**3. Results**

The automated design of wire antennas using SED has been applied to several PDAs, with different maximum sizes, number of elements, and operation frequencies, and with different requirements both on Gain and input matching, always obtaining very good results.

We present here only a few examples, chosen also to show the flexibility of SED. All designed antennas have been compared with known antennas. However, since our antennas are wide-band 3D structures, it has been difficult to device a suitable comparison antenna. To get a meaningful comparison, we decided to compare our designed antennas with an antenna of comparable size.

The first presented antenna (Casula et al., 2009), shown in Fig.4a, has been obtained by constraining the evolution of each individual only in two directions (i.e. horizontally and vertically). This limitation is a hard limitation, and significantly affects the antenna performances. This compromise leads anyway to antennas easy to realize, and with good performances.

The designed antenna works at the operation frequency of 800 MHz, and the requested bandwidth is of 70 MHz (i.e. 9%, from 780 MHz to 850 MHz). The best designed antenna is represented in Fig.1a. The antenna size is 0.58λ0 x 0.67λ0 x 1.2λ0, λ0 being the space wavelength at the operation frequency of 800 MHz, its gain is above 11.6 dB (see Fig.5) and its SWR is less than 2 in the whole bandwidth of 70 MHz (see Fig.1b). No additional matching network is therefore required.

126 Genetic Programming – New Approaches and Successful Applications

with a penalty directly proportional to the excess size.

knowledge of the application problem domain.

investigations.

**3. Results** 

antenna of comparable size.

performances.

through the relative weights.

The process requires, as inputs, the required frequency range of the antenna, the number of frequency points NF to be evaluated, the metal conductivity and the maximum size of the antenna. Actually, the generated antenna can overcome the bounding box dimensions, but

The proposed fitness functions try to perform a trade-off between contrasting objectives,

In this sense, we can say that the selected individuals are the best adapted to the (present) antenna requirements. However, a different view would be the association of each

In fact, generic evolutionary algorithms, like SED, PSO, DE, GA are a very powerful tool for solving difficult single objective problems, but they can also be applied to solving many multi-objective problems. Actually, real-world problems usually include several conflicting objectives, and a suitable trade-off must be found. An interesting topic is therefore the study of Multi-Objective optimization methods (Chen, 2009), and in solving such multi-objective problems the adopted optimization method must provide an approximation of the Pareto set such that the user can understand the trade-off between overlapped and conflicting objectives, in order to make the final decision. Usually, a decomposition method is implemented to convert a multi-objective problem into a set of mono-objective problems, and an optimal Pareto front is approximated by solving all the sub-problems together (Carvalho, 2012), and this requires insight not only of the algorithmic domain, but also

In design methods dealing with a set of individuals, like SED, such point of view could lead to better ways to explore the solution space, and is a promising direction for future

The automated design of wire antennas using SED has been applied to several PDAs, with different maximum sizes, number of elements, and operation frequencies, and with different

We present here only a few examples, chosen also to show the flexibility of SED. All designed antennas have been compared with known antennas. However, since our antennas are wide-band 3D structures, it has been difficult to device a suitable comparison antenna. To get a meaningful comparison, we decided to compare our designed antennas with an

The first presented antenna (Casula et al., 2009), shown in Fig.4a, has been obtained by constraining the evolution of each individual only in two directions (i.e. horizontally and vertically). This limitation is a hard limitation, and significantly affects the antenna performances. This compromise leads anyway to antennas easy to realize, and with good

requirements both on Gain and input matching, always obtaining very good results.

(different) requirement to a different fitness, thus leading to a multi-objective design.

The chosen comparison antenna has been a 4-elements dipole array, with the same H-plane size of our antenna. This array, shown in Fig.4b, is composed of 4 vertical elements, with a length of 1.2λ0 and spacing of 0.58λ0 in the H-plane and of 0.67λ0 in the E-plane, and its gain is within +/- 1 dB with respect to our antenna. The latter, therefore, uses in an effective way its size. However, it must be stressed that our antenna has a single, and well-matched, feed point, while the array needs a BFN to produce the correct feeding currents of the array elements, which have also a quite large Q. The array realization is therefore more complex.

**Figure 4.** a) SED designed antenna; b) Reference Planar Array with 4 elements and the same size (in the H-plane).

**Figure 5.** Gain and SWR of the GP designed antenna compared to the Gain of the reference Planar Array with 4 elements and the same size (in the H-plane).

Note that we have considered the antenna made of perfectly conducting (PEC) wires. The VSWR constraint has prevented to fall in a super-directive solution, but the robustness of a designed ideal antenna respect to conductor losses has not been checked.

Structure-Based Evolutionary Design Applied to Wire Antennas 129

with perfect conductors, but with a larger size. On the other hand, antennas designed assuming perfect conductors are characterized by collected and closer branches and tend to

**Figure 6.** a) Antenna 2A, designed using perfect conductors; b) Antenna 2B, designed using finite metal

be super-directive.

conductivity.

**Figure 7.** SWR of the antennas 2A and 2B.

The second example removes the constraints of right-angle junctions made in the first example, and will be used also to evaluate the role of the conductor losses on the SED performances. As a matter of fact, this can be easily done by designing an optimal antenna assuming PEC (Antenna 2A) and another one, assuming a finite conductivity σ (antenna 2B), in this case equal to that of pure copper (σ=5.8\*107 S/m). Then the first antenna is analysed by including also the finite conductivity of the wires (Casula et al., 2011b).

For the 2A antenna, at the operation frequency of 500 MHz, requiring a bandwidth of 60 MHz (i.e. 12%, from 470 MHz to 530 MHz), SED designs the antenna shown in Fig.6a. The performances of the antenna 2A are shown in Table 1.

Antenna 2A has been analysed also assigning to the conductors a finite conductivity equal to the pure copper (σ=5.8\*107 S/m). The results show a significant degradation of the antenna performances, since even using a very good conductor as material, the dissipations due to the finite conductivity are very large, making the antenna unusable (in fact NEC2 gives similar values for the SWR, but a very low efficiency). In other words, such antenna is actually close to a super-directive one.

On the other side, asking SED to design an antenna with the same specifications of antenna 2A, but assuming σ=5.8\*107 S/m, we obtain an antenna with similar performances with respect to the 2A antenna, but with a larger size (Antenna 2B). The designed antenna is shown in Fig.6b, and, since the losses affect the antenna gain, the finite conductivity effect is already included in the fitness. The performances of the antenna 2B are shown in Table 1.

This antenna shows similar performances with respect to the antenna shown in Fig.6a, but it has a larger size (0.1833λ03 with respect to 0.03λ03). Nevertheless, unlike the antenna shown in Fig.6a, it is feasible.


**Table 1.** Performances of the antennas 2A and 2B.

The frequency responses of both antennas are shown in Fig. 7 and 8. Also from these responses, we easily deduce that antenna 2A (designed and analysed using PEC) is almost superdirective.

The presented results show that the introduction of a finite value of metal conductivity allows to obtain antennas with similar performances with respect to the antennas designed with perfect conductors, but with a larger size. On the other hand, antennas designed assuming perfect conductors are characterized by collected and closer branches and tend to be super-directive.

**Figure 6.** a) Antenna 2A, designed using perfect conductors; b) Antenna 2B, designed using finite metal conductivity.

**Figure 7.** SWR of the antennas 2A and 2B.

128 Genetic Programming – New Approaches and Successful Applications

performances of the antenna 2A are shown in Table 1.

**Design** 

(PEC) Fig.6a 0.33λ0x 0.22λ0 x

actually close to a super-directive one.

in Fig.6a, it is feasible.

**Antenna Conductivity**

2A +<sup>∞</sup>

2B

superdirective.

σ **(S/m)** 

5.8\*107 (pure copper)

**Table 1.** Performances of the antennas 2A and 2B.

Note that we have considered the antenna made of perfectly conducting (PEC) wires. The VSWR constraint has prevented to fall in a super-directive solution, but the robustness of a

The second example removes the constraints of right-angle junctions made in the first example, and will be used also to evaluate the role of the conductor losses on the SED performances. As a matter of fact, this can be easily done by designing an optimal antenna assuming PEC (Antenna 2A) and another one, assuming a finite conductivity σ (antenna 2B), in this case equal to that of pure copper (σ=5.8\*107 S/m). Then the first antenna is

For the 2A antenna, at the operation frequency of 500 MHz, requiring a bandwidth of 60 MHz (i.e. 12%, from 470 MHz to 530 MHz), SED designs the antenna shown in Fig.6a. The

Antenna 2A has been analysed also assigning to the conductors a finite conductivity equal to the pure copper (σ=5.8\*107 S/m). The results show a significant degradation of the antenna performances, since even using a very good conductor as material, the dissipations due to the finite conductivity are very large, making the antenna unusable (in fact NEC2 gives similar values for the SWR, but a very low efficiency). In other words, such antenna is

On the other side, asking SED to design an antenna with the same specifications of antenna 2A, but assuming σ=5.8\*107 S/m, we obtain an antenna with similar performances with respect to the 2A antenna, but with a larger size (Antenna 2B). The designed antenna is shown in Fig.6b, and, since the losses affect the antenna gain, the finite conductivity effect is already included in the fitness. The performances of the antenna 2B are shown in Table 1.

This antenna shows similar performances with respect to the antenna shown in Fig.6a, but it has a larger size (0.1833λ03 with respect to 0.03λ03). Nevertheless, unlike the antenna shown

**(SWR<2)** 

70 MHz

90 MHz

**MAX Directivity Gain (dBi)** 

(14%) 26 100

(18%) 20 90.09

**Efficiency (%)** 

**Shown Antenna Size Bandwidth**

0.4λ<sup>0</sup>

1.3λ<sup>0</sup>

The frequency responses of both antennas are shown in Fig. 7 and 8. Also from these responses, we easily deduce that antenna 2A (designed and analysed using PEC) is almost

The presented results show that the introduction of a finite value of metal conductivity allows to obtain antennas with similar performances with respect to the antennas designed

Fig.6b 0.47λ0 x 0.3λ0 x

analysed by including also the finite conductivity of the wires (Casula et al., 2011b).

designed ideal antenna respect to conductor losses has not been checked.

Structure-Based Evolutionary Design Applied to Wire Antennas 131

= ⋅ (3)

probability of local maxima of the fitness, which trap the evolution process. The robustness

We choose to maximize gain as the main goal of the fitness. Since we want to maximize the

 *take into account the desired main lobe amplitude)* 

Our goal is the maximization of the gain in the region 1 while minimizing the gains in the other 3 regions, with all the gains expressed in dB. Since we want to optimize the antenna in a certain frequency bandwidth, we start computing a suitable weighted average gain GAW1

1

1 *NF AW i Ei F i G wG N* <sup>=</sup>

wherein the average is taken over the NF frequency points, spanning the whole bandwidth of interest. In (3.1) GEi is the endfire gain and wi depends on the input impedance of the

*) or (Re (ZIN) > 400* 

Ω*)]* 

*)] and [(Im(ZIN) >Re (ZIN))]* 

1

Ω

≤ *400* Ω

αi is a weight proportional to the difference between the imaginary part XINA and the real

The average gains over all other regions, namely GBGR in the back direction, GFGR in the front region and GRGR in the rear region, are then computed. An "effective" endfire gain GAW is

respect to realization errors is also evaluated and taken into account in the fitness.

end-fire gain, the radiation pattern has been divided into 4 regions:

1. The endfire direction:

2. The back direction:

3. The FRONT region:

Δϑ *and* Δϕ

4. The REAR region:

≤ *|*θ*|* ≤

Δθ*; 0° +2*

Δϕ *<* ϕ ≤ *90°* 

> ≤ *|*ϕ*|* ≤ *180°;*

 *180°; 90°* 

 *|*θ*| > 90°+2*

θ *= 90°;* ϕ *= 0°* 

θ *= 90°;* ϕ *= 180°* 

 *(where* 

 *0°* 

on region 1:

PDA:

0.2

 = 

1

part RINA of the array input impedance.

*If [(35*

then obtained properly weighting each gain:

*If [(Re (ZIN) < 35* 

Ω ≤ *Re(ZIN)*

*wi i* α

**Figure 8.** Gain of the antennas 2A and 2B.


**Table 2.** Performances of the antenna designed using pure copper (shown in Fig.6b) for different values of conductivity.

In Table 2, the antenna shown in Fig.6b, designed supposing the metal to be copper, has been analysed for different values of conductivity. While the maximum directivity is almost constant with respect to σ, the efficiency rapidly decreases. It is therefore required to take into account in SED the actual conductivity of the antenna material, but, doing so, the designed antennas will show similar performances to the antenna designed using copper, with an acceptable value for the efficiency.

The last presented antenna (Casula et al., 2011a) is a broadband parasitic wire array for VHF-UHF bands with a significant gain, showing significant improvements over existing solutions (Yagi and LPDA) for the same frequency bands. In order to fulfil these strict requirements, we had to devise a quite complicate fitness function, composed by several secondary objectives overlapped to the main goal; these objectives are expressed by appropriate weights modelling trade-offs between different goals. These relative weights have been modelled by linear relations to avoid discontinuities and thus reducing the probability of local maxima of the fitness, which trap the evolution process. The robustness respect to realization errors is also evaluated and taken into account in the fitness.

We choose to maximize gain as the main goal of the fitness. Since we want to maximize the end-fire gain, the radiation pattern has been divided into 4 regions:

1. The endfire direction:

θ *= 90°;* ϕ *= 0°* 

130 Genetic Programming – New Approaches and Successful Applications

**Figure 8.** Gain of the antennas 2A and 2B.

with an acceptable value for the efficiency.

of conductivity.

**Material Conductivity** σ **(S/m) Efficiency (%) Max Directivity Gain (dB)** 

**Table 2.** Performances of the antenna designed using pure copper (shown in Fig.6b) for different values

In Table 2, the antenna shown in Fig.6b, designed supposing the metal to be copper, has been analysed for different values of conductivity. While the maximum directivity is almost constant with respect to σ, the efficiency rapidly decreases. It is therefore required to take into account in SED the actual conductivity of the antenna material, but, doing so, the designed antennas will show similar performances to the antenna designed using copper,

The last presented antenna (Casula et al., 2011a) is a broadband parasitic wire array for VHF-UHF bands with a significant gain, showing significant improvements over existing solutions (Yagi and LPDA) for the same frequency bands. In order to fulfil these strict requirements, we had to devise a quite complicate fitness function, composed by several secondary objectives overlapped to the main goal; these objectives are expressed by appropriate weights modelling trade-offs between different goals. These relative weights have been modelled by linear relations to avoid discontinuities and thus reducing the

PEC +∞ 100 20.35 Copper 5.8\*107 90.09 20.3 Aluminium 3.77\*107 87.71 20.29 Stainless Steel 0.139\*107 34.84 20.01 2. The back direction:

θ *= 90°;* ϕ *= 180°* 

3. The FRONT region:

 *|*θ*| > 90°+2*Δθ*; 0° +2*Δϕ *<* ϕ ≤ *90°* 

 *(where* Δϑ *and* Δϕ *take into account the desired main lobe amplitude)* 

4. The REAR region:

 *0°* ≤ *|*θ*|* ≤ *180°; 90°* ≤ *|*ϕ*|* ≤ *180°;* 

Our goal is the maximization of the gain in the region 1 while minimizing the gains in the other 3 regions, with all the gains expressed in dB. Since we want to optimize the antenna in a certain frequency bandwidth, we start computing a suitable weighted average gain GAW1 on region 1:

$$\mathbf{G}\_{AW1} = \frac{1}{\mathcal{N}\_F} \sum\_{i=1}^{N\_F} \mathbf{w}\_i \cdot \mathbf{G}\_{Ei} \tag{3}$$

wherein the average is taken over the NF frequency points, spanning the whole bandwidth of interest. In (3.1) GEi is the endfire gain and wi depends on the input impedance of the PDA:

$$w\_i = \begin{cases} 0.2 & \text{if } \{ (\text{Re (ZIN)} \prec 35 \,\,\Omega \) \text{ or } (\text{Re (ZIN)} \succ 400 \,\,\Omega) \} \\\\ \alpha\_i & \text{if } \{ (35 \,\Omega \le \text{Re(ZIN)} \prec 400 \,\,\Omega) \} \quad \text{and } \{ (\text{Im(ZIN)} \succ \text{Re(ZIN)}) \} \\\\ 1 & \text{if } \{ (35 \,\Omega \le \text{Re(ZIN)} \prec 400 \,\,\Omega \}) \text{ or } (\text{Im(ZIN)} \succ \text{Re(ZIN)}) \} \end{cases}$$

αi is a weight proportional to the difference between the imaginary part XINA and the real part RINA of the array input impedance.

The average gains over all other regions, namely GBGR in the back direction, GFGR in the front region and GRGR in the rear region, are then computed. An "effective" endfire gain GAW is then obtained properly weighting each gain:

$$\begin{aligned} \text{G}\_{AW} &= \text{G}\_{\text{AW1}} \cdot \frac{1}{1 + \alpha\_{\text{BGR}} \, ^\ast \text{G}\_{\text{BGR}}}\\ &\cdot \frac{1}{1 + \alpha\_{\text{FGR}} \, ^\ast \text{G}\_{\text{FGR}}} \cdot \frac{1}{1 + \alpha\_{\text{RGR}} \, ^\ast \text{G}\_{\text{RGR}}} \end{aligned} \tag{4}$$

The weigths αBGR, αFGR and αRGR are chosen through a local tuning in order to get the maximum gain in the end-fire direction and an acceptable radiation pattern in the rest of the space. In our case, we obtained the following values: αBGR=0.08, αFGR=0.14 and αRGR=0.02.

In order to design a wideband antenna, we must add some parameters taking into account the antenna input matching, and therefore we introduced suitable weights connected to the antenna input impedance. Holding gain weights fixed, the other parameters concerning input matching are added one by one choosing each weight through a further local tuning.

The GAW is therefore furthermore modified taking into account:


according to the following guidelines:


At this point we have a modified average gain GM, expressed by:

$$\mathbf{G}\_{M} = \mathbf{G}\_{AW} \cdot \left(\frac{1}{1 + a\_{\text{\tiny XR}}}\right) \cdot \left(\frac{1}{1 + a\_{\text{\tiny XR}} \left| \mathbf{X}\_{\text{IN}}^{A} \right|}\right) \tag{5}$$

Structure-Based Evolutionary Design Applied to Wire Antennas 133

where σR2 and σX2 are the normalized variance of RINA and of XINA, respectively.

graded according to their f2 value.

**Figure 9.** Designed Antenna Structure.

above 18 dB.

The difference GR-GM (where GR is a suitably high gain, needed only to work with positive fitness values) is then modulated taking into account both the Q factor (obtained as the ratio between the imaginary part and the real part of the array input impedance at the central frequency) and the structure size to get a particular fitness f1. The individual generated by the genetic process associated to a fitness f1 higher or very close to the best fitness obtained as yet, are then perturbed (assigning random relocations to array elements) and analysed to assess their robustness respect to random modification of the structure. Two different random perturbed antennas are considered for each individual, and the final fitness f2 is the partial fitness f1 averaged over all the initial and perturbed configurations. This random relocation allows getting robust structures respect to both constructive errors and bad weather conditions (for example movements due to wind effect). On the other hand, this robustness test is quite time-consuming. Therefore it is performed only on antennas already showing good performances. The final population is

The antenna designed using the fitness expressed by (3.3) is a PDA with 20 elements: 1 reflector, 1 driven element and 18 directors. The operation frequency is 500 MHz, and the requested bandwidth is of 70 MHz (i.e. 14%, from 475 MHz to 545 MHz). The best antenna is represented in Fig.9, and its shape is typical of all antennas designed using our SED optimization technique. The antenna size is very small, since it fits in a box large 1.72 λ0 x 0.03 λ0 x 0.57 λ0, being λ0 the space wavelength at the operation frequency of 500 MHz. Its SWR is less than 2 in the whole bandwidth of 70 MHz, and its gain is

The 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 Fig.10, and

it appears that 300 generations are enough to reach convergence.

$$\cdot \cdot \left(\frac{1}{1 + \alpha\_{\text{NN}} \cdot \frac{\mathbf{R}\_{\text{IN}}^{A} - \left| \mathbf{X}\_{\text{IN}}^{A} \right|}{\mathbf{R}\_{\text{IN}}^{A}}}\right) \cdot \left(\frac{1}{1 + \alpha\_{\text{NN}} \cdot \frac{\left| \mathbf{R}\_{\text{IN}}^{A} \cdot 300 \right|}{\mathbf{R}\_{\text{IN}}^{A}}}\right) \cdot \left(\frac{1}{1 + \alpha\_{\text{NN}} \cdot \sigma\_{\text{R}}^{2}}\right) \cdot \left(\frac{1}{1 + \alpha\_{\text{NN}} \cdot \sigma\_{\text{X}}^{2}}\right) \cdot \left(\frac{1}{1 + \alpha\_{\text{SNR}}}\right)$$

where σR2 and σX2 are the normalized variance of RINA and of XINA, respectively.

132 Genetic Programming – New Approaches and Successful Applications

1

⋅ ⋅

α

The GAW is therefore furthermore modified taking into account:

b. The SWR over all the required bandwidth

structures with an |XINA| as small as possible;

part of the input impedance (as long as it is lower than 300 Ω);

evolution in areas of the evolution space with good SWR values.

At this point we have a modified average gain GM, expressed by:

*G G*

RX RR A

 α

1 + 1 + \*

*A A IN IN A IN*

*R X R*

α

according to the following guidelines:

up structures with RINA> |XINA|;

low Q factor;

<sup>1</sup> = 1 + \*

*G G AW AW <sup>G</sup>*

⋅

1 1 1 + \* 1 + \*

BGR BGR

(4)

FGR FGR RGR RGR

α

The weigths αBGR, αFGR and αRGR are chosen through a local tuning in order to get the maximum gain in the end-fire direction and an acceptable radiation pattern in the rest of the space. In our case, we obtained the following values: αBGR=0.08, αFGR=0.14 and αRGR=0.02.

In order to design a wideband antenna, we must add some parameters taking into account the antenna input matching, and therefore we introduced suitable weights connected to the antenna input impedance. Holding gain weights fixed, the other parameters concerning input matching are added one by one choosing each weight through a further local tuning.

1. A step is introduced, with a weight αXR=50 if |XINA|> RINA, and αXR=0 otherwise, to boost

2. A weight αXX=0.03 is introduced, related to |XINA|, forcing the evolution process to

3. A weight αRX=0.1 is introduced, related to RINA-|XINA|, to advantage structures with a

4. A weight αRR=0.055 is introduced, related to RINA, to boost up structures with a high real

5. Weights αVR=αVX=0.015 are introduced, inversely related to the normalized variance of

6. A sequence of small steps, related to the SWR (with a weight αSWR between 30 for an SWR>20 and 0.005 for an SWR<4), is introduced to first boost up and then hold the

1 1 = 1 + 1 + *M AW <sup>A</sup>*

⋅ ⋅ ⋅⋅⋅ <sup>−</sup> ⋅ ⋅

α

IN

R

XR XX

⋅ ⋅

1 1 1 11

α*X*

A 2 2

ασ

R - 300 1 + 1 + 1 +

*IN*

IN VX R VR X SWR

 ασ

(5)

α

RINA and XINA, to advantage structures with a regular impedance behaviour;

a. The values of RINA, XINA (averaged over the BW), and their normalized variance;

*G G*

 α The difference GR-GM (where GR is a suitably high gain, needed only to work with positive fitness values) is then modulated taking into account both the Q factor (obtained as the ratio between the imaginary part and the real part of the array input impedance at the central frequency) and the structure size to get a particular fitness f1. The individual generated by the genetic process associated to a fitness f1 higher or very close to the best fitness obtained as yet, are then perturbed (assigning random relocations to array elements) and analysed to assess their robustness respect to random modification of the structure. Two different random perturbed antennas are considered for each individual, and the final fitness f2 is the partial fitness f1 averaged over all the initial and perturbed configurations. This random relocation allows getting robust structures respect to both constructive errors and bad weather conditions (for example movements due to wind effect). On the other hand, this robustness test is quite time-consuming. Therefore it is performed only on antennas already showing good performances. The final population is graded according to their f2 value.

The antenna designed using the fitness expressed by (3.3) is a PDA with 20 elements: 1 reflector, 1 driven element and 18 directors. The operation frequency is 500 MHz, and the requested bandwidth is of 70 MHz (i.e. 14%, from 475 MHz to 545 MHz). The best antenna is represented in Fig.9, and its shape is typical of all antennas designed using our SED optimization technique. The antenna size is very small, since it fits in a box large 1.72 λ0 x 0.03 λ0 x 0.57 λ0, being λ0 the space wavelength at the operation frequency of 500 MHz. Its SWR is less than 2 in the whole bandwidth of 70 MHz, and its gain is above 18 dB.

**Figure 9.** Designed Antenna Structure.

The 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 Fig.10, and it appears that 300 generations are enough to reach convergence.

Structure-Based Evolutionary Design Applied to Wire Antennas 135

**Figure 11.** (a) Gain and (b) SWR comparison between the PDA Designed Antenna and a standard Yagi with the same size (and 9 elements); (b): SWR comparison between the PDA Designed Antenna and a

**Figure 12.** (a) Gain and (b) SWR comparison between the PDA Designed Antenna and a standard Yagi

**Figure 13.** (a) Gain and (b) SWR of the PDA Designed Antenna with a fitness pushing towards a larger

a) b)

a) b)

SWR bandwidth

with the same number of elements, 20, and a far larger size (6 λ0 vs 1.72 λ0).

a) b)

standard Yagi with the same size (and 9 elements).

**Figure 10.** Plot of convergence of the designed antenna in Fig.9.

To assess the performances of our designed PDA, we need a comparison antenna. The best candidate is an existing Yagi but its choice is by no means obvious. Since, for a parasitic antenna, an increase in the number of elements adds little to the antenna complexity, we think that the most significant comparison is a gain comparison with a standard Yagi with the same size of our PDA (about 1.72λ0 in the endfire direction), and a size comparison with an Yagi with the same number of elements as our PDA. The first standard Yagi is composed of only 9 elements, and its gain and SWR, compared to our optimized PDA, are shown in Fig. 11. The standard Yagi bandwidth (SWR<2) is about 35 MHz (7% compared to 14%) with a gain between 12 and 13 dB, i.e. at least 5 dB less than ours, over the whole bandwidth.

A standard Yagi antenna with the same number of elements than our PDA, i.e., 20 has been selected for the second comparison. Though this antenna is very large (its size is about 6λ0x 0.5λ0), it has (see Fig.12) a quite narrow bandwidth (its gain is above 15 dB in a bandwidth smaller than 10%, and even its SWR is less than 2 in a bandwidth of about 9%) if compared with our PDA.

The PDA antenna of Fig. 11 and 12 has been designed choosing a fitness which pushes individuals toward higher Gain giving a smaller importance to input matching. As a further example, it is possible, by suitably choosing the fitness weights, to design a PDA antenna which favours individuals with better input matching. The performances of such an antenna are shown in Fig.13. The bandwidth (with SWR<2) has increased to 150 MHZ (30%), and its gain is only a few dB less than the first optimized PDA antenna. It is important to highlight that the size of the antenna with a larger input bandwidth is the same of the antenna with a higher gain.

In Fig. 14 we show also the F/B ratio of both the PDA designed antennas, which is very close also to standard Yagis' F/B. This comparison shows that, though the PDA we have designed appear to be more difficult to realize than a standard Yagi, they allow significantly better performances in a larger bandwidth, both on input matching, gain and F/B ratio. Furthermore, it is significantly smaller than standard Yagis.

**Figure 10.** Plot of convergence of the designed antenna in Fig.9.

 **1000**

 **10000**

**Fitness Value [Logscale]**

with our PDA.

higher gain.

To assess the performances of our designed PDA, we need a comparison antenna. The best candidate is an existing Yagi but its choice is by no means obvious. Since, for a parasitic antenna, an increase in the number of elements adds little to the antenna complexity, we think that the most significant comparison is a gain comparison with a standard Yagi with the same size of our PDA (about 1.72λ0 in the endfire direction), and a size comparison with an Yagi with the same number of elements as our PDA. The first standard Yagi is composed of only 9 elements, and its gain and SWR, compared to our optimized PDA, are shown in Fig. 11. The standard Yagi bandwidth (SWR<2) is about 35 MHz (7% compared to 14%) with a gain between 12 and 13 dB, i.e. at least 5 dB less than ours, over the whole bandwidth.

 **0 50 100 150 200 250 300**

**Generations**

A standard Yagi antenna with the same number of elements than our PDA, i.e., 20 has been selected for the second comparison. Though this antenna is very large (its size is about 6λ0x 0.5λ0), it has (see Fig.12) a quite narrow bandwidth (its gain is above 15 dB in a bandwidth smaller than 10%, and even its SWR is less than 2 in a bandwidth of about 9%) if compared

The PDA antenna of Fig. 11 and 12 has been designed choosing a fitness which pushes individuals toward higher Gain giving a smaller importance to input matching. As a further example, it is possible, by suitably choosing the fitness weights, to design a PDA antenna which favours individuals with better input matching. The performances of such an antenna are shown in Fig.13. The bandwidth (with SWR<2) has increased to 150 MHZ (30%), and its gain is only a few dB less than the first optimized PDA antenna. It is important to highlight that the size of the antenna with a larger input bandwidth is the same of the antenna with a

In Fig. 14 we show also the F/B ratio of both the PDA designed antennas, which is very close also to standard Yagis' F/B. This comparison shows that, though the PDA we have designed appear to be more difficult to realize than a standard Yagi, they allow significantly better performances in a larger bandwidth, both on input matching, gain and F/B ratio.

Furthermore, it is significantly smaller than standard Yagis.

**Figure 11.** (a) Gain and (b) SWR comparison between the PDA Designed Antenna and a standard Yagi with the same size (and 9 elements); (b): SWR comparison between the PDA Designed Antenna and a standard Yagi with the same size (and 9 elements).

**Figure 12.** (a) Gain and (b) SWR comparison between the PDA Designed Antenna and a standard Yagi with the same number of elements, 20, and a far larger size (6 λ0 vs 1.72 λ0).

**Figure 13.** (a) Gain and (b) SWR of the PDA Designed Antenna with a fitness pushing towards a larger SWR bandwidth

Structure-Based Evolutionary Design Applied to Wire Antennas 137

Finally, we consider the computational issue. The computational cost of SED, like that of many other random optimization techniques, is the computational cost required to evaluate each individual. Therefore different techniques, such as SED and standard GA, can have different cost as long as they evaluate a different number of individuals, or more complex

For the example presented in Fig.10, SED requires 3\*105 NEC evaluations of individuals. GA with comparable antenna size (such as the one described in (Jones & Joines, 1997)) requires a likely, or even larger, number of NEC evaluations. Since also the number of NEC unknown is more or less the same for both approaches, depending essentially on the antenna size, we can conclude that SED has a computational cost comparable, or slightly larger than standard GA. On the other hand, SED allows to explore a far larger solution space. If we consider as computational effectiveness of a design approach the size of the solution space explored for a given computational cost, we can conclude that SED is computationally more effective and

A comparison between SED and other algorithms like Particle Swarm Optimization and Differential Evolution, shows that both the computational cost and the complexity are of the same order of magnitude, also in these cases. But, again, the performances obtained by them

In Table 3, we show the results obtained by our PDA, designed using SED, compared with

(Baskar et al., 2005), who used PSO to optimize the element spacing and lengths of a Yagi–

(Goudos et al., 2010) who used Generalized Differential Evolution applied to Yagi-Uda

(Li, 2007), who used Differential Evolution to optimize the geometric parameters of Yagi-

(Yan et al., 2010), who designed a wide-band Yagi-Uda antenna with X-shape driven dipoles and parasitic elements using differential evolution algorithm, obtaining a

**frequency (dB)** 

**SED** 20 0.57x1.72 λ<sup>0</sup> 21 1.4 30%

**Table 3.** Comparison between the performances reached by SED, PSO and DE in the design of a

**VSWR at center frequency** 

16.4 1.05 -

17.58 1.1 -

12.5 1.8 20%

16.59 1.085 -

**Bandwidth (VSWR<2)** 

**Elements Size Gain at center** 

λ0

λ0

λ0

λ0

ones.

with more performing antennas than GA.

the results obtained by:

Uda antenna;

antenna design;

Uda antennas;

**Baskar 2005** 

**Goudos 2010** 

**Yan 2010** 

**Li 2007** 

Parasitic Wire Dipole Array.

bandwidth of 20%.

**N°** 

**(PSO)** 15 0.239x4.115

**(DE)** 15 0.239x4.943

**(DE)** 11 0.527x1.391

**(DE)** 15 0.459x4.664

are not as good as the ones obtained using SED.

**Figure 14.** F/B ratio comparison between the PDA Designed Antenna with a fitness pushing towards a larger Gain bandwidth and one towards a larger SWR bandwidth.

In order to demonstrate that the inclusion of the antenna robustness into the fitness using our simple device works well, we have tested a hundred random perturbations of the reference antenna of Fig.9. These have been obtained perturbing the ends of each arm of the antenna with a random value between -2 and 2 mm. The standard deviations of the SWR and gain are shown in Fig.15 and are expressed in percentage with respect to the values of the unperturbed antenna shown in Fig.9. Despite of such huge perturbation, the designed PDA is so robust that the behaviour of all perturbed antennas is essentially the same of the unperturbed one. Therefore, despite of its (relative) low computational cost, the approach we have devised to include robustness in the fitness allows to design antennas which are very robust respect to realization errors.

**Figure 15.** Standard Deviation of SWR and Gain of the PDA Designed Antenna in Fig.9, considering 100 randomly perturbed configurations.

Finally, we consider the computational issue. The computational cost of SED, like that of many other random optimization techniques, is the computational cost required to evaluate each individual. Therefore different techniques, such as SED and standard GA, can have different cost as long as they evaluate a different number of individuals, or more complex ones.

136 Genetic Programming – New Approaches and Successful Applications

larger Gain bandwidth and one towards a larger SWR bandwidth.

very robust respect to realization errors.

100 randomly perturbed configurations.

**Figure 14.** F/B ratio comparison between the PDA Designed Antenna with a fitness pushing towards a

In order to demonstrate that the inclusion of the antenna robustness into the fitness using our simple device works well, we have tested a hundred random perturbations of the reference antenna of Fig.9. These have been obtained perturbing the ends of each arm of the antenna with a random value between -2 and 2 mm. The standard deviations of the SWR and gain are shown in Fig.15 and are expressed in percentage with respect to the values of the unperturbed antenna shown in Fig.9. Despite of such huge perturbation, the designed PDA is so robust that the behaviour of all perturbed antennas is essentially the same of the unperturbed one. Therefore, despite of its (relative) low computational cost, the approach we have devised to include robustness in the fitness allows to design antennas which are

**Figure 15.** Standard Deviation of SWR and Gain of the PDA Designed Antenna in Fig.9, considering

For the example presented in Fig.10, SED requires 3\*105 NEC evaluations of individuals. GA with comparable antenna size (such as the one described in (Jones & Joines, 1997)) requires a likely, or even larger, number of NEC evaluations. Since also the number of NEC unknown is more or less the same for both approaches, depending essentially on the antenna size, we can conclude that SED has a computational cost comparable, or slightly larger than standard GA. On the other hand, SED allows to explore a far larger solution space. If we consider as computational effectiveness of a design approach the size of the solution space explored for a given computational cost, we can conclude that SED is computationally more effective and with more performing antennas than GA.

A comparison between SED and other algorithms like Particle Swarm Optimization and Differential Evolution, shows that both the computational cost and the complexity are of the same order of magnitude, also in these cases. But, again, the performances obtained by them are not as good as the ones obtained using SED.

In Table 3, we show the results obtained by our PDA, designed using SED, compared with the results obtained by:



**Table 3.** Comparison between the performances reached by SED, PSO and DE in the design of a Parasitic Wire Dipole Array.

Both (Baskar et al., 2005), (Goudos et al., 2010) and (Li, 2007) decide to perform the optimization only at the center frequency, and this is a simpler task and can lead to better results than an optimization over the whole antenna bandwidth, which is the choice we made in our SED design. Nonetheless, the results obtained by SED are better than the ones obtained by PSO and DE even at the center frequency.

Structure-Based Evolutionary Design Applied to Wire Antennas 139

Work supported by Regione Autonoma della Sardegna, under contract number CRP1\_511, with CUP F71J09000810002, titled *"Valutazione e utilizzo della Genetic Programming nel progetto* 

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In fact we are able to get a wideband antenna with a very high gain, i.e. we both maximize antenna gain and minimize SWR and antenna size within the whole bandwidth (which is a wide bandwidth, equal to 30%).

Therefore, SED can lead to better results if compared with PSO and DE, both in terms of performances and of overall size. This is probably due to the fact that the solution space of SED is larger than the corresponding solution spaces of PSO and DE, and hence a proper choice of the fitness function can push the evolution process to more performing antennas.
