The proposed quasi-analytical method undertakes the shaping of a desired footprint as a composition of several phi-symmetric circular Taylor patterns exhibiting flat-topped beams. The method starts by uniformly dividing the desired footprint pattern in angular sectors. For each sector the average beamwidth is calculated and a phi-symmetric circular Taylor pattern with a similar beamwidth is synthesized. The pattern is composed of a set of these circular footprints and the corresponding circular aperture can be obtained analytically. The final pattern is obtained after sampling this circular aperture for application to rectangular grid-circular boundary planar arrays.

The method has been applied to synthesize a footprint pattern with a coverage area of continental Europe. The desired footprint pattern was obtained using 200 sectors and 10 different real flat-topped circular patterns with 2 to 11 filled nulls, all of them with -25 dB sidelobe level and ±0.1 dB ripple level. A real circular aperture of radius 100ƒÜ was used and later sampled for a planar array of 64128 elements using a rectangular grid of 0.7£Ωf. In order to reduce the number of elements as well as the dynamic range ratio |Imax/Imin| of amplitude excitation to 50, we remove those weakly excited elements from the array, yielding a planar array of 15169 elements only, without reducing the pattern performance of the initial array. The final pattern (Fig. 1) has a -20.0 dB sidelobe level and ±0.6 dB ripple level in the coverage area.

The use of Global Search Optimization Methods to solve electromagnetic problems has been widely extended. Among them the most popular for the antenna community are Genetic Algorithms (GA) and recently Particle Swarm Optimization (PSO). Algorithms based on these methods have been used to afford the design of arrays and other types of antennas.

Another family of global search algorithms also based on social behaviour is the Ant Colony Optimization (ACO) that was introduced by Dorigo in 1991 and it is based on the ant colonies behaviour to obtain food and carry it back to the nest. It is a “shortest path” based algorithm. When ants are looking for food they give off pheromone on the ground. The pheromone are chemical substances that increase the probability of other ants to follow the way chosen by the previous ants. The shorter the trail from the nest to the food source, the higher the pheromone concentration level and thus the probability of ants choosing that path. Although these algorithms have been applied in many different problems their use by the antenna community has been limited. So, it is the purpose of this work to show the application of an algorithm based on ACO (using real numbers) to different antenna problems. Firstly, to a classical electromagnetic problem: array synthesis both linear and planar and with different design criteria. In all the cases the algorithm performance and convergence was demonstrated to be as powerful as GA or PSO. Secondly, we have successfully applied the algorithm to design different monopolar Ultra Wide Band microstrip antennas. The last application was in the design of a planar truncated periodic structure to reduce E-plane mutual coupling in a multilayer patch antennas array (Figure 1 and 2). Antennas work at 3GHz and the achieved mutual coupling reduction was of more than 10dB (having a final mutual coupling smaller than -35dB for elements separated 0.75wavelengths).

In the last two examples we have used the CST Microwave Studio program as the analysis tool to calculate our "desirability " functions, i.e., our algorithm interacts with CST. Results are very promising and the application field unlimited as for the other evolutionary algorithms so this can help this type of algorithms to join these other popular evolutionary optimization techniques (GA, PSO,...) as a useful tool for the antenna designer. The algorithm has common features with all of them as they perform a population-based search with probabilistic transition rules. The ACO simplicity is one major advantage.

An innovative method for linear arrays beam configuration is presented. In the proposed method, every element of the linear array is connected to its feed through a switch, so it can be active (with a voltage given by V_n) or passive, as Fig. 1 shows. Pattern reconfigurability is achieved by appropriately switching on or off the array elements. The optimal configuration of the switches for each of the radiated patterns as well the common voltages of the N elements are calculated by using a genetic algorithm. For each configuration, the currents in the driven and parasitic elements are determined via their self and mutual impedances, by inversion of the impedance matrix. In the example presented, we considered a linear array composed by 20 lambda/2-dipoles parallel to the z axis and equispaced lambda/2 along the x axis, with a ground plane located lambda/4 behind the array. This antenna will switch the power pattern from a pencil to a flat-topped broadside beam, using the switches configurations shown in Table 1 ('1' means a driven element or "switched on", whereas '0' means a parasitic element or "switched off"). Fig. 2 shows the patterns radiated by this antenna: a pencil beam with -16.3 dB sidelobe level and a flat-topped beam pattern with 42 deg. beamwidth measured at −3 dB, ±0.75 dB of ripple and SLL=−17.5 dB.

Introduction. The analysis of antennas by the Time-Domain Integral-Equation (TDIE) method becomes popular recently. Exciting the antenna by a narrow Gaussian pulse, TDIE can produce values of an observed quantity on all the frequencies covered by the pulse spectrum. Compared to the method of moments, which has to be run separately on each harmonics of interest, TDIE can provide higher efficiency [1].

Problems. Due to the frequency domain nature of antenna parameters, time responses of computed quantities have to be converted to the frequency domain, and here, the objective function is formulated. In order to eliminate the necessity of Fourier transforming time responses in each iteration step, the objective function has to be composed in the time domain directly [2]. Since several requirements on antenna parameters are conflicting, the optimization procedure is asked to provide sets of Pareto-optimal solutions in the global sense.

Methods Used. So far, the multi-objective cost function was composed in the frequency domain, and Pareto-optimal solutions were computed using evolutionary algorithms (EA) [3] and particle swarm optimization (PSO) [4]. In the proposed method, TDIE is applied to analyze planar scatterers; low-pass filters process time-domain responses to eliminate dispersion errors, and the filtered responses are used for computing mean-valued time-domain directivity patterns, gains and scattering parameters. A multi-objective cost function combines the specified objectives, and the Pareto-optimal set of solutions is obtained by the combination of EA and PSO.

Original Results. The algorithm of the time-domain multi-objective optimization of planar antennas is the principal original idea presented in the paper. Combination of PSO and EA for obtaining mean-valued time-domain Pareto-optimal solutions is another original contribution of the paper.

Verification. The planar antennas of interest are analyzed by the method of moments in the frequency domain, and computed parameters are understood as a reference solution. The reference solution is compared to time-domain ones, which are mapped into the frequency domain by FFT.

Conclusion. The developed time-domain multi-objective optimization of planar antennas can be used for broadband synthesis of planar radiating structures. The proposed synthesis method exhibits good efficiency, and satisfactory accuracy. The accuracy can be increased by tuning the solution in the frequency domain.

References

[1] POLJAK, D., THAM, C. Y. Integral Equation Techniques in Transient Electromagnetics. Southampton: WIT Press, 2003

[2] RAIDA, Z. et al. Broadband characterization of antennas: suppression of analysis inaccuracies in the time domain In Proc. of the 9th International Conference on Electromagnetics in Advanced Applications ICEAA 2005. Torino: Polytecnico di Torino, 2005, p. 201–204

[3] DEB, K. Multi-Objective Optimization using Evolutionary Algorithms. Chichester: J. Wiley & Sons, 2002.

[4] LUKEŠ, Z., RAIDA, Z. Multi-objective optimization of wire antennas: genetic algorithms versus particle swarm optimization. Radioengineering, 2005, vol. 14, no. 4, p. 91-97.

The application of genetic algorithms (GAs) as optimization tools for the design of antennas has been an active field of research in the past decade. The main reasons for this interest are related to their robustness, enabling the solution of optimization problems for which local techniques of optimization are not effective, as well as their versatility, permitting the successful use of the same schemes to different problems. There are, however, inherent restrictions to the applicability of the GAs. As a consequence of their structure, the problems, for which high computational times are needed to accurately simulate each possible solution, remain yet excessively costly. To overcome this problem, several efforts have been devoted to find more efficient optimization schemes, resulting not only in improved versions of the genetic algorithms, e.g. micro-genetic algorithms and hybrid taguchi genetic algorithms, but also in new global techniques of optimization derived from different philosophies, e.g. particle swarm optimizations and ant colony optimizations. These improvements, along with the increasing capability of computers and the development of parallel codes, have led to satisfactory solutions for more complex problems. Moreover, in problems for which the accuracy of the optimized solution is not critical, a usual procedure for decreasing the total computational time is to reduce the computational burden of the models by, for example, using a coarse meshing of the computational grid in simulators based on finite element methods or decreasing the number of basis functions in codes applying the method of moments. Unfortunately, an estimation of the error introduced by these approaches is often difficult to make. In this communication, an additional stage is introduced in the optimization procedure to assure the accuracy of the final result. This additional step, based on the space-mapping (SM) technique, allows GA operators to employ coarse models in the simulations to find an approximate solution of the problem. Once attained, SM provides an accurate solution of the problem with a relatively low computational cost. As an example of optimization the hybrid algorithm GA-SM is used to select the lengths and feeding points of an array of 3x3 patch antennas on a finite ground plane. The results show the advantages of applying this scheme. REFERENCES [1] Y. Rahmat-Samii and E. Michielssen, Eds., Electromagnetic Optimization by Genetic Algorithms. New York, N.Y.: John Wiley & Sons, 1999. [2] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. Bakr, K. Madsen, and J. Søndergaard, "Space mapping: The state of the art," IEEE Trans. Microwave Theory Tech., vol. 52, no. 1, pp. 337-361, Jan. 2004.

The usefulness and limitations of several real-valued particle swarm optimisation (PSO) based schemes, when applied to linear array synthesis are presented in this work. PSO with synchronous and asynchronous updates of the swarm as well as both, global and local topologies have been considered for the algorithm. First of all, a brief description of the four PSO based schemes investigated will be presented along with a description of the synthesis problem considering both, isotropic and directive element patterns.

A preliminary parametric study considering complex synthesis has been carried out to find out the best configuration for the PSO optimiser, including an analysis of the influence of the fitness function, the effect of the inertial weight (w) and acceleration constants (c1,c2), the swarm size (P) and the maximum speed allowed for particles (Vmax). Results of the analysis demonstrate that the PSO with asynchronous updates of the swarm and global topology (PSO-AG), with w=0.729, c1=c2=1.49445 and a Vmax equal to the dynamic range of the search space exhibits the most accurate results and the best efficiency in terms of computational cost for this problem.

Once the heuristic method has been properly tuned, the PSO-AG scheme has been applied to a wide variety of linear synthesis problems. Both, complex and amplitude-only synthesis have been carried out starting from a predefined amplitude distribution or a randomly generated one. Phase-only synthesis starts from an amplitude predefined distribution. Moreover, every synthesis problem presented above has been tested considering different number of elements and radiation pattern masks. For instance, the optimiser has been checked against different objective patterns including main beam pointing, sidelobe level reduction, predefined width and depth nulls placement.

Illustrative results of this study will be presented along with a discussion of the pros and cons of the optimisation method for each case.

Furthermore, real-valued genetic algorithms (GA) have also been applied to the synthesis problem presented and some details comparing the pros and cons of both heuristic optimisation methods will be presented. In short, the relatively new PSO technique, with an easier implementation, becomes a versatile and attractive alternative to other heuristic methods such as GA.

We report our efforts to design circular arrays of log-periodic (LP) antennas for broadband applications. LP antennas are nearly frequency independent over wide bands of frequency. Therefore, circular arrangements of identical LP antennas are also expected to operate nearly independent of frequency. Such an array is desirable for its beam-steering ability, which can be achieved by feeding the elements of the array appropriately. On the other hand, mutual couplings between the LP antennas affect the radiation characteristics significantly and deteriorate the frequency-independence property. Consequently, it becomes essential to place the LP antennas correctly and determine the best excitations for the antennas to obtain both frequency-independence and beam-steering properties. We solve this complex problem by taking advantage of a powerful electromagnetic simulation environment.

Fig. 1(a) shows a representative configuration for a three-element LP array located on the x-y plane. Each element of the array is actually a trapezoidal-tooth LP antenna, which is designed for broadband operation in 300-800 MHz range and constructed by 38 monopole pairs attached to two separate feed lines connected to a voltage source. (Ö. Ergül and L. Gürel, "Nonplanar trapezoidal-tooth log-periodic antennas: design and electromagnetic modelling," Radio Science, vol. 40, Oct. 2005.) To analyze the array accurately, we consider all the electromagnetic interactions by formulating the problem with the electric-field integral equation. For efficiency, the resulting matrix equation is solved iteratively, where the matrix-vector products are accelerated by multilevel fast multipole algorithm. To reduce the effects of the mutual couplings and obtain broadband operation, we maximize the directivity of the array by finding the best combination for the excitations of the antennas. The required optimization is achieved by genetic algorithms and by using superposition techniques on the complex far-field radiations. Similar optimizations are employed to find the excitations that provide the maximum directive gain in a given direction, and hence, to add the beam-steering ability to the array.

As an example, Figs. 1(b)-(d) present the far-field radiation patterns of the array in Fig. 1(a) at 400 MHz, 550 MHz, and 700 MHz. The directive gain in the -x direction is optimized and a broadband operation is achieved. In the presentation, we will provide more examples on the circular arrays of LP antennas.

We present an algorithm for the design of large slotted waveguide arrays which is alternative to the methods based on the equivalent circuit of the slot [1]. The core of the technique is the full-wave model of the antenna, recently proposed by ourselves [2], whose main feature is a suitable segmentation of the antenna into two fundamental parts: one `internal`, formed by the waveguide circuit, and the other `external`, consisting of the half space above the plate containing the radiating slots. The internal part is further segmented into several blocks, connected to each other via their Generalized Admittance Matrices (GAM's). The latter are calculated by a commercial full-wave tool, such as Ansoft HFSS, in such a way that even the case of waveguides with arbitrary cross-section can be easily treated. The front plate is taken as a whole and analyzed by the moment method, in terms of modes of the slots. The latter are considered as pieces of waveguides (whose length is just the conductor thickness) which provide the connection between internal and external parts.

Instead of computing the GAM of any specific block forming the array, it is more efficient to build a data base containing the GAM's, computed for discrete sets of geometrical parameters (typically, displacements and lengths . The GAM corresponding to any actual block is then obtained by interpolating the stored values.

The next step consists of building, by standard algebra, the GAM of the whole array, thus obtaining both the scattering parameters at input ports and the slot modal voltages V(d₁,...,d_N,l₁,...,l_N), required for the design. Note that all the actual effects such as mutual couplings, thickness and truncation are taken into account. We are now able to perform the design, which, as usual, starts from the choice of a set of ideal Voltages V ^id, corresponding to the desired radiation pattern. The geometry corresponding to the above voltage distribution is calculated by minimizing the goal functional F, which is defined as:
F(d₁,...,d_N,l₁,...,l_N) =∑ |V_k(d₁,...,d_N,l₁,...,l_N)- V_k ^id|

In the cases considered, we have seen that the optimisation of the above functional does not depend considerably on the choice starting point, although a prudent choice, for instance one deriving from classical design formulas, is always recommended. Finally, the optimization strategy is based on a classical deterministic Newton method, which converges when the functional reaches a local minimum.

The latter condition occurs when the gradient of the functional tends to zero. This is a very key point, as it can be realized by noting that the gradient gives also a measurement of the sensitivity of the solution with respect to errors on the geometrical parameters and the most robust solution just occurs when the functional is flat. This last consideration, together with the high accuracy of the full-wave approach, makes the method very suitable for the design of slotted arrays.

References

[1] R. S. Elliott, "On the Design of Travelling -Wave-Fed Longitudinal Shunt Slot Arrays", IEEE TAP, Vol 27, 1979.

[2] A. Morini , T. Rozzi, and G. Venanzoni, "Full-Wave Analysis of Slotted Waveguide Arrays", IEEE TAP, Vol.54, 2006.

Particle Swarm Optimization (PSO) is a rather novel global stochastic optimization technique in which the parameter space of the problem to be optimized is searched by agents (particles) whose behavior simulates that of a swarm, or flock, of beings, controlled according to a swarm- or flock-like set of social interaction rules. The position of each particle is used to evaluate the value of the function to be optimized. Individual particles are then attracted, with a stochastic-varying strength, by both the position of their best past performance and by the position of the global best performance of the whole swarm [1].

In [2,3] some variations of the standard PSO algorithm have been proposed in order to increase the efficiency of the searching algorithm over the solution space with a negligible overhead in the algorithm complexity and speed. The basic idea of the variations proposed there is that of using several independent but mutually interacting swarms to better and faster explore the whole space domain without being trapped in local minima.

In this contribution these new algorithms will be tested over a planar array synthesis problem, seeking for an optimal distribution of excitations and element placement to satisfy a given array mask with the fewest possible elements. Array optimization problems have been recently successfully addressed with conventional PSO techniques [4], but in most cases one dimensional arrays have been considered.

References

[1] J. Kennedy, R.C. Eberhart, Swarm Intelligence, Morgan Kaufmann: San Francisco, CA (2001).
[2] L. Matekovits, M. Mussetta, P. Pirinoli, S. Selleri, R.E. Zich, "Improved PSO algorithms for electromagnetic optimization", Proc. of 2005 APS Symposium, July 2005.
[3] S. Selleri, M. Mussetta, P. Pirinoli, R.E. Zich, L. Matekovits, "Some insight over new variations of the particle swarm optimization method," IEEE Antennas Wireless Propagat. Lett., in press.
[4] D.W. Boeringer, D.H. Werner, "Particle swarm optimization versus genetic algorithms for phased array synthesis", IEEE Trans. Antennas Propagat., 52, pp. 771-779, (2004).

Introduction. The design of electromagnetic (EM) structures is usually based on exploiting their numerical models. Numerical models request high computational power. Their evaluation has to be repeated many times in the optimization cycle: each update of the optimized parameters has to be followed by a new analysis. Replacing a numeric model of the designed EM structure by a neural one is one of ways to reduce CPU-time demands [1].

Problem. At the present, two basic approaches are used to optimize neural models of EM structures, conventional neural optimization and several new methods described in detail in [2]. Conventional approach exploiting feed-forward neural networks (NN) and recurrent ones which are trained by gradient algorithms is used for automated development of EM models in most cases. Unfortunately, this technology exhibits several imperfections:

1. The training has to be repeated several times in order to reveal a global minimum by local (gradient) optimization techniques. If the local algorithm is replaced by global one (genetic, particle-swarm (PSO) [3], etc.), the training consumes enormous CPU power.

2. Developed neural models can suffer from over-training: the training process minimizes errors in training patterns, but interlaying points exhibit relatively high error.

In the paper, we propose possible solutions of the above-listed problems, and apply them to modeling EM structures below.

Methods Used. Training abilities of feed-forward and recurrent NN are enhanced by using a hybrid learning algorithm. The proposed algorithm combines (PSO) as a representative of global method and Levenberg-Marquardt method as a local method which reduces CPU-power demands. In order to reduce over-training, the number of neurons in hidden layers is changed during the neural model creation.

Original Results. The proposed technology of the automated creation of neural models of EM structures applied to optimization some planar microwave filters and planar antennas is the original contribution of the paper.

Verification. The results are verified by comparing parameters of developed neural models of EM structures and their numeric models in Zeland IE3D. The developed neural models exhibit better training abilities and better immunity to over-training compared to so-far existing approaches.

Conclusion. Developed approach brings improvement in designing EM structures. Original combination of above methods makes it possible to fast and high accuracy modeling EM structures. It decreases CPU time demands and it brings automation in EM structures developing process.

References
[1] RAIDA, Z. Modeling EM Structures in the Neural Network Toolbox of MATLAB. IEEE Antennas & Propagation Magazine. 2002, vol. 44, no. 6, p. 46-67.
[2] RAYAS-SÁNCHEZ, J., E. EM-Based Optimization of Microwave Circuits Using Artificial Neural Networks: The State-of-the-Art. IEEE Transactions on Micrwave Theory and Techniques. 2004. no. 1, p. 402-434.
[3] RAHMAT-SAMII, Y., GIES, D., ROBINSON, J. Particle Swarm Optimization (PSO): A Novel Paradigm for Antenna Design. The Radio Science Bulletin. 2003, no. 305, p. 14-22.