# An Enhanced Genetic Algorithm for Parameter Estimation of Sinusoidal Signals
## 5 Parameter Estimation of Sinusoidal Signals
### The Voice Dataset  
The voice dataset gives the magnitudes of the voice when the vowel 'ooh' was sung at a pitch of 290 Hz. The frequencies and amplitudes found in the signal are used to determine the phonetic vowel, and are of interest in voice synthesis, therapy, and training. In the present work, a ten-parameter model is utilized for the signal as expressed as follows:  
\[y(t)=K+\sum_{i=1}^{3}A_{i}\sin(\omega_{i}t+\varphi_{i}), \tag{8}\]  
where \(K\) is the offset, \(A\) denotes the amplitude of sinusoid signal, \(\omega\) and \(\varphi\) are the frequency and phase, respectively. The parameter initialization is set as \(K\in[-1,1]\), \(A_{i}\in[0,100]\), \(\omega_{i}\in[-\pi/2,\pi/2]\), and \(\varphi_{i}\in[-\pi/2,\pi/2]\). The corresponding parameter configurations for the algorithms were the same as the ones in Table 1, except for the crossover and mutation parameters of the EGA. The new values were set as \([P_{c1},P_{c2}]=[0.8,0.7]\), and \([P_{m1},P_{m2}]=[0.05,0.22]\), respectively. The same \(N=200\), and \(MaxGen=500\) were assigned for all the algorithms as well. Figure 7 shows the original data as well as the fitting curves, from which one can observe that all the curves follow the pattern well. For having an insight into the difference of the algorithms, the estimated parameters, and the coefficients of determination \(\left(R^{2}\right)\) were examined, listed in Table 5. As can be seen from the table, the estimated parameters are very close, it is difficult to arrive at a conclusion. However, by comparing the values of the \(R^{2}\), the performance of the EGA can be determined, which is the best among the algorithms. In other literature, [PERSON] et al. [47] considered \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**SGA**} & \multicolumn{2}{c}{**PSO**} & \multicolumn{2}{c}{**CS**} & \multicolumn{2}{c}{**CMGA**} & \multicolumn{2}{c}{**EGA**} \\ \cline{2-10}  & \(G_{max(min)}\) & \(G_{mean}\) & \(G_{max(min)}\) & \(G_{mean}\) & \(G_{max(min)}\) & \(G_{max(min)}\) & \(G_{mean}\) & \(G_{max(min)}\) & \(G_{mean}\) \\ \hline \(f_{1}(x)\) & \(45(7)\) & \(22.94\) & \(136(26)\) & \(64.10\) & \(41(10)\) & \(34.66\) & \(38(6)\) & \(28.16\) & \(36(5)\) & \(18.28\) \\ \(f_{2}(x)\) & \(54(8)\) & \(26.40\) & \(190(9)\) & \(101.86\) & \(65(27)\) & \(37.84\) & \(40(23)\) & \(35.26\) & \(44(12)\) & \(25.84\) \\ \(f_{3}(x)\) & \(200(13)\) & \(189.88\) & \(200(108)\) & \(135.40\) & \(187(30)\) & \(148.54\) & \(159(10)\) & \(121.63\) & \(139(7)\) & \(108.44\) \\ \(f_{4}(x)\) & \(200(200)\) & \(200.00\) & \(200(200)\) & \(200.00\) & \(200(120)\) & \(141.40\) & \(200(138)\) & \(120.33\) & \(\mathbf{156(51)}\) & \(\mathbf{100.72}\) \\ \(f_{5}(x)\) & \(200(200)\) & \(188.34\) & \(200(200)\) & \(200.00\) & \(200(120)\) & \(167.75\) & \(200(30)\) & \(\mathbf{69.93}\) & \(200(51)\) & \(182.08\) \\ \(f_{6}(x)\) & \(200(48)\) & \(1809.98\) & \(200(200)\) & \(189.58\) & \(200(108)\) & \(165.04\) & \(181(44)\) & \(140.71\) & \(169(6)\) & \(\mathbf{124.98}\) \\ \(f_{7}(x)\) & \(200(39)\) & \(160.36\) & \(200(200)\) & \(200.00\) & \(200(15)\) & \(178.25\) & \(161(37)\) & \(153.24\) & \(\mathbf{153(39)}\) & \(\mathbf{119.04}\) \\ \(f_{8}(x)\) & \(200(19)\) & \(161.82\) & \(200(200)\) & \(200.00\) & \(200(42)\) & \(124.76\) & \(200(62)\) & \(\mathbf{133.08}\) & \(200(\mathbf{18})\) & \(140.10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the convergence speeds of SGA, PSO, CS, CMGA, and EGA.