# A Novel Evolutionary Algorithm for Designing Robust Analog Filters
## 4 Experiments and Results
### Evolving Robust Analog Filters Using Genetic Algorithms: The Classical Robust Design  
In this experiment, the G3 PCX-GA [52] is used to improve the robustness of the best filters evolved using standard GP through parameter tuning while keeping functional structure unchanged. As we can see from Figures 2 and 4, these two evolved filters are very complex, with 39 and 27 parameters to search. As the objective function of this optimization is highly multi-modal, this is hard optimization even for G3 PCX-GA, as the experiment demonstrates. The running parameters for this experiment are summarized in Table 2.  
Figure 6: Frequency responses of the best lowpass and highpass filters evolved using GP, GPGARMS, GPRMS, and GPRPE in the perturbed-parametric environment: (**a**) Frequency response distribution of the best filter evolved using standard GP. (**b**) Frequency response distribution of the best filter evolved using standard GPGARMS. (**c**) Frequency response distribution of the best filter evolved using standard GPRMS. (**d**) Frequency response distribution of the best filter evolved using standard GPRPE. Figure 7: Robustness of highpass filters evolved by GP, GPGARMS, GPRMS, and GPRPE. For (**a,b**), lower values correspond to higher robustness. (**a**) The average Type-I robustness defined in Section 3.5. (**b**) The average Type-II robustness defined in Equation (5). Evolving Robust Analog Filters Using Genetic Programming: Open-Ended Topology Innovation for Robust Design  
In the following experiment, we try to evolve robust analog filters that have a higher tolerance of the variation in component values and have graceful performance degradation. Two GP algorithms for evolving robust solutions are used: GPRMS and GPRPE. The configurations for this experiment are the same as those used in Section 4.1. Figure 8 shows the best lowpass and highpass filters evolved using GPRMS with robustness requirements. Compared with the filters evolved using standard GP in Figures 2 and 4, these two filters use fewer components, while the functional performance remains similar. The robustness of these two filters is compared to that of filters evolved using GP and GPGARMS in Figure 6 for the highpass filter problem. As can be seen, both GP methods with robustness requirements (GPRMS and GPRPE) evolved filters with higher robustness, as shown by their much less dispersed frequency response distributions. Statistical evidence is presented in Section 4.4. Compared to GPGARMS, one distinguishing characteristic of GPRMS and GPRPE is that they evolved more robust filters without significant degradation of functional performance. This observation can also be made in Figure 7, which shows that GPGARMS has the lowest performance variation (standard deviation) across all perturbation levels but also has the highest average-101-point deviation from the target frequency response curve. Figure 8: Evolved robust lowpass and highpass filters with complexity much lower than the filters evolved without considering robustness in the fitness in Figures 2 and 4. (**a**) Topology of the most robust lowpass filter with only 13 evolved components using GPRMS. (**b**) Topology of the most robust lowpass filter with only 16 evolved components using GPRMS. (**c**) Equivalent analog circuit of the most robust lowpass filter in (**b**). From Figure 7a, however, we find that filters evolved by both GPRMS and GPRPE, in a trade-off for higher robustness, have an average 101-point deviation higher than that evolved by standard GP for all perturbation levels less than 0.3. However, when the perturbation level is greater than 0.3, GPRMS achieves an average-101-point deviation that is lower than that achieved by GP, and GPRPE's performance remains worse. This means that GP with the multiple-simulation method, compared with the perturbed fitness evaluation method, evolved designs with a higher tolerance to parametric perturbation.