## 3 Proposed Graphic Equalizer Designs
### Bark Band Equalizer  
This section tackles the design of a GEQ with a Bark frequency division of bands using symmetric band filters. The design principles were primarily the same as in the third-octave design above, with a few modifications. The main challenge of the Bark GEQ design was the fact that the Bark center frequencies were not equidistantly spaced on the logarithmic frequency scale, as was the case with octave and third-octave designs. Figure 6 illustrates this difference by showing the center frequency of each band on a logarithmic frequency scale. Figure 6a shows that the third-octave center frequencies \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(m\) & **23** & **24** & **25** & **26** & **27** \\ \hline \(q_{1,m}\) & 0.00166 & 0.00295 & 0.00544 & 0.0105 & 0.0214 \\ \(q_{3,m}\) & 8.09\(\times 10^{-6}\) & 1.25\(\times 10^{-5}\) & 1.91\(\times 10^{-5}\) & 2.84\(\times 10^{-5}\) & 4.08\(\times 10^{-5}\) \\ \hline \(m\) & **28** & **29** & **30** & **31** & \\ \hline \(q_{1,m}\) & 0.0456 & 0.103 & 0.257 & 0.754 & \\ \(q_{3,m}\) & 5.46\(\times 10^{-5}\) & 6.27\(\times 10^{-5}\) & 3.68\(\times 10^{-5}\) & \(-1.18\times 10^{-4}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Polynomial coefficients for the Nyquist gain of band filters \(H_{m}(z)\), \(m=23,24,\ldots 31\), in the third-octave design. Figure 5: Nyquist gain of the different band filters as a function of the peak gain for the **(a)** third-octave and **(b)** Bark band case. All curves are antisymmetric with respect to the 0 dB peak gain. were located on a straight line, whereas in Figure 6b, the Bark center frequencies formed a nonlinear curve. The numerical data for these plots are given in Tables 1 and 3. In addition to the Bark center frequencies, Table 3 gives the bandwidths selected for the band filters. Note that while the center frequencies were selected according to the standard definition [23], the bandwidths were not. This was due to the nature of the least-squares GEQ design: by choosing the bandwidths in Table 3, the user could control the GEQ at the Bark center frequencies to obtain greater overall accuracy. For best accuracy, the band filter's right bandwidth boundaries had to coincide with the center frequency of the adjacent band filter. This was achieved by setting the bandwidths of the first 23 bands to \(B_{m}=\omega_{\text{c,}m+1}-(\omega_{\text{c,}m}^{2})/(\omega_{\text{c,}m+1})\). Since this method could not be applied to the last band filter, its left edge was matched with the center frequency of the penultimate band filter. The resulting band filter responses are shown in Figure 4b. The Bark EQ design utilized the same filters as the third-octave EQ, and thus, the values for the same parameters had to be optimally chosen. Since the uneven spacing of the Bark center frequencies required the band filters of the GEQ to have different bandwidths, a consequent implication was that the parameter \(c\) had to be frequency dependent. Additionally, this complication led to more iteration rounds, a different initial prototype gain, and a non-constant weighting matrix for good accuracy. The parameter optimization was performed in the same way as in the third-octave case above. The only difference was that instead of 4096 random extreme cases (command gains at zero or