**2.3. Fitness function**

The fitness function must measure how closely the design meets the desired requirements. To achieve our design goal, a fitness should be developed, which is to direct the evolution process on a structure with reduced size, with the highest end-fire gain, and with an input match as better as possible in the widest frequency range. Actually, the increase in a parameter (i.e. the gain) usually results in a reduction in the other ones (i.e. frequency bandwidth and input matching), thus the algorithm must manage an elaborate trade-off between these conflicting goals. Therefore, the form of the fitness function can be a critical point, since only a suitable fitness can lead the design process to significant results. Moreover, depending on the used fitness, the computation time can be largely reduced (i.e. a good result can be obtained with less generations).

After evaluation of different fitness structures, we have chosen a fitness function composed by three main terms suitably arranged as:

$$Fitness = \left(F\_M + F\_G\right) \cdot F\_S \tag{1}$$

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 term (FS) takes into account the antenna size.

In (2.1):

124 Genetic Programming – New Approaches and Successful Applications

*(StretchAlongZ 1.3315124586134857 (Wire 0.42101090906114413 1.0* 

 *(RotateWithRespectTo\_Y 0.3577743384222999 END)))))* 

 *(StretchAlongX 0.5525837649288541 (StretchAlongY 1.4819461053740617* 

*(Wire 0.5581593081319647 1.0 (RotateWithRespectTo\_X -0.44260816356142224* 

 *(StretchAlongX 1.42989629787443 (StretchAlongZ 1.346598788775623* 

*(Wire 0.3707701115469606 1.0 (RotateWithRespectTo\_X 0.5262591815805174* 

*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* 

 *(RotateWithRespectTo\_Z 0.08068272691709244 (StretchAlongZ 0.7166185389610261* 

 *(RotateWithRespectTo\_Z -0.7423883999218206 (RotateWithRespectTo\_Z 0.07210315212202911* 

The fitness function must measure how closely the design meets the desired requirements. To achieve our design goal, a fitness should be developed, which is to direct the evolution process on a structure with reduced size, with the highest end-fire gain, and with an input match as better as possible in the widest frequency range. Actually, the increase in a parameter (i.e. the gain) usually results in a reduction in the other ones (i.e. frequency bandwidth and input matching), thus the algorithm must manage an elaborate trade-off between these conflicting goals. Therefore, the form of the fitness function can be a critical point, since only a suitable fitness can lead the design process to significant results. Moreover, depending on the used fitness, the computation time can be largely reduced (i.e. a good result can be obtained with less

*S-expression:* 

*Tree 0:* 

*Tree 1:* 

*Tree 2:* 

 *END))))))* 

 *END))))* 

*GX 4 001* 

**2.3. Fitness function** 

generations).

*GE* 

The corresponding NEC-2 input file is:

$$F\_M = \left| 1 - \overline{\text{SWR}} \right| \cdot \alpha\_M; \qquad F\_G = \left| \frac{G\_{MAX}}{\overline{G}} \right| \cdot \alpha\_G; \qquad F\_S = 1 + \frac{D\_{REAL} - D\_{MAX}}{D\_{MAX}} \cdot \alpha\_S \tag{2}$$

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 maximum allowed size for the antenna.

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 additional cost.

The evaluation procedure for each individual (i.e. for each antenna) can be described by the flowchart in Fig.3.

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

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 with a penalty directly proportional to the excess size.

Structure-Based Evolutionary Design Applied to Wire Antennas 127

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

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

**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).

matching network is therefore required.

H-plane).

The proposed fitness functions try to perform a trade-off between contrasting objectives, through the relative weights.

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

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 knowledge of the application problem domain.

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 investigations.
