## 3 Proposed Graphic Equalizer Designs
### Bark Band Equalizer
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \(m\) & \(f_{\text{c}}\) (Hz) & \(f_{\text{B}}\) (Hz) & \(m\) & \(f_{\text{c}}\) (Hz) & \(f_{\text{B}}\) (Hz) & \(m\) & \(f_{\text{c}}\) (Hz) & \(f_{\text{B}}\) (Hz) \\ \hline
**1** & 50 & 133.3 & **9** & 1000 & 315.5 & **17** & 3400 & 1110 \\
**2** & 150 & 160.0 & **10** & 1170 & 370.8 & **18** & 4000 & 1467 \\
**3** & 250 & 171.4 & **11** & 1370 & 426.9 & **19** & 4800 & 1828 \\
**4** & 350 & 177.8 & **12** & 1600 & 466.2 & **20** & 5800 & 2194 \\
**5** & 450 & 214.7 & **13** & 1850 & 558.1 & **21** & 7000 & 2735 \\
**6** & 570 & 235.9 & **14** & 2150 & 651.0 & **22** & 8500 & 3619 \\
**7** & 700 & 256.7 & **15** & 2500 & 744.8 & **23** & 10,500 & 5333 \\
**8** & 840 & 294.4 & **16** & 2900 & 926.5 & **24** & 13,500 & 6000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bark center frequencies [23]\(f_{\text{c}}\) and their adjusted filter bandwidths \(f_{\text{B}}\) used in the proposed Bark GEQ. Note that these bandwidths used for filter design are different from the standard Bark bandwidth definitions. Figure 6: (a) Third-octave center frequencies vs. (b) Bark center frequencies. \(\pm 12\) dB), only 2048 such gain configurations were used since the results from the random integer gain configurations were eventually emphasized. Based on the optimization, the parameter \(c\) was set to \(c=0.36\) for the first band and to \(c=0.42\) for bands 2-24. This way, the leakage of the relatively wide first band filter to its neighboring bands was better controlled. An additional step to improve the accuracy of the Bark GEQ at low frequencies applied frequency-dependent weighting values to the WLS solution. The best accuracy, from among the many choices tested, was achieved when the weighting matrix in (3) was:  
\[\mathbf{W}=\text{diag}(1\ 0.5\ 0.5\ \ldots\ 0.5). \tag{17}\]  
Finally, the Bark EQ design used two iteration steps. The initial interaction matrix was first constructed with a prototype gain of 1 dB, after which (3) was calculated thrice: the interaction matrix for the second and third iteration rounds was obtained using the optimized gains from the first and second WLS solution, respectively. The final solution yielded the optimized band filter dB gains that were then converted to linear gain factors that were used to update the band filter coefficients. Similarly to the third-octave case, the filter design required the Nyquist gain. In order to have the filter peak gain as the sole input (since the center frequencies and bandwidths were set constants), the Nyquist gain was again approximated with a third-order polynomial based on the peak gain (see (16)). Excessively oversampled versions of the Bark band filters were designed, and their responses were evaluated at 22.05 kHz in order to obtain the necessary data for MATLAB's curve fitting tool. The Nyquist gains of the first 18 bands were observed to be always approximately zero and thus did not need to be changed. The Nyquist gain relative to the peak gain of bands 19-24, however, were approximated: their target shapes are shown in Figure 5b, and the corresponding polynomial coefficients are given in Table 4. Note that similarly to the third-octave case, coefficients \(q_{0}\) and \(q_{2}\) were zero for all filters. For the Bark EQ, the polynomial approximation of the Nyquist gains was highly accurate with the error being less than \(\pm 0.2\) dB for all bands in comparison to the target curves in Figure 5b.