Structure-Based Evolutionary Programming Design of BroadbandWire Antennas

A design technique for wire antennas, based on the Structure-Based Evolutionary Programming, is used to design a broadband antenna with an end-�re radiation pattern and a very simple geometry, operating in the 3–16GHz frequency band, namely, from the S band to the Ku band. e antenna has been analyzed with NEC-2 during the evolutionary process, looking for high gain, good input match, and robustness with respect to realization tolerances. e outcome of our design procedure shows a very good performance.

e traditional approach to the design of wire antennas starts by choosing a well-de�ned structure, whose parameters need to be suitably optimized.is requires, as a prerequisite, the choice of an antenna model which has proven able to comply with the design speci�cations.Successful proposals of broadband antennas are self-similar, so for those antennas the chosen model must be a self-similar one, either continuous, like a conical antenna, or discrete.e latter can be implemented either as an array [12] or as a prefractal antenna [19].Of course, the �nal antenna could not ful�ll exactly the model [13], but also such differences must be chosen externally.As a matter of fact, the choice of the model heavily constraints the whole design process.As a consequence, a signi�cant skilled human interaction is required in the initial choice of the structure.e model could also be �ne-tuned as the design proceeds, but this requires a skilled human interaction, too.e traditional approach is quite expensive, and therefore design techniques without human interaction are of interest, as long as they provide equal, or better, results.is can be achieved only when no initial structure is assumed, since this choice (which in a fully automated procedure cannot be further modi�ed) can constraint too strongly the �nal solution.
Up to now, such a general tool has been sought for among random search procedures.Some of them are inspired by natural processes, either looking for an effective cooperative scheme, like Particle Swarm Optimization (PSO) [20], or aiming at exploring very different solution sets, like Genetic Algorithms (GA) [21][22][23].As a matter of fact, the latter could be a good candidate for a valuable general tool, being inspired by Darwinian natural selection, since the variety of all living forms is known to everyone.However, despite of their strong premises, GA cannot ful�ll this purpose since they still assume a completely de�ned antenna structure, and only an handful of parameters remains to be optimized.Borrowing the biological language, we can say that GA works at the nucleotide (i.e., bit) level and this strongly limits its effectiveness as a design tool.
On the other hand, the full power of evolution-inspired methods in the antenna design has been highlighted only when Genetic Programming (GP) or, more precisely, Structure-based Evolutionary Design (SED) [24] approaches have been proposed for simple wire antennas [25,26], arrays [24], and FSS [27].Evolutionary programming does not require any antenna model, neither asks for a structure locked from the beginning.Instead, it considers a (virtually) in�nite solution space, de�ned only by very loose constraints.If we require a wire antenna, SED is able to look for the �nal design among all possible wire antennas.As a matter of fact, SED can be used to automatically, and effectively, search, in this very huge solution space, for novel antenna con�gurations, which can be signi�cantly more performant than antennas developed using standard techniques.
e strength of the SED resides in its description of each element of the solution space as the set of instructions needed to realize it.is description can be translated in a highly effective tree description [25] for computer implementation and allows the standard genetic operators (crossover, mutation) to reach an unparallel power.As a matter of fact, SED works at the organ level, so that crossover is the exchange of whole working parts of the individuals, while mutation, working on a subtree root, affects the whole sub-tree (as happens in the true natural selection).SED requires also a suitable �tness function, tailored to the problem at hand, and a time-effective analysis procedure.
To assess SED as a viable tool for robust design of broadband antennas, we consider here the design of a wideband wire antenna with an end-�re radiation pattern and a very simple geometry, operating in a range from S to Ku frequency bands, namely, from 3 GHz to 16 GHz, and reasonably matched at the input port.Apart from these simple "constraints, " SED does not assume any other a priori information on the antenna structure.Rather, SED builds up the structure of the individual antennas as the procedure evolves.erefore, SED solution space has the power of the continuum and allows exploring, and evaluating, general wire antenna con�gurations.
e solution space, namely, the set of admissible solutions in which the procedure looks for the optimum, is composed, in our case, of every broadband antenna with no limit on the number of wire segments, nor on the size or orientation, represented as real numbers.On the other hand, GA works on a given antenna model, so that its solution space is a discrete one and therefore is a very small subset of the SED solution space.
e most common broadband wire antennas are the log-periodic dipole arrays (LPDAs) [28][29][30], used in a wide range of applications due to their very wide bandwidth and their relatively narrow-beam characteristics.However, a wire LPDA operating in the bandwidth of the antenna proposed in this paper (namely, C, X, and Ku frequency bands) cannot be realized, because the corresponding dipoles would be too short at these frequencies.On the other hand, the broadband antenna designed here with SED can be easily scaled in order to work also at lower frequencies, keeping the same performances both in terms of input matching and of gain.
In order to compute the �tness, an analysis of each wire antenna generated by SED is needed.is has been performed using NEC-2 [31], a well-known, time-effective and well-assessed Method of Moments code.is soware has been successfully used to model a wide range of wire antennas, with high accuracy (see, e.g., [32,33]), and is now considered as one of the reference e.m. soware (see, e.g., [ 26,34]).For this reason, it has been used here.However, sometimes the SWR data of NEC-2 could have a reduced accuracy, therefore the �nal output of the design procedure has been validated, by testing it with HFSS [35], a commercial FEM code, since it has been shown that the results of this soware are in very good agreement with experiment (see, e.g., [36]).As a matter of fact, although NEC is faster by orders of magnitude, NEC and HFSS results are in very good agreement, thus assessing our choice for the �tness evaluation.

Antenna Design and Fitness Function
e initial structure of each SED individual is depicted in Figure 1.Each individual of the population (antenna) is composed by a principal vertical wire (the main dipole in Figure 1), connected to the feeding port on its bottom side, and by a number N (chosen by SED) of wires connected to the upper side of the main dipole with an arbitrary length and orientation in space.At the remote end of each of the N wires, we connect zero, one, or more further wires, still with arbitrary length and orientation, and so on, in an iterative manner.e structure is �nally mirrored with respect to the horizontal plane, as indicated in Figure 1.
Each individual is built up using only four operations: (a) add a wire according to the present directions and length; (b) transform the end of the last added wire in a branching point; (c) modify the present directions and length; (d) stretch (or shrink) the last added wire.
In the �rst step of the evolutionary design, N individuals are randomly built.en, an iterative procedure starts, where the �tness of each individual is evaluated, and the next generation of the population is built assigning a larger probability of breeding to the individuals with the highest �tness.e iterative procedure ends when suitable stopping rules are met (i.e., when the individual antenna ful�lls, within a predetermined tolerance, the speci�ed requirements).
Aer each antenna has been generated, its geometrical coherency is veri�ed, and incoherent antennas (e.g., an antenna with two elements too close, or even intersecting) are discarded.en it is analysed by NEC-2 and its �tness is computed.e SED approach has been implemented in Java, while the analysis of each individual has been implemented in C++ (using the freeware source code Nec2cpp) and checked using the freeware tool 4nec2 [31].
e performance of each individual (antenna) of the population is evaluated by a proper �tness function, which is strongly dependent on the problem at hand, namely, by the electromagnetic behavior of the designed antenna, and must measure how closely the actual antenna meets the design speci�cations.
In the speci�c case of the broadband wire antenna of this paper, the �tness function has been selected in order to lead the evolution process toward a structure with a good input match in a frequency range as wide as possible (within S, C, X, and �u bands), while keeping the highest end-�re gain and a reduced size.
Since improving one parameter usually results in worsening the other ones, the design technique has to handle a complicate trade-off between the con�icting ob�ectives.e �tness structure is therefore a critical point in the design procedure, since only an appropriate choice can lead the design process to performing results, while largely reducing the computation time.
e chosen �tness has been built from the desired antenna performances [24] as where  SWR and  GAIN are suitable weights (whose values depend also on the input impedance of the actual antenna), SWR, and  are, respectively, the mean values of the SWR and gain over the bandwidth of interest,  ANT represents the actual antenna size, and  MAX is the maximum allowed size for the antenna.Finally,  SIZE is an appropriate weight which takes into account the requirement of a small size of the antenna.e values for the �tness weights have been obtained aer a suitable local tuning, following an approach similar to the one described in detail in [24].e weight  GAIN in the �tness function (1) has the following expression: where  Back is the gain computed in the back direction ( = 90 ∘ ,  = 10 ∘ ),  Front is the average gain computed in the front region (||  90 ∘ + 2Δ, 0 ∘ + 2Δ    90 ∘ , where Δ and Δ indicate the main lobe amplitude), and  Rear is the average gain computed in the rear region (0 ∘ ≤ || ≤ 10 ∘ , 90 ∘ ≤ || ≤ 10 ∘ ).e weights  Back ,  Front and  Rear are chosen through a local tuning in order to get the maximum gain in the end-�re direction and an acceptable radiation pattern in the rest of the space.In the performed evolutionary process these parameters have the following values:  Back = 0.12,  Front = 0.17 and,  Rear = 0.06.e weight  SWR in the �tness function ( 1) is expressed using suitable parameters strictly related to the antenna input impedance, which are individually tuned.e resulting expression for  SWR is gives a signi�cant penalization to antennas with a large imaginary part of the input impedance, but it has a step-like behavior.erefore, in order to get a further, smooth penalization to antennas with a large , we have added also the term with   .We have observed that a combination of the two terms is more effective than either one separately.
e inclusion of ohmic losses into the gain computation, as well as the requirement of a good input match over all the required bandwidth, prevents from selecting superdirective solutions.
On the other hand, the requirement of a robust solution is not taken into account in the �tness, since we have found a different approach more efficient.e individuals associated with the highest �tness values, or very close to the best �tness value obtained so far, are perturbed (assigning random relocations to the elements) and analysed to assess their robustness with respect to random modi�cations of the structure.is random relocation allows to get robust structures with respect to both constructive errors and bad weather conditions (e.g., movements due to wind effect).

Results
In order to test the procedure, we have designed a broadband wire antenna, with equal wire diameter (0.665 mm) and conductivity (     6 S/m).We have required that the antenna has an input impedance of 200 Ω in the whole bandwidth, which is a typical characteristic impedance of bi�lar lines �37�.e use of a bi�lar line as a feeding networ� avoids the need of a balun to connect the balanced wire antenna to an unbalanced input, as provided for example by standard coaxial cables, having a typical input impedance of 50 Ω.On the other hand, the designed antenna can also be connected to a standard coaxial line using a commercial balun with an impedance transformation ratio of 4 : 1.In this case, since a balun with such large bandwidth (3-16 GHz) cannot be obtained, the operating bandwidth must be divided into a number of subbands, and an appropriate balun must be used in each subband.
e antenna has been designed using a population size of 1000 individuals, with a crossover rate set to 60%, and a  mutation rate set to 40%.Its convergence plot is shown in Figure 2, and it appears that 250 generations are enough to reach convergence.e best individual of the evolutionary process, obtained aer several runs (e.g., a few tens) of the code, is shown in Figure 3, and the cartesian coordinates of each wire are reported in Table 1.is antenna is very easy to realize, consisting in only 12 metallic wires, and can be produced with a very low cost by the same technology used for Yagi and LPDA arrays.
In Figure 4 we show the input frequency response of the designed antenna using both NEC-2 and HFSS.It is clear that the antenna bandwidth (S < − dB) extends from 3 GHz up to well beyond 16 GHz.It is also clear that both NEC-2 and HFSS give essentially comparable results.is is also true for the radiation pattern.So, the use of NEC-2 in the design is fully assessed, and we will show only the NEC-2 results in the following.In Figure 5 the end-�re Gain is reported.In the bandwidth 3-16 GHz, the mean Gain of the antenna is equal to 12.7 dB and the mean F/B ratio is about 11.6 dB.
In Table 2 the antenna gain, front-to-back ratio, and efficiency in the operating bandwidth are shown.
It is worth noting that the inclusion of ohmic losses into the gain computation is very important, since this prevents from selecting superdirective antennas during the evolution.As a matter of fact, the efficiency of the designed antenna is very good (greater than 97%, and with a mean value of 98.05%), despite of the relatively small electrical conductivity of the metal (     6 S/m).Besides, while the maximum directivity is almost constant with respect to , the efficiency rapidly decreases [38].It is therefore required to take into account in SED the actual conductivity of the antenna material, in order to discard individuals with low efficiency, which can result in unusable antennas.
Finally, the NEC-2 Far-Field-pattern in the operating frequency bandwidth is plotted in Figure 6  radiation patterns con�rm that the useful bandwidth of the designed antenna is 3-16 GHz, where the input matching is very good and the far-�eld is essentially end �re, with a good Gain and F/B ratio.In order to evaluate the performance improvement of the broadband antenna proposed in this paper over standard solutions, we can compare it with wire log-periodic dipole arrays (LPDAs), the most popular broadband wire antennas [28][29][30].However, a wire LPDA in the  frequency band (and beyond) cannot be realized because the dipole lengths would be too small with increasing frequency.On the other hand, T 1: Cartesian coordinates of the ends of the wires for the antenna in Figure 2 the proposed antenna can be easily scaled in order to work at low frequencies, without degrading its performances.In Figures 7 and 8 we show, respectively, the return loss and the Gain, plotted with respect to the normalized frequency, of the antenna designed with SED within S, C, X, and Ku bands and of the same antenna scaled at the center frequency of 2 GHz.e simulations, performed with NEC-2, show a very similar behavior in the whole operating bandwidth, con�rming that the proposed antenna can be easily scaled to work at any lower frequency.
Following [29,30], a wire LPDA with the same bandwidth of our broadband antenna scaled at 2 GHz (Figures 7 and 8) will require at least 20 elements to get an average gain of only 8.5 dB, with the log period  equal to 0.9.However,  should be kept lower than 0.85 in order to ensure a good behaviour of the LPDA [30], and this constraint limits the gain of a standard LPDA to a value below 8 dB.is comparison shows that the proposed antenna allows signi�cantly better performances with respect to standard LPDAs, with a little bit more complicated structure, but without requiring the typical twisted-cable feeding network of the LPDAs.

Conclusion
A new design technique for a wideband wire antenna has been presented.It is based on the Structure-based Evolutionary Design (SED), which exploits the concept of the Evolutionary Programming.Since no a priori structure is assumed, a suitable �tness function allows to reach signi�cant electrical performances with a simple geometry.Extension to multiobjective �tness is under consideration, but the results reported here show that its use is not required, except perhaps for very complicate requirements.
Inclusion of the ohmic losses and of a suitable robustness test leads to a small antenna size, while preventing from super directive solutions.e proposed approach can therefore be effectively employed also for different sets of requirements.
��n��c� �f �n�eres�s e authors declare that there is no con�ict of interests.

F 2 : 3 F 3 :
Plot of convergence of the designed antenna shown in Figure Geometry of the designed wire antenna.

F 4 :
Frequency response of the designed antenna.

F 6 :
Simulated (NEC-2) normalized Far-Field pattern of the designed antenna.Blue line: E-Plane; red line: H-Plane.

F 7 :
. For each frequency, the E-Plane and the H-Plane are shown.e reported International Journal of Antennas and Propagation Simulated (NEC-2) Return Loss of the designed antenna at the center frequency of 8 GHz (a) and of the designed antenna scaled at the center frequency of 2 GHz (b).

F 8 :
Simulated (NEC-2) Gain of the designed antenna at the center frequency of 8 GHz (a) and of the designed antenna scaled at the center frequency of 2 GHz (b).
IN in the antenna required bandwidth.e two weights  IN and   are both connected to the  factor of the antenna.However,  IN .