Optimisation of an unequally spaced dual-band printed base station antenna array using a marginal distribution tech n iqu e

P.L. Starke and G.G. Cook

Abstract: An efficient optimisation algorithm is presented for printed antenna design which aims to reduce the number of simulation runs required to find an optimal solution. Such efficiency is important in antenna design because of the hgh computational cost of each simulation run. The algorithm makes use of marginal distributions to guide the optimisation process, and its real-valued (rather than binary) representation of optimisation parameters makes it directly applicable to a wide range of antenna design problems. The particular application presented is for the design of an unequally spaced dual-band printed cellular base station array. The optimisation algorithm successfully chooses spacing and element length values for the array to meet radiation pattern requirements in both the GSM900 (88G960 MHz) and DCS1800 (1710-1880 MHz) bands. Theoretical results for the optimised printed wire array are compared with measurements.

1 Introduction

In the design of base station antennas for modern mobile communications systems, a large number of design goals and constraints must be satisfied. These include multiband capabilities (e.g. GSM900, DCSl800 and UMTS), tight restrictions on radiation pattern, and minimum visual impact. Satisfying all of these requirements simultaneously is a challenging design problem, and the use of optimisation algorithms to aid in the design process becomes highly attractive.

For a number of years, optimisation algorithms have been used in the design of antennas and scatterers because of their ability to find solutions to complex nonlinear problems and, in array design, classic examples include the thinning of large arrays [l] and the design of Yagi-Uda arrays [2]. Further publications demonstrate the ability of optimisation algorithms in sidelobe reduction [3] and general array pattern synthesis [4].

In cases where the number of optimisation parameters is large, stochastic optimisation algorithms (such as the genetic algorithm) are considered to be superior to traditional gradient-based techniques, which are more likely to become trapped in local minima [5]. As a result, the genetic algorithm (GA) is now widely used in electromagnetic design and comprehensive reviews of early work in this area are available in the literature [6, 71. The GAS used in published electromagnetics work employ either a binary or real-valued encoding of optimisation parameters, and make use of traditional crossover operators (single or multi-point crossover for binary-coded GAS and some form of linear crossover for real-coded implementations). While these techniques have proved to be robust, they are not

necessarily efficient and may require an excessive number of iterations before converging on a good design. Ths is particularly problematic when printed antennas are in- volved due to the significant computational expense involved in each simulation run [8]. Developments in optimisation algorithm research suggest that more efficient techniques could be achieved if statistical information from the optimisation process is used directly to guide the search for improved designs [9, 101.

In this paper we introduce a novel and robust optimisa- tion algorithm making use of such statistical information. The algorithm uses a direct real-valued encoding of design parameters and is suitable, amongst other things, for the optimisation of multi-variable antenna problems. The particular application presented is the optimisation of a dual-band printed-wire base station array. Dual-band antenna design has been addressed in numerous publica- tions, but these have generally focused on single elements rather than arrays [I 1-13]. A recent publication presents a dual-band printed base station array for use in the Japanese cellular telephone network [14]. However, at 5.4m ths is too long for use in the GSM network and the paper does not address E-plane (vertical) radiation pattern shaping. Here we optimise a nonuniformly spaced printed array antenna for dual-band operation with shaped E-plane (vertical) radiation pattern. The optimisation of nonuniform spacing is presented as an alternative to the standard approach in whch element amplitudes and phases are varied to control the pattern of a uniformly spaced array. This approach has the potential to simplify the design of the feed network. Added novelty is derived from the use of an original printed wire method of moments (MOM) code as the electromagnetics engine.

0 IEE, 2002 IEE Proceedings online no. 20020539 DOI: IO. 1049/ip-map:20020539 Paper first received 15th August 2001 and in revised form 22nd April 2002

The authors are with the Communications and Radar Group, Department of Electronic and Electrical Engineering, University of Shefiield, Mappin Street, Sheffield SI 3JD, UK

2 Optimisation using marginal distributions

The optimisation using marginal distributions (OMD) algorithm makes use of the simple observation that a good solution to a particular design problem is likely to be similar (at least in some respects) to designs that have previously shown potential. This observation forms the basis of other

IEE Proc.-Micr~iw Antennus Propug., Vol. 149, No. 4, August 2002 21 1

stochastic search algorithms, such as the GA [7], but OMD is unique in the way in which it makes explicit use of distribution statistics to guide the search for an improved design.

In a design problem involving N real-valued parameters (the element lengths and spacings of an antenna array, for example), each potential design can be treated as a location within an N-dimensional search space. We then expect that a good solution (for example, a design having a radiation pattern close to the specification) is likely to be found near to previously promising solutions. OMD makes use of this hypothesis by ensuring that the intensity with which it searches a particular region of the search space is proportional to the density of promising designs already found in that region. This weighted exploration of the search space is marginalised by considering the densities along each of the N axes (i.e. design parameters) independently when selecting new parameter values for the next generation of design candidates. Initially, of course, the algorithm will have no information about the location of promising designs, so the first step is to generate a set of design candidates spread randomly across the search space. These candidates are then evaluated (using a printed wire MOM code in this case), and the A4 best candidates are retained for use in the next stage. These candidates form an elite set.

Up to thls point, the algorithm is very similar to the standard GA. The difference now comes in the manner in whch the set of elite candidates is used to generate the next set, or population, of design candidates. In the GA, this is done using a crossover operator applied to randomly selected pairs of elite candidates. In OMD, the marginal distributions of the elite set over the search space are used. Essentially, each parameter value (e.g. the length of a particular array element) for the new population is generated according to a statistical distribution that approximates the marginal distribution of that parameter value amongst the members of the elite set. The size of the elite set thus clearly affects the ability of the algorithm to model the design problem being optimised. In practice, of course, it is difficult to quantify the complexity of a design problem, but experience suggests that values of M between 30 and 100 will suffice. The population size may also be chosen in this range, and is generally given the same value as M. The selection process for the updated elite set then involves selecting the best M candidates from the pool comprising the existing M elite members plus the M members of the newly generated population.

Fig. 1 illustrates a particular distribution of elite candi- dates for a two-dimensional optimisation problem. By way of example, the parameters to be optimised are chosen to be the lengths (L,, L2) of elements in a two-element array (the spacing between the two having been fixed). In thls case, the elite set might be chosen to include candidates producing the hghest gain, say, using the MOM code to evaluate performance. The marginal probability density functions associated with this elite set are shown alongside, and it is clear how the peaks of these functions correspond to concentrations of elite candidates in the search space. When it comes to generating a new array design, the length of the first element will be chosen using a random number generator with probability density function @df) matching pdf(L,). A similar procedure will then be used to choose b, t h s time using pdf(b). In this way it is ensured that the length values chosen for the new population are most likely to be close to the more often occurring length values of the elite set. Thus with OMD statistical information about each element length, taken independently over the entire elite set,

L2

0-

marginal probability density functions

0 u \ N Li

marginal probability density functions

0 \ N

L1

Fig. 1 distribution of elite cundidutes

Marginal distribution functions associated with LI particular

is used in determining the element lengths of the next population. In contrast, a GA would generate element lengths using a crossover operator applied to randomly selected pairs of elite candidates. It should be pointed out that because marginal distributions, rather than multivariate distributions (such as in MIMIC [9]) are used, the values for LI and L2 may each correspond to different clusters within the search space. Ths makes it possible for good aspects of different elite clusters to be combined in new candidates, potentially resulting in designs with even better perfor- mance.

After generating the new population, candidates are evaluated using the MOM code, and the elite set is updated to reflect the best candidates so far found. The whole process is then repeated, and the algorithm terminates as soon as a satisfactory design is found - one having the specified gain, for example. To discourage early conver- gence on sub-optimal parameter values (i.e. a sub-optimal design), an extra degree of variance is introduced by adding a Gaussian distributed zero-mean random number to each design parameter of each new candidate. The variance of this random number (also referred to as the mutation level or forgetting factor) is varied adaptively [ 151.

In terms of underlying philosophy, OMD is similar to the univariate marginal distribution breeder genetic algorithm (UMDBGA) [lo]. UMDBGA also makes use of marginal distributions to generate new candidates; however, it assumes that the marginal probability density functions are zero outside the domain of selected candidates (i.e. the elite set). In contrast, OMD ensures that the marginal probability density functions are nonzero (even if small) over the entire search space. This ensures that those regions of the search space that have not yet produced elite candidates are not completely excluded from the future search. The principal difference between OMD (as well as other distribution based algorithms) and standard GAS is that, in the latter, only two elite candidates are used to produce each new candidate. The genes of these two are combined using a crossover operator to form the new candidate. In contrast, OMD makes use of information from the entire elite set (namely, its marginal distributions) when generating each new candidate.

3 MOM code for evaluation of candidates

To apply the OMD algorithm to a real world design problem, it is necessary to have available some means of

212 IEE Pruc-Micruiv. Aiireniurs Prupq., Vol. 149, Nu. 4, Augicsl 2002

evaluating each of the candidate designs generated during the optimisation process. Since, in the present case, we wish to consider the optimisation of a printed wire array, we have developed a highly efficient MOM code specifically to analyse the printed collinear array structure. Ths helps to reduce the computational burden of evaluating the large number of arrays considered during each optimisation run. The MOM code makes use of a moment method solution for thin printed wire antennas, and assumes an infinite ground plane [16]. Thus assumption makes the code computationally efficient and is reasonable for the predic- tion of the E-plane pattern of a collinear printed dipole array, since finite ground plane artefacts will decrease with array length. Further computational efficiency is achieved by pre-computing the Sommerfeld-type integrals contained within the Green's functions of the MOM impedance matrix calculations. A polynomial interpolation routine allows all of the required integral values to be interpolated from a relatively small set of pre-computed values. The code also examines the wire geometry and identifies any symmetry present. Ths symmetry information is then used to eliminate all redundant calculations from the MOM impedance matrix computation. To further expedite the analysis of each array, the feed network is not included in the array model.

4 Dual-band base station array optimisation

The particular design problem addressed in this paper is that of a dual-band base station array for use in the GSM900 (880-960 MHz) and DCS1800 (1710-1880MHz) bands of the digital cellular telephone network. We have chosen a collinear printed wire dipole array as the basic design to be optimised, and a section of this array is illustrated in Fig. 2, whch shows the wire elements printed on a grounded dielectric substrate.

Fig. 2 Optimisation parameters are shown

Section from an arruy of printed dipole elements

4. I Fitness function Design objectives for the optimisation process are derived from typical requirement specifications given for base transceiver station (BTS) arrays in the GSM mobile telephone network. In particular, the vertical radiation pattern specifications presented in Table 1 for both the

IEE Proc-Micron;. Antennas Propag , Vol. 149, No. 4, August 2002

Table 1: Summary of vertical radiation pattern require- ments for dual-band base-station antennas

GSMSOO requirements

Frequency range

Peak gain

Beamwidth

Side lobes in 20" sector above main lobe

Side lobes in 90" sector below main lobe

Radiation in rear sector

All other side lobes

880-960 MHz 15.5 dBi

< 10.5" to -3dB points

<17.5" to -10dB points

I> 18dB down on main lobe

unrestricted

>20dB below main lobe

7 13dB below main lobe

DCS1800 requirements

Frequency range

Peak gain

Beamwidth

Side lobes in 20" sector above main lobe

Side lobes in 90" sector below main lobe

Radiation in rear sector

All other side lobes

171C-1880MHz

17.5 dBi

<6O to -3dB points

<11.5" to -10dB points

18dB down on main lobe

unrestricted

>20dB below main lobe

13dB below main lobe

GSM900 and DCS1800 bands are used as the basis for a fitness function that produces a numerical measure of how closely a particular design meets these requirements. The optimisation process involves selecting the lengths and spacings of the elements of the array, which the MOM code then uses to calculate antenna performance metrics such as radiation pattern and input impedance. These metrics are subsequently input to the fitness function for evaluation. The fitness function used in the array optimisation consists of a weighted sum of penalty values, each of whch reflects the difference between the actual and desired performance of the array. For example, one penalty value is the absolute difference between the desired main lobe gain and the computed gain for a particular array (in dB). Other penalty values relate to the beamwidth of the mainlobe, the sidelobe levels in the 20" sector above the main lobe and elsewhere, and the input impedance of each element. Since the aim is to design a dual-band antenna, these penalty values are calculated at both 920 MHz and 1800 MHz and then added together. A total penalty value of zero indicates that the antenna meets all the requirements, although a minimum nonzero value may be the best attainable. The input impedance penalty value is calculated as the average ratio of element reactance to element resistance,

This encourages OMD to choose dipole lengths that are near resonant in the two operating bands.

4.2 Optimisation process The printed wire array is optimised using OMD in conjunction with the fitness function and MOM code. The design parameters given to the optimisation algorithm are the lengths and spacings of the elements in the array. However, to constrain the dimensionality of the search

213

space, all elements are of equal length, so that only one length parameter needs to be included. It is decided, based on gain considerations, to use 11 elements in the array, and there are thus 11 design parameters to optimise, i.e. the single element length and 10 inter-element spacings, producing 11 marginal distribution axes.

Fig. 3 illustrates the optimisation process. OMD chooses length and spacing values for each design candidate, using the marginal distributions of the elite set, and the MOM code then calculates the radiation pattern and input impedance values at both 920 MHz and 1800 MHz. These performance measures are used to compute a fitness penalty value for each design, and OMD then compares the fitness values of the current designs with those of designs included in the elite set. Any new designs that outperform elite set members (i.e. have lower penalty values) are inserted into the elite set and the superseded designs are removed. The process repeats until an array design is found that completely meets the requirements (i.e. zero penalty value), or some maximum number of iterations is completed.

Optimisation runs are performed on a 700 MHz Pentium I11 and comprise 70 generations of 40 candidates each. The number of candidates in the elite set is also 40 and the initial mutation level is set at 5%. One run takes - 15 h. For the dielectric, a 20 mm thick PTFE substrate ( E ~ = 2.5) is chosen and the printed elements have an equivalent wire diameter of 0.5". A 5" electrical downtilt is imposed on each design by varying the phase of the element feeds using point source theory, prior to fitness evaluation. The element driving phases are not therefore optimisation parameters, and all driving voltages have unit amplitude.

I

n begin

generate new array .-i distributions designs from marginal

I I

calculate phasing for required downtilt

compute radiation patterns and input impedances at 920 MHz and 1800 MHz

calculate fitness values for each array

update elite set 4 no yes

is the best design satisfactory?

Fig. 3 Array optimisation process

5 Results

5. I Simulations Figs. 4 and 5 show E-plane radiation patterns computed by the MOM code using the optimised array element lengths

90

2o h 60

v - 6 0

-90

Fig. 4 E-plane radiation pattern jor optitnised squinted array at 920 MHz Pattern for a uniformly spaced array is also shown - optimised . . . 'uniform ~~- requirements

90

2o ?--'-.. 60 15

10

E 5 r^ '1 0

-5

-10

-15

D

- 90

Fig. 5 E-plane radiation pattern for optimised squinted array at I800 MHz Pattern for a uniformly spaced array is also shown - optimised . . . 'uniform

_ _ _ requirements

IEE Proc-Microw Antennas Propug., Vol. 149, No. 4, August 2002 214

and spacings at the centre frequencies of the two operating bands. The design specifications, as well as the radiation pattern for a uniformly spaced array (optimised only for two parameters instead of 11 - a common element length and spacing), are also displayed, and it is clear that the optimisation process has successfully chosen parameter values that meet the radiation pattern requirements of Table 1. It is also clear that, without optimisation of individual element spacing, a simple uniform array would not meet these requirements. As shown in Figs. 6 and 7, further simulation of the optimised antenna using the MOM code reveals that the radiation patterns remain within specification across both bands, apart from two minor violations of the sidelobe restrictions. Typical calculated element impedances for the printed dipoles are 10-j40 Q at 920MHz and 170&j880Q at 1800MHz, and they do not vary significantly between elements.

90

0

-90

Fig. 6 across GSM900 band - 880 MHz . . . . 920 MHz

~ _ _ requirements

E-plane radiation patterns for optimised squinted army

- . . - 960 MHz

5.2 Measurements To facilitate antenna measurements, a second design is optimised without beam squinting and using a cheaper polypropylene substrate (E, = 2.2). A prototype of this optimised array has been fabricated on a copper-backed block of the polypropylene (dimensions: 20 x 500 x 1900 111111). Dipoles are made from 0.25 mm diameter tinned copper wire, and are fed through the ground plane and substrate, as shown in Fig. 8. A microstrip feed network fabricated on standard FR4 PCB material is then used to distribute the signal to the 11 element feeds at the back of the ground plane. Measured radiation patterns are compared with computer simulations in Figs. 9 and 10.

6 Discussion

The OMD algorithm has been particularly successful in reducing sidelobes in the critical sector above the mainlobe,

IEE Proc.-Microrv. Antennus Propay., VoL 149, No. 4, August 2002

90

h 60 15

10

5 5

.s 0 m

- 5

-10

-1 5 -

0

v L 6 0

-90

Fig. 7 E-plane radiation patterns for optimised squinted army across DCSlSOO band

1710MHz . . . . 1800 MHz ~. . _ 1880 MHz _ _ _ requirements

-

wire dipole twin wire feed

A , radius 0.25 mm)

+ polypropylene L

I copper groundplane

Fig. 8 prototype

Twin-wire feed conjiqurution for dipole elements in untenna

and in meeting the other specified E-plane radiation pattern requirements in simulation. However, the vastly different matching requirements of the printed dipole elements in the two operating bands would make it difficult to develop efficient feed networks for the optimised arrays. This problem can be overcome by replacing the simple printed dipole elements with true dual-band printed patch elements, which is the subject of future work. Referring to Table 2, it is seen that the optimised designs have inter-element spacings that are smaller near the middle of the array and larger towards the ends. Ths agrees with an observation made in the design of larger arrays that tapering element density towards the ends of an array helps to reduce the near-in sidelobes [l]. This is clearly evident in the optimised 1800 MHz radiation pattern of Fig. 5.

The measured array patterns display reasonable corre- spondence with those simulated around the main lobe, but show some higher than expected sidelobe levels on either side. This is due primarily to phase errors in the feed network, which had to be fabricated as seven separate PCBs and then soldered together. These errors were particularly severe at 1800MHz. More precise fabrication of the feed network, together with a better anechoic measurement environment, would improve agreement. The use of element

215

p- 60

- 90

Fig. 9 920 MHz for non-squinted array prototype - theoretical ’ . . .measured

Computed and mecisured E-plane rudiution patterns at

90

60

-5

9 -10 .- m n,

-15 ._ 2 g -20 -

b -25

- 30

- 90

Fig. 10 1800 MHz for non-squinted array prototype - theoretical . . ‘measured

Computed and measured E-plane radiation patterns at

spacing for pattern shaping simplifies the feed network design to an extent, since all elements are uniformly excited for a nonsquinted array. The measurements, though not yet satisfactory, do nonetheless serve as an indication that the optimised design should be realistically achievable with better manufacturing tolerances, such as those one might find commercially.

216

Table 2: Array geometries for the two optimised antennas

Optimised Value, cm

parameter Squinted Nonsquinted

L1, L2, ..., L11

s1,2

s2.3

s.4

s4.5

s5.6

s6,7

%,8

S8.9

.%lo

SlO.11

11.13

20.6

20.0

16.6

13.8

13.8

13.4

15.5

12.9

21.0

20.6

11.23

20.9

20.1

15.6

15.1

14.0

14.1 14.2

13.0

18.5

21.0

7 Conclusion

A new statistical optimisation technique driving an efficient MOM code is presented and applied to the synthesis of a dual-band 900 MHz and 1800 MHz cellular base station array antenna as an illustration of its potential. The algorithm is able to successfully specify an array that will meet prescribed radiation pattern requirements in the two bands, by variation of element spacings and a common element length. The printed dipole elements used in this work do not easily facilitate a practical dual-band matchng arrangement, although the concept of tailoring the radiation pattern by varying element spacings rather than driving functions should simplify feed network design. Future work will look to overcome this problem through the use of stacked patch elements and a printed patch MOM code. The use of the printed patch code will also allow an integrated feed network to be included in the optimisation process, though at significant computational expense.

8 Acknowledgments

Funding from Vodafone supports this work. The authors would also like to thank A.J.T. Whitaker, D. Cox and M. Sidik for their assistance during the fabrication and measurement of the prototype antenna.

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