# An Enhanced Genetic Algorithm for Parameter Estimation of Sinusoidal Signals
## 4 Benchmark Function Study  
Benchmark functions [14; 30; 34; 39; 43] are widely adopted to demonstrate the performance of algorithms. Thus, in the present work, benchmark based numerical experiments are performed for the EGA as well. Here, eight unimodal and multimodal functions [40] are selected (see Appendix A). As a comparison, studies using the SGA [44] as well as the PSO [11; 29], the CS [28; 29], and the CMGA [30] are also performed for the functions; the corresponding parameter configurations are listed in Table 1. Without favoritism, all the algorithms are assigned the same size initial population, \(N=50\), and the maximum iteration number, \(MaxGen=200\). The optimization results are plotted in Figures 5 and 6. To have an insight into how well these algorithms performed, all the vertical coordinates are displayed on a logarithmic scale. From Figures 5 and 6, it can be seen directly that the performance of the EGA is superior to that of the SGA, PSO, CS, and CMGA for most of the benchmark functions, except for the Zakharov function (\(f_{2}(x)\)). Examining the optimization results of \(f_{2}(x)\) shown in Figure 5b, it is noted that the evolutionary curve of the SGA displays in part. This is caused by the estimating value of zero (equaling to the optimum), and the logarithmic \(y\)-axis. In this regard, the SGA performs better than the EGA. However, when compared to other results, the proposed EGA shows overwhelming advantages against others, especially for [PERSON]'s No. 5 function (\(f_{8}(x)\)). Different from other functions, \(f_{8}(x)\) contains uniform distribution terms, which increases not only the number of local optima but also the uncertainty, and hence, the optimization becomes much more difficult. Taking a look at the results in Figure 6d, frequent vibrations are observed in the evolution curves of the SGA and PSO, while the EGA shows an unremitting evolution trend, demonstrating its robustness against noise. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**SGA** & **PSO** & **CS** & **CMGA** & **EGA** \\ \hline \(N=50\) & \(N=50\) & \(N=50\) & \(N=50\) & \(N=50\) \\ \(MaxGen=200\) & \(MaxGen=200\) & \(MaxGen=200\) & \(MaxGen=200\) & \(MaxGen=200\) \\ \(P_{c}=0.7\) & \(C_{1}=2,C_{2}=2\) & \(Pa=0.25\) & \(m_{1}\cdot t_{1}=1/600,m_{1}=0.1\) & \([P_{c1},P_{c2}]=[0.6,0.7]\) \\ \(P_{m}=0.01\) & \(w_{1}=0.9,w_{2}=0.4\) & \(-\) & \(m_{2}\cdot t_{2}=1/600,m_{2}=0.1\) & \(Figure 5: Evolutionary curves of unimodal benchmark functions. (**a**) Sphere function. (**b**) Zakharov function. (**c**) Rosenbrock function. (**d**) Ackley function. Figure 6: _Cont._  
To further examine the performance, the obtained best and mean solutions as well as the convergence speed of each function were compared and listed in Tables 2-4, respectively. All the results were obtained over 50 Monte-Carlo simulations. Through Table 2, it is evident that the EGA outperforms the SGA, PSO, CS, and CMGA for most cases. Looking at the optimization results for \(f_{2}(x)\), although only the SGA achieves the global optimum, the EGA still prevails over the others when comparing the rest optimizations. The results listed in Table 3 also confirm this point. In terms of the convergence speed, the maximum, the minimum, and the average termination iterations were measured, where the termination criterion was defined by \(\left|O_{opt}-O_{obt}\right|<10^{-4}\), \(O_{opt}\) is the optimum of a given problem, and \(O_{obt}\) is the obtained optimized result, respectively. From Table 4, one can find that the convergence speed of the proposed EGA is fast, presenting a strong competitiveness over its counterparts. From these comparisons, it can be concluded that the EGA shows an overwhelming superiority over the competitors in terms of accuracy, convergence speed, and robustness against noise.