# A Novel Evolutionary Algorithm for Designing Robust Analog Filters
## 4 Experiments and Results
### Evolving Robust Analog Filters Using Genetic Algorithms: The Classical Robust Design
From Figure 7b, we find that GPGARMS shows a clear advantage in terms of \(f_{robust11}\), which is, however, achieved by sacrificing functional performance. In other words, the performance is more stable with regard to its average frequency response. Nevertheless, the total deviation from the target response becomes larger. To further compare the characteristics of the evaluated algorithms, we did an extensive analysis of the performances of all filters evolved by the five algorithms, and this analysis is summarized in Table 3. For each algorithm, 15 runs were performed to evolve 15 lowpass filters. Each filter was then subjected to 10,000 simulations with random perturbations of their component parameters. We calculated the average 101-point deviation (\(f_{robust1}\)) and its standard deviation (\(f_{robust11}\)) from the target frequency response, for each filter. The upper half of the table shows the average of \(f_{robust1}\) and its standard error over 15 runs for each algorithm under different perturbation environment. The lower half shows the average of \(f_{robust11}\) and its standard error over 15 runs for each algorithm under different perturbation environments. The perturbation magnitudes are represented by the magnitudes of the Gaussian perturbation as a percentage of component parameter values. In the top half of the table, we can see that the GPRPE method evolved filters with the highest average performance or \(f_{robust1}\). These filters are not only best at a perturbation level of 0.2, which is applied during the evolution process, but also best across all other perturbation levels tested in the \"evaluation\" phase. This implies that ideal analog filters have continuous performance landscapes with satisfactory degradation properties. What is surprising here is that not only are the \(f_{robust1}\) values of the filters evolved by GPRPE the best but also their standard error values are lowest at all perturbation levels. This means that the GPRPE has the maximum number of fitness evaluations; thus, the genetic operations algorithm evolves high-performance filters with the most reliability. Interestingly, there is a complementary relationship between the performances of GPRPE and GPRMS. Examining the lower half of Table 3, the GPRMS turns out to be the best algorithm to evolve robust filters with the highest robustness in terms of evaluation criterion \(f_{robust11}\), with minimum variation in the frequency response, and this advantage holds true for all perturbation levels. Comparing GPRPE and GPRMS, it appears that GPRMS with its multi-simulation method can evolve more stable solutions due to its superior estimation of the true robustness property of a candidate solution. GPRPE, instead, suffers from its robustness estimation error but enjoys a higher amount of genetic operations. These results show that there is no simple conclusion about which method (of GPRPE and GPRMS) is better than the other. The choice depends on the design objectives.