mimo array capacity optimization using a genetic algorithm · ericsson mobile communications do...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 6, JULY 2011 2471

MIMO Array Capacity OptimizationUsing a Genetic Algorithm

Manuel O. Binelo, André L. F. de Almeida, Member, IEEE, and F. Rodrigo P. Cavalcanti, Member, IEEE

Abstract—This paper discusses the design of multiple-input–multiple-output (MIMO) antenna systems and proposesa genetic algorithm to obtain the position and orientation ofeach MIMO array antenna that maximizes the ergodic capacityfor a given propagation scenario. One challenging task in theMIMO system design is to accommodate the multiple antennasin the mobile device without compromising the system capacity,due to spatial and electrical constraints. Based on an interfacebetween the antenna model and the propagation channel model,the ergodic capacity is considered as the objective function of theMIMO array optimization. Simulation results corroborate theimportance of polarization and antenna pattern diversitiesfor MIMO in small terminals. Our results also show that theelectromagnetic coupling effect can be exploited by the optimizerto decrease signal correlation and increase MIMO capacity. Acomparison among a uniform linear array (ULA), a uniformcircular array (UCA), and a genetic algorithm (GA)-optimizedarray is also carried out, showing that the topology given by theoptimizer is superior to that of the standard ULA and UCA forthe considered propagation channel.

Index Terms—Array optimization, capacity, genetic algorithm(GA), multiple-input–multiple-output (MIMO).

I. INTRODUCTION

MULTIPLE-input–multiple-output (MIMO) systems havebeen an important research topic due to their capability

of providing a significant increase in channel capacity pro-portional to the number of transmit and/or receive antennas.This is generally achieved by exploiting spatial diversity atthe transmit and receive branches [1], [2]. When consideringrealistic antenna models, the channel capacity will dependon three main variables, namely, the transmit antenna arrayconfiguration, the receive antenna array configuration, and theenvironment configuration, i.e., the distribution of the channelscatterers [3].

Manuscript received November 29, 2010; revised March 16, 2011 andMay 18, 2011; accepted May 21, 2011. Date of publication June 2, 2011;date of current version July 18, 2011. This work was supported in partby the Innovation Center, Ericsson Telecomunicações S.A., Brazil, by SonyEricsson Mobile Communications do Brasil Ltda, and by Conselho Nacionalde Desenvolvimento Científico e Tecnológico under Grant 312845/2009-0.The review of this paper was coordinated by Prof. E. Bonek.

M. O. Binelo is with the Wireless Telecom Research Group, Federal Univer-sity of Ceará, 60455-760 Fortaleza-CE, Brazil, and also with the University ofCruz Alta, 98025-810 Cruz Alta-RS, Brazil (e-mail: [email protected]).

A. L. F. de Almeida and F. R. P. Cavalcanti are with the Wireless TelecomResearch Group, Federal University of Ceará, 60455-760 Fortaleza-CE, Brazil(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2158460

Multiplexing and diversity gains in MIMO systems can beobtained, for instance, by resorting not only to spatial diversitybut to polarization diversity as well. Whereas spatial diversitycan be achieved by ensuring enough antenna separation, polar-ization diversity can be achieved by exploiting different antennaorientations in the array configuration [4]. In cellular systems,a reasonable degree of spatial diversity can be obtained at thebase station. However, at the mobile terminal, this situation iscompletely different since a good separation of the antennasensuring spatial diversity is not always possible. Moreover,electromagnetic coupling between terminal antennas is a prac-tical problem that affects system performance. Therefore, underthese constraints, optimizing the antenna placement at the mo-bile terminal to yield a satisfactory performance is a challengingproblem [5], [6].

The work in [7] shows some important aspects about im-plementation of MIMO transceivers in small terminals. In [8],planar inverted-F antenna (PIFA) compact arrays are comparedwith dipole arrays with the same antenna separation. That workstates that PIFA antennas are an ideal candidate for compactarray designs due to the low interference between antennascaused by the proximity to the ground plane. Although find-ing the globally optimal antenna set was not in the scope of[8], the work allows one to conclude about the importanceof empirical inference from the designer. The dipoles wereuniform linear arrays (ULAs), whereas the PIFAs had a 180◦

difference in orientation between the two elements. The PIFAarrays outperformed the dipoles. The said paper attributes thisresult to the nearly omnidirectional pattern of the PIFA, butit does not justify the position and orientation of the usedPIFA antennas and does not investigate the influence of theorientation.

In [9], the relationship between MIMO capacity and arrayconfiguration is investigated for different outdoor propagationscenarios. The work concludes that, for a small angular spreadof 8◦, the best separation of antenna array elements wouldbe four wavelengths to approximate the uncorrelated channelperformance, whereas for 68◦, the optimum distance is thetypical 1/2 wavelength. It is also noted in that work that, withhigher angle spreads, the system was less sensitive to arrayorientation changes. In general, these works corroborate therelevance of the propagation environment characteristics in thedesign of the MIMO array.

Another very important MIMO diversity technique is truepolarization diversity (TPD). In TPD, instead of typical al-ternated 0◦ and 90◦ polarization, we can use any angle tochange polarization. This is particularly useful for small mobileterminals that have small space for exploring spatial diversity.

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2472 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 6, JULY 2011

The study in [4] shows that TPD can significantly improveMIMO capacity. However, now, the designer has to deal notonly with antenna positioning in the array but also with an-tenna orientation, bringing more complexity to the transceiverdesign. Not only the orientation of individual array elementsis important, but the orientation of the hole array also has animportant impact on MIMO capacity, particularly for smallangle spreads, typically in outdoor environments, according to[10]. In [11], a model for antenna correlation using arbitraryantenna orientation is used, and the work also shows that usingdifferent orientations for the antennas increases MIMO capac-ity. In [12], the challenges of antenna configurations in smallmobile terminals are discussed. The work uses a measurementsystem to evaluate antenna configurations. The conclusions ofthe work are that small changes in antenna configuration cancause drastic changes in capacity and that the directive proper-ties of the antennas and propagation channel cause importantfluctuations on performance.

In [13], the question is whether UCAs are better for MIMOcapacity than ULAs. It turns out that ULAs are better forsome direction-of-arrival (DOA) configurations whereas UCAshave a more stable behavior. A similar analysis is made in[14]. However, the question on the MIMO transceiver designis much wider. Which array arrangement is better for a givenpropagation scenario or for a given set of scenarios? Whicharray configuration, from a MIMO capacity viewpoint, is morestable?

Reviewing these works, we can see that the possibilitiesfor the MIMO system designer are endless, and just studyingreference cases might not be enough to find a solid solution.Using optimization tools that can take all these possibilities intoaccount may yield more robust MIMO designs.

Suppose that the designer knows the antenna model and thenumber of antenna elements to be used in a mobile termi-nal. Given an available space at the terminal for placing theantennas, a pertinent question is now in order. What are thebest antenna configurations to be used in a given propagationenvironment? To answer this question, a possible road to takeis to make use of experimental results or intuition to find aprobable good antenna configuration for a certain propagationscenario and then use simulation or prototype measurements tosee if the design fulfills the project requisites.

The second approach is to use an optimization method forfinding an optimal (or suboptimal) solution to the antennaarray configuration problem and then check if this solutionis satisfactory. In this case, we jump from a computer-aideddesign paradigm to a computer design one. This approach hasbecome increasingly common, as shown in [15].

Genetic algorithm (GA)-based optimization has been usedwith success in various engineering problems. In [16], a GAis used to optimize antenna arrays used for channel charac-terization, i.e., determination of multipath DOAs. The systemstarts with big regular 2-D or 3-D array of antennas (morethan 20 antennas), and then, the algorithm changes individualantenna position and orientation by trying to find the arraythat minimizes the parameter estimation error. Although theobtained antenna positioning patterns may seem random andsometimes not intelligible for the designer, they offer better

precision in terms of channel characterization than usuallyregular (or uniformly spaced) patterns.

In [17], the authors resort to GA-based optimization to findchannel parameters such as multipath attenuations and delays.In [18], a GA is used for blind channel estimation. The studyreports that the GA method can offer better channel estimationaccuracy than traditional methods. A similar approach was alsoproposed in [19]. Recently, a GA has been used to find goodantenna element positions in sparse MIMO radar arrays [20] byminimizing the sidelobes of the radar pattern. Another recentwork [21] used a GA to find the optimal distribution of a 3 ×3 MIMO system for an indoor propagation channel. An inter-esting aspect of that work is the inclusion of electromagneticcoupling in the model. However, the work does not show eitherwhich distributions were found or how the distributions changeaccording to different multipath channel parameters.

The work in [22] defends the idea of using nature-inspiredmethods for MIMO antenna design, but the works mentionedin there deal with the problem of antenna geometry definitionand not antenna array topology for different propagation envi-ronments. In [23], a method of moments (MOM) is proposedto optimize MIMO antenna position and orientation. The op-timization is done by minimizing antenna cross correlation byconsidering an independent and identically distributed (i.i.d.)propagation scenario. Although antenna cross correlation de-grades capacity, we cannot say that the configuration thatminimizes antenna correlation is the same configuration thatmaximizes MIMO capacity in non-i.i.d. propagation scenarios.A similar approach is used in [24]. The work in [25] solves theproblem of MIMO PIFA placement for an i.i.d. channel, alsominimizing antenna cross correlation, but using the infinitesi-mal dipole method and invasive weed optimization, which isanother nature-inspired method.

In this paper, we address the antenna array capacity opti-mization problem by resorting to a GA method. The goal isto find an optimal or a suboptimal configuration for antennaposition and orientation that maximizes the ergodic channelcapacity. Assuming an array of dipoles and a channel model thatinterfaces the propagation environment with the antenna arrayresponse pattern, the GA manages to find, for each antenna,the best position and orientation subject to a space constraint.Our results show that the proposed antenna array optimizationmethod can be a valuable tool for small MIMO terminal design,and it can also give insights about different strategies to achieveMIMO diversity. Due to the nature of GAs, the proposedmethod is very general. It can incorporate different types ofantenna models, and it can be also used in different propagationchannel models. Our simulations take into account differentsets of antennas and constraints in terms of available space andconsider electromagnetic coupling effects. We also comparethe capacity provided by the optimized MIMO array with thatof standard linear and circular arrays. The results corroboratethe importance of jointly exploiting polarization and antennapattern diversities, as have been recently shown in [26] and [27].

In Section II, we present the channel model that is exploitedby the proposed algorithm. In Section III, we present the GAused for the optimizations and detail how the population isrepresented, how reproduction occurs, and how we used the

BINELO et al.: MIMO ARRAY CAPACITY OPTIMIZATION USING A GA 2473

Fig. 1. Geometric channel model used in simulations.

ergodic channel capacity as the fitness function of the GA.Section IV presents simulation results and results discussion.Finally, in Section V, we draw conclusions.

II. CHANNEL MODEL

In Fig. 1, the channel geometric model used to generatethe plane waves and to interface it to the antenna arrays isillustrated. We assume that the distance between antenna arraysand the scattering clusters is much higher than the distancebetween the array elements. In this case, we can assume thatthe DOA and the direction of departure (DOD) of a givenplane wave are the same for all the antenna elements of thearray. Each cluster is modeled by a finite set of plane wavesand has a main DOA/DOD, both in azimuth and elevation.Angle spread and polarization spread within a cluster followa Gaussian distribution.

The channel double-directional impulse response, which isassociated with the DOA pair (φRx, θRx) and the DOD pair(φTx, θTx), is given by the contribution of a finite number ofdominant multipath components [28], i.e.,

A(φRx, θRx, φTx, θTx) =L∑

l=1

Al(φRx,l, θRx,l, φTx,l, θTx,l)

(1)

where L is the number of arriving paths, and φ and θ denote theazimuth and elevation angles, respectively. The contribution ofeach path Al can be expanded as follows:

Al(φRx, θRx, φTx, θTx) = Wlejφl

× δ(φRx − φRx,l)δ(θRx − θRx,l)

× δ(φTx − φTx,l)δ(θTx − θTx,l) (2)

where φl = j2πfc, and Wl is the polarimetric transmissionmatrix, which is defined as

Wl =[

γHH γVH

γHV γVV

]. (3)

The (m,n)th entry hm,n of MIMO channel matrix H (M ×N) can be expressed in terms of the directional channel impulse

response according to the following expression [28]:

hmn =L∑

l=1

gTTx(φTx,n,l, θTx,n,l, rTx,n)

× Al(φRx,l, θRx,l, φTx,l, θTx,l)× gRx(φRx,m,l, θRx,m,l, rRx,m)× exp (j [k(φRx,l, θRx,l) · xRx,m])× exp (j [k(φTx,l, θTx,l) · xTx,n]) (4)

where gRx is the antenna pattern response to the direction(φRx,m,l, θRx,m,l), and gTx is the antenna pattern response tothe direction (φTx,m,l, θTx,m,l) at the transmitter. The responseof the antenna considers the impact of mutual electromagneticcoupling of nearby antennas. We calculate this effect by inte-grating the numerical electromagnetics code (NEC) in our sim-ulation. The (2 × 1) vector gRx is the product of the complexscalar gain gRx (phase and amplitude) of the receiver antennaand the unitary (2 × 1) polarization vector pRx composed ofvertical and horizontal responses, i.e.,

gRx(φRx,n,l, θRx,n,l, rRx,n)=gRx(φRx,n,l, θRx,n,l, rRx,n)pRx.(5)

Similarly, for transmitter antenna pattern response gTx, wehave

gTx(φTx,n,l, θTx,n,l, rTx,n)=gTx(φTx,n,l, θTx,n,l, rTx,n)pTx.(6)

Vector rRx,m is the antenna orientation, k is the wave vector,xRx is the relative position of the mth receiver antenna, andxTx is the relative position of the nth transmitting antenna. Theinner product of vector wave k (arriving or departing wave)with an antenna position (transmitter or receiver) is defined by

k(φ, θ)·x=2π

λ(x cos θ cos φ+y cos θ sin φ+z sin θ sinφ).

(7)

Fig. 2 shows the interface between the antenna and the propa-gation environment, which is used to build the effective MIMOchannel response. Note that (4) is an entry of the effectiveMIMO radio channel, which includes the effect of antennaresponses in the channel impulse response. As we will see later,the position and orientation of each receive antenna, which aredefined by vectors xRx and rRx, respectively, are variables tobe optimized by the proposed optimization algorithm. VectorrRx is composed of elements α, β, and γ, which represent therotations around the x-, y-, and z-axes, respectively. In thispaper, we limit ourselves to the optimization of the receive arrayparameters. The joint optimization of the receive and transmitarray parameters xTx and rTx is left to future work.

A. Electromagnetic Coupling

Electromagnetic coupling has a strong effect when the anten-nas are separated by small distances (typically less than λ/2).With handheld telecommunication devices, this is often, if notalways, the case. Instead of implementing an electromagneticcode from the ground up, in this paper, we chose to use an


Fig. 2. Antenna–propagation environment interface.

available and well-established code, which is integrated to ourchannel model. We use the NEC [29], which is a public domainsoftware. The version we chose to work with is NEC2C, whichis a C language implementation of the NEC2 Fortran originalcode. The NEC uses the MOM to solve the electromagneticfield problem. One of its main qualities is the low computationalcost of the solutions, since MOM codes are much faster than,e.g., finite-element-method-based codes.

The mutual coupling of the antennas affect the far-fieldresponse of the antennas in two ways. First, it changes the mag-nitude of the response, modifying the directivity of the antenna.Second, it perturbs the phase response of the antenna. Fig. 3shows both effects in a dipole antenna under the influence ofmutual coupling with a nearby dipole antenna. It is worth notingthat the effect of mutual coupling will not always degradeMIMO capacity. Depending on the considered channel modeland on the antenna configuration, it may well increase capacityby decorrelating the signal (true MIMO gain) or by increasingthe directive gain toward a cluster direction. The work in [10]explains how electromagnetic coupling can decrease antennacorrelation by means of pattern diversity. Since in small termi-nals it is not possible to use spatial separation of antennas toachieve diversity, it is important to consider mutual coupling asa possible source of pattern diversity and not just a downside inMIMO antenna design.

Fig. 4 shows the capacity of a 2 × 2 MIMO system as afunction of antenna separation for a channel with one cluster.When the antennas are very close, there is a strong gain incapacity for the system with antenna mutual coupling. Thisis caused not only by the signal decorrelation effect but alsoby a directive gain toward the cluster. The drop in capacityfor a distance of λ/2 and its raise at 0.8λ are caused bycorrelation and decorrelation effects of mutual coupling. Ourresults corroborate the works in [5] and [6], where the samepattern of the raise in correlation (drop in capacity) at λ/2 andthe drop in correlation (raise in capacity) at 0.2λ and 0.8λ isfound.

Fig. 3. Effect of mutual coupling in the far field of a dipole antenna.

Fig. 4. Effect of mutual coupling on MIMO capacity.

III. GENETIC ALGORITHM OPTIMIZATION

A GA works by analogy to genetic inheritance and differen-tiation that occurs in biology that permits a species to fit itselfto the environment in an adaptation process. We can think aboutthe channel characteristics as the environment and the antennaarray as the biologic species that needs to fit in the environment(the channel). The fitness of the antennas to the environmentcan be measured by the ergodic capacity. As a bird can developa beak more adapted to a specific kind of seed, by analogy, anantenna array could develop characteristics better suited for aspecific channel or kind of channel.

We cannot produce antennas capable of self-reproduction,but we can simulate that in a computer using digital represen-tations of antennas and channels. The genetic code is a digitalcode that represents the individual characteristics that can beinherited from previous generations and be passed by to thenext generations. Our individual is the antenna array, and our


Fig. 5. Fluxogram of the employed GA.

genetic code is the channel array model stored in the computermemory. A population, or generation, is a collection of antennaarrays. The genes that define an array are the antenna type,position, and orientation. We start with a random generation,where each antenna has a position and an orientation assignedto it by a random variable with a uniform distribution, withinthe limits of the desired volume space for the antennas. Thenumber of individuals in the generation is increased by crossingand mutation. The crossing operation is the reproduction ofnew individuals that inherit part of the characteristics fromone individual and other part from other individuals, e.g., theparents. Which characteristic will come from each parent isdecided by an aleatory factor. The mutation operation is analeatory small change in the genes.

The next step is to select the individuals better suited tothe environment using the fitness function and then repeat thereproduction step with this selected group. The reproductionand selection steps are repeated until an optimization criterionis met or a certain number of generations are met. Our fitnessfunction is defined by the channel ergodic capacity, as will bedetailed later.

We can see the GA as a method that explores a search space.The randomness of mutation makes the optimizer explore dif-ferent regions escaping from local maxima, whereas the fitnessfunction gives a direction to the search. One advantage of theproposed GA is its low sensibility for initial conditions whencompared with traditional numerical methods. A very usefulaspect of the GA is that, as a searching algorithm, it is notlimited to numerical field operations and can use any operationthat can be expressed in the algorithm. One example of thisfeature would be the search for the best kind of antenna for agiven scenario or even to merge different antennas into the samesolution. A limitation of this method is that it can stop in a localmaximum, and in some cases, it is not possible to know if thesolution is local or global, or if the algorithm is capable or notof leaving the local maxima. Fig. 5 shows a general diagram ofthe GA optimization method that is applied to our problem.

A. Population and Reproduction

The antenna is represented in the system by the tuple(kind, position, orientation). The kind is an integer token thatidentifies the antenna far-field pattern. In this paper, we considerideal half-wave dipoles, although more than one kind of antennacould be used. Position vector x = [x, y, z]T and orientation

vector r = [α, β, γ]T (yaw, pitch, roll). A collection of antennas[a1, a2, . . . , aM ] defines an array. An array is one individual inthe population. The GA needs a finite set into which to search;thus, the position and orientation need to be both limited andquantized. The degrees of freedom of the position are limited bya limited volume defined prior to the simulation. The availablevolume is generally a practical constraint of the antenna arraydesign in small terminals. The quantization is naturally imposedby the computer quantization of the floating-point numbers.Defining the antenna geometry by a collection of 3-D points S,its radiation pattern is rotated by the following transformation:

S′ = ( c1 c2 c3 )S (8)

where

c1 =

⎛⎝ cos γ cos β cos α − sin γ sin α

− sin γ cos β cos α − cos γ sin αsin β cos α

⎞⎠ (9)

c2 =

⎛⎝ cos γ cos β sin α + sin γ cos α

− sin γ cos β sin α + cos γ cos αsin β sin α

⎞⎠ (10)

c3 =

⎛⎝− cos γ sin β

sin γ sin βcos β

⎞⎠ . (11)

The genetic code of each individual in the collection isformed by the antennas’ tuples. The reproduction is done bycombining portions of the deoxyribonucleic acid from twoparents. A new array is derived by choosing antennas from twoancestor arrays. A pseudorandom function is used to choosefrom which parent each antenna will be copied for the newindividual in the population. After the reproduction, a smallpseudorandom change is made in each antenna parameter. Sucha change defines the mutation procedure. It is worth noting thatthe amount and extent of the mutation have a strong impacton the algorithm performance. Small changes can make thealgorithm converge faster, but it is more prone to get stuck ina local maximum, whereas stronger changes make it leave thelocal maximum for better maxima but makes the system lessstable. Therefore, a tradeoff between convergence speed andstability exists, as usual in numerical optimization methods.

Another parameter to take into account is the size of theoffspring in each generation. A small offspring provides fastercomputation and less memory usage, at the expense of moreiterations necessary to solve the problem. Fig. 6 shows thereproduction and mutation operations, where α1 to α6 denote arandom displacement.

B. Fitness Function

To select the individuals for the next generation, it is neces-sary to use a fitness function, which is always problem related.The interface between the fitness function and the GA makesit possible to choose any kind of fitness function as long as thefunction respects the inputs and outputs of the interface. Theinput for the fitness function has to be the current generationof individuals: in our case, the various array configurations


Fig. 6. Crossing and mutation procedure.

Fig. 7. Fitness function used in the GA.

obtained by the crossing and mutation operations. The outputof the function has to be some value attached to each individual,making it possible to classify it. In our case, that value corre-sponds to the ergodic channel capacity (in bits per channel use),which is given by [30]

C =1

Nq

Nq∑q=1

log2 det[INr +

SNRNt

HqHHq

](12)

where Nq is the number of realizations to compute the expec-tation statistics, and Hq represents the qth channel realization.Note that, according to (4), each entry hmn of Hq (M × N)depends on relative antenna positions xRx,1,...,M and antennafar-field patterns according to their orientations rRx,1,...,M at agiven channel realization. Recall that only the receive antennasare the object of the present investigation. The antennas at thetransmitter (i.e., the base station for the downlink) are supposedto have fewer placement constraints and are not optimized here.Therefore, the objective function of the GA is to solve thefollowing problem:

arg maxxRx,1,...,M ,rRx,1,...,M

C(xRx,1,...,M , rRx,1,...,M ). (13)

The diagram given in Fig. 7 illustrates the fitness function.Index i refers to the ith array configuration in the generationpopulation, whereas index k refers to the kth generation in thesimulation. These two pieces of information generate a set ofplane waves that are stored and remain constant during all thesimulations. This is important because each generation is nota leap in time. The succession of generations is time relatedin nature but has no relation to the time variation of the radiochannel. The collection of plane waves has two dimensions. Werefer to w as the wave index and q as the realization index.A number of wave realizations are first generated and storedand further used to compute the statistics of ergodic channel

Fig. 8. Capsule collision detection.

capacity. The same random realizations have to be applied foreach individual of the generation; otherwise, a noise wouldbe added to the solution. It is interesting to note that thisarchitecture makes it possible to use any channel environmentdefinition model that delivers a set of plane waves as the output.Even real site channel characterization measurements mightbe used.

The NEC is integrated to our optimization software in thefitness function, as shown in Fig. 7. For each iteration, and foreach array in the population of possible solutions, the inputfiles that define the geometry and electric characteristics ofthe antennas are created. Then, the NEC is invoked, and itssolver calculates the far-field results of the antennas consideringthe mutual coupling effect. The NEC then writes the far-fieldresults to its output files. Our software then reads the NECoutput files and proceeds with the channel simulation for thefitness function.

One problem initially faced during the simulations was thefact that, when two antennas had an intersection, the NECwould see a short circuit between the antennas, generatingincorrect far-field results. The problem was solved by applyinga collision test during the phase of reproduction of antennas.Every antenna wire was protected by a capsule. The systemmeasures the minimum distance between antennas’ wires; ifthis distance is greater than the sum or than capsule radiosr1 and r2, the antennas are considered to be in collision, andthe array is dropped from the population. This idea is shownin Fig. 8.

IV. SIMULATION RESULTS

Here, a set of computer simulation results is presented forsome propagation scenarios and system configurations. We aimat investigating the link between the GA-optimized antennas’positions and orientations to the propagation environment inquestion. We also evaluate the theoretical channel capacityobtained by optimizing the antenna array configurations usingthe proposed GA algorithm.

A. One Cluster, 3 × 3 MIMO

In Fig. 9, we consider a 3 × 3 MIMO system using half-wavelength dipoles. The transmitter antennas are spaced by


Fig. 9. Evolved 3 × 3 MIMO configuration. One cluster with a fixed maindirection. SNR = 20 dB. Volume = (2λ)3.

Fig. 10. Histogram for the evolved 3 × 3 MIMO configuration. One clusterwith a fixed main direction. SNR = 20 dB. Volume = (2λ)3.

two wavelengths. The propagation channel is characterized bya single scattering cluster with high angle spread and totalpolarization diversity. Since there is a large space for the systemto exploit, any solution with antenna separation greater thanλ/2 and polarization diversity will be a good solution. Wemade 20 simulations, and all results presented a pattern similarto that of Fig. 9. The lines in the figure present a samplingof the DOAs, and the line color represents its polarization,ranging from fully horizontal to fully vertical. Fig. 10 showsthe histogram of the ergodic capacity of the initial randomantenna positions and orientation, as well as the ergodic capac-ities obtained after optimization. The initial ergodic capacityhad a mean of 9.48 bits/s/Hz with a standard deviation of1.01 bits/s/Hz. The GA improved it to a mean of 14.14 bits/s/Hzwith a standard deviation of 0.43 bit/s/Hz.

Fig. 11. First evolved 3 × 3 MIMO configuration. One cluster with a fixedmain direction. SNR = 20 dB. Volume = (0.2λ)3.

Fig. 12. Second evolved 3 × 3 MIMO configuration. One cluster with a fixedmain direction. SNR = 20 dB. Volume = (0.2λ)3.

B. One Cluster, 3 × 3 MIMO, Small Volume

Figs. 11 and 12 show two results that emerged for a 3 × 3MIMO system with one cluster with DOAs illustrated by thecolored lines in the figure. The difference in this scenario is thatthe receiver antennas were limited to a search space of 0.2λ. Wemade 20 simulations; 16 of them showed a pattern equivalentto Fig. 11, with a mean of 11.15 bits/s/Hz and a standarddeviation of 0.1390 bit/s/Hz. However, four of them presenteda pattern similar to that of Fig. 12, with resulting ergodic


Fig. 13. Evolved 3 × 3 MIMO configuration. One cluster with a uniformlydistributed main direction. SNR = 20 dB. Volume = (0.2λ)3.

capacity of mean 11.89 bits/s/Hz and standard deviation of0.33 bit/s/Hz. The initial conditions of the antennas had a meanof 6.76 bits/s/Hz and a standard deviation of 0.92 bit/s/Hz.

Fig. 13 shows the simulation results for a 3 × 3 MIMOsystem with a search space limited to 0.23λ. It also considersone cluster, but this time, the cluster main direction is not fixedbut uniformly distributed around the space. The cluster hasthe same angle spread of other simulations but is not plottedin the figure due to its main direction distribution. In thiscase, all 20 simulations had shown results similar to that ofFig. 11. Since the available space is too small to achieve signaldiversity through antenna spacing, the optimizer made use oftwo strategies, i.e., orthogonal polarizations and orthogonal pat-terns. According to Fig. 11, antenna 3 is orthogonally polarizedto antennas 1 and 2. Antennas 1 and 2 are placed in parallel.The electromagnetic coupling effect makes antennas 1 and 2get directional gains in opposite directions. Fig. 14 shows howthe optimizer made use of polarization and pattern diversity, ex-ploiting the electromagnetic mutual coupling effect, to produceMIMO diversity.

Fig. 15 shows the histogram for all 20 simulations for onecluster with a fixed main direction. Fig. 16 shows the histogramfor one cluster with a uniformly distributed main direction.Fig. 17 shows the evolution of the solutions for this last sim-ulation scenario. We can see the ability of the system to escapefrom a local maximum.

C. Array Topology Comparison

In [13], an ULA is compared with an UCA. The work in [31]shows the impact of DOA over correlation for the ULA, and thecorrelation degrades MIMO capacity. The capacity is calculatedfor one cluster for different DOAs using a 4 × 4 MIMO

Fig. 14. Resulting antenna pattern for the evolved 3 × 3 MIMO configuration.One cluster with a fixed main direction. SNR = 20 dB. Volume = (0.2λ)3.

Fig. 15. Histogram for the evolved 3 × 3 MIMO configuration. One clusterwith a fixed main direction. SNR = 20 dB. Volume = (0.2λ)3.

system. They have concluded that the ULA achieves very highcapacities for some DOAs but also very low capacities for otherDOAs. The UCA could not achieve the ULA top capacity buthad a much more stable behavior, showing the same capacitydespite the DOA. In our work, we simulate a channel with acluster having 15◦ of angle spread with a normal multipathdistribution. The schematic is shown in Fig. 18. We also useour GA to evolve a 2-D topology solution for the problem.For this problem, the fitness function was the average ergodiccapacity, considering the cluster to be in a different DOA ateach statistical realization. The constraint for the evolved arraywas that the distance between elements should not be greaterthan λ/2. The array resulted for the GA is shown in Fig. 19.As shown in Fig. 20, we found the results for the ULA andUCA to be similar to [13], with the UCA being more stable.The ULA had a higher peak and an average capacity but hadvery strong capacity losses for some DOAs, agreeing with [31].The evolved GA solution was better than both. It had the highestaverage ergodic capacity while being much more stable than theULA and with all capacities above the UCA solution. Fig. 21


Fig. 16. Histogram for the evolved 3 × 3 MIMO configuration. One clus-ter with a uniformly distributed main direction. SNR = 20 dB. Volume =(0.2λ)3.

Fig. 17. A 3 × 3 MIMO array evolution in a small volume.

Fig. 18. Schematic of the ULA and UCA.

Fig. 19. Array topology resulted from GA optimization.

Fig. 20. Performance comparison of ULA, UCA, and GA array topologies.

Fig. 21. Histogram for the evolved 4 × 4 MIMO configuration. Uniformcluster main direction distribution. SNR = 20 dB. Volume = (0.2λ)3.

shows the histogram for the solutions. Although there weredifferent resulting capacities, all solutions are geometricallysimilar. We can also see that, in some simulations, the algorithm


could not escape from a local optimum. This behavior indicatesthat the proposed algorithm can be further improved to avoidsuch local optima.

V. CONCLUSION

The proposed GA-based optimization algorithm for antennaarray positioning has proved to be successful in finding goodMIMO antenna schemes for a given propagation scenario.Some solutions found by the GA optimizer were very subtle,and a human designer would have difficulties in trying to iden-tify the best location and orientation for the antennas accordingto the specified propagation environment. The comparison ofthe ULA, the UCA, and the array evolved by our method showsthat it is a much more efficient engineering method than theintuitive and trial-and-error approach.

The results so far have shown that pattern and polarizationdiversities play an important (if not the most important) role inMIMO capacity when there is little space available to positionthe antennas. One important aspect of the proposed method isits generality, as it can be adapted to be used with differentantenna and propagation models.

As a perspective of this work, we should consider the use ofdifferent types of antennas: preferably practical mobile terminalantennas.

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Manuel O. Binelo received the B.Sc. degree incomputing science from the University of Cruz Alta,(UNICRUZ), Cruz Alta, Brazil, in 2004 and theM.Sc. degree in mathematical modeling from theRegional University of Northwest of Rio Grandedo Sul State, Ijuí, Brazil, in 2007. He is currentlyworking toward the Ph.D. degree in teleinformaticsengineering with the Federal University of Ceará,Fortaleza, Brazil.

He is currently a Researcher with the WirelessTelecom Research Group, Federal University of

Ceará, and a Professor of computing science with UNICRUZ.


André L. F. de Almeida (M’08) received the B.Sc.and M.Sc. degrees in electrical engineering from theFederal University of Ceará, Fortaleza, Brazil, in2001 and 2003, respectively, and the double Ph.D.degree in sciences and teleinformatics engineeringfrom the University of Nice-Sophia Antipolis, Nice,France, and the Federal University of Ceará in 2007.

In 2002, he was a Visiting Researcher withEricsson Research, Stockholm, Sweden. He was aPostdoctoral Fellow with the Computer Science, Sig-nals and Systems Laboratory, Centre National de la

Recherche Scientifique, University of Nice-Sophia Antipolis, from January toDecember 2008. He is currently an Assistant Professor with the Departmentof Teleinformatics Engineering. Federal University of Ceará. He is also aResearcher with the Wireless Telecom Research Group, where he has workedon several funded research projects. His research interests include blind equal-ization and source separation, multiple-antenna techniques, multilinear algebra,and applications of tensor modeling to wireless communication systems.

F. Rodrigo P. Cavalcanti (M’10) received the B.Sc.and M.Sc. degrees in electrical engineering fromFederal University of Ceará, Fortaleza, Brazil, in1994 and 1996, respectively, and the D.Sc. degreein electrical engineering from the State University ofCampinas, Campinas, Brazil, in 1999.

Upon graduation, he joined the Federal Univer-sity of Ceará, where he is currently an AssociateProfessor and holds the Wireless CommunicationsChair with the Department of Teleinformatics En-gineering. In 2000, he founded and, since then, has

directed the Wireless Telecom Research Group (GTEL), which is a researchlaboratory based on Fortaleza, which focuses on the advancement of wirelesstelecommunications technologies. At GTEL, he manages a program of researchprojects in wireless communications sponsored by the Ericsson InnovationCenter in Brazil. He has edited one book, published over 100 conference andjournal papers, and deposited three international patents dealing with subjectssuch as radio resource management, cross-layer algorithms, service qualityprovisioning, transceiver architectures, and signal processing algorithms.

Prof. Cavalcanti is a Distinguished Professor of the Brazilian NationalScientific and Technological Development Council for his technology devel-opment and innovation track record. He also holds a Leadership and Manage-ment professional certificate from the Massachusetts Institute of Technology,Cambridge.

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