**2.1. SED applied to the design of wire antennas**

120 Genetic Programming – New Approaches and Successful Applications

of the SED, which calls for a regularization procedure.

and of the main steps of the evolutionary process.

The rest of this chapter is organised as follows:

for each individual is described.

fitness for the problem at hand.

**Figure 1.** Flowchart of the Evolutionary Design.

flowchart of Fig.1:

obtained by SED are significantly higher.

typical evolution operators work on actual physical structures, rather than on sequences of bits with no intuitive link to the structure shape. The enormous power of SED fully allows the exploration of more general shapes for the structure. The main drawback is the ill-posedness

• Section 2 starts with a general description of the Structure-based Evolutionary Design,

• SED is then specifically applied to the design of broadband parasitic wire arrays (Sections 2.1-2.3): a suitable tree representation of wire antennas is devised, appropriate antenna requirements are set, a suitable fitness is derived and the evaluation procedure

• In Section 3 several examples are presented: for each set of requirements, a suitable fitness function must be derived, and some suggestions are given to choose the best

• The results obtained with SED are finally compared with other algorithms like Particle Swarm Optimization and Differential Evolution, showing that the performances

SED is a global random search procedure, looking for individuals best fitting a given set of specifications. These individuals are described as instruction sets, and internally represented as trees. The main steps of the whole evolutionary design can be summarized in the

**2. Description of the Structure-based Evolutionary Design** 

The Structure-Based Evolutionary Design, based on evolutionary programming, has been devised and applied to the design of broadband parasitic wire arrays for VHF-UHF bands. This requires first to devise a suitable tree representation of wire antennas, well tailored to the SED meta-operators, and then suitable antenna requirements. We consider only antennas with a symmetry plane, and with all element centres on a line. Therefore, each "wire" is actually a symmetric pair of metallic trees, and only one of them must be described.

In antenna design, the most intuitive fitness function can be built as the "distance" between actual and required far-field behaviour (Franceschetti et al., 1988) or, even more simply, as the antenna gain or SNR (Lo et al., 1966). However, this is not the case for SED. The solution space, i.e., the set of admissible solution in which the procedure looks for the optimum, is composed, in our case, of every Parasitic Dipole Array (PDA) antenna with no limit on the number of wire segments, nor on the size or orientation, represented as real numbers. The design problem is therefore strongly ill-conditioned and, in order to stabilize it, appropriate suitable antenna requirements must be set. Far-field requirements are unable to stabilize the problem, since the far-field degrees of freedom are orders of magnitude less than those of

the solution space (Bucci & Franceschetti, 1989), so that a huge number of different antennas gives the same far field. As a matter of fact, a wire segment whose length is a small fraction of the wavelength can be added or eliminated without affecting the far field. We must therefore revert to near-field requirements. Among them, the easiest to implement, and probably the most important, is a requirement on the input impedance over the required bandwidth. Since this constraint is a "must-be" in order to get a usable solution, we get the required stabilization at virtually no additional cost. As a further advantage, a low input reactance over the bandwidth prevents from superdirective solutions (Collin, 1985) even when a reduced size is forced as a constraint.

Structure-Based Evolutionary Design Applied to Wire Antennas 123

Each node of the tree is an operator belonging to one of the following classes:

This mixed representation largely increases the power of the standard genetic operations (mutation and cross-over), since each element can evolve independently from the others. Of course, after each complete PDA is generated, its geometrical coherency is verified, and incoherent antennas (e.g., an antenna with two elements too close, or even intersecting) are

The 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. The integration with NEC-2 has mainly been achieved through three

1. a parser for the conversion of the s-expressions, represented as n-ary trees, in the

2. a NecWrapper which writes the NEC listing to a file, launches a NEC2 instance in a

In order to better grasp the representation chosen, the S-expression for the simple Yagi

3. an Evaluator which calculates the fitness using the output data generated by NEC.

separate process, and parses the output generated by NEC;

**Figure 2.** Antenna Structure corresponding to the S-expression of the example

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

equivalent NEC input files;

antenna of Fig.2 follows.

discarded.

classes:

The performances of each individual (antenna) of the population are evaluated by its fitness function. The details of the fitness function we have chosen for PDA design are widely described in the next section. However, at this point it must be stressed that the fitness function depends in an essential way on the electromagnetic behaviour of the individual.

Since we are interested in assessing SED as a viable, and very effective, design tool, we accurately try to avoid any side-effect stemming out from the electromagnetic analysis of our individuals. Therefore we rely on known, well-established and widely used antenna analysis programs. Since our individuals are wire antennas, our choice has fallen on NEC-2 (Burke et al., 1981).

The Numerical Electromagnetics Code (NEC-2) is a MoM-based, user-oriented computer code for the analysis of the electromagnetic response of wire antennas and other metallic structures (Lohn et al., 2005). It is built around the numerical solution of the integral equations for the currents induced on the structure. This approach allows taking well into account the main second-order effects, such as conductor losses and the effect of lossy ground on the far field. Therefore we are able to evaluate the actual gain, and not the array directivity, with a two-fold advantage. First of all, the gain is the far-field parameter of interest and, second, this prevents from considering superdirective antennas, both during the evolution and as final solution, which is even worse. NEC has been successfully used to model a wide range of antennas, with high accuracy (Burke & Poggio, 1976a, 1976b, 1981; Deadrick et al., 1977) and is now considered as one of the reference electromagnetic software (Lohn et al., 2005; Linden & Altshuler, 1996, 1997). However, since SED is by no means linked, or tailored, to NEC, a different, and most effective, EM software could be used, to reduce the total computational time, further improving the accuracy of the simulation.

## **2.2. Construction and evaluation of each parasitic dipole array**

Each PDA is composed of a driven element and a fixed number of parasitic elements. In order to get transverse dimensions close to those of Yagi and LPDA, and to ease the realization, the centers of the elements are arranged on a line, with the driven element at the second place of the row. In Yagi terminology, we use a single reflector. We actually have experimented with more reflectors but, exactly as in standard Yagi, without any advantage over the single-reflector configuration. Each element is symmetric w.r.t its center, and the upper part is represented, in the algorithm, as a tree.

Each node of the tree is an operator belonging to one of the following classes:


122 Genetic Programming – New Approaches and Successful Applications

when a reduced size is forced as a constraint.

(Burke et al., 1981).

the solution space (Bucci & Franceschetti, 1989), so that a huge number of different antennas gives the same far field. As a matter of fact, a wire segment whose length is a small fraction of the wavelength can be added or eliminated without affecting the far field. We must therefore revert to near-field requirements. Among them, the easiest to implement, and probably the most important, is a requirement on the input impedance over the required bandwidth. Since this constraint is a "must-be" in order to get a usable solution, we get the required stabilization at virtually no additional cost. As a further advantage, a low input reactance over the bandwidth prevents from superdirective solutions (Collin, 1985) even

The performances of each individual (antenna) of the population are evaluated by its fitness function. The details of the fitness function we have chosen for PDA design are widely described in the next section. However, at this point it must be stressed that the fitness function depends in an essential way on the electromagnetic behaviour of the individual.

Since we are interested in assessing SED as a viable, and very effective, design tool, we accurately try to avoid any side-effect stemming out from the electromagnetic analysis of our individuals. Therefore we rely on known, well-established and widely used antenna analysis programs. Since our individuals are wire antennas, our choice has fallen on NEC-2

The Numerical Electromagnetics Code (NEC-2) is a MoM-based, user-oriented computer code for the analysis of the electromagnetic response of wire antennas and other metallic structures (Lohn et al., 2005). It is built around the numerical solution of the integral equations for the currents induced on the structure. This approach allows taking well into account the main second-order effects, such as conductor losses and the effect of lossy ground on the far field. Therefore we are able to evaluate the actual gain, and not the array directivity, with a two-fold advantage. First of all, the gain is the far-field parameter of interest and, second, this prevents from considering superdirective antennas, both during the evolution and as final solution, which is even worse. NEC has been successfully used to model a wide range of antennas, with high accuracy (Burke & Poggio, 1976a, 1976b, 1981; Deadrick et al., 1977) and is now considered as one of the reference electromagnetic software (Lohn et al., 2005; Linden & Altshuler, 1996, 1997). However, since SED is by no means linked, or tailored, to NEC, a different, and most effective, EM software could be used, to reduce the total computational time, further improving the accuracy of the simulation.

Each PDA is composed of a driven element and a fixed number of parasitic elements. In order to get transverse dimensions close to those of Yagi and LPDA, and to ease the realization, the centers of the elements are arranged on a line, with the driven element at the second place of the row. In Yagi terminology, we use a single reflector. We actually have experimented with more reflectors but, exactly as in standard Yagi, without any advantage over the single-reflector configuration. Each element is symmetric w.r.t its center, and the

**2.2. Construction and evaluation of each parasitic dipole array** 

upper part is represented, in the algorithm, as a tree.

This mixed representation largely increases the power of the standard genetic operations (mutation and cross-over), since each element can evolve independently from the others. Of course, after each complete PDA is generated, its geometrical coherency is verified, and incoherent antennas (e.g., an antenna with two elements too close, or even intersecting) are discarded.

The 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. The integration with NEC-2 has mainly been achieved through three classes:


In order to better grasp the representation chosen, the S-expression for the simple Yagi antenna of Fig.2 follows.

**Figure 2.** Antenna Structure corresponding to the S-expression of the example

*S-expression:* 

*Tree 0:* 

```
(StretchAlongZ 1.3315124586134857 (Wire 0.42101090906114413 1.0 
 (StretchAlongX 0.5525837649288541 (StretchAlongY 1.4819461053740617 
 (RotateWithRespectTo_Y 0.3577743384222999 END)))))
```
*Tree 1:* 

```
(Wire 0.5581593081319647 1.0 (RotateWithRespectTo_X -0.44260816356142224
```
 *(RotateWithRespectTo\_Z 0.08068272691709244 (StretchAlongZ 0.7166185389610261* 

Structure-Based Evolutionary Design Applied to Wire Antennas 125

*MAX*

 α

( ) *<sup>M</sup> G S Fitness F F F* = +⋅ (1)

After evaluation of different fitness structures, we have chosen a fitness function composed

The first term (FM) takes into account the input matching of the antenna, the second term (FG) takes into account the antenna gain including the effect of ohmic losses, and the last

> 1 ; ; 1 *MAX REAL MAX M M G GS S*

wherein αM, αG and αS are suitable weights, while *SWR* and *G* are the mean values of SWR and gain over the bandwidth of interest, DREAL represents the real antenna size and DMAX is

The requirement of a given, and low, VSWR all over the design bandwidth is obviously needed to effectively feed the designed antenna. However it has an equally important role. The VSWR requirement (a near-field requirement) stabilizes the problem, at virtually no

The evaluation procedure for each individual (i.e. for each antenna) can be described by the

*G D*

<sup>−</sup> = − ⋅ = ⋅ =+ <sup>⋅</sup> (2)

*<sup>G</sup> D D F SWR <sup>F</sup> <sup>F</sup>*

αα

**Figure 3.** Flowchart of the evaluation procedure for each individual of the population.

by three main terms suitably arranged as:

term (FS) takes into account the antenna size.

the maximum allowed size for the antenna.

In (2.1):

additional cost.

flowchart in Fig.3.

```
 (StretchAlongX 1.42989629787443 (StretchAlongZ 1.346598788775623
```
 *END))))))* 

*Tree 2:* 

```
(Wire 0.3707701115469606 1.0 (RotateWithRespectTo_X 0.5262591815805174
```

```
 (RotateWithRespectTo_Z -0.7423883999218206 (RotateWithRespectTo_Z 0.07210315212202911
```
 *END))))* 

```
The corresponding NEC-2 input file is:
```

```
GW 1 17 0.00E00 0.00E00 0.00E00 -1.34E-02 1.44E-02 1.33E-01 1.36E-03
```

```
GW 2 22 -1.38E-01 0.00E00 0.00E00 -1.25E-01 0.00E00 1.66E-01 1.36E-03
```

```
GW 3 15 1.21E-01 0.00E00 0.00E00 1.21E-01 0.00E00 1.18E-01 1.36E-03
```
*GX 4 001* 

*GE* 
