## 3 Proposed Graphic Equalizer Designs
### Third-Octave GEQ using Symmetric Band Filters  
The used third-octave center frequencies (spaced equidistantly on a logarithmic scale starting from 1 kHz in each direction) and the optimized bandwidths leading to the best overall performance of the GEQ are shown in Table 1. The nominal bandwidths for the proposed third-octave equalizer were obtained with \(B_{m}=(\sqrt[3]{2}-1/\sqrt[3]{2})\omega_{c,m}\approx 0.4662\omega_{c,m}\) so that the boundaries of the bands coincided with the center frequencies of the neighboring bands. However, the bandwidths marked with an asterisk in Table 1 were modified from the nominal values in order to account for the frequency warping near the Nyquist limit. The modifications resulted in optimized bandwidths with regard to target magnitude shapes obtained with oversampling. The resulting band filter shapes are seen in Figure 4a, where the squares indicate the gain at the bandwidth boundaries as specified by the parameter \(c\). Appropriate parameter values must be selected for the WLS design to be accurate. These are the number of iterations, the prototype gain used to obtain the initial interaction matrix \(\mathbf{B}\), the parameter \(c\) defining the gain at the bandwidth boundaries, and the weighting factors for each band [3, 6, 22]. The value for each of these parameters was chosen jointly with an optimization loop in MATLAB. For the optimization, 8192 command gain configurations were applied: half of these were random extreme gain settings with each command gain being zero or \(\pm 12\) dB, and the other half were random settings, where the command gains were integers in the range \([-12\,12]\). Different parameter combinations were used to design a GEQ for each command gain configuration, and the maximum approximation error was calculated (see Section 5 for the error definition). Next, the maximum value and the mean of the maximum errors were computed for each setting combination, as also the number of cases where the error exceeded the allowed error of \(\pm 1\) dB. Finally, the optimal parameter values were selected based on the best obtained results, giving more weight to the random integer cases than to the extreme command gain configuration. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(m\) & \(f_{\mathbf{c}}\) (Hz) & \(f_{\mathbf{B}}\) (Hz) & \(m\) & \(f_{\mathbf{c}}\) (Hz) & \(f_{\mathbf{B}}\) (Hz) & \(m\) & \(f_{\mathbf{c}}\) (Hz) & \(f_{\mathbf{B}}\) (Hz) & \(m\) & \(f_{\mathbf{c}}\) (Hz) & \(f_{\mathbf{B}}\) (Hz) \\ \hline
**1** & 19.69 & 9.178 & **9** & 125.0 & 58.28 & **17** & 793.7 & 369.7 * & **25** & 5040 & 2338 * \\
**2** & 24.80 & 11.56 & **10** & 157.5 & 73.43 & **18** & 1000 & 465.8 * & **26** & 6350 & 2943 * \\
**3** & 31.25 & 14.57 & **11** & 198.4 & 92.51 & **19** & 1260 & 586.8 * & **27** & 8000 & 3704 * \\
**4** & 39.37 & 18.36 & **12** & 250.0 & 116.6 & **20** & 1587 & 739.3 * & **28** & 10,080 & 4638 * \\
**5** & 49.61 & 23.13 & **13** & 315.0 & 146.9 & **21** & 2000 & 930.6 * & **29** & 12,700 & 5684 * \\
**6** & 62.50 & 29.14 & **14** & 396.9 & 185.0 & **22** & 2520 & 1172 * & **30** & 16,000 & 6803 * \\
**7** & 78.75 & 36.71 & **15** & 500.0 & 233.1 & **23** & 3175 & 1476 * & **31** & 20,160 & 4117 * \\
**8** & 99.21 & 46.25 & **16** & 630.0 & 293.7 & **24** & 4000 & 1857 * & & & \\ \hline \end{tabular}  
* Adjusted (reduced) bandwidths due to frequency warping close to the Nyquist limit.  
\end{table} 
Table 1: Standard 31 third-octave center frequencies \(f_{\mathrm{c}}\) and adjusted bandwidths \(f_{\mathrm{B}}\) used in the proposed symmetric third-octave GEQ. Based on the parameter optimization, the number of iterations was set to one; the prototype gain was set to 11 dB; the band-boundary parameter was set to \(c=0.38\); but no weighting factors were used, i.e., the weight for all bands was one. The value for the parameter \(c\) was reduced slightly compared to the third-octave GEQ in [3], which resulted in narrower peak filters.