# A Novel Evolutionary Algorithm for Designing Robust Analog Filters
## 4 Experiments and Results
### Evolving Analog Filters Using GP without Considering Robustness in the Fitness Function  
In this experiment, 15 analog lowpass and highpass filters are evolved using standard GP without incorporating a robustness criterion in the fitness function Equation (3). The following common running parameters (Table 1) are used throughout all GP experiments in this paper:  
Note that the maximum tree size here is 8 rather than 17, used in many other works. This parameter was selected by considering the reduced tree sizes due to our simplified parameter representation method and available computational resources. The maximum tree depth of 8 allows for the synthesis of analog filters with up to 100 components, which is sufficient for our purpose. We select the evolved filter with the highest performance to test its noise tolerance over the degradation or variation in the component parameters with different perturbation magnitudes. As described above, the evaluation of robustness with respect to parameter perturbation is conducted by running 10,000 simulations of the configurations of the Gaussian parameter perturbations. Figures 2 and 4 show the topology of the evolved lowpass and highpass filters with the highest performance out of 15 runs. The frequency responses of these filters are summarized in Figure 5 along with the results of evolved robust filters. The evolved best lowpass and highpass filters have 39 \begin{table}
\begin{tabular}{l l} \hline \hline
**Total Population Size: 2000 (400/400/400/400/400)** & **Number of Subpopulations: 5** \\ \hline Migration interval: 5 generations & Migration size: 30 individuals \\ Max tree depth: 8 & Crossover probability: 0.7 \\ InitTreeDepth: 3–5 & Standard mutation probability: 0.1 \\ Flag bit mutation rate: 0.1 & Swapping-tree mutation rate: 0.1 \\ Tournament size: 7 & Parametric mutation probability: 0.5 \\ Max evaluations: 1,000,000 & Flag mutation probability: 0.3 \\ Pool size of elite individuals: 20 & Elite pool update frequency: 5 generations \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental parameters for analog filter synthesis without robustness criterion. and 27 components, respectively. The lowpass and highpass filters approximate the ideal frequency response closely, with the sum of deviations over 101 points being only 6.43 and 0.32, respectively. Figure 5a,b show the frequency responses of the four best lowpass and four highpass filters evolved using GP, GPGARMS, GPRMS, and GPRPE. The performances of the evolved filters in an ideal condition without parametric perturbation were compared. Here, the \"best\" means the highest fitness of each approach out of 15 runs. Only standard GP has no consideration of robustness. The other three methods considered robustness in their fitness function definition. From Figure 5, first, one can see that the frequency responses of the filters evolved by GP, GPRMS, and GPRPE are very similar. That means introducing robustness requirements does not necessarily decrease performance with nominal parameters significantly. The only exception is GPGARMS, which distinguishes itself from the others with much worse nominal performance (larger deviations). After checking the standard deviation of the sum of deviations of 10,000 runs with 20%-level parametric perturbation, it appears that the nominal performance is sacrificed by the GA for better robustness. One can also observe that the filters with highest performance for both filter types are evolved using GPRPE. This can be explained by the actual number of genetic topological and parametric operations. Here, standard GP only ran 500,000 fitness/function evaluations during which genetic operations were conducted (we did not run 1,000,000 function evaluations for GP because standard GP always  
Figure 4: The topology of the best highpass analog filters evolved with standard GP with 500,000 function evaluations without considering a robustness requirement in the fitness function(parameters are omitted).