Wave propagation and optimal antenna layout using a genetic algorithm

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Wave propagation and optimal antenna layout using a genetic algorithm

CHOPARD, Bastien, BAGGI, Yves, LUTHI, Pascal

Abstract

This paper discusses the computational approach we have used to devise an optimal mobile communication system (cellular phones) in an urban environment. We rst present a lattice Boltzmann model for simulating wave propagation and taking into account the absorption and reeection of waves over obstacles. Second, we combine this wave propagation model with a genetic algorithm in order to nd the antennas locations which ensure an optimal coverage of a given region of a city. We have considered a parallel implementation of our computation.

CHOPARD, Bastien, BAGGI, Yves, LUTHI, Pascal. Wave propagation and optimal antenna layout using a genetic algorithm. Speedup, 1997, vol. 11, no. 2, p. 42-47

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http://archive-ouverte.unige.ch/unige:121361

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Wave Propagation and Optimal Antenna Layout using a Genetic Algorithm

B.Chopard, Y. Baggi and P. Luthi

CUI, University of Geneva, 1211 Geneva 4, Switzerland

J.-F.Wagen

Swiss Telecom PTT, FE 412, 3000 Berne 29

Abstract

This paper discusses the computational approach we have used to devise an optimal mobile communication system (cellular phones) in an urban en- vironment. We rst present a lattice Boltzmann model for simulating wave propagation and taking into account the absorption and reection of waves over obstacles. Second, we combine this wave propagation model with a ge- netic algorithm in order to nd the antennas locations which ensure an optimal coverage of a given region of a city. We have considered a parallel implemen- tation of our computation.

1 Introduction

The planning of a wireless communication system like a cellular phone network re- quires to determine the optimal position of base stations (or servers) which are the xed antenna receiving and transmitting the signal to the mobile user. This plan- ning is based on accurate predictions of radio-wave propagation in heterogeneous media such as urban environments. Computing how radio waves are absorbed, re- ected, diracted and scattered on the buildings of a city is a dicult problem, studied by various authors [2, 9, 14] but beyond analytical calculations.

The region where the intensity received from a server is larger than some given threshold is called an urban cell. Its size depends on the emitting power, the an- tenna height above the ground and the nature of obstacles in the vicinity. The determination of the coverage region of an antenna is a crucial question because the base stations forming the network must be placed in appropriate locations so that a complete coverage of the city is guaranteed. A minimum number of cells is expected but each of them should be no larger than what is allowed by trac or propagation requirements. An overlap between adjacent cells is necessary to ensure continuity (handover) of a mobile call but the area of this overlapping region must be kept to an acceptable value.

Finding such an antenna layout is a hard optimizationproblem. Here we consider the so-called Genetic Algorithm (GA) technique [7] to address this question. The approach consist of evolving a \population" of possible solutions, each solution being a set of possible server positions across the city. According to the above criteria (number of cells, quality of the coverage, overlap between pairs of cells), a score can be assigned to each solution. The best solutions are then selected and combined in order to form better solutions. This process is iterated as long as necessary to obtain an acceptable antenna distribution. This technique is very much in the same spirit as the evolution of living systems which, according to Darwinian theory, develops towards tter organisms.

Before the GA method can be applied we rst have to compute the urban cell associated with each antenna location. For this purpose, we have developed a new physical model to simulate wave propagation in a city. Our technique is based on the Lattice Boltzmann (LB) method, which was rst proposed for solving uid ow problems [3, 13]. Our numerical approach provides a direct solution of the intensity pattern around an antenna. Its advantages are the simplicity of the numerical scheme, the ease to implement various boundary conditions (reection, absorption) and its intrinsic parallel formulation which makes an implementationon a massively parallel computer straightforward.

This paper is organized as follows: section 2 is devoted to a description of our wave propagation model and a discussion of the performance obtained on various platforms. In section 3, we present the genetic algorithm implementation whereas section 4 discuss the results of our approach.

2 The wave model

Our wave propagation method is based on a lattice Boltzmann model and con- siders, as basic components, a set a ux traveling along the main directions of a two-dimensional square lattice of spacing r. Due to this level description, various behaviors can be assigned to a lattice site: free propagation, reection, absorption and propagation through a building wall. This simulation tool has been parallelized on various MPPs and give good propagation predictions when compared with out- door measurements [6].

In what follows we shall restrict the discussion to the main aspects of our wave model. More information can be found in [10, 6, 5, 3, 4]. At time^teach lattice site

r is characterized by ve real-valued quantities ^{f0; f1; f2;f3;}and ^f4 representing the ux entering the site with velocity^v0 = 0,^v1,^v2,^v3and^v4, respectively. The ux are moving synchronously on the lattice according to a discrete clock of time step .

The quantity^f0 is a stationary ux and the other ^v_i are of modulus^v = _r⁼ so that, in one time step, the ux travel one lattice spacing either east, north, west or south. Figure 1 illustrates this dynamics.

The evolution of the system is specied by the following rule

fi(^r+^vi^;t+) = 1

r^f

i(0)(^r;t) +

1^? 1

r

fi(^r;t) (1) where r is a parameter and ^f_i⁽⁰⁾ the local equilibrium distribution. These two quantities are yet to be dened properly in order to observe wave propagation. We introduce

=^Xfi ^J=^Xfi^vi (2)

f

i(0)=^a +^v2^iv2J ifⁱ⁶= 0^; and f⁰⁽⁰⁾ = a⁰ (3)

With^a0+ 4^a= 1 andr = 1⁼2, it can be shown [6, 3] that obeys the following equation

@

t2 ^?2^av2r2 = 0

which is the wave equation with propagation speed ^c =^vp2^a. The highest speed

c

0 is dened as the speed when no stationary ux is present in the model. This is achieved with^a0 = 0 so that^c0=^{v =p}2. Thus, by tuning the quantity^a, one can

2

v₁ v₂

v₃

v₄

f₁

f₀ f₁ f₂

f₃

f₄

(a) (b)

Figure 1: (a) The velocity vectors and (b) Illustration of the LB wave dynamics with one incoming ux ^f1. An intuitive interpretation of this evolution rule is a discrete form of the Huygens principle which states that each wave front point is a source of secondary spherical wavelets.

model various refraction indexesⁿ=^c0=c= 1^=p4^aand, when describing a random media, even have a dierent index for each lattice site. The evolution (1) then reads

fi(^r+^vi^;t+) =

2nn^{22?2 ?f0} ifⁱ= 0^;

1

2n^{2 ?f}i⁺² else (ⁱ+ 2 is wrapped onto 1, 2, 3, 4). (4) This rule is particularly easy to implement on a massively parallel computer: the right-hand side can be computed simultaneously for each lattice site and, then, the left-hand side is obtained with regular, local interprocessor communications.

Due to the linearity of these equations we can dene a 55 propagation matrix

Wand rewrite the evolution in the following linear form

fi(^r+^vi^;t+) =^X4

j⁼⁰

Wij^fj(^r;t)^:

Provided ⁿ= 1 and^f0 = 0,^Wreduces to the so called Transmission Line Matrix (TLM) which has been used in a quite dierent context [8] (see [11] for a more detailed discussion).

2.1 Source and boundary conditions

In order to compute the coverage of an antenna in an urban area, a crucial point is to introduce a wave source (transmitters) and model absorption or reection over buildings. Within the present lattice Boltzmann framework, these boundary con- ditions are naturally implemented. In what follows, we shall restrict the discussion to the case ⁿ= 1.

We dene a sinusoidal wave source node as a lattice site where all the ux ^fi

take the same value: ^fi=sin(2 ^t=T), where^tis the iteration time,^T the period of the source anda normalization factor which depends on^T. Due to the discrete nature of the system, it is necessary to have a large enough ^T; it is observed that choosing^{T <}6 leads to poor propagation predictions [11].

In our model, the buildings are mapped on particular lattice sites having special propagation matrices, according to their physical properties. We consider several behaviors: partial or complete reection and absorption. Reecting nodes return all incoming ux with opposite sign and direction. An attenuation factor is heuristically xed in order to account for the reection coecient relating incident to reected energy at a particular boundary. On each lattice site belonging to a building we set the propagation matrix ^R such that ^P_j^Rij^fj = ^fi⁺², with possibly a dierent value of²[^?1^;0[.

Attenuation nodesconsist of the regular propagation matrix with a multiplicative factor in front: ^A=^W where ²]0^;1[. Attenuation nodes do not prevent a wave from being transmitted through the obstacle but they lower its energy. Wave propagation at an attenuation node is always accompanied with reection, due to the breaking of spatial homogeneity.

Note that, within our approach, obstacles or boundaries are dened at the level of a site. A site does not need to know whether its neighbor is an obstacle or not.

Therefore, reection on a line, a parabola or any complicated shape is treated iden- tically. It is through the collective eect of several adjacent reecting or absorbing sites that the global behavior will emerge. This feature allows us to handle scanned, vectorized maps or any obstacle layouts with little pre-processing.

Our simulation of wave propagation takes place in two dimensions, although the system we study is three-dimensional. Our numerical scheme can easily be extended to 3D but the CPU time required for a full 3D simulation would be much higher. We gave evidences [6] that a 2D model produces correct results provided the intensity

I received at site^r

I(^r) = 20log¹⁰

0

@ v

u

u

t 1 2^T

t^X0+T

t⁼t⁰[^f1+^f2+^f3+^f4]²

1

A

is suitably renormalized according to the distance separating^rfrom the source.

2.2 Parallel performance

When implemented on a distributed memory parallel computer, the run time^Tp of the above wave propagation simulations can be expressed by the following perfor- mance model [10]

Tp=^an2

p

+^bn (5)

whereⁿ²is the problem size (number of lattice sites) and^pthe number of processors.

The quantity ^a and^b are machine dependent but are constant when varying ⁿor

p on the same computer. Thus, they provide a good way of comparing dierent platforms. They can measured accurately by tting relation (5) for various ⁿand

p.

Table 1 summaries the results obtained for dierent machines. The Connection Machine CM200 is considered as a whole, i.e. we set ^p = 1. The PC machine is a network of 10 Pentium Pro 200 MHz, interconnected with a crossbar switch and fast ethernet links. It runs Linux and PVM. As we can see, it achieves a ver good performance.

We have also tested the SGI Origin-200, which is a shared memory machine. It does not t the performance model (5). A single processor of the Origin (R10000) is faster than a SP2 thin node by a factor 1.5. A two-processor run gives a signicant speedup, yet smaller than 2. The gain of adding more processor is low, however.

Used with 4 processors, the Origin-200 is slower than four SP2 nodes.

3 The Genetic Algorithm

The genetic algorithm (GA) we have developed to nd an optimal server location operates as follows: (1) it chooses, at random, an initial population of possible solutions (each solution being a specic antenna layout); (2) it evaluates the quality (tness) of these solutions; (3) it selects the good ones and (4) combines them with genetic operators to create a new population.

4

CM200 IBM SP2 PCs Cray T3D a 5^:110^?8 5^:810^?7 6^:910^?7 1^:010^?6 b 4^:010^?6 6^:010^?6 2^:010^?5 1^:610^?5

Table 1: Performance comparison between dierent parallel machine for the wave propagation code. We consider a simulation where a source is located in the middle of a square domain. The value of^areects the CPU performance of a node (time to perform a single site computation) whereas ^b indicates the time to perform an interprocessor communication.

For the sake of simplicity, we assume that only a set of^N pre-assigned locations

ri are admissible for the antenna positions (160 for the part of Geneva we have considered, chosen as the corners of the buildings). As a pre-computation, a wave propagation simulation, using the model discussed in section 2 is performed for each possible antenna position. The intensity map is then stored for further processing by the GA.

An individual in the genetic population is dened as a selection ofⁿtransmitters locations among the ^N possible ones. It is coded as boolean vector of size ^N with a value 1 in the ^ith position to indicate the presence of the antenna at location^r_i. The intensity pattern for a particular layout is obtained by the superposition of the corresponding patterns with one antenna.

3.1 Fitness denition

Depending on the antenna positions, each point in the propagation space (the city) can be reached by zero, one or several servers. The quality of any particular ar- rangement is obtained by computing a tness function. Below we propose some denitions from which a reasonable tness can be built.

Our procedure is the following: if, for some point^r the intensity received by a transmitter^T_jis less than the quantity^I_ccalled thecoverage threshold, we say that that there is no coverage and that ^rdoes not belong to the cell of the transmitter.

In practice, ^I_c =^?80 dB.

Another threshold^It, called the transition thresholdis introduced. It is larger than ^Ic (typically ^It = ^?60 dB). The purpose of this threshold is to dene an intensity interval ^Ic ^{I I}t that will be used to specify the overlap between adjacent cells.

We also dene thedegree of coverageof a site^ras the number^kof transmitters whose emitting power is larger than ^Ic at ^r. There issinglecoverage at a location

r if the coverage degree is 1. There isdouble coverage if the degree is two. If the degree is larger than 2, we say that there is multiple coverage.

We call^S, ^Dand^M the total areas of the regions with single, double and mul- tiple coverage, respectively. These areas can be further split into two contributions, according to the second threshold ^It

S=^S>+^S< ^D=^D>+^D< ^M =^M>+^M<

The subscript ^<(^>) denotes the area of the regions where the intensity is smaller (larger) than^It.

Let^S0 be the total area of the region we want to cover with a set of antennas.

S

0 is typically computed as the area of all streets in the city. We can characterize a server layout by the following quantities

=^S+^D+^M

= ^S>

2= ^D<

D

m =^M

S

0

A good solution may be dened as (1) ^{0 !} 1: all points ^r receive an intensity larger than^Icfrom at least one transmitter; (2)^1!1: the cells dened by^{I I}t

around each antenna tessellate the entire space; (3)^2!1: all points^rwith double coverage is in the overlapping regions where both transmitters are received with an intensity level within the dened transition interval^Ic^{I I}t; (4)m ^!0: there is no multiple coverage.

From these four conditions (which may not be independent), we dene heuristi- cally the tness function

F =^b00+^b11+^b22+^bm(1^?m) (6) which should be maximized in order to nd an optimal antenna distribution. The coecients ^b0, ^b1,^b2 and^bm are weights that have been empirically determined as

b

0= 6,^b1= 26,^b2= 4 and^bm = 4. The maximal tness value is the 40.

Finally, notice that we do not impose any restriction on the cell size. This could be done if the trac of communications is not homogeneous across the various part of the city. The tness function (6) could be modied to account for this kind of constraints.

3.2 Genetic evolution

Starting from an initial population of possible solutions^Ai, the GA will evolve it by creating new solutions out of the old ones. This process is analogous to a Darwinian evolution for living organisms. The genome of an individual is composed of its antenna locations. Two individual ^A1 and ^A2 can be combined to produce two new solutions^A12 and ^A21 by using thecrossoveroperation which swaps portions of the genome between the individuals. In our particular problem, the crossover is implemented as follows: a rectangular regions ^R in the city map is randomly chosen. The antenna locations of ^A1 within the region ^R are combined with the antenna locations outside ^Rbelonging to^A2. This gives a rst ospring^A12. The second ospring,^A21 is obtained similarly by interchanging the role of^A1 and^A2. In order to put a selective pressure which should drive the evolution towards better solutions, the parents ^A1and^A2 are chosen with a probability proportional to their tness ^F (roulette wheel selection[7]). In order to keep a xed population size, the ospring will replace two other individuals. Our strategy is to eliminate two individual chosen with a probabilityinverselyproportional to their tness.

Mutations can also be considered. The mutation operator acts on a single indi- vidual, not a pair as the crossover. With low probability (1/1000 in our case) an antenna which is present in a solution will disappear while others may appear at the empty admissible locations. Mutations prevents the genetic search from being trapped in some local region of maximal tness.

A cycle composed of selection, crossover, mutation and replacement is called a generation and is the basic time evolution unit of a GA. The population size, mutation rate, number of generations and number of crossover by generation are parameters that can adjusted by the user. There is currently no general theory to determine the best parameters, i.e. those leading to the faster convergence. In this study, we have considered several hundreds of generations for a population of

P = 200 individuals. The number of crossovers occurring during each generation is varying from 4 to 15. More details concerning the GA implementation and the software environment we have developed can be found in [1].

6

700 [m]

Figure 2: Best server map corresponding to the transmitters locations obtained by the genetic algorithm. The black dots indicate the positions of 13 servers and the colored regions show the propagation cell associated to each of them.

3.3 Parallelization: the island model

The tness computation is often the most consuming part of the algorithm. A parallelization of the GA can signicantly speed up the process. The tness calcu- lation of the^P individuals can be distributed among^pprocessors [12], while a single master process takes care of the population management (selection, crossover,...).

A related technique is to evolve concurrently and asynchronously several smaller populations of size ^P=p on^pprocessors. From time to time, individuals from one population are allowed to migrate to another population in order to increase genetic diversity. This approach is often referred to as the asynchronous island model and is believed to converge faster than the GA acting upon a single population of size

P. It is also simpler to implement because it basically consists in replicating the sequential code on several nodes. The above island model has been implemented on a IBM SP2, using the PVM message passing library.

4 Results and conclusions

The results we mention describe the typical antenna distribution returned by a GA run. We shall not discuss the performance of the algorithm in term of speed and quality because, in real life, the antenna location is determined empirically and manually, making a comparison dicult.

In gure 2 we show a part of the city of Geneva and the result of a GA search whose tness is ^F = 27^:2. The thresholds ^Ic and ^It we have used are -40 dB and -30 dB, respectively. This are not the values mentioned earlier but they allows us to generate smaller cells and test the GA on smaller systems. In this example we have restricted the propagation to the streets, not allowing the wave to pass over the buildings. This corresponds to a situation where the antenna are placed not too high above the ground level. This gure gives thebest server mapcorresponding to the antenna locations found by the GA. The color given to each pixel ^r is chosen

according the transmitter which is received with the largest power so that each colored region represents an urban cell. The best server map is thus the analogous of the Voronoi diagram associated to the antenna positions, except that here the metrics is not given by the Euclidean distance but by the wave intensity.

The frame shown in gure 2 delineates the region in which the tness function is computed. Wave propagation takes place on an extended area (not fully shown here) to avoid problems with the domain boundaries. We observe that the wide open- air regions and the main streets are well covered, although about 1/3 of the total area remains empty. Globally, the antennas are well distributed over the complete region, except for the clearly undesirable overlap occurring in the lower left corner of the image. Such a situation, where two server are nearby could possibly be avoided by changing the weights in the tness denition or introducing penalties when two sources are to close to each other.

This study should be taken as a preliminary investigation of the GA approach.

At this stage, we may conclude that this method is certainly promising to solve the present optimization problem. However it should be improved in order to provide exploitable results. The tness function given by expression (6) capture some es- sential parts of the problem but many parameters can still be adjusted and new constraints could be taken into account. Also, we may envisage to combine the GA solution with an interactive tool allowing the user to move manually the servers positions.

References

[1] Y. Baggi. Procede de planication en cellules urbaines. Technical report, University of Geneva, CUI, 1996. Master disertation.

[2] H. L. Bertoni, W. Honcharenko, L. R. Maciel, and H. Xia. Uhf propagation prediction for wireless personal communications. InIEEE Proceedings, Vol 82, No. 9, pages 1333{1359, 1994.

[3] B. Chopard and M. Droz. Cellular Automata Modeling of Physical Systems. Cambridge University Press. to appear.

[4] B. Chopard and P.O. Luthi. Lattice boltzmann computations and applications to physics. Theoretical Computer Science. to appear.

[5] B. Chopard and P.O. Luthi. A lattice boltzman model and its application. In S. Bandini and G. Mauri, editors,ACRI'96 Proceedings of the Second Confer- ence on Cellular Automata for Research and Industry, pages 13{24. Springer- Verlag, London, 1997.

[6] B. Chopard, P.O. Luthi, and Jean-Frederic Wagen. A lattice boltzmann method for wave propagation in urban microcells. IEE Proceedings - Microwaves, An- tennas and Propagation, 1997. to appear.

[7] G. Goldberg.Genetic Algorithms in Search, Optimization and Machine Learn- ing. Addison Wesley, Reading, MA, 1989.

[8] W. J. R. Hoeer. The transmission-line matrix method. theory and ap- plications. IEEE Transaction on microwave theory and techniques, MTT- 33(10):882{893, October 1985.

[9] T. Kurner, D. J. Cichon, and W. Wiesbeck. Concepts and results for 3d digital terrain-based wave propagation models: an overview. IEEE Jour. on Selected Areas in Communications, 11(7):1002{1012, Sept. 1993.

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[10] P. O. Luthi, B. Chopard, and J.-F. Wagen. Wave propagation in urban mi- crocells: a massively parallel approach using the TLM method. In WAS- NIEWSKI Jerzy DONGARRA J, MADSEN Kaj, editor, APPLIED parallel computing: computations in physics, chemistry and engineering science, pages 429{435. Berlin Springer, 1996. Lecture notes in computer science; vol. 1041.

[11] P. O. Luthi and Bastien Chopard. Wave propagation with transmission line matrix. Technical report, University of Geneva and Swiss Telecom PTT, 1994.

[12] M. Oussaidene, B. Chopard, O.V. Pictet, and M. Tomassini. Parallel genetic programming and its application to trading model induction. Parallel Com- puting, 1997. to appear.

[13] Y.H. Qian, S. Succi, and S.A. Orszag. Recent advances in lattice boltzmann computing. In D. Stauer, editor, Annual Reviews of Computational Physics III, pages 195{242. World Scientic, 1996.

[14] K. Rizk, J.-F. Wagen, and F. Gardiol. Ray tracing based path loss prediction in two microcellular environments. In Proceedings IEEE PIMRC'94, pages 210{214, The Hague Netherlands, September 1994.