EuCAP 2006 - European Conference on Antennas & Propagation

Session: Session 5A10A - Optimisation Techniques and Synthesis (15j)
Type: Oral Antenna
Date: Friday, November 10, 2006
Time: 08:30 - 12:20
Room: Gallieni B

Seq   Time   Title   Abs No
1   08:30   Optimization Using Taguchi Method for Electromagnetic Applications
Elsherbeni, A.; Weng, w; Yang, F
The University of Mississippi, UNITED STATES

This paper presents a novel electromagnetic optimization technique based on Taguchi method. Taguchi method using the concept of the orthogonal array (OA) effectively reduces the number of tests required in an optimization process. Although this method has been successfully applied in many fields such as chemical engineering, mechanical engineering, IC manufacture, power electronics etc., it is not well known to the electromagnetic community, and only limited applications are available. It is the goal of this study to introduce Taguchi method to the electromagnetic community and demonstrate its great potential in electromagnetic optimizations.

The implementation procedure of the Taguchi method is presented in Fig. 1. The selection of a proper orthogonal array (OA), design of input parameters using OA, and process of creating a response table are addressed in this work. The proposed optimization procedure has been applied in designing linear antenna arrays, and desired antenna patterns are successfully achieved as shown in Fig. 2. Compared to other optimization techniques, such as the genetic algorithm (GA) and particle swarm optimization (PSO), Taguchi method is easy to implement, and can quickly converge to the optimum solutions.

Fig. 1. Flow chart of Taguchi method.

Fig. 2. Array factor of a 20-element linear array with a null controlled pattern.

2   08:50   Mode Converter Synthesis by the Particle Swarm Optimization
Bogdashov, A.A.; Rodin, Y.V.
Institute of Applied Physics, RUSSIAN FEDERATION

Particle Swarm Optimization (PSO) is an effective, simple and promising method intended for the fast search in multi-dimensional space [1]. Besides special testing problems a number of engineering tasks of electrodynamics were solved with the PSO successfully [2]. Scattering matrix technique is a very fast and accurate method for mode converter design. We illustrate PSO by a number of converters intended for high-power microwaves control: a TE01 miter bend, a matching horn for maser section, a design of built-in converter for multi-charged ion source.

PSO Algorithm
The PSO optimization procedure is based on the agent (particle) motion in the space of parameters. Let a system be described by a set of N parameters. Each set of parameters has respective fitness value F. The search for a global best state (max F) with particle swarm optimization involves a set of M virtual particles moving through the N-dimensional space. Particle positions and velocities are updated in accordance with the simple formulas [1].
In accordance with multiple PSO investigations Cp=2.041, Cg=0.948, w=0.729 were selected. Introduction of the damping factor w < 1 accelerates convergence of algorithm to the best fitness value.

Mode converter optimization (TE11-HE11)
We consider axially symmetric converters and use the fast scattering matrix technique for converter profile optimization. Smooth profile is approximated by a spline function with Np nodes (longitudinal coordinate j-th node position zj and radius Rj ). Input and output radii of converter are fixed (9.5 and 27.5mm). Fitness value calculated as a coupling coefficient of output field Eyi and desired one Ad, and the reflected power is taken into account.

Fig.1. PSO convergence for Np=5.

Fig.2. Mode content along the optimized converter (f=30GHz).

The mode converter was manufactured. Field intensity patterns were measured in a few 50mm separated cross-sections by the scanning detector at low power level. The phase was recovered by the Katzenelenbaum - Semenov algorithm. The coupling coefficient of the measured field pattern (with recovered phase) and the desired one (Ad) is 98.7%. Measured reflection coefficient is less than -25dB in the frequency range 30±0.5GHz.

Particle Swarm Optimization for design of TE01 miter bend
Preliminary reported design of the high-efficient TE01 miter bend (34.272GHz) was used as a testing task for PSO investigation [3].

[1] Kennedy J. and Eberhart R., "Particle Swarm Optimization", Proc. of the 1995 IEEE International Conference on Neural Networks, pp. 1942-1948, IEEE Press, 1995.
[2] J. Robinson, Y. Rahmat-Samii "Particle Swarm Optimization in Electromagnetics," IEEE Trans. Antennas Propagat., Vol. 52, No.2, pp.397-407, Feb. 2004.
[3] A. Bogdashov, G. Denisov, D. Lukovnikov, Y. Rodin and J. Hirshfeld. "Ka-Band Resonant Ring for Testing Components for a High-Gradient Linear Accelerator". IEEE Transactions on Microwave Theory and Techniques, vol.53, No. 10, October 2005, pp. 3152-3154

3   09:10   Constrained Least-Squares Optimization in Phase Synthesis of Conformal Array Antennas
Vaskelainen, L.
VTT Technical Research Centre of Finland, FINLAND

Constrained optimization problem for the unknown element excitations W in a conformal array antenna can be written in matrix form

Y contains the goal gain values in directions ) is the individual element gain value, rj is the place-vector of element and is the unit direction vector to direction . Constraint equations can be used to set some exact gain-values or to set any other linear constraints for W. The solution of this problem is known when the phases of Y and g are fixed. In this paper the minimized function used in the least-squares solution is studied. That part of this function, which depends on phases of Y and g, can be written in form

where , Qr,H and Qg are calculated from X, x and (diagonal matrix including weights for all directions used). Optimized phases can now be found by searching such phases for Y and g which minimize (2). An optimized coefficient to set the amplitudes of Y and g compatible must also be found. This optimization problem is fast to solve, because the matrix sizes are small (only the non-zero values of Y and g are necessary to use).

Phase synthesis problem can be solved by using x=1 (unit matrix) and as g the predefined values of W. In Fig. 1 left an 14x16 array is set on an elliptical cone and a super-elliptical contoured beam is synthesized using optimized goal-function phases. In Fig. 1 right the same synthesis problem is solved by using Gaussian distribution for element amplitudes and phase-synthesis for element excitation phases.

Fig. 1 Power-synthesis with goal-function phase optimization (left) and phase-synthesis with gaussian element amplitude distribution (right)

4   09:30   An Effective Hybrid Approach in Some Array Synthesis Problems
D'Urso, M.1; Isernia, T.2
1University, ITALY;
2Universita' Mediterranea di Reggio Calabria, ITALY

In the last years, a large interest has been devoted to the exploitation of global optimization techniques in electromagnetic design. Simulated Annealing, Genetic Algorithms and Particle Swarm Optimization have been widely applied to different problems, as design of lens, inverse scattering and to the synthesis of arrays, including the design of non-uniformly spaced arrays and to the so-called 'optimal compromise amongst sum and difference patterns' problem.

On the other side, the diffused enthusiasm for a wide these 'physically inspired' optimization techniques has induced to neglect, in a number of cases, some characteristics of the problems at hand which may be useful in the global optimization process. Global optimization techniques are in fact limited in their performances from their computational burden which arises very rapidly with the number of unknowns. As a consequence, a sub-optimal solution (rather than the globally optimal one) will be generally achieved in large scale problems. Such a circumstance, which is known to researchers involved in inverse scattering and non linear inverse problems, is usually underestimated in design problems, also on the basis that synthesis problems, opposite to reconstruction problems, may have many different satisfactory solutions

Moreover, as we will show during the conference, the (sub-optimal) solutions obtained in such a way can be significantly worse than the actually (globally) optimal solutions. For example, recent results achieved by the authors in the synthesis of pencil beams by means of non-uniformly spaced arrays and in the 'optimal compromise amongst sum and difference patterns' problem show that usual implementation of global optimization based procedures may keep significantly far from the actually globally optimal solutions.

Starting from the above, a new hybrid approach to array synthesis problems which exploit convexity of the problem with respect to a part of the degrees of freedom has been recently proposed and discussed by the authors. In particular, this strategy takes definite advantage from the convexity of the problem with respect to a part of the unknowns thus reducing the problem to a global optimization of a cost functional which is just a function of a part of the total number of unknowns. The hybrid approach has been successfully applied to the synthesis of pencil beams by means of non-uniformly spaced arrays and to the 'optimal compromise amongst sum and difference patterns problem', by achieving increased performances with respect to previous synthesis techniques

In this communication, we first recall in a unitary fashion the above mentioned approach. Then, we extend the already available synthesis procedure proposed for the synthesis of pencil beams by means of linear non-uniformly spaced arrays to the synthesis of difference patterns and pencil beams by means of planar arrays. Finally, the capability of the proposed hybrid procedure to get effective design solutions, by means of the concept of interleaved arrays, to the problem of realizing a dual beam array antenna with individually steerable patterns and arrays performing jammer rejection at the physical layer is also presented.

5   09:50   Study on the Reciprocity for the Maximum Directive Gains of a Transmitting Array and a Receiving Array
Kagoshima, K.; Obote, S.
Ibaraki University, JAPAN

This paper describes the reciprocity of a transmitting array and a receiving array when they have maximum directivities. This study may be useful and applicable to an optimum design of the transmitting array or the receiving array with interference incidence.

The optimum excitation coefficients of the transmitting array with arbitrary elements which maximize its directivity have been obtained by Inagaki et al. However, if null constraints in some directions are required, a solution for the excitation coefficients has not been achieved yet.

On the other hand, the maximum directivity of the receiving array may principally be obtained by using the knowledge of an adaptive array algorithm which determines its weight for the maximum SINR of the receiving array. Since the analysis of the maximum SINR has been carried out with consideration of mutual coupling between elements of the receiving array, it is expected that we can optimize the directivity of the transmitting array with null constraints through the reciprocal relation between the transmitting array and the receiving array. However, the directional patterns of the transmitting and the receiving arrays without interference incidence, which are equivalent to be designed for maximum directivities, did not coincide each other. Then, we investigated the reason of this discrepancy of the directional patterns between the transmitting and the receiving arrays.

To achieve the maximum directivity in the receiving array, we found that the load impedance of each element should be matched to the output terminal impedance of the array. We derived the equation for determining the matched load impedances and solved it for the case of a two element array. Comparing the calculated result of the receiving antenna patterns with two kinds of load impedances ,we have confirmed that the matched load impedances are necessary to realize the maximum directivity of the receiving array.

In the case of the three element array, the directive patterns do not coincide, even though the matched load impedances were used. To our knowledge of the receiving array, this is unexpected and difficult to be cleared. The key to solve this problem is the fact that the currents at the element terminals should be summed to obtain the receiving array pattern, instead of the voltages across the load impedances. If the load impedances are the same, the sum of the voltages are equal to that of the currents. However, since the matched load impedances are not all the same in the case of three elements, the voltages across the loads cannot be summed directly. We should insert impedance transformers which transform the terminal impedances into the same load impedances, for instance, 50 before summing. The calculated result shows the effect of the impedance transformers in the case of three elements, where the receiving pattern with maximum directivity agrees well to the transmitting one with the excitation coefficients for the maximum directivity.

Using the reciprocal relation described above carefully, we can realize the optimum design of the array with more practical and sophisticated requirements for the recent wireless communications.

6   10:40   Very Fast Subarray Position Calculation for Minimizing Sidelobes in Sparse Linear Phased Arrays
Lee, C.1; Leigh, D.1; Ryall, K.1; Miyashita, H.2; Hirata, K.2
1Mitsubishi Electric Research Laboratories, UNITED STATES;
2Mitsubishi Electric Corporation, JAPAN

We have developed a set of software tools which allow an antenna designer to interactively explore the space of subarray positions in a linear phased array with the goal of minimizing the maximum sidelobe level of the array response. The designer can use the tools to quickly visualize the response as he or she manipulates the subarray positions. The software aids this task by indicating which specific manipulations will most rapidly improve the response toward the goal. The user can also command the software to automatically optimize the array to the nearest local minimum starting from a specific subarray configuration. We combined the local search with a randomized starting point to search for globally optimal configurations. The results were consistent with our previous work which performed exhaustive searches to find the globally optimum solution for certain samples of the array parameters. In all cases we were able to find the global optimum, or a suboptimal solution a fraction of a decibel from it, within a few minutes on a common desktop computer.

We used the software and results above to seed a machine learning algorithm which optimized a model of the subarray configurations over an entire range of design parameters, providing a simple way to calculate a near-optimal subarray configuration given those parameters. The model used was a set of low-order polynomials which provided the position of each subarray within the physically realizable space of the array. The domain of each polynomial is one or more variable design parameters, such as the total length of the phased array. Each polynomial yields the position of one subarray within the array. The learning algorithm gradually refines the polynomials so that the subarray positions they give produce increasingly lower values of the maximum sidelobe level over the entire domain.

For the cases of sparse, linear phased arrays with total lengths ranging from 60 to 170 wavelengths, and with between 6 and 11 subarrays (each 10 wavelengths long and approximately uniformly illuminated) we were able to produce sets of third-order polynomials describing subarray positions that yielded results less than one decibel RMS from the sampled global optima. Because of the simplicity of the resulting models, it is possible to calculate near-optimal subarray configurations of a reconfigurable linear phased array using only the computational capabilities of a simple microcontroller. It is also possible for the designer to use the model for greater understanding of how changes in array parameters will affect the best achievable result for the array.

We discuss future work where these results can be applied, such as quickly calculated sidelobe minimization for electronic beam steering, reconfigurable arrays and approximate solutions for system planning purposes.

7   11:00   Optimal Design of Switched Beam Antenna Arrays Using Particle Swarm Optimization
Papadopoulos, K. A.1; Papagianni, C. A.1; Foukarakis, I.E.2; Kaklamani, D. I.1; Venieris, I. S.1
1National Technical University of Athens, GREECE;

Adaptive antennas techniques are emerging as a promising solution to a number of problems in applications related to mobile communications. This paper presents a method for optimal design of switched beam planar antenna arrays based on the Particle Swarm Optimization (PSO) algorithm. Switched beam antenna arrays are a subset of smart antennas that can enhance the capacity of a cellular system. PSO is a population based algorithm that exploits a set of potential solutions to the optimization problem, which are determined on the basis of social interactions between independent agents. The synthesis problem addressed is to find spatial and feeding configuration of array elements that conform to certain design constraints enforced on multiple diagrams of differentiated directive gains and values (beam forming). Two different fitness functions were evaluated on this basis; the proposed methodology is very flexible allowing the incorporation of additional factors such as coupling characteristics and feeding network constraints.

The design problem consists of finding the complex excitations of antenna elements as well as estimating their appropriate positioning along the x-y plane in order to produce certain radiation patterns according to established error criteria. This problem can not be handled by analytical methods due to difficult nonlinear constraints imposed. On the contrary, stochastic methods have proven to be more efficient since they do not require to approximate gradients of the error criteria. Employing stochastic algorithms such as PSO for the particular design problem requires the determination of an appropriate Fitness Function (FF) that quantifies the optimality of a solution. The designated pattern characteristics chosen to establish a suitable error criterion are beam steering angle, suppression of side lobes, half power beam-width, and beam-width at side lobes level. Based on these criteria two objective functions were used; the first one minimizes the deviation of the synthesized pattern from the design goals, while the second takes into consideration coupling effects (through voltage optimization) and additionally checks periodically side lobe level variation outside the main lobe.

The following figures present the results of the proposed optimization technique (FF 2) in the synthesis of an antenna consisting of five half-wave dipoles, for four radiation patterns covering the x-y plane with the main beams pointing at 0, 90, 180 and 270 degrees. Obtained patterns exhibited good agreement with the design goals and the approximately symmetrical positioning of the elements confirms these findings.

This design problem is of high complexity as for the case of 5-element array it demands the determination of 50 independent variables (10 for each radiator). Fitness Distance Ratio variant of the optimization algorithm which introduces a particle attraction factor towards nearby particles' best prior positions, was found to produce better results for the array synthesis problem. Also reflective boundary conditions were used to limit solutions within the given search space. The whole process involved the fine tuning of parameters such as population size.

8   11:20   Design of Frequency-Selective Radomes Using Parallel Particle Swarm Optimisation
Sabielny, M.
EADS - Defence Electronics, GERMANY

Particle Swarm Optimisation (PSO) is very much in the focus of interest inside the EM-community at present. PSO - developed by Kennedy and Eberhart while trying to model the swarm behaviour of bees for example has surprisingly revealed interesting capabilities as an optimiser in general and recently in the area of EM-optimisation. Particularly the aspect of designing Frequency Selective Surfaces (FSS) as part of a radome is quite difficult to handle due to the inherent nature of this kind of problem: multi-modality and multi-dimensionality. In order to attack these problems Genetic Algorithms (GA) have been used in the past in different implementations (GA, parallel GA, MicroGA, etc.). All these GA's offer a lot of tuneable system parameters affecting overall performance of the optimiser very much. This makes it rather difficult to find a suitable and / or optimal parameter setting of the GA. PSO in contrast is equipped with an appealing small number of available system parameters. This circumstance makes PSO very interesting for any application engineer.

The goal of this contribution is to demonstrate the successful application of PSO as a state-of-the-art tool in the design process of a planar and unlimited pass-band radome. The first part of the paper will cover the underlying FSS-calculation-kernel as well as the applied concatenation technique for the overall radome structure, consisting of an arbitrary number of dielectric and FSS layers. As opposed to other FSS-kernels this one works in the spatial domain using RWG basis functions in order to model any possible shape of FSS-elements, either rounded or straight up to a high level of accuracy. By this procedure the well-known stair case effect occurring in FSS-discretisations made by roof-top basis functions does not play an important role anymore. The main optimisation parameters are subsequently exposed. In contrast to latest developments of FSS-elements together with GA's only well known and not stochastic FSS-elements are used since the EM-performance of these FSS-elements is much more predictable in this way.

The second part of the paper will explain the implementation of the PSO which has been programmed from scratch and exclusively as an MPI-parallelised computer code running on a distributed computer cluster system. This part of the contribution will also include a comparison of different possible cost (or target) functions and their respective performance in the context of the parallel PSO. A suitable and representative design example will further underline the inherent capabilities of this approach.

Finally weaknesses of this PSO-concept are outlined which have been identified during this study and which have to be attacked in future developments. Design experiences of the author are added - if appropriate - throughout the paper.

9   11:40   Fast Optimization of 3D Focusing System Using Genetic Algorithm
Ney, M.; Jehamy, E.; Landrac, G.

Artificial constrained lenses have received much interest for potential use in millimeter-wave ACC radar systems. For next generation radar, it is considered to utilize high gain antennas with beam reconfiguration that allows both long and short-range operations. For instance, primary sources distributed in the focal plane can be switched to perform beam scanning over a wide angle for short-range operation. In this case, a good resolution can be obtained and the system can also work for long-range operation with no additional complex hardware.

Artificial constrained lenses used as focusing element has obvious advantages, including low cost process when implemented in foam technology [1]. Primary sources are distributed in the focal plane to allow beam tilting and can be easily implemented in planar technology. However, as the beam is tilted from normal angle, aberrations increase yielding a gain reduction and higher dissymmetric side lobes. These affect the performance of the radar system. As a result, the lens profiles must be modified to decrease aberrations for off-normal beam operation. Full-wave analysis cannot be considered for direct optimization as the antenna system is about 12 wavelengths in dimension, thus requiring exhaustive computer cost. Instead, a technique based on ray-tracing is used to compute the phase error produced by the focusing element. Different techniques are available to minimise the phase error. For instance, variational methods have been proposed [2]. However, in addition to complex mathematical manipulations that they require, there are severe limitations concerning the degrees of freedom that would be required to obtain satisfactory results.

In this paper, these limitations are discussed and one proposes instead an alternate approach: First, genetic algorithms are used to optimize the lens profiles. Such algorithms have been already proposed for two-dimensional lenses [3]. It proposed to extend the technique to the general three-dimensional case. Examples of optimisation with GA will be shown and validate the adequacy of the technique as compared to the variational approach. Once the profiles have been obtained, a full-wave analysis based on Boundary Element Method is performed to the whole structure to account all other phenomena such as coupling with primary source and diffraction interactions between plates that compose the lens (see figure 1). Finally, measurements made in a millimetre-wave anechoic chamber will be shown to validate the model and optimization technique.

Figure 1: Artificial lens elements and primary source for long-range testing in foam technology.

[1] E. Jehamy, G. Landrac, S. Pinel, B. Della, F. Galle, M. Ney, "A compact constrained metal plate lens for anti-collision radar at 76 GHz, Proc. Int. Conf. on Antenna INICA, Berlin, October 2003.
[2] E. Jehamy, M.Ney, G.Landrac, "Rduction des aberrations des lentilles dilectrique artificiel", 15mes Journe nationale des micro-ondes, Nantes, France, 11-13 May 2005.
[3] E. Jehamy, G. Landrac , M. M. Ney, M. Salaun, "Genetic algorithm and full wave design applied to metal plate lenses at 76GHz", ANTEM-2005, Saint-Malo, France, 15-17 June 2005.
10   12:00   Waveguide Antenna Synthesis According to the Amplitude Radiation Characteristics in the Frequency Band
Andriychuk, M.; Zamorska, O.
Inst. of Appl. Probl. of Mech. & Math., NASU, UKRAINE

A) Analysis Problem. The antennas with waveguide excitation, namely the plane waveguide array and resonant waveguide antenna, are considered in the paper. The geometrical parameters of array are presented in Fig. 1. We write down the radiation pattern (RP) of such array using expression for a far field of radiating flat aperture with elliptic polarization [1]. The field in a far zone of antenna with flat aperture can be presented accordingly to [2]. Using considerations similar to the case of waveguide array, we receive the expression for the resonant antenna RP. B) Synthesis Problem. During the process of statement of the synthesis problems in the given frequency range, as a rule, it is required to provide the best approximation to the prescribed amplitude RP in an operating frequency range and the minimal values of the front-to-rear factor (FRF) outside of this range. For the waveguide array, the functional which allows to satisfy such requirements was introduced in [2] It is necessary to determine the factors of excitation currents in separate waveguides of array, which minimize the functional [2].

In the case of resonant antenna the least value of FRF in the operating range is specified as criterion of optimization, and this value is maximized by a choice of parameters, which describe the geometry of antenna. We use the minimax functional as the criterion of optimization.

C) Method of Solution. Minimization of the functional [2] allows to receive the system of nonlinear equations for the unknown excitation factors . Having substituted this expression into formula for calculation of the RP, we receive the nonlinear integral equation for synthesized RP. This equation is solved numerically by the method of successive approximations. Optimization of this functional is carried out by the especially modified coordinate descent method.

D) Numerical Modeling, Conclusions. The numerical calculations for modeling the optimal RP of array with various namber M of the exciting waveguidwes in the wide frequency range were implemented. The results of numerical optimization of parameters for resonant antenna were carried out for the problem of the FRF factor maximization in range of 5% in neighborhood of the central frequency.

The considered optimization problems of waveguide antennas give the possibility to take into account the various requirements to the both RP and FRF in the operating frequency range. The developed algorithms enable to achieve the minimal mean-square deviation of the prescribed and synthesized amplitude RPs, and to optimize (to maximize or minimize) the values of FRF for one range, as well as for the several frequency ranges.

[1] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 1997.
[2[ M. I. Andriychuk, N. N. Voitovich, P. A. Savenko, V. P. Tkachuk, The antenna synthesis according to the amplitude radiation pattern. Numerical methods and algorithms. Kiev: Nauk. Dumka, 1993. (In Russian).