Dublin,Ireland. Prior to joining Bell Labs in 2009, he wasa Senior Research Officer with the Optical AccessLaboratory, University of Essex, Colchester, U.K.

He has been supervising a number of Ph.D. students and has publishedmore than 60 technical papers in peer-reviewed journals and conferences. Inaddition, he has authored five book chapters and has more than 17 filed patentapplications. His current research work involves developing algorithms forlarge-scale, distributed, self-organizing networks for the next generation ofwireless networks (4G and beyond) and small-cell flat cellular networks. Inaddition, he has a particular interest in modeling the business impact of newtechnologies.

utility fair optimization of antenna tilt angles in lte networks

Documents

convex optimization

adjustment of tilt angle

optimization problem

optimization isconvex

optimalantenna tilt

index termsantenna tilt

user utility

modern base stations